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+
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+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #68462 (https://www.gutenberg.org/ebooks/68462)
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-The Project Gutenberg eBook of Einstein and the Universe, by Charles
-Nordmann
-
-This eBook is for the use of anyone anywhere in the United States and
-most other parts of the world at no cost and with almost no restrictions
-whatsoever. You may copy it, give it away or re-use it under the terms
-of the Project Gutenberg License included with this eBook or online at
-www.gutenberg.org. If you are not located in the United States, you
-will have to check the laws of the country where you are located before
-using this eBook.
-
-Title: Einstein and the Universe
-       A popular exposition of the famous theory
-
-Author: Charles Nordmann
-
-Translator: Joseph McCabe
-
-Contributor: Viscount Haldane
-
-Release Date: July 5, 2022 [eBook #68462]
-
-Language: English
-
-Produced by: deaurider and the Online Distributed Proofreading Team at
-             https://www.pgdp.net (This file was produced from images
-             generously made available by The Internet Archive)
-
-*** START OF THE PROJECT GUTENBERG EBOOK EINSTEIN AND THE
-UNIVERSE ***
-
-
-
-
-
-Transcriber’s Notes:
-
-  Underscores “_” before and after a word or phrase indicate _italics_
-    in the original text.
-  Equal signs “=” before and after a word or phrase indicate =bold=
-    in the original text.
-  Small capitals have been converted to SOLID capitals.
-  Typographical and punctuation errors have been silently corrected.
-
-
-
-
-                     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
-
-
-  _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._
-
-“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
-
-
-  _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._
-
-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
-
-
-  _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._
-
-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
-
-
-  _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._
-
-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
-
-
-  _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._
-
-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
-
-
-  _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._
-
-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?
-
-
-  _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._
-
-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
-
-
-  _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._
-
-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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-<p style='text-align:center; font-size:1.2em; font-weight:bold'>The Project Gutenberg eBook of Einstein and the Universe, by Charles Nordmann</p>
-<div style='display:block; margin:1em 0'>
-This eBook is for the use of anyone anywhere in the United States and
-most other parts of the world at no cost and with almost no restrictions
-whatsoever. You may copy it, give it away or re-use it under the terms
-of the Project Gutenberg License included with this eBook or online
-at <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you
-are not located in the United States, you will have to check the laws of the
-country where you are located before using this eBook.
-</div>
-
-<p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Title: Einstein and the Universe</p>
-<p style='display:block; margin-left:2em; text-indent:0; margin-top:0; margin-bottom:1em;'>A popular exposition of the famous theory</p>
-<p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Author: Charles Nordmann</p>
-<p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Translator: Joseph McCabe</p>
-<p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Contributor: Viscount Haldane</p>
-<p style='display:block; text-indent:0; margin:1em 0'>Release Date: July 5, 2022 [eBook #68462]</p>
-<p style='display:block; text-indent:0; margin:1em 0'>Language: English</p>
-  <p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em; text-align:left'>Produced by: deaurider and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive)</p>
-<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK EINSTEIN AND THE UNIVERSE ***</div>
-<hr class="chap x-ebookmaker-drop" />
-<h1>EINSTEIN AND THE UNIVERSE</h1>
-
-<p class="f150 space-below2"><i>A Popular Exposition of the Famous Theory</i></p>
-
-<p class="f150"><i>By</i> CHARLES NORDMANN</p>
-<p class="f120 space-below2"><i>Astronomer to the Paris Observatory.</i></p>
-
-<p class="f120 space-below2"><i>Translated by</i> JOSEPH McCABE</p>
-
-<p class="center space-below2"><i>With a Preface by the Rt. Hon.</i><br />
-THE VISCOUNT HALDANE, O.M.</p>
-
-<p class="f120">T. FISHER UNWIN LTD.</p>
-<p class="center space-below2">LONDON: ADELPHI TERRACE</p>
-
-<p class="center space-below2"><i>First published in English<span class="ws2">April 1922</span></i><br />
-<i>Second Impression<span class="ws4">June 1922</span></i></p>
-
-<p class="center"><i>All rights reserved</i></p>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_5">[Pg 5]</span></p>
-
-<h2 class="nobreak" id="PREFACE">PREFACE</h2>
-</div>
-
-<p>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
-<span class="pagenum" id="Page_6">[Pg 6]</span>
-check. A new school of mathematical physicists, still, however, small
-in number, is beginning to appreciate this.</p>
-
-<p>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.</p>
-
-<p>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 <i>Begriffe</i>. The language is that of
-the <i>Vorstellung</i>, 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.
-<span class="pagenum" id="Page_7">[Pg 7]</span></p>
-
-<p>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 <a href="#CHAPTER_V">V</a> and <a href="#CHAPTER_VI">VI</a>,
-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
-<a href="#CHAPTER_III">third chapter</a> 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.</p>
-
-<p>This, however, is the fault, not of M. Nordmann himself, but of a phase
-<span class="pagenum" id="Page_8">[Pg 8]</span>
-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.</p>
-
-<p class="author fontsize_150"><i>Haldane</i></p>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_9">[Pg 9]</span></p>
-<h2 class="nobreak">CONTENTS</h2>
-</div>
-
-<table class="fontsize_120 no-wrap" border="0" cellspacing="0" summary="TOC" cellpadding="0" >
-  <tbody><tr>
-      <td class="tdl"><span class="smcap">Preface by the Rt. Hon. Viscount Haldane, O.M.</span></td>
-      <td class="tdr"><a href="#PREFACE">pp.&nbsp;&emsp;5-8</a></td>
-   </tr><tr>
-      <td class="tdl"><span class="smcap">Introduction</span></td>
-      <td class="tdr"><a href="#INTRO">pp. 13-15</a></td>
-   </tr><tr>
-      <td class="tdc_space-above1" colspan="2"><b>CHAPTER I</b><br />THE METAMORPHOSES OF<br /> SPACE AND TIME</td>
-   </tr><tr>
-      <td class="tdl"><p class="neg-indent">
-           <i>Removing the mathematical difficulties—The pillars<br />
-            of knowledge—Absolute time and space, from<br />
-            Aristotle to Newton—Relative time and space,<br />
-            from Epicurus to Poincaré and Einstein—<br />
-            Classical Relativity—Antinomy of stellar<br />
-            aberration and the Michelson experiment</i></p></td>
-    <td class="tdr_bott"><a href="#CHAPTER_I">pp. 17-31</a></td>
-   </tr><tr>
-      <td class="tdc_space-above1" colspan="2"><b>CHAPTER II</b><br />SCIENCE IN A NO-THOROUGHFARE</td>
-   </tr><tr>
-      <td class="tdl"><p class="neg-indent">
-              <i>Scientific truth and mathematics—The precise function<br />
-              of Einstein—Michelson’s experiment, the Gordian knot<br />
-              of science—The hesitations of Poincaré—The strange,<br />
-              but necessary, Fitzgerald-Lorentz hypothesis—The<br />
-              contraction of moving bodies—Philosophical and<br />
-              physical difficulties</i> </p></td>
-      <td class="tdr_bott"><a href="#CHAPTER_II">pp. 32-52</a></td>
-   </tr><tr>
-      <td class="tdc_space-above1" colspan="2"><b>CHAPTER III</b><br />EINSTEIN’S SOLUTION</td>
-   </tr><tr>
-      <td class="tdl"><p class="neg-indent">
-             <i>Provisional rejection of ether—Relativist<br />
-              interpretation of Michelson’s experiment—New aspect<br />
-              of the speed of light—Explanation of the contraction<br />
-              of moving bodies—Time and the four dimensions of<br />
-              space—Einstein’s “Interval” the only material<br />
-              reality</i></p></td>
-      <td class="tdr_bott"><a href="#CHAPTER_III">pp. 53-72</a><span class="pagenum" id="Page_10">[Pg 10]</span></td>
-   </tr><tr>
-      <td class="tdc_space-above1" colspan="2"><b>CHAPTER IV</b><br />EINSTEIN’S MECHANICS</td>
-   </tr><tr>
-      <td class="tdl"><p class="neg-indent">
-                <i>The mechanical foundation of all the sciences—<br />
-                 Ascending the stream of time—The speed of<br />
-                 light an impassable limit—The addition of speeds<br />
-                 and Fizeau’s experiment—Variability of mass—<br />
-                 The ballistics of electrons—Gravitation and light<br />
-                 as atomic microcosms—Matter and energy—<br />
-                 The death of the sun</i></p></td>
-      <td class="tdr_bott"><a href="#CHAPTER_IV">pp. 73-100</a></td>
-   </tr><tr>
-      <td class="tdc_space-above1" colspan="2"><b>CHAPTER V</b><br />GENERALISED RELATIVITY</td>
-   </tr><tr>
-      <td class="tdl"><p class="neg-indent">
-           <i>Weight and inertia—Ambiguity of the Newtonian<br />
-           law—Equivalence of gravitation and accelerated<br />
-           movement—Jules Verne’s projectile and the principle<br />
-           of inertia—Why rays of light are subject to<br />
-           gravitation—How light from the stars is<br />
-            weighed—An eclipse as a source of light</i></p></td>
-      <td class="tdr_bott"><a href="#CHAPTER_V">pp. 101-123</a></td>
-   </tr><tr>
-      <td class="tdc_space-above1" colspan="2"><b>CHAPTER VI</b><br />THE NEW CONCEPTION<br /> OF GRAVITATION</td>
-   </tr><tr>
-      <td class="tdl"><p class="neg-indent">
-            <i>Geometry and reality—Euclid’s geometry and<br />
-             others—Contingency of Poincaré’s criterion—<br />
-             The real universe is not Euclidean but Riemannian— <br />
-             The avatars of the number π—The point of view of<br />
-             the drunken man—Straight and geodetic lines—The<br />
-             new law of universal attraction—Explanation of the<br />
-             anomaly of the planet Mercury—Einstein’s theory<br />
-             of gravitation</i></p></td>
-      <td class="tdr_bott"><a href="#CHAPTER_VI">pp. 124-147</a></td>
-   </tr><tr>
-      <td class="tdc_space-above1" colspan="2"><b>CHAPTER VII</b><br />IS THE UNIVERSE INFINITE?</td>
-   </tr><tr>
-      <td class="tdl"><p class="neg-indent">
-               <i>Kant and the number of the stars—Extinct stars and<br />
-                  dark nebulæ—Extent and aspect of the astronomical<br />
-                  universe—Different kinds of universes—Poincaré’s<br />
-                  calculation—Physical definition of the infinite—<br />
-                  The infinite and the unlimited—Stability and<br />
-                  curvature of cosmic space-time—Real and virtual<br />
-                  stars—Diameter of the Einsteinian universe—<br />
-                  The hypothesis of globes of ether</i></p></td>
-      <td class="tdr_bott"><a href="#CHAPTER_VII">pp. 148-159</a><span class="pagenum" id="Page_11">[Pg 11]</span></td>
-   </tr><tr>
-      <td class="tdc_space-above1" colspan="2"><b>CHAPTER VIII</b><br />SCIENCE AND REALITY</td>
-   </tr><tr>
-      <td class="tdl"><p class="neg-indent">
-           <i>The Einsteinian absolute—Revelation by science—<br />
-              Discussion of the experimental bases of Relativity—<br />
-              Other possible explanations—Arguments in favour<br />
-              of Lorentz’s real contraction—Newtonian space<br />
-              may be distinct from absolute space—The real is<br />
-              a privileged form of the possible—Two attitudes<br />
-              in face of the unknown</i></p></td>
-      <td class="tdr_bott"><a href="#CHAPTER_VIII">pp. 160-172</a></td>
-   </tr><tr>
-      <td class="tdc_space-above1" colspan="2"><b>CHAPTER IX</b><br />EINSTEIN OR NEWTON?</td>
-   </tr><tr>
-      <td class="tdl"><p class="neg-indent">
-            <i>Recent discussion of Relativism at the Academy<br />
-               of Sciences—Traces of the privileged space<br />
-               of Newton—The principle of causality,<br />
-               the basis of science—Examination of M.<br />
-               Painlevé’s objections—Newtonian arguments<br />
-               and Relativist replies—M. Painlevé’s formulas<br />
-               of gravitation—Fecundity of Einstein’s theory—<br />
-               Two conceptions of the world—Conclusion</i></p></td>
-      <td class="tdr_bott"><a href="#CHAPTER_IX">pp. 173-185</a></td>
-   </tr>
- </tbody>
-</table>
-
-<p><span class="pagenum" id="Page_12">[Pg 12]</span></p>
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_13">[Pg 13]</span></p>
-<h2 class="nobreak" id="INTRO">INTRODUCTION</h2>
-</div>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_14">[Pg 14]</span>
-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?</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.”</p>
-
-<p>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
-<span class="pagenum" id="Page_15">[Pg 15]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p class="author fontsize_120"><span class="smcap">Charles Nordmann.</span></p>
-
-<p><span class="pagenum" id="Page_16">[Pg 16]</span></p>
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_17">[Pg 17]</span></p>
-<p class="f150"><b>EINSTEIN AND THE UNIVERSE</b></p>
-</div>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<h2 class="nobreak" id="CHAPTER_I">CHAPTER I</h2>
-<p class="f120">THE METAMORPHOSES OF<br /> SPACE AND TIME</p>
-</div>
-
-<p class="neg-indent"><i>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.</i></p>
-
-<p>“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?”</p>
-
-<p>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.</p>
-
-<p>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 <i>integration</i> and <i>differentiation</i>.
-<span class="pagenum" id="Page_18">[Pg 18]</span></p>
-
-<p>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.</p>
-
-<p>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.<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a>
-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.</p>
-
-<hr class="tb" />
-
-<p>All our ideas, all science, and even the whole of our practical life,
-<span class="pagenum" id="Page_19">[Pg 19]</span>
-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.</p>
-
-<p>So felt Leconte de Lisle when, addressing himself to “divine death,” he
-wrote, in his profound, philosophic way:</p>
-
-<div class="poetry-container">
-<div class="poetry">
-  <div class="stanza">
-    <div class="verse indent0">Free us from time, number, and space:</div>
-    <div class="verse indent0">Grant us the rest that life hath spoiled.</div>
-  </div>
-</div>
-</div>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_20">[Pg 20]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_21">[Pg 21]</span>
-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.</p>
-
-<hr class="tb" />
-
-<p>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).</p>
-
-<p>“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
-<span class="pagenum" id="Page_22">[Pg 22]</span>
-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.”<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a> 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.</p>
-
-<p>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:</p>
-
-<p class="blockquot">“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.”<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a></p>
-
-<p>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.”</p>
-
-<p>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
-<span class="pagenum" id="Page_23">[Pg 23]</span>
-space, taken by themselves and without any reference to an external
-object, independent by their very nature.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_24">[Pg 24]</span>
-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.</p>
-
-<p>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:</p>
-
-<p>“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 <i>here</i> 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
-<span class="pagenum" id="Page_25">[Pg 25]</span>
-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.”</p>
-
-<p>Poincaré works out his idea in this way:</p>
-
-<p class="blockquot">“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.”</p>
-
-<p>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
-<span class="pagenum" id="Page_26">[Pg 26]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_27">[Pg 27]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_28">[Pg 28]</span>
-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.</p>
-
-<p>That is what is called the Principle of Classic Relativity.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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
-<big><b>V</b></big>.</p>
-
-<p>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
-<span class="pagenum" id="Page_29">[Pg 29]</span>
-tried with the same result—has shown that the question must be
-answered in the negative.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_30">[Pg 30]</span>
-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 <big><b>V</b></big> + 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 <i>nil</i> if
-the train and the cyclist were travelling at the same rate.</p>
-
-<p>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 <big><b>V</b></big> +
-30 kilometres.</p>
-
-<p>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.
-<span class="pagenum" id="Page_31">[Pg 31]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<i>the ether shares the motion of the earth</i>. We have, however, seen
-that other experiments, not less precise, had settled that <i>the ether
-does not share the motion of the earth</i>.</p>
-
-<p>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.</p>
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_32">[Pg 32]</span></p>
-<h2 class="nobreak" id="CHAPTER_II">CHAPTER II</h2>
-<p class="f120">SCIENCE IN A NO-THOROUGHFARE</p>
-</div>
-
-<p class="neg-indent"><i>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.</i></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>However paradoxical this may seem to people who regard mathematics as
-<i>of itself</i> 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
-<span class="pagenum" id="Page_33">[Pg 33]</span>
-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 <i>given facts</i>.
-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.</p>
-
-<p>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: <i>Naturæ non imperatur nisi parendo.</i></p>
-
-<p>Are not Einstein’s theories, as some imperfectly informed writers have
-suggested, only a play of mathematical formulæ (taking the word in the
-<span class="pagenum" id="Page_34">[Pg 34]</span>
-meaning given to it by both mathematicians and philosophers)? If they
-were only a towering mathematical structure in which the <i>x</i>’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.</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_35">[Pg 35]</span></p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>We may resume in the few following words the story of the origin, the
-starting-point, of Einstein’s system.</p>
-
-<div class="blockquot">
-<p class="neg-indent">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.</p>
-
-<p class="neg-indent">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.</p>
-
-<p class="neg-indent">3. Michelson’s experiment seems to prove, on the
-contrary, that the earth bears the ether with it in its movement.</p>
-</div>
-
-<p>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.</p>
-
-<p>In order to understand how that was done—which is the vital point of
-<span class="pagenum" id="Page_36">[Pg 36]</span>
-the whole system—we must retrace our steps a little and examine the
-precise conditions of Michelson’s famous experiment.</p>
-
-<p>I pointed out in the <a href="#CHAPTER_I">preceding chapter</a> 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.</p>
-
-<p>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.</p>
-
-<p>An analogy will help us to understand the matter. Suppose a violent
-<span class="pagenum" id="Page_37">[Pg 37]</span>
-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?</p>
-
-<p>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.</p>
-
-<p>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 <i>in the wind</i> 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.</p>
-
-<p>On the other hand, the aviator who flies eastward will never reach his
-<span class="pagenum" id="Page_38">[Pg 38]</span>
-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 <i>in the
-wind</i> an infinite distance.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_39">[Pg 39]</span>
-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 <i>vice
-versa</i>. 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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.
-<span class="pagenum" id="Page_40">[Pg 40]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>Let us now see what this Fitzgerald-Lorentz hypothesis, as strange as
-it is simple, really is.</p>
-
-<p>But we must first glance at a preliminary matter of some importance.
-<span class="pagenum" id="Page_41">[Pg 41]</span>
-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 <i>a
-priori</i>. 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,<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_42">[Pg 42]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>Hence, whatever may have been said to the contrary, the issue of
-<span class="pagenum" id="Page_43">[Pg 43]</span>
-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.</p>
-
-<p>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.”</p>
-
-<p>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:</p>
-
-<p class="blockquot">“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?”</p>
-
-<p>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.”</p>
-
-<p>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
-<span class="pagenum" id="Page_44">[Pg 44]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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?</p>
-
-<hr class="tb" />
-
-<p>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
-<span class="pagenum" id="Page_45">[Pg 45]</span>
-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 <i>experience</i>, 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 <i>a priori</i> from the problematical
-identity of absolute space and the ether; but this negative result does
-not justify us in denying the identity <i>a posteriori</i>.</p>
-
-<p>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.</p>
-
-<p>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, <i>this is because the two mirrors
-approached each other in the second part of the experiment</i>. To put
-it differently, <i>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</i>.</p>
-
-<p>When we repeat the experiment with all kinds of different apparatus, we
-<span class="pagenum" id="Page_46">[Pg 46]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>But it is far less easy to admit that this alteration may be exactly
-<span class="pagenum" id="Page_47">[Pg 47]</span>
-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.</p>
-
-<p>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?</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>There is a further reason for this. Even if Michelson’s apparatus were
-<span class="pagenum" id="Page_48">[Pg 48]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_49">[Pg 49]</span>
-translation of the immortal <i>Principia</i>. 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.</p>
-
-<p>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 <big><b>V</b></big> as the velocity of
-light and <i>v</i> 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</p>
-
-<table class="fontsize_120 no-wrap" border="0" cellspacing="0" summary=" " cellpadding="0" >
-  <tbody><tr>
-      <td class="tdl">&nbsp;</td>
-      <td class="tdc" rowspan="3"><span class="fontsize_200">√</span></td>
-      <td class="tdc bt" rowspan="3"><span class="fontsize_200">&nbsp;(</span></td>
-      <td class="tdc bt" rowspan="3"><span class="fontsize_150">1</span></td>
-      <td class="tdc bt">&nbsp;<i>v</i>²</td>
-      <td class="tdc bt" rowspan="3"><span class="fontsize_200">)</span></td>
-      <td class="tdc" rowspan="3">&nbsp;.</td>
-   </tr><tr>
-      <td class="tdl">1 to&nbsp;&nbsp;</td>
-      <td class="tdc">&nbsp;&nbsp;——</td>
-   </tr><tr>
-      <td class="tdl">&nbsp;</td>
-      <td class="tdc">&nbsp;V²</td>
-   </tr>
- </tbody>
-</table>
-
-<p class="no-indent">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</p>
-
-<table class="fontsize_120 no-wrap" border="0" cellspacing="0" summary=" " cellpadding="0" >
-  <tbody><tr>
-      <td class="tdl">&nbsp;</td>
-      <td class="tdc" rowspan="3"><span class="fontsize_200">√</span></td>
-      <td class="tdc bt" rowspan="3"><span class="fontsize_200">&nbsp;(</span></td>
-      <td class="tdc bt" rowspan="3"><span class="fontsize_150">1</span></td>
-      <td class="tdc bt">1</td>
-      <td class="tdc bt" rowspan="3"><span class="fontsize_200">)</span></td>
-      <td class="tdc" rowspan="3">&nbsp;.</td>
-   </tr><tr>
-      <td class="tdl">1 to&nbsp;&nbsp;</td>
-      <td class="tdc">&nbsp;&nbsp;——————</td>
-   </tr><tr>
-      <td class="tdl">&nbsp;</td>
-      <td class="tdc">100,000,000</td>
-   </tr>
- </tbody>
-</table>
-
-<p class="no-indent">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.</p>
-
-<p>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
-<span class="pagenum" id="Page_50">[Pg 50]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_51">[Pg 51]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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,
-<span class="pagenum" id="Page_52">[Pg 52]</span>
-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.</p>
-
-<p>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?</p>
-
-<p>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.</p>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_53">[Pg 53]</span></p>
-<h2 class="nobreak" id="CHAPTER_III">CHAPTER III</h2>
-<p class="f120">EINSTEIN’S SOLUTION</p>
-</div>
-
-<p class="neg-indent"><i>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.</i></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>We will now follow his example. We shall no longer, in the course of
-<span class="pagenum" id="Page_54">[Pg 54]</span>
-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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_55">[Pg 55]</span>
-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 <i>nil</i>.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>Unfortunately—perhaps we ought to say fortunately, because, after all,
-<span class="pagenum" id="Page_56">[Pg 56]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_57">[Pg 57]</span></p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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 <i>simultaneously</i>.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_58">[Pg 58]</span>
-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.</p>
-
-<p>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 <i>yes</i>. Yet according to
-Einstein the answer is <i>no</i>.</p>
-
-<p>Here is the simple proof, as we deduce it from Einstein’s general idea.</p>
-
-<p>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
-<span class="pagenum" id="Page_59">[Pg 59]</span>
-coming to me from the blue peg. This front ray reaches my eye <i>at
-the same time</i> 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 <i>after</i> 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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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,
-<span class="pagenum" id="Page_60">[Pg 60]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_61">[Pg 61]</span>
-He cannot learn how it behaves during its passage across the
-intermediate space, in which there is no matter.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_62">[Pg 62]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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 <i>relations</i> of these dimensions which were real.</p>
-
-<p>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.</p>
-
-<p>Spatial relations themselves are relative, in a space which is already
-relative.
-<span class="pagenum" id="Page_63">[Pg 63]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>The fact that these alterations of shape owing to velocity are
-reciprocal is one of the most curious consequences of all this. A man
-<span class="pagenum" id="Page_64">[Pg 64]</span>
-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.</p>
-
-<p>Which is right, and which wrong? Both. Each point of view is accurate,
-but there are only personal points of view.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_65">[Pg 65]</span>
-watch it pass, though our watches may be just the same as
-theirs.<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a>
-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.</p>
-
-<p>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!</p>
-
-<hr class="tb" />
-
-<p>Formerly, before the Einsteinian <i>hegira</i>, before the Relativist
-Era opened, everybody was convinced that the portion of <i>space</i>
-occupied by an object was sufficiently and explicitly defined by its
-dimensions—length, breadth, and height. These are what are called
-the three <i>dimensions</i> 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.
-<span class="pagenum" id="Page_66">[Pg 66]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>The dimensions of an object, its shape, the apparent <i>space</i>
-occupied by it, depend upon its velocity: that is to say, upon the
-<i>time</i> which the observer takes to traverse a certain distance
-relatively to the object. Here we have <i>space</i> already depending
-upon <i>time</i>. In addition, the observer measures the time with
-a chronometer, the seconds of which are more or less accelerated
-according to his velocity.</p>
-
-<p>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 <i>Encyclopédie</i>, in
-the article “Dimension”:</p>
-
-<p>“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.
-<span class="pagenum" id="Page_67">[Pg 67]</span>
-The idea may not be admitted, but it seems to be not without merit, if
-it be only the merit of originality.”</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>Thus velocity (velocity relatively to the things observed, we must
-always remember) acts in a sense as a double brake lessening durations
-<span class="pagenum" id="Page_68">[Pg 68]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_69">[Pg 69]</span>
-eye the whole chronological and spatial series of moves that may follow
-the first move, with all their effects on the board. He <i>sees the
-whole series simultaneously</i>.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_70">[Pg 70]</span>
-world without space and time than we can observe certain microbes under
-the microscope without first injecting colouring matter into them.</p>
-
-<p>Are time and space, then, merely hallucinations? And, if so, what
-<i>is</i> real?</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_71">[Pg 71]</span>
-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.<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a></p>
-
-<p>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.</p>
-
-<p>It is this quantity which Einstein has called the <i>Interval</i>
-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.”</p>
-
-<p>The sole reality accessible to man in the external world, the one
-really objective and impersonal thing which is comprehensible, is the
-Einsteinian <i>Interval</i> as we have defined it. The <i>Interval</i>
-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.
-<span class="pagenum" id="Page_72">[Pg 72]</span></p>
-
-<p>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.</p>
-
-<p>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 <i>Interval</i> 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.</p>
-
-<p>It must, in fine, be clearly understood that the <i>Einsteinian
-Interval</i> 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.</p>
-
-<p>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.</p>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_73">[Pg 73]</span></p>
-<h2 class="nobreak" id="CHAPTER_IV">CHAPTER IV</h2>
-<p class="f120">EINSTEIN’S MECHANICS</p>
-</div>
-
-<p class="neg-indent"><i>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.</i></p>
-
-<p class="space-above1">When Baudelaire wrote:</p>
-
-<div class="poetry-container">
-<div class="poetry">
-  <div class="stanza">
-    <div class="verse indent0">I hate the movement that displaces lines,</div>
-  </div>
-</div>
-</div>
-
-<p class="no-indent">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.</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_74">[Pg 74]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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 <i>a priori</i> 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,
-<span class="pagenum" id="Page_75">[Pg 75]</span>
-there is the mechanical element—exclusive in some, the principal
-element in others.</p>
-
-<p>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.</p>
-
-<p>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 <i>bodies</i>. 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.</p>
-
-<p>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 <i>fiancée</i>
-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
-<span class="pagenum" id="Page_76">[Pg 76]</span>
-compare the delicious dishes which they respectively make from it for
-the nourishment of the mind.</p>
-
-<p>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:</p>
-
-<div class="poetry-container">
-<div class="poetry">
-  <div class="stanza">
-    <div class="verse indent0">unde omnia credita pendent,</div>
-  </div>
-</div>
-</div>
-
-<p class="no-indent">or the material facts, is going to judge Einstein’s
-system for us.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_77">[Pg 77]</span>
-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 <i>ascending</i> the stream of time.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>The experiment we have imagined could not very well be carried out
-with the actual resources of our laboratories. We cannot attain these
-<span class="pagenum" id="Page_78">[Pg 78]</span>
-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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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
-<big><b>V</b></big>, what will it do if the action of the force is prolonged for
-<span class="pagenum" id="Page_79">[Pg 79]</span>
-another second? According to classical mechanics it will take on the
-velocity <b>2V</b>.<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a>
-Let us imagine an observer who is travelling at the velocity
-<big><b>V</b></big>, 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 <big><b>V</b></big>, and, as the observer
-already has the velocity <big><b>V</b></big>, the absolute velocity of
-the body will be <b>2V</b>. In the same way it will be <b>3V</b> at
-the end of three seconds, <b>4V</b> 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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_80">[Pg 80]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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, <i>relatively to him</i>, 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 <a href="#CHAPTER_II">Chapter II</a>, which gives
-us the value of the contraction due to velocity. We then easily find that, if we
-<span class="pagenum" id="Page_81">[Pg 81]</span>
-have two velocities, <i>v</i> and <i>v</i>₂, and if we call the
-resultant <i>w</i>, classical mechanics stated that</p>
-
-<p class="f150"><i>w</i> = <i>v</i>₁ + <i>v</i>₂</p>
-
-<p>The Einstein mechanics says that this is not correct, and that what we
-really have (C being the velocity of light) is</p>
-
-<table class="fontsize_150 no-wrap" border="0" cellspacing="0" summary=" " cellpadding="0" >
-  <tbody><tr>
-      <td class="tdl">&nbsp;</td>
-      <td class="tdc">&nbsp;<i>v</i>₁</td>
-      <td class="tdc">+</td>
-      <td class="tdc"><i>v</i>₂</td>
-   </tr><tr>
-      <td class="tdl"><i>w</i> =&nbsp;</td>
-      <td class="tdc" colspan="3">&nbsp;&nbsp;————</td>
-   </tr><tr>
-      <td class="tdc">&nbsp;</td>
-      <td class="tdc" rowspan="2">1</td>
-      <td class="tdc" rowspan="2">+</td>
-      <td class="tdc bb"><i>v</i>₁<i>v</i>₂</td>
-   </tr><tr>
-      <td class="tdl">&nbsp;</td>
-      <td class="tdc">C²</td>
-   </tr>
- </tbody>
-</table>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p><span class="pagenum" id="Page_82">[Pg 82]</span></p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_83">[Pg 83]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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
-<span class="pagenum" id="Page_84">[Pg 84]</span>
-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.</p>
-
-<p>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 <i>inertia</i>. 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.</p>
-
-<p>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 <i>mass</i> of
-bodies, which is the measure of their inertia.</p>
-
-<p>If in our example the locomotive produces in the second case an
-acceleration only half as great, we express this by saying that the
-<span class="pagenum" id="Page_85">[Pg 85]</span>
-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.</p>
-
-<p>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.”</p>
-
-<p>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 <i>mass</i>. 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!</p>
-
-<p>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
-<i>coefficient</i>.</p>
-
-<p>Here again, unfortunately, we have to recant—or, perhaps, we should
-say fortunately, as there is no pleasure like that of novelty.</p>
-
-<p>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
-<span class="pagenum" id="Page_86">[Pg 86]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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 <i>the
-mass of bodies is not constant: it depends upon their velocity, and
-increases with an increase of velocity</i>.</p>
-
-<p>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 <i>perceptibly</i> constant, and believed they
-might conclude that it was <i>absolutely</i> constant. In the case of
-greater velocities that is not so.</p>
-
-<p>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.</p>
-
-<p>Mass, therefore, increases rapidly with velocity. It becomes infinite
-when the velocity equals that of light. No body whatever can attain or
-<span class="pagenum" id="Page_87">[Pg 87]</span>
-surpass the velocity of light, because, in order to pass that limit, it
-would need to overcome an infinite resistance.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_88">[Pg 88]</span>
-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 <i>electrons</i>.</p>
-
-<p>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.</p>
-
-<p>But how was it possible to measure the speed of these projectiles?</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_89">[Pg 89]</span>
-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.</p>
-
-<p>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).</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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 <i>self-induction</i>. 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
-<span class="pagenum" id="Page_90">[Pg 90]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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. <i>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.</i> 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
-<span class="pagenum" id="Page_91">[Pg 91]</span>
-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.</p>
-
-<p>The experiment has been made repeatedly by physicists of distinction.
-The result of it is surprising: the real mass is <i>nil</i>, 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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_92">[Pg 92]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_93">[Pg 93]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_94">[Pg 94]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>Experience, “sole source of truth,” has given its verdict.</p>
-
-<p>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”
-<span class="pagenum" id="Page_95">[Pg 95]</span>
-has no more reality than the time and space in which we thought we
-located it! Our solid realities were but phantoms.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>There is, in fine, another fundamental idea of mechanics, that of
-<i>energy</i>, which takes on a new aspect in the light of Einstein’s
-theory: an aspect which, in turn, is largely justified by experiment.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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,
-<span class="pagenum" id="Page_96">[Pg 96]</span>
-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 <i>positive nucleus</i>, 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.</p>
-
-<p>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 <i>positive nucleus</i> 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.</p>
-
-<p>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.”
-<span class="pagenum" id="Page_97">[Pg 97]</span></p>
-
-<p>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!</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_98">[Pg 98]</span>
-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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>To finish with Einstein’s mechanics, let me reproduce a very suggestive
-application of these ideas about the identity of energy and mass.</p>
-
-<p>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
-<span class="pagenum" id="Page_99">[Pg 99]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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
-<span class="pagenum" id="Page_100">[Pg 100]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>“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.”</p>
-
-<p>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.”</p>
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_101">[Pg 101]</span></p>
-<h2 class="nobreak" id="CHAPTER_V">CHAPTER V</h2>
-<p class="f120">GENERALISED RELATIVITY</p>
-</div>
-
-<p class="neg-indent"><i>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.</i></p>
-
-<p class="space-above1">We are now on the threshold of the great
-mystery of gravitation.</p>
-
-<p>In the <a href="#CHAPTER_IV">preceding chapter</a> 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.</p>
-
-<p>The theory of Relativity, however, has up to the present not touched a
-phenomenon that is fundamental, essential, ubiquitous in our cosmos. I
-<span class="pagenum" id="Page_102">[Pg 102]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.”</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<i>freely</i> in space?
-<span class="pagenum" id="Page_103">[Pg 103]</span></p>
-
-<p><i>Because of the principle of inertia.</i> 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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>We shall see later how the principle of inertia of Einstein’s
-<span class="pagenum" id="Page_104">[Pg 104]</span>
-generalised mechanics has a strange affinity to this idea, and at the
-same time to the curious declination, the <i>clinamen</i>, which the
-great and profound Lucretius attributed to the free path of the atoms.
-But we must not anticipate.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>I am aware that Newton grounded his principle on astronomical as
-well as terrestrial observations. He noticed that, <i>apart from any
-attraction by other celestial bodies</i>, 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
-<i>freely</i>; that they are governed in their motions by a force which
-he called universal attraction.</p>
-
-<p>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.
-<span class="pagenum" id="Page_105">[Pg 105]</span></p>
-
-<p>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 <i>petitio principii</i>—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.</p>
-
-<p>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 <i>Hypotheses non fingo</i>, 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
-<i>a priori</i> easier to admit—that there is a special force of
-gravitation.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_106">[Pg 106]</span>
-synthesis, and enabled us to see more clearly the sublime unity of the world.</p>
-
-<hr class="tb" />
-
-<p>To tell the truth—let us premise this before we go further into the
-profound and marvellous truths of General Relativity—it is <i>a
-priori</i> evident that Newton’s law of universal attraction can no
-longer be considered satisfactory.</p>
-
-<p>It says: <i>Bodies attract each other in direct proportion to their
-masses and in inverse proportion to the square of their distances.</i>
-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?</p>
-
-<p>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
-<span class="pagenum" id="Page_107">[Pg 107]</span>
-distances vary, as we saw with Einstein, according to the relative
-velocity of the observer.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>Laplace wrote, in his <i>Exposition du Système du Monde</i>: “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.</p>
-
-<hr class="tb" />
-
-<p>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)
-<span class="pagenum" id="Page_108">[Pg 108]</span>
-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.</p>
-
-<p>Let us return to the illustration which I used in a <a href="#CHAPTER_IV">previous
-chapter</a> 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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_109">[Pg 109]</span>
-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:</p>
-
-<div class="poetry-container">
-<div class="poetry">
-  <div class="stanza">
-    <div class="verse indent4">Nulli, de nulla parte, neque ullo</div>
-    <div class="verse indent0">Tempore, inane potest vacuum subsistere rei,</div>
-    <div class="verse indent0">Quin sua quod natura petit concedere pergat.</div>
-    <div class="verse indent0">Omnia quapropter debent per inane quietum</div>
-    <div class="verse indent0">Æque ponderibus non æquis concita ferri.<a id="FNanchor_8" href="#Footnote_8" class="fnanchor">[8]</a></div>
-  </div>
-</div>
-</div>
-
-<p>Now if weight were a <i>force</i> 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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_110">[Pg 110]</span>
-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:</p>
-
-<div class="poetry-container">
-<div class="poetry">
-  <div class="stanza">
-    <div class="verse indent0">An thou wilt, let us dream a dream.</div>
-  </div>
-</div>
-</div>
-
-<p>Let us suppose our dream fulfilled. The lift falls from above with
-precisely the accelerated velocity of an object falling in a vacuum.</p>
-
-<p>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:</p>
-
-<div class="poetry-container">
-<div class="poetry">
-  <div class="stanza">
-    <div class="verse indent2">No blade of grass had felt</div>
-    <div class="verse indent0">The light traces of her steps.</div>
-  </div>
-</div>
-</div>
-
-<p>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.
-<span class="pagenum" id="Page_111">[Pg 111]</span></p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>But let us overlook these trifles and return to the admirable
-illustration he has prophetically provided for our convenience in
-explaining Einstein’s system.</p>
-
-<p>Let us take the projectile when it begins to fall freely toward the
-moon.<a id="FNanchor_9" href="#Footnote_9" class="fnanchor">[9]</a>
-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.</p>
-
-<p>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
-<span class="pagenum" id="Page_112">[Pg 112]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_113">[Pg 113]</span>
-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.</p>
-
-<p>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.”</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_114">[Pg 114]</span>
-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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>This now enables us to establish very simply the following fundamental
-fact, unknown only a few years ago, but now brilliantly proved by
-experiment: <i>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.</i></p>
-
-<p>We showed in one of the <a href="#CHAPTER_IV">preceding chapters</a>
-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.</p>
-
-<p>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.</p>
-
-<p>In point of fact, if something in the universe is an <i>invariant</i>,
-as physicists say, or invariable, for the observers who move at
-<span class="pagenum" id="Page_115">[Pg 115]</span>
-different speeds, this something must <i>naturally</i> 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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_116">[Pg 116]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_117">[Pg 117]</span>
-travels in a straight line, is that on account of the enormous velocity
-of light its trajectory is only very slightly curved by weight.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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,
-<span class="pagenum" id="Page_118">[Pg 118]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_119">[Pg 119]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_120">[Pg 120]</span>
-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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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 <i>was</i> proved, and very
-decisively, by the observations.</p>
-
-<p>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
-<span class="pagenum" id="Page_121">[Pg 121]</span>
-night and far from the sun. As everybody knows, the sun passes
-successively, in its annual course, through the different
-constellations of the zodiac.</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_122">[Pg 122]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_123">[Pg 123]</span>
-interpretation, therefore, is that the phenomena are due to the effect
-of weight upon light.</p>
-
-<p>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.</p>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_124">[Pg 124]</span></p>
-<h2 class="nobreak" id="CHAPTER_VI">CHAPTER VI</h2>
-<p class="f120">THE NEW CONCEPTION OF GRAVITATION</p>
-</div>
-
-<p class="neg-indent"><i>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.</i></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_125">[Pg 125]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.<a id="FNanchor_10" href="#Footnote_10" class="fnanchor">[10]</a>
-We may therefore say that practically a straight line is the path
-followed by light in a homogeneous medium.</p>
-
-<p>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?</p>
-
-<p>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
-<span class="pagenum" id="Page_126">[Pg 126]</span>
-Euclid, but they are based upon different axioms and postulates—in a
-word, different definitions.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>A question that we may legitimately ask is: Does the real universe
-correspond to the classical geometry of Euclid or to that of Riemann?</p>
-
-<p>It was long believed that it corresponded to Euclid’s geometry.
-Poincaré himself, speaking of Euclid’s system, said:</p>
-
-<p class="blockquot">“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.”</p>
-
-<p>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
-<span class="pagenum" id="Page_127">[Pg 127]</span>
-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.</p>
-
-<p>“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.</p>
-
-<p>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 <i>convenient</i> representation
-of the real world.</p>
-
-<p>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? <i>The answer to this question is in the negative.</i></p>
-
-<p>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
-<span class="pagenum" id="Page_128">[Pg 128]</span>
-realise the first condition of parallelism—coexistence—in the same
-local plane.</p>
-
-<p>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.</p>
-
-<p>Kant regarded the truths—to be accurate, the deductive
-affirmations—of the Euclidean geometry as “synthetic judgments <i>a
-priori</i>,” 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.</p>
-
-<p>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.</p>
-
-<p>There were many philosophers who ridiculed Gauss and his experiments.
-With the <i>a priori</i> dogmatism that one so often encounters amongst
-<span class="pagenum" id="Page_129">[Pg 129]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>There are many other reasons why the universe, in consequence of
-gravitation, does not conform to the laws of Euclid’s geometry.</p>
-
-<p>For instance, in the Euclidean geometry the extent of the circumference
-has a well-known proportion to its diameter, and this is indicated by
-<span class="pagenum" id="Page_130">[Pg 130]</span>
-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?<a id="FNanchor_11" href="#Footnote_11" class="fnanchor">[11]</a>
-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.</p>
-
-<p>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.</p>
-
-<p>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,
-<span class="pagenum" id="Page_131">[Pg 131]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_132">[Pg 132]</span></p>
-
-<p>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.</p>
-
-<p>The difference is generally very slight, it is true. But there
-<i>is</i> 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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_133">[Pg 133]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_134">[Pg 134]</span></p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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:</p>
-
-<p>Since the “Interval” of two events is the same for two observers moving
-at uniform and different velocities, it is <i>natural</i> 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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_135">[Pg 135]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.<a id="FNanchor_12" href="#Footnote_12" class="fnanchor">[12]</a>
-The houses, for instance, will seem narrower
-in proportion to their height. Nevertheless the general aspect and
-<span class="pagenum" id="Page_136">[Pg 136]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>Yet there is something in the aspect of the world that must be common
-to the drunkard and the drinker of water.</p>
-
-<p>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.
-<span class="pagenum" id="Page_137">[Pg 137]</span></p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_138">[Pg 138]</span>
-proportions and substituted irregular and wavy lines for straight lines.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>Here is an illustration which will, I think, lead us to the heart of
-the problem of gravitation and to the solution of it.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>But we have reached the hilly district. It is now impossible for us to
-<span class="pagenum" id="Page_139">[Pg 139]</span>
-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 <i>geodetical</i> of the surface covered.</p>
-
-<p>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 <i>geodetical</i>, the shortest route between two points, is the
-arc of a great circle passing through the two points.</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_140">[Pg 140]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_141">[Pg 141]</span>
-the previously separated sisters, inertia and weight: <i>Every moving
-body freely left to itself in the universe describes a geodetic.</i></p>
-
-<p>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!</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_142">[Pg 142]</span>
-that the curvature of the universe is caused at each point by the
-presence or vicinity of material masses.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_143">[Pg 143]</span></p>
-
-<hr class="tb" />
-
-<p>It is surprising that Newton and Einstein agree in expressing the
-movements of gravitating stars in an <i>almost</i> identical form,
-because their starting-points are very different.</p>
-
-<p>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.</p>
-
-<p>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 <i>almost</i> identical with Newton’s.</p>
-
-<p>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.</p>
-
-<p>But there is another divergence between the two laws. According to the
-Newtonian law the planets revolving round the sun describe ellipses
-<span class="pagenum" id="Page_144">[Pg 144]</span>
-which—neglecting the small perturbations due to the other
-planets—have a rigorously fixed position.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_145">[Pg 145]</span></p>
-
-<p>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.</p>
-
-<p>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!</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_146">[Pg 146]</span></p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>Whatever modifications it may undergo in the future—for everything in
-<span class="pagenum" id="Page_147">[Pg 147]</span>
-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.</p>
-
-<p>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.</p>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_148">[Pg 148]</span></p>
-
-<h2 class="nobreak" id="CHAPTER_VII">CHAPTER VII</h2>
-<p class="f120">IS THE UNIVERSE INFINITE?</p>
-</div>
-
-<p class="neg-indent"><i>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.</i></p>
-
-<p class="space-above1">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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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;
-<span class="pagenum" id="Page_149">[Pg 149]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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?</p>
-
-<p>There are those who deny <i>a priori</i> 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.</p>
-
-<p>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
-<span class="pagenum" id="Page_150">[Pg 150]</span>
-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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_151">[Pg 151]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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?</p>
-
-<p>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
-<span class="pagenum" id="Page_152">[Pg 152]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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?</p>
-
-<p>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.”</p>
-
-<p>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
-<span class="pagenum" id="Page_153">[Pg 153]</span>
-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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_154">[Pg 154]</span>
-exact for very remote masses; and it cannot be finite except on the
-condition that it is ephemeral in point of time.</p>
-
-<p>For Newton, indeed, the <i>stellar</i> 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.</p>
-
-<hr class="tb" />
-
-<p>These difficulties and obscurities disappear in great part when we
-consider space, or space-time, from the Einsteinian standpoint of
-General Relativity.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_155">[Pg 155]</span>
-(a spherical surface), to see how the universe may be at one and the same
-time finite and unlimited.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_156">[Pg 156]</span>
-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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_157">[Pg 157]</span>
-rising to giddy heights that were far beyond the reach of the golden
-wings of fantasy.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>We may therefore prefer not to commit ourselves without further
-instructions. We may not feel ourselves obliged to choose between the
-<span class="pagenum" id="Page_158">[Pg 158]</span>
-two conceptions, and we may leave the benefit of the doubt to whichever
-of the two is false.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_159">[Pg 159]</span>
-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.</p>
-
-<p>“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!</p>
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_160">[Pg 160]</span></p>
-<h2 class="nobreak" id="CHAPTER_VIII">CHAPTER VIII</h2>
-<p class="f120">SCIENCE AND REALITY</p>
-</div>
-
-<p class="neg-indent"><i>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.</i></p>
-
-<p class="space-above1">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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>Up to this point the theory of Relativity well deserves its name. But
-<span class="pagenum" id="Page_161">[Pg 161]</span>
-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 <i>absolute</i> 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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_162">[Pg 162]</span>
-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.</p>
-
-<hr class="tb" />
-
-<p>But before we go further into it, before we turn the raw light upon it,
-let us make one observation.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_163">[Pg 163]</span>
-everything. If you ram the ground floor of a house and bring it down,
-you bring down also the upper stories.</p>
-
-<p>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.</p>
-
-<p>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?</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>In sum, the whole Einsteinian synthesis flows from the issue of the
-Michelson experiment, or at least from a particular interpretation of
-that issue.</p>
-
-<p>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
-<span class="pagenum" id="Page_164">[Pg 164]</span>
-the air we breathe; for without this medium the heat of the sun, mother
-and nurse of all terrestrial life, would never reach us.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_165">[Pg 165]</span>
-direction of the movement. This contraction increases with their
-velocity in the ether, which explains the negative result of the
-Michelson experiment.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>Hence the two immediate possible explanations of the negative result of
-the Michelson experiment are:</p>
-
-<div class="blockquot">
-<p>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.</p>
-
-<p>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.</p>
-</div>
-<p><span class="pagenum" id="Page_166">[Pg 166]</span></p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>It is like a boat crossing the Lake of Geneva at a certain speed. About
-<span class="pagenum" id="Page_167">[Pg 167]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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:
-<span class="pagenum" id="Page_168">[Pg 168]</span></p>
-
-<div class="blockquot">
-<p>1. The contraction of bodies by velocity is real (Lorentz).</p>
-
-<p>2. The contraction of bodies by velocity is only an appearance due
-to the laws of the propagation of light (Einstein).</p>
-
-<p>3. The contraction of bodies by velocity is neither real nor
-apparent: there is no such thing (hypothesis of super-ether connected
-with ether).</p>
-</div>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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?</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_169">[Pg 169]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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 <a href="#CHAPTER_VII">preceding chapter</a>,
-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.”</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_170">[Pg 170]</span>
-grammar; and such things have never succeeded in revolutionising science.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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:</p>
-
-<p>“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
-<span class="pagenum" id="Page_171">[Pg 171]</span>
-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.’</p>
-
-<p>“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.</p>
-
-<p>“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.”</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_172">[Pg 172]</span>
-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.”</p>
-
-<p>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.”</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="chapter">
-<p><span class="pagenum" id="Page_173">[Pg 173]</span></p>
-<h2 class="nobreak" id="CHAPTER_IX">CHAPTER IX</h2>
-<p class="f120">EINSTEIN OR NEWTON?</p>
-</div>
-
-<p class="neg-indent"><i>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.</i></p>
-
-<p class="space-above1">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?</p>
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_174">[Pg 174]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_175">[Pg 175]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>One might also give a different definition of the privileged system,
-<span class="pagenum" id="Page_176">[Pg 176]</span>
-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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.
-<span class="pagenum" id="Page_177">[Pg 177]</span></p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>M. Painlevé observes, with some reason, that in this form the principle
-<span class="pagenum" id="Page_178">[Pg 178]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>To the critical discussion of Relativity which he so brilliantly
-<span class="pagenum" id="Page_179">[Pg 179]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="tb" />
-
-<p>As far as the principles are concerned, the controversy will go on. It
-must end in something like the following dialogue:
-<span class="pagenum" id="Page_180">[Pg 180]</span></p>
-
-<p><i>The Newtonian</i>: 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.</p>
-
-<p><i>The Relativist</i>: 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.</p>
-
-<p><i>The Newtonian</i>: 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.</p>
-
-<p><i>The Relativist</i>: 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.</p>
-
-<p><i>The Newtonian</i>: For my part, I can easily imagine a single
-material point in the universe having a certain position and a certain
-velocity in it.
-<span class="pagenum" id="Page_181">[Pg 181]</span></p>
-
-<p><i>The Relativist</i>: 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.</p>
-
-<p><i>The Newtonian</i>: That is not my opinion.</p>
-
-<p><i>The Impartial Spectator</i>: 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?</p>
-
-<p><i>The Newtonian and the Relativist</i> (together): No, unhappily.</p>
-
-<p><i>The Metaphysician</i> (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.</p>
-
-<p><i>The Newtonian and the Relativist</i> (together): In that case we are
-quite sure we shall never learn anything further about it than we know
-or believe now.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_182">[Pg 182]</span>
-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.</p>
-
-<p>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.”</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_183">[Pg 183]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_184">[Pg 184]</span>
-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.</p>
-
-<hr class="tb" />
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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
-<span class="pagenum" id="Page_185">[Pg 185]</span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<p class="center"><i>Printed in Great Britain by<br />
-Hazell, Watson &amp; Viney, Ld.,<br />
-London and Aylesbury.</i></p>
-
-<hr class="chap x-ebookmaker-drop" />
-
-<div class="footnotes">
-<p class="f150"><b>Footnotes:</b></p>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a>
-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.</p>
-</div>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a>
-<i>Physics</i>, bk. iv, ch. xiv.</p>
-</div>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a>
-<i>De Natura Rerum</i>, bk. i, vv. 460 ff.</p>
-</div>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a>
-It is assumed that the ship is not rolling or pitching,
-and that there is no vibration in the train.</p>
-</div>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a>
-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.</p>
-</div>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a>
-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.</p>
-</div>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a>
-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.</p>
-</div>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_8" href="#FNanchor_8" class="label">[8]</a>
-<i>De Natura Rerum</i>, bk. ii, vv. 235-40.</p>
-</div>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_9" href="#FNanchor_9" class="label">[9]</a>
-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.</p>
-</div>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_10" href="#FNanchor_10" class="label">[10]</a>
-It goes without saying that in all this we assume that the
-luminous ray travels in a homogeneous medium.</p>
-</div>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_11" href="#FNanchor_11" class="label">[11]</a>
-We are, of course, imagining the earth as perfectly
-circular, without irregularities.</p>
-</div>
-
-<div class="footnote"><p class="no-indent">
-<a id="Footnote_12" href="#FNanchor_12" class="label">[12]</a>
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