,transcriber's ,notes3 ,underscores 80 before and after a word or phrase indicate italics in the original text4 ,equal signs 80 before and after a word or phrase indicate bold in the original text4 ,small capitals have been converted to ,,solid capitals4 ,typographical and punctuation errors have been silently corrected4 ,,einstein ,,and ,,the ,,universe ,a ,popular ,exposition of the ,famous ,theory ,by ,,charles ,,nordmann ,astronomer to the ,paris ,observatory4 ,translated by ,,joseph ,mc,c,a,b,e ,with a ,preface by the ,rt4 ,hon4 ,,the ,,viscount ,,haldane1 ,o4,m4 ,t4 ,,fisher ,,unwin ,,ltd4 ,,london3 ,,adelphi ,,terrace ,first published in ,english ,april #aibb ,second ,impression ,june #aibb ,all rights reserved ,,preface ,a distinguished ,german authority on mathematical physics1 writing recently on the theory of ,relativity1 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 formula4 ,such success is conceivable4 ,mathematical methods present1 however1 two advantages4 ,their terminology is precise and concentrated1 in a fashion which ordinary language cannot afford to adopt4 ,further1 the symbols which result from their employment have implications which1 when brought to light1 yield new knowledge4 ,this is deductively reached1 but it is none the less new knowledge4 ,with greater precision than is usual1 ordinary language may be made to do some1 if not a great deal1 of this work for which mathematical methods are alone quite appropriate4 ,if ordinary language can do part of it an advantage may be gained4 ,the difficulty that attends mathematical symbolism is the accompanying tendency to take the symbol as exhaustively descriptive of reality4 ,now it is not so descriptive4 ,it always embodies an abstraction4 ,it accordingly leads to the use of metaphors which are inadequate and generally untrue4 ,it is only qualification by descriptive language of a wider range that can keep this tendency in check4 ,a new school of mathematical physicists1 still1 however1 small in number1 is beginning to appreciate this4 ,but for ,english and ,german writers the new task is very difficult4 ,neither ,anglo-,saxon nor ,saxon genius lends itself readily in this direction4 ,nor has the task as yet been taken in hand completely1 so far as ,i am aware1 in ,france4 ,still1 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 ,germans4 ,lucidity in expression is an endowment which the best ,french writers possess in a higher degree than we do4 ,some of us have accordingly awaited with deep interest ,french renderings of the difficult doctrine of ,einstein4 ,m4 ,nordmann1 in addition to being a highly qualified astronomer and mathematical-physicist1 possesses the gift of his race4 ,the ,latin capacity for eliminating abstractness from the description of facts is everywhere apparent in his writing4 ,individual facts take the places of general conceptions1 of ,begriffe4 ,the language is that of the ,vorstellung1 in a way that would hardly be practicable in ,german4 ,nor is our own language equal to that of ,france in delicacy of distinctive description4 ,this book could hardly have been written by an ,englishman4 ,but the difficulty in his way would have been one as much of spirit as of letter4 ,it is the lucidity of the ,french author1 in combination with his own gift of expression1 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 contains4 ,the rendering seems to me1 after reading the book both in ,french and in ,english1 admirable4 ,m4 ,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 in4 ,remembering ,goethe's maxim that he who would accomplish anything must limit himself1 he has not aimed at covering the full field to which ,einstein's teaching is directed4 ,but he succeeds in making many abstruse things intelligible to the layman4 ,perhaps the most brilliant of his efforts in this direction are ,chapters ,v and ,,vi1 in which he explains with extraordinary lucidity the new theory of gravitation and of its relation to inertia4 ,i think that ,m4 ,nordmann is perhaps less successful in the courageous attack he makes in his third chapter on the obscurity which attends the notion of the 8,interval40 ,but that is because the four-dimensional world1 which is the basis of experience of space and time for ,einstein and ,minkowski1 is in itself an obscure conception4 ,mathematicians talk about it gaily and throw its qualities into equations1 despite the essential exclusion from it of the measurement and shape which actual experience always in some form involves4 ,they lapse on that account into unconscious metaphysics of a dubious character4 ,this does not destroy the practical value of their equations1 but it does make them very unreliable as guides to the character of reality in the meaning which the plain man attaches to it4 ,here1 accordingly1 we find the author of this little treatise to be a good man struggling with adversity4 ,if he could make the topic clear he would4 ,but then no one has made it clear excepting as an abstraction which works1 but which1 despite suggestions made to the contrary1 cannot be clothed for us in images4 ,this1 however1 is the fault1 not of ,m4 ,nordmann himself1 but of a phase of the subject4 ,with the subject in its other aspects he deals with the incomparable lucidity of a ,frenchman4 ,i know no book better adapted than the one now translated to give the average ,english reader some understanding of a principle1 still in its infancy1 but destined1 as ,i believe1 to transform opinion in more regions of knowledge than those merely of mathematical physics4 ,haldane ,,contents ,,preface ,,by ,,the ,,rt4 ,,hon4 ,,viscount ,,haldane1 ,o4,m4 pp4 #e-h ,,introduction pp4 #ac-ae ,,chapter ,i ,,the ,,metamorphoses ,,of ,,space ,,and ,,time ,removing the mathematical difficulties--,the pillars of knowledge--,absolute time and space1 from ,aristotle to ,newton--,relative time and space1 from ,epicurus to ,poincar and ,einstein--,classical ,relativity--,antinomy of stellar aberration and the ,michelson experiment pp4 #ag-ca ,,chapter ,,ii ,,science ,,in ,a ,,no-,,thoroughfare ,scientific truth and mathematics--,the precise function of ,einstein--,michelson's experiment1 the ,gordian knot of science--,the hesitations of ,poincar--,the strange1 but necessary1 ,fitzgerald-,lorentz hypothesis--,the contraction of moving bodies--,philosophical and physical difficulties pp4 #cb-eb ,,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 8,interval0 the only material reality pp4 #ec-gb ,,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 pp4 #gc-ajj ,,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 pp4 #aja-abc ,,chapter ,,vi ,,the ,,new ,,conception ,,of ,,gravitation ,geometry and reality--,euclid's geometry and others--,contingency of ,poincar's criterion--,the real universe is not ,euclidean but ,riemannian--,the avatars of the number --,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 pp4 #abd-adg ,,chapter ,,vii ,,is ,,the ,,universe ,,infinite8 ,kant and the number of the stars--,extinct stars and dark nebul--,extent and aspect of the astronomical universe--,different kinds of universes--,poincar'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 pp4 #adh-aei ,,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 pp4 #afj-agb ,,chapter ,,ix ,,einstein ,,or ,,newton8 ,recent discussion of ,relativism at the ,academy of ,sciences--,traces of the privileged space of ,newton--,the principle of causality1 the basis of science--,examination of ,m4 ,painlev's objections--,newtonian arguments and ,relativist replies--,m4 ,painlev's formulas of gravitation--,fecundity of ,einstein's theory--,two conceptions of the world--,conclusion pp4 #agc-ahe ,,introduction ,this book is not a romance4 ,nevertheless4444 ,if love is1 as ,plato says1 a soaring toward the infinite1 where shall we find more love than in the impassioned curiosity which impels us1 with bowed heads and beating hearts1 against the wall of mystery that environs our material world8 ,behind that wall1 we feel1 there is something sublime4 ,what is it8 ,science is the outcome of the search for that mysterious something4 ,a giant blow has recently been struck1 by a man of consummate ability1 ,albert ,einstein1 upon this wall which conceals reality from us4 ,a little of the light from beyond now comes to us through the breach he has made1 and our eyes are enchanted1 almost dazzled1 by the rays4 ,i propose here to give1 as simply and clearly as is possible1 some faint reflex of the impression it has made upon us4 ,einstein's theories have brought about a profound revolution in science4 ,in their light the world seems simpler1 more co-ordinated1 more in unison4 ,we shall henceforward realise better how grandiose and coherent it is1 how it is ruled by an inflexible harmony4 ,a little of the ineffable will become clearer to us4 ,men1 as they pass through the universe1 are like those specks of dust which dance for a moment in the golden rays of the sun1 then sink into the darkness4 ,is there a finer or nobler way of spending this life than to fill one's eyes1 one's mind1 one's heart with the immortal1 yet so elusive1 rays8 ,what higher pleasure can there be than to contemplate1 to seek1 to understand1 the magnificent and astounding spectacle of the universe8 ,there is in reality more of the marvellous and the romantic than there is in all our poor dreams4 ,in the thirst for knowledge1 in the mystic impulse which urges us toward the deep heart of the ,unknown1 there is more passion and more sweetness than in all the trivialities which sustain so many literatures4 ,i may be wrong1 after all1 in saying that this book is not a romance4 ,i will endeavour in these pages to make the reader understand1 accurately1 yet without the aid of the esoteric apparatus of the technical writer1 the revolution brought about by ,einstein4 ,i will try also to fix its limits2 to state precisely what1 at the most1 we can really know to-day about the external world when we regard it through the translucent screen of science4 ,every revolution is followed by a reaction1 in virtue of the rhythm which seems to be an inherent and eternal law of the mind of man4 ,einstein is at once the ,sieys1 the ,mirabeau1 and the ,danton of the new revolution4 ,but the revolution has already produced its fanatical ,marats1 who would say to science3 8,thus far and no farther40 ,hence we find some resistance to the pretensions of over-zealous apostles of the new scientific gospel4 ,in the ,academy of ,sciences ,m4 ,paul ,painlev takes his place1 with all the strength of a vigorous mathematical genius1 between ,newton1 who was supposed to be overthrown1 and ,einstein4 ,in my final pages ,i will examine the penetrating criticisms of the great ,french geometrician4 ,they will help me to fix the precise position1 in the evolution of our ideas1 of ,einstein's magnificent synthesis4 ,but ,i would first expound the synthesis itself with all the affection which one must bestow upon things that one would understand4 ,science has not completed its task with the work of ,einstein4 ,there remains many a depth that is for us unfathomable1 waiting for some genius of to-morrow to throw light into it4 ,it is the very essence of the august and lofty grandeur of science that it is perpetually advancing4 ,it is like a torch in the sombre forest of mystery4 ,man enlarges every day the circle of light which spreads round him1 but at the same time1 and in virtue of his very advance1 he finds himself confronting1 at an increasing number of points1 the darkness of the ,unknown4 ,few men have borne the shaft of light so deeply into the forest as has ,einstein4 ,in spite of the sordid cares which harass us to-day1 amid so many grave contingencies1 his system reveals to us an element of grandeur4 ,our age is like the noisy and unsubstantial froth that crowns1 and hides for a moment1 the gold of some generous wine4 ,when all the transitory murmur that now fills our ears is over1 ,einstein's theory will rise before us as the great lighthouse on the brink of this sad and petty twentieth century of ours4 ,,charles ,,nordmann4 ,,einstein ,,and ,,the ,,universe ,,chapter ,i ,,the ,,metamorphoses ,,of ,,space ,,and ,,time ,removing the mathematical difficulties--,the pillars of knowledge--,absolute time and space1 from ,aristotle to ,newton--,relative time and space1 from ,epicurus to ,poincar and ,einstein--,classical ,relativity--,antinomy of stellar aberration and the ,michelson experiment4 8,have you read ,baruch80 ,la ,fontaine used to cry1 enthusiastically4 ,to-day he would have troubled his friends with the question 8,have you read ,einstein80 ,but1 whereas one needs only a little ,latin to gain access to ,spinoza1 frightful monsters keep guard before ,einstein1 and their horrible grimaces seem to forbid us to approach him4 ,they stand behind strange moving bars1 sometimes rectangular and sometimes curvilinear1 which are known as 8co-ordinates40 ,they bear names as frightful as themselves--8contravariant and covariant vectors1 tensors1 scalars1 determinants1 orthogonal vectors1 generalised symbols of three signs10 and so on4 ,these strange beings1 brought from the wildest depths of the mathematical jungle1 join together or part from each other with a remarkable promiscuity1 by means of some astonishing surgery which is called integration and differentiation4 ,in a word1 ,einstein may be a treasure1 but there is a fearsome troop of mathematical reptiles keeping inquisitive folk away from it2 though there can be no doubt that they have1 like our ,gothic gargoyles1 a hidden beauty of their own4 ,let us1 however1 drive them off with the whip of simple terminology1 and approach the splendour of ,einstein's theory4 ,who is this physicist ,einstein8 ,that is a question of no importance here4 ,it is enough to know that he refused to sign the infamous manifesto of the professors1 and thus brought upon himself persecution from the ,pan-,germanists4#a ,mathematical truths and scientific discoveries have an intrinsic value1 and this must be judged and appreciated impartially1 whoever their author may chance to be4 ,had ,pythagoras been the lowest of criminals1 the fact would not in the least detract from the validity of the square of the hypotenuse4 ,a theory is either true or false1 whether the nose of its author has the aquiline contour of the nose of the children of ,sem1 or the flattened shape of that of the children of ,cham1 or the straightness of that of the children of ,japhet4 ,do we feel that humanity is perfect when we hear it said occasionally3 8,tell me what church you frequent1 and ,i will tell you if your geometry is sound40 ,truth has no need of a civil status4 ,let us get on4 #a ,albert ,einstein1 born in #ahgi1 is a ,german ,jew of ,wrtemberg4 ,he studied in ,switzerland1 and was an engineer there until #aiji1 when he became professor at ,zurich ,university4 ,in #aiaa he passed to ,prague ,university1 in #aiab to the ,zurich ,polytechnic1 and in #aiad to the ,prussian ,academy of ,science4 ,he refused to give his name to the manifesto in which ninety-three professors of ,germany and ,austria defended ,germany's war-action4--,trans4 99 99 99 99 99 ,all our ideas1 all science1 and even the whole of our practical life1 are based upon the way in which we picture to ourselves the successive aspects of things4 ,our mind1 with the aid of our senses1 chiefly ranges these under the headings of time and space1 which thus become the two frames in which we dispose all that is apparent to us of the material world4 ,when we write a letter1 we put at the head of it the name of the place and the date4 ,when we open a newspaper1 we find the same indications at the beginning of each piece of telegraphic news4 ,it is the same in everything and for everything4 ,time and space1 the situation and the period of things1 are thus seen to be the twin pillars of all knowledge1 the two columns which sustain the edifice of men's understanding4 ,so felt ,leconte de ,lisle when1 addressing himself to 8divine death10 he wrote1 in his profound1 philosophic way3 ,free us from time1 number1 and space3 ,grant us the rest that life hath spoiled4 ,he inserts the word 8number0 only in order to define time and space quantitatively4 ,what he has finely expressed in these famous and superb lines is the fact that all that there is for us in this vast universe1 all that we know and see1 all the ineffable and agitated flow of phenomena1 presents to us no definite aspect1 no precise form1 until it has passed through those two filters which are interposed by the mind1 time and space4 ,the work of ,einstein derives its importance from the fact that he has shown1 as we shall see1 that we have entirely to revise our ideas of time and space4 ,if that is so1 the whole of science1 including psychology1 will have to be reconstructed4 ,that is the first part of ,einstein's work1 but it goes further4 ,if that were the whole of his work it would be merely negative4 ,once he had removed from the structure of human knowledge what had been regarded as an indispensable wall of it1 though it was really only a frail scaffolding that hid the harmony of its proportions1 he began to reconstruct4 ,he made in the structure large windows which allow us now to see the treasures it contains4 ,in a word1 ,einstein showed1 on the one hand1 with astonishing acuteness and depth1 that the foundation of our knowledge seems to be different from what we had thought1 and that it needs repairing with a new kind of cement4 ,on the other hand1 he has reconstructed the edifice on this new basis1 and he has given it a bold and remarkably beautiful and harmonious form4 ,i have now to show in detail1 concretely1 and as accurately as possible1 the meaning of these generalities4 ,but ,i must first insist on a point which is of considerable importance3 if ,einstein had confined himself to the first part of his work1 as ,i have described it1 the part which shatters the classical ideas of time and space1 he would never have attained the fame which now makes his name great in the world of thought4 ,the point is important because most of those--apart from experts--who have written on ,einstein have chiefly1 often exclusively1 emphasised this more or less 8destructive0 side of his work4 ,but1 as we shall see1 from this point of view ,einstein was not the first1 and he is not alone4 ,all that he has done is to sharpen1 and press a little deeper between the badly joined stones of classical science1 a chisel which others1 especially the great ,henri ,poincar1 had used long before him4 ,my next point is to explain1 if ,i can1 the real1 the immortal1 title of ,einstein to the gratitude of men3 to show how he has by his own powers rebuilt the structure in a new and magnificent form after his critical work4 ,in this he shares his glory with none4 99 99 99 99 99 ,the whole of science1 from the days of ,aristotle until our own1 has been based upon the hypothesis--properly speaking1 the hypotheses--that there is an absolute time and an absolute space4 ,in other words1 our ideas rested upon the supposition that an interval of time and an interval of space between two given phenomena are always the same1 for every observer whatsoever1 and whatever the conditions of observation may be4 ,for instance1 it would never have occurred to anybody as long as classical science was predominant1 that the interval of time1 the number of seconds1 which lies between two successive eclipses of the sun1 may not be the fixed and identically same number of seconds for an observer on the earth as for an observer in ,sirius 7assuming that the second is defined for both by the same chronometer74 ,similarly1 no one would have imagined that the distance in metres between two objects1 for instance the distance of the earth from the sun at a given moment1 measured by trigonometry1 may not be the same for an observer on the earth as for an observer in ,sirius 7the metre being defined for both by the same rule74 8,there is10 says ,aristotle1 8one single and invariable time1 which flows in two movements in an identical and simultaneous manner2 and if these two sorts of time were not simultaneous1 they would nevertheless be of the same nature4444 ,thus1 in regard to movements which take place simultaneously1 there is one and the same time1 whether or no the movements are equal in rapidity2 and this is true even if one of them is a local movement and the other an alteration4444 ,it follows that even if the movements differ from each other1 and arise independently1 the time is absolutely the same for both40#b ,this ,aristotelic definition of physical time is more than two thousand years old1 yet it clearly represents the idea of time which has been used in classic science1 especially in the mechanics of ,galileo and ,newton1 until quite recent years4 ,it seems1 however1 that in spite of ,aristotle1 ,epicurus outlined the position which ,einstein would later adopt in antagonism to ,newton4 ,to translate liberally the words in which ,lucretius expounds the teaching of ,epicurus3 8,time has no existence of itself1 but only in material objects1 from which we get the idea of past1 present1 and future4 ,it is impossible to conceive time in itself independently of the movement or rest of things40#c #b ,physics1 bk4 iv1 ch4 xiv4 #c ,de ,natura ,rerum1 bk4 i1 vv4 #dfj ff4 ,both space and time have been regarded by science ever since ,aristotle as invariable1 fixed1 rigid1 absolute data4 ,newton thought that he was saying something obvious1 a platitude1 when he wrote in his celebrated ,scholion3 8,absolute1 true1 and mathematical time1 taken in itself and without relation to any material object1 flows uniformly of its own nature4444 ,absolute space1 on the other hand1 independent by its own nature of any relation to external objects1 remains always unchangeable and immovable40 ,the whole of science1 the whole of physics and mechanics1 as they are still taught in our colleges and in most of our universities1 are based entirely upon these propositions1 these ideas of an absolute time and space1 taken by themselves and without any reference to an external object1 independent by their very nature4 ,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 boats1 flowing on just the same whether there were phenomena or not4 ,space1 similarly1 was rather like the bank of the river1 indifferent to the ships that passed4 ,from the time of ,newton1 however1 if not from the time of ,aristotle1 any thoughtful metaphysician might have noticed that there was something wrong in these definitions4 ,absolute time and absolute space are 8things in themselves10 and these the human mind has always regarded as not directly accessible to it4 ,the specifications of space and time1 those numbered labels which we attach to objects of the material world1 as we put labels on parcels at the station so that they may not be lost 7a precaution that does not always suffice71 are given us by our senses1 whether aided by instruments or not1 only when we receive concrete impressions4 ,should we have any idea of them if there were no bodies attached to them1 or rather to which we attach the labels8 ,to answer this in the affirmative1 as ,aristotle1 ,newton1 and classical science do1 is to make a very bold assumption1 and one that is not obviously justified4 ,the only time of which we have any idea apart from all objects is the psychological time so luminously studied by ,m4 ,bergson3 a time which has nothing except the name in common with the time of physicists1 of science4 ,it is really to ,henri ,poincar1 the great ,frenchman whose death has left a void that will never be filled1 that we must accord the merit of having first proved1 with the greatest lucidity and the most prudent audacity1 that time and space1 as we know them1 can only be relative4 ,a few quotations from his works will not be out of place4 ,they will show that the credit for most of the things which are currently attributed to ,einstein is1 in reality1 due to ,poincar4 ,to prove this is not in any way to detract from the merit of ,einstein1 for that is1 as we shall see1 in other fields4 ,this is how ,poincar1 whose ideas still dominate the minds of thoughtful men1 though his mortal frame perished years ago1 expressed himself1 the triumphant sweep of his wings reaching further every day3 8,one cannot form any idea of empty space4444 ,from that follows the undeniable relativity of space4 ,any man who talks of absolute space uses words which have no meaning4 ,i am at a particular spot in ,paris--the ,place du ,panthon1 let us suppose--and ,i say3 #8,i will come back here to-morrow4' ,if anyone asks me whether ,i mean that ,i will return to the same point in space1 ,i am tempted to reply1 #8,yes4' ,i should1 however1 be wrong1 because between this and to-morrow the earth will have travelled1 taking the ,place du ,panthon with it1 so that to-morrow the square will be more than #b1jjj1jjj kilometres away from where it is now4 ,and it would be no use my attempting to use precise language1 because these #b1jjj1jjj kilometres are part of our earth's journey round the sun1 but the sun itself has moved in relation to the ,milky ,way1 and the ,milky ,way in turn is doubtless moving at a speed which we cannot learn4 ,thus we are entirely ignorant1 and always will be ignorant1 how far the ,place du ,panthon shifts its position in space in a single day4 ,what ,i really meant to say was3 #8,to-morrow ,i shall again see the dome and faade of the ,panthon4' ,if there were no ,panthon1 there would be no meaning in my words1 and space would disappear40 ,poincar works out his idea in this way3 8,suppose all the dimensions of the universe were increased a thousandfold in a night4 ,the world would remain the same1 giving the word #8same' the meaning it has in the third book of geometry4 ,nevertheless1 an object that had measured a metre in length will henceforward be a kilometre in length2 a thing that had measured a millimetre will now measure a metre4 ,the bed on which ,i lie and the body which lies on it will increase in size to exactly the same extent4 ,what sort of feelings will ,i have when ,i awake in the morning1 in face of such an amazing transformation8 ,well1 ,i shall know nothing about it4 ,the most precise measurements would tell me nothing about the revolution1 because the tape ,i use for measuring will have changed to the same extent as the objects ,i wish to measure4 ,as a matter of fact1 there would be no revolution except in the mind of those who reason as if space were absolute4 ,if ,i have argued for a moment as they do1 it was only in order to show more clearly that their position is contradictory40 ,it would be easy to develop ,poincar's argument4 ,if all the objects in the universe were to become1 for instance1 a thousand times taller1 a thousand times broader1 we should be quite unable to detect it1 because we ourselves--our retina and our measuring rod--would be transformed to the same extent at the same time4 ,indeed1 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 fashion1 drawing it out as if it were rubber--we should have no means of knowing the fact4 ,there could be no better proof that space is relative1 and that we cannot conceive space apart from the things which we use to measure it4 ,when there is no measuring rod1 there is no space4 ,poincar 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 supposition1 but not a truer hypothesis1 unless we imply the existence of absolute space4 ,it may be remembered that certain unwary controversialists have tried to infer from ,poincar's argument that the condemnation of ,galileo was justified4 ,nothing could be more amusing than the way in which the distinguished mathematician-philosopher defended himself against this interpretation1 though one must admit that his defence was not wholly convincing4 ,he did not take sufficiently into account the agnostic element4 ,poincar1 in any case1 is the leader of those who regard space as a mere property which we ascribe to objects4 ,in this view our idea of it is only1 so to say1 the hereditary outcome of those efforts of our senses by means of which we strive to embrace the material world at a given moment4 ,it is the same with time4 ,here again the objections of philosophic ,relativists were raised long ago1 but it was ,poincar who gave them their definitive shape4 ,his luminous demonstrations are1 however1 well known1 and we need not reproduce them here4 ,it is enough to observe that1 in regard to time as well as space1 it is possible to imagine either a contraction or an enlargement of the scale which would be completely imperceptible to us2 and this seems to show that man cannot conceive an absolute time4 ,if some malicious spirit were to amuse itself some night by making all the phenomena of the universe a thousand times slower1 we should not1 when we awake1 have any means of detecting the change4 ,the world would seem to us unchanged4 ,yet every hour recorded by our watches would be a thousand times longer than hours had previously been4 ,men would live a thousand times as long1 yet they would be unaware of the fact1 as their sensations would be slower in the same proportion4 ,when ,lamartine appealed to time to 8suspend its flight10 he said a very charming1 but perhaps meaningless1 thing4 ,if time had obeyed his passionate appeal1 neither ,lamartine nor ,elvire would have known and rejoiced over the fact4 ,the boatman who conducted the lovers on the ,lac du ,bourget would not have asked payment for a single additional hour2 yet he would have dipped his oars into the pleasant waters for a far longer time4 ,i venture to sum up all this in a sentence which will at first sight seem a paradox3 in the opinion of the ,relativists it is the measuring rods which create space1 the clocks which create time4 ,all this was maintained by ,poincar and others long before the time of ,einstein1 and one does injustice to truth in ascribing the discovery to him4 ,i am quite aware that one lends only to the rich1 but one does an injustice to the wealthy themselves in attributing to them what does not belong to them1 and what they need not in order to be rich4 ,there is1 moreover1 one point at which ,galileo and ,newton1 for all their belief in the existence of absolute space and time1 admitted a certain relativity4 ,they recognised that it is impossible to distinguish between uniform movements of translation4 ,they thus admitted the equivalence of all such movements1 and therefore the impossibility of proving an absolute movement of translation4 ,that is what is called the ,principle of ,classic ,relativity4 99 99 99 99 99 ,an unexpected fact served to bring these questions upon a new plane1 and led ,einstein to give a remarkable extension to the ,principle of ,relativity of classic mechanics4 ,this was the issue of a famous experiment by ,michelson1 of which we must give a brief description4 ,it is well known that rays of light travel across empty space from star to star1 otherwise we should be unable to see the stars4 ,from this physicists long ago concluded that the rays travelled in a medium that is devoid of mass and inertia1 is infinitely elastic1 and offers no resistance to the movement of material bodies1 into which it penetrates4 ,this medium has been named ether4 ,light travels through it as waves spread over the surface of water at a speed of something like #ahf1jjj miles a second3 a velocity which we will express by the letter ,v4 ,the earth revolves round the sun in a veritable ocean of ether1 at a speed of about #ah miles a second4 ,in this respect the rotation of the earth on its axis need not be noticed1 as it pushes the surface of the globe through the ether at a speed of less than two miles a second4 ,now the question had often been asked3 ,does the earth1 in its orbital movement round the sun1 take with it the ether which is in contact with it1 as a sponge thrown out of a window takes with it the water which it has absorbed8 ,experiment--or rather1 experiments1 for many have been tried with the same result--has shown that the question must be answered in the negative4 ,this was first established by astronomical observation4 ,there is in astronomy a well-known phenomenon discovered by ,bradley which is called aberration4 ,it consists in this3 when we observe a star with a telescope1 the image of the star is not precisely in the direct line of vision4 ,the reason is that1 while the luminous rays of the star which have entered the telescope are passing down the length of the tube1 the instrument has been slightly displaced1 as it shares the movement of the earth4 ,on the other hand1 the luminous ray in the tube does not share the earth's motion1 and this gives rise to the very slight deviation which we call aberration4 ,this proves that the medium in which light travels1 the ether which fills the instrument and surrounds the earth1 does not share the earth's motion4 ,many other experiments have settled beyond question that the ether1 which is the vehicle of the waves of light1 is not borne along by the earth as it travels4 ,now1 since the earth moves through the ether as a ship moves over a stationary lake 7not like one floating on a moving stream71 it ought to be possible to detect some evidence of this speed of the earth in relation to the ether4 ,one of the devices that may be imagined for the purpose is the following4 ,we know that the earth turns on itself from west to east1 and travels round the sun in the same way4 ,it follows that in the middle of the night the revolution of the earth round the sun means that ,paris will be displaced1 in the direction from ,auteuil toward ,charenton1 at a speed of about thirty kilometres a second4 ,during the day1 of course1 it is precisely the opposite4 ,paris changes its place round the sun in the direction from ,charenton toward ,auteuil4 ,well1 let us suppose that at midnight a physicist at ,auteuil sends a luminous signal4 ,a physicist receiving this ray of light at ,charenton1 and measuring its velocity1 ought to find that the latter is ,v #cj kilometres4 ,we know that1 as a result of the earth's motion1 ,charenton recedes before the ray of light4 ,consequently1 since light travels in a medium1 the ether1 which does not share the earth's motion1 the observer at ,charenton ought to find that the ray reaches him at a less speed than it would if the earth were stationary4 ,it is much the same as if an observer were travelling on a bicycle in front of an express train4 ,if the express travels at thirty metres a second and the cyclist at three metres a second1 the speed of the train in relation to the cyclist will be #cj-c #bg metres a second4 ,it would be nil if the train and the cyclist were travelling at the same rate4 ,on the other hand1 if the cyclist were going toward the train1 the speed of the train in relation to him would be #cj #c #cc metres a second4 ,similarly1 when the physicist at ,charenton sends out a luminous message at midnight1 and the physicist of ,auteuil receives it1 the latter ought to find that the ray of light has a velocity of ,v #cj kilometres4 ,all this may be put in a different way4 ,suppose the distance between the observer at ,auteuil and the man at ,charenton were exactly twelve kilometres4 ,while the ray of light emitted at ,auteuil speeds toward ,charenton1 that town is receding before it to a small extent4 ,it follows that the ray will have to travel a little more than twelve kilometres before it reaches the man of science at ,charenton4 ,it will travel a little less than that distance if we imagine it proceeding in the opposite direction4 ,now the ,american physicist ,michelson1 borrowing an ingenious idea from the ,french physicist ,fizeau1 succeeded1 with a high degree of accuracy1 in measuring distances by means of the interference-bands of light4 ,every variation in the distance measured betrays itself by the displacement of a certain number of these bands1 and this may easily be detected by a microscope4 ,let us next suppose that our two physicists work in a laboratory instead of between ,charenton and ,auteuil4 ,let us suppose that they are1 by means of the interference-bands1 measuring the space traversed by a ray of light produced in the laboratory1 according as it travels in the same direction as the earth or in the opposite direction4 ,that is ,michelson's famous experiment1 reduced to its essential elements and simplified for the purpose of this essay4 ,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 direction4 ,but no such difference was found4 ,contrary to all expectation1 and to the profound astonishment of physicists1 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 velocity4 ,it is an undeniable consequence of this that the ether shares the motion of the earth4 ,we have1 however1 seen that other experiments1 not less precise1 had settled that the ether does not share the motion of the earth4 ,out of this contradiction1 this conflict of two irreconcilable yet indubitable facts1 ,einstein's splendid synthesis1 like a spark of light issuing from the clash of flint and steel1 came into being4 ,,chapter ,,ii ,,science ,,in ,a ,,no-,,thoroughfare ,scientific truth and mathematics--,the precise function of ,einstein--,michelson's experiment1 the ,gordian knot of science--,the hesitations of ,poincar--,the strange1 but necessary1 ,fitzgerald-,lorentz hypothesis--,the contraction of moving bodies--,philosophical and physical difficulties4 ,it would be foolish to pretend that we can penetrate the most obscure corners of ,einstein's theories without the aid of mathematics4 ,i believe1 however1 that we can give in ordinary language--that is to say1 by means of illustrations and analogies--a fairly satisfactory idea of these things1 the intricacy of which is usually due to the infinitely subtle and supple play of mathematical formul and equations4 ,after all1 mathematics is not1 never was1 and never will be1 anything more than a particular kind of language1 a sort of shorthand of thought and reasoning4 ,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 medival slowness of the syllogisms expressed in our words4 ,however paradoxical this may seem to people who regard mathematics as of itself a means of discovery1 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 equations4 ,if ,i may use a somewhat trivial illustration1 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 meats4 ,no spectator could have1 or would wish to have1 eaten the animal alive1 but in the form in which it issues from the machine it can at once be digested and assimilated4 ,yet the meat is merely the animal conveniently prepared4 ,that is what mathematics does4 ,by means of a marvellous machinery the mathematician extracts the valuable marrow from the given facts4 ,it is a machinery that is particularly useful in cases where the wheels of verbal argument1 the chain of syllogisms1 would soon be brought to a halt4 ,does it follow that1 properly speaking1 mathematics is not a science8 ,does it follow at least that it is only a science in so far as it is based upon reality1 and fed with experimental data1 since 8experience is the sole source of truth40 ,i refrain from answering the question1 as ,i am one of those who believe that everything is material for science4 ,still1 it was worth while to raise the question because many are too much disposed to regard a purely mathematical education as a scientific education4 ,nothing could be further from the truth4 ,pure mathematics is1 in itself1 merely an abbreviated form of language and of logical thought4 ,it cannot1 of its own nature1 teach us anything about the external world2 it can do so only in proportion as it enters into contact with the world4 ,it is of mathematics in particular that we may say3 ,natur non imperatur nisi parendo4 ,are not ,einstein's theories1 as some imperfectly informed writers have suggested1 only a play of mathematical formul 7taking the word in the meaning given to it by both mathematicians and philosophers78 ,if they were only a towering mathematical structure in which the x's shoot out their volutes in bewildering arabesques1 with swan-neck integrals describing ,louis ,,xv patterns1 they would have no interest whatever for the physicist1 for the man who has to examine the nature of things before he talks about it4 ,they would1 like all coherent schemes of metaphysics1 be merely a more or less agreeable system of thought1 the truth or falseness of which could never be demonstrated4 ,einstein's theory is very different from that1 and very much more than that4 ,it is based upon facts4 ,it also leads to facts--new facts4 ,no philosophical doctrine or purely formal mathematical construction ever enabled us to discover new phenomena4 ,it is precisely because it has led to such discovery that ,einstein's theory is neither the one nor the other4 ,that is the difference between a scientific theory and a pure speculation1 and it is that which1 ,i venture to say1 makes the former so superior4 ,like some suspension bridge boldly thrown across an abyss1 ,einstein's theory rests1 on the one side1 on experimental phenomena1 and it leads1 at the other side1 to other1 and hitherto unsuspected1 phenomena1 which it has enabled us to discover4 ,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 bridge4 ,it is that1 and nothing but that4 ,but the arrangement of the beams and bars might have been different1 and the bridge--though less light and graceful1 perhaps--still have been able to join together the two sets of facts on which it rests4 ,in a word1 mathematical reasoning is only a kind of reasoning in a special language1 from experimental premises to conclusions which are verifiable by experience4 ,now there is no language which cannot in some degree be translated into another language4 ,even the hieroglyphics of ,egypt had to give way before ,champollion4 ,i am therefore convinced that the mathematical difficulties of ,einstein's theories will some day be replaced by simpler and more accessible formul4 ,i believe1 indeed1 that it is even now possible to give by means of ordinary speech an idea1 rather superficial perhaps1 but accurate and substantially complete1 of this wonderful ,einsteinian structure which ranges all the conquests of science1 as in some well-ordered museum1 in a new and superb unity4 ,let us try4 99 99 99 99 99 ,we may resume in the few following words the story of the origin1 the starting-point1 of ,einstein's system4 #a4 ,observation of the stars proves that interplanetary space is not empty1 but is filled with a special medium1 ether1 in which the waves of light travel4 #b4 ,the fact of aberration and other phenomena seems to prove that the ether is not displaced by the earth during its course round the sun4 #c4 ,michelson's experiment seems to prove1 on the contrary1 that the earth bears the ether with it in its movement4 ,this contradiction between facts of equal authority was for years the despair and the wonder of physicists4 ,it was the ,gordian knot of science4 ,long and fruitless efforts were made to untie it until at last ,einstein cut it with a single blow of his remarkably acute intelligence4 ,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 experiment4 ,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 east3 that is to say1 in the direction in which the earth itself moves1 at a speed of about eighteen miles a second1 as it travels round the sun1 or in the opposite direction4 ,as a matter of fact1 ,michelson's experiment was rather more complicated than that1 and we must return to it4 ,four mirrors are placed at an equal distance from each other in the laboratory1 in pairs which face each other4 ,two of the opposing mirrors are arranged in the direction east-west1 the direction in which the earth moves in consequence of its revolution round the sun4 ,the other two are arranged in a plane perpendicular to the preceding1 the direction north-south4 ,two rays of light are then started in the respective directions of the two pairs of mirrors4 ,the ray coming from the mirror to the east goes to the mirror in the west1 is reflected therefrom1 and returns to the first mirror4 ,this ray is so arranged that it crosses the path of the light which goes from north to south and back4 ,it interferes with the latter light1 causing 8fringes of interference0 which1 as ,i said1 enable us to learn the exact distance traversed by the rays of light reflected between the pairs of mirrors4 ,if anything brought about a difference between the length of the two distances1 we should at once see the displacement of a certain number of interference-fringes1 and this would give us the magnitude of the difference4 ,an analogy will help us to understand the matter4 ,suppose a violent steady east wind blew across ,london1 and an aviator proposed to cross the city about twelve miles from extreme west to east and back3 that is to say1 going with the wind on his outward journey and against it on the return journey4 ,suppose another aviator1 of equal speed1 proposed at the same time to fly from the same starting-point to a point twelve miles to the north and back1 the second aviator will fly both ways at right angles to the direction of the wind4 ,if the two start at the same time1 and are imagined as turning round instantaneously1 will they both reach the starting-point together8 ,and1 if not1 which of them will have completed his double journey first8 ,it is clear that if there were no wind1 they would get back together1 as we suppose that they both do twenty-four miles at the same speed1 which we may roughly state to be #bjj yards a second4 ,but it will be different if1 as ,i postulated1 there is a wind blowing from east to west4 ,it is easy to see that in such circumstances the man who flies east to west will take longer to complete the journey4 ,in order to get it quite clearly1 let us suppose that the wind is travelling at the same speed as the aviator 7#bjj yards a second74 ,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 north4 ,he will therefore have traversed in the wind a real distance equal to the diagonal of a square measuring twelve miles on each side4 ,instead of flying twenty-four miles1 he will really have flown thirty-four in the wind1 the medium in relation to which he has any velocity4 ,on the other hand1 the aviator who flies eastward will never reach his destination1 because in each second of time he is driven westward to precisely the same extent as he is travelling eastward4 ,he will remain stationary4 ,to accomplish his journey he would need to cover in the wind an infinite distance4 ,if1 instead of imagining a wind equal in velocity to the aviator 7an extreme supposition in order to make the demonstration clearer71 ,i had thought of it as less rapid1 we should again find1 by a very simple calculation1 that the man who flies north and south has less distance to cover in the wind than the man who flies east and west4 ,now take rays of light instead of aviators1 the ether instead of the wind1 and we have very nearly the conditions of the ,michelson experiment4 ,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 mirrors4 ,therefore the ray of light which travels between these two mirrors1 forth and back1 must cover a longer distance in ether than the ray which goes from the south mirror to the north and back4 ,but how are we to detect this difference8 ,it is certainly very minute1 because the speed of the earth is ten thousand times less than the velocity of light4 ,there is a very simple means of doing this3 one of those ingenious devices which physicists love1 a differential device so elegant and precise that we have entire confidence in the result4 ,let us suppose that our four mirrors are fixed rigidly in a sort of square frame1 something like those 8wheels of fortune0 with numbers on them that one sees in country fairs4 ,let us suppose that we can turn this frame round as we wish1 without jerking or displacing it1 which is not difficult if it floats in a bath of mercury4 ,i then take a lens and observe the permanent interference-fringes which define the difference between the paths traversed by my two rays of light1 north-south and east-west4 ,then1 without losing sight of the bands or fringes1 ,i turn the frame round a quarter of a circle4 ,owing to this rotation the mirrors which were east-west now become north-south1 and vice versa4 ,the double journey made by the north-south ray of light has now taken the direction east-west1 and has therefore suddenly been lengthened2 the double journey of the east-west ray has become north-south1 and has been suddenly shortened4 ,the interference-fringes1 which indicate the difference in length between the two paths1 which has suddenly changed1 must necessarily be displaced1 and that1 as we can calculate1 to no slight extent4 ,well1 we find no change whatever6 ,the fringes remain unaltered4 ,they are as stationary as stumps of trees4 ,it is bewildering1 one would almost say revolting1 because the delicacy of the apparatus is such that1 even if the earth moved through the ether at a rate of only three kilometres a second 7or ten times less than its actual velocity71 the displacement of the fringes would be sufficient to indicate the speed4 99 99 99 99 99 ,when the negative result of this experiment was announced1 there was something like consternation amongst the physicists of the world4 ,since the ether was not borne along by the earth1 as observation had established1 how could it possibly behave as if it did share the earth's motion8 ,it was a ,chinese puzzle4 ,more than one venerable grey head was in despair over it4 ,it was absolutely necessary to find a way out of this inexplicable contradiction1 to end this paradoxical mockery which the facts seemed to oppose to the most rigorous results of calculation4 ,this the men of science succeeded in doing4 ,how8 ,by the method which is generally used in such circumstances--by means of supplementary hypotheses4 ,hypotheses in science are a kind of soft cement which hardens rapidly in the open air1 thus enabling us to join together the separate blocks of the structure1 and to fill up the breaches made in the wall by projectiles1 with artificial stuff which the superficial observer presently mistakes for stone4 ,it is because hypotheses are something like that in science that the best scientific theories are those which include least hypotheses4 ,but ,i am wrong in using the plural in this connection4 ,in the end it was found that one single hypothesis conveniently explained the negative result of the ,michelson experiment4 ,that is1 by the way1 a rare and remarkable experience4 ,hypotheses usually spring up like mushrooms in every dark corner of science4 ,you get a score of them to explain the slightest obscurity4 ,this single hypothesis1 which seemed to be capable of extricating physicists from the dilemma into which ,michelson had put them1 was first advanced by the distinguished ,irish mathematician ,fitzgerald1 then taken up and developed by the celebrated ,dutch physicist ,lorentz1 the ,poincar of ,holland1 one of the most brilliant thinkers of our time4 ,einstein would no more have attained fame without him than ,kepler would without ,copernicus and ,tycho ,brahe4 ,let us now see what this ,fitzgerald-,lorentz hypothesis1 as strange as it is simple1 really is4 ,but we must first glance at a preliminary matter of some importance4 ,a number of able men have declared--after the issue1 let it be said--that the result of the ,michelson experiment could only be negative a priori4 ,in point of fact1 they argue 7more or less71 the ,classic ,principle of ,relativity1 the principle known to ,galileo and ,newton1 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 it4 ,thus1 when two ships or two trains pass each other1#d it is impossible for the passengers to say which of the two is moving1 or moving the more rapidly4 ,all that they can perceive is the relative speed of the trains or ships4 #d ,it is assumed that the ship is not rolling or pitching1 and that there is no vibration in the train4 ,the men of science to whom ,i have referred say that1 if ,michelson's experiment had had a positive result1 it would have given us the absolute velocity of the earth in space4 ,this result would have been contrary to the ,principle of ,relativity of classical philosophy and mechanics1 which is a self-evident truth4 ,therefore the result could only be negative4 ,this is1 as we shall see1 ambiguous4 ,there is1 if ,i may say so1 a flaw in the argument which has escaped the notice even of distinguished men of science like ,professor ,eddington1 the most erudite of the ,english ,einsteinians4 ,it was he who organised the observations of the solar eclipse of ,may #bi1 #aiai1 which have1 as we shall see1 furnished the most striking verification of ,einstein's deductions4 ,in the first place1 if ,michelson's experiment had had a positive result1 what it would have indicated is the velocity of the earth in relation to the ether4 ,but1 for this to be an absolute velocity1 the ether would have to be identical with space4 ,this is so far from being necessary that we can easily conceive a space--to put it better1 a discontinuity--between two stars that contains no ether and across which neither light nor any other known form of energy would travel4 ,when ,eddington says that 8it is legitimate and reasonable10 that it is 8inherent in the fundamental laws of nature10 that we cannot detect any movement of bodies in relation to ether1 and that this is certain 8even if the experimental evidence is inadequate10 he affirms something which would be evident only if space and ether were evidently identical4 ,but this is far from being the case4 ,if ,michelson's experiment had had a positive result1 if we had detected a velocity on the part of the earth1 should we have discovered a velocity in relation to an absolute standard8 ,certainly not4 ,it is quite possible that the stellar universe which is known to us1 with its hundreds of thousands of galaxies which it takes light millions of years to cross1 may be contained in a sphere of ether that rolls in an abyss which is devoid of ether1 and is sown here and there with other universes1 other giant drops of ether1 from which no ray of light or anything else may ever reach us4 ,it is1 at all events1 not inconceivable4 ,and in that case1 assuming that the ether has the properties attributed to it by classic physics1 even if we had detected the movement of the earth in relation to it1 we should not have discovered an absolute movement1 but at the most a movement in relation to the centre of gravity of our particular universe1 a standard which we could not refer to some other which would be absolutely stationary4 ,the ,classical ,principle of ,relativity would not be violated4 ,hence1 whatever may have been said to the contrary1 the issue of ,michelson's experiment might1 in these hypotheses1 be either positive or negative without any detriment to ,classical ,relativism4 ,as a matter of fact1 it was negative1 so nothing further need be said4 ,experiment has pronounced1 and it alone had the right to pronounce4 ,these distinctions were not unknown to ,poincar1 and he wrote3 8,by the real velocity of the earth ,i understand1 not its absolute velocity1 which is meaningless1 but its velocity in relation to the ether40 ,therefore the possibility of the existence of a velocity discoverable in relation to the ether was not regarded as an absurdity by ,poincar4 ,he said3 8,any man who speaks of absolute space uses a word that has no meaning40 ,it is worth while noticing that in all this the development of ,poincar's ideas betrays a certain hesitation4 ,speaking of experiments analogous to those of ,michelson1 he said3 8,i know that it will be said that we are not measuring its absolute velocity1 but its velocity in relation to the ether4 ,that is scarcely satisfactory4 ,is it not clear that1 if we conceive the principle in this fashion1 we can make no deductions whatever from it80 ,from this it is evident that ,poincar1 in spite of himself and all his efforts to avoid it1 was disposed to find the distinction between space and ether 8scarcely satisfactory40 ,i must admit that ,poincar's own argument seems to me not wholly satisfactory1 or at least not convincing4 8,nature10 says ,fresnel1 8cares nothing about analytical difficulties40 ,i imagine that it cares just as little about philosophical or purely physical difficulties4 ,it is hardly an incontestable criterion to suppose that a conception of phenomena is so much nearer to reality the more 8satisfactory0 it is to us1 or the better it is found adapted to the weakness of the human mind4 ,otherwise we should have to hold1 whether we liked or no1 that the universe is necessarily adapted to the categories of the mind2 that it is constituted with a view to giving us the least possible intellectual trouble4 ,that would be a strange return to anthropocentric finalism and conceit6 ,the fact that vehicles do not pass there1 and that pedestrians have to turn back1 does not prove that there are no such things as no-thoroughfares in our towns4 ,it is possible1 even probable1 that the universe also1 considered as an object of science1 has its no-thoroughfare4 ,clearly one may reply to me that it is not the universe that is adapted to our mind1 but the mind that has become adapted to the universe in the evolutionary course of their relations to each other4 ,the mind needs in its evolution to adapt itself to the universe1 in conformity with the principle of minimum action formulated by ,fermat3 perhaps the most profound principle of the physical1 biological1 and moral world4 ,in that respect the simplest and most economical ideas are the nearest to reality4 ,yes1 but what proof is there that our mental evolution is complete and perfect1 especially when we are dealing with phenomena of which our organism is insensible8 99 99 99 99 99 ,experiment alone has proved1 and had the right to prove1 that it is impossible to measure the velocity of an object relatively to the ether4 ,at all events1 this is now settled4 ,after all1 since it is evidently in the very nature of things that we cannot detect an absolute movement1 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 it8 ,possibly2 but it cannot be proved4 ,if it is so--which is not at all certain--it is in the last resort experience1 the one source of truth1 which thus tends to prove1 indirectly1 that the ether is really identical with space4 ,in that case1 however1 a space devoid of ether1 or one containing spheres of ether1 would no longer be conceivable1 and there can be nothing but a single mass of ether with stars floating in it4 ,in a word1 the negative result of ,michelson's experiment could not be deduced a priori from the problematical identity of absolute space and the ether2 but this negative result does not justify us in denying the identity a posteriori4 ,let us return to our proper subject1 the ,fitzgerald-,lorentz hypothesis which explains the issue of the ,michelson experiment1 and which was in a sense the spring-board for ,einstein's leap4 ,the hypothesis is as follows4 ,the result of the experiment is that1 whereas when the path of a ray of light between two mirrors is transverse to the earth's motion through ether1 and it is then made parallel to the earth's motion1 the path ought to be longer1 we actually find no such lengthening4 ,according to ,fitzgerald and ,lorentz1 this is because the two mirrors approached each other in the second part of the experiment4 ,to put it differently1 the frame in which the mirrors were fixed contracted in the direction of the earth's motion1 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 detected4 ,when we repeat the experiment with all kinds of different apparatus1 we find that the result is always the same 7no displacement of the fringes74 ,it follows that the character of the material of which the instrument is made--metal1 glass1 stone1 wood1 etc4--has nothing to do with the result4 ,therefore all bodies undergo an equal and similar contraction in the direction of their velocity relatively to the ether4 ,this contraction is such that it exactly compensates for the lengthening of the path of the rays of light between two points of the apparatus4 ,in other words1 the contraction is greater in proportion as the velocity of bodies relatively to the ether becomes greater4 ,that is the explanation proposed by ,fitzgerald4 ,at first it seemed to be very strange and arbitrary1 yet there was1 apparently1 no other way of explaining the result of ,michelson's experiment4 ,moreover1 when you reflect on it this contraction is found to be less extraordinary1 less startling1 than one's common sense at first pronounces it4 ,if we throw some non-rigid object1 such as one of those little balls with which children play1 quickly against an obstacle1 we see that it is slightly pushed in at the surface by the obstacle1 precisely in the same sense as the ,fitzgerald-,lorentz contraction4 ,the ball is no longer round4 ,it is a little flattened1 so that its diameter is shortened in the direction of the obstacle4 ,we have much the same phenomenon1 though in a more violent form1 when a bullet is flattened against a target4 ,therefore1 if solid bodies are thus capable of deformation--as they are1 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 shape4 ,but it is far less easy to admit that this alteration may be exactly the same1 in the given conditions1 for all bodies1 whatever be the material of which they are composed4 ,the little ball we referred to would by no means be flattened so much if it were made of steel instead of rubber4 ,moreover1 there is in this explanation something quite improbable1 something that shocks both our good sense and that caricature of it which we call common sense4 ,is it possible to admit that the contraction of bodies always exactly compensates for the optic effect which we seek1 whatever be the conditions of the experiment 7and they have been greatly varied78 ,is it possible to admit that nature acts as if it were playing hide-and-seek with us8 ,by what mysterious chance can there be a special circumstance1 providentially and exactly compensating for every phenomenon8 ,clearly there must be some affinity1 some hidden connection1 between this mysterious material contraction of ,fitzgerald and the lengthening of the light path for which it compensates4 ,we shall see presently how ,einstein has illumined the mystery1 revealed the mechanism which connects the two phenomena1 and thrown a broad and brilliant light upon the whole subject4 ,but we must not anticipate4 ,the contraction of the apparatus in ,michelson's experiment is extremely slight4 ,it is so slight that if the length of the instrument were equal to the diameter of the earth--that is to say1 #h1jjj miles--it would be shortened in the direction of the earth's motion by only six and a half centimetres6 ,in other words1 the contraction would be far too small to be in any way measurable in the laboratory4 ,there is a further reason for this4 ,even if ,michelson's apparatus were shortened by several inches--that is to say1 if the earth travelled thousands of times as rapidly as it does round the sun--we could not detect and measure it4 ,the measuring rods which we would use for the purpose would contract in the same proportion4 ,the deformation of any object by a ,fitzgerald-,lorentz contraction could not be established by any observer on the earth4 ,it could be discovered only by an observer who did not share the movement of the earth3 an observer on the sun1 for instance1 or on a slow-moving planet like ,jupiter or ,saturn4 ,micromegas would1 before he left his planet to visit us1 have been able to discover1 by optical means1 that our globe is shortened by several inches in the direction of its orbital movement2 supposing that ,voltaire's genial hero were provided with trigonometrical apparatus infinitely more delicate than that used by our surveyors and astronomers4 ,but when he reached the earth1 ,micromegas1 with all his precise apparatus1 would have found it impossible to detect the contraction4 ,he would have been greatly surprised--until he met ,einstein and heard1 as we shall hear1 the explanation of the mystery4 ,i have1 unfortunately1 neither the time nor the space--it is here1 especially1 that space is relative1 and is constantly shortened by the flow of the pen--to give the dialogue which would have taken place between ,micromegas and ,einstein4 ,perhaps1 indeed1 if we are to be faithful to the ,voltairean original1 the dialogue would have been very superficial1 for--to speak confidentially--,i believe that ,voltaire never quite understood ,newton1 though he wrote much about him1 and ,newton was less difficult to understand than ,einstein is4 ,neither did ,mme4 du ,chtelet1 for all the praise that has been lavished upon her translation of the immortal ,principia4 ,it swarms with meaningless passages which show that1 whether she knew ,latin or no1 she did not understand ,newton4 ,but all this is another story1 as ,kipling would say4 ,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 made4 ,as the compensation is always precise1 we may try to calculate the exact law which governs the contraction as a function of velocities1 and makes it1 as we find1 a precise compensation for the latter4 ,lorentz has done this4 ,taking ,v as the velocity of light and v as the velocity of the body moving in ether1 ,lorentz found that1 in order to have compensation in all cases1 the length of the moving body must be shortened1 in the plane of its progress1 in the proportion of 7 v 7 #a to 7#a ---------------- 74 7 ,v 7 ,if we take by way of illustration the case of the orbital movement of the earth1 where v is equal to thirty kilometres1 we find that the earth contracts in the plane of its orbit in the proportion 7 #a 7 7#a ---------------------- 74 7 #ajj1jjj1jjj 7 ,the difference between these two numbers is _/111 and the two hundred millionth part of the earth's diameter is equal to #f centimetres4 ,it is the figure we had already found4 ,this formula1 which gives the value of the contraction in all cases1 is elementary4 ,even the inexpert can easily see the meaning of it4 ,it enables us to calculate the extent of contraction for every rate of velocity4 ,we can easily deduce from it that if the earth's orbital motion were1 not #cj kilometres1 but #bfj1jjj kilometres a second1 it would be shortened by one-half its diameter in the plane of its motion 7without any change in its dimensions in the perpendicular74 ,at that speed a sphere becomes a flattened ellipsoid1 of which the small axis is only half the length of the larger axis2 a square becomes a rectangle1 of which the side parallel to the motion is twice as small as the other4 ,these deformations would be visible to a stationary spectator1 but they would be imperceptible to an observer who shares the movement1 for the reason already given4 ,the measuring rods and instruments1 and even the eye of the observer1 would be equally and simultaneously altered4 ,think of the distorting mirrors which one sees at times in places of amusement4 ,some show you a greatly elongated picture of yourself1 without altering your breadth4 ,others show you of your normal height1 but grotesquely enlarged in width4 ,try1 now1 to measure your height and breadth with a rule1 as they are given in these deformed reflections in the mirror4 ,if your real height is #e feet #f inches1 and your real width #b feet1 the rule will1 when you apply it to the strange reflection of yourself in the glass1 merely tell you that this figure is #e feet #f inches in height and #b feet in breadth4 ,the rule as seen in the mirror undergoes the same distortion as yourself4 ,hence it is that1 even if the globe of the earth had the fantastic speed which we suggested above1 its inhabitants would have no means of discovering that they and it were shortened by one-half in the plane east to west4 ,a man #e feet #f inches in height1 lying in a large square bed in the direction north-south1 then changing his position to east-west1 would1 quite unknown to himself1 have his length reduced to #b feet #i inches4 ,at the same time he would become twice as stout as before1 because previously his breadth was orientated from east to west4 ,but the earth travels at the rate of only thirty kilometres a second1 and its entire contraction is only a matter of a few centimetres4 ,in contrast with the earth's velocity1 the speed of our most rapid means of transport is only a small fraction of a kilometre a second4 ,an aeroplane going at #cfj kilometres an hour has a speed of only #ajj metres a second4 ,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 us4 ,that is why--that is the only reason why--the solid objects with which we are familiar seem to keep a constant shape1 at whatever speed they pass before our eyes4 ,it would be quite otherwise if their speed were hundreds of thousands of times greater4 ,all this is very strange1 very surprising1 very fantastic1 very difficult to admit4 ,yet it is a fact1 if there really is this ,fitzgerald-,lorentz contraction1 which has so far proved the only possible explanation of the ,michelson experiment4 ,but we have already seen some of the difficulties that we find in entertaining the existence of this contraction4 ,there are others4 ,if all that we have just said is true1 only objects which are stationary in the ether would retain their true shapes1 for the shape is altered as soon as there is movement through the ether4 ,hence1 amongst the objects which we think spherical in the material world 7planets1 stars1 projectiles1 drops of water1 and so on71 there would be some that really are spheres1 whilst others would1 on account of the speed or slowness of their movements1 be merely elongated or flattened ellipsoids1 altered in shape by their velocity4 ,amongst the various square objects1 some would be really square1 while others1 travelling at different speeds relatively to the ether1 would be rather rectangles1 shortened on their longer sides owing to their velocity4 ,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 otherwise1 because1 as the ,michelson experiment proves1 we cannot detect a velocity relatively to the ether4 ,this we utterly decline to believe1 say the ,relativists4 ,there are too many difficulties about the matter4 ,why speak persistently1 as ,lorentz does1 of velocities in relation to the ether1 when no experiment can detect such a velocity1 yet experiment is the sole source of scientific truth8 ,why1 on the other hand1 admit that some of the objects we perceive have the privilege of appearing to us in their real shape1 without alteration1 while others do not8 ,why admit such a thing when it is1 of its very nature1 repugnant to the spirit of science1 which is always opposed to exceptions in nature--science deals only with general laws--especially when the exceptions are imperceptible8 ,that was the state of affairs--very advanced from the point of view of the mathematical expression of phenomena1 but very confused1 deceptive1 contradictory1 and troublesome from the physical point of view--when 8at length ,malherbe arrived0 444 ,i mean ,einstein4 ,,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 8,interval0 the only material reality4 ,einstein's first act of intelligent audacity was that1 without relegating the ether to the category of those obsolete fluids1 such as phlogiston and animal spirits1 which obstructed the avenues of science until ,lavoisier appeared--without denying all reality to ether1 for there must be some sort of support for the rays which reach us from the sun--he observed that1 in all that we have as yet seen1 there is always question of velocities relatively to the ether4 ,we have no means whatever of establishing such velocities1 and perhaps it would be simpler to leave out of our arguments this entity1 real or otherwise1 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 difficulties4 ,the first point is then3 ,einstein begins1 provisionally1 by omitting the ether from his line of reasoning4 ,he neither denies nor affirms its existence4 ,he begins by ignoring it4 ,we will now follow his example4 ,we shall no longer1 in the course of our demonstration1 speak about the medium in which light travels4 ,we shall consider light only in relation to the beings or material objects which emit or receive it4 ,we shall find that our progress becomes at once much easier4 ,for the moment we will relegate the ether of the physicists to the store of useless accessories1 along with the suave1 formless1 vague--but so precious artistically--ether of the poets4 99 99 99 99 99 ,shortly1 what does ,michelson's experiment prove8 ,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 west4 ,let us imagine two similar guns in the middle of a plain1 both firing at the same moment1 in calm weather1 and discharging their shells with the same initial velocity1 but one toward the west and the other toward the east4 ,it is clear that the two shells will take the same time to traverse an equal amount of space1 one going toward the west and the other toward the east4 ,the rays of light which we produce on the earth behave in this respect1 as regards their progress1 exactly as the shells do4 ,there would therefore be nothing surprising in the result of the ,michelson experiment1 if we knew only what experience tells us about the luminous rays4 ,but let us push the comparison further4 ,let us consider the shell fired by one of the guns1 and imagine that it hits a target at a certain spot1 and that1 when it reaches the target1 the residual velocity of the shell is1 let us say1 fifty metres a second4 ,i imagine the target mounted on a motor tractor4 ,if the latter is stationary the velocity of the shell in relation to the target will be1 as we said1 fifty metres a second at the point of impact4 ,but let us suppose that the tractor and the target are moving at a speed of1 for instance1 ten metres a second toward the gun1 so that the target passes to its preceding position exactly at the moment when the shell strikes it4 ,it is clear that the velocity of the shell relatively to the target at the moment of impact will not now be fifty metres1 but #ej #aj #fj metres a second4 ,it is equally evident that the speed will fall to #ej-aj #dj metres a second if 7other things being equal7 the target is travelling away from the gun1 instead of toward it4 ,if1 in the latter case1 the velocity of the target were equal to that of the shell1 it is clear that the relative velocity of the shell would now be nil4 ,so much is clear enough4 ,that is how jugglers in the music-halls can catch eggs falling from a height on plates without breaking them4 ,it is enough to give the plate1 at the moment of contact1 a slight downward velocity1 which lessens by so much the velocity of the shock4 ,that is also how skilled boxers make a movement backward before a blow1 and thus lessen its effective force1 whereas the blow is all the harder if they advance to meet it4 ,if the luminous rays behaved in all respects like the shells1 as they do in the ,michelson experiment1 what would be the result8 ,when one advances very rapidly to meet a ray of light1 one ought to find its velocity increased relatively to the observer1 and lessened if the observer recedes before it4 ,if this were the case1 all would be simple2 the laws of optics would be the same as those of mechanics2 there would be no contradiction to sow discord in the peaceful army of our physicists1 and ,einstein would have had to spend the resources of his genius on other matters4 ,unfortunately--perhaps we ought to say fortunately1 because1 after all1 it is the unforeseen and the mysterious that lend some charm to the way of the world--this is not the case4 ,both physical and astronomical observation show that1 under all conditions1 when an observer advances rapidly toward luminous waves or recedes rapidly from them1 they still show always the same velocity relatively to him4 ,to take a particular case1 there are in the heavens stars which recede from us and stars which approach us2 that is to say1 stars from which we recede1 or which we approach1 at a speed of tens1 and in some cases hundreds1 of miles a second4 ,but an astronomer1 de ,sitter1 has proved that the velocity of the light which reaches us is1 for us1 always exactly the same4 ,thus1 up to the present it has proved quite impossible for us1 by any device or movement1 to add to or lessen in the least the velocity with which a ray of light reaches us4 ,the observer finds that the rate of speed of the light is always exactly the same relatively to himself1 whether the light comes from a source which rapidly approaches or recedes from him1 whether he is advancing toward it or retreating before it4 ,the observer can always increase or lessen1 relatively to himself1 the speed of a shell1 a wave of sound1 or any moving object1 by pushing toward or moving away from the object4 ,when the moving object is a ray of light1 he can do nothing of the kind4 ,the speed of a vehicle cannot in any case be added to that of the light it receives or emits1 or be subtracted from it4 ,this fixed speed of about #ahf1jjj miles a second1 which we find always in the case of light1 is in many respects analogous to the temperature of #bgc below zero which is known as 8absolute zero40 ,this also is1 in nature1 an impassable limit4 ,all this proves that the laws which govern optical phenomena are not the same as the classic laws of mechanical phenomena4 ,it was for the purpose of reconciling these apparently contradictory laws that ,lorentz1 following ,fitzgerald1 gave us the strange hypothesis of contraction4 99 99 99 99 99 ,but we shall now find ,einstein showing us1 in luminous fashion1 that this contraction is seen to be perfectly natural when we abandon certain conceptions--perhaps erroneous1 though classical--which ruled our habitual and traditional way of estimating lengths of space and periods of time4 ,take any object--a measuring rod1 for instance4 ,what is it that settles for us the apparent length of the rod8 ,it is the image made upon our retina by the two rays that come from the two ends of the rod1 and which reach our eye simultaneously4 ,i italicise the word1 because it is the key of the whole matter4 ,if the rod is stationary before us1 the case is simple4 ,but if it is moved while we are looking at it1 the case is less simple4 ,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 identical1 and independent of the velocities of the object and the observer4 ,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 case4 ,i stand on the bank by the side of a railway4 ,on the line is a handsome ,pullman car1 in which it is so pleasant to think that space is relative1 in the ,galileian sense of the word4 ,close to the line ,i have two pegs fixed1 one blue1 the other red1 and they exactly mark the ends of the coach and indicate its length4 ,then1 without leaving my observation-post on the bank1 my face turned towards the middle of the coach1 ,i give orders for the coach to be drawn back and coupled to a locomotive of unheard-of power1 which is to carry the coach past me at a fantastic speed1 millions of times faster than the speed any mere engineer could provide4 ,such is the potential superiority of the imagination over sober reality6 ,i assume further that my retina is perfect1 and is so constituted that the visual impressions will remain on it only as long as the light which causes them4 ,these somewhat arbitrary suppositions count for nothing in the essence of the demonstration4 ,they are only for the sake of convenience4 ,now for the question4 ,will the coach 7which ,i assume to be of some rigid metal71 as it passes before me at full speed1 seem to me to be exactly the same length as it did when it was at rest8 ,to put it differently1 at the moment when ,i see its front end coincide with the blue peg ,i had planted1 shall ,i see its back end coincide at the same time with the red peg8 ,to this question ,galileo1 ,newton1 and all the supporters of classic science would reply yes4 ,yet according to ,einstein the answer is no4 ,here is the simple proof1 as we deduce it from ,einstein's general idea4 ,i am1 recollect1 on the edge of the track1 at an equal distance from both pegs4 ,when the front end of the coach coincides with the blue peg1 it sends toward my eye a certain ray of light 7which1 for convenience1 we will call the front ray71 and this coincides with the luminous ray coming to me from the blue peg4 ,this front ray reaches my eye at the same time as a certain ray that comes from the back end of the coach 7which we will call the back ray74 ,does the back ray coincide with the ray which comes to me from the red peg8 ,clearly not4 ,the front ray leaves the front end of the coach at the same speed as the back ray leaves the back end2 as any observer in the coach would find who cared to try the ,michelson experiment on them4 ,but the front end of the coach is receding from me while the back end is approaching me4 ,hence the front ray travels toward my eye more slowly than the back ray1 though ,i cannot perceive this1 as1 when they reach me1 ,i find that they both have the same velocity4 ,hence the back ray1 which reaches my eye at the same time as the front ray1 must have left the back end of the coach later than the front ray left the front end of the coach4 ,therefore1 when ,i see the front end of the coach coincide with the blue peg1 ,i at the same time see the back end of the carriage after it has passed the red peg4 ,therefore the length of a coach travelling at full speed1 and such as it appears to me1 is shorter than the distance between the two pegs1 which indicated the length of the coach at rest4 ,q4,e4,d4 ,very little attention is needed for any person to understand this argument1 though its elementary simplicity has not been attained without difficulty4 ,it is part of ,einstein's mathematical argument and of his conception of simultaneity4 99 99 99 99 99 ,it follows that the coach1 or1 in general1 any object1 seems to be contracted in virtue of its velocity1 and in the direction of that velocity1 relatively to the spectator4 ,the same thing happens1 obviously1 if the observer moves in relation to the object1 because we can know only relative velocities1 in virtue of the ,classical ,principle of ,relativity of ,newton and ,galileo4 ,in this new light the ,lorentz-,fitzgerald contraction becomes intelligible1 or at least admissible4 ,the contraction1 thus considered1 is not the cause of the negative result of the ,michelson experiment3 it is an effect of it4 ,it is now quite clear1 and we see that there was something wrong with the classical way of estimating the instantaneous dimension of objects4 ,certainly the fact that luminous rays1 starting out from their sources at different speeds1 should have the same speed when they reach our eye1 is strange4 ,it upsets our habitual way of looking at things4 ,if ,i may venture to use a comparison simply for the purpose of provoking reflection1 not at all in the way of explanation1 we have here something analogous to what happens with the bombs of aviators4 ,bombs of a given type1 whether released at a height of #e1jjj or of #aj1jjj metres1 which therefore have very different downward velocities at #e1jjj metres from the ground1 have always the same residual velocity when they reach the ground4 ,this is due to the moderating and equalising influence of the atmospheric resistance1 which prevents the speed from increasing indefinitely1 and makes it constant when it has attained a certain value4 ,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 light8 ,who knows8 ,but perhaps such questions have no meaning for the physicist4 ,he can know nothing about the behaviour of light except when it leaves its source or when it reaches the eye1 whether armed with instruments or no4 ,he cannot learn how it behaves during its passage across the intermediate space1 in which there is no matter4 ,indeed1 the more deeply we study the new physics1 the more we see that it derives almost all its strength from its systematic disdain of all that is beyond phenomena1 all that cannot fall under experimental observation4 ,it is because it is solely based upon facts 7however contradictory they may be7 that our proof of the necessary contraction of objects owing to their velocity relatively to the observer is so strong4 99 99 99 99 99 ,we must understand the profound significance of the ,fitzgerald-,lorentz contraction4 ,this apparent contraction is by no means due to the movement of objects relatively to the ether4 ,it is essentially the effect of the movements of objects and observers relatively to each other1 or relative movements in the sense of the older mechanics4 ,the greatest relative velocities to which we are accustomed in our daily life are less than a few kilometres a second4 ,the initial velocity of the shell fired by 8,bertha0 was only about #a1cjj metres a second4 ,for movements so slow as this the ,relativist contraction is entirely negligible4 ,hence1 as the classical mechanics had never observed such contraction1 it regarded the shapes and dimensions of rigid objects as independent of systems of reference4 ,it was very nearly true2 and that makes all the difference between true and false4 ,to say that #iii1iij #i #a1jjj1jjj1 is to say something that is very nearly true1 and is therefore false4 ,when it was discovered that the earth was round no change was made in their procedure by architects4 ,they continued to build as if the direction indicated by the plumb-line was always parallel to itself4 ,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 velocities4 ,what does it matter8 ,the practical point of view is not1 and cannot be1 that of science except indirectly4 ,so much the worse if there is no indirect influence1 or if it is slow in coming4 ,some years ago1 however1 we discovered things which move at speeds1 relatively to us1 of tens or hundreds of thousands of kilometres a second2 the projectiles of the cathode rays and of radium4 ,in this case the ,relativist contraction is very considerable4 ,we shall see how it has been observed4 ,but let us first recapitulate what we have seen4 ,objects seem to alter their shape in the direction of their movement and not in the direction perpendicular to this4 ,therefore their forms1 even if they be composed of an ideal and perfectly rigid material1 depend on their velocity relatively to the observer4 ,this is the essentially new point of view which ,einstein's 8,special ,relativity0 superimposes upon the ,relativity of classical mechanics and philosophers4 ,for these the absolute dimensions of a rigid object or a geometrical figure were not absolute2 it was only the relations of these dimensions which were real4 ,the new point of view is that these relations are themselves relative1 because they are a function of the velocity of the observer4 ,it is a sort of ,relativity in the second degree1 of which neither the philosophers nor the classic physicists had dreamed4 ,spatial relations themselves are relative1 in a space which is already relative4 ,in the case of our ,pullman car and the two pegs which mark its length when it is stationary1 an observer situated in the carriage would find the distance between the two pegs shortened as he passes them4 ,the coach would seem to him longer than the distance between the pegs4 ,i who remain beside the pegs observe the contrary4 ,yet ,i have no means of proving to the passenger that he is wrong4 ,i see quite plainly that the ray of light which comes from the back peg runs behind the coach1 and has therefore1 relatively to it1 a speed of less than #ahf1jjj miles a second4 ,i know that this is the reason for the passenger's error1 but ,i have no means of convincing him that he is wrong4 ,he will always say1 and rightly3 8,i have measured the speed at which this ray reaches me1 and ,i have found it #ahf1jjj miles a second40 ,each of us is really right4 ,in very rapid motion a square would seem to the observer a rectangle2 a circle would appear to be an ellipse4 ,if the earth travelled some thousands of times faster round the sun1 we should see it elongated1 like a giant lemon suspended in the heavens4 ,if an aviator could fly at a fantastic speed over ,trafalgar ,square1 in the direction of the ,strand--and if the impressions on his retina were instantaneous--he would see the ,square as a very flattened rectangle4 ,if he flew in a diagonal line about it1 he would find it shaped like a lozenge4 ,if the same aviator flew across a road on which fat cattle were being driven to the slaughter-house1 he would be astonished1 for the beasts would seem to him extraordinarily lean1 while there would be no change in their length4 ,the fact that these alterations of shape owing to velocity are reciprocal is one of the most curious consequences of all this4 ,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 #agj1jjj miles an hour1 though there would be no limit--would find that his fellows had become dwarfs only half as large as himself4 ,would he have become a giant1 a sort of ,gulliver amongst the ,lilliputians8 ,not in the least4 ,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 himself1 and who are quite sure of the contrary4 ,which is right1 and which wrong8 ,both4 ,each point of view is accurate1 but there are only personal points of view4 ,again1 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 movement4 ,if ,i might venture to relieve this sober exposition by a reflexion rather less austere than is usual in physics1 ,i would say that the new system affords a supreme justification of egoism1 or1 rather1 of egocentricism4 ,it is the same with time as with space4 ,by similar reasoning to that which has shown us how the distance of things in space is connected with their velocity relatively to the observer1 it can be shown that their distance in time likewise depends upon this4 ,it would be useless to reproduce here the whole of the ,einsteinian argument as to duration4 ,it is analogous to that which we have used in regard to length1 and even simpler4 ,the result is as follows4 ,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 pass1 though our watches may be just the same as theirs4#e ,similarly1 all the gestures of men who are on moving vehicles will seem to a stationary observer slowed down1 and therefore prolonged1 and vice versa4 ,but the velocity would1 as in the case of variation in length1 have to be fantastic to make these variations in time perceptible4 #e ,the best definition of the second that can be given is the following3 it is the time which light takes to cover #ahf1jjj miles in empty space and far from any strong gravitational field4 ,this definition1 the only strict definition1 is further justified by the fact that there is no better means of regulating clocks than luminous or ,hertzian 7which have the same speed7 signals4 ,it is not less true that the time between the birth and the death of any creature1 its life1 will seem longer if the creature moves rapidly and fantastically relatively to the observer4 ,in this world1 where appearance is almost everything1 this is not without importance1 and it follows that1 philosophically speaking1 to move on is to last longer2 but for others1 not for oneself2 just as others may seem to me to last longer4 ,a striking1 a profound1 an unforeseen justification of the words of the sage3 immobility is death6 99 99 99 99 99 ,formerly1 before the ,einsteinian hegira1 before the ,relativist ,era opened1 everybody was convinced that the portion of space occupied by an object was sufficiently and explicitly defined by its dimensions--length1 breadth1 and height4 ,these are what are called the three dimensions of an object2 just as we speak1 to use a different expression1 of the longitude1 latitude1 and altitude of each of its points1 or as we speak in astronomy of its right ascension1 declination1 and distance4 ,it was quite understood that we had1 in addition1 to indicate the epoch1 the moment1 to which these data correspond4 ,if ,i define the position of an aeroplane by its longitude1 latitude1 and altitude1 these indications are only correct for a certain moment1 because the aeroplane is moving relatively to the observer1 and the moment also must be indicated4 ,in this sense it has long been known that space depends upon time4 ,but the ,relativist theory shows that it depends upon time in a much more intimate and deeper manner1 and that time and space are as closely connected as those twin monsters which the surgeon cannot separate without killing both4 ,the dimensions of an object1 its shape1 the apparent space occupied by it1 depend upon its velocity3 that is to say1 upon the time which the observer takes to traverse a certain distance relatively to the object4 ,here we have space already depending upon time4 ,in addition1 the observer measures the time with a chronometer1 the seconds of which are more or less accelerated according to his velocity4 ,hence it is impossible to define space without time4 ,that is why we now say that time is the fourth dimension of space1 or that the space in which we live has four dimensions4 ,it is remarkable that there were able men in the past who had a more or less clear intuition of this4 ,thus we find ,diderot1 in #aggg1 writing in the ,encyclopdie1 in the article 8,dimension03 8,i have already said that it is impossible to conceive more than three dimensions4 ,a learned man of my acquaintance1 however1 believes that one might regard duration as a fourth dimension1 and that the product of time by solidity would be1 in a sense1 a product of four dimensions4 ,the idea may not be admitted1 but it seems to be not without merit1 if it be only the merit of originality40 ,it was algebra1 undoubtedly1 that gave rise to the idea of a space with more than three dimensions4 ,since1 in point of fact1 lines or spaces of one dimension are represented by algebraical expressions of the first degree1 surfaces or spaces of two dimensions by formul of the second degree1 and volumes or spaces of three dimensions by expressions of the third degree1 it was natural to ask oneself if formul of the fourth and higher degrees are not also the algebraical representation of some form of space with four or more dimensions4 ,the four-dimensional space of the ,relativists is1 however1 not quite what ,diderot imagined4 ,it is not the product of time by extension1 for a diminution of time is not compensated in it by an increase of space4 ,quite the contrary4 ,take two events1 such as the successive passage of our ,pullman car through two stations4 ,for a passenger in the car the distance between the two stations1 measured by the length of the track covered1 is1 as we saw1 shorter than for a person who is standing stationary beside the line4 ,the time between passing through the two stations is likewise less for the first observer4 ,the number of seconds and fractions of seconds marked by his chronometer is smaller for him1 as we saw4 ,in a word1 distance in time and distance in space diminish simultaneously when the velocity of the observer increases1 and both increase when the velocity of the observer lessens4 ,thus velocity 7velocity relatively to the things observed1 we must always remember7 acts in a sense as a double brake lessening durations and shortening lengths4 ,if a different illustration be preferred1 velocity enables us to see both spaces and times more obliquely1 at an increasingly sharp angle4 ,space and time are therefore only changing effects of perspective4 ,can we conceive space of four dimensions8 ,that is to say1 can we imagine or visualise it8 ,even if we cannot1 it proves nothing as regards the reality of such space4 ,during ages no one conceived such a thing as the ,hertzian waves1 and even to-day we have no direct sense-impression of them4 ,they exist none the less4 ,as a matter of fact1 we find it difficult to conceive space of three dimensions4 ,if it were not for our muscular changes1 we should know nothing about it4 ,a paralysed and one-eyed man1 that is to say1 a man without the sensation of relief which we get from binocular vision--and even this is1 in the first place1 a muscular sensation--would1 with his single eye1 see all objects on the same plane1 as on the drop-scene of a theatre4 ,he could have no perception of three-dimensional space4 ,i believe there are people who can form an idea of four-dimensional space4 ,the successive appearances of a flower in its various phases of growth1 from the day when it is but a frail green bud until the time when its exhausted petals fall sadly to the ground1 and the successive changes of its corolla under the influence of the wind1 give us a globular image of the flower in four-dimensional space4 ,are there any who can see all this together8 ,i believe that there are1 especially amongst good chess-players4 ,when a skilful player plays well1 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 move1 with all their effects on the board4 ,he sees the whole series simultaneously4 ,the words ,i have italicised look contradictory4 ,that is because we are in a province where it is all but impossible to express the fine shades of things in words4 ,one might just as well attempt to define verbally all that there is in a symphony of ,beethoven4 8,the translator is a traitor40 ,if there is any truth in the proverb1 it is because words are the organ of translation4 99 99 99 99 99 ,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 ruins4 ,we had regarded time and space as hooks solidly fastened to the wall behind which lurks reality1 and on these we hang our floating ideas of the material world1 just as we hang our coats on the rack4 ,now they lie1 torn down and crumpled1 amongst the rubbish of ancient theories1 victims of the hammer-blows of the new physics4 ,we knew quite well1 of course1 that the souls of men were inscrutable to us1 but we did think that we saw their faces4 ,now1 as we approach them1 we find that it is only masks we saw4 ,the material world1 as ,einstein shows it to us1 is a sort of masked ball1 and1 by a deceptive irony1 it is we ourselves who have made the black velvet masks and the gay costumes4 ,instead of revealing reality to us1 space and time are1 according to ,einstein1 only moving veils1 woven by ourselves1 which hide it from us4 ,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 them4 ,are time and space1 then1 merely hallucinations8 ,and1 if so1 what is real8 ,no4 ,once the ,relativist has thrown down the tottering ruins1 he begins to reconstruct4 ,behind the veils1 now torn down and trodden under foot1 a new and more subtle reality is about to appear4 ,if we describe the universe in the usual way1 in separate categories of space and time1 we see that its aspect depends upon the observer4 ,happily1 it is not the same when we describe it in the unique category of the four-dimensional continuum in which ,einstein locates phenomena1 and in which space and time are inseparably united4 ,if ,i may venture to use this illustration1 time and space are like two mirrors1 one convex1 the other concave1 the curvature of which is accentuated in proportion to the velocity of the observer4 ,each of these mirrors gives us1 separately1 a distorted picture of the succession of things4 ,but this is fortunately compensated for by the fact that1 when we combine the two mirrors so that one reflects the rays received by the other1 the picture of the succession of things is restored in its unaltered reality4 ,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 increases4 ,we have shown that4 ,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 space4 ,to be precise1 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 side1 which remains invariable4#f #f ,in the geometrical calculus or representation that may be substituted for this the hypotenuse of the triangle is the distance in time1 each second being represented by #cjj1jjj kilometres4 ,taking this third side for base1 the other two will describe1 above it1 a triangle more or less elevated according as the velocity of the observer is more or less reduced4 ,this fixed base of the triangle1 of which the other two sides--the spatial distance and the chronological distance--vary simultaneously with the velocity of the observer1 is1 therefore1 a quantity independent of the velocity4 ,it is this quantity which ,einstein has called the ,interval of events4 ,this 8,interval0 of things in four-dimensional space-time is a sort of conglomerate of space and time1 an amalgam of the two4 ,its components may vary1 but it remains itself invariable4 ,it is the constant resultant of two changing vectors4 ,the 8,interval0 of events1 thus defined1 gives us for the first time1 according to ,relativist physics1 an impersonal representation of the universe4 ,in the striking words of ,minkowski1 8space and time are mere phantoms4 ,all that exists in reality is a sort of intimate union of these entities40 ,the sole reality accessible to man in the external world1 the one really objective and impersonal thing which is comprehensible1 is the ,einsteinian ,interval as we have defined it4 ,the ,interval of events is to ,relativists the sole perceptible part of the real4 ,apart from that there is something1 perhaps1 but nothing that we can know4 ,strange destiny of human thought6 ,the principle of relativity has1 in virtue of the discoveries of modern physics1 spread its wings much farther than it did before1 and has reached summits which were thought beyond the range of its soaring flight4 ,yet it is to this we owe1 perhaps1 our first real perception of our weakness in regard to the world of sense1 in regard to reality4 ,einstein's system1 of which we have now to see the constructive part1 will disappear some day like the others1 for in science there are merely theories with 8provisional titles10 never theories with 8definitive titles40 ,possibly that is the reason of its many victories4 ,the idea of the ,interval of things will1 no doubt1 survive all these changes4 ,the science of the future must be built upon it4 ,the bold structure of the science of our time rises upon it daily4 ,it must1 in fine1 be clearly understood that the ,einsteinian ,interval tells us nothing about the absolute1 about things in themselves4 ,it1 like all others1 shows us only relations between things4 ,but the relations which it discloses seem to be real and unvarying4 ,they share the degree of objective truth which classic science attributed1 with1 perhaps1 unfounded assurance1 to the chronological and spatial relations of phenomena4 ,in the view of the new physics these were but false scales4 ,the ,einsteinian ,interval alone shows us what can be known of reality4 ,einstein's system1 therefore1 takes pride in having lifted for all future time a corner of the veil which conceals from us the sacred nudity of nature4 ,,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 sun4 ,when ,baudelaire wrote3 ,i hate the movement that displaces lines1 he thought only1 like the physicists of his time1 of the static deformations which have been known as long as there have been men to observe them4 ,what we have seen about ,einsteinian time and space has taught us that there must be1 in addition to these1 kinematic deformations1 to which every material object1 however rigid it seems1 is liable4 ,movement1 therefore1 displaces lines much more than ,baudelaire supposed1 even the lines of the hardest of marble statues4 ,this kind of deformation1 which is pleasant rather than hateful1 since it brings us nearer to the heart of things1 has upset the whole of mechanics4 ,mechanics is at the foundation of all the experimental sciences1 because it is the simplest1 and because the phenomena it studies are always present--if not exclusively present--amongst the phenomenal objects of the other sciences1 such as physics1 chemistry1 and biology4 ,the converse of this is not true4 ,for instance1 there is not a single phenomenon in chemistry or biology in which one has not to study bodies in movement1 objects endowed with mass and giving out or absorbing energy4 ,on the other hand1 the peculiar aspects of a biological1 chemical1 or physical phenomenon1 such as the existence of a difference of potential1 an oxidation1 or an osmotic pressure1 are not always found in the study of the movements of a ponderable mass and of the forces which act upon and through it4 ,compared with mechanics1 the sciences of physics1 chemistry1 and biology have1 in the order in which we name them1 objects of increasing complexity and generality1 or1 to put it better1 of decreasing universality4 ,these sciences are mutually dependent in the way that the trunk1 branches1 leaves1 and flowers of a tree are4 ,they are to some extent related to each other as are the various parts of the jointed masts on which military telegraphists fix their antenn4 ,the lower part of the mast1 the larger part1 sustains the whole2 but it is the upper parts which bear the delicate and complicated organs4 ,the object of the great synthetists in science has always been1 and is1 to reduce all phenomena to mechanical phenomena1 as ,descartes attempted4 ,whether these attempts are well-grounded or no1 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 elements1 is a question that has been1 and will continue to be1 much discussed4 ,but1 however thinkers may differ on that point1 they are agreed on this3 in all natural phenomena1 in all phenomena that are objects of science1 there is the mechanical element--exclusive in some1 the principal element in others4 ,all this leads to the conclusion that whatever modifies mechanics1 modifies at the same time the whole structure of ideas founded thereon--that is to say1 the other sciences1 the whole of science1 our entire conception of the universe4 ,but we are now going to see that ,einstein's theory1 as a direct effect of what it teaches in regard to space and time1 completely upsets the classical mechanics4 ,it is in this way1 particularly1 that it has shaken the rather somnolent frame of traditional science1 and the vibration is not yet over4 ,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 realities1 of bodies4 ,here we can compare theory and reality1 the mathematical premises and the substantial verifications2 and we shall be pleased to see what the facts1 given in experience1 have to say on the matter4 ,we shall be able to make our choice1 with informed minds and sound criteria1 between the old and the new ideas4 ,in a word1 if ,i may use this illustration1 as long as we were dealing with ideas of space and time--which are empty frames in themselves1 vases that would interest us chiefly by the liquids they contain--we were rather like the young men who have to choose a fiance solely by the description of her which has been given them4 ,we are now going to see with our own eyes1 and see at work the two aspirants to our affection3 classical science and ,einstein's theory4 ,we shall see both of them take up the paste of facts1 and we shall be able to compare the delicious dishes which they respectively make from it for the nourishment of the mind4 ,theories have no value except as functions of facts4 ,those which1 like so many in metaphysics1 have no real criterion by which we may test them1 are all of the same value4 ,experience1 the sole source of truth of which ,lucretius said long ago3 unde omnia credita pendent1 or the material facts1 is going to judge ,einstein's system for us4 99 99 99 99 99 ,the result of the ,michelson experiment1 the impossibility of proving any velocity of the earth in relation to the medium in which light is propagated1 amounts to this3 we have no means whatever of detecting a speed higher than that of light4 ,this consequence of the ,michelson experiment will be better understood1 perhaps1 if we put it in a tangible form4 ,here is an illustration that will serve our purpose4 ,in some astronomical novel an imaginary observer is supposed to recede from the earth at a speed greater than that of light--at #cjj1jjj miles a second1 let us say--yet to keep his eyes 7armed with prodigious glasses7 steadily fixed on this little globe of ours4 ,what will happen8 ,evidently1 our observer will see the train of earthly events in inverse order1 because in the course of his voyage he will catch up in succession the luminous waves which left the earth before him4 ,the farther away they are1 the longer it must be since they left the earth4 ,after a time our man1 or our superman1 will witness the ,battle of the ,marne4 ,he will first see the field strewn with the dead4 ,gradually the dead men will rise and join their regiments1 and presently they will be seen in groups in ,gallieni's taxis1 which will travel backwards at full speed to ,paris1 arriving in the midst of a population that is extremely anxious about the issue of the struggle1 and the soldiers will1 naturally1 be unable to give them any news4 ,in a word1 our observer will1 if he recedes from the earth at a speed greater than that of light1 see terrestrial events happening as if he were ascending the stream of time4 ,it would be very different if the observer remained stationary1 and the earth receded from him at a speed of #cjj1jjj miles a second4 ,what would happen then8 ,it is clear that in this case our observer will see terrestrial events1 not in inverse order1 but as they are3 except that they would seem to him to take place with majestic slowness1 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 event4 ,in sum1 the phenomena observed by him being essentially different in the two cases1 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 him2 to detect the real movement of the event through space4 ,this means1 of course1 movement relatively to the medium of the propagation of light1 not necessarily1 as we saw1 movement in relation to absolute space4 ,the experiment we have imagined could not very well be carried out with the actual resources of our laboratories4 ,we cannot attain these fantastic speeds1 and even if we could the observer would not distinguish much4 ,but we have chosen a colossal instance1 and the results of it would be colossal1 as there would be question of nothing less than a reversal of the order of time4 ,if we were to use more modest means1 the results will be more modest1 but according to the older theories they ought to be recorded in our instruments4 ,but the ,michelson experiment--a miniature version of what we have just described--shows that the differences we should expect are not observed4 ,therefore the premise we laid down--that there can be velocities greater than that of light in empty space--does not harmonise with reality4 ,hence this velocity of light is a wall1 a limit that cannot be passed4 99 99 99 99 99 ,now let us see what follows4 ,there is at the base of classical mechanics1 as it was founded by ,galileo1 ,huyghens1 and ,newton1 and as it is taught everywhere1 a principle which is in the long run1 like all the principles of mechanics1 grounded upon experience4 ,it is the principle of the composition of velocities4 ,if a boat1 which makes ten miles an hour in smooth water1 sails down a river which flows at five miles an hour1 the speed of the boat in relation to the bank will be1 as we may find by actual measuring1 equal to the sum of the two speeds1 or fifteen miles an hour4 ,this is the rule of the addition of velocities4 ,in a more general way1 if a body starts from a state of rest1 and under the action of some force takes on in a second the velocity ,v1 what will it do if the action of the force is prolonged for another second8 ,according to classical mechanics it will take on the velocity #b,v4#g ,let us imagine an observer who is travelling at the velocity ,v1 yet thinks he is at rest4 ,it will seem to him1 at the end of the first second1 that the body is at rest 7because it has the same velocity as the observer74 ,in virtue of the ,classical ,principle of ,relativity1 the apparent movement of the body must be the same for our observer as if the rest were real4 ,this means that at the end of the second second the relative velocity of the body in reference to the observer will be ,v1 and1 as the observer already has the velocity ,v1 the absolute velocity of the body will be #b,v4 ,in the same way it will be #c,v at the end of three seconds1 #d,v at the end of four seconds1 and so on4 ,could it increase indefinitely if the force continues to act long enough8 ,classical mechanics says 8yes40 ,einstein says 8no10 because there cannot be a greater velocity than that of light4 #g ,as an example of an identical force acting during periods of time successively equal to #a1 #b1 or #c1 we may take three guns of the same calibre1 but of lengths equal to #a1 #b1 and #c1 and of which the charges1 or rather1 their propulsive forces1 are identical and constant4 ,it is found that the initial velocities of the shells are1 in relation to each other1 #a1 #b1 and #c4 ,we have imagined an observer who has the velocity ,v relatively to us1 and who believes that he is at rest4 ,for him the body observed was likewise at rest at the beginning of the second second1 because its velocity was the same as that of the observer4 ,from the fact that the apparent movement of the body is for the observer1 during the second second1 the same as it was for us during the first1 classical mechanics concluded that its velocity doubles during the second second4 ,it did not know what ,einstein has now taught us3 that the time and space of this observer are different from ours4 ,what is a velocity8 ,it is the space traversed in the course of a second4 ,but the space thus measured by our moving observer1 which he believes to be of a certain length1 is in reality1 for us who are stationary1 smaller than he thinks1 because the rules he uses are1 as ,einstein has shown1 shortened by velocity without his perceiving it4 ,therefore the velocities are not added together in equal proportions and indefinitely for a given observer1 as classical mechanics maintained4 ,under the action of the same force1 the old mechanics said1 a body will always experience the same acceleration1 whatever be the velocity already acquired4 ,under the action of the same force1 the new mechanics says1 the motion of the body will be accelerated less and less in proportion to its velocity4 ,take1 for instance1 some movable object having1 relatively to me1 a velocity of #bjj1jjj kilometres a second4 ,let us place an observer on this object4 ,the observer will then start1 in the same direction and under the same conditions as we have done1 a second movable object1 which will thus have1 relatively to him1 a speed of #bjj1jjj kilometres4 ,the ,relativist says that the resultant velocity of the second object relatively to us will not be1 as the classical addition of velocities would make it1 #bjj1jjj #bjj1jjj #djj1jjj kilometres a second4 ,it will be only #bgg1jjj kilometres a second4 ,what the second moving observer took to be #bjj1jjj kilometres 7because his measuring rod was shortened owing to velocity7 was really only #gg1jjj of our kilometres4 ,how is it possible to calculate that8 ,simply by using the formula of ,lorentz which ,i gave in ,chapter ,,ii1 which gives us the value of the contraction due to velocity4 ,we then easily find that1 if we have two velocities1 v and v1 and if we call the resultant w1 classical mechanics stated that w v v ,the ,einstein mechanics says that this is not correct1 and that what we really have 7,c being the velocity of light7 is v v w -------------------------- 7 vv 7 7#a ------------------7 7 ,c 7 ,i apologise for again introducing--it shall be the last time--an algebraical formula into my work4 ,but it spares me a large number of words1 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 it4 ,the formula expresses in the first place the fact that the resultant of the velocities1 however great it may be1 cannot be greater than the speed of light4 ,it conveys also that1 if one of the component velocities is that of light1 the resultant velocity must have the same value4 ,it means1 in fine1 that in the case of the slight velocities we have to do with in actual life 7that is to say1 when the component velocities are much smaller than that of light7 the resultant is very nearly equal to the sum of the two components1 as the classical mechanics says4 ,the classical mechanics was1 we must remember1 founded upon experience4 ,we understand how1 in those circumstances1 ,galileo and his successors1 dealing only with relatively slowly moving bodies1 reached a principle which seemed to be true for them1 but is only a first approximation4 ,for instance1 the resultant of two velocities1 each equal to a hundred kilometres a second 7which is far higher than any velocities obtainable by ,galileo and ,newton71 amounts to1 not #bjj kilometres1 but #aiiiiiigh kilometres4 ,the difference is scarcely twenty-two millimetres in #bjj kilometres6 ,we can quite understand that the earlier experimenters could not detect differences even less minute than that4 99 99 99 99 99 ,amongst the verifications of the new law of composition of velocities we may quote one1 the outcome of an early experiment of the great ,fizeau1 which is very striking4 ,imagine a pipe full of some liquid1 such as water1 and a ray of light travelling along it4 ,we know the speed of light in water3 it is much lower than in air or in empty space4 ,suppose1 further1 that the water is not stationary1 but flows through the pipe at a certain speed4 ,what will be the velocity of the ray of light when it leaves the pipe after traversing the moving liquid8 ,that was what ,fizeau1 with many variations of the conditions of the experiment1 tried to ascertain4 ,the velocity of light in water is about #bbj1jjj kilometres a second4 ,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 mechanics4 ,now the results of ,fizeau's experiment are in complete harmony with ,einstein's formula1 and are not in harmony with that of the older mechanics4 ,many observers1 including1 recently1 the ,dutch physicist ,zeeman1 have repeated ,fizeau's experiment with the greatest care1 but the result was the same4 ,when ,fizeau made the experiment in the last century1 attempts were made to interpret his results in the light of the older theories4 ,this1 however1 led to very improbable hypotheses4 ,fresnel1 for instance1 trying to explain ,fizeau's results1 had been compelled to admit that the ether is partially borne along by the water as it flows1 and that this partial displacement varies with the length of the luminous waves sent through1 or that it is not the same for the blue as for the red waves6 ,a very startling deduction1 and one very difficult to admit4 ,the new law of composition of velocities given to us by ,einstein1 on the other hand1 immediately and with perfect accuracy explains ,fizeau's results4 ,they are opposed to the classical law4 ,the facts1 the sovereign judges and criteria1 show in this case that the new mechanics corresponds to reality2 the earlier mechanics does not1 at least in its traditional form4 ,here is something1 therefore1 which enables us to see at once the profound truth 7scientific truth being what is verifiable71 the beauty1 of the doctrine of ,einstein3 something which shows us1 superbly1 how a scientific1 a physical1 theory differs from an arbitrary and more or less consistent philosophical system4 ,experience1 the supreme judge1 decides in favour of the ,einsteinian mechanics against the older mechanics4 ,we shall see further examples2 and we shall not find a single case in which the verdict is the other way4 99 99 99 99 99 ,let us turn now to a different matter4 ,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 used4 ,up to this we have spoken only of velocities and movements4 ,let us see how these things look when we at the same time examine the particular qualities of the moving objects1 of bodies1 of matter4 ,everybody knows that the characteristic feature of matter is what we call inertia4 ,if matter is at rest1 a force is needed to set it in motion4 ,if it is in motion1 it needs a force to stop it4 ,it needs one to accelerate the movement and one to alter the direction4 ,this resistance which matter offers to the forces which tend to modify its condition of rest or movement is what we call inertia4 ,but different bodies may offer a different degree of resistance to these forces4 ,if a force is applied to an object1 it will give it a certain acceleration4 ,but the same force applied to another object will1 as a rule1 give it a different acceleration4 ,a race-horse making a supreme effort will get along much more quickly under a small jockey than under a man of fifteen stone4 ,a draught-horse will run more quickly if the cart it draws is empty than if it is full of goods4 ,you can start a perambulator with a push that would be useless in the case of a heavy truck4 ,when a locomotive with a few coaches suddenly starts1 the velocity imparted to the train during the first second is what we call its acceleration4 ,if the same locomotive starts1 in the same conditions1 with a much longer train1 we see that the acceleration is less4 ,hence the idea1 introduced into science by ,newton1 of the mass of bodies1 which is the measure of their inertia4 ,if in our example the locomotive produces in the second case an acceleration only half as great1 we express this by saying that the mass of the second train is double that of the first4 ,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 metal1 we see that the two trains are equal in mass4 ,in a word1 the masses of bodies are conventional data defined by the fact that they are proportional to the accelerations caused by one and the same force4 ,to put it differently1 the mass of a body is the quotient of the force which acts upon it by the acceleration given to it4 ,poincar used to say picturesquely3 8,masses are coefficients which it is convenient to use in calculations40 ,if there is one property of bodies which comes within the range of our senses1 a property of which every man has some sort of instinct or intuition1 it is mass4 ,yet careful analysis shows us that we are unable to define it otherwise than by disguised conventions4 ,poincar's definition seems paradoxical in its admission of powerlessness4 ,but it is correct4 ,mass is only a 8coefficient10 a conventional outcome of our weakness6 ,nevertheless1 something remained upon which we thought we could base1 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 phenomena4 ,we believed in the constancy of mass1 of this convenient and so clearly defined coefficient4 ,here again1 unfortunately1 we have to recant--or1 perhaps1 we should say fortunately1 as there is no pleasure like that of novelty4 ,the older mechanics taught us that mass is constant in one and the same body1 and is therefore independent of the velocity which the body acquires4 ,from which it followed1 as we have already explained1 that1 if a force continues to act1 the velocity acquired at the end of a second will be doubled at the end of two seconds1 tripled at the end of three seconds1 and so on indefinitely4 ,but we have just seen that the velocity increases less during the second second than during the first1 and so on1 continuously diminishing until1 when the velocity of light is attained1 that of the moving body can increase no further1 whatever force may act upon it4 ,what does that mean8 ,if the velocity of a body increases less during the second second1 it must be because it offers an increasing resistance to the accelerating force4 ,everything happens as if its inertia1 its mass1 had changed6 ,which amounts to saying that the mass of bodies is not constant3 it depends upon their velocity1 and increases with an increase of velocity4 ,in the case of feeble velocities this influence is imperceptible4 ,it was because the founders of classical mechanics1 an experimental science1 had experience only of relatively feeble velocities that they found that mass was perceptibly constant1 and believed they might conclude that it was absolutely constant4 ,in the case of greater velocities that is not so4 ,similarly1 in the case of feeble velocities1 in the new mechanics as well as the old1 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 direction1 to give a curve to their trajectories4 ,in the case of great velocities that is not so4 ,mass1 therefore1 increases rapidly with velocity4 ,it becomes infinite when the velocity equals that of light4 ,no body whatever can attain or surpass the velocity of light1 because1 in order to pass that limit1 it would need to overcome an infinite resistance4 ,in order to make it quite clear1 let us give certain figures which show how mass varies with velocity4 ,the calculation is easy1 thanks to the formula which we have previously seen1 giving the values of the ,fitzgerald-,lorentz construction4 ,a mass of #a1jjj grammes will weigh an additional two grammes at the velocity of #a1jjj kilometres a second4 ,it will weigh #a1jfj grammes at the velocity of #ajj1jjj kilometres a second2 #a1cda grammes at the velocity of #bjj1jjj kilometres a second2 #b1jjj grammes 7or double7 at the velocity of #bei1hjf kilometres a second2 #c1ije grammes at the velocity of #bij1jjj kilometres a second4 99 99 99 99 99 ,that is what the new theory tells us4 ,but how can we verify it8 ,it would have been impossible only fifty years ago1 when the only velocities known were those of our vehicles and projectiles1 which then did not rise1 even in the case of shells1 above one kilometre a second4 ,the planets themselves are far too slow for the purpose of verification4 ,mercury1 for instance1 the swiftest of them1 travels at a speed of only a hundred kilometres a second1 which is not enough4 ,if we had at our disposal no higher velocities than these1 we should have no means of settling which was right1 the classical mechanics with its constancy of mass or the new mechanics with its assertion of variability4 ,it is the cathode rays and the ,beta rays of radium which have provided us with velocities great enough for the purpose of verification4 ,these rays consist of an uninterrupted bombardment by small and very rapid projectiles1 each of a mass less than the two-thousandth part that of an atom of hydrogen1 and charged with negative electricity4 ,they are the electrons4 ,the cathode tubes of radium give out a continuous bombardment of these minute projectiles1 charged1 not with melinite1 but electricity3 far smaller than the shells of our artillery1 but animated with infinitely greater initial speeds4 ,the velocity of 8,bertha's0 shells is contemptible in comparison4 ,but how was it possible to measure the speed of these projectiles8 ,we know that electrified bodies act upon each other4 ,they attract or repel each other4 ,now our electrons are charged with electricity4 ,if1 therefore1 we put them in an electric field1 between two plates connected at the edges by an electrical machine or an induction coil1 they will be subjected to a force that will cause them to change their direction4 ,the cathode rays1 in other words1 will change their direction under the influence of an electric field4 ,the amount of diversion will depend upon the speed of the projectiles and upon their mass2 that is to say1 upon the resistance of inertia which the mass opposes to the causes which tend to divert it4 ,but this is not all4 ,the electric charges borne by the projectiles are in movement1 even rapid movement4 ,now1 electricity in movement is an electric current1 and we know that currents are diverted by magnets or magnetic fields4 ,therefore the cathode rays will be diverted by the magnet4 ,this diversion will1 like the former1 depend upon the velocity and the mass of the projectile2 but not quite in the same way4 ,other things being equal1 the magnetic diversion will be greater than the electrical diversion1 if the velocity is high4 ,as a matter of fact1 the magnetic diversion is due to the action of the magnet on the current4 ,it will be greater in proportion to the intensity of the current2 and the current will be more intense in proportion to the height of the velocity1 since it is the movement of the projectile which causes the current4 ,on the other hand1 the trajectory of our little projectiles will be less influenced by the electrical attraction in proportion as the velocity of the projectile is great4 ,hence it is easy to see that when we subject a cathode ray to the action of an electric field1 then to that of a magnetic field1 we may1 by comparing the two deviations1 measure at one and the same time the velocity of the projectile and its mass 7related to the known electric charge of the electron74 ,in this way we find enormous velocities1 rising from a few tens of kilometres to #aej1jjj kilometres a second1 and even more4 ,as to the ,beta rays of radium1 they are still more rapid4 ,in cases they attain velocities not far short of that of light1 and higher than #bij1jjj kilometres a second4 ,here are just the velocities we need in order to test whether or no mass increases with them4 99 99 99 99 99 ,in order to understand clearly the progress of the experiments1 it remains to say a few words about the curious phenomenon of electrical inertia which is called self-induction4 ,when we want to set up an electric current1 we find a certain initial resistance which ceases as soon as the current begins4 ,if afterwards we want to break the current1 it tends to maintain itself1 and we have just the same trouble to stop it as to stop a vehicle in motion4 ,it is a matter of daily experience4 ,sometimes the trolley of a tramcar leaves for a moment the wire which conducts the current1 and we then see sparks4 ,why8 ,there was a current passing from the wire to the trolley1 and if the trolley breaks away from the wire for a moment1 leaving an interval of air which obstructs the passage of electricity1 the current will not stop4 ,it has been set going1 as it were1 and it leaps the obstacle in the form of a spark4 ,this phenomenon is what we call self-induction4 ,self-induction--or 8self0 as the electrical workers call it--is a real inertia4 ,the surrounding medium offers resistance to the force which tends to establish an electric current1 and to that which tends to stop a current already set up2 just as matter resists the force which tends to cause it to pass from rest to movement1 or from movement to rest4 ,there is1 therefore1 a real electrical inertia as well as mechanical inertia4 ,but our cathodic projectiles1 our electrons1 are charged4 ,when they begin to move1 they start an electric current2 when they come to rest1 the current ceases4 ,besides mechanical inertia1 then1 they must also have electrical inertia4 ,they have1 so to speak1 two inertias2 that is to say1 two inert masses1 a real and mechanical mass1 and an apparent mass due to the phenomena of electro-magnetic self-induction4 ,by studying the two deviations1 electric and magnetic1 of the ,beta rays of radium or of the cathode rays1 it is possible to determine the respective parts of each of these masses in the total mass of the electron4 ,the electro-magnetic mass due to the causes which we have explained varies with the velocity1 according to certain laws which we gather from the theory of electricity4 ,hence1 by observing the relation between the total mass and the velocity1 we can see what part belongs to the real and invariable mass and what to the apparent mass of electro-magnetic origin4 ,the experiment has been made repeatedly by physicists of distinction4 ,the result of it is surprising3 the real mass is nil1 and the whole mass of the particle is of electro-magnetic origin4 ,here is something that is calculated to modify entirely our ideas of the essence of what we call matter4 ,but that is another story4 ,physicists then asked themselves--this is what we were coming to1 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 ,relativity4 ,the result of the experiments is absolutely clear and consistent1 and some of them have dealt with ,beta rays corresponding to a mass-value ten times greater than the original mass4 ,this result is3 mass varies with velocity1 and in exact accord with the numerical laws of ,einstein's dynamics4 ,here is a new and valuable experimental confirmation4 ,this in turn tends to show that classical mechanics was merely a rough approximation1 valid at the most only for the comparatively slight velocities with which we have to deal in the very restricted course of daily life4 ,thus the mass of bodies1 the ,newtonian property which was believed to be the very symbol of constancy1 the equivalent of what loyalty to treaties is in the moral order of things1 is now merely a small coefficient1 variable1 undulating1 and relative to the point of view4 ,in virtue of the reciprocity which we have described1 when there is question of contraction due to velocity1 the mass of an object increases in the same way1 not only if the object is displaced1 but if the observer is displaced1 and without any other observer1 connected with the object1 being able to detect the difference4 ,for instance1 a measuring rod that moves at a velocity of about #bfj1jjj kilometres a second will not only have its length shortened by one-half1 but will have its mass doubled at the same time4 ,hence its density1 which is the relation of its mass to its volume1 will be quadrupled4 ,the physical ideas which were believed to be most solidly established1 most constant1 most unshakeable1 have been uprooted by the storm of the new mechanics4 ,they have become soft and plastic things moulded by velocity4 99 99 99 99 99 ,further confirmations of the new formula1 quite independent of the one we have just described1 have recently been provided by physicists4 ,one of the most astonishing of these is given in spectroscopy4 ,as is well known1 when we cause a ray of sunlight1 admitted through a narrow slit1 to pass through the edge of a glass prism1 the ray expands1 as it issues from the prism1 like a beautiful fan1 the successive blades of which consist of the different colours of the rainbow4 ,when we examine closely this coloured fan1 we notice certain fine discontinuities1 narrow lines or gaps1 in which there is no light4 ,they look like cuts made with a pair of scissors in our polychrome fan4 ,they are the dark lines of the solar spectrum4 ,each of these lines1 or each group of them1 corresponds to a special chemical element1 and serves to identify this1 whether in our laboratories or in the sun and the stars4 ,it was explained long ago that these lines are due to electrons which revolve rapidly round the nuclei of the atoms4 ,their sudden changes of velocity give rise to a wave 7like those caused in water when you drop a pebble into it7 in the surrounding medium1 and this is one of the characteristic luminous waves of the atom4 ,it reveals itself in one of the lines of the spectrum4 ,the ,danish physicist ,bohr has recently developed this theory in detail1 and has shown that it accurately explains the various spectral lines of the different chemical elements4 ,these1 ,i may note1 differ from each other in the number and arrangement of the electrons which revolve within their atoms4 ,now ,sommerfeld has argued as follows4 ,the electrons which gravitate near the centre of an atom must have a higher velocity than those which revolve in its outer part2 just as the smaller planets1 ,mercury and ,venus1 revolve round the sun far more rapidly than the larger planets1 ,jupiter and ,saturn4 ,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 electrons3 appreciably greater1 as the former revolve with enormous velocities4 ,we can calculate that1 in those conditions1 each line in the spectrum of a chemical element must in reality consist of a number of fine lines joined together4 ,this is precisely what ,paschen afterwards 7#aiaf7 found4 ,he discovered that the structure of the fine lines is strictly such as ,sommerfeld had predicted4 ,it was an astonishing confirmation of an hypothesis3 a proof of the soundness of the new mechanics4 ,but that is not all4 ,we know that the ,x-rays are vibrations analogous to light1 the same in origin1 but consisting of much shorter waves1 or waves with a far higher frequency4 ,hence1 while light comes from the external electrons of the miniature solar system which we call an atom1 the ,x-rays come from the most rapid electrons--those nearest to the centre4 ,it follows that the special structure of the fine lines1 due to the variation of the mass of the electron with its velocity1 must be much more marked in the case of the ,x-rays than in the case of the spectral lines of light4 ,this1 again1 was confirmed by experiment4 ,the figures expressing the observed facts correspond exactly with the calculations of the new mechanics1 as regards the predicted variation of mass with velocity4 ,it is therefore settled that the phenomena which take place in the microcosm of each atom are subject to the laws of the new mechanics1 not the old1 and that1 in particular1 masses in motion vary as the new mechanics demands4 ,experience1 8sole source of truth10 has given its verdict4 ,we are now very far from the ideas which were once prevalent4 ,lavoisier taught us that matter can neither be created nor destroyed4 ,it remains always the same4 ,what he meant was that mass is invariable1 as he proved by means of scales4 ,now it appears that1 perhaps1 bodies have no mass at all--if it is entirely of electro-magnetic origin--and that1 in any case1 mass is not invariable4 ,this does not mean that ,lavoisier's law has now no meaning4 ,there remains something that corresponds to mass at low velocities4 ,our idea of matter is1 however1 revolutionised4 ,by matter we particularly meant mass1 which seemed to us to be at once the most tangible and most enduring of its properties4 ,now this 8mass0 has no more reality than the time and space in which we thought we located it6 ,our solid realities were but phantoms4 ,the reader must pardon me for whatever difficulties he finds in this exposition4 ,the new mechanics opens out to us such strange new horizons that it is worth far more than a rapid and superficial glance4 ,if you want to see a vast prospect in an unexplored world1 you must not hesitate to do some rough climbing1 however breathless it may leave you for the time4 99 99 99 99 99 ,there is1 in fine1 another fundamental idea of mechanics1 that of energy1 which takes on a new aspect in the light of ,einstein's theory3 an aspect which1 in turn1 is largely justified by experiment4 ,we saw that a body charged with electricity and in motion makes a certain resistance to interference1 on account of the electrical inertia which is known as self-induction4 ,calculation and experiment show that1 if we reduce the dimensions of a body that is charged with a certain quantity of electricity1 without altering the charge1 the electrical inertia increases4 ,as a matter of fact1 in our hypotheses1 and if the inertia is entirely electro-magnetic in origin1 the electrons are now merely a sort of electric trails moving in the propagating medium of electrical and luminous waves which we call ether4 ,the electrons are no longer anything in themselves4 ,they are merely1 in the words of ,poincar1 a sort of 8holes in ether10 round which the ether presses much as a lake makes eddies which check the progress of a boat4 ,in that case1 however1 the smaller the holes in the ether are1 the more important will be the agitation of the ether round them2 and1 consequently1 the greater will be the inertia of the 8hole in ether0 which represents the corpuscle under investigation4 ,what will follow8 ,we know from measurements we have made that the mass of the tiny sun of each atom1 the positive nucleus1 round which the planet-electrons revolve1 is greater than that of an electron4 ,if this mass and the corresponding inertia are electro-magnetic in origin1 it follows that the positive nucleus of the atom is much smaller than the electron4 ,let us consider the atom of hydrogen1 the lightest and simplest of the gases4 ,we know that it consists of one planet only1 one single negative electron revolving round the minute central sun1 the positive nucleus4 ,we know also that the mass of the electron is two thousand times as small as that of the hydrogen atom4 ,it follows1 as we can calculate1 that the positive nucleus must have a radius two thousand times smaller than that of the electron4 ,now1 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 nucleus4 ,we conclude that the latter is in reality much smaller than the electron1 as theory predicted4 ,all this irresistibly compels us to think that the inertia of the various component parts of atoms--that is to say1 of all matter--is exclusively electro-magnetic in origin4 ,there is now no matter4 ,there is only electrical energy1 which1 by the reactions of the surrounding medium upon it1 leads us to the fallacious belief in the existence of this substantial and massive something which hundreds of generations have been wont to call 8matter40 ,and from all this it also follows1 by calculation and by the simple and elegant reasoning of ,einstein1 of which ,i here convey only the faintest adumbration1 that mass and energy are the same thing1 or are at least the two different sides of one and the same coin4 ,there is1 then1 no longer a material mass4 ,there is nothing but energy in the external universe4 ,a strange--in a sense1 an almost spiritual--turn for modern physics to take6 ,according to all this the greater part of the 8mass0 of bodies must be due to a considerable and concealed internal energy4 ,it is this energy which we find gradually dissipated in radio-active bodies1 the only reservoirs of atomic energy which have as yet opened externally4 ,if this is true1 if energy and mass are synonymous1 if mass is merely energy1 it follows that free energy must possess the property of mass4 ,as a matter of fact1 light1 for instance1 has mass4 ,careful experiments have shown that when a ray of light strikes a material object1 it exerts upon it a pressure which has been measured4 ,light has mass2 therefore it has weight1 like all masses4 ,when we come to consider the new form given by ,einstein to the problem of gravitation1 we shall see a further and beautiful proof that light has weight4 ,we can calculate that the light received from the sun by the earth in the space of a year is rather more than #eh1jjj tons4 ,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 luminary4 ,the reason for the difference is that1 when we produce heat from a certain amount of coal1 we use only a small proportion of its total energy1 its chemical energy4 ,its intra-atomic energy is inaccessible to us4 ,it is a pity1 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 ,england6 ,how many problems that would simplify6 ,when humanity emerges from the ignorance and the clumsy barbarism in which it lives to-day--that is to say1 in some hundreds of centuries--this will be accomplished4 ,yes1 it will one day be done4 ,it will be a glorious spectacle1 one in which we may justly rejoice in advance4 ,meantime1 our sun1 like all the other stars1 like every incandescent body1 loses its weight in proportion as it radiates4 ,but this happens so slowly that we need not fear to see it disappear at some early date1 like the ephemeral things which die because they gave themselves too freely4 99 99 99 99 99 ,to finish with ,einstein's mechanics1 let me reproduce a very suggestive application of these ideas about the identity of energy and mass4 ,there is in chemistry a well-known elementary law which is called 8,prout's ,law40 ,it states that the atomic masses of all the elements must be whole multiples of the mass of hydrogen4 ,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 element1 the atom of hydrogen4 ,this supposed unity of matter seems to be more and more confirmed by the facts4 ,on the one hand1 it is proved that the electrons which come from different chemical elements are identical4 ,on the other hand1 in the transformation of radio-active bodies we find heavy atoms simplifying themselves by successively emitting atoms of helium gas4 ,lastly1 the great ,british physicist ,sir ,ernest ,rutherford showed in #aiai that by bombarding the atoms of nitrogen gas1 in certain circumstances1 by means of radium emanation1 we can detach hydrogen atoms from them4 ,this experiment1 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 hypothesis4 ,yet1 when we accurately measure and compare the atomic masses of the various chemical elements1 we find that they do not strictly conform to ,prout's ,law4 ,for instance1 while the atomic mass of hydrogen is #a1 that of chlorium is #cedf1 which is not a whole multiple of #a4 ,but we can calculate that1 if the formation of complex atoms from hydrogen upwards is accompanied1 as is probable1 by variations of internal energy1 as a consequence of the radiation of a certain amount of energy during the combination1 it necessarily follows 7since the lost energy has weight7 that there will be variations in the mass of the body composed1 and these will explain the known departures from ,prout's ,law4 99 99 99 99 99 ,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 difficult4 ,it is because1 since we could not use terminology and technical formul which would be unsuitable in this work1 we have had to be content with bold and rapid moves into the districts we wished to reconnoitre4 ,perhaps they have sufficed to enable the reader to understand what a revolution in the very bases of science1 what an explosion amidst its age-old foundations1 the brilliant synthesis of ,einstein has caused4 ,new light now streams upon all who slowly climb the slopes of knowledge3 upon all who1 wisely renouncing the desire to know 8why10 would at least learn the 8how0 in many things4 ,a little before his death1 foreseeing1 with the intuition of genius1 that a new era opened in mechanics1 ,poincar 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 mechanics4 8,it is10 he added1 8with ordinary mechanics that their life is concerned3 it is that alone that they will ever have to apply4 ,whatever speed our motor-cars may attain1 they will never reach a speed at which the old mechanics ceases to be true4 ,the new is a luxury1 and we must think of luxuries only when it can be done without injury to necessaries40 ,i would appeal from ,poincar's text to the man himself4 ,for him this luxury1 the truth1 was a necessary4 ,on the day in question1 it is true1 he thought of the young4 ,but do men ever cease to be children8 ,to that the master1 too early taken from us1 would have replied1 in his grave1 smiling manner3 8,yes--at all events1 it is better to suppose so40 ,,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 light4 ,we are now on the threshold of the great mystery of gravitation4 ,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 light4 ,they had hitherto been separate and anarchic provinces of the universe4 ,we now know that the same laws govern mechanics and optics4 ,if for a time it appeared otherwise1 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 speeds4 ,it is in its power of synthesis that ,einstein's mechanics is so splendid4 ,thanks to it1 we perceive more unity1 more harmony1 more beauty1 than formerly in this astounding universe1 in which our thoughts and our anxieties are so ephemeral4 ,the theory of ,relativity1 however1 has up to the present not touched a phenomenon that is fundamental1 essential1 ubiquitous in our cosmos4 ,i mean gravitation1 the mysterious property of bodies which rules the tiny atom no less than the most gigantic star1 and directs their paths in majestic curves4 ,the universal attraction which1 as far as earth is concerned1 we call weight was a kind of steep-cliffed island in the sea of phenomena1 something unrelated to the rest of natural philosophy4 ,the ,einsteinian mechanism1 as we have described it up to now1 passed by this island1 taking no notice of it4 ,for that reason it was1 in this form1 known as 8the theory of ,special ,relativity40 ,in order to convert it into a perfect instrument of synthesis1 the phenomenon of gravitation had to be introduced4 ,it is thus that ,einstein crowned his work1 and his system assumed the form which is well called 8the theory of ,general ,relativity40 ,einstein has drawn gravitation from its 8splendid isolation10 and has annexed it1 docile and vanquished1 to the triumphal chariot of his mechanics4 ,he has1 moreover1 given ,newton's famous law a more correct form1 and experiment1 the supreme judge1 has declared this the only just form4 ,how he did this1 by what subtle and powerful chain of reasoning1 by what calculations based upon facts1 ,i will now endeavour to tell2 and ,i will again do my best to avoid the network of barbed wire of mathematical terminology4 ,why did ,newton1 followed by the whole of classical science1 believe that gravitation1 the fall of bodies1 did not belong to the mechanics of which he formulated the laws8 ,why1 in a word1 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 space8 ,because of the principle of inertia4 ,this principle1 the foundation of the whole ,newtonian mechanics1 may be expressed thus3 a body which is not acted upon by any force maintains its velocity and direction unchanged4 ,why do we equip steam-engines with the heavy wheels which we call 8fly-wheels10 which work nothing8 ,because the principle of inertia is certainly nearly true4 ,when the engine experiences a sudden and sharp check1 or an acceleration1 the fly-wheel serves to keep it steady4 ,driven by the speed it has acquired1 and driving the engine in its turn1 it tends to preserve its velocity1 and it prevents or modifies accidental checks or accelerations4 ,the principle is therefore based upon experience1 especially on the experiments of ,galileo1 who verified it by rolling balls down planes inclined at different angles4 ,for instance1 we find that a ball set in motion on a highly polished horizontal plane keeps its direction1 and would preserve its velocity if the resistance of the atmosphere and the friction of the plane did not gradually reduce it to zero4 ,we find that1 in proportion as we reduce the friction1 the ball tends to maintain its speed so much the longer4 ,newton's principle of inertia is based upon a number of these experiments4 ,it is by no means in the nature of a self-evident mathematical truth4 ,this is so true that ancient thinkers believed1 contrary to classical mechanics1 that the movement ceases as soon as the cause of it is removed4 ,certain of the ,greek philosophers even thought that all bodies travel in a circle1 if nothing interferes with them1 because the circular is the noblest of all movements4 ,we shall see later how the principle of inertia of ,einstein's generalised mechanics has a strange affinity to this idea1 and at the same time to the curious declination1 the clinamen1 which the great and profound ,lucretius attributed to the free path of the atoms4 ,but we must not anticipate4 99 99 99 99 99 ,this belief1 that an object left freely to itself and not acted upon by any force preserves its velocity and direction1 cannot pretend to be more than an experimental truth4 ,but the observations on which it is based1 especially those of ,galileo1 but any that may be imagined by physicists1 could not possibly be conclusive1 because in practice it is impossible to protect a moving body from every external force1 such as atmospheric resistance1 friction1 or other4 ,i am aware that ,newton grounded his principle on astronomical as well as terrestrial observations4 ,he noticed that1 apart from any attraction by other celestial bodies1 and as far as we can see1 the planets seem to maintain their direction and velocity relatively to the vault of heaven4 ,but ,relativists think that the words ,i have italicised in the preceding sentence1 which reflect ,newton's idea1 really beg the question4 ,his argument assumes that the planets do not circulate freely2 that they are governed in their motions by a force which he called universal attraction4 ,we shall see how ,einstein came to think that this is not a force1 and in that case the issue of the argument is very different4 ,however that may be1 the classical principle of inertia is a truth based upon 7imperfect7 experience1 and it is therefore subject to the constant control of facts4 ,all that we can say about it is that practically--that is to say1 approximately--it harmonises with what we find4 ,newton did not regard it as such1 not as a more or less precise approximation1 but as a strict truth4 ,that is why1 when he saw that the planets do not travel in straight lines but in curved orbits1 he concluded--which is a petitio principii--that they were subject to a central force1 gravitation4 ,that is why heavy bodies did not seem to him amenable to the mechanical laws which he had formulated for bodies left freely to themselves4 ,that is why1 in a word1 ,newton's law of gravitation and his laws of dynamics are two distinct and separate things4 ,the great genius1 the mind which had no equal1 was nevertheless human4 ,the immortal ,descartes put forward strange statements and very occult hypotheses 7about the pineal gland and animal spirits71 after he had expressly resolved to affirm nothing that he did not perceive clearly and distinctly4 ,in the same way ,newton1 after laying down as his principle ,hypotheses non fingo1 put the hypotheses of absolute time and space at the very basis of his mechanics4 ,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 gravitation4 ,these are weaknesses which the greatest of men do not escape4 ,they ought to make us admire all the more the finer aspects of their work4 ,so deep is the furrow ploughed by these great students of the unknown that1 even when it is not straight1 it takes two centuries and a half before men dream of inquiring afresh whether ,newton's distinction between purely mechanical and gravitational phenomena was just4 ,it is the signal distinction of ,einstein that he successfully accomplished this3 that1 after erasing many things which were supposed to be finally settled1 he blended mechanics and gravitation in a superb synthesis1 and enabled us to see more clearly the sublime unity of the world4 99 99 99 99 99 ,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 satisfactory4 ,it says3 ,bodies attract each other in direct proportion to their masses and in inverse proportion to the square of their distances4 ,what does that mean8 ,we saw that the mass of a body varies with its velocity4 ,when1 for instance1 we introduce the mass of our planet into calculations which involve ,newton's law1 what precisely do we mean8 ,do we mean the mass which the earth would have if it did not revolve round the sun8 ,or do we mean the larger mass which it has in virtue of its motion8 ,this motion1 however1 is not always of the same speed1 because the earth travels in an ellipse1 not a circle4 ,what value shall we give to this variable mass in the calculation8 ,that which corresponds to perihelion or aphelion1 the period when the earth travels most rapidly or most slowly8 ,moreover1 ought we not also to take into account the velocity of translation of the solar system1 which in turn increases or diminishes according to the season8 ,again1 under ,newton's law what shall we make the distance from the earth to the sun8 ,is it to be the distance relatively to an observer on the earth or on the sun1 or to a stationary observer in the middle of the ,milky ,way who does not share the motion of our system across it8 ,here again we shall have different values in each case1 because spatial distances vary1 as we saw with ,einstein1 according to the relative velocity of the observer4 ,hence ,newton's law is1 in spite of its simple and artistic form1 ambiguous and far from clear4 ,i am aware that the differences ,i have just noted are not very important1 but our calculations show that they are by no means negligible4 ,einsteinians therefore regard it as indisputable1 apart from the considerations which we shall see presently1 that ,newton's law1 in its classical form1 is obscure1 and must be modified and completed4 ,these preliminary remarks will serve to at least put us in the frame of mind that is required of iconoclasts2 and in science the iconoclasts are often the makers of progress4 ,the particular idols at which we are preparing to deal a few audacious blows are the conception of the ,newtonian law and gravitation4 ,laplace wrote1 in his ,exposition du ,systme du ,monde3 8,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 distance40 ,nothing can better show us than this sentence of the great mathematician the importance of the step taken by ,einstein when he1 as we shall see1 improved what had been regarded as the very type1 the most perfect example1 of scientific truth3 the famous ,newtonian law4 99 99 99 99 99 ,gravitation1 or weight1 has this in common with inertia1 that it is a quite general phenomenon4 ,all material objects1 whatever may be their physical and chemical condition1 are both inert 7that is to say1 according to their mass they resist forces which tend to displace them7 and heavy 7they fall when they are left to themselves74 ,but it is a strange thing1 noted by ,newton1 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 weight4 ,this figure is the mass of the body4 ,let us return to the illustration which ,i used in a previous chapter in dealing with ,einstein's mechanics4 ,if two trains drawn by two similar locomotives start in the same conditions1 and if the velocity communicated to the first train at the end of a second is double that communicated to the second1 we conclude that the inertia1 the inert mass1 of the second train 7leaving out of account the friction with the rails7 is twice as great as that of the first4 ,if we afterwards weigh our two trains1 we find that the weight of the second is similarly twice as great as that of the first4 ,this experiment1 though crude enough in our illustration1 has been made with great precision by physicists1 who used delicate methods which we need not describe here4 ,the result was the same4 ,the inert mass and the weight of bodies are exactly expressed by the same figures4 ,newton saw in this a mere coincidence4 ,einstein found in it the key to the hermetically sealed and inviolate dungeon in which gravitation was isolated from the rest of nature4 ,let us see how4 ,there is one remarkable feature of weight or gravitation3 whatever be the nature of the objects1 they always fall at the same speed 7apart from atmospheric resistance74 ,this is easily proved by causing a number of different objects to fall1 in the same period of time1 down a long tube in which a vacuum has been created4 ,they all reach the bottom of the tube at the same time4 ,a ton of lead and a sheet of paper will1 if they are launched into the void simultaneously from the summit of a tower1 reach the ground simultaneously1 with a velocity the acceleration of which is1 near the ground1 #iha centimetres a second4 ,this fact was known to ,lucretius4 ,two thousand years ago that profound and immortal poet wrote3 ,nulli1 de nulla parte1 neque ullo ,tempore1 inane potest vacuum subsistere rei1 ,quin sua quod natura petit concedere pergat4 ,omnia quapropter debent per inane quietum que ponderibus non quis concita ferri4#h #h ,de ,natura ,rerum1 bk4 ii1 vv4 #bce-dj4 ,now if weight were a force analogous to electrical attraction1 to the propulsion of a locomotive1 or even to the propulsive action of a charge of powder1 this ought not to be the case4 ,the velocities which it communicates to different masses ought to be different from each other4 ,the two trains of unequal mass in our illustration receive unequal accelerations from the same locomotive4 ,nevertheless1 if a great trench suddenly opened before them1 they would fall into it with the same velocity4 ,from this it is only one step to conclude that gravitation is not a force1 as ,newton thought1 but simply a property of space in which bodies move freely4 ,einstein took this step without hesitation4 ,imagine the cable of the lift in some colossal skyscraper suddenly breaking4 ,the lift will fall with an accelerated movement1 though less rapidly than it would in a vacuum1 on account of the atmospheric resistance and the friction of the cage of the apparatus4 ,but let us suppose1 further1 that the electrical engine which works the lift has its commutator reversed at the same time1 and this accelerates the fall to such an extent that the velocity of the descent increases #iha centimetres in every second4 ,it would be quite easy for our engineers to carry out this experiment1 though the interest of it has not up to the present seemed great enough to justify it4 ,but we have the right1 when it is necessary to make a subject clear1 to say with the poet3 ,an thou wilt1 let us dream a dream4 ,let us suppose our dream fulfilled4 ,the lift falls from above with precisely the accelerated velocity of an object falling in a vacuum4 ,if the passengers have kept cool enough in their giddy rush downward to observe what happens1 they will notice that their feet cease to press against the floor of the lift4 ,they can imagine themselves like ,la ,fontaine's charming and poetic princess3 ,no blade of grass had felt ,the light traces of her steps4 ,our passengers' purses will1 even if they are full of gold1 no longer be heavy in their pockets--which may give them a momentary anxiety4 ,if their hats are released from their hands1 they will remain suspended in the air beside them4 ,if they happen to have scales with them1 they will notice that the pans remain poised at equal height1 even if various weights are put in one pan4 ,all this is because the objects1 as a natural effect of their weight1 fall toward the ground with the same velocity as the lift itself4 ,their weight has disappeared4 99 99 99 99 99 ,jules ,verne described this state of things in the projectile which he imagined taking his heroes from the earth to the moon1 at the moment when the romantic projectile reaches the 8neutral point03 that is to say1 the point where it leaves the earth's sphere of gravitation1 but has not yet entered that of the moon4 ,we might add that ,jules ,verne perpetrated a few little scientific heresies in connection with his projectile4 ,in particular1 he forgot that1 in compliance with what is most conspicuously evident in the principle of inertia1 the unfortunate passengers ought to have been flattened like pancakes against the bottom of the projectile when the charge was fired4 ,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 attraction1 that of the earth and that of the moon4 ,but let us overlook these trifles and return to the admirable illustration he has prophetically provided for our convenience in explaining ,einstein's system4 ,let us take the projectile when it begins to fall freely toward the moon4#i ,it is evident that from this point onward1 until it lands on the moon1 it will behave exactly like the lift which we have described4 #i ,it is obvious that we assume the projectile to be without rotation3 that is to say1 the ,columbia cannon must not1 in our hypotheses1 be rifled4 ,this is indispensable1 for if the projectile turned1 there would be centrifugal effects which would greatly complicate both the phenomena and our argument4 ,during this fall upon the moon the passengers1 if they have miraculously escaped being flattened at the start1 will see the various objects about them suddenly deprived of their weight1 floating in the air1 and1 at the slightest shake1 adhering to the walls or the vaulted roof of the projectile4 ,they will feel themselves extraordinarily light1 and they will be able to make prodigious leaps without any effort4 ,this is because they and all the objects about them fall toward the moon with the same velocity as the projectile4 ,hence the disappearance of weight or gravitation1 which vanish as if spirited away by some magician4 ,the magician is the properly accelerated movement1 the unimpeded fall of the observers4 ,in a word1 to get rid of the apparent effects of gravitation in any place whatever it is enough for the observer to acquire a properly accelerated velocity4 ,that is what ,einstein calls the 8principle of equivalence03 equivalence of the effects of weight and of an accelerated movement4 ,the one cannot be distinguished from the other4 ,let us imagine ,jules ,verne's projectile and its unfortunate passengers transported a long distance from the moon1 the earth1 and the sun1 to some deserted and glacial region of the ,milky ,way where there is no matter1 and so remote from the stars that there is no longer any weight or attraction4 ,let us suppose that our projectile is abandoned there1 and motionless4 ,it is clear that in these circumstances there will be no such thing as high or low--no such thing as weight--for the passengers4 ,they will find themselves relieved of every inconvenience of weight4 ,they may1 if they choose1 stand on the inner wall of the upper part of the projectile or on the floor1 as it was when they were falling upon the moon4 ,now let us suppose that the wizard ,merlin quietly approaches and1 fastening a cord to the ring on the top of the projectile1 begins to drag it with a uniformly accelerated movement4 ,what will happen to the passengers8 ,they will notice that they have suddenly recovered their weight1 and that they are riveted to the floor of the projectile1 much as they were drawn to the surface of our planet before they left it4 ,indeed1 if the motion of ,merlin is accelerated #iha centimetres a second1 they will have exactly the same sensations of weight as they had on the earth4 ,they will notice that if they throw a plate into the air at a given moment1 it will fall upon the floor and be broken4 8,this is10 they will think1 8because we are again subject to weight4 ,the plate falls in virtue of its weight1 its inert mass40 ,but ,merlin will say3 8,the plate falls because1 on account of its inertia1 it has retained the increasing velocity which it had at the moment when it was thrown4 ,immediately afterwards1 as ,i drew the projectile with an accelerated movement1 the ascending velocity of the projectile was greater than that of the plate4 ,that is why the bottom of the projectile1 in its accelerated ascending course1 knocked against the plate and broke it40 ,this proves that the weight or gravitation of a body is indistinguishable from its inertia4 ,inert mass and heavy mass are not1 as ,newton supposed1 two things which happen by some extraordinary coincidence to be equal2 they are identical and inseparable4 ,the two things are really one4 ,and we are thus led to believe that the laws of weight and the laws of inertia1 the laws of gravitation and those of mechanics1 must be identical1 or must at least be two modalities of one and the same thing3 much as the full face and the profile of the same man are the same face seen under two different angles4 ,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 them1 it will not alter their illusion4 ,they will believe that they are at rest and floating at a point of space where weight has been restored3 that is to say1 in the language of the experts1 at a point of space where there is a 8gravitational field40 ,this phrase is analogous to the familiar 8magnetic field10 which refers to a part of space in which there is magnetic action1 a part in which the needle of the compass has a definite direction imposed upon it4 ,in sum1 we can at any point replace a gravitational field1 or the effects of weight1 by a properly accelerated movement of the observer1 and vice versa4 ,there is a complete equivalence between the effects of weight and those of an appropriate movement4 99 99 99 99 99 ,this now enables us to establish very simply the following fundamental fact1 unknown only a few years ago1 but now brilliantly proved by experiment3 ,light does not travel in a straight line in those parts of the universe where there is gravitation1 but its path is curved like that of heavy objects4 ,we showed in one of the preceding chapters that in the four-dimensional continuum in which we live1 which we might call 8space-time0 but which we more simply call the universe1 there is something that remains constant1 identical for observers who move at given and different velocities4 ,it is the 8,interval0 of events4 ,it is natural to suppose that this 8,interval0 will remain identical even if the velocity of the observers changes--even if it is accelerated like the velocity of the lift in our illustration1 or of ,jules ,verne's projectile1 during their fall4 ,in point of fact1 if something in the universe is an invariant1 as physicists say1 or invariable1 for the observers who move at different speeds1 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 observer1 and who is therefore in a state of uniformly accelerated movement4 ,from this we deduce certain consequences of a fundamental character4 ,in the first place1 one thing is evident1 and is unanimously admitted by physicists3 in a vacuum1 and in a region of space where there is no force acting and no such thing as weight1 light travels in a straight line4 ,that is certain for many reasons--in the first place1 on the mere ground of symmetry1 because in a region of isotropic vacuum a ray which is uninfluenced will not depart from its rectilinear path in any direction whatever4 ,that is evident1 whatever hypothesis we adopt as to the nature of light1 and even if1 like ,newton1 we suppose that it consists of ponderable particles4 ,admitting that1 let us now suppose that at some point in the universe where there is weight--at the moon's surface1 for instance--there is a remarkable gun which can fire a ball that has and retains 7along its whole path7 the velocity of light4 ,the trajectory of this ball will be very extensive1 on account of its great velocity1 yet curved toward the surface of the moon on account of its weight4 ,as we may make our choice in the field of hypotheses1 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 trail4 ,there were projectiles of this character during the ,great ,war4 ,as the ball advances1 it also falls every second toward the moon's surface1 to the same extent as any other projectile would which was fired at any velocity whatever1 or had no velocity4 ,all objects near the surface of the ground 7in a vacuum7 fall at the same vertical velocity1 and this is independent of their motion in the horizontal direction4 ,that is1 in fact1 the reason why the paths of projectiles are the more curved the less initial speed they have4 ,seen from the windows of ,jules ,verne's projectile 7which is itself falling toward the moon71 the trajectory of the ball will seem to the passengers to be a straight line1 because it falls with the same velocity as they4 ,now let us suppose that a luminous ray1 from the flame of the gun1 starts at the same time and in the same direction as the ball4 ,this luminous ray will obviously be rectilinear for the passengers in the projectile1 because light travels in a straight line when there is no weight4 ,consequently1 since it has the same form1 direction1 and velocity as the luminous ball1 the passengers will see the ray of light coincide in its whole course with the trajectory of the ball4 ,it further follows that the 8,interval0 7both in time and space7 of the luminous ray and of the ball is1 and remains1 zero4 ,now this 8,interval0 must remain the same1 whatever be the velocity of the observer4 ,hence1 if ,jules ,verne's projectile ceases to fall1 and is stopped at the moon's surface1 its passengers will continue to see the luminous ray coincide at every point with the trajectory of the ball4 ,this trajectory is1 as they now notice1 curved on account of weight4 ,therefore1 the luminous ray is similarly curved in its path on account of weight4 ,this shows that light does not travel in a straight line1 but falls1 under the influence of gravitation1 like all other objects4 ,the reason why this was never known before1 and it was always thought that light travels in a straight line1 is that on account of the enormous velocity of light its trajectory is only very slightly curved by weight4 ,that is easy to understand4 ,at the earth's surface1 for instance1 light must fall 7like all other objects7 with a velocity equal to #iha centimetres at the end of a second4 ,now by the end of a second a luminous ray has travelled #cjj1jjj kilometres4 ,suppose we could observe a horizontal luminous ray #cjj kilometres long near the earth's surface--a very far-fetched supposition--during the thousandth part of a second1 which it will take the ray to pass from one observer to the other1 it will fall to the extent of only about the five-thousandth of a millimetre4 ,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 rectilinear4 ,is there no means of verifying whether light is or is not bent out of its path by gravitation8 ,there is such a means in astronomy1 as we shall now see4 99 99 99 99 99 ,it is impossible to detect the curvature of a luminous ray travelling from one point to another on the earth's surface1 mainly because weight on the earth is too slight to bend the ray much4 ,a further reason is that our planet is so ridiculously small that we cannot follow the light over a sufficient distance4 ,but what cannot be done on this little globule of ours1 the entire diameter of which light can cover in the twenty-fifth of a second1 may possibly be done in the gigantic laboratory of celestial space4 ,we have1 almost within our reach--a mere matter of #ic1jjj1jjj miles away1 that is to say--a star on which weight is twenty-seven times greater than on the earth4 ,we mean the sun4 ,on the sun a body left to itself falls #acb metres in the first second4 ,its fall is twenty-seven times as rapid as on the earth4 ,hence1 near the sun1 light will be much more bent out of its path by gravitation4 ,the deviation will be all the greater from the fact that the sun is #hjj1jjj miles in diameter1 and a luminous ray needs a much longer time to cover this distance than to travel the length of the earth's diameter4 ,hence gravitation acts upon the ray of light during a much longer time than upon a ray that reaches the earth1 and it will be all the more curved4 ,take a luminous ray that comes from a star at a great distance behind the sun4 ,if it reaches us after passing near to the sun1 it will behave like a projectile4 ,its path will no longer be rectilinear4 ,it will be slightly curved toward the sun4 ,in other words1 the ray will deviate from a straight line1 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 star4 ,it has been diverted4 ,calculation shows that this deviation1 though very slight1 can be measured4 ,it is equal to an angle of a second and three-quarters3 an angle which the delicate methods of our astronomers are able to measure4 ,certainly such an angle is very far from considerable1 for it takes #cbd1jjj angles of one second to make a right angle4 ,in other words1 an angle of one second is that at which we should see the two ends of a rod1 a metre in length1 fixed in the ground1 at a distance of #bjf kilometres4 ,if our eyes were sharp enough to see a man of normal height standing #bjj kilometres away from us1 our glance1 in passing from his head to his feet1 would have a very small angle of deviation4 ,well1 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 sun4 ,minute as this angle is1 the methods of the astronomer are so delicate and precise that he can determine it4 ,the tiny measurement is by no means to be despised4 ,disdain of the men who devote themselves to such refined subtleties is very much out of place1 because our modern science has been revolutionised by this measurement4 ,einstein is right1 and ,newton wrong1 because we have been able to measure this minute angle and establish the curvature of light4 ,a great difficulty arose when we wished to verify this4 ,how can we observe in full daylight a ray of light that comes to us from a star and passes close to the sun8 ,it cannot be done4 ,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 correctly1 in the light which is diffused by our atmosphere4 ,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 universe4 ,in literary symbolism1 in politics1 the light of day is the very symbol of progress and knowledge3 night is the symbol of ignorance4 ,what folly6 ,it is a blasphemy against night1 the sweetness of which we ought rather to venerate4 ,i do not refer to its romantic charm1 but to the mighty progress in knowledge which it has enabled us to make4 ,midnight is not merely the hour of crime4 ,it is also the hour of prodigious flight toward remote worlds4 ,during the day we see only one sun3 by night we see millions of suns4 ,the blinding veil which the sunlight draws across the heavens may be woven of the most brilliant rays1 but it is none the less a veil1 for it makes us as blind as the moths which1 in a strong light1 can see no further than the tips of their wings4 ,in order to solve our problem1 therefore1 we have to observe in complete darkness stars which are nevertheless near the edge of the sun's disk4 ,is that impossible8 ,no4 ,nature has met our need by providing total eclipses of the sun which may at times be seen from various stations on the earth4 ,at those times the bright disk is hidden for a few minutes behind the disk of the moon4 ,midday is turned into midnight4 ,we see stars shine out close to the masked face of the sun4 99 99 99 99 99 ,fortunately1 a total eclipse1 visible in ,africa and ,south ,america1 was due on ,may #bi1 #aiai1 shortly after ,einstein had1 on the strength of an argument like that we have just expounded1 announced the deviation of the light of the stars when it passed the sun4 ,two expeditions were organised by the astronomers of ,greenwich and ,oxford4 ,one proceeded to ,sobral1 in ,brazil1 the other to the small ,portuguese island ,principe1 in the ,gulf of ,guinea4 ,some of the ,english astronomers were rather sceptical about the issue4 ,how could we1 until it was proved1 admit that ,newton was wrong1 or had at least failed to formulate a perfect law8 ,but this was proved1 and very decisively1 by the observations4 ,these observations consisted in taking a certain number of photographs during the few minutes of total eclipse of the stars near the sun4 ,they had been photographed with the same instruments some weeks before1 at a time when the region of the sky in which they shine was visible at night and far from the sun4 ,as everybody knows1 the sun passes successively1 in its annual course1 through the different constellations of the zodiac4 ,if the light of the stars which were photographed were not bent out of its path in passing the sun1 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 previously4 ,but if the light from them were bent out of its course during the eclipse by the gravitational influence of the sun1 it would be quite otherwise4 ,the reason is as follows4 ,when the moon rises on one of our plains1 it is not round1 as everybody will have noticed1 but flattened at top and bottom1 somewhat like a giant tangerine lifted above the horizon for some magic supper4 ,the moon has1 of course1 not ceased to be round4 ,it merely seems to be flattened because the rays which come from its lower edge1 and have to pass through a thick stratum of the atmosphere before they reach us1 are bent toward the ground by the refraction of the denser atmosphere much more than are the rays coming from the moon's upper edge1 which pass through a less dense mass of air4 ,our eyes see the edge of the moon in the direction from which its rays come to us1 not in the direction from which they started4 ,that is why the lower edge of the moon seems to us to be raised higher above the horizon than it really is4 ,this deviation is due to refraction4 ,in the same way a star situated a little to the east of the sun 7the rays in this case being curved by weight1 not by refraction7 will seem to us further away from it4 ,it will look as if it were further east than it really is4 ,similarly1 a star to the west of the sun will seem to us still further from the sun's western edge4 ,hence the stars on either side of the sun will1 if ,einstein is right1 be more widely separated from each other in the negatives taken during the eclipse4 ,in their normal position1 on the photographs taken during the night1 they will seem nearer to each other4 ,this is precisely what was found when the photographs taken at ,sobral and ,principe were studied with the aid of the micrometer4 ,not only was it thus proved that the light of the stars is bent out of its path by the sun1 but it was found that the deviation had exactly the extent which had been predicted by ,einstein4 ,it amounts to an angle of one second and three-quarters 7#age7 in the case of a star that is quite close to the sun's disk1 and the angle decreases rapidly in proportion to the distance of stars from the sun4 ,it was a great triumph for the theory of ,einstein1 and for the first time it gave us some connecting link between light and gravitation4 ,on the preceding page ,i compared the curvature of light owing to its weight with the deviation that is caused by atmospheric refraction4 ,as a matter of fact1 there were astronomers who wondered whether the agreement between ,einstein's theory and the results obtained during the eclipse was not merely a coincidence3 whether the deviation that was recorded was not due to refractive action by the sun's atmosphere4 ,it seems impossible to admit this4 ,sometimes we see comets passing quite close to the surface of the sun during their journey through space4 ,their movement would be considerably disturbed if the sun's atmosphere were refractive enough to account for the deviations observed at ,sobral and ,principe4 ,perturbations of cometary orbits of this nature1 near the sun1 have never been recorded4 ,the only possible interpretation1 therefore1 is that the phenomena are due to the effect of weight upon light4 ,thus the light of the stars1 weighed in a balance of the most exquisite delicacy1 has given us a decisive confirmation of ,einstein's theoretical deductions4 ,by its fruit we know the tree4 ,,chapter ,,vi ,,the ,,new ,,conception ,,of ,,gravitation ,geometry and reality--,euclid's geometry and others--,contingency of ,poincar's criterion--,the real universe is not ,euclidean but ,riemannian--,the avatars of the number --,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 gravitation4 ,does the universe conform to the laws of geometry8 ,it is a question that has been much discussed by philosophers and scholars1 but the deviation of light owing to its weight now enables us to approach it with confidence4 ,in our schools we are taught a magnificent series of geometrical theorems1 all solidly interconnected1 the principal of which were created by the great ,greek genius1 ,euclid4 ,that is why classical geometry is known as ,euclidean geometry4 ,its theorems are based upon a certain number of axioms and postulates1 though these are really only affirmations or definitions4 ,the most important of these definitions is3 8,a straight line is the shortest distance between two points40 ,that seems to schoolboys quite simple1 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 tape2 and at the sports ground one is not in a mood or has not time to bother about the validity of the axioms of geometry4 ,what is the precise meaning of this definition of a straight line8 ,there has been a great deal of discussion of that point4 ,henri ,poincar has written a number of fine and profound pages on it1 yet his conclusions are not entirely without an element of uncertainty4 ,in practice we all know what we mean by a straight line3 it is the line that we make by means of a good ruler4 ,but how do we know that a ruler is good and correct8 ,by holding it up before the eye1 and seeing that both ends of it and all the intermediate points in its edge merge together when we look along it4 ,that is how a carpenter tells if a board is smoothly planed4 ,in a word1 in practice we mean by a straight line the line which is taken by the eye of the rifleman looking along his sights4 ,all this amounts to saying that a straight line is the direction in which a ray of light travels4 ,however we look at the matter1 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 light4#aj ,we may therefore say that practically a straight line is the path followed by light in a homogeneous medium4 #aj ,it goes without saying that in all this we assume that the luminous ray travels in a homogeneous medium4 ,and that gives rise to a question4 ,is the world in which we live1 the universe1 in conformity with ,euclid's geometry8 ,is it ,euclidean8 ,it must be understood that ,euclid's geometry is not the only one that has been created4 ,in the nineteenth century there were bold and profound mathematicians--,riemann1 ,bolyay1 ,lobatchewski1 even ,poincar--who founded new and different and rather strange geometries4 ,they are just as logical and coherent as the classical geometry of ,euclid1 but they are based upon different axioms and postulates--in a word1 different definitions4 ,for instance1 8parallels0 are said to be two straight lines1 being in the same plane1 which can never meet4 ,the geometry which we learned in our boyhood says3 8,through a given point there can be only one straight line parallel to a given straight line40 ,this is said to be ,euclid's postulate4 ,riemann1 however1 does not admit this and wishes to replace it by3 8,through a given point there cannot be any straight line parallel to a given straight line0--that is to say1 any line which never meets it4 ,upon this ,riemann founds a quite consistent system of geometry4 ,who will venture to say that ,euclid's geometry is true and that of ,riemann false8 ,as theoretical ideal constructions they are both equally true4 99 99 99 99 99 ,a question that we may legitimately ask is3 ,does the real universe correspond to the classical geometry of ,euclid or to that of ,riemann8 ,it was long believed that it corresponded to ,euclid's geometry4 ,poincar himself1 speaking of ,euclid's system1 said3 8,it is1 and will remain1 the most convenient1 7#a7 because it is the simplest2 7#b7 because it agrees very well with the properties of natural solids1 the bodies with which our limbs and our eyes are concerned1 and out of which we make our measuring instruments40 ,when people used to say in earlier ages that the earth is flat1 they argued pretty much as ,poincar does3 8,this theory is the most convenient1 7#a7 because it is the simplest2 7#b7 because it agrees very well with the properties of the natural objects with which we are in contact40 ,but when men came into touch with more remote objects1 when navigators and astronomers multiplied these remote objects1 the idea of a flat earth ceased to be the most convenient1 the simplest1 and the best suited to the facts of experience4 ,then appeared the idea that the earth is round1 and this was found infinitely more convenient1 simpler1 and better adapted to the material universe4 8,convenience10 which ,poincar makes a criterion of scientific truth1 is a contingent and elastic thing4 ,a point of view may be convenient in ,london and not in ,bedford4 ,a theory may be convenient in an area of a hundred yards and no longer convenient for an area of a hundred million miles4 ,the hypothesis of a flat earth has been replaced by the theory of the earth's rotundity4 ,the stationary earth has been replaced by a revolving globe4 ,in the same way1 it seems that in our time ,euclid's geometry must give way to another as a convenient representation of the real world4 ,can there be1 in our universe1 our space1 a parallel to a straight line8 ,that is to say1 is it true that two straight lines being in the same plane will never meet8 ,the real meaning of the question is3 ,is it impossible for two luminous rays1 travelling in empty space and being in what 7for each fraction of the rays7 we will call the same plane1 ever to meet8 ,the answer to this question is in the negative4 ,as these two luminous rays are bent out of their paths in space by the gravitation of the stars1 and as they are differently affected in this way because they are at different distances from the stars1 it follows necessarily that they will cease to be parallel 7in the ,euclidean sense of the word7 and will finally meet2 or at least that they cease to realise the first condition of parallelism--coexistence--in the same local plane4 ,in a word1 if we consider the matter1 not within the ridiculously limited field of experiment in the laboratory1 but in the vast field of celestial space1 the real universe is not ,euclidean1 because in it light does not travel in a straight line4 ,kant regarded the truths--to be accurate1 the deductive affirmations--of the ,euclidean geometry as 8synthetic judgments a priori10 or self-evident propositions4 ,as we have seen1 ,kant was wrong1 not only from the point of view of theoretical geometry1 but also from the point of view of real geometry4 ,the etymology of the word 8geometry0 7which means 8measuring the earth07 is enough of itself to show that it was originally1 and chiefly1 a practical science4 ,that is a sufficient justification for our asking which geometry is most in accord with the real universe4 ,gauss1 a profound thinker1 asked the question long ago1 in the last century1 and he made certain delicate experiments to measure if the sum of the angles of a triangle is really equal to two right angles1 as the ,euclidean geometry says4 ,with this view he took a vast triangle1 the apices of which were formed by the highest peaks of three widely separated mountains4 ,one of them was the famous ,brocken4 ,with his assistants he took simultaneous sights of each peak in relation to the other two1 and he found that the sum of the three angles of the triangle only differed from #ahj degrees to an extent that might be put down to error in observation4 ,there were many philosophers who ridiculed ,gauss and his experiments4 ,with the a priori dogmatism that one so often encounters amongst these people they said that his measurements1 even if they had had a different result1 would have proved nothing to the detriment of ,euclid's theorems1 but would merely have shown that some disturbing cause bent the luminous rays between the three apices of the triangle4 ,this is true1 but it does not matter4 ,if ,gauss had found that the sum of the angles of the triangle in question was larger than two right angles1 it would have proved that real geometry is not the geometry of ,euclid4 ,the question which ,gauss asked was profound and reasonable4 ,the philosophers who ridiculed it might have been challenged to define real straight lines1 natural straight lines1 in any other terms than those of the passage of light4 ,gauss did not find the sum of the angles different from two right angles because his measurements were not sufficiently precise4 ,if they had been much more rigorous1 or if he could have used a much larger triangle--with the earth1 ,jupiter in opposition1 and another planet as its apices--he would have found a considerable difference4 ,the real universe is not ,euclidean4 ,it is only approximately ,euclidean in those parts of space where light travels in a straight line3 that is to say1 in the parts which are far from any gravitational mass1 such as that in which1 on an earlier page1 we left ,jules ,verne's projectile4 ,there are many other reasons why the universe1 in consequence of gravitation1 does not conform to the laws of ,euclid's geometry4 ,for instance1 in the ,euclidean geometry the extent of the circumference has a well-known proportion to its diameter1 and this is indicated by the ,greek letter 4 ,this proportion1 expressing how many times the diameter is contained in the circumference1 is equal to #cadaeibfe 444 etc41 but ,i pass over the rest1 as has an infinite number of decimals4 ,we then ask3 ,in practice is the proportion of circumferences to their diameters really equal to the classic value of 8 ,for instance1 is this precisely the proportion of the earth's circumference to its diameter8#aa ,einstein says that it is not1 and he gives us the following proof4 ,imagine two very clever and quick and wizard-like surveyors setting out to measure the circumference and diameter of the earth at the ,equator4 ,they both use the same scales of measurement4 ,they begin measuring at the same moment1 and they start from the same point on the ,equator4 ,but one goes westward and the other eastward1 and their speeds are equal1 and such that the one who goes westward keeps up with the earth's rotation1 and thus sees the sun all day long stationary at the same height above the horizon4 ,in music-halls1 for instance1 one sometimes sees an acrobat walking on a rolling ball and keeping to the top of the ball1 because the pace of his steps is exactly equal and contrary to the displacement of the spherical surface4 #aa ,we are1 of course1 imagining the earth as perfectly circular1 without irregularities4 ,a stationary observer in space--on the sun1 let us say--would thus see our surveyor who is going westward1 stationary right opposite to him4 ,on the other hand1 the surveyor who goes eastward will seem to him to go round the earth1 and twice as quickly as if he had remained at the starting-point4 ,when each of our surveyors1 both going at the same speed1 has finished his task of measuring the round of the earth1 will they both have the same result8 ,evidently not4 ,as the super-observer in the sun will see1 the yard of the surveyor who travels eastward is shortened by velocity in virtue of the ,fitzgerald-,lorentz contraction4 ,on the other hand1 the yard of the surveyor who travels westward does not experience this contraction1 as the super-observer on the sun1 in reference to whom he remains stationary1 would see4 ,consequently the two surveyors reach different figures for the earth's circumference1 the one who travels westward finding a result a few yards less than that of the other4 ,yet it is obvious that when they proceed to measure the earth's diameter1 travelling at the same speed1 the two observers will reach the same figure for it4 ,hence the 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 direction4 ,therefore1 as the real values of are different1 they cannot be the unique and quite definite figure of classical geometry4 ,therefore the real universe does not conform to this geometry4 ,these differences1 in the illustration we have given1 are due to the earth's rotation4 ,from the standpoint of gravitation the earth's rotation has centrifugal effects which modify the centripetal influence of weight4 ,we have seen1 moreover1 that for the surveyor whose speed equals that of the earth's rotation the value of is smaller than for the observer whose speed seems to be double that of the rotation4 ,thus the effects of weight being the reverse of those of rotation1 or of centrifugal force1 it follows 7it would be just as easy to prove this as the preceding7 that the effect of weight is to give something less than its classical value4 ,in a word1 in the universe real circumferences traced upon gravitating masses1 such as stars1 are1 in proportion to their diameters1 less than they are in the ,euclidean geometry4 ,the difference is generally very slight1 it is true4 ,but there is a difference4 ,if we put a mass of a thousand kilogrammes in the centre of a circle that is ten metres in diameter1 the figure will differ in reality from its ,euclidean value by less than one-thousand-million-billionth4 ,in the neighbourhood of such formidable masses of matter as the stars are1 the difference may be far greater1 as we shall see4 ,this is the origin of the divergences between ,newton's law of gravitation and that of ,einstein3 divergences which observation has settled in favour of the latter4 ,but we will not anticipate4 99 99 99 99 99 ,we showed in a previous chapter that the real universe of the ,relativists is a four-dimensional continuum--not three-dimensional1 as classic science thought--and that in this continuum distances in time and space are relative4 ,the only thing that has a value independent of the conditions of observation--that has an absolute1 or at least objective1 value--is what we called the 8,interval0 of events1 the synthesis of the spatial and chronological data4 ,yet1 in spite of its four dimensions1 the universe1 as we discussed it in connection with the ,michelson experiment and the ,special ,relativity which this discloses1 was nevertheless a ,euclidean continuum1 in which the classical geometry was verified1 and light travelled in a straight line4 ,as we have just seen1 we have to recant this4 ,the universe not only has four dimensions1 but it is not ,euclidean4 ,with what geometry does the universe accord best--or most conveniently1 to use the language of ,poincar8 ,probably that of ,riemann4 ,when we take the compasses and draw a small circle on a sheet of paper spread on the table1 the radius of the circle is found by the distance between the points of the compasses1 and the circle is ,euclidean4 ,but if we draw the circle on an egg1 the fixed point of the compasses being stuck in the top of the egg1 and again get the radius by the distance between the points1 the circle we have now drawn is not ,euclidean4 ,the proportion of the circumference to the radius as thus defined is smaller than 1 just as it is smaller than when the circle is traced round a massive star4 ,well1 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 egg1 taking into account the fact that these surfaces have only two dimensions while the universe has four4 ,two-dimensional space may be flat like the sheet of paper or curved like the surface of the egg4 ,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 geometry4 ,in just the same way space with more than two dimensions may or may not be ,euclidean4 ,as a matter of fact1 the universe is1 as we saw1 only approximately ,euclidean in those regions which are remote from all heavy masses4 ,it is not ,euclidean1 but curved or warped in the vicinity of the stars2 and the curvature is the greater in proportion as we approach the stars4 ,hence the geometry of curved space1 as founded by ,riemann1 seems to be the best adapted to the real universe4 ,it is the one used by ,einstein in his calculations4 99 99 99 99 99 ,when we sought to prove1 on a previous page1 that rays of light fall just as projectiles of the same velocity would1 we used the following argument3 ,since the 8,interval0 of two events is the same for two observers moving at uniform and different velocities1 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 say1 whose velocity is uniformly accelerated4 ,there is1 in fact1 no reason why the passengers in a train which runs at a uniform speed of sixty miles an hour should observe an 8invariant0 element in phenomena just as do those in another train moving at half the speed1 yet this 8invariant0 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 second4 ,to admit the contrary would be to grant a privileged position in the universe to the first two and others like them4 ,if there is any estate in the world that has had its unjust privileges suppressed by the new physics1 it is the study of the material world4 ,this privilege of observers moving at a uniform velocity would be the less justified as1 if we go to the root of the matter1 it is very difficult to say exactly what a uniform movement is4 ,what do we mean when we say that a train has a uniform velocity of sixty miles an hour8 ,we mean that the train has this velocity in reference to the rails or the ground4 ,but in reference to an observer in a balloon1 or who passes in another train1 the velocity has not the same value1 and it may cease to be a uniform velocity4 ,we know only relative movements1 or1 to be quite accurate1 movements relative to some material object or other4 ,according to our choice of this object1 this standard of comparison1 the same velocity may be uniform or accelerated4 ,in the long run1 it is clear1 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 accelerated4 ,that is the profound reason why the ,einsteinian 8,interval0 of things1 the invariable quantity or 8,invariant10 must be the same for all observers whatever be their velocity1 and in particular for observers moving at velocities equivalent1 in a given place1 to the effects of gravitation4 ,but in that case the inferences we draw from the ,michelson experiment1 in regard to the aspect of phenomena for observers in uniform different movements of translation1 no longer suffice to explain to us the whole of reality4 ,they need to be completed in such fashion that the universal invariant1 the 8,interval0 of things1 remains the same for an observer who is moving in any way whatever4 ,if ,i pass along a street at some unheard-of speed1 but with a uniform motion1 its general aspect may1 on account of the contraction caused by my velocity1 be a little different from what it would seem to me if ,i were stationary4#ab ,the houses1 for instance1 will seem narrower in proportion to their height4 ,nevertheless the general aspect and proportions of objects will be much the same in both cases1 and they will have something in common4 ,thus the gas-lights will seem to me thinner1 but they will be straight4 #ab ,it goes without saying that we assume the observer to have a retina with instantaneous impressions4 ,it will be quite otherwise if the observer's movements are varied3 if1 for instance1 we imagine him a drunken giant1 reeling about at a prodigious speed4 ,for such an observer the street will have quite a new aspect4 ,the gas-jets will no longer be straight1 but zigzag1 reproducing in an inverse way the zigzags which he himself makes as he reels along4 ,this is so true that caricaturists generally represent the trees and lamp-posts and houses seen by a drunken man by ridiculously waving lines4 ,our observer will be convinced that objects really have the zigzag forms which he sees1 and that the forms change at every step he takes4 ,try to tell him that it is he who is dancing1 not the objects2 that it is he who is not walking straight1 not the dog he has on leash4 ,he will not believe it--and from the point of view of ,general ,relativity he is neither more nor less right than you4 ,yet there is something in the aspect of the world that must be common to the drunkard and the drinker of water4 ,if the whole universe were suddenly plunged in a mass of gelatine which has set1 and one were to squeeze or alter the shape in any way of this gelatinous mass1 there would still be something unchanged in the coagulated stuff4 ,what is this something8 ,and what is the calculus to use for it8 ,the answer to these questions was the last stage for ,einstein to cover in order to establish the equations of gravitation and ,general ,relativity4 99 99 99 99 99 ,here it was the penetrating genius of ,henri ,poincar that indicated the path4 ,it is very necessary to insist on this1 as justice has not been done in the matter to the great ,french mathematician4 ,if all the bodies in the universe were to be simultaneously dilated1 and to an identical extent1 we should have no means of knowing it4 ,our instruments and our own bodies being similarly dilated1 we should not perceive this formidable historical and cosmic event4 ,it would not distract us for a moment from the trivialities of the hour4 ,what is more1 not only will it be unrecognisable if worlds are modified in such a fashion as to alter the scale of lengths and time1 but it would be impossible to distinguish between two worlds1 if one single point of the first corresponds to each point of the second2 if to each object or event of the one world there corresponds one of the same character1 placed exactly in the same position1 in the other4 ,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 view4 ,poincar has the distinction of first calling our attention to this and proving that the relativity of things must be understood in this very broad sense4 ,the amorphous and plastic continuum in which we place the universe has a certain number of properties which are exempt from all idea of measurement4 ,the study of these properties is the work of a special geometry1 a qualitative geometry4 ,the theorems of this geometry have this peculiarity1 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 lines4 ,this is the geometry which1 as ,poincar ably indicated1 must be used for the four-dimensional and1 according to its regions1 more or less ,euclidean continuum which is the ,einsteinian universe4 ,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-drinker4 ,it is along this route1 or a route analogous to this1 that ,einstein at last reached success4 ,the universe being a more or less warped continuum1 he proposed to apply to it the geometry created by ,gauss for the study of surfaces of variable curvature3 a geometry generalised by ,riemann4 ,it is by means of this special geometry that we express the fact that the 8,interval0 of events is an invariant4 ,here is an illustration which will1 ,i think1 lead us to the heart of the problem of gravitation and to the solution of it4 99 99 99 99 99 ,let us consider a surface of variable curvature--for instance1 the surface of any large district with its hills1 mountains1 and valleys4 ,when we travel in this region1 we can proceed in a straight line as long as we are on the level plain4 ,a straight line on a level plain has the remarkable feature of being the shortest distance between two points4 ,it has also this peculiarity1 that it is the only one of its kind and its length1 whereas we may draw a great number of lines that are not straight uniting the two points1 longer than the straight line but all of equal length4 ,but we have reached the hilly district4 ,it is now impossible for us to follow a straight line from one point to another if there is a hill between them4 ,whatever path we take1 it will be curved4 ,but amongst the various possible paths which lead from one point to the other on the farther side of the hill1 there is one--and only one1 as a rule--which is shorter than any of the others1 as we could prove by means of a tape4 ,this shortest path1 the only one of its kind1 is what is called the geodetical of the surface covered4 ,in the same way no vessel can go in a straight line if it is sailing from ,lisbon to ,new ,york4 ,it must follow a curved path1 because the earth is round4 ,but amongst the possible curved paths there is a privileged one which is shorter than the others3 the one which follows the direction of the great circle of the earth4 ,in going from ,lisbon to ,new ,york1 though they are nearly in the same latitude1 vessels are careful not to head straight westward1 in the direction of the parallels4 ,they sail a little to the north-west1 so that when they reach ,new ,york they come from the north-east1 having followed pretty closely a terrestrial great circle4 ,on our globe1 as on all spheres1 the geodetical1 the shortest route between two points1 is the arc of a great circle passing through the two points4 ,now the 8,interval0 of two points in the four-dimensional universe precisely represents the geodetical1 the minimum path of progress between the two points traced in the universe4 ,where the universe is curved1 the geodetic is a curved line4 ,where the universe is approximately ,euclidean1 it is a straight line4 ,i may be told that it is very difficult to imagine as curved a three-dimensional space1 and still more a four-dimensional4 ,i agree4 ,we have already seen that it is difficult enough to imagine four-dimensional space even when it is not curved4 ,but what does that prove8 ,there are many other things in nature which we cannot visualise or form a mental picture of4 ,the ,hertz waves1 the ,x-rays1 and the ultra-violet waves exist all the same1 though we cannot imagine them1 or at least only by giving them a visible form which does not belong to them4 ,it is just one of our human infirmities that we cannot conceive what we cannot picture to ourselves4 ,hence our tendency to--if one may use an inelegant but expressive word--visualise everything4 ,let us therefore return to our geodetics4 ,these we can very well picture to ourselves1 because in the universe1 in spite of its four dimensions1 they are lines of only one dimension1 like all other lines that we know4 99 99 99 99 99 ,the existence of geodetics1 of shortest-distance lines1 will now beautifully explain to us the connection between inertia and weight1 which did not appear in the ,euclidean world of classic science4 ,hence the ,newtonian distinction between the principle of inertia and the force of gravitation4 ,we ,relativists find this distinction no longer necessary4 ,material masses1 like light1 travel in a straight line when they are far from a gravitational field1 and in a curved line when they are near gravitational masses4 ,in virtue of symmetry a free material point can only follow a geodetic in the universe4 ,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 themselves1 we are driven to the following conclusion1 which unites in a simple way the previously separated sisters1 inertia and weight3 ,every moving body freely left to itself in the universe describes a geodetic4 ,far from the massive stars this geodetic is a straight line1 because there the universe is almost ,euclidean4 ,near the stars it is a curved line1 because there the universe is not ,euclidean4 ,a fine conception1 combining in a single rule the principle of inertia and the law of gravitation6 ,a brilliant synthesis of mechanics and gravitation1 putting an end to the schism which so long kept them separate and non-corresponding sciences6 ,in this bold and simple theory gravitation is not a force4 ,the planets have curved paths because near the sun1 just as in the neighbourhood of every concentration of matter the universe is curved or warped4 ,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 distances4 ,if we could follow the line over millions of miles1 and during a sufficient period1 we should find it curved4 ,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 universe1 just as at a sports ground cyclists have no need to turn the handles when they reach the corner1 but pedal straight on1 because the slope of the ground compels them of itself to turn4 ,in the sports ground1 as in the solar system1 the curvature is greater in proportion as the machine is nearer to the inner edge of the track4 ,all that now remains is to assign to the universe1 to space-time1 such a curvature at its various points that the geodetics will exactly represent the paths of the planets and of falling bodies1 admitting that the curvature of the universe is caused at each point by the presence or vicinity of material masses4 ,in this calculation we have to take into account the fact that the 8,interval0--that is to say1 the part of the geodetic between two points that are very near each other--must be an invariant whoever may be the observer4 ,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 observer4 ,the length of the line is the same1 whether it appears straight or curved4 ,taking all this into account1 and doing prodigies of mathematical skill of which we have sufficiently indicated the object1 ,einstein has succeeded in expressing the law of gravitation in a completely invariant form4 ,in calculating1 on the ground of ,newton's law1 the 8,interval0 of two astronomical events--for instance1 the successive falls of two meteorites into the sun--we should find that the 8,interval0 has not precisely the same value for observers who are moving at different velocities4 ,with the new form given to the law by ,einstein the difference disappears4 ,the two laws1 however1 differ little from each other1 as was to be expected in view of the accuracy with which astronomers found ,newton's law verified during a couple of centuries4 ,the improvement made in ,newton's law by ,einstein means1 in a word 7and to use the old language of the ,euclidean universe71 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 approached4 99 99 99 99 99 ,it is surprising that ,newton and ,einstein agree in expressing the movements of gravitating stars in an almost identical form1 because their starting-points are very different4 ,newton starts from the hypothesis of absolute space1 the empirical laws of the motions of the planets expressed in ,kepler's laws1 and the belief that gravitational attraction is a force proportional to mass4 ,einstein1 on the other hand1 in making his calculations starts from the conditions of invariance which we indicated4 ,he starts1 in a sense1 from the philosophical principle or postulate or impulse to hold that the laws of nature are invariant and independent of the point of view--irrelative1 if ,i may use the word4 ,einstein even abandons the hypothesis which ascribed the curving of gravitational paths to a distinct force of attraction4 ,yet1 starting from a point of view so different from that of ,newton1 and one that seems at first less overloaded with hypotheses1 ,einstein reaches a law of gravitation which is almost identical with ,newton's4 ,this 8almost0 is of immense interest1 because it enables us to test which is the accurate law1 that of ,newton or that of ,einstein4 ,they give the same results when there is question of velocities that are feeble in comparison with that of light1 but their results differ a little when there is question of very high velocities4 ,we have already seen that1 near the sun1 light itself is bent out of its course in exact conformity with ,einstein's law1 and in a way that ,newton's law did not predict as such4 ,but there is another divergence between the two laws4 ,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 position4 ,suppose we put on a table a slice of lemon cut through the longer diameter of the fruit1 and imagine that the chief stars1 the northern constellations1 are painted on the vaulted roof of the vast hemispherical room in the middle of which we place our table4 ,the slice of lemon has very nearly the form of an ellipse1 and1 if we take one of the pips to represent the sun1 it will stand for the orbit of one of our planets4 ,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 revolve4 ,this means that the slice of lemon remains stationary4 ,einstein's law says1 on the contrary1 that the orbital ellipse turns very slowly amongst the stars while the planet traverses it4 ,this means that our slice of lemon must turn slightly on the table1 in such wise that the two ends of the lemon do not remain opposite the same stars painted on the wall4 ,if we calculate1 in virtue of ,einstein's law1 the extent to which the elliptical orbits of the planets must thus turn1 we find it so small as to be impossible of observation except in the case of one planet1 the swiftest of all1 ,mercury4 ,mercury revolves completely round the sun in about eighty-eight days1 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 7#dc7 at the end of a century4 ,small as this quantity is1 the refined methods of the modern astronomer can easily measure it4 ,as a matter of fact1 it had been noticed during the last century that ,mercury was the only one of the planets to show a slight anomaly in its movements1 which could not be explained by ,newton's law4 ,le ,verrier made prodigious calculations in connection with it1 as he thought that the anomaly might be due to the attraction of an unknown body lying between ,mercury and the sun4 ,he hoped that he would thus discover1 by calculation1 an intra-,mercurial planet1 just as he had discovered the trans-,uranian planet ,neptune4 ,but no one ever observed his planet1 and the anomaly of ,mercury continued to be the despair of astronomers4 ,now1 in what did the anomaly consist8 ,precisely in an abnormal rotation of the planetary orbit2 a rotation which ,le ,verrier's calculations showed to be forty-three seconds of an arc in a century4 ,that is exactly the figure that we deduce1 without using any hypothesis1 from ,einstein's law of gravitation6 ,it is true that1 according to the recent calculations of ,grossmann1 the astronomical observations collected by ,newcomb give as the recorded value of the secular displacement of the perihelion of ,mercury1 not #dc as ,le ,verrier believed1 but #ch at the most4 ,the agreement with ,einstein's theoretical result is1 therefore1 not perfect 7which would have been extraordinary71 but it is striking1 and is within the limits of possible error of observation4 ,einstein's law is just as exact as ,newton's for the slower planets4 ,for faster bodies1 the motion of which can be observed with a higher degree of precision1 ,newton's law is wrong1 and ,einstein's triumphs once more4 99 99 99 99 99 ,this improvement of what had been considered perfect--the work of ,newton--is a great victory for the human mind4 ,astronomy and celestial mechanics derive additional precision and power of forecast from it4 ,we can now follow the golden orbs1 on the triumphal wings of calculation1 better than we could before1 or antedate their movements by centuries4 ,but there is another test of ,einstein's law of gravitation4 ,if it is sound1 the duration of a phenomenon increases1 according to ,einstein1 when the gravitational field becomes more intense4 ,it follows that the duration of the vibration of a given atom must be longer on the sun than on the earth4 ,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 earth4 ,recent observations tend to confirm this1 but the verification is less satisfactory than in the case of ,mercury because other causes may intervene to modify the wave-lengths4 ,on the whole1 the powerful synthesis which ,einstein calls the theory of ,general ,relativity1 which we have here rapidly outlined1 is a lofty and beautiful mental construction as well as a superb instrument of exploration4 ,to know is to forecast4 ,this theory forecasts1 and better than its predecessors did4 ,for the first time it combines gravitation and mechanics4 ,it shows how matter imposes upon the external world a curvature or warping of which gravitation is but a symptom3 just as the weeds one sees floating on the sea are but indications of the current which bears them along4 ,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 nature4 ,but ,i have sufficiently shown that if ,i have succeeded in enabling the reader to understand--to feel1 at least--these matters without invoking the aid of the pure light which geometry pours upon the invisible4 ,,chapter ,,vii ,,is ,,the ,,universe ,,infinite8 ,kant and the number of the stars--,extinct stars and dark nebul--,extent and aspect of the astronomical universe--,different kinds of universes--,poincar'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 ether4 ,is the universe infinite8 ,it is a question that men have asked in all ages1 though they have not defined its meaning very accurately4 ,the theory of ,relativity enables us to approach it from a new and subtle point of view4 ,kant--the genial grumbler who found it so horribly monotonous to see the same sun shining1 and the same spring blossoming1 every year--took his stand on metaphysical considerations when he affirmed that space is infinite1 and is sown with similar stars in all parts4 ,it is1 perhaps1 better to confine ourselves in such a matter to the results of recent observation1 and close the doors of our debating-room against the fog of metaphysics4 ,indeed1 the latter would compel us to define pure space1 about which we know nothing--not even if there is such a thing4 ,the proof that we know little about it is the fact that the ,newtonians believe in it1 while the ,einsteinians regard it merely as an inseparable attribute of material things4 ,they define space by matter2 and they then have to define the latter4 ,descartes1 on the contrary1 defined matter in terms of extension1 which is the same thing as space4 ,it is a vicious circle4 ,it is therefore better to leave ,kant's metaphysical arguments out of our discussion1 and adhere strictly to experience1 to what is measurable4 ,to simplify matters1 we will admit the reality of this continuum in which the stars float1 which is traversed by their radiations1 which common sense calls space4 ,if there were stars everywhere--if they were infinite in number--there would also be space and matter everywhere4 ,newtonians might find this a triumph equally with ,einsteinians4 ,those who believe in absolute space and those who deny it--,absolutists and ,relativists--would equally rejoice4 ,it would be fortunate if astronomical observation were to show that the number of the stars is infinite1 and thus the holders of contrary opinions could both chant a victory in their writings4 ,but what does astronomical observation actually report8 ,there are those who deny a priori that the number of the stars can be infinite4 ,that number1 they said1 is capable of increase2 it is therefore not infinite1 because nothing can be added to the infinite4 ,the argument is specious1 but unsound2 although ,voltaire himself was seduced by it4 ,one need not be a great mathematician to see that it is always possible to add to an infinite number1 and that there are infinite quantities which are themselves infinitely small in comparison with others4 ,let us get on to the facts4 ,if the stellar universe has no limits1 there is no visual line drawn from the earth to the heavens which will not encounter one of the stars4 ,the astronomer ,olbers has said that the whole nocturnal sky would in that case shine with the brilliance of the sun4 ,but the total brilliance of all the stars put together is only three thousand times greater than that of a star of the first magnitude1 or thirty million times less than the light of the sun4 ,but that proves nothing1 as ,olbers' argument is wrong1 for two reasons4 ,on the one hand1 there are necessarily a good many extinct or dark stars in the heavens4 ,some of them have been closely studied1 even weighed4 ,they betray their existence by periodically eclipsing brighter stars1 with which they revolve4 ,on the other hand1 it was discovered some time ago that celestial space is occupied over large stretches by dark gaseous masses and clouds of cosmic dust1 which absorb the light of more distant stars4 ,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 night4 99 99 99 99 99 ,now let us put on our spectacles--our telescopes1 ,i mean--and turn from the province of possibility to that of reality1 and we shall see that recent astronomical observation has yielded a number of remarkable facts which lead irresistibly to the following conclusions4 ,the number of the stars is not1 as was long supposed1 limited by the range of our telescopes alone4 ,as we get further away from the sun1 the number of stars contained in a unity of space1 the frequence of the stars1 the density of the stellar population1 do not remain uniform1 but decrease in proportion as we approach the limits of the ,milky ,way4 ,the ,milky ,way is a vast archipelago of stars1 our sun lying in its central region4 ,this mass of stars1 to which we belong1 has1 roughly1 the shape of a watch-case1 the thickness being only about half the width of the structure4 ,light1 which travels from the earth to the moon in little over a second1 from the earth to the sun in eight minutes1 and from the earth to the nearest star in three years1 needs at least #cj1jjj years--three hundred centuries--to pass from end to end of the ,milky ,way4 ,the number of stars in the ,milky ,way is something between #ejj and #a1ejj millions4 ,it is a small number3 scarcely equal to the human population of the earth1 much smaller than the number of molecules of iron in a pin's head4 ,in addition to these we have discovered dense masses of stars1 such as the ,magellanic ,clouds1 the cluster in ,hercules1 and so on1 which seem to belong to the fringes of our ,milky ,way--to be suburbs of it1 so to say4 ,these suburbs seem to stretch a considerable distance1 particularly on one side of the ,milky ,way4 ,the furthest away is1 perhaps1 not less than #bjj1jjj light-years from us4 ,beyond these1 space seems to be deserted1 devoid of stars over expanses which are enormous in comparison with the dimensions of our galactic universe as we have described it4 ,what is beyond this8 ,well1 beyond this we find those strange bodies1 the spiral nebul1 lying like silver snails in the garden of the stars4 ,we have discovered several hundred thousand of them4 ,some astronomers believe that these spiral masses of stars may be annexes of the ,milky ,way1 reduced models of it4 ,most astronomers incline to think1 for very good reason1 that the spiral nebul are systems like the ,milky ,way1 and comparable to it in their dimensions4 ,if the former view is correct1 the entire system of stars accessible to our telescopes could be traversed by light in some hundreds of thousands of years4 ,on the second hypothesis the dimensions of the stellar universe to which we belong must be multiplied by ten1 and light would take at least millions of years to traverse it4 ,on the first view the entire stellar universe1 in so far as it is accessible to us1 consists of the ,milky ,way and its annexes3 that is to say1 a local concentration of stars1 beyond which we can see nothing4 ,the stellar universe is1 in other words1 practically limited1 or at least finite4 ,on the second view our ,milky ,way is simply one of the myriads of spiral universes we see4 ,the spiral nebula 7with its hundreds of millions of stars7 plays the same part in this vaster universe that a star has in the ,milky ,way4 ,we have the same problem as before1 but on a vaster scale3 if the ,milky ,way consists of a concentration of a finite number of stars1 as observation proves1 does the accessible universe consist of a finite number of spiral nebul8 ,experience has as yet not pronounced on this point4 ,but in my opinion it is probable that1 when our instruments are powerful enough to tackle such a problem--in several centuries1 perhaps--science will answer 8yes40 ,if it were otherwise1 if the spiral nebul were fairly evenly distributed as we go outward1 we can show by calculation that1 attraction being in inverse proportion to the square of the distance1 gravitation would have an infinite intensity in such a universe1 even in the part in which we live4 ,but this is not the case4 ,it follows that1 either the attraction of two masses decreases at great distances rather more rapidly than in inverse proportion to the square of the distance 7which is not wholly impossible71 or that the number of stellar systems and stars is finite4 ,personally ,i favour the second hypothesis1 but it is incapable of proof4 ,in such matters there is always an alternative1 always a way of escaping in accordance with one's bias1 and there is really nothing that compels us to say that the stars are finite in number4 99 99 99 99 99 ,starting from the mean value1 as it has been observed1 of the proper motions of the nearer stars1 ,henri ,poincar has calculated that the total number of stars in the ,milky ,way must be about one thousand million4 ,the figure agrees fairly well with the results of the star-gauges effected by astronomers by means of photographic plates4 ,he has also shown that the proper motions of stars would be greater if there were many more stars than those which we see4 ,thus ,poincar's calculations are opposed to the hypothesis of an indefinite extension of the stellar universe1 as the number of stars 8counted0 agrees fairly closely with the number 8calculated40 ,we should add1 however1 that these calculations prove nothing if the law of attraction is not quite the inverse proportion of the square at enormous distances4 ,on the other hand1 if the universe is finite in space as it is conceived in classic science1 the light of the stars1 and isolated stars themselves1 would gradually drift away into the infinite1 and the cosmos would disappear4 ,our mind resents this consequence1 and astronomical observation discovers no trace whatever of such a dislocation4 ,in a word1 in the space of the 8,absolutists0 the stellar universe can only be infinite if the law of the square of distances is not quite exact for very remote masses2 and it cannot be finite except on the condition that it is ephemeral in point of time4 ,for ,newton1 indeed1 the stellar universe might be finite within an infinite universe1 because in his view there can be space without matter4 ,for ,einstein1 on the contrary1 the universe and the material or stellar universe are one and the same thing1 because there is no space without matter or energy4 99 99 99 99 99 ,these difficulties and obscurities disappear in great part when we consider space1 or space-time1 from the ,einsteinian standpoint of ,general ,relativity4 ,what is the meaning of the sentence1 8,the universe is infinite08 ,from either the ,einsteinian1 the ,newtonian1 or the ,pragmatist point of view it means3 ,if ,i go straight ahead1 going on eternally1 ,i shall never get back to my starting-point4 ,is it possible8 ,newton is compelled to say yes1 because in his view space stretches out indefinitely1 independent of the bodies that occupy part of it1 whether the number of the stars is or is not limited4 ,but ,einstein says no4 ,for the ,relativist the universe is not necessarily infinite4 ,is it therefore limited1 fenced in by some sort of railings8 ,no4 ,it is not limited4 ,a thing may be unlimited without being infinite4 ,for instance1 a man who moves on the surface of the earth may travel over it indefinitely in every direction without ever reaching a limit4 ,the surface of the earth1 thus regarded1 or the surface of any sphere whatsoever1 is therefore both finite and unlimited4 ,well1 we have only to apply to space of three dimensions what we find in two-dimensional space 7a spherical surface71 to see how the universe may be at one and the same time finite and unlimited4 ,we saw that1 in consequence of gravitation1 the ,einsteinian universe is not ,euclidean1 but curved4 ,it is1 as we said1 difficult1 if not impossible1 to visualise a curvature of space4 ,but the difficulty exists only for our imagination1 which is restricted by our life of sense1 not for our reason1 which goes farther and higher4 ,it is one of the commonest of errors to suppose that the wings of the imagination are more powerful than those of reason4 ,if one wants proof of the contrary1 one has only to compare what the most poetic of ancient thinkers made of the starry heavens with what modern science tells about the universe4 ,here is the way to approach our problem4 ,let us not notice for the moment the rather irregular distribution of stars in our stellar system1 and take it as fairly homogeneous4 ,what is the condition required for this distribution of the stars under the influence of gravitation to remain stable8 ,calculation gives us this reply3 ,the curvature of space must be constant1 and such that space is bent like a spherical surface4 ,rays of light from the stars may travel eternally1 indefinitely1 round this unlimited1 yet finite1 universe4 ,if the cosmos is spherical in this way1 we can even imagine the rays which emanate from a star--the sun1 for instance--crossing the universe and converging at the diametrically opposite point of it4 ,in such case we might expect to see stars at opposite points in the heavens1 of which one would be the image1 the spectre1 the 8double0 of the other--in the sense which the ancient ,egyptians gave to the word4 ,properly speaking1 this 8double0 would represent1 not the generating star as it is1 but as it was at the time when it emitted the rays which form the double1 or millions of years earlier4 ,if we observe the original and the double star1 the reality and the mirage1 simultaneously from some remote part of the stellar system1 such as our planet1 we shall see a great difference between them1 since the 8copy0 will show us the original as it was thousands of centuries before4 ,it may1 in fact1 happen that the second star is more brilliant than the first1 because in the meantime the first has gradually cooled1 and may even be extinct4 99 99 99 99 99 ,it is improbable that we should find many of these phantom-stars1 or virtual stars1 luminous and unreal daughters of heavy suns4 ,the reason is that the rays in their passage through the universe will generally be diverted by the stars near which they pass4 ,concentration or convergence of them at the antipodes of the real star must be rare4 ,moreover1 the rays are to some extent absorbed by the cosmic stuff they meet in space4 ,it is1 however1 not impossible that the astronomers of the future may discover such phenomena4 ,it is1 in fact1 not impossible that we have already observed such things without knowing it4 ,in any case1 what observers have not done in the past they may very well do in the future1 thanks to the suggestions of the new science4 ,possibly it is going to have a great effect on observational astronomy and induce it to furnish brilliant new verifications of theory4 ,there may be astonishing results1 unforeseen by our folly1 of the new conceptions1 surpassing in their fantastic poetry the most romantic constructions of the imagination4 ,reality1 or at least the possible1 is rising to giddy heights that were far beyond the reach of the golden wings of fantasy4 ,i spoke on a previous page of the millions of years which light takes to travel round our curved universe4 ,starting from the fairly well-ascertained value of the quantity of matter comprised in the ,milky ,way1 it is possible to calculate the curvature of the world and its radius4 ,we find that the radius has a value equal to at least #aej1jjj1jjj light-years4 ,it therefore takes light at least #ijj1jjj1jjj years1 at a speed of #ahf1jjj miles a second1 to travel round the universe1 assuming that it consists only of the ,milky ,way and its annexes4 ,the figure is quite consistent with the figures we get from astronomical observation for the dimensions of the galactic system1 and also with the much larger figures which we find if we regard the spiral nebul as ,milky ,ways4 ,thus for the ,relativist the universe may be unlimited without being infinite4 ,as to the ,pragmatist1 who goes straight ahead--who follows what he calls a straight line1 or the path of light--he will get back in the end to the body from which he started1 provided that he has time enough at his disposal4 ,he will then say that1 if that is the nature of things1 the universe is not infinite4 ,hence the question of the infinity or finiteness of the universe can be controlled by experience1 and some day it will be possible to prove whether the whole cosmos and space are ,newtonian or ,einsteinian4 ,unfortunately1 it will have to be a very long experience1 with various little practical difficulties to overcome4 ,we may therefore prefer not to commit ourselves without further instructions4 ,we may not feel ourselves obliged to choose between the two conceptions1 and we may leave the benefit of the doubt to whichever of the two is false4 99 99 99 99 99 ,moreover1 there is perhaps a third issue3 if not for the ,pragmatist1 at least for the philosopher--,i mean1 seeing that in ,england physics comes under the head of 8,natural ,philosophy10 for the physicist4 ,here it is4 ,if all the heavenly bodies we know belong to the ,milky ,way1 other and very remote universes may be inaccessible to us because they are optically isolated from us2 possibly by the phenomena of the cosmic absorption of light1 to which we have already referred4 ,but this might also be due to something else which will1 perhaps1 shock ,relativists1 but will seem to ,newtonians quite possible4 ,the ether1 the medium that transmits the luminous waves1 and which ,einstein has ended by admitting once more 7refusing1 however1 to give it its familiar kinematic properties71 and matter seem more and more to be merely modalities4 ,we explained this1 on the strength of the most recent physical discoveries1 in a previous chapter4 ,there is nothing to prove that these two forms of substance are not always associated4 ,does this not give me the right to think that perhaps our whole visible universe1 our local concentration of matter1 is only an isolated clump or sphere of ether8 ,if there is such a thing as absolute space 7which does not mean that it is accessible to us71 it is independent of ether as well as matter4 ,in that case there would be vast empty spaces1 devoid of ether1 all round our universe4 ,possibly other universes palpitate beyond these2 and for us such worlds would be for ever as if they did not exist4 ,no ray of knowledge would ever reach us from them4 ,nothing could cross the black1 dumb abysses which environ our stellar island4 ,our glances are confined for ever within this giant--yet too small--monad4 8,are there1 then10 some will cry in astonishment1 8things which exist1 yet we will never know them80 ,naive pretension--to want to embrace everything in a few cubic centimetres of grey brain-stuff6 ,,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 unknown4 ,we approach the end of our work4 ,has reality1 seen through the prism of science1 changed its aspect with the new theories8 ,yes1 certainly4 ,the ,relativist theory claims to have improved the achromatism of the prism and by this means improved the picture it gives us of the world4 ,time and space1 the two poles upon which the sphere of empirical data turned1 which were believed to be unshakeable1 have been dislodged from their strong positions4 ,instead of them ,einstein offers us the continuum in which beings and phenomena float3 four-dimensional space-time1 in which space and time are yoked together4 ,but this continuum is itself only a flabby form4 ,it has no rigidity4 ,it adapts itself docilely to everything4 ,there is nothing fixed1 because there is no definite point of reference by means of which we could distribute phenomena2 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 vessels4 ,up to this point the theory of ,relativity well deserves its name4 ,but now1 in spite of it and its very name1 there rises something which seems to have an independent and determined existence in the external world1 an objectivity1 an absolute reality4 ,this is the 8,interval0 of events1 which remains constant and invariable through all the fluctuations of things1 however infinitely varied may be the points of view and standards of reference4 ,from this datum1 which1 speaking philosophically1 strangely shares the intrinsic qualities with which the older absolute time and absolute space were so much reproached1 the whole constructive part of ,relativity1 the part which leads to the splendid verifications we described1 is derived4 ,thus the theory of ,relativity seems to deny its origin1 even its very name1 in all that makes it a useful monument of science1 a constructive tool1 an instrument of discovery4 ,it is a theory of a new absolute3 the ,interval represented by the geodetics of the quadri-dimensional universe4 ,it is a new absolute theory4 ,so true is it that even in science you can build nothing on pure negation4 ,for creation you need affirmation4 ,the theory of ,relativity has won brilliant victories1 crowned by the decisive sanction of facts4 ,we have given some astonishing instances of these in our earlier chapters4 ,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 standpoint4 ,it would also--there is real danger in this--be to close against the mind other paths where there are still flowers to cull4 ,we will not do that4 ,it is therefore important1 in spite of its successes--nay1 on account of them--to turn the light of criticism upon the foundations of the new doctrine4 ,even ,csar1 as he mounted the ,capitol1 had to listen to the jokes of the soldiers round his chariot and lower his pride4 ,the theory of ,relativity also1 as it advances in all its magnificence along the ,triumphal ,way1 must learn that it has its limits1 perhaps its weaknesses4 99 99 99 99 99 ,but before we go further into it1 before we turn the raw light upon it1 let us make one observation4 ,whatever be the obscurities of physical theories1 whatever be the eternal and fated imperfection of science1 one thing may be positively laid down here3 scientific truths are the best established1 the most certain1 the least doubtful of all the truths we can know in regard to the external world4 ,if science cannot reveal to us the nature of things in its entirety1 there is nothing else that can do it as well4 ,the truths of sentiment1 of faith1 of intuition1 have nothing to do with those of science as long as they remain strictly truths of the interior world4 ,they are on another plane4 ,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 reality1 to the scientific investigation of the truth4 ,it is therefore nonsense to speak of a 8bankruptcy of science0 as contrasted with the certainty which other disciplines may give us respecting the external world4 ,the bankruptcy of one would make all the others bankrupt4 ,when it is not a question of the intimate oasis in which the serene realities of sentiment flourish1 but of the arid and imperfectly explored desert of the material world1 the scientific facts are the basis of all constructions4 ,destroy those and you destroy everything4 ,if you ram the ground floor of a house and bring it down1 you bring down also the upper stories4 ,to say the truth1 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 phenomena1 and that finds expression in the laws of science4 ,is not this science which shows us the vast universe well-ordered1 coherent1 harmonious1 mysteriously united1 organised like a great mute symphony1 dominated by law instead of caprice1 by irrefragable rules instead of individual wills--is this not a revelation8 ,there you have the only means of reconciling the minds which are devoted to external realities and those which bow to metaphysical mystery4 ,to talk of bankruptcy of science--if it means anything more than to point out human weakness1 which is1 alas6 obvious enough--is really to calumniate that part of the divine which is accessible to our senses1 the part which science reveals4 99 99 99 99 99 ,in sum1 the whole ,einsteinian synthesis flows from the issue of the ,michelson experiment1 or at least from a particular interpretation of that issue4 ,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 sun4 ,this medium is known to physicists as ether4 ,lord ,kelvin1 who was honoured by being buried in ,westminster ,abbey not far from the tomb of ,newton1 rightly regarded the existence of interstellar ether as proved as fully as the existence of the air we breathe2 for without this medium the heat of the sun1 mother and nurse of all terrestrial life1 would never reach us4 ,in his theory of ,special ,relativity1 ,einstein1 as we saw1 interprets phenomena without introducing the ether1 or at least without introducing the kinematic properties which are usually attributed to it4 ,in other words1 ,special ,relativity neither affirms nor denies the existence of the classic ether4 ,it ignores it4 ,but this indifference to or disdain of the ether disappears in the theory of ,general ,relativity4 ,we saw in a previous chapter that the trajectories of gravitating bodies and of light are directly due1 on this theory1 to a special curvature and the non-,euclidean character of the medium which lies close to massive bodies in the void--that is to say1 ether4 ,this1 therefore1 though ,einstein does not give it the same kinematic properties as classic science did1 becomes the substratum of all the events in the universe4 ,it resumes its importance1 its objective reality4 ,it is the continuous medium in which spatio-temporal facts evolve4 ,hence in its general form1 and in spite of the new kinematic attitude which is ascribed to it1 ,einstein's general theory admits the objective existence of ether4 ,stellar aberration shows that this medium is stationary relatively to the orbital motion of the earth4 ,the negative result of ,michelson's experiment tends1 on the contrary1 to prove that it shares the earth's motion4 ,the ,fitzgerald-,lorentz hypothesis solves this antinomy by admitting that the ether does not really share the earth's motion1 but saying that all bodies suddenly displaced in it are contracted in the direction of the movement4 ,this contraction increases with their velocity in the ether1 which explains the negative result of the ,michelson experiment4 ,lorentz's explanation seemed to ,einstein inadmissible on account of certain improbabilities which we pointed out1 and especially because it assumes that there is in the universe a system of privileged references which recalls ,newton's 8absolute space40 ,einstein1 taking his stand on the principle that all points of view are equally relative1 does not admit that there are in the universe privileged spectators--spectators who are stationary in the ether--who could see things as they are1 whereas these things would be deformed for every other observer4 ,then1 while preserving the ,lorentz contraction and the formul in which it is expressed1 ,einstein says that this contraction1 while it really exists1 is only an appearance1 a sort of optical illusion1 due to the fact that the light which shows us objects does not travel instantaneously1 but with a finite velocity4 ,this spread of light follows laws of such a nature that apparent space and time are changed in precise accordance with the formul of ,lorentz4 ,that is the foundation of ,einstein's ,special ,relativity4 ,hence the two immediate possible explanations of the negative result of the ,michelson experiment are3 #a4 ,moving objects are contracted in the stationary ether1 the fixed substratum of all phenomena4 ,this contraction is real1 and it increases with the velocity of the body relatively to the ether4 ,that is ,lorentz's explanation4 #b4 ,moving objects are contracted relatively to any observer whatsoever4 ,this contraction is only apparent1 and is due to the laws of the propagation of light4 ,it increases with the velocity of the moving body relatively to the observer4 ,that is ,einstein's explanation4 99 99 99 99 99 ,but there is at least one other possible explanation4 ,it introduces new and strange hypotheses1 but they are by no means absurd4 ,indeed1 it is especially in physics that truth may at times seem improbable4 ,this explanation will show how we may account for the result of the ,michelson experiment apart from either ,lorentz or ,einstein4 ,this third explanatory hypothesis is as follows4 ,every material body bears along with it1 as a sort of atmosphere1 the ether that is bound up with it4 ,there is1 in addition1 a stationary ether in the interstellar spaces2 an ether insensible to the motion of the material bodies that move in it1 and which we may1 to distinguish it from the ether bound up with bodies1 call the 8super-ether40 ,this super-ether occupies the whole of interstellar space1 and near the heavenly bodies it is superimposed upon the ether which they bear along4 ,the ether and the super-ether interpenetrate each other just as they penetrate matter1 and the vibrations they transmit spread independently4 ,when a material body sends out series of waves in the ether which surrounds it1 these move relatively to it with the constant velocity of light4 ,but when they have traversed the relatively thin stratum of ether bound up with the material body1 which merges gradually in the super-ether1 they spread in the latter1 and it is relatively to this that they progressively take their velocity4 ,it is like a boat crossing the ,lake of ,geneva at a certain speed4 ,about the middle of the lake it has this speed relatively to the narrow current which the ,river ,rhone makes there1 and then it resumes it relatively to the stationary lake4 ,in the same way the luminous rays of the stars1 although they come from bodies which are approaching or receding from us1 have the same velocity when they reach us1 and this will be the common velocity which the super-ether imposes upon them4 ,thus also1 on the other hand1 the stellar rays that reach our telescopes will be transmitted to us by the super-ether1 without the very thin stratum of mobile ether bound up with the earth being able to disturb their propagation4 ,these hypotheses explain and reconcile all the facts3 7#a7 the fact of stellar aberration1 because the rays which reach us from the stars are transmitted to us unaltered by the super-ether2 7#b7 the negative result of the ,michelson experiment1 because the light which we produce in the laboratory travels in the ether that is borne along by the earth1 where it originates2 7#c7 the fact that1 in spite of the approach or recession of the stars1 their light reaches us with the common velocity which it had acquired in the super-ether1 shortly after it started4 ,however strange this explanation may seem1 it is not absurd1 and it raises no insurmountable difficulty4 ,it shows that1 if the result of the ,michelson experiment is a sort of no-thoroughfare1 there are other ways out of it besides ,einstein's theory4 ,to resume the matter1 we have offered to us three different ways of escaping the difficulties1 the apparent contradictions1 involved in our experience--the antinomy arising from aberration and the ,michelson result--and they are reduced to these alternatives3 #a4 ,the contraction of bodies by velocity is real 7,lorentz74 #b4 ,the contraction of bodies by velocity is only an appearance due to the laws of the propagation of light 7,einstein74 #c4 ,the contraction of bodies by velocity is neither real nor apparent3 there is no such thing 7hypothesis of super-ether connected with ether74 ,this shows that the ,einsteinian explanation of phenomena is by no means imposed upon us by the facts1 or is at least not absolutely imposed by them to the exclusion of any other explanation4 99 99 99 99 99 ,is it at least imposed by reason1 by principles1 by the evidential character of its rational premises1 or because it does not conflict with our good sense and mental habits as the others do8 ,one would suppose this at first1 when one compares it with the teaching of ,lorentz2 and1 in order to relieve this discussion1 ,i will for the moment leave out of account the third theory which ,i sketched1 that of a super-ether4 ,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 velocity1 not in any way upon their nature2 that it was supposed to be the same for all bodies1 no matter what was their chemical composition or physical condition4 ,a little reflexion shows that this strange suggestion is not so clearly inadmissible4 ,we know that the atoms are all formed of the same electrons1 and they differ1 and differentiate bodies1 only in their number and arrangement4 ,if1 then1 the electrons common to all matter and their relative distances experience simultaneously a contraction due to velocity1 it is natural enough to suppose that the result maybe the same for all objects4 ,when an iron grating of a given length is dilated by heat1 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 yard1 provided they are identical4 ,hence it is not really here that we find the improbability which caused ,relativists to reject the ,lorentz theory4 ,it is in the principles of the theory4 ,it is because the theory admits in nature a system of privileged reference--the stationary ether relatively to which bodies move4 ,let us examine this more closely4 ,it has been said that ,lorentz's stationary ether is merely a resuscitation of ,newton's absolute space1 which the ,relativists have so vigorously attacked4 ,that is very far from the truth4 ,if1 as we supposed in the preceding chapter1 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 space1 which would then be the real 8absolute space40 ,from this standpoint the ,lorentzian ether cannot be identified with absolute space4 ,to do so amounts to saying that the space called 8absolute0 by ,newton does not deserve the name4 ,if ,newtonian space is only the physical continuum in which the events of our universe happen1 it is anything but stationary4 ,in that case the whole fault one has to find with ,newton is that he used a wrong expression3 that he called something absolute which is merely privileged for a given universe4 ,it would be a quarrel about grammar2 and such things have never succeeded in revolutionising science4 ,but the ,relativists--at least those impenitent ,relativists1 the ,einsteinians--will not be content with that4 ,it is not enough for them that the ,newtonian space with all its privileges may not be absolute space4 ,our conception of the universe1 as a moving island of ether1 is well calculated to reconcile the pre-eminence of ,newtonian space with that agnosticism which forbids us to hope to attain the absolute4 ,but this again is not enough for the ,einsteinians4 ,what they mean to do is to strip of all its privileges the ,newtonian space on which the structure of classical mechanics has been reared4 ,they mean to reduce this space to the ranks1 to make it no more than analogous to any other spaces that can be imagined and which move arbitrarily in reference to it4 99 99 99 99 99 ,from the agnostic1 the sceptical1 point of view this is a fine and strong attitude4 ,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 reserves4 ,it is legitimate to call into question even the denials of doubters1 because1 after all1 they are really themselves affirmations4 ,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 this3 8,yes1 everything is possible2 or1 rather1 many things are possible1 but all things are not4 ,yes1 if ,i go into a strange house1 the drawing-room clock may be round1 square1 or octagonal4 ,but once ,i have entered the house and seen that the clock is square1 ,i have a right to say3 #8,the clock is square4 ,it has the privilege of being square4 ,it is a fact that it is neither round nor octagonal4' 8,it is the same in nature4 ,the physical continuum which contains1 like a vase1 all the phenomena of the universe1 might have1 relatively to me--and as long as ,i have not observed it--any forms or movements whatever4 ,but as a matter of fact1 it is what it is4 ,it cannot be different things at the same time4 ,the drawing-room clock cannot at one and the same time be composed entirely of gold and entirely of silver4 8,there is therefore one privileged possibility amongst the various possibilities which we imagine in the external world4 ,it is that which has been effectively realised3 that which exists40 ,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 it1 as far as space and time are concerned4 ,the ,newtonians1 on the other hand1 say that we can recognise real space and real time by special signs4 ,we will analyse these signs later4 ,in a word1 the pure ,relativists have tried to escape the necessity of supposing that reality is inaccessible4 ,it is a point of view that is at once more modest and much more presumptuous than that of the ,newtonians1 the ,absolutists4 ,it is more modest because according to the ,einsteinian we cannot know certain things which the ,absolutist regards as accessible3 real time and space4 ,it is more presumptuous because the ,relativist says that there is no reality except that which comes under observation4 ,for him the unknowable and non-existent are the same thing4 ,that is why ,henri ,poincar1 who was the most profound of ,relativists before the days of ,einstein1 used to repeat constantly that questions about absolute space and time have 8no meaning40 ,one might sum it up by saying that the ,einsteinians have taken as their motto the words of ,auguste ,comte3 8,everything is relative1 and that is the only absolute40 ,newton1 whose spatio-temporal premises ,henri ,poincar vigorously refused to admit1 and classical science take up an attitude1 on the contrary1 which ,newton himself well described when he wrote3 8,i am but a child playing on the shore1 rejoicing that ,i find at times a well-polished pebble or an unusually fine shell1 while the great ocean of truth lies unexplored before me40 ,newton says that the ocean is unexplored1 but he says that it exists2 and from the features of the shells he found he deduced certain qualities of the ocean1 especially those properties which he calls absolute time and space4 ,einsteinians and ,newtonians are agreed in thinking that the external world is not in our time entirely amenable to scientific research4 ,but their agnosticism differs in its limits4 ,the ,newtonians believe that1 however external to us the world may be1 it is not to such an extent as to make 8real time and space inaccessible to us40 ,the ,einsteinians hold a different opinion4 ,what separates them is only a question of degree of scepticism4 ,the whole controversy is reduced to a frontier quarrel between two agnosticisms4 ,,chapter ,,ix ,,einstein ,,or ,,newton8 ,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 ,m4 ,painlev's objections--,newtonian arguments and ,relativist replies--,m4 ,painlev's formul of gravitation--,fecundity of ,einstein's theory--,two conceptions of the world--,conclusion4 ,what are these 8special signs0 by which the ,newtonian conception of nature recognises that we are in touch with the privileged space which ,newton called absolute space1 and which seemed to him the real1 intrinsic1 exclusive frame of phenomena8 ,these signs or criteria are implicitly at the root of the development of classic science1 but they for a time remained in the shades of the discussions provoked by ,einstein's theory4 ,leaving aside for a moment other1 and perhaps less noble1 cares1 ,m4 ,paul ,painlev1 addressing the ,academy of ,sciences at ,paris1 has with brilliant success drawn attention to the arguments1 ancient yet ever robust1 which constitute the strength of the ,newtonian conception of the world4 ,let us from this point speak of the absolute time and space of ,newton and of ,galileo as privileged space and privileged time1 in order not to expose our flanks further to the metaphysical objections--not without justification--which the qualification 8absolute0 provokes4 ,why is classical science1 the mechanics of ,galileo and ,newton1 founded upon privileged space and privileged time8 ,why do they refer all phenomena to these unique standards1 and consider them adequate to reality8 ,it is on account of the principle of causality4 ,the principle may be formulated thus3 ,identical causes produce identical effects4 ,that means that the initial conditions of a phenomenon determine its ulterior modalities4 ,it is briefly a statement of the determinism of phenomena1 and without that science is impossible4 ,it is1 of course1 possible to be captious on the point4 ,conditions entirely identical with given initial conditions can never be reproduced or discovered at a different time or in a different place4 ,there is always some circumstance that will be different2 for instance1 the fact that in the interval between the two experiments the ,nebula in ,andromeda will have come several thousand miles nearer to us4 ,and we have no influence on the ,nebula in ,andromeda4 ,happily--this saves the situation--distant bodies have1 it seems1 only a negligible influence on our experiments4 ,that is why we can repeat them4 ,for instance1 if we to-day put a gramme of sulphuric acid in ten grammes of soda-solution 7one-tenth71 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 pressure2 in spite of the fact that meantime ,marshal ,foch sailed for the ,united ,states4 ,thus the principle of causality 7like causes1 like effects7 is always verified1 and never found at fault4 ,it is therefore an empirical truth1 but in addition to this it imposes itself on our mind with irresistible force4 ,it even imposes itself upon animals4 8,the scalded cat avoids hot water10 is proof enough4 ,in any case1 not science only but the whole life of man and animals is based upon it4 ,it is a consequence of the principle that if the initial conditions of a movement present a symmetry1 this will appear again in the movement4 ,m4 ,paul ,painlev insisted strongly on this in the course of the recent discussion of ,relativism at the ,academy of ,sciences4 ,the principle of inertia in particular follows from this statement3 a body left to itself far from any material mass will1 by reason of symmetry1 remain at rest or travel in a straight line4 ,it will certainly follow a straight line for a given observer 7or for observers moving with uniform velocities relatively to the first74 ,the ,newtonians say that the space of these observers is privileged4 ,on the other hand1 for another observer who is1 relatively to them1 moving at an accelerated velocity1 the path of the moving body will be a parabola1 and will no longer be symmetrical4 ,therefore the space of this new observer is not privileged space4 ,it seems to me that the ,relativists might reply to this as follows4 ,you have no right to define the initial conditions for a given observer1 then the subsequent movement for another observer who is moving with accelerated velocity4 ,if you thus define your initial conditions relatively to the latter1 the moving body at the moment when it is released is not free for this observer1 but falls in a gravitational field4 ,it is therefore not surprising that the motion produced seems to him accelerated and dissymmetrical4 ,the principle of causality is not wrong for either observer4 ,one might also give a different definition of the privileged system1 saying3 it is that relatively to which light travels in a straight line in an isotropic medium4 ,but in that case the rays from the stars travel in a spiral for an observer fixed on a turning earth1 and the ,newtonians would infer from this that the earth turns relatively to their privileged space4 ,einsteinians will reply that the space in which the rays travel is not isotropic1 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 rotation4 ,they will always find an escape which will leave the principle of causality intact4 ,it seems difficult1 therefore1 to give unanswerable proof of the existence of the privileged system when we start from the principle of causality4 ,each party retains its position4 99 99 99 99 99 ,on the other hand1 there is evidential value1 a keen and convincing penetration1 in the second part of the criticism which ,m4 ,painlev directs against the principles of ,einstein's theory4 ,let us sum up the argument of the distinguished geometrician4 ,you1 he says to the ,einsteinians1 deny all privilege to any system of reference whatever4 ,but when you want to deduce1 by calculation1 the law of gravity from your general equations1 you cannot do it1 and you really do not do it1 except by introducing scarcely disguised ,newtonian hypotheses and privileged axes of reference4 ,you only reach the result of your calculation by sharply separating time and space as ,newton does1 and by referring your gravitating moving objects to purely ,newtonian privileged axes1 in the case of which certain conditions of symmetry are realised4 ,to this fine and profound criticism which ,m4 ,painlev raises may be added that of ,wiechert1 who has pointed out various other hypotheses introduced by ,einstein in the course of his calculations4 ,in a word1 ,einstein seems not to have kept entirely clear of the ,newtonian premises which he repudiates4 ,he has not the disdain for them that one would suppose1 and he does not hesitate to have recourse to them occasionally for the purpose of helping out his calculations4 ,that is rather to pay a little reverence to the idols you have burned4 ,in reply the ,einsteinians will doubtless say that1 if they introduce ,newtonian axes in the course of their arguments1 it is to make the results of calculation comparable to the result of experimental measurements4 ,the axes introduced into their equations have for the ,relativists the sole privilege of being those to which experimenters refer their measurements4 ,but we must admit that that is no small privilege4 99 99 99 99 99 ,that is not all4 ,the principle of ,general ,relativity amounts to this3 ,all systems of reference are equivalent for expressing natural laws1 and these laws are invariant to any system of reference to which they are related4 ,that means in effect3 ,there are relations between objects of the material world which are independent of the one who observes them1 and particularly of his velocity4 ,thus1 when a triangle is drawn on paper1 there is something in the triangle which characterises it and which is identical1 whether the observer passes very quickly or very slowly1 or at any speed and in any direction whatever1 beside the paper4 ,m4 ,painlev observes1 with some reason1 that in this form the principle is a sort of truism4 ,it is a severe verdict1 yet it expresses a certain fact4 ,the real relations of external objects cannot be altered by the standpoint of the observer4 ,einstein replies that it is at all events something to have provided a sieve by which we may sift the laws and formul which serve to represent the phenomena that have been empirically observed3 a criterion which they must pass before they are recognised as correct4 ,this is true4 ,newton's law1 in its classic form1 did not meet this criterion4 ,this proves that it was not quite so obvious4 ,a truth that was unknown yesterday has become to-day a truism4 ,so much the better4 ,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 8heuristic0 value4 ,but it is none the less true1 as ,m4 ,painlev points out with great force and clearness1 that the principle of ,general ,relativity1 considered in this light1 would be unable to provide precise laws4 ,it would be quite consistent with a law of gravity in which the attraction would be in inverse proportion1 not to the square1 but to the seventeenth or hundredth power1 or any power whatever1 of the distance4 ,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 wit1 that relatively to any observer whatsoever light travels locally with the same velocity in every direction4 ,we have also to add various hypotheses which ,m4 ,painlev regards as ,newtonian4 ,to the critical discussion of ,relativity which he so brilliantly presented at the ,academy of ,sciences ,m4 ,paul ,painlev added a valuable mathematical contribution of which the chief result is the following3 ,it is possible to excogitate other laws of gravitation than that offered by ,einstein1 and all of them will fulfil the ,einsteinian conditions4 ,the learned ,french geometrician indicated several of these1 especially one of which the formula differs considerably from that of ,einstein1 yet equally and precisely explains the motions of the planets1 the displacement of the perihelion of ,mercury1 and the deviation of rays of light near the sun4 ,this new formula corresponds to a space that is independent of time1 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 spectrum4 ,the verification or non-verification of this consequence of ,einstein's equation1 of which we pointed out the difficulties 7perhaps insurmountable7 in a previous chapter1 thus acquires a new importance4 ,it is a remarkable thing that many of the formul of gravitation given by ,m4 ,painlev lead to the conclusion1 differently from that of ,einstein1 that space remains ,euclidean even near the sun1 in the sense that measures are not necessarily contracted4 ,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 gravitation4 ,there are great days--or1 rather1 great nights--in store for the astronomer4 99 99 99 99 99 ,as far as the principles are concerned1 the controversy will go on4 ,it must end in something like the following dialogue3 ,the ,newtonian3 ,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 line8 ,if so1 you recognise the existence of privileged observers--those for whom the line is straight4 ,for another observer the line is a parabola4 ,therefore his point of view is wrong4 ,the ,relativist3 ,yes1 ,i grant it2 but in point of fact there is no point in the universe where there is no influence of distant material masses4 ,therefore your moving object left to itself is a mere fiction1 and ,i am not going to base science upon an unverifiable piece of imagination4 ,the whole aim of the ,relativist is to rid science of everything that has no experimental significance4 ,as to the observer who sees the moving object in question describe a parabola1 he will interpret his observation to mean that the object is in a gravitational field4 ,the ,newtonian3 ,you are therefore compelled to admit that far away from all matter1 far from all heavenly bodies1 there can be what you call a gravitational field1 that it varies according to the velocity of the observer1 and that it can be very intense in spite of the distance of the heavenly bodies1 and even1 at times1 increase with that distance4 ,these are strange and absurd hypotheses4 ,the ,relativist3 ,they are strange1 but ,i defy you to prove that they are absurd4 ,they are less absurd than to localise and set in motion a point that is isolated and independent of any material mass4 ,the ,newtonian3 ,for my part1 ,i can easily imagine a single material point in the universe having a certain position and a certain velocity in it4 ,the ,relativist3 ,for my part1 on the contrary1 if such a material point existed1 it would be absurd and impossible to speak of its position and its motion4 ,it would have neither position nor motion nor rest4 ,such things can exist only with reference to other material points4 ,the ,newtonian3 ,that is not my opinion4 ,the ,impartial ,spectator3 ,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 universe4 ,can you try this experiment8 ,the ,newtonian and the ,relativist 7together73 ,no1 unhappily4 ,the ,metaphysician 7coming up like the third thief in the fable73 ,then1 gentlemen1 ,i advise you to return to your telescopes1 your laboratories1 and your tables of logarithms4 ,the rest is my affair4 ,the ,newtonian and the ,relativist 7together73 ,in that case we are quite sure we shall never learn anything further about it than we know or believe now4 99 99 99 99 99 ,meantime1 it is impossible to exaggerate the importance of the new light thrown on the question of ,relativity by the intervention of ,m4 ,paul ,painlev at the ,academy of ,sciences4 ,it will have a lasting and prodigious echo4 ,will ,einstein's fine synthesis be defeated8 ,shall we see it sink in the controversies1 doubts1 and obscurities of which we have given a short account8 ,i think not4 ,when ,christopher ,columbus discovered ,america1 it was all very well to tell him that his premises were wrong1 and that if he had not believed that he was sailing for the ,indies he would never have reached a new continent4 ,he might have replied1 after the style of ,galileo3 8,i discovered it1 for all that40 ,the method that gives good results is always a good method4 ,when we have to plunge into the depths of the unknown to discover something new1 when we have to learn more and better1 the end justifies the means4 ,when he reminds us of optics1 mechanics1 and gravitation1 now bound up together in a new sheaf1 of the deviation of light by gravity which he foretold against all expectation1 of the anomalies of ,mercury which he was the first to explain1 and of his improvement of the ,newtonian law1 ,einstein has the right to say1 with some pride3 8,there is what ,i have done40 ,it is said that the paths by which he attained all these fine results are not devoid of unpleasant false turns and quagmires4 ,well1 there are many ways to ,rome and to truth1 and some of them are not perfect4 ,the main thing is to get there4 ,and in this case the truth means ancient facts brought into a new harmony1 and new facts set forth in prophetic equations and verified in the most surprising manner4 ,if discussion of principles--if theory1 which is only the servant of knowledge--shrugs its servile and disloyal shoulders a little over ,einstein's work1 at all events experience1 the sole source of truth1 has justified him4 ,brilliant formul that ,einstein had not foreseen are now discovered to explain the anomaly of ,mercury and the deviation of light4 ,it is good3 but we must not forget that the first of these correct formul1 that of ,einstein1 went boldly in advance of the verification4 ,new trenches have been won in the war against the eternal enemy1 the unknown4 ,certainly we have now to organise them and create more direct roads to them4 ,but to-morrow we shall have to advance again1 to gain more ground4 ,we shall have1 by any theoretical device that we can1 to state other new facts1 unknown but verifiable facts4 ,that is what ,einstein did4 ,if it is a weakness of ,einstein's teaching to deny all objectivity1 all privilege1 to any system of reference whatever1 while utilising such a system for the necessities of calculation1 it was at all events a weakness shared by the great ,poincar4 ,to the day of his death he rebelled energetically against the ,newtonian conception4 ,the support of such a genius1 whom one finds involved in all our modern discoveries1 is enough to secure some respect for the ,relativist theory4 ,if we have on the one side ,newton and his ardent and persuasive apologist1 equipped with a fine mathematical genius1 ,paul ,painlev1 we have on the other side ,einstein and ,henri ,poincar4 ,even in earlier history we have ,aristotle against ,epicurus1 ,copernicus against the ,scholastics1 at the same barricade4 ,it is an eternal war of ideas1 and it may be endless if1 as ,poincar believed1 the ,principle of ,relativity is at the bottom only a convention with which experience cannot quarrel because1 when we apply it to the entire universe1 it is incapable of verification4 ,it is the fertility of the ,einsteinian system which proves that it is strong and sound4 ,are the new beings with which it has peopled science--the discoveries predicted by it--legitimate children8 ,the ,newtonians say that they are not4 ,but in properly ordered science1 as in an ideal ,state1 it is the children that matter1 not their legitimacy4 ,at all events the vigorous counter-offensive of ,m4 ,painlev has driven back to their lines the over-zealous apostles of the new gospel1 who thought that they had pulverised classic science beyond hope of recovery4 ,each side now remains in its positions4 ,there is no longer any question of regarding the ,newtonian conception of the world as a piece of childlike barbarism4 ,a different conception is now opposed to it--that is all4 ,the war between them is as yet undecided1 and may remain for ever undecided1 as the weapons with which it might be possible to bring it to an issue are sealed up for ever in the arsenal of metaphysics4 99 99 99 99 99 ,whatever may happen1 ,einstein's teaching has a power of synthesis and prediction which will inevitably incorporate its majestic system of equations in the science of the future4 ,m4 mile ,picard1 perpetual secretary of the ,academy of ,sciences1 and one of the luminous and profound thinkers of our time1 has asked if it is an advance 8to try1 as ,einstein has done1 to reduce physics to geometry40 ,without lingering over this question1 which may be insoluble1 like all speculative questions1 we will conclude with the distinguished mathematician that the only things which matter are the agreement of the final formul with the facts and the analytic mould in which the theory casts the phenomena4 ,considered from this angle1 ,einstein's theory has the solidity of bronze4 ,its correctness consists in its explanatory force and in the experimental discoveries predicted by it and at once verified4 ,what changes in theories are the pictures we form of the objects between which science discovers and establishes relations4 ,sometimes we alter these pictures1 but the relations remain true1 if they are based upon observed facts4 ,thanks to this common fund of truth1 even the most ephemeral theories do not wholly die4 ,they pass on to each other1 like the ancient runners with their torch1 the one accessible reality3 the laws that express the relations of things4 ,to-day it happens that two theories together clasp the sacred torch4 ,the ,einsteinian and the ,newtonian vision of the world are two faithful reflections of it3 just as the two images1 polarised in opposite directions1 which ,iceland spar shows us in its strange crystal both share the light of the same object4 ,tragically isolated1 imprisoned in his own 8self10 man has made a desperate effort to 8leap beyond his shadow10 to embrace the external world4 ,from this effort was born science1 and its marvellous antenn subtly prolong our sensations4 ,thus we have in places approached the brilliant raiment of reality4 ,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 float4 ,einstein has discovered new light for us in the depths of the unknown4 ,he is1 and will remain1 one of the light-houses of human thought4 ,printed in ,great ,britain by ,hazell1 ,watson amp2 ,viney1 ,ld41 ,london and ,aylesbury4