A CRITICISM OF EINSTEIN AND HIS PROBLEM NEW YORK CHAS. H. DANIELS 49 WEST 55 STREET A CRITICISM OF EINSTEIN AND HIS PROBLEM BY W. H. V. READE, M.A. TUTOR OF KEBLE COLLEGE, OXFORD OXFORD BASIL BLACKWELL BROAD STREET MD CCCC XXII By the same Author: THE MORAL SYSTEM OF DANTE'S INFERNO (1909) AN ESSAY ON THE POLITICAL THEORY OF DANTE (1916) THE REVOLT OF LABOUR AGAINST CIVILIZATION (Blackwell, 1919) CONTENTS CHAPTER PAGE I. The Case of the Swimmer and the Stream i II. '1 he Case of the Passenger and the Train 16 III. The Addition of Velocities . . . . 27 IV. Light and the First Principle of Motion 40 V. The Unique Position of Light . . . . 65 VI. Euclid, Velocity and Direction . . . . 83 VII. Non-uniform Motion and Gravitation .. 109 In re naturaliter obscura, qui in exponendo plura quam necesse est superfundit, addit tenebras, non adimit densitatem. MACROBIUS. CHAPTER I THE CASE OF THE SWIMMER AND THE STREAM A COMMON infirmity of mathematicians is failure to express themselves in a manner intelligible to the vulgar. Many books devoted to this new problem of ' relativity ' have been composed with the sincere intention of avoiding technicalities ; but always, after some three or four chapters, the author wanders away to the familiar mysteries, leaving his profane com- panions to wait outside the temple. When a mathematician declares that he is going to write for the general public, he means, apparently, that his pages will not wholly be covered with equations, and that good, sound, dictionary words will be freely employed. He forgets how deeply his own mind is saturated with assumptions to which the layman has no kind of clue. This little foible, to be sure, is not peculiar to mathematicians. Most of us who chance to have mastered a technical lan- guage are slow to unlearn it again for the benefit of others. We chafe at their innocent questions, and too hastily acquit them of brains. It is only that mathematics, more than any 2 A CRITICISM OF EINSTEIN other science, is forced by its nature to dwell among distant abstractions, to which the pas- sage is too narrow and arduous to be traversed without an affable guide. The object of these few pages is to review the elements of Einstein's problem without pretence of mathe- matical intricacy, for which, indeed, the author can boast no qualifications. Even less will the reader find here a book of metaphysics, or an attempt to catalogue the many senses of ' relativity ' which careful analysis might dis- close. The most definite presupposition of the argument is a belief that difficulties, like the entities of Ockham, should not be multiplied beyond what is necessary. Without an elabor- ate use of symbols the manifold development of mathematical principles would doubtless be impossible ; but, if the principles themselves be not amenable to simple expression, we may fairly be excused for doubting their truth. My own interest in the subject was excited chiefly by accounts of the Michelson-Morley experiment, or rather, by the analogy, so often quoted in that context, between the experience of a swimmer in any running stream and the behaviour of light in the problematical ' ether.' On the supposition that it takes a swimmer, capable of a certain velocity in still water, a longer time to swim a measured distance, half THE SWIMMER AND THE STREAM 3 up and half down stream, than to swim the same distance at right angles to the current, some important conclusions were based on the failure to detect by the famous experiment any difference in velocity between two rays of light, so directed as to travel an identical distance upon two different axes, at right angles to one another. Postponing for the present all refer- ence to the special question of light and the et'Ker, I wish first to investigate a little more closely the variable fortunes of the swimmer in the stream. The figures justifying the ordinary doctrine may easily be found (as, for example, in the first chapter of Professor Eddington's Time, Space and Gravitation), nor do I propose to question their accuracy. The result, nevertheless, is a little perplexing, and one can imagine an amateurish critic, with a certain instinctive taste for probability, hankering after a very different conclusion. Given that a swimmer has a normal velocity in still water, it seems clear enough that some- thing must be subtracted from his ordinary speed when he has to struggle against a cur- rent, and something added when he travels with its aid. But does it not also seem obvious that the stream must repay on the downward course exactly as much as it borrowed on the upward? If x be the normal velocity, and 4 A CRITICISM OF EINSTEIN x y the effect of retardation, must not x + y be the figure of the greater velocity achieved in the downward journey? But if so, the average velocity for the whole performance will be x, which, by hypothesis, is the man's normal rate in still water. On the other hand, when he swims across the stream, there must always be some retardation, no matter how slight it may be. Hence at no time in crossing the river can he attain to the velocity x. His progress, therefore, on the up and down journey should always be better than on the other. Now this reasoning is admirable except for one omission. It is important to note a change in the manner of reckoning, as compared with the usual way of presenting the facts. The difference is that, in one case, the trial of speeds is relative to a fixed distance in space, in the other, to a fixed period of time. If a man with a velocity in still water of one mile an hour is commanded to swim for thirty minutes against a stream that halves his velocity, he will accomplish just a quarter of a mile in the time. If he then turns about and swims for the next thirty minutes with the current to help him, his distance will be neither one quarter of a mile, nor two, but three. His total, there- fore, will be exactly one mile in the hour, which THE SWIMMER AND THE STREAM 5 represents his normal velocity. On the other hand, when he swims across the stream in either direction, he will continuously be subject to some retardation (no matter what the precise figure may be), with the result that he never can attain to that same normal velocity, and so must cover less distance in an hour. This argument is quite unassailable; and, though it may, in some sense, be compatible with the other, it does somehow leave a different impression on the mind, and is provocative of further reflec- tion. The same point may conveniently be illustrated from the case of a pedestrian. A man may decide one day to walk to a place four miles away on the top of a hill, and thence to return by the same route. During the out- ward and uphill journey he walks at only two miles an hour, and so arrives there in just two hours. Returning downhill he raises his pace to four miles an hour, and thus needs but one hour to get home. Now here, though he has travelled one half of the total distance at two miles an hour, the other half at four, his average speed has not been three miles to the hour. For he has taken three hours altogether, and the distance was only eight miles. Another day, perhaps, he decides not to walk to any fixed destination, but merely to take his exer- cise for two hours. In the first hour the hill is 6 A CRITICISM OF EINSTEIN mostly against him, and he travels no more than two miles. For the rest of his way, how- ever, he has the ground in his favour, and in the same time covers double the distance. Two miles in the first hour, and four in the second, gives him exactly an average of three, which, perhaps, he is wont to regard as his normal velocity when walking on level ground. What answer, then, will he give to the question, whether it is easier to accomplish a journey bisected into uphill and downhill halves, or the same journey spread out on the flat? While he is standing perplexed for a moment, a smart young relativist may drop in and offer to help. ' The truth is,' he will say, ' that, on the former occasion, the first four miles of your walk con- tained 30 minutes apiece, the second four only 15; whereas, in the later exhibition of your velocity, the first hour consisted of only two miles, the second of four.' Admiring this in- genious ' transformation,' our pedestrian friend may still catch himself wondering how many minutes there usually are in a mile, and how many yards in an hour. But let us leave him and return to the stream. When the swimmer's course is measured in spatial distance, his time for the two successive trials appears to vary; but when the course is a fixed period of time, and he is again asked THE SWIMMER AND THE STREAM / to compare the journey across the stream with the journey upstream and down, the variation appears in space-distance. We note, however, that he covers a greater distance (i.e. preserves a higher average velocity) when he divides the time equally between upstream and down than when he travels all the while across the current. How is this fact to be reconciled with the result of the space-course, where the cross-journey appears to take less time than the other ? But first let us try a third experiment. Having laid the course, first in space, then in time, let us try the effect of laying it in velocity. Here, perhaps, we may find ourselves puzzled. Com- parison of velocities has, hitherto, been the whole object of the swimming performance. How then, if the velocity is to be invariable, can the problem be said to exist? In itself, however, there is nothing improbable in the thought of a human swimmer who can preserve the same velocity at various angles to the force of the stream. Usually, it is true, a man will diminish his normal speed, as measured in a tank or pond, when he has to swim in the teeth of a current. But when called upon to swim a race of, say, 100 yards against a moderate stream, he may well succeed, by dint of special effort, in maintaining his pond-velocity over the course. So too, when the race is finished, and 8 A CRITICISM OF EINSTEIN he turns to drift back to the starting-point, he will leave the stream to do most of the work for him, and thus once more may travel with his average speed. Human beings, however, are incalculable creatures, and psychological causes had best be excluded from physics. Suppose, then, you were to hear of a mechanical swimmer, which travelled up, down, or across any current with unaltered velocity. At first you would refuse to believe it, but the testimony of impeccable witnesses might induce you to look into the matter. Hav- ing failed to convict the automaton of trickery, or to detect any error in the measurements of velocity, you would cast about next for some rational explanation. The hypothesis of a new and mysterious ' force ' might tempt you, but we at least, with a view to the character of our whole enquiry, must firmly expel that god from the machine. There is no room for ' force ' in the argument, if this third trial is to be comparable with the other two. What possible explanation, then, could be offered of a velocity unaffected by change of direction in a stream running 5 or 10 miles an hour? Two, and only two (we may think at first) could account for the phenomenon, and both of them sound absurd. In the first place, you might lose your temper THE SWIMMER AND THE STREAM 9 and protest that the thing did not move at all. But if this seemed too flatly opposed to the evidence, the only remaining alternative would be to infer that time and space themselves must shrink and expand. Remote from each other as these two suggestions must seem, we shall find occasion, as the argument proceeds, to examine them both. We shall have to en- quire whether so queer a dilemma can actually be forced on us, and even whether it is neces- sary to destroy it by accepting both horns at once. At this stage, however, it is only the second of the two explanations that calls for attention. We have noticed already, when reflecting on the swimmer and the walker, the alternate variation of time and of space. When the swimmer's distance was the same in the two courses, his times were different; when the period of time was fixed, his distances were unequal ; and in either case, of course, his velocity was affected. But since the compound of time and space is velocity, is there not a third possibility, that, when his velocity is con- stant, the actual times and actual distances must be the variable factors? Are the mile and the hour quite the stolid old conservatives we have always supposed them to be? Ob- serve, too, that it would be useless to endow only one of them with elastic dimensions; for 10 A CRITICISM OF EINSTEIN then the velocity might not always be constant. The variation must be possible in either. But where can we find a swimmer with constant velocity in all circumstances? We shall not have far to go. The conventional doctrine, that less time is required to swim 100 yards across a river than to swim 50 upstream and then 50 down, must credit the swimmer with a certain stan- dard velocity in still water, and this he must be supposed to retain. In other words, all the subsequent calculations depend upon the assumption that a certain normal speed is diminished or augmented by the current in which he is to swim. Without this the whole problem collapses. So many yards a minute (accomplished by so many similar strokes in motionless water) must be regarded as his private possession, or there is no sense at all in proceeding to calculate the varying effects of the stream. Now, when this much is con- ceded, and when it is found that the man, swim- ming the same distance twice, spends more time on one course than on the other, only one explanation is possible without falling into open absurdity. The normal velocity is, by hypothesis, unaltered ; the times are uneven ; the distances, therefore, are not the same. I have no thought of impugning the accuracy of THE SWIMMER AND THE STREAM II the space-measurements made on the bank, or the respectability of the watches used for the timing. The precision of these I take to be absolute, but I submit that the swimmer is sub- jected to two unequal trials. He is asked to swim, first a shorter, then a longer course, as though their length were the same. A more scientific way, perhaps, of expressing my pro- position would be to assert at once that ' retar- dation ' is a myth. But as that might be rather too sudden a shock, we must turn aside for some preliminary reflections. Every swimmer, like every oarsman, will assure you that going upstream is harder work than coming down. Every swimmer, there- fore, and every oarsman will be wrong. There is no difference whatever between the two pro- cesses, so long as you stick to the relevant facts. The common belief depends primarily upon the ambition to reach a certain point on the bank. It depends also on physiological or psycho- logical causes, which have nothing to do with the question before us. When we laugh at the thought of a swimmer who always travels exactly the same distance in a given time, re- gardless of current and direction, this is chiefly because we insist on measuring the distance by the irrelevant bank, instead of by the water itself; and again, because we remember our 12 A CRITICISM OF EINSTEIN physical exhaustion after battling against the stream. But why this superfluous battle? Simply because we were aiming at some point on the bank. Abandon this idle preference; be content to make your normal number of normal strokes in a minute, and the measurable result of them, expressed in what we may call water-distance, will never vary so much as the breadth of a hair. There is nothing mysterious about * water-distance ' ; it is just the same as distance on land. I introduce the term only as a gentle reminder that swimming is usually done in the water, not on the bank. As a simple illustration of the point, place a long stick in a river, arrange that no waves or eddies shall disturb its orderly progress, and allow it to float in a line with the current. It will then be, for the purposes of the experiment, an in- tegral part of the river, with the river's exact velocity and no other. Now place a swimmer at one end of it, and bid him swim with his normal effort (i.e. that which produces his velocity in still water) to the other end. Time him carefully, and then repeat the trial in the opposite direction, stipulating that he shall make exactly the same number of strokes, with exactly the same vigour, as before. Beyond a shadow of doubt, he will accomplish his task in exactly the same time as before ; and this he THE SWIMMER AND THE STREAM 13 will do over and over again, at whatever angle to the current you choose to arrange the stick, and no matter what comments may be offered by loafers on the bank. In actual rivers, no doubt, it would be difficult to secure the re- quired conditions for a satisfactory experiment with a stick; but, in effect, this is what every swimmer does in the test so freely quoted in connection with the Michelson-Morley experi- ment. If, then, he takes more time to go up- stream than to return to the same point on the bank, it can only be because he swims beyond the end of the stick before he returns. Simi- larly, if the time across the stream (i.e. once across) is shorter than on the upward course, but longer than on the downward, the corres- ponding ratio between the distances must hold good. And this it is perfectly easy to prove in an analogous case to which I shall shortly proceed. The sensation of effort when one struggles against a current of water, an adverse wind, or, for that matter, the gradient of a hill, is so familiar that the theory of ' retardation ' and f resistance ' will not easily be abandoned. Imagine, however, the plight of a swimmer set in the midst of an ocean on a night of absolute and impenetrable darkness. No friendly bank is there to arouse his longing, no pole-star to 14 A CRITICISM OF EINSTEIN guide him by its beams. Better still would it be if, like a fish, he could swim under water. But if he is to remain with his head on the surface, not even the buffeting waves must be allowed to give him a hint. All must be dark and sleek and oily, as he cleaves his lonely way. Man is vexed with imagination, and I can easily credit such a swimmer with various emotions. He may even begin to wonder how long it will be before he must permanently cast in his lot with the fish. But I challenge him to have the slightest sense of direction ; I challenge him to find it easier or harder to go one way than another, or (unless it be from the rush of the air) to guess that he is being swept along north- wards, perhaps, or southwards with a velocity of 20 miles an hour. Should he succeed in persuading himself that one route is easier than another, this will be merely the work of fancy ; and, while he supposes himself to be moving steadily in one direction, he will almost cer- tainly be swimming round and round the cir- cumference of a circle. Precisely the same would be the case of the swimmer between ordinary banks, could he but banish his prefer- ence for some particular direction, his queru- lous anxiety to reach a point on the shore. All this I set down in the fond hope of diminishing initial prejudice against the reception of an THE SWIMMER AND THE STREAM 15 elementary truth. Strictly speaking, the psychological aspect is irrelevant to the physical problem ; and in the next chapter the proof that a swimmer obliged to conform to a rule imposed by the bank does swim further upstream than down will be set forth without reference to the swimmer's emotions. CHAPTER II THE CASE OF THE PASSENGER AND THE TRAIN To Einstein belongs the credit of raising an important question, even if it is difficult to believe that he has given the right answer. In his valuable little book, translated by Dr. Law- son, of Sheffield University, he discusses the case ( 6, 9 and 10) of a man who walks along the corridor of a moving train, in the direction of the engine (though the direction is not here important), and covers so many yards in a cer- tain number of seconds. The question is whether the measurements of time and space made within the train can be simply identified with the like measurements made on the em- bankment. The man walks, perhaps, at the rate of 2 yards per second within the train. Does it follow that he is likewise walking at 2 yards per second as judged by and from the embankment? If the 2 yards be accepted as the space-distance, does it follow that these yards are covered in what we may call i em- bankment second? Or again, if we take i second as the given time, does it follow that the distance covered is equal to 2 embankment THE PASSENGER AND THE TRAIN I/ yards? Einstein's answer is negative. He discovers here the relativity of both time and distance, and forthwith sweeps us away to the ' Lorentz transformation ' and to speculations about the mysterious behaviour of clocks and measuring-rods. All this I believe to be entirely superfluous, because it all rests on a false presupposition, derived from, or akin to, the error made in the Michelson-Morley experiment, or rather in the interpretation of the case of the swimmer. We have first to consider, as Einstein him- self has remarked, that a passenger walking up, down or across the corridor of a moving train is exactly analogous to a man swimming up, down or across a flowing stream. The banks are the same in both cases, the passenger is the swimmer, the train is the stream. What Einstein, so far as I can judge, has failed to perceive is the solution of the swimming pro- blem thus plainly afforded. When the pas- senger walks towards the tail of the train, that is to say, in the opposite direction to that of the train, as judged from the bank, he is strictly analogous to the man swimming upstream. When he reverses his direction, he is walking downstream, and, when he walks to and fro across the corridor or carriage, he is swimming across the current.. Why, then, does he lack 1 8 A CRITICISM OF EINSTEIN the swimmer's varying sensations? Solely because he is not trying to reach a point on the bank. He is travelling, perhaps, in the carriage nearest the engine, and the restaurant- car is at the tail of the train. In due course he desires his luncheon, and walks along the cor- ridor to get it. While he walks at his normal 3 miles an hour in one direction, it worries him not at all if the train is carrying him at, per- haps, 50 or 60 miles an hour in the opposite direction. He is not thinking about the em- bankment, but is brooding on his imminent lunch. Nor, I suppose, will anyone dream of disputing his ability to walk with the same velocity, without varying his effort, no matter what his direction within the train. In point of fact, he can easily get sensations analogous to those of swimming upstream, if he tries to return to a point on the embankment already passed by the train. To do this would often be impossible, but, if the train happened to be moving at only some 5 or 6 miles an hour, it could be done with an effort, that is to say, by walking or running with more than the pas- senger's normal velocity. Let us proceed, then, to examine the case of the passenger with the help of a diagram. Trains usually go faster than men, nor does the speed of the train affect the principle of the THE PASSENGER AND THE TRAIN IQ argument ; but, for the sake of a closer analogy to a swimmer who can make some actual pro- gress upstream, as judged from the bank, I shall slow down the train till it is crawling along at 3 yards a second, while the passenger shall be hurried up to 5 yards a second; so that, were he to alight and walk side by side with trie train, he would gain on it at the rate of 2 yards a second. Take the following diagram : A H C C' < c v T S 1 T! Direction ' A U D 1 The outer pair of parallel lines represent the embankment, the inner pair the train. The line AB is drawn at right angles to the embank- ment and the train ; CD is the interior width of the carriage ; AH, ST and BK are each equal to CD. With these data in his posses- sion, as well as the actual velocity of the train and the normal velocity of the passenger ' in still water,' the observer on the bank sets out to enquire how long it will take the passenger, first to walk along CD and back to C, then to 2O A CRITICISM OF EINSTEIN walk from S to T, and back once more to S. Arguing on the basis of the ' retardation ' theory, the observer will calculate that more time will be required to walk from C to D while the train is moving than when it is at rest; for by CD he will mean a certain portion of the straight line AB, and the train, he will argue, must all the while be carrying the pas- senger ' downstream.' Referring to his data, he will come to the conclusion that the man's velocity along CD or DC will be reduced from 5 yards a second to 4. Similarly, he will cal- culate the ' upstream ' velocity as 2 yards per second, the ' downstream ' as 8. But here I must interpolate an important note on the general character of the argument, namely, that it makes not the slightest difference whether the figures calculated for the retarda- tion and acceleration are accurate or not. That is to say, it makes no difference in principle, however much it may affect the value of the result in a particular case. Instead of my figures 4, 2 and 8 yards per second, any others may be taken the more absurd the better- to represent the different velocities, without touching the principle of the argument by which this problem of ' relativity ' is solved. Well then, to return to the case before us, we must now name a definite figure for the length THE PASSENGER AND THE TRAIN 21 of CD. Four yards will be the most con- venient, because that is the distance, according to the embankment calculation, to be accom- plished in one second by the passenger when he is walking across the current of the train. The passenger will thus be allowed two seconds to walk from C to D and back again to C. But now the fun begins. For the passenger walks at 5 yards per second in any direction without the smallest difficulty, and is forbidden by his principles to depart from his standard velocity. To put it another way, he must walk 5 yards, not 4, if he is to remain on the line CD, as viewed from the embankment, so as to arrive at D at the end of one second. His course, therefore, in the diagram must be CD' in one direction, DC' in the other, and the length of each of those lines will be 5 yards. Further, since each is the hypotenuse of a right-angled triangle, the length of the remaining side (CC 1 or DD 1 } will be 3 yards, by Euclid I., 47. Our passenger, then, walks 10 yards in the first 2 seconds, while the scientific gentlemen on the bank are prepared to swear that he has walked only 8. Next comes the journey from S 1 to T. According to the bank calculation, the man will walk this 4 yards at only 2 yards per second, and so will require 2 seconds for his task. But during this time the passenger, 22 A CRITICISM OF EINSTEIN who knows nothing of ' retardation ' will walk 10 yards up the train, which gives us the length of ST'. At T 1 he will appear to them to be standing at T, and his last stage, according to them, will be to return from T to 5 with a speed of 8 yards per second. For this they will allow him just half a second, but he, as before, will preserve his constant velocity, and will travel only 2\ yards in the time ; and thus we get the length of T' S ' . Both parties now proceed to add up their totals. On the bank they find that the man has walked 16 yards in 4^ seconds, and that it took him 2\ seconds to do the 8 yards up train and down, as against 2 seconds for the same distance across the stream and back. He, on the other hand, knows that in their 4^ seconds he has walked 22^ yards, and that his four stages, in yards, were 5, 5, 10, and 2\. The number of yards walked by the passenger, in this particular example, must always be 5 times the number of seconds calculated on the bank. Had his velocity been 6, 10, 100, or any other number of yards per second we should have had to multiply their number of seconds by his actual number of yards per second, instead of by 5, but the principle would be always the same. The velocity of the train in no way affects his velocity. There is neither retardation nor THE PASSENGER AND THE TRAIN 2J acceleration, and exactly the same is true of the swimmer in the stream. He merely swims greater or smaller distances, according to the number of seconds he occupies in swimming. Perhaps it will be useful to exhibit the principle in the alternative way. In the first trial a certain number of seconds were, so to say, allotted to the passenger, and he translated them into his own number of yards. But now suppose they ask him to walk 4 yards across the train, 4 back again, 4 up-train, and 4 down. As before they will calculate his time as 4^ seconds ; but he, feeling bound on this occasion to stick to their distances, will only require 3^ seconds for the whole journey ; the number of their yards in this case being 5 times the number of his seconds. Nothing new is re- vealed in this second presentation of the facts. The only difference is that the track is mea- sured in time on one occasion, in space on the other. In both cases, the philosophers of the embankment, with all the requisite data at their disposal, fell into the same error, because they would talk about ' retardation,' instead of grasping the cardinal fact, that the velocity of the train could have no possible effect on the constant velocity of the passenger, as he walks to and fro within the train. There still remains the last, and not the least 24 A CRITICISM OF EINSTEIN instructive, of the three possibilities, when the track is, so to speak, laid in velocity, not in space or in time. The observers on the bank, we now assume, know the passenger's velocity, and have somehow persuaded themselves that it remains constant in all circumstances. Their object is to discover the width of the carriage (that is to say, the length of CD) and the length of time required for crossing it. The time they hope to discover by observation, and thence to deduce the distance between C and D. Keeping a sharp watch, and choosing a favourable occasion, they observe him leave C and walk along the line to D, which they, of course, see as part of the line AB. After exactly one second he is observed to arrive at D, whence they infer that the distance must be 5 yards. But though it is true that he has walked 5 yards in one second, his actual course has been from C to D'. They have thus pro- vided themselves with a false measuring-rod, and therewith will roam about the world com- puting times and distances and getting them wrong every time. For they will measure with their own CD, a distance of 4 yards, which the passenger would traverse in f of a second, and will count it as 5 yards, or, expressed in time, as one second. Other varieties of this error could be constructed, but the general principle THE PASSENGER AND THE TRAIN 25 will always be the same. In all cases, whether they start from a known distance between two points or from the observation of a time actu- ally occupied by the passenger in covering a distance not yet measured, their genuine know- ledge of his velocity will be the very cause of their subsequent errors. Reviewing the results of the three alter- native trials, we note that three factors are involved, distance in space, distance in time, and velocity; and we find that, whenever the bank party have correct information about any one of the three, they must go astray about the other two. Yet all three varieties of the error spring from a common source, their failure to detect the actual course of the traveller. D and D 1 are points in the train 3 yards apart. There is nothing mysterious about them ; they are not one confused identity, but separate points which might be marked with chalk and observed by anybody who took the trouble to visit them. The passenger goes always to one; the observers suppose him to go always to the other. Similarly, his actual return across the carriage is always at the angle repre- sented by DC 1 in the diagram; and again, when they see him at T he is at T', and, when they believe him to have returned to S, he has stopped short at S'. Exactly the same things 26 A CRITICISM OF EINSTEIN happen in the case of the swimmer in the stream. The spectators see him swim across a straight line from bank to bank, but in order to afford them that spectacle, he has to swim aslant across the water. He actually swims along the hypotenuse of a right-angled triangle, while they firmly believe him to be travelling along one of the other sides. There is no ' retar- dation ' when he goes across or up the stream, no ' acceleration ' or additional velocity when he comes down with the current. He keeps always to his constant and uniform velocity, just as though he were disporting himself in his private tank. Consideration of these facts, which appear to be indisputable, sets us wondering whether the whole imposing fabric of ' relativity ' is not already tottering rather ominously on its base. Further light will be thrown on that question in the next chapter. Meanwhile our attention has been drawn to a very important fact, which might provisionally be called ' relativity of direction.' For since the passenger can only walk along the line CD, regarded as part of the line AB, by actually walking along the dif- ferent line CD', we have the materials for a very pretty dispute about ' straight lines ' and * shortest distances.' This subject, however, is reserved for a later chapter, where its full significance will be more clearly discerned. CHAPTER III THE ADDITION OF VELOCITIES IN THE sixth section of his book Einstein states the classical theorem of the ' addition of velocities,' and in due course proceeds to ques- tion its truth. His most important argument refers to the constant velocity of light and Fizeau's experiment, but this it will be expedient to postpone until we have examined the simpler, but supposedly analogous, case ot a man walking along the corridor of a train in the direction of the train's journey, that is to say, towards the engine. The train, perhaps, is travelling at 60 miles an hour, and the man is walking at 3. Hence, according to the classical doctrine, he ought to be advancing at the rate of 63 miles an hour relatively to the embankment. But this, Einstein proceeds to argue, is an untenable conclusion, and conse- quently the classical theorem can no longer be upheld. We thus resume the enquiry stated already at the beginning of the preceding chapter, whether times and distances, as measured within the train, are absolutely the same as when the embankment is the reference- 28 A CRITICISM OF EINSTEIN body, or whether a ' transformation ' is re- quired. Here, to begin with, there seems to be a surprising confusion between two distinct questions, the one relating to the actual point on the embankment at which the passenger arrives in a given time, the other to the addition of velocities. The answer to the first presents no difficulties, and indeed admits of no doubt; the answer to the second, or at least the first part of the answer, is that ' addition of velocities ' is an extremely careless and un- critical expression. The easier point can be determined by the following example. Before the train leaves London a chalk line is drawn across the corridor, at right angles to the sides of the train. This line is extended to a point A on the embankment (or platform), which it meets at right angles. We thus have A marked as the starting-point of both the train and the man. We place the man firmly on the chalk line, and start the train. In an hour the train travels 60 miles and stops. Meanwhile, at some time during the journey, no matter when, the man walks 20 yards towards the engine, chalks another line parallel to the first, and stands on it till the train comes to rest. When the train stops, the original and the later chalk lines are extended, respectively, to points B THE ADDITION OF VELOCITIES 2Q and C on the embankment. The train, there- fore, as represented by the first line, has travelled a distance AB, which is 60 miles, and the man has arrived at C. What is the length of BC? The only conceivable answer is 20 yards, and the fact can be tested any day of the week. Now, if the train's journey had been shorter, or the man's walk in the train much longer, we might have persuaded him to move continuously, and with uniform velocity, so as to arrive at the second line exactly when the train finished its journey. The result, of course, as regards the eventual distance from A (and from B) is just the same in principle whether he walks throughout the train's journey or only during some part of it ; nor would there be any change in the nature of the problem, if the direction of the walk were towards the tail of the train. There is even a sense (an impor- tant sense, as we shall presently see) in which the man need never have been in the train at all. We might have placed him at B, with orders to get to C when and how he pleased, so long as he arrived before the train pulled up at B. Any spare time he could then have occupied in wondering whether Achilles would ever catch the tortoise, if the hero were obliged to stop every time his rival felt disposed for a rest. 3