Opening…

THE EVOLUTION OF SCIENTIFIC THOUGHT FROM NEWTON TO EINSTEIN

By

A. D’ABRO

NEW YORK BONI & LIVERIGHT 1927

COPYRIGHT 1927 :: BY BONI & LIVERIGHT, INC. PRINTED IN THE UNITED STATES

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PREFACE

ALTHOUGH in the course of the last three centuries scientific theories have been subject to all manner of vicissitude and change, the governing motive that has inspired scientists has been ever the same—a search for unity in diversity, a desire to bring harmony and order into what might at first sight appear to be a hopeless chaos of experimental facts.

In this book the essential features of Newton’s great discoveries, the apparent inevitableness of absolute space and time in classical science, are passed in review. Then we come to Riemann, that great mathematician who wrested the problem of space from the dogmatic slumber where it had rested so long. Finally we see how Einstein succeeded in transporting to the realm of physics the ideas that Riemann had propounded, giving us thereby that supreme achievement of modern thought, the theory of relativity.

Although I have used non-technical language, great care has been given to an accurate presentation of facts. In certain parts, however, notably in those devoted to non-Euclidean geometry and to the principle of Action, a looseness of presentation has appeared unavoidable owing to the extreme technicality of the subjects discussed. But as it was a question of presenting these subjects loosely or leaving them out of the picture entirely, it appeared preferable to sacrifice accuracy to general comprehensiveness.

Here, however, the reader may be reminded that even for those who are interested solely in trends of thought or in the evolution of ideas, no popular or semi-popular book can ever aspire to take the place of the highly technical mathematical works. The superiority of the latter lies not in the bare mathematical formulæ which they contain. Rather does it reside in the power the mathematical instrument has of giving us a deeper insight into the problems of nature, revealing unsuspected harmonies and extending our survey into regions of thought whence the human intelligence would otherwise be excluded. Thus the sole rôle a semi-popular book can hope to perform is to serve as a general introduction, to whet the appetite for further knowledge, if a craving for knowledge is within us. To presume, as the philosophers do, that a vague understanding of a highly technical subject, gleaned from semi-popular writings, or from the snatching at a sentence here and there in a technical book, should enable them to expound a theory, criticise it, and, worse still, ornament it with their own ideas, is an opinion which has done much to create a spirit of distrust towards their writings. The answer Euclid gave to King Ptolemy, “There is no royal road, no short cut to knowledge,” remains true to-day, still [Pg vi] truer than in the days of ancient Alexandria, when science had not yet grown to the proportions of a mighty tree.

I wish to take this opportunity to express my gratitude to Prof. Leigh Page of Yale University for his kindness in looking over the manuscript and offering many valuable suggestions.

A. D’ Abro.

NEW YORK, 1927.

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FOREWORD

“And now, in our time, there has been unloosed a cataclysm which has swept away space, time and matter, hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope, and entailing a deeper vision.”

H. WEYL (“Space, Time and Matter”).

THE theory of relativity represents the greatest advance in our understanding of nature that philosophy has yet witnessed.

If our interest is purely philosophical, we may wish to be informed briefly of the nature of Einstein’s conclusions so as to examine their bearing on the prevalent philosophical ideas of our time. Unfortunately for this simplified method of approach, it is scarcely feasible. The conclusions themselves involve highly technical notions and, if explained in a loose, unscientific way, are likely to convey a totally wrong impression. But even assuming that this first difficulty could be overcome, we should find that Einstein’s conclusions were of so revolutionary a nature, entailing the abandonment of our ideas on space, time and matter, that their acceptance might constitute too great a strain on our credulity. Either we would reject the theory altogether as a gigantic hoax, or else we should have to accept it on authority, and in this case conceive it in a very vague and obscure way.

From a perusal of the numerous books that have been written on the subject by a number of contemporary philosophers, the writer is firmly convinced that the only way to approach the theory, even if our interest be purely philosophical, is to study the scientific problem from the beginning. And by the beginning we refer not to Einstein’s initial paper published in 1905; we must go back much farther, to the days of Maxwell and even of Newton. Here we may mention that however revolutionary the theory of relativity may appear in its philosophical implications, it is a direct product of the scientific method, conducted in the same spirit as that which inspired Newton and Maxwell; no new metaphysics is involved. Indeed Einstein’s theory constitutes but a refinement of classical science; it could never have arisen in the absence of that vast accumulation of mathematical and physical knowledge which had been gathered more especially since the days of Galileo. Under the circumstances it is quite impossible to gain a correct impression of the disclosures of relativity unless we first acquaint ourselves with the discoveries of classical science prior to Einstein’s time. It will therefore be the aim of this book, before discussing Einstein’s theory proper, to set forth as simply as possible this necessary preliminary information.

Modern science, exclusive of geometry, is a comparatively recent creation and can be said to have originated with Galileo and Newton. [Pg x] Galileo was the first scientist to recognise clearly that the only way to further our understanding of the physical world was to resort to experiment. However obvious Galileo’s contention may appear in the light of our present knowledge, it remains a fact that the Greeks, in spite of their proficiency in geometry, never seem to have realised, the importance of experiment (Democritus and Archimedes excepted).

To a certain extent this may be attributed to the crudeness of their instruments of measurement. Still, an excuse of this sort can scarcely be put forward when the elementary nature of Galileo’s experiments and observations is recalled. Watching a lamp oscillate in the cathedral of Milan, dropping bodies from the leaning tower of Pisa, rolling balls down inclined planes, noticing the magnifying effect of water in a spherical glass vase, such was the nature of Galileo’s experiments and observations. As can be seen, they might just as well have been performed by the Greeks. At any rate, it was thanks to such experiments that Galileo discovered the fundamental law of dynamics, according to which the acceleration imparted to a body is proportional to the force acting upon it.

The next advance was due to Newton, the greatest scientist of all time if account be taken of his joint contributions to mathematics and physics. As a physicist, he was of course an ardent adherent of the empirical method, but his greatest title to fame lies in another direction. Prior to Newton, mathematics, chiefly in the form of geometry, had been studied as a fine art without any view to its physical applications other than in very trivial cases.[1] But with Newton all the resources of mathematics were turned to advantage in the solution of physical problems. Thenceforth mathematics appeared as an instrument of discovery, the most powerful one known to man, multiplying the power of thought just as in the mechanical domain the lever multiplied our physical action. It is this application of mathematics to the solution of physical problems, this combination of two separate fields of investigation, which constitutes the essential characteristic of the Newtonian method. Thus problems of physics were metamorphosed into problems of mathematics.

But in Newton’s day the mathematical instrument was still in a very backward state of development. In this field again Newton showed the mark of genius, by inventing the integral calculus. As a result of this remarkable discovery, problems which would have baffled Archimedes were solved with ease. We know that in Newton’s hands this new departure in scientific method led to the discovery of the law of gravitation. But here again the real significance of Newton’s achievement lay not so much in the exact quantitative formulation of the law of attraction, as in his having established the presence of law and order at least in one important realm of nature, namely, in the motions of heavenly bodies. Nature thus exhibited rationality and was not mere blind chaos [Pg xi] and uncertainty. To be sure, Newton’s investigations had been concerned with but a small group of natural phenomena (planetary motions and falling bodies), but it appeared unlikely that this mathematical law and order should turn out to be restricted to certain special phenomena; and the feeling was general that all the physical processes of nature would prove to be unfolding themselves according to rigorous mathematical laws.

It would be impossible to exaggerate the importance of Newton’s discoveries and the influence they exerted on the thinkers of the eighteenth century. The proud boast of Archimedes was heard again—“Give me a lever and a resting place, and I will lift the earth.”—But the boast of Newton’s successors was far greater—“Give us a knowledge of the laws of nature, and both future and past will reveal their secrets.”

To-day these hopes appear somewhat childish, but this is because we have learnt more of nature than was ever dreamt of by Newton’s contemporaries. Nevertheless, although we recognise that we can never be demigods, the mathematical instrument in conjunction with the experimental method, still constitutes our most fruitful means of progress.

Now Newton, in his application of mathematics to the problems of physics, had been concerned only with the very simplest of physical problems—planetary motions, mechanics, propagation of sound, etc. But when it came to applying the mathematical method to the more intricate physical problems, a considerable advance was necessary in our scientific knowledge, both mathematical and empirical. Thanks to the gradual accumulation of physical data, and thanks to the efforts of Newton’s great successors in the field of pure mathematics (Euler, Lagrange, Laplace), conditions were ripe in the first half of the nineteenth century for a systematic mathematical attack on many of nature’s secrets.

The mathematical theories constructed were known under the general name of theories of mathematical physics. In so far as they represented a mere application of mathematics to natural phenomena, they had their prototype in Newton’s celestial mechanics. The only difference was that they dealt with a wide variety of physical phenomena (electric, hydrostatic, etc.), no longer with those of a purely mechanical nature. The most celebrated of these theories (such as those of Maxwell, Boltzmann, Lorentz and Planck) were concerned with very special classes of phenomena. But with Einstein’s theory of relativity, itself a development of mathematical physics, the scope of our investigations is so widened that we are appreciably nearer than ever before to the ideal of a single mathematical theory embracing all physical knowledge. This fact in itself shows us the tremendous philosophical interest of the theory of relativity.

Now in all these theories of mathematical physics, the same type of procedure is invariably followed. Experimenters establish certain definite facts and detect precise numerical relationships between magnitudes, for example, between the intensity of an electric current flowing along a wire and the intensity and orientation of the magnetic field surrounding the wire. The mathematical physicist then enters upon the scene, assigns certain letters of the alphabet to the [Pg xii] physical entities involved (in the present case electric current designated by and magnetic intensity designated by ) and by this means translates the numerical relationships discovered by the experimenter into mathematical form. He thus obtains a mathematical relationship or equation which is assumed to constitute the mathematical image of the concrete physical phenomenon . His task will now be to extract from his mathematical equation or equations all their necessary mathematical consequences. In this way, provided his technique does not fail him, he may be led to new equations . These new equations , when translated back from the mathematical to the physical, will express new physical relationships .

The mathematician assumes that just as his equations were the necessary mathematical consequences of his original equations , so also must the physical translation of constitute a physical phenomenon , which follows as a necessary consequence of the existence of the physical phenomenon . If occurs, must ensue.

We thus understand the significance of a theory of mathematical physics. Its utility is to allow us to foresee and to foretell physical phenomena. In this way it suggests precise experiments which might never have been thought of, and permits us to anticipate new relationships and new laws and to discover new facts. From a philosophical point of view, by establishing a rational connection between seemingly unconnected phenomena, it enables us to detect the harmony and unity of nature which lie concealed under an outward appearance of chaos.

Of course the experimenter in the first place must be very careful to give accurate information to the mathematician; for if by any chance his information should be only approximately correct, the mathematical translation would likewise be lacking in accuracy, and the mathematical consequences of a might be still further at variance with the world of physical reality. It is as though, when firing at a distant target, we were to point the rifle a wee bit too far to one side; the greater the range, the wider would be the divergence. Dangers of this sort are of course inevitable, for human observations are necessarily imperfect. In any case, therefore, the mathematician’s physical anticipations will always require careful checking up by subsequent experiment. Obviously, however, something much deeper is at stake than mere accuracy of observation.

Mathematical deductions are mind-born; they pertain to reason and are not dependent on experience. When, therefore, we assume that our mathematical deductions and operations will be successful in portraying the workings of nature, we are assuming that nature also is rational, and that therefore a definite parallelism or correspondence exists between the two worlds, the mathematical and the physical. A priori, there appears to be no logical necessity why any such parallelism should exist. Here, however, we are faced with a situation over which it is useless to philosophise. Success has attended the efforts of mathematical physicists in so large a number of cases that, however marvellous it may appear, we can scarcely escape the conclusion that nature must be rational and susceptible to mathematical law. In [Pg xiii] fact, were this not the case, prevision would be impossible and science non-existent.

It may be that nature is only approximately rational; it may be that her appearance of rationality is due to the very crudeness of our observations and that more refined experiments would yield a very different picture. Eddington goes so far as to suggest that the difficulties which confront us in the study of quantum phenomena may indeed be due to the fact that nature is found to be irrational when we seek to examine her processes in a microscopic way. This is a possibility which we cannot afford to reject. But at any rate, as long as our theories appear to be verified by experiment, we must proceed as though nature were rational, and hope for the best.

Now it must not be thought that the introduction of the mathematical instrument into our study of nature creates any essential departure from the commonplace method of ordinary deductive and inductive reasoning. There is no particular mystery about mathematical analysis; its only distinguishing feature is that it is more trustworthy, more precise, and permits us to proceed farther and along safer lines.

Consider, for example, the well-known change of colour from red to white displayed by the light radiated through an aperture made in a heated enclosure, as the temperature increases. From this elementary fact of observation Planck, thanks to mathematical analysis, was able to deduce the existence of light quanta and thence the possibility that all processes of change were discontinuous, and that a body could only rotate with definite speeds. Obviously, commonplace reasoning unaided by mathematics would never have led us even to suspect these extraordinary results.

Now when we say that a theory of mathematical physics is correct, all we mean is that the various mathematical consequences we can extract from its equations call for the existence of physical phenomena which experiment has succeeded in verifying. On the other hand, if our mathematical anticipations do not tally with experimental verification, we must recognise that our theory is incorrect. This does not mean that it is incorrect from a purely mathematical point of view, for in any case it exemplifies a possible rational world; but it is incorrect in that it does not exemplify our real world. We must then assume that our initial equations were in all probability bad translations of the physical phenomena they were supposed to represent.

In a number of cases, however, it has been found unnecessary to abandon a theory merely because one of its anticipations happened to be refuted by experiment. Instead, it is often possible to assume that the discrepancy between the mathematical anticipation and the physical result may be due to some contingent physical influence, which, owing to the incompleteness of the physical data furnished us by the experimenters, our equations have failed to take into consideration. A case in point is afforded by the discovery of Neptune.

The Newtonian mathematical treatment of planetary motions assigned a [Pg xiv] definite motion to the planet Uranus. Astronomical observation then proved that the actual motion of Uranus did not tally with these mathematical anticipations. Yet it was not deemed necessary to abandon Newton’s law; Adams and Leverrier suggested the possibility that an unknown planet lying beyond the orbit of Uranus might be responsible for the deviations in Uranus’ motion. Taking the existence of this unknown planet into consideration in his mathematical calculations, Leverrier succeeded in determining the exact position which it would have to occupy in the heavens at an assigned date. As is well known, at the precise spot calculated, the elusive planet (presently named Neptune) was discovered with a powerful telescope.

This procedure of ascribing discrepancies in our mathematical anticipations to the presence of contingent influences rather than to the falsity of our theory is only human. There is no inclination, merely because the hundredth case turns out to be an exception, to abandon a theory which has led to accurate anticipations in 99 cases out of 100. But we must realise that this procedure of appealing to foreign influences, while perfectly legitimate in a tentative way, must be applied with a certain amount of caution; in every particular case it must be justified by a posteriori determination of fact. Thus Leverrier was also the first to discover certain irregularities in the motion of the planet Mercury. As in the case of Uranus, he attempted to ascribe these discrepancies to the presence of an interior planet which he called Vulcan and which he assumed to be moving between the orbit of Mercury and the sun. Astronomers have, however, failed to find the slightest trace of Vulcan, and a belief in its existence has been abandoned. If contingent influences are to be invoked for Mercury’s anomalies, we must search for them in some other direction.

In this particular case all other suggestions were equally unsatisfactory. Hence even before the advent of Einstein’s theory, doubts had been raised as to the accuracy of Newton’s law of gravitation. The procedure of patching up a mistaken theoretical anticipation with hypotheses ad hoc has not much to commend it. Yet when, as was the case with Vulcan and Neptune, the influence we appeal to is of a category susceptible of being observed directly, the method is legitimate. But when our hypothesis ad hoc transcends observation by its very nature, and when, added to this, its utility is merely local, accounting for one definite fact and for no other, it becomes worse than useless.

This abhorrence of science for the unverifiable type of hypothesis ad hoc so frequently encountered in the speculations of the metaphysicians is not due to a mere phenomenalistic desire to eliminate all that cannot be seen or sensed. It arises from a deeper motive entailing the entire raison d’être of a scientific theory. Suppose, for instance, that our theory had led us to anticipate a certain result, and that experiment or observation should prove that in reality a different result was realised. We could always adjust matters by arbitrarily postulating some local invisible and unverifiable influence, which we might ascribe to the presence of a mysterious [Pg xv] medium—say, the ether . We should thus have added a new influence to our scheme of nature.

If we should now take this new influence into consideration, the first numerical result would, of course, be explained automatically, since our ether was devised with this express purpose in view. But we should now be led to anticipate a different numerical result for some other phenomenon. If this second anticipation were to be disproved by experiment we could invoke some second unverifiable disturbing influence to account for the discrepancy, while leaving the first result unchanged. Let us call this new influence the ether . We might go on in this way indefinitely.

But it is obvious that our theory of mathematical physics whose object it was to allow us to foresee and to foretell would now be useless. No new phenomenon could be anticipated, since past experience would have shown us that unforeseen influences must constantly be called into play if theory were to be verified by experiment. Under these circumstances we might just as well abandon all attempts to construct a mathematical model of the universe.

Suppose now that by modifying once and for all our initial premises we are led to a theory which allows us to foresee and foretell numerical results that are invariably verified with the utmost precision by experiment, without our having to call to our assistance a number of foreign hypotheses. In this case we may assume that the new theory is correct, since it is fruitful; and that our former theory was incorrect, because it led us nowhere.

The considerations we have outlined have an important bearing on the understanding of the outside world as shared by the vast majority of scientists. If we hold that the simplest of all the mathematical theories which finds itself in accord with experiment constitutes the correct theory, giving us the correct representation of the real world, we shall recognise that it would be a dangerous procedure to saddle ourselves with a number of hypothetical presuppositions at too early a stage of our investigations.

To be sure, we may have to make a certain number of fundamental assumptions, but we must regard these as mere working hypotheses which may have to be abandoned at a later stage if peradventure they lead to too complicated a synthesis of the facts of experiment. We shall see, for instance, that relativity compels us to abandon our traditional understanding of space and time. It is this fact more than any other which has been responsible for the cool reception accorded to the theory by many thinkers. When, however, we become convinced that Einstein’s synthesis is the simplest that can be constructed if due account be taken of the results of ultra-refined experiment, and when we realise that a synthesis based on the classical understanding of separate space and time would be possible only provided we were willing to introduce a host of entirely disconnected hypotheses ad hoc which would offer no means of direct verification, we cannot easily contest the soundness of Einstein’s conclusions.

There are some, however, who argue that we have an a priori intuitional understanding of space and time which is fundamental, and [Pg xvi] that we should sacrifice simplicity of mathematical co-ordination if it conflicts with these fundamental intuitional notions. Needless to say, no scientist could subscribe to such views. Quite independently of Einstein’s discoveries, mathematicians had exploded these Kantian opinions on space and time many years ago. As Einstein very aptly remarks in his Princeton lectures:

“The only justification for our concepts and system of concepts is that they serve to represent the complex of our experiences; beyond this they have no legitimacy. I am convinced that the philosophers have had a harmful effect upon the progress of scientific thinking in removing certain fundamental concepts from the domain of empiricism where they are under our control to the intangible heights of the a priori. For even if it should appear that the universe of ideas cannot be deduced from experience by logical means but is in a sense a creation of the human mind without which no science is possible, nevertheless this universe of ideas is just as little independent of the nature of our experiences as clothes are of the form of the human body. This is particularly true of our concepts of time and space which physicists have been obliged by the facts to bring down from the Olympus of the a priori in order to adjust them and put them in a serviceable condition.”

In the passage just quoted, Einstein argues from the standpoint of the physicist, but the opinions he expresses will certainly be endorsed by pure mathematicians. They, more than all others, have been led to realise how cautious we must be of the dictates of intuition and so-called common sense. They know that the fact that we can conceive or imagine a certain thing only in a certain way is no criterion of the correctness of our judgment. Examples in mathematics abound. For example, before the discoveries of Weierstrass, Riemann and Darboux the idea that a continuous curve might fail to have a definite slant at every point was considered absurd; and yet we know to-day that the vast majority of curves are of this type. In the same way our intuition would tell us that a line, whether curved or straight, being without width, would be quite unable to cover an area completely; yet once again, as Peano and others have shown, our intuition would have misled us. Many other examples might be given, but they are of too technical a nature and need not detain us. At all events, mathematicians, as a whole, refused to question the soundness of Einstein’s theory on the sole plea that it conflicted with our traditional intuitional concepts of space and time, and we need not be surprised to find Poincaré, one of the greatest mathematicians of the nineteenth century, lending full support to Einstein when the theory was so bitterly assailed in its earlier days.

We have now to consider in a very brief way certain of the philosophical problems which antedate Einstein’s discoveries, but with which his theory is intimately connected. Long before the advent of Einstein, problems pertaining to the relativity of motion through empty space had occupied the attention of students of nature. There were some who held that empty space, and with it all motion, must [Pg xvii] be relative; that states of absolute motion or absolute rest through empty space were meaningless concepts. According to these thinkers, in order to give significance to motion and rest, it was necessary to refer the successive positions of the body to some other arbitrarily selected body taken as a system of reference. We should thus obtain relative rest or motion of matter, with respect to matter. All these views were in full accord with our visual perceptions and they were expressed by what is known as the visual or kinematic principle of the relativity of motion. Other thinkers preferred to uphold the opposing philosophy. They assumed that space was absolute; that all motion must be absolute; that there was meaning to the statement that a body was in motion or at rest in space, regardless of the presence of other bodies to be used as terms of comparison. The controversy mighty have continued indefinitely, had it not been for the appearance of the scientist with his empirical methods of investigation.

Galileo and Newton were the first to recognise in a clear way that, provided certain very plausible assumptions were made, the dynamical evidence adduced from mechanical experiments proved the relativistic philosophy to be untenable. Classical science was therefore compelled to recognise the absoluteness of space and motion. It is true that many philosophers still defended the relativity of all motion. But their failure to take into consideration the real obstacles that seemed to bar the way to a relativistic conception of motion, coupled with the looseness of their scientific arguments, precluded their opinions from exercising any influence on scientific thought. Now it is to be noted that notwithstanding the absolute nature which Newton attributed to all states of motion and of rest in empty space, a certain type of absolute motion called Galilean[2] or again uniform translationary motion (defined by an absolute velocity but no absolute acceleration), was recognised by him as being incapable of detection, so far as experiments of a mechanical nature were concerned. This complete irrelevancy of absolute velocity or absolute Galilean motion to mechanical experiments was expressed in what is known as the Galilean or Newtonian or classical or dynamical principle of the relativity of Galilean motion through empty space. The existence of such a principle of relativity created a duality in the physical significance of motion, hence of space, but the philosophical importance of the principle, as referring to the problem of space, was lessened by the fact that this relativity applied solely to experiments of a mechanical nature. It was confidently assumed that electromagnetic and optical experiments would be successful in revealing the absolute Galilean motions which had eluded mechanical tests.

Such was the state of affairs when Einstein, in 1905, published his celebrated paper on the electrodynamics of moving bodies. In this he remarked that the numerous difficulties which surrounded the equations of electrodynamics, together with the negative experiments [Pg xviii] of Michelson and others, would be obviated if we extended the validity of the Newtonian principle of the relativity of Galilean motion (which applied solely to mechanical phenomena) so as to include all manner of phenomena: electrodynamic, optical, etc.

When extended in this way the Newtonian principle of relativity became Einstein’s special principle of relativity. Its significance lay in its assertion that absolute Galilean motion or absolute velocity must ever escape all experimental detection. Henceforth absolute velocity should be conceived of as physically meaningless, not only in the particular realm of mechanics, as in Newton’s day, but in the entire realm of physical phenomena. Einstein’s special principle, by adding increased emphasis to this relativity of velocity, making absolute velocity metaphysically meaningless, created a still more profound distinction between velocity and accelerated or rotational motion. This latter type of motion remained absolute and real as before. From this we see that the special principle was far from justifying the ultra-relativistic belief in the complete relativity of all motion, as embodied in the kinematical or visual principle of relativity.

It is most important to understand this point and to realise that Einstein’s special principle is merely an extension of the validity of the classical Newtonian principle to all classes of phenomena. Apart from this extension there is no essential difference between the two principles, since both deal exclusively with Galilean motion or velocity. Much of the criticism that has been directed against Einstein’s special theory, and in particular against its so-called paradoxes, can be traced to a confusion on the part of the critic between the very wide visual or kinematical principle of the relativity of all motion, including acceleration as well as velocity, and the much more restricted special principle which deals solely with velocity.

Owing to this initial error it has been assumed by some that the theory of relativity professes to have established the complete relativity of all motion. So far as the special theory is concerned, this assumption is obviously untenable; and a very marked distinction is maintained, as in classical science, between the significance of acceleration and of velocity. This distinction is by no means new, since it forms part and parcel of classical science; but in view of its importance to any one who wishes to obtain any insight into the philosophical implications of Einstein’s more difficult theory, we have deemed it advisable in the course of this book to devote a preliminary chapter to Newtonian mechanics. We may add that Einstein’s paper created an uproar, for it compelled us to reorganise in a radical way those fundamental forms of human representation which we call space and time.

After discussing the special theory we shall proceed to examine Einstein’s general theory, which he began to develop in 1912 and which he completed in 1916. The general theory deals more especially with gravitation, though it also sheds new light on the problem of the relativity of motion. It is our opinion that the reader will [Pg xix] save himself much unnecessary trouble if he realises from the start that Einstein, even in the general theory, does not succeed in establishing the complete relativity of all motion. While it is true that acceleration loses much of its absoluteness, we are still far from being able to subscribe to the very wide visual or kinematical principle which the complete relativity of all motion would necessitate.

Following the general theory, Einstein entered into cosmological considerations on the form of the universe as a whole. This part of the theory is still highly speculative; and until such time as astronomical observations conducted on the globular clusters and the Milky Way have given their verdict, nothing definite can be said.

From the standpoint of the relativity of motion, this last part of Einstein’s theory is of fascinating interest; for should it be proved correct, the relativity of all motion might finally be justified and the bugaboo of absolute rotation dispelled forever. We should then be led to a modified form of the visual or kinematical principle of the complete relativity of motion; namely, to Mach’s mechanics. On the other hand, should astronomical observation deny the correctness of Einstein’s cosmological views, the theory would have failed to establish the complete relativity of all motion.

In addition to all this preliminary physical information with which it is necessary to be acquainted (Newtonian mechanics, electrodynamics) before proceeding to a study of Einstein’s theory, a considerable amount of purely mathematical knowledge must be mastered. Here we refer not to the actual technique of calculation, but to the general significance of the mathematical doctrines developed by the great mathematicians of the past.

Einstein’s theory, more especially the second part (the general theory), is intimately connected with the discoveries of the non-Euclidean geometricians, Riemann in particular. Indeed, had it not been for Riemann’s work, and for the considerable extension it has conferred upon our understanding of the problem of space, Einstein’s general theory could never have arisen. As Weyl expresses it:

“Riemann left the real development of his ideas in the hands of some subsequent scientist whose genius as a physicist could rise to equal flights with his own as a mathematician. After a lapse of seventy years this mission has been fulfilled by Einstein.”

In a general way it may be said that prior to the discovery of non-Euclidean spaces two conflicting philosophies held the field. Some thinkers inclined to Berkeley’s views and maintained that the concept of space arose from the complex of our experiences, and chiefly from a synthesis of our visual and tactual impressions. Others, such as Kant, argued that the concept of three-dimensional Euclidean space was antecedent to all reason and experience and was essentially a priori, a form of pure sensibility. As in the case of the relativity of motion, discussions might have gone on indefinitely had it not been for the work of the psycho-physicists and mathematicians. [Pg xx] The latter settled the question by proving that Berkeley had guessed correctly, at least in a general way. The essence of Riemann’s discoveries consists in having shown that there exist a vast number of possible types of spaces, all of them perfectly self-consistent. When, therefore, it comes to deciding which one of these possible spaces real space will turn out to be, we cannot prejudge the question. Experiment and observation alone can yield us a clue. To a first approximation, experiment and observation prove space to be Euclidean, and this accounts for our natural belief in the truth of the Euclidean axioms, accepted as valid merely by force of habit. But experiment is necessarily inaccurate, and we cannot foretell whether our opinions will not have to be modified when our experiments are conducted with greater accuracy. Riemann’s views thus place the problem of space on an empirical basis excluding all a priori assertions on the subject.

Of course, these discoveries on the part of mathematicians precede Einstein’s theory by fully seventy years. They are the direct outcome of non-Euclidean geometry and would in no wise be affected by the fate of Einstein’s theory. But, on the other hand, the relativity theory is very intimately connected with this empirical philosophy; for, as will be explained later, Einstein is compelled to appeal to a varying non-Euclideanism of four-dimensional space-time in order to account with extreme simplicity for gravitation. Obviously, had the extension of the universe been restricted on a priori grounds, by some ukase, as it were, to three-dimensional Euclidean space, Einstein’s theory would have been rejected on first principles. On the other hand, as soon as we recognise that the fundamental continuum of the universe and its geometry cannot be posited a priori and can only be disclosed to us from place to place by experiment and measurement, a vast number of possibilities are thrown open. Among these the four-dimensional space-time of relativity, with its varying degrees of non-Euclideanism, finds a ready place.

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PART I PRE-RELATIVITY PHYSICS

[Pg 23]

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CHAPTER I MANIFOLDS

IN this chapter we shall be concerned solely with manifolds and their dimensionality; and not with their metrics, which pertains to a different problem entirely.

We all possess a certain instinctive understanding of what is meant by continuity. We notice, for example, that sounds, colours or tactual sensations merge by insensible gradations into other sounds, colours or tactual sensations, without any abrupt transitions. An aggregate of such continuous sensations constitutes what is called a sensory continuum or continuous manifold. That continuity is a concept which springs from experience can scarcely be doubted, and it can be accounted for by the inability of our crude senses to differentiate between impressions which are almost alike.

Consider, for example, the succession of musical notes exhibited in the chromatic scale on the piano. Here we are not in the presence of a sensory continuum, for the successive sounds do not merge into one another by insensible degrees. Even an untrained ear can differentiate between a and the -sharp immediately following it. But we can conceive of a piano in which a sufficient number of semitones and intermediary notes have been interposed so that every note would be indistinguishable from its immediate successor and immediate predecessor, although we should still be able to differentiate between non-contiguous notes. It would thus be possible for us to pass through a continuous chain of sounds from any one musical sound to any other without our ear’s ever being able to detect a sudden jump; and this is what we mean by calling our aggregate of sounds a sensory continuum.

Suppose now that we were to remove any one of these notes from our piano (excluding the two extreme ones). The continuity of our chain of sounds would be broken, for when we reached the missing note we should detect a sudden variation in pitch as we passed from the sound immediately preceding the removed note to the one immediately following it. In short, the removal of one of the notes would cut our continuous chain of sounds in two.

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The dimensionality of our continuum of sounds is obviously unity, for we can assign successive numbers to the successive notes, starting from some standard note, and by this means determine them without ambiguity.

Let us complicate matters somewhat by assuming that every individual note may be sounded with various intensities, but always in such a way that a note of given intensity can never be distinguished from the one sounded just a little louder or just a little softer. Once again it will be possible for us to pass in a continuous way from a note of feeble intensity to the same note sounded with louder intensity, without our ear’s ever being able to detect a variation in intensity between two successive sounds. More generally we shall be able to pass in a continuous way from a note of given pitch and given intensity to one of some other pitch and some other intensity.

In the case assumed we should be dealing with a two-dimensional continuum or continuous manifold of sounds; for in order to locate a definite sound it would be necessary for us to designate it by two numbers, one specifying its pitch and the other its intensity. We may also notice that whereas in the first case, by removing one of the notes, we were able to cut the manifold of successive pitches into two parts, the removal of one particular note of definite pitch and intensity will now be incapable of effecting this separation.

For instance, if we were to remove the note sounded with a definite intensity it would still be possible to pass in a continuous way from any one note of given intensity to any other note of our continuum by circumscribing the missing note; namely, by choosing some route of transfer which would pass through a differing in intensity from that of the we had removed.

In the present case, if we wished to divide our two-dimensional manifold into two parts, it would be necessary to remove some one-dimensional continuum of sounds, for example the one-dimensional continuum formed by all the notes of a given pitch , but varying in intensity, or again of given intensity but varying in pitch. If this were done, it would be impossible for us to pass in a continuous way from a note of definite pitch and intensity of our two-dimensional continuum to a note of any other definite pitch and intensity; for we could never get past the removed line of sounds without our ear’s detecting a sudden change.

We might complicate matters still further by taking into consideration variations in tonality, as for instance the variation which our ear can detect between two given notes of the same pitch and intensity sounded by two different instruments, such as a violin and an organ. Assuming that every one of our notes of given pitch and intensity in our two-dimensional manifold could also vary in tonality by imperceptible degrees, we should be dealing with a three-dimensional sensory continuum in which every note of given pitch, intensity and tonality could be defined unambiguously by the choice of three numbers.

As before, we should find that the removal of a single note of given pitch, intensity and tonality, or even the removal of a one-dimensional continuum of notes such as all those of given intensity and pitch, but varying in tonality, was quite insufficient to effect a separation in our three-dimensional manifold. In the present case we should have [Pg 25] to remove some two-dimensional continuum—say, all notes of given intensity but varying in pitch and tonality. Only then should we have effected a separation between any given element and any other one, rendering it impossible for a continuity of sound impressions to extend between the two elements. By proceeding in this way indefinitely it is obvious that we can conceive of sensory continua of any number of dimensions; there is no need to limit ourselves to three.

In a general way we may say, therefore, that a sensory continuum is -dimensional when, in order to render a path of continuous passage impossible between any two of its elements, it is necessary to remove a sub-continuum of dimensions. This sub-continuum itself is known to be dimensional because we can separate it into two parts only by removing from it a sub-continuum of dimensions, etc., till we finally get a sub-sub-sub ... continuum, which can be separated by removal of a single element. But such an element no longer constitutes a continuum, since passage in it is excluded; its dimensionality is then zero; so that the continuum it separates in two is obviously one-dimensional.

A sensory continuum would also be given by a succession of weights placed on one’s hand, each only slightly heavier than the preceding one. Again our tactual impressions might yield a sensory continuum. In view of the fact that it is possible to account for the rise of the concept of space, even in the consciousness of a blind man, through the sole means of his tactual impressions, it may be of interest to discuss briefly an illustration of a tactual continuum—that obtained by exploring the surface of our skin by means of pinpricks. If these pinpricks are sufficiently close to one another, it will be impossible for us to differentiate between them and we shall always experience the sensation of one solitary pinprick. We can thus consider the sensory continuum obtained by some definite chain of pinpricks extending, let us say, from our elbow to our hand. This particular chain of sensations exhibits all the characteristics of a one-dimensional sensory continuum, since every one of its elements is indistinguishable from its immediate neighbours and since the removal of one of these elements (pinpricks) would create a hiatus rendering it impossible for us to pass in a continuous way from elbow to hand, along the chain.

But if we should now consider all the possible chains of pinpricks extending from a point on our elbow to a point on our hand, the mere removal of one particular pinprick from one particular chain would be insufficient to interfere with the continuous passage. We might always follow one of the other chains, or even, following the same chain up to the missing element, skip round the latter without sensory continuity being interfered with. In the present case the only way to render this continuous passage impossible would be to remove some continuous chain of pinpricks—say, those circling round our wrist. The sensory skin-continuum would now be divided into two parts, and as the continuum removed was one-dimensional, we should conclude that the [Pg 26] skin continuum as manifesting its sensitivity to tactual stimuli was two-dimensional.

In a similar way, in crude geometry we recognise a wire as one-dimensional, since by removing a point of the wire our finger cannot pass in a continuous way from one extremity to the other. Likewise, a surface is regarded as two-dimensional because only by cutting it along a line is it possible to interrupt the smooth passage of our finger from any one point to any other. The mere removal of a point on the surface would not interfere with the continuous passage as it did in the case of the wire. It is the same for a volume. Only a surface can divide it in two; hence volume is three-dimensional.[3]

When we seek to determine the dimensionality of perceptual space, itself a sensory continuum produced by the superposition of the visual, the tactual and the motive continua, the problem is more difficult. It would be found, however, that perceptual space has three dimensions; but as the necessary explanations would require several chapters we must refer the reader to Poincaré’s profound writings for more ample information.

Summarising, we may say that our belief in the tri-dimensionality of space can be accounted for on the grounds of sensory experience.

Now the subject of our investigations up to the present point has been the dimensionality of sensory continua and the general characteristics of sensory continuity; considerations relating to measurement, or to the extensional equality of two continuous stretches in our continua, have not been entered upon. Neither has any definition of what is meant by a straight line been introduced at this stage. As a result, metrical geometry, which deals with measurements, and projective geometry, which deals with the projections of points, cannot be discussed. The only type of geometry we can consider at this stage is that purely qualitative non-metrical type called Analysis Situs, which deals solely with problems of connectivity.

Connectivity relates to the types of paths of continuous passage from one part of a continuum to another. Manifolds may possess the same dimensionality and yet differ in connectivity. Thus, the connectivity of a sphere differs from that of a torus or doughnut; since the doughnut, in contrast to the sphere, presents a hole or discontinuity through its centre. Yet both sphere and doughnut are two-dimensional surfaces.

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In Analysis Situs, metrical considerations obviously play no part. From a metrical point of view, although a sphere differs in shape from an ellipsoid, yet the connectivity or Analysis Situs of the two surfaces is exactly the same. We may add that there exists an Analysis Situs for every continuous manifold, so that we may conceive of an Analysis Situs of dimensions corresponding to an -dimensional manifold.

Our next task is to determine how a metrics can be established in a sensory continuum. Consider, for example, a continuous stretch of shades of grey passing from white to black. What do we mean exactly when we say that some definite shade is twice as dark as another? Obviously no definite meaning can be assigned to this statement until we have posited some convention permitting us to establish comparisons.

As another instance, take the case of a continuous stream of sounds varying in pitch. What do we mean by saying that some particular musical note is twice as high in pitch as some other, or that the interval between two notes is equal to the interval between two others? If we were to be guided solely by our ear we might assert that as certain musical notes, though differing in pitch, yet appear to present a certain undefinable similarity (the successive octaves), the intervals between these successive similar notes should be considered equal or congruent. We should thus define as equal the extension of notes subtending the successive octaves of a given musical note.

But if now we had learnt to measure the frequencies of vibration of the various musical sounds, a new type of measurement would immediately suggest itself. Starting from any musical note—say, the middle of the piano, we should find that the octave of was vibrating twice as fast, the following three times as fast, and the second octave of our original four times as fast. It would then appear plausible to define equal intervals between musical notes by the differences in their rates of vibration, and we should infer that the distance between and its first octave was equal to the distance between this last note and the following , and equal again to the distance between this and the following superoctave .

We should thus have obtained a definition of equal stretches of sounds which was at variance with our original definition, in which all octave intervals were regarded as equal or congruent. In view of these conflicting results, we could not well escape the conclusion that a sensory continuum of itself offers us no precise means of defining equal stretches, and that whatever definition we might finally select would be a mere matter of choice, an arbitrarily posited convention.

And yet in the case of space, itself a sensory continuum, men have found no difficulty in agreeing on a common system of measurements. As we shall see in the following chapters, the definition of the equality of different stretches of space to which men were unavoidably led was imposed upon them by the behaviour of certain bodies located in space, bodies which were deemed to remain rigid, hence to occupy equal volumes and equal lengths of space wherever they were displaced. For the present, however, we may leave these metrical considerations aside and confine our attention to the general concept of mathematical or geometrical space, which is the subject of study of the pure mathematician.

The concept of a sensory continuum, hence of perceptual space, as presented to us by crude experience, contains certain contradictions and peculiarities which it was necessary to eliminate before it could be subjected to rigorous mathematical treatment. In the first [Pg 28] place, this perceptual space is not homogeneous, and the principle of sufficient reason demands that pure empty conceptual space be homogeneous and isotropic, the same everywhere and the same in all directions.

This homogeneity of space permits us to foresee that it must be unbounded, since a boundary would suggest a discontinuity of structure, defining an inside and an outside, hence a lack of homogeneity. Prior to Riemann’s discoveries it was thought that the absence of a boundary would necessitate the infiniteness of space. To-day we know that this belief is unjustified, for a space can be finite and yet unbounded; and two major varieties of such spaces have been discovered by mathematicians.

But the inherent inconsistencies which endure in all sensory continua constituted a still more important reason for compelling mathematicians to idealise perceptual space. In a sensory continuum, as we have seen, a sensation cannot be distinguished from its immediate successor, the sensation ; neither can be differentiated from . Yet no difficulty is experienced in differentiating from . Expressed mathematically, these facts yield the inconsistent series of relations ; ; . Now, an inconsistency of this sort precludes all mathematical treatment. In mathematics magnitudes cannot be both equal and unequal; they must either be one or the other. The mathematician is therefore compelled to idealise the sensory continuum of experience by assuming that were it not for the crudeness of our senses, the points or sensations , and would all be distinguishable, and that in place of ; and , we should have ; ; .

But it is obvious that a continuum idealised in this way becomes atomic or discrete, since between and , as between and , no intermediary points have been mentioned. In order to re-establish continuity the mathematician is forced to postulate that between any two points and there exist an indefinite number of intermediary points, such that no one of these points has an immediate neighbour. In other words, the continuum is infinitely divisible.

Thus the magnitudes 1 and 2 are not neighbours, since a number of rational fractions separate them. And no two of these fractional numbers are immediate neighbours, since whichever two such numbers we choose to select, we can always discover an indefinite number of other fractional numbers existing between them. Between any two points on a line in our continuum, however close together they may be, we have thus interposed an indefinite number of rational fractions defining points; yet, despite this fact, we have by no means eliminated gaps between the various points along our line.

The Greek mathematician Pythagoras was the first to draw attention to this deficiency after studying certain geometrical constructions. He remarked, for instance, that if we considered a square whose sides were of unit length, the diagonal of the square (as a result of his famous geometrical theorem of the square of the hypothenuse) would be equal [Pg 29] to . Now, is an irrational number and differs from all ordinary fractional or rational numbers. Hence, since all the points of a line would correspond to rational or ordinary fractional numbers, it was obvious that the opposite corner of the square would define a point which did not belong to the diagonal. In other words, the sides of the square meeting at the opposite corner to that whence the diagonal had been drawn, would not intersect the diagonal; and we should be faced with the conclusion that two continuous lines could cross one another in a plane and yet have no point in common.

The only way to remedy this difficulty was to assume that the point corresponding to and in a general way points corresponding to all irrational numbers (such as , and radicals) were after all present on a continuous mathematical line. Accordingly, mathematical continuity along a line was defined by the inclusion of all numbers whether rational or irrational, and a similar procedure was followed for a mathematical continuum of any number of dimensions. In this way mathematicians obtained what is known as the Grand Continuum, or Mathematical Continuum.[4]

Now, it is obvious that although the mathematical continuum is still called a continuum, it differs considerably from the popular conception of a continuum, where every element merges into its neighbour. However this may be, the mathematical continuum, and with it mathematical continuity, are as near an approach to the sensory continuum and to sensory continuity as it is possible for mathematicians to obtain. The sensory continuum itself is barred from mathematical treatment owing to its inherent inconsistencies.

And here an important point must be noted. In a sensory continuum considered as a chain of elements, an understanding of nextness or contiguity, hence an understanding of order, was imposed upon us by judgments of identity in our sensory perceptions. But the same no longer holds in the case of a mathematical aggregate of points, owing to the absence of that merging condition which guided us in the sensory manifolds. Theoretically, we may with equal justification order these points in whatever way we choose, and by varying the order in which we pass from point to point, we should find that the dimensionality of the aggregate varied in consequence.[5]

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Dimensionality is thus a property of order, and order must be imposed before dimensionality can be established.

In practice, the geometrician will retain the type of ordering relation imposed by our sensory experience and will conceive, by abstraction, of a mathematical space of points which will manifest itself as three-dimensional when this ordering relation is adhered to. There is nothing to prevent him, however, from conceiving of mathematical spaces of any whole number of dimensions, either by modifying the ordering relation or again by modelling his mathematical manifold on some -dimensional sensory continuum.

Let us suppose, then, that we have conceived of a three-dimensional mathematical space, obtained as an abstraction from the three-dimensional space of common experience. Just as was the case with our sensory continuum, the mathematical continuum will be amorphous; no intrinsic metrics will be inherent in it, hence it will present us with no definite geometry. The definition of congruence, that is, of the equality of two spatial stretches and more generally of two volumes, and the identity of shapes and sizes, will remain as conventional as before; and it will be only after we have introduced measuring conventions into an otherwise indifferent mathematical space that Metrical Geometry as opposed to Analysis Situs will be possible. However, the discussion of these points will be reserved for the next chapters.

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CHAPTER II THE BIRTH OF METRICAL GEOMETRY

IN the preceding chapter, we mentioned that the concept of continuity was suggested by our sense experiences; and that our understanding of space as a three-dimensional continuum had arisen from a synthesis of our various sense impressions. In the present chapter, we shall be concerned more especially with the geometry of the space continuum; and we shall show how measurement in turn would appear to have arisen from experience, more particularly visual experience.

For this purpose, let us consider the case of a motionless observer, rooted to the earth ever since his birth—a species of man-plant. Viewing the world whence he stood, he would notice that whereas certain visual impressions manifested a property of what he would recognise as “permanence,” others would appear as squirming forms moving across his field of vision. In order to simplify this discussion, we shall omit to take into consideration any awareness of focussing efforts, on the part of our observer, as also any appreciation on his part of the convergence of his eyes. Under the circumstances, his visual perceptions would reveal a world of two dimensions, “up and down, right and left”; the third dimension with which we are all familiar, i.e., “away from and towards,” would be lacking. As a result, the squirming forms passing through his field of vision would be interpreted as betraying a two-dimensional world of changing forms, which would in no wise be connected with the existence of a third dimension. In particular, there would be no reason for him to attribute these changes to the variations in the distances of rigid material objects from his post of observation.

But suppose, now, that concomitant with the activity of his will, our observer were to become aware of certain muscular exertions that accompanied a variation in the shapes of those forms which had hitherto remained fixed and undeformed in his field of vision. In ordinary parlance, our observer would be displacing his post of observation, that is to say, “walking.” He would no longer remain fixed like a tree. As a result of these displacements, which he would end by recognising as such, not only would forms erstwhile fixed in shape appear to vary, but, vice versa, certain forms hitherto squirming could be made to maintain an unchanging appearance. Eventually, he would recognise that in those cases where variations of shape and size could be counteracted by suitable displacements of his post of observation, he had been observing rigid bodies varying their distances along a third dimension with respect to him. In this way, there would arise an understanding both of rigid bodies and of a third dimension. Furthermore, owing to [Pg 33] his being able to repeat his experiments here as there, a realisation of the homogeneity of space would ensue.

We see, then, that the three-dimensional space of experience appears to have arisen as the result of a synthesis of private views, each one of which would be that of an observer unable to move from a certain fixed spot. This synthesis would be extremely complex; unfortunately we have no time to mention the various conditions that would have to be taken into consideration. Suffice it to say that our senses of sight, of touch, of muscular effort, of sound and of smell, to which should also be added the action of the semi-circular canals, would all play a part, dovetailing one into the other. Further considerations would also show that there is nothing mysterious in the fact that these various data should yield concordant results, rather than an incompatible set of conflicting spaces.

All we wish to point out is that by the physical space of experience, we do not merely wish to imply space with the objects located therein, such as it would appear from some definite point of observation. We do not mean the private vision in which rails converge and distant objects appear smaller; we mean a synthesis of these private perspectives, yielding us a common public space.

One private perspective with its converging rails taken by itself and considered without reference to other perspectives could not contain sufficient data to enable us to conceive of three-dimensional space, homogeneous and isotropic.[6] That this synthesis has been arrived at without the conscious effort of reason is granted. Nevertheless, though instinctive, the co-ordination of private experiences and perspectives is of great complexity; and it would not be impossible to conceive of this co-ordination as having followed other lines, just as an aggregate of books may be arranged in alphabetical order, or in order of size or of content, and so forth. With a change in our ordering relation we might have obtained a space of a greater number of dimensions. Undoubtedly, however, when account is taken of the facts of experience, the three-dimensional co-ordination is by far the simplest; hence there is no reason to be surprised at its having imposed itself with such force. These too brief indications must suffice for the present.

And now let us return to the problem of measurement. We have mentioned that in certain cases it would be possible, by changing our post of observation, to counteract the apparent modifications in the visual shapes of bodies that had suffered a displacement in our field of vision.

For instance, if we received a visual impression corresponding to a circle, and if this impression were followed by one corresponding to a triangle, and if it were impossible for us to re-establish the circular impression, we should have to assume that the body had [Pg 34] changed in shape; whereas, if it were possible to re-establish the circular impression by exerting certain efforts (which would finally be interpreted as a displacement of our point of observation), we should end by assuming that we had witnessed a partial rotation of a rigid cone-shaped object.

This discovery of rigid objects in nature is of fundamental importance. Without it, the concept of measurement would probably never have arisen and metrical geometry would have been impossible. But with the discovery of objects which were recognised as rigid, hence as maintaining the same size and shape wherever displaced, it was only natural to appeal to them as standards of spatial measurement.

Measurements conducted in this way would soon have proved that between any two points a certain species of line called the straight line would yield the shortest distance; and this in turn would have suggested the use of straight measuring rods. Henceforth, two straight rods would be considered equal or congruent if, when brought together, their extremities coincided. As for a physical definition of straightness, it could have been arrived at in a number of ways, either by stretching a rope between two points or by appealing to the properties of these rigid bodies themselves. For instance, two rods would be recognised as straight if, after coinciding when placed lengthwise, they continued to coincide when one rod was turned over on itself. Finally, parallelograms would be constructed by forming a quadrilateral with four equal rods, and parallelism would thus have been defined.

Equipped in this way, the first geometricians (those who built the Pyramids, for instance) were able to execute measurements on the earth’s surface and later to study the geometry of solids, or space-geometry. Thanks to their crude measurements, they were in all probability led to establish in an approximate empirical way a number of propositions whose correctness it was reserved for the Greek geometers to demonstrate with mathematical accuracy. Thus there is not the slightest doubt that geometry in its origin was essentially an empirical and physical science, since it reduced to a study of the possible dispositions of objects (recognised as rigid) with respect to one another and to parts of the earth. In fact, the very word geometry proves this point conclusively.

Now, an empirical science is necessarily approximate, and geometry as we know it to-day is an exact science. It professes to teach us that the sum of the three angles of a Euclidean triangle is equal to 180°, and not a fraction more or a fraction less. Obviously no empirical determination could ever lay claim to such absolute certitude. Accordingly, geometry had to be subjected to a profound transformation, and this was accomplished by the Greek mathematicians Thales, Democritus, Pythagoras, and finally Euclid.

The difficulty that Euclid had to face was to succeed in defining exactly what he meant by a straight line and by the equality of two distances in space. So long as geometry was in its empirical stage these definitions were easy enough. All that men had to say was, “Two solid rods will be recognised as straight if after turning one of them [Pg 35] over they still remain in perfect contact,” or again, “The distance between the two extremities of a material rod remains the same by definition wherever we may transport the rod.”

Euclid, however, could not appeal to such approximate empirical definitions; for perfect rigour was his goal. Accordingly he was compelled to resort to indirect methods. By positing a system of axioms and postulates, he endeavoured to state in an accurate way properties which were presented only in an approximate way by the solids of nature. Euclid’s geometry was thus the geometry of perfectly rigid bodies, which, though idealised copies of the bodies commonly regarded as rigid in the world of experience, were yet defined in such a manner as to be untainted by the inaccuracies attendant on all physical measurements.

But this empirical origin of Euclid’s geometrical axioms and postulates was lost sight of, indeed was never even realised. As a result Euclidean geometry was thought to derive its validity from certain self-evident universal truths; it appeared as the only type of consistent geometry of which the mind could conceive. Gauss had certain misgivings on the matter, but did not have the courage to publish his results owing to his fear of the “outcry of the Bœotians.” At any rate, the honour of discovering non-Euclidean geometry fell to Lobatchewski and Bolyai.

To make a long story short, it was found that by varying one of Euclid’s fundamental assumptions, known as the Parallel Postulate, it was possible to construct two other geometrical doctrines, perfectly consistent in every respect, though differing widely from Euclidean geometry. These are known as the non-Euclidean geometries of Lobatchewski and of Riemann.

Euclid’s parallel postulate can be expressed by stating that through a point in a plane it is always possible to trace one and only one straight line parallel to a given straight line lying in the plane. Lobatchewski denied this postulate and assumed that an indefinite number of non-intersecting straight lines could be drawn, and Riemann assumed that none could be drawn.

From this difference in the geometrical premises important variations followed. Thus, whereas in Euclidean geometry the sum of the angles of any triangle is always equal to two right angles, in non-Euclidean geometry the value of this sum varies with the size of the triangles. It is always less than two right angles in Lobatchewski’s, and always greater in Riemann’s. Again, in Euclidean geometry, similar figures of various sizes can exist; in non-Euclidean geometry, this is impossible.

It appeared, then, that the universal absoluteness of truth formerly credited to Euclidean geometry would have to be shared by these two other geometrical doctrines. But truth, when divested of its absoluteness, loses much of its significance, so this co-presence of conflicting universal truths brought the realisation that a geometry was true only in relation to our more or less arbitrary choice of a system of geometrical postulates. From a purely rational point of [Pg 36] view, there was no means of deciding which of the several consistent sets was true. The character of self-evidence which had been formerly credited to the Euclidean axioms was seen to be illusory.

However, there are a number of rather delicate points to be considered, and these we shall now proceed to investigate. Euclid’s parallel postulate and the alternative non-Euclidean postulates reduce to indirect definitions of what we intend to call a straight line in the respective geometries mentioned. If there existed such a universal as absolute straightness, represented, let us say, by a Euclidean straight line, we might claim that Euclidean geometry constituted the true geometry, since its straight line conformed to the ideal of absolute straightness. But this existence of a universal representing absolute straightness is precisely one of the metaphysical cobwebs of which the discovery of non-Euclidean geometry has purged science. To illustrate this point more fully, let us assume that we think we know what is implied by a straight line. Whether we merely imagine a straight line or endeavour to realise one concretely, we are always faced with the same difficulty. For instance, we consider that a rod is straight when it can be turned over and superposed with itself, or else we place our eye at one of its extremities and note that no bumps are apparent. Again, we may realise straightness by stretching a string, viewing a plumb line or the course of a billiard ball. We may also execute measurements with our rigid rods; the straight line between any two points will then be defined by the shortest distance. But whatever method we adopt, it is apparent that our intuitive recognition of straightness in any given case will always be based on physical criteria dealing with the behaviour of light rays and material bodies. We may close our eyes and think of straightness in the abstract as much as we please, but ultimately we should always be imagining physical illustrations.

Suppose, then, that material bodies, including our own human body, were to behave differently when displaced. If corresponding adjustments were to affect the paths of light rays, we should be led to credit rigidity to bodies which from the Euclidean point of view would be squirming when set in motion. As a result, our straight line, that is, the line defined by a stretched rope, our line of sight, the shortest path between two points, would no longer coincide with a Euclidean straight line. From the Euclidean standpoint our straight line would be curved, but from our own point of view it would be the reverse; the Euclidean straight line would now manifest curvature both visually and as a result of measurement. A super-observer called in as umpire would tell us that we were arguing about nothing at all. He would say: “You are both of you justified in regarding as straight that which appears to you visually as such and that which measures out accordingly. It will be to your advantage, therefore, to reserve your definitions of straightness for lines which satisfy these conditions. But you are both of you wrong when you attribute any absolute significance to the concept, for you must realise that your opinions will always be [Pg 37] contingent on the nature of the physical conditions which surround you.”

Incidentally, we are now in a position to understand why the Euclidean axioms appeared self-evident or at least imposed by reason. They represented mathematical abstractions derived from experience, from our experience with the light rays and material bodies among which we live. We shall return to these delicate questions in a subsequent chapter. For the present, let us note that since our judgment of straightness is contingent on the disclosures of experience, even the geometry of the space in which we actually live cannot be decided upon a priori. To a first approximation, to be sure, this geometry appears to be Euclidean; but we cannot prophesy what it may turn out to be when nature is studied with ever-increasing refinement. It was with this idea in view that Gauss, who had mastered in secret the implications of non-Euclidean geometry, undertook triangulations with light rays over a century ago. Furthermore, even were the geometry to be established for one definite region of space, we could not assert that our understanding of straightness, hence of geometry, might not vary from place to place and from time to time; hence we cannot assert with Kant that the propositions of Euclidean geometry possess any universal truth even when restricting ourselves to this particular world in which we live.

Such discussions might have appeared to be merely academic a few years ago; and non-Euclidean geometry, though of vast philosophical interest, might have seemed devoid of any practical importance. But to-day, thanks to Einstein, we have definite reasons for believing that ultra-precise observation of nature has revealed our natural geometry arrived at with solids and light rays to be slightly non-Euclidean and to vary from place to place. So although the non-Euclidean geometers never suspected it (with the exception of Gauss, Riemann and Clifford), our real world happens to be one of the dream-worlds whose possible existence their mathematical genius foresaw.

Now, all these investigations initiated by attempts to prove the correctness of the parallel postulate led mathematicians to further discoveries.

A more thorough study of Euclid’s axioms and postulates proved them to be inadequate for the deduction of Euclid’s geometry. Euclid himself had never been embarrassed by the incompleteness of his basic premises, for the simple reason that although he failed to express the missing postulates explicitly, he appealed to them implicitly in the course of his demonstrations. The great German mathematician Hilbert and others succeeded in filling the gap by stating explicitly a complete system of postulates for Euclidean and non-Euclidean geometries alike. Among the postulates missing in Euclid’s list was the celebrated postulate of Archimedes, according to which, by placing an indefinite number of equal lengths end to end along a line, we should eventually pass any point arbitrarily selected on the line. Hilbert, by denying this postulate, just as Lobatchewski and Riemann had denied Euclid’s parallel postulate, succeeded in constructing a new geometry known [Pg 38] as non-Archimedean. It was perfectly consistent but much stranger than the classical non-Euclidean varieties. Likewise, it was proved possible to posit a system of postulates which would yield Euclidean or non-Euclidean geometries of any number of dimensions; hence, so far as the rational requirements of the mind were concerned, there was no reason to limit geometry to three dimensions.

Incidentally, we see to what rigour of analysis and to what profound introspection the mathematical mind must submit; for the implicit postulates appealed to unconsciously by Euclid are so inconspicuous that it is only owing to the dialectics of modern mathematicians that their presence was finally disclosed and the deficiency remedied by their explicit statement.

From all this rather long discussion on the subject of postulates and axioms we see that the axioms or postulates of geometry are most certainly not imposed upon us a priori in any unique manner. We may vary them in many ways and, as regards real space, our only reason for selecting one system of postulates rather than another (hence one type of geometry in preference to another) is because it happens to be in better agreement with the facts of observation when solid bodies and light rays are taken into consideration. Our choice is thus dictated by motives of a pragmatic nature; and the Kantians were most decidedly in the wrong when they assumed that the axioms of geometry constituted a priori synthetic judgments transcending reason and experience.

[Pg 39]

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CHAPTER III RIEMANN’S DISCOVERIES AND CONGRUENCE

THE procedure of presentation of non-Euclidean geometry which we have followed to this point hinges on the parallel postulate, hence on the definition of the straight line. In many respects, a much deeper method of investigation was that pursued by Riemann, founded on the concept of congruence. By congruence we mean the equality of two distances and more generally of two volumes in space. As we have explained elsewhere, the two methods lead to the same results. Indeed, once a metrical geometry has been defined, whether by the method of postulates or by any other means, a corresponding definition of a straight line and of equal or congruent distances is entailed thereby.

Thus, with Euclidean geometry, congruent lengths at different places are exemplified by the lengths spanned by a material rod transported from one place to another. Congruent or rigid objects having thus been defined, a straight line is given by the axis of rotation of a material body, two of whose points are fixed, or again by the shortest distance between two points measured with our rigid rod.

Nevertheless, although the various methods of presentation are equivalent, it may be of advantage to make the definition of congruence fundamental rather than that of the straight line. Such was the procedure followed by Riemann.

When we revert to experience for an understanding of congruence, we find it exemplified in the rigid bodies of nature, whose geometrical dispositions yield, more or less precisely, the results of pure Euclidean geometry. If we idealise congruence, as thus defined, and express it mathematically, we may say that perfectly rigid bodies are those whose measurements would yield Euclidean results with absolute precision. But though the mathematician has thereby eliminated from his definitions the inaccuracies attendant on physical measurements, his understanding of congruence reduces to a mere idealised copy of the behaviour of special bodies found in nature. While he has thus obtained a possible mathematical definition of congruent bodies (that given in nature), it remains to be seen whether other types of congruence would not also be rationally possible. His aim must therefore be to define congruence mathematically, without appealing to experience.

When, however, we discard the empirical criterion which prompted us to define material bodies as rigid, we find that a unique mathematical definition of rigidity eludes us. For to say that a body remains rigid or congruent with itself during displacement means that the spatial distance between its extremities remains ever the same. But our [Pg 40] only means of disclosing this fact is by measuring the body with an admittedly rigid rod at successive intervals of time and noting the continued identity in our numerical results. Hence it follows that the value of these results would be nullified were we to cast any doubt on the maintenance of the rigidity of our measuring rod. And how could we ever justify its rigidity unless we were to compare it with some other rod regarded as rigid, and so on ad infinitum? From all this it appears that a body can be regarded as rigid only with respect to our measuring rod; and in order to ascribe any significance to rigidity we must first admit that our measuring rod is rigid by definition or by convention. We have no other means of establishing this rigidity.

We should reach the same conclusions were we to compare two lengths and situated in different parts of space. We could not say that the two lengths were equal or congruent in any absolute sense merely because our measuring rod could be made to coincide now with , now with . A definition of this sort would obviously presuppose that our measuring rod had remained undeformed or congruent with itself when displaced from to . It would reduce to testing the congruence of and by presupposing that we knew how to recognise the maintenance of congruence in our rod during displacement. In thus defining congruence in terms of congruence our argument would be circular.

The whole trouble arises from the continuity of space which precludes us from attributing any absolute meaning to the statement that two lengths situated in different parts of space are equal or unequal. There is no absolute significance in stating that there is as much space between and as between and . We cannot compare lengths by counting the number of points that they contain, since there are just as many points between the extremities of an inch as between the extremities of a mile—an infinite number in either case.[7]

In short, we see that in mathematical continua, just as in sensory ones, measurement and comparison of lengths can be considered only as a result of some convention; there is nothing in the continua themselves to suggest any definite metrics or geometry. This is what is meant by saying that space is amorphous and presents us with no means of determining absolute shape and size.

[Pg 41]

Now, it might appear from the preceding discussions that so far as the mathematician is concerned, since a definition of equal distances is purely conventional, any conceptual rod might be chosen for this purpose. We might take a rod which, when moved about, would squirm like a worm, and elevate it to the position of a standard rod to which all other lengths would have to be compared. But this would be too extreme; for although mathematical space possesses no inherent metrics, yet certain requirements are demanded of rods susceptible of being considered as remaining congruent when displaced. It is these requirements, as postulated by Riemann, which we shall now proceed to discuss. Riemann assumed that the necessary requirements would be as follows:

In the first place, he assumed that if we restricted ourselves to infinitesimal volumes of space, congruence would be established by means of Euclidean solids and measuring rods. This postulate is called the postulate of Euclideanism in the infinitesimal.

Furthermore, if our rods remained congruent and we made certain constructions with them (pyramids, etc.), we should be able to transfer these rods and make exactly the same constructions, presenting exactly the same numerical relationships, in any other part of space. This postulate is a mere reflection of our belief in the homogeneity and isotropy of measurements in space, the same everywhere and the same in any direction. When presented in a slightly different form, it is often referred to as the postulate of free mobility.

[Pg 42]

We may also say that if two rods coincide in a certain region of space, they should continue to coincide when separated, then brought together again in any other region.[8]

With these restrictions imposed, Riemann discovered that the Euclidean type of measurement was only one among others. Two other types, namely, the Lobatchewskian and the Riemannian, were also possible.

Sophus Lie considered the same problem, though from a different standpoint. He argued that a rigid body would be such that when one of its points was fixed, any other of its points would describe a surface. When one of its lines was fixed, any other of it's points would describe a curve, and finally, when three points were fixed, the rigid body would be unable to move.

As a result of these investigations it was proved that there existed bodies which, as contrasted with Euclidean solids, would squirm when displaced. Yet, in spite of this fact, these non-Euclidean bodies would present all the mathematical requirements of congruent bodies. These alternative types of bodies may be called Riemannian bodies and Lobatchewskian bodies, respectively.

If we select Euclidean congruence for the purpose of measuring space, non-Euclidean bodies will appear to squirm as they move; if we select non-Euclidean congruence, it will be the Euclidean bodies which appear to vary in form when displaced. It is usual, however, to reserve the term rigid for the Euclidean bodies; but this is only in order to conform to the ordinary understanding of rigidity as derived from experience. If we omit to take into consideration the physical behaviour of material bodies which has forced a certain conception of rigidity upon us, there appears to be no mathematical reason for assigning greater importance to Euclidean measurements than to non-Euclidean ones. It follows that the non-Euclidean bodies are just as much entitled to the appellation “rigid” as are the better-known Euclidean ones. When it comes to deciding which of the types of rigidity represents conservation of absolute shape and size, the problem appears to be entirely meaningless. Accordingly we shall refer to the various types of bodies as Euclideanly or as non-Euclideanly rigid.

Now it is perfectly obvious that when we measure lengths and ratios of lengths with one type of standard rods or another, we shall obtain conflicting numerical ratios and values. Thus, if we fix one extremity of our standard rod and allow it to rotate in all directions, its free extremity will describe points on a spherical surface regardless of the type of rod or geometry with which we are dealing. In a similar way we might obtain a circumference. If, then, we abide by the Euclidean system of measurement, the value of the ratio of the lengths of circumference and diameter would remain ever the same however great the diameter; we should always obtain the same constant number, known as —equal to 3.141592 ..., first calculated by Archimedes. On the other hand, with Riemannian or Lobatchewskian measurements, we should obtain a variable value for this ratio, always smaller than in the case of Riemannian geometry, decreasing as the diameter increased; and always greater than in the event of our having selected [Pg 43] Lobatchewskian measurements. Similar discrepancies would attend the measurement of all other geometrical figures and angles.

And here there is a further point to be considered. Even when we have defined a particular type of measurement, be it Euclidean or non-Euclidean, we have by no means fixed the behaviour of our rods and bodies in any absolute sense; and an understanding of absolute rigidity still escapes us. We can only define Euclidean bodies, for example, as those whose laws of disposition yield Euclidean numerical results. The following illustration will show us that these bodies might all vary in absolute shape when displaced, yet still yield the same Euclidean results.

All we have to do is to consider a transparent plane and a man executing geometrical constructions with Euclidean rods on its surface. If we consider a point-source of light casting the shadows of these rods on some other surface, whether plane or curved, these shadows, considered as measuring rods in turn, will still yield exactly the same Euclidean results, their laws of disposition will remain unchanged, and hence they will be rigid Euclidean rods exactly to the same extent as the original ones. Yet as contrasted with these original rods the shadows would squirm when displaced varying in shape and size, and the Euclidean straight lines would appear curved.

In short we see that absolute shape, straightness, size and rigidity in conceptual mathematical space escape us completely, and the significance of rigidity as portraying the maintenance of an unchanging volume of space, even when considering Euclidean bodies, is indeterminate. In its most extended sense, this is what mathematicians mean by the relativity of space. The sole justification for the general acceptance of the concept of rigidity in the popular sense is due to the presence of material bodies in our universe which we agree to accept as standards. These, incidentally, yield Euclidean results.

Summarising, we see that mathematical space is amorphous; it has no particular metrics, no particular geometry. According to our methods of measurement, we may obtain one geometry or another in the same space. It is often convenient to express all these results by saying that space is Euclidean, Riemannian or Lobatchewskian; but we must be careful to note that space itself has very little to do with the matter.

Let us now examine another aspect of the problem. Until such time as we have fixed our choice on a system of measurements, the amorphous nature of space forbids us to attach any determinate significance to a distance between two points. This, in turn, prevents us from attaching any significance to what may be considered the shortest distance between two points. However, when we have adopted one of the three measuring conventions, the significance of a shortest distance becomes determinate and a definite line joining the two points will be found to embody this shortest distance. Lines of this type are known under the name of geodesics; and in any of the geometries with which we shall be dealing, they play the part taken by the straight line in [Pg 44] Euclidean geometry.[9]

Corresponding to every one of the three congruence definitions we have mentioned, there exists a definite type of straight line or geodesic between any two points. With Euclidean congruence we obtain the Euclidean straight line which satisfies Euclid’s parallel postulate, and vice versa. Again, if we adopt one of the two non-Euclidean types of congruence we are led to the non-Euclidean straight lines or geodesics, those which satisfy the non-Euclidean postulates, and vice versa. In short, we see that the two methods of presenting non-Euclidean geometry, either through the metrical or congruence method or through the parallel-postulate method, are in the main equivalent.

In a more thorough treatment of non-Euclidean geometry other non-metrical methods of presentation are sometimes adhered to. One of these is based on the theory of groups and projective geometry; another is due to the discoveries of Levi-Civita and Weyl and depends on the fundamental concept of an infinitesimal parallel displacement; it opens the way to Weyl’s still more general geometry. Still another mathematical method of exploring space is that of continuous tracks which was investigated by Eisenhart and Veblen. In this book we shall refrain from discussing these more difficult methods, for they present too technical an aspect;[10] but it would be a great mistake to assume that these imaginative flights of the pure mathematicians were of no utility to physical science. Apart from the deep philosophical light which they throw on the entire problem of space, we know from past experience that what was at its inception a pure mathematical dream has more than once become at some later date the image of physical reality. Non-Euclidean geometry, and possibly Weyl’s still stranger geometry, are cases in point.

We have now to consider a number of questions pertaining to location and motion in space. Mathematical space, as we have seen, is -dimensional and amorphous. But in order to bring it into closer contact with the physical space of experience we may assume the necessary postulates of order and contiguity to have been specified. We thus obtain three-dimensional mathematical space. Now mathematical space, in view of its amorphous nature, is essentially relative. The very homogeneity of space, its sameness “here” as “there,” precludes our being able to differentiate one absolute position from another. It will follow that absolute motion, i.e., a variation of absolute position in empty amorphous mathematical space, can have no significance.

Our sole means of giving significance to position and motion will be by selecting some three-dimensional frame of reference as standard. For instance, we might consider three mutually perpendicular axes [Pg 45] meeting at a point. A more concrete illustration would be afforded by considering the two adjoining walls and the floor of our room. Then, after having selected the standard rods which we intend to employ, and also a clock, we should proceed to measure the position and change of position of an object with respect to our room. This would yield us the relative position and motion of the object (relative to our room).

Now, if our choice of a frame of reference, and our methods of measurement, were not arbitrary, but were imposed in some inevitable way by the texture or nature of space, we might still claim that position and motion, though relative to our frame, were yet absolute, since they could always be determined without ambiguity. But this inevitability of a particular frame of reference is precisely what the amorphous nature of mathematical space renders impossible, for there is nothing in mathematical space which permits us to single out any particular frame. Hence no absolute unique determination of position and motion can be contemplated. In fine, we must conclude that in mathematical space position and motion, like distance and shape, are entirely relative.

Thus far we have been considering mathematical space; and mathematical space is but an abstraction from the space of experience. It remains to be seen, therefore, whether our conclusions will apply without modification when we consider the physical space of experience. Here we may state that so far as absolute position is concerned there is nothing to change in our conclusions. Thus, in everyday life, when we speak of returning to the same point in space, we do not really mean the same point in empty space. We mean rather the same point on the earth’s surface; and we do not stop to consider whether the earth has moved through space in the interval. As a matter of fact it would be totally impossible for us to discover exactly what distance the earth had travelled. Granting that we might ascertain the displacement of the earth along its orbit round the sun, all further progress would be arrested by our ignorance of the velocity of the sun through space. We might even measure the velocity of the sun with respect to the stars, but this again would not yield us its velocity through empty space; for we should still remain in complete ignorance of the velocity of the stars as a whole through space. So we see that when we speak of the same point in physical space we are always referring to the same relative position with respect to some observable frame of reference. It is not absolutely necessary that the frame be material, and we might perfectly well conceive of it as being defined by three perpendicular light rays. The essential requirement is that it should be observable, so that all men should agree on the frame they were talking about.

Owing to the important part the earth’s crust occupies in our daily lives, the impossibility of defining the same point at different times without the help of a frame of reference is somewhat obscured. But it is imperative to master this aspect of the elusiveness of space before proceeding to the study of the more complex questions raised by the [Pg 46] theory of relativity.

When, however, we consider the problem of motion, we shall find that in contradistinction to what holds for mathematical space, motion in physical space appears to be absolute. While it is true that from a purely visual standpoint the motion of a body will depend essentially on the frame of reference we select, and will vary when we change our frame from the earth to, say, the moon or the sun, yet owing to the generation of dynamical effects it is possible to decide in an absolute way whether the body is rotating or not. This is due to the fact that these telltale dynamical effects are present just the same regardless of our choice of a frame of reference. It was owing to these empirically discovered dynamical facts that Newton felt compelled to accept the absolute significance of motion, hence of space. We may gather from what has been said that we may not extend lightly to the physical space of experience the conclusions derived from conceptual mathematical space.

We now come to a last point. Suppose that in mathematical space we wish to measure the distance between two objects moving in various ways through our frame. For our measurements to have any significance, we must of course specify that the positions of the two objects correspond to the same instant of time. If, therefore, a change in the relative motion of our frame of reference were assumed to cause a modification in our understanding of the sameness of a time at two different places, the distance between the two objects would vary with the frame in which we were reckoning it. But here we are discussing pure mathematical space, which we may always conceive of as divorced from time. Hence we may assume that the sameness of a time at two different places remains unaffected by any alteration in the choice of our frame.

Classical science believed that the space of experience would also satisfy these conditions. It thought that time was absolute, that its rate of flow was ever the same, and that the concept of the sameness of a time at two different places was an absolute and unambiguous one. This assumption was tantamount to divorcing space from time and regarding them as two essentially different categories. It has been one of the triumphs of the relativity theory to disprove this fundamental thesis, and to show that in the real world of experience a continuum of space by itself is but a shadow. Henceforth the continuum of reality is space-time, a four-dimensional continuum of events, and no longer, as in classical science, a three-dimensional one of points, coupled with a one-dimensional continuum of instants. But inasmuch as the necessity for these new conceptions will be justified at length in future chapters, we shall refrain from discussing them at this stage.

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CHAPTER IV THE PROBLEM OF PHYSICAL SPACE

THUS far we have discussed more especially that abstraction from the space of experience which we called mathematical space; we have seen it to be entirely amorphous. In the present chapter we shall have to consider the space of experience and determine to what extent it differs from conceptual mathematical space. An important difference as regards the problem of motion has already been noted, for we remember that in physical space, motion often manifests itself as absolute. For the present, however, we shall confine our attention to the problem of congruence.

Mathematical space is amorphous; it possesses no intrinsic metrics, and our choice of standards of measurement is largely arbitrary. As a result, absolute shape, size and straightness are meaningless concepts. But in physical space, it is a matter of common knowledge that men have no difficulty in agreeing at least approximately on the sameness of two shapes or of two sizes. They agree that a stone remains undeformed when displaced, hence declare the stone to be rigid, whereas they recognise that an object behaving like a worm is of the squirming variety.

So here there appears to exist an important difference between mathematical space, where no particular definition of congruence is suggested, and physical space, where a definite type seems to impose itself naturally and is accepted unanimously. After all, there is nothing very mysterious about this unanimous agreement; for had men refused to be guided in their definition of congruence by their sense of sight, they would have been led into all manner of difficulties. They would have had to assume that a stone carried in their hands, though appearing unchanged visually, was yet squirming, and, vice versa, that bodies which appeared to squirm were yet rigid. The difficulties in reaching some common understanding of measurement would thus have been hopelessly great. In fact, the definition of congruence, which we have mentioned, imposes itself so irresistibly that it is only since the discovery of non-Euclidean geometry that its absolute validity has been refuted.

Viewing the situation as it now stands, we may say that material rods, visually recognised as rigid, will be taken as norms of measurement in physical space. In other words, it will be assumed that rods which coincide when brought together will continue to remain congruent when transferred, independently of one another, to other regions of space. Of course, the physicist will guard himself as much as possible against local contingent influences, such as variations of pressure and temperature, which might influence the behaviour of his rods. But when all these elementary precautions have been observed, the geometry determined by our rods will automatically become the [Pg 48] geometry of space. This physical definition of congruence may be termed practical congruence, as distinguished from theoretical congruence, which is embodied by the mathematical types we have discussed.

With his standard rigid bodies defined in this way by physical objects, the physicist can perform measurements in the space of his frame and in consequence obtains the numerical results of three-dimensional Euclidean geometry. Inasmuch as the more careful he is to guard against such contingent influences as variations of temperature, the more accurately will his numerical results approximate to those of pure Euclidean geometry, he feels justified in stating that rigid objects behave like rigid Euclidean bodies and that the space of our experience is rigorously Euclidean.

We may also recall that his rigid bodies having been defined, the definition of a straight line as the axis of rotation of a revolving solid, two of whose points are fixed, or as the shortest distance between two points in space, follows immediately; and of course the straight line thus defined satisfies Euclid’s parallel postulate, since it is derived from the behaviour of Euclidean solids.

At this stage, a number of popular exponents of non-Euclidean geometry have fallen into a rather unfortunate error. They have argued that material bodies under perfect conditions must necessarily behave like Euclidean solids, for if they behaved like non-Euclidean bodies when displaced they would squirm and change in shape. As it would be inadmissible to credit any such distorting influence to that void which we call empty space, a non-Euclideanism of material bodies would be debarred on first principles. But these men overlook the fact that Riemann’s and Lobatchewski’s geometries do not in any way refer to bodies which squirm and are distorted in any absolute sense as they move about. The non-Euclidean bodies are merely distorted when contrasted with Euclidean bodies taken as standards; but it would be equally true to state that Euclidean bodies likewise would squirm when displaced if we were to contrast them with non-Euclidean bodies taken as standards. In any case, both Euclidean and non-Euclidean bodies behave in a homogeneous way throughout space.[11] By this we mean that wherever they might be situated in empty space, measurements computed with them would yield the same numerical results. As for Euclidean congruence and Euclidean rigidity, it is by no means more representative of real rigidity than are the non-Euclidean varieties. There is therefore no reason to appeal to a distorting effect of empty space in order to account for a possible non-Euclidean behaviour of our material solids when displaced from point to point. Non-Euclideanism may or may not exist in real space, but this is a point for physical measurement and not for philosophy or mathematics to decide.

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All we can say is that the principle of sufficient reason compels us to credit empty space with a sameness throughout, and that our measuring rods and material bodies must also behave homogeneously and isotropically, as indeed they do in the three geometries discussed. Only if measurements undertaken with our rods in different parts of space yielded variable non-homogeneous numerical results should we have to assume that space was not really empty and that our rods were subject to local influences.

At any rate, the early non-Euclidean geometers, realising that space as a result of measurement might turn out to be non-Euclidean, busied themselves with devising means of settling the question once and for all. Now, as a result of their measurements with material rods there was no doubt that space was very approximately Euclidean. But here it must be realised that the two non-Euclidean geometries as opposed to the Euclidean variety are not unique. We may conceive of various intensities of non-Euclideanism of both types, merging by insensible gradations into Euclidean geometry. It was therefore still an open question whether space, in spite of its apparent Euclidean characteristics, might not betray a slight trace of non-Euclideanism. A simple illustration will make this point clearer.

We saw that in Euclidean geometry the ratio of the length of a circumference to its diameter was always the same number, . In Riemann’s geometry this number was always smaller than , and decreased progressively from the value to the value zero as the diameter of the circle increased. But there was nothing definite about this rate of decrease; it might be very rapid, just as it might be exceedingly slight. We must conceive, therefore, of varying intensities of non-Euclideanism, or of departure from Euclideanism. Hence, if the non-Euclideanism of real space were exceedingly slight, it might require measurements extending over a circumference of gigantic proportions, reaching as far as the stars in order to detect it; and measurements conducted in restricted areas could not be considered conclusive. The only means of disclosing slight traces of non-Euclideanism would therefore be obtained by having recourse to measurements conducted over cosmic distances.

Of course, in an attempt of this sort, measurements with material rods were out of the question and it was necessary to appeal to other methods of exploration. These were obtained by taking advantage of the propagation of light rays in empty space. It was argued that the principle of sufficient reason precluded light rays from deviating to the right or to the left from their course along the straightest path through empty space; this belief was also in accord with the important physical principle of Least Action, as deduced from the laws of mechanics.[12] Accordingly, light rays would follow geodesics in empty space; and, as we have seen, a knowledge of the geodesics or straight lines reveals as much about the geometry of space as congruence itself.

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Gauss appears to have been the first to undertake space explorations of this sort, when he conducted experiments on light rays transmitted from one mountain top to another. But his observations were too crude and executed over too small an area to detect any trace of non-Euclideanism. Lobatchewski suggested astronomical observations conducted on the course of rays of starlight through interstellar space. For instance, if two light rays emitted from a very distant star and striking the earth at two different points of its orbit appeared to manifest converging directions, we should know that space was Riemannian. If the two rays appeared to diverge from a common point, space would be Lobatchewskian; and, finally, if for very distant stars these two directions appeared identical, space would be truly Euclidean. Yet the most refined astronomical measurements of stellar parallaxes failed to reveal the slightest trace of non-Euclideanism. Hence it was assumed that if any trace of non-Euclideanism was present in real space it was without doubt exceedingly slight, so that for all practical purposes the geometry of space might be regarded as Euclidean. Such were the results obtained by a physical exploration of space.

From all this we see that the physicist, basing his exploration of space on empirical methods, is perfectly justified in stating that its geometry can be determined, that a true definition of congruence can be arrived at, and that the equality of two lengths and of two spatial configurations has a definite significance in nature.

And yet, when we submit all these various examples to a critical analysis, we cannot help but see that this determination of the geometry of space is essentially physical and is, therefore, contingent on the behaviour of material objects and of rays of light. Had the behaviour of material bodies when displaced been regulated by other physical laws, had rays of light followed different courses, the geometry we should have attributed to space might have been entirely different. And we may well wonder what the behaviour of physical objects should have to do with the geometry of space. We shall return to this aspect of the problem later.

Also, it has sometimes been argued that our recognition of shape and size must possess a much deeper significance and cannot be attributed merely to the laws of behaviour of material objects and of rays of light. For instance, it is pointed out that even a child who knows nothing of measurement judges, on simple visual inspection, that a coin (when viewed from a perpendicular direction) is round and an egg oval. He does not feel it necessary to verify this fact by applying a ruler. However, regardless of what opinions we may eventually defend on the subject of a geometry intrinsic to physical space, it can scarcely be held that this last argument of the critic proves his point in the slightest degree.

It may be instructive to consider this illustration in greater detail. In the first place, we must realise that all we are in a position to appreciate when merely viewing an object, is its image on our retina. Indeed, were we to interpose a microscope or a deforming lens between object and eye, the object would suffer no change; but its image cast on our retina would, of course, be modified. As a result, our judgment [Pg 51] of its shape and size would vary. Hence, when we decide unhesitatingly that the coin is round, the only inference to be drawn is that the coin’s image on the retina is judged by the brain (or the mind, or whatever we decide to call it) to be circular. We do not wish to intrude on the ground of the eye-specialist or physiologist, who is better equipped than we are to proceed farther in this analysis, but we may point out that the problem which we are now considering is of an entirely different nature from the one whence we started. There we were considering whether shape in empty space was absolute; here we are considering whether an image cast on our retina should impress itself upon our recognition with any definite shape.

The difference amounts to just this: In mathematical space, and even in physical space, absolute measurement seemed to elude us, since in view of the continuity of space it appeared impossible to proceed with an enumeration of points. But in the case of the retina, its surface is no longer homogeneous; it possesses a heterogeneous structure like all tissues, probably a discrete one forming a pattern. In all such cases a definite metrics suggests itself naturally, just as on a net, in the absence of a ruler, we would compare lengths instinctively by counting the holes separating our points.

Whether or not, in the case of the retina, our appreciation of the coin’s roundness comes from some unconscious counting process is a problem for the specialist to decide; but at all events, when we consider that the retina and the eye are Euclidean bodies just like our rulers, the concordance between our computations of shape and size, as determined by rods, and our direct visual appreciation of shape does not appear to present much of a mystery.

But there is still a further point to be considered. Were we to have a direct intuition of congruence and of absolute shape or length, our measurements with rods would have to be adjusted so as to conform to this intuition. The introduction of rods would thus be contemplated merely as an adjunct, in order to obtain greater definiteness. But it so happens that such is not the case.

Our intuitive visual appreciations yield results which differ from results obtained with rods, not merely accidentally as a result of the imperfection of human observation, but systematically.

A coin that measures out as round will appear flattened to the eye.[13] This phenomenon is illustrated by the well-known optical illusion wherein two rigid rulers which coincide when placed side by side, appear of unequal magnitude when placed horizontally and vertically, respectively. It is a well-known fact that the vertical appears to be longer than the horizontal. For this reason, vertical stripes on a cloth cause the wearer to appear thinner and taller, whereas horizontal stripes produce the reverse effect. Now the mere fact that we have agreed to accept such discrepancies as due to optical illusions rather [Pg 52] than to the untrustworthiness of our rods proves that we deliberately reject our intuitive judgment of shape and size in favour of more sophisticated rules of measurement. In other words, we have abandoned direct intuition for physical determinations, hence for convenient but conventional standards.

In short, we may state that we possess no direct intuition of shape and length, and that what little we appear to have is traceable in the final analysis to the properties of material bodies and light rays.

Let us consider a last example relating to distance. We realise, without resorting to measurement, that the distance across the street is less than the entire length of the street. A number of different reasons conspire to account for this conviction. First, the muscular sensations which accompany the convergence of the eyes and the focussing of images on the retina would in themselves suggest the origin of our appreciation of distance. In addition, we know from experience that it requires a greater effort and a longer time to travel along the entire length of the street than to step across from one sidewalk to the other. And if we enquired why it took a shorter time to cross the street, we should find that it was due to the fact that we advanced with definite steps.

Although in mathematical space the two distances would be equal or unequal according to our measuring conventions, in practical life we have unwittingly posited our measuring convention by walking. Our successive steps henceforth define in our estimation congruent distances; and under the circumstances distances become measurable in terms of these steps. But these steps are themselves controlled by our human frame; hence all we have done has been to measure space in terms of our legs. That measurement originated in this way is indicated clearly by such words as foot and cubit, or again by the definition of a yard as expressed by the distance between the tip of a certain king’s nose and the extremities of his fingers. And it is because our human limbs manifest the same type of congruence as the material bodies around us that this type of measurement once again imposes itself so strongly upon us. Indeed, measurements as computed with human limbs appear to be instinctive and are exemplified in children who, having grown, are surprised to find the rooms and buildings they have not seen for some years, appear considerably smaller. Instinctively, the child is measuring size in terms of his own height.

Thus, in whatever way we examine the matter, there appears to be nothing mysterious in our natural belief in the absoluteness of shape and size or in a definite geometry pertaining to space. None of the examples mentioned thus far entitle us to maintain that physical space manifests a definite metrics and that congruence is other than conventional.

If we consider the problem in its present state, we see that it is the physical behaviour of material bodies and light rays which is in the final analysis responsible for our natural belief in absolute shape. But this realisation brings with it the assurance that space itself has eluded us entirely in our discussions. Such was indeed Poincaré’s stand. He maintained that though for purposes of convenience it [Pg 53] was only natural for us to measure space as we do, yet if needs be we could disregard the behaviour of material bodies entirely, adopt non-Euclidean standards of measurement, and proceed as before.

In spite of this change we could construct exactly the same engineering works, in fact rewrite the whole of physics. Needless to say, everything would be extremely complex, all our known laws would be disfigured, and hypotheses ad hoc would have to be introduced. But when all is said and done, the task would be theoretically possible; so that if we disregard the criteria of convenience and simplicity, there is nothing to choose between the various types of measurements, any more than between the metric system and the British units.

As a further illustration of the elusiveness of absolute shape and size, Poincaré asks us to conceive of a hollow spherical volume placed anywhere in space, and to assume that the temperature in the sphere decreases progressively from the centre, becoming absolute zero at the surface. He assumes that this hollow sphere is peopled by imaginary beings whose bodies expand and contract with the temperature and that all material bodies in the sphere behave in a similar manner. If we should supplement these suppositions by assuming that the refractive index of the medium in the sphere’s interior varies in a certain definite way, the rays of light in this hypothetical world would describe circles.

This closed universe would of course appear infinite to its inhabitants, since as they proceeded from the centre to the surface their bodies would grow smaller, their steps shorter, so that it would be impossible for them to reach its boundary however long they walked. The geometricians of this imaginary world would feel justified in proceeding exactly as we have done ourselves. They would define as remaining congruent when displaced, hence as rigid, those bodies which appeared to them to remain the same wherever they carried them. Owing to the paths devised for the light rays and to the sameness in the reduction of the sizes of all objects as the centre was left behind, the expanding and contracting bodies of this universe would present all the characteristics of rigidity. On conducting measurements with their rigid rods the hypothetical beings would obtain non-Euclidean results, their entire world would appear to them as non-Euclidean, and non-Euclidean geometry would be as inevitable to them as Euclidean geometry is to the average layman. Some Kant among the hypothetical beings would surely arise and explain that non-Euclidean space was the a priori form of pure sensibility, transcending reason and experience. Then eventually some great mathematician would come along, sweep all those cobwebs aside, and prove that there existed other perfectly consistent types of geometries and that the ingrained preference of his fellow citizens for non-Euclidean geometry was due to the dictates of common experience and constituted by no means the a priori form of pure sensibility.

It is not merely in its philosophical aspect that Poincaré’s illustration is interesting. The major point is the following: The hypothetical beings would be just as much entitled to assert that [Pg 54] space was non-Euclidean as we are to assert that it is Euclidean. It is true that if we could look into their world we should say that they had a wrong understanding of measurement, and were totally in error when they assumed that their bodies were rigid, since we could see them getting smaller and smaller as they neared the surface. But we must not forget that the imaginary beings, in turn, could they but view our Euclidean bodies, would return the compliment and accuse us of having a wrong understanding of rigidity and measurement.

From all this it follows that by a mere variation in physical conditions the same space would be considered non-Euclidean or Euclidean. Obviously, by reason of this contradiction, space itself can have nothing to do with the problem; the type of space which physicists are discussing reduces therefore to a relational synthesis of physical results. Space itself remains amorphous.

Poincaré develops analogous arguments when he discusses the parallax observations conducted on the rays of starlight. Euclidean geometry, for instance, regarded purely as a system of measurement, is from a mathematical point of view the simplest type of geometry for the same reason that a monosyllable is simpler than a polysyllable. It is therefore obviously to our interest to retain it if possible. Of course if, as in the hypothetical world discussed previously, material bodies behaved like non-Euclidean solids and if light rays followed appropriate courses, we should have to abandon Euclidean geometry for reasons of practical convenience. But since, in the world we live in, our habitual solids behave to a high order of approximation as do Euclidean solids, our preference for Euclidean geometry seems perfectly legitimate even from the standpoint of physics.

Suppose now that the parallaxes of the very distant stars turned out to be negative: would Euclidean geometry and Euclidean space have to be abandoned? As Poincaré points out, this would by no means be necessary or even advisable. It is true that if we assumed, as is the custom, that rays of starlight follow geodesics through space, negative parallaxes would imply a trace of Riemannianism in space; but the primary point to decide is, “How do we know that rays of light follow geodesics?” Obviously this is capable neither of proof nor of disproof. An empirical proof that such a contention was correct or incorrect would be possible only were we to know beforehand how the geodesics of space were situated, for then we could determine by observation whether rays of light followed them or not. But how could we establish the way the geodesics lie unless we were already apprised of this geometry which we now proposed to determine? Obviously, our procedure would be circular. Can we at least assume that rays of light must inevitably follow geodesics? Would any other assumption be impossible? Certainly not. A denial of the assumption would modify our understanding of optical phenomena; but what if it did? We could always get out of the difficulty by varying the laws of optical transmission, and still retain Euclidean geometry. In other words, the geometry the [Pg 55] physicist credits to space is contingent on his acceptance of a number of physical laws; and by varying these laws in an appropriate way he could still account for observed facts and credit corresponding types of geometry to space. Since all these various systems of physical laws would account for the facts of experience, how can we ever hope to decide which one of these systems corresponds to reality? And under the circumstances, what use is there in discussing the real geometry of space? All we can discuss is expediency.

In other words, Poincaré, by divorcing space from its material content, geometry from physics, places space and its geometry beyond the control of experiment; so that there is really nothing left for the physicist to argue about.

As a matter of fact, the entire conflict is more apparent than real; it centres round the meaning we wish to ascribe to the word reality, whether we conceive of reality in the pragmatic scientific sense or in the metaphysical sense as embodying “true being.” For the theoretical physicist, “reality” means that hypothesis which will permit him to co-ordinate natural phenomena with the maximum of simplicity. He knows of no other test for reality and would probably evince very little interest in the unattainable reality of the metaphysician. If, therefore, a co-ordination of the facts of experience presents greater simplicity when he assumes space to be Euclidean or non-Euclidean, then space is Euclidean or non-Euclidean in spite of the fact that phenomena might just as well have been co-ordinated (though in a less simple way) had some other hypothesis been selected.

Riemann expresses the selective rôle of the criterion of simplicity when he writes:

“Nevertheless it remains conceivable that the measure relations of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry], and, in fact, we should have to assume that they are not if, by doing so, we should ever be enabled to explain phenomena in a more simple way.”[14]

So much for the scientific understanding of the word. But if, on the other hand, when discussing “reality” we are referring to the reality of the metaphysicians, as happens to be the case with Poincaré, then it appears quite impossible to dispute his stand; at least it is scarcely credible that any scientist would feel inclined to do so. For, assuming even that such a thing as metaphysical reality has any meaning at all, why should it be connected with simplicity of co-ordination? At any rate, Einstein himself, whose entire theory centres round a definite geometry being ascribed to the spatio-temporal background, writes: “Sub specie œternitatis, Poincaré in my opinion is right.”

[Pg 56]

If any criticism is to be directed against Poincaré’s stand, it should be on the ground that he showed himself a poor prophet when he claimed that it would always be simpler to retain Euclidean geometry. Einstein has proved the contrary. At all events, in what is to follow, we will concern ourselves solely with the real space of the physicist, that is, with the space to which he is led when he seeks to co-ordinate phenomena with the maximum of simplicity. With this understanding of space in our minds, a first reason for rejecting the concept of an amorphous space arises when we find that a large number of different methods of investigation all point to the same definite metrics for space. Thus, the various material bodies we encounter are by no means identical in nature; some are light, others are heavy, and their chemical and molecular constitutions are certainly not the same. And yet in every case, whether our rods be of wood, of stone, or of steel, we obtain the same Euclidean results provided we operate as far as possible under the same conditions of temperature and pressure. In other words, there appears to be a sameness in our determinations of congruence regardless of the material bodies to which we appeal.

This uniqueness of the geometry of space is still further exemplified in the following example: Here are two totally different methods of exploring space, one with material rods giving us a physical definition of congruence, and one with light-ray triangulations giving us a physical definition of geodesics. In either case we are led to the same Euclidean geometry, and this concordance appears rather strange, for we might have expected that if the geometry we credited to space were irrelevant to space, the type of geometry obtained would have varied according to the physical exploration method considered. Besides, if space were amorphous, hence possessed no geodesics, it would be inconceivable that a free body or a light pulse should know how and where to move. The very definiteness and Euclidean straightness of the paths of free bodies and light rays, when referred to a certain frame of reference, would seem to indicate that space had a structure and was not amorphous.

To be sure, in view of modern discoveries there is nothing very strange in the fact that the courses of free bodies should coincide with the paths of light waves, since light has been proved to possess momentum just as matter does. But even so, it appears strange that the courses defined by moving bodies should yield the same geometry as measurements conducted with bodies at rest.

Then again, there are the dynamical properties of space, which we cannot afford to neglect. If physical space were amorphous, all paths through space should be equivalent, and yet centrifugal force and forces of inertia manifest themselves for certain paths and motions and not for others. Whence could these forces arise if not from the structure of space itself? Such was indeed Newton’s contention.

In view of all these occurrences, difficult to account for if we believe in the amorphous nature of space, unless we appeal to some miraculous pre-established harmony, it appears as though space must be credited with a definite structure or metrics which, in the [Pg 57] light of experiment, turns out to be Euclidean, at least to a first approximation. Expressed in a different way, real space appears to be permeated by an invisible field, the Metrical Field, endowing it with a metrics or structure.

Now, in view of the fact that the mathematical requirements of that void we call empty space preclude it from having a metrical field (i.e., a structure or a metrics per se), the simplest way out of the difficulty is to assume that real space is not truly empty, but is filled with some mysterious physical medium, which we may call the “ether,”[15] and that it is this physical medium and not space itself which possesses a Euclidean structure. Henceforth it will be this ether structure which will be responsible for the apparent metrics or metrical field of space, which will cause material bodies to settle into definite shapes, and which will regulate the courses of light rays and free bodies far from matter.

For the physicist, however, this ether will be inseparable from real space; so that, to all intents and purposes, when the physicist discusses real space he will be referring to space together with its ether content. Were this ether structure to vanish, space would become amorphous, bodies would not know what shape to take, and light rays would not know where and how to move.

At this stage we must mention the premonitions of Riemann on the subject of the metrical field of real space. Riemann did not attribute this structure of space to the presence of some invisible medium, the ether, possessing a structure of its own. According to him the origin of the metrical field should be sought elsewhere. He felt that the metrical field of space should be compared to a magnetic or an electric field pervading space. And just as a magnetic field exists in the space surrounding a magnet, Riemann searched for the physical cause of the metrical field. With characteristic boldness, he found it in the matter of the universe; the metrical field thus became a species of material field. If Riemann’s ideas are accepted we can understand how a redistribution of the star matter in the universe, altering as it would the lay of the metrical field, would produce deformations in the shape of a given body and variations in the paths of light rays. As Weyl tells us, a spherical ball of clay compressed into any other form might again be made to appear spherical were all the matter in the universe to be redistributed in a suitable way.

It would also follow that were all the matter in the universe to be annihilated, and as a result the metrical field to vanish, space (assuming that any physical meaning were left the term) would become completely amorphous, just like mathematical space; light rays would not know where to move, all geodesics having disappeared; and were one lone material body to be introduced into otherwise empty space, it would not know what shape to take. Without the metrical field, physical space would be unthinkable.

Still further important consequences follow from this matter-moulding [Pg 58] hypothesis of Riemann. Prior to these views, the principle of sufficient reason appeared to imply that physical space would always turn out to be homogeneous—the same in all places. This does not necessarily mean Euclidean, for, as we know, Riemann’s and Lobatchewski’s geometries also correspond to homogeneous spaces; but all varying degrees of non-Euclideanism from place to place were thought to be excluded a priori. Of the vast realm of possible types of geometries or spaces discovered by Riemann, only the homogeneous types survived; a situation which Weyl finds appropriately expressed by the classical line: “Parturiunt montes, nascetur ridiculus mus.”

But with the new views advocated by Riemann the situation changes entirely; for now the texture, structure or geometry of space is defined by the metrical field, itself produced by the distribution of matter. Any non-homogeneous distribution of matter would then entail a variable structure or geometry for space from place to place.

Now the question arises, What is the nature of this metrical field? Is it merely a name we are giving to the structure of space caused by matter? This view appears impossible, for space of itself, being a mere void, is not amenable to structure. Is the metrical field a direct emanation from matter, a rarefied form of matter? Or again is it a reality of a category differing from matter (call it the ether), which in the absence of matter would be amorphous, and which could only be forced into a structure by the influences due to matter and transmitted from place to place through the ethereal substance? In this case we should again be led to the view that what we commonly call the structure or geometry of real space reduces to the structure of the matter-moulded ether-filling space.

Riemann’s exceedingly speculative ideas on the subject of the metrical field were practically ignored in his day, save by the English mathematician Clifford, who translated Riemann’s works, prefacing them to his own discovery of the non-Euclidean Clifford space. Clifford realised the potential importance of the new ideas and suggested that matter itself might be accounted for in terms of these local variations of the non-Euclideanism of space, thus inverting in a certain sense Riemann’s ideas. But in Clifford’s day this belief was mathematically untenable. Furthermore, the physical exploration of space, even in the interior of liquids, seemed to yield unvarying Euclideanism. And here the remarkable irony of the whole situation must be noted. Although experimenters had utilised the most refined apparatus for detecting a possible non-Euclideanism of space and had failed in their efforts, it was reserved for the theoretical investigator Einstein, by a stupendous effort of rational thought, based on a few flimsy empirical clues, to unravel the mystery and to lead Riemann’s ideas to victory.[16] Nor were Clifford’s hopes disappointed, for the varying non-Euclideanism of the continuum was to reveal the mysterious secret of gravitation, and perhaps also of matter, motion and electricity.

[Pg 59]

Before solving the problem, however, Einstein had been led to recognise that space of itself was not fundamental. The fundamental continuum whose non-Euclideanism was to be investigated was therefore not one of space but one of Space-Time, a four-dimensional amalgamation of space and time possessing a four-dimensional metrical field governed by the matter distribution. Einstein accordingly applied Riemann’s ideas to space-time instead of to space, and attempted to explore the geometry of space-time by a purely rational co-ordination of known empirical facts. He discovered that the moment we substitute space-time for space (and not otherwise), and assume that free bodies and rays of light follow geodesics no longer in space but in space-time, the long-sought-for local variations in geometry become apparent. They are all around us, in our immediate vicinity; and yet we had never realised it. We had called their effects gravitational effects, ascribing them to forces foreign to the geometry of the extension, and never suspecting that they were the result of those very local variations in the geometry for which our search had ever been vain even though we had extended our observations to the depths of the universe. Indeed, it may be said that the theory of relativity is the theory of the space-time metrical field.

While we are on the subject, we may mention that this mysterious metrical field which moulds both space and time appears to be conditioned entirely by the matter of the universe. Such at least are the conclusions which the existence of Einstein’s cylindrical universe would suggest. The problem is, however, still extremely obscure. It is still possible to believe with Eddington and de Sitter that the metrical field or space-time ether-structure might subsist in the absence of matter, contrary to the views of Riemann.

Finally let us consider how these new views will affect the problem of absolute shape and size. We may say that space-time possesses a definite geometry but that this geometry is subject to local variations both in space and in time, as the masses of the universe modify their positions. Yet, as we have mentioned on several occasions, a geometry, even when fully determined and unchanging, does not imply absolute shape and size. It merely regulates the relationships or defines the laws of the mutual dispositions of bodies, and these laws may remain unmodified even though the absolute shape and size of the bodies vary. Restricting ourselves to the problem of absolute size, we shall find that with the finite universe the metrical field yields a universal standard of length given by the radius of the finite universe. As referred to this natural gauge, an object presents a definite size, but obviously there is nothing absolute about the gauge itself. Not only will its magnitude be governed by the total amount of matter in the universe, an amount determined presumably by accident, but furthermore there would be no means of determining the magnitude of this gauge otherwise than by adopting some other gauge, and so on ad infinitum.

[Pg 60]

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CHAPTER V AN ALTERNATIVE VIEW OF THE NON-EUCLIDEAN GEOMETRIES

THE type of geometry we obtain is dependent, as we know, on our definition of congruence. If we define as congruent displacements the displacements of those bodies which in ordinary life we consider rigid and undeformable, we obtain Euclidean geometry. If, on the other hand, we define as congruent the displacements of those bodies which in ordinary life we should regard as of changeable shape, we generally obtain some non-Euclidean type of geometry.

In order to simplify what is to follow, we shall first consider the particular case of two-dimensional geometry. Consider, then, a plane surface. If on this plane surface we effect measurements with Euclidean rods, we obtain Euclidean results; if we employ rods which according to the Euclidean point of view squirm in an appropriate way when displaced, we obtain the geometry of Riemann or of Lobatchewski, as the case may be. It is to be noted that the plane is the same in all three cases, and yet the geometry we obtain on its surface may be Euclidean or non-Euclidean. It is not the plane itself but our methods of measurement conducted thereon which have changed.

Now it may be mentioned that there exist a number of alternative ways of representing both Euclidean and non-Euclidean geometries. Which way is to be preferred depends largely on the scope of our investigations. One of these alternative procedures has often been appealed to in popular writings (by Helmholtz in particular) because it enables us to visualise the sequence of theorems involved, without abandoning thereby our habitual Euclidean representations.

Thus three-dimensional Euclidean space is taken as a starting point. It is then proposed to show how, in this three-dimensional Euclidean space, a two-dimensional non-Euclidean geometry can arise. The fundamental space which we are here postulating being Euclidean, distances in this space must be computed with rigid Euclidean rods. Then it is shown that if we apply tiny Euclidean rods to the surface of a sphere and conduct measurements on this surface, we shall obtain a series of numerical results which are representative of Riemann’s geometry. The chief advantage of this method of presentation is that it allows us to foresee at a rapid glance that Riemann’s geometry must be consistent,[17] since in the present case it reduces to Euclidean geometry on a sphere, which is a particular case of Euclidean geometry in three-dimensional space; and Euclidean geometry is known to be [Pg 61] consistent. But aside from this advantage the procedure is to be avoided, for it tends to obscure the philosophical importance of non-Euclidean geometry.

In the first place it is not always possible to follow this method. Consider, for example, the case of Lobatchewski’s geometry. Just as Riemann’s geometry was that of the spherical surface, so Lobatchewski’s geometry turns out to be that of a peculiar saddle-shaped surface called the pseudosphere, as was proved by Beltrami. Hilbert, however, has shown that there cannot exist a surface free from singularities which would represent the total spread of Lobatchewski’s plane geometry; hence here is a first reason for being on our guard against a number of unsuspected difficulties.

But this is not all. If we adopt the method of representing two-dimensional non-Euclidean geometry as the geometry obtained by taking Euclidean measurements on a curved surface, how are we to conceive of three-dimensional Riemannian geometry? Obviously we must start with a four-dimensional Euclidean space in which a three-dimensional spherical surface is embedded. But as a four-dimensional Euclidean space transcends our immediate experience and is in the nature of a mathematical fiction, we should be led to the erroneous conclusion that three-dimensional non-Euclidean geometry must likewise be a purely conceptual construction having nothing in common with measurements which might be conducted with material rods.

We should be led into still greater difficulties if we wished to represent a three-dimensional non-Euclidean space possessing various degrees of non-Euclideanism from place to place. Reasoning by analogy, we should be tempted to say that just as an irregularly curved surface embedded in three-dimensional Euclidean space yielded a two-dimensional geometry of varying degrees of non-Euclideanism, so now all we should have to do would be to conceive of an irregularly curved three-dimensional surface embedded in a four-dimensional Euclidean space. But the analogy would be deceptive; for calculation shows that it would be impossible to represent an arbitrarily curved three-dimensional surface in a four-dimensional Euclidean space. In the general case we should have to situate our variously bumped three-dimensional surface in a six-dimensional Euclidean space. In the same way, a four-dimensional non-Euclidean space of variable curvature could be represented only in a ten-dimensional Euclidean space; and so on.[18]

Now in view of these difficulties, in view of the fact that to represent a non-Euclidean space of three dimensions in terms of Euclidean space, we may be compelled to appeal to a Euclidean space of six dimensions, it is certainly exalting Euclidean space unduly to regard it as The Space. This is especially obvious when we remember that we could have conceived of our three-dimensional [Pg 62] non-Euclidean space directly, as a result of measurements with squirming rods, without any reference to Euclidean geometry, and without ever having had to introduce any greater number of dimensions.

At this juncture the beginner is often inclined to argue as follows: “You say that non-Euclidean geometry of two dimensions is the geometry obtained by executing Euclidean measurements on a suitably curved surface. But your concept of curvature is meaningless unless we conceive it as contrasted with some pre-existing standard of straightness. Does not your presentation prove, therefore, that Euclidean space as representative of flatness or straightness is logically antecedent to non-Euclidean space, which connotes curvature?”

This argument is radically incorrect, and arises from too loose an understanding of non-Euclidean geometry. In the first place, to assume that the concept of curvature presupposes the concept of straightness is no more inevitable than to assume that the concept of straightness presupposes that of curvature. For if it is true to say that that which is curved is that which is not straight, it is equally true to say that that which is straight is that which is not curved; so that in the absence of curvature, straightness would in turn be meaningless. The question of deciding, for example, whether a circle or a straight line is the more fundamental is strictly a matter of opinion. The Greeks held circular motion to be the noblest of all motions; and between the designations “noblest” and “most fundamental,” the distinction is exceedingly slight. And even this is not all, for thus far we have been arguing as though Euclidean space were truly flat, and non-Euclidean space truly curved. But we must remember that this curvature of which we speak is not to be construed as representing anything absolute; it arises solely from the particular type of representation we have agreed to select.

Thus, in the presentation of non-Euclidean geometry which we have just been discussing, we started from three-dimensional Euclidean space as a basis and considered curved surfaces embedded in this space. This form of presentation was equivalent, as we know, to stating that Euclidean measurements must be adhered to on the curved surface. But we might have proceeded otherwise. We might have started with three-dimensional non-Euclidean space and considered the geometry of curved surfaces (of a sphere, for example) embedded in this space. This would have been equivalent to applying non-Euclidean rods on our curved surface.

Under suitable conditions the geometry of our curved surface (or sphere) would then become Euclidean. Euclideanism itself would thus be linked with curvature, while non-Euclideanism, which would here be the type of geometry obtained on the plane embedded in our three-dimensional non-Euclidean space, would accordingly be linked with flatness. In short, we must not allow the word “curvature” to mislead us into forming a false impression about non-Euclidean geometry. In many respects Riemann’s choice of the word “curvature” has proved unfortunate and the more scientific appellation non-Euclideanism [Pg 63] should be adhered to.[19]

There is still another point which we must mention. It might be argued that in Riemann’s geometry of two dimensions, that is to say, in the geometry of the spherical surface when Euclidean measurements are adhered to, a straight line on the sphere, being a great circle, constitutes a closed curve, whereas in Euclidean geometry a straight line can never constitute a closed curve. Here, however, the difficulty which arises does not pertain so much to Euclideanism and non-Euclideanism as to another branch of geometry, Analysis Situs, on which we shall make a few brief remarks in a note at the end of the chapter. The fact is that the geometry of the cylinder embedded in Euclidean space is Euclidean to the same extent as the geometry of the plane, and yet on a cylinder certain straight lines, namely, the rings which surround the cylinder, constitute closed curves.

It might be objected that these rings are not Euclidean straight lines. But here the critic would be confusing geometry and dimensionality. From the standpoint of two-dimensional Euclidean geometry, the rings on the cylinder are perfectly straight lines, and it would never enter the mind of the flat being moving over the cylindrical surface to view them in any other light. Only when we represent the cylinder’s surface in a three-dimensional Euclidean space must we regard the rings as curves. In short, lines that would be straight in two dimensions might be curved when viewed from the standpoint of a higher dimensionality. By analogy, lines which would be regarded as straight in three-dimensional space might be recognised as curved were we to appeal to a fourth [Pg 64] dimension. Thus, whichever way we choose to investigate the concept of straightness, we find that it eludes us more and more.

It appears unnecessary to dwell further on these rather special problems, for they would lead us too far afield.

Now, finally, there is still another reason, on which we have lightly touched, which must make us very wary of being led astray by faulty analogies. When, for instance, we consider the two-dimensional geometry of a spherical surface embedded in a three-dimensional Euclidean space, we realise that the surface divides space into an outside and an inside. We should then be inclined to argue that a three-dimensional Riemannian or spherical space must also divide four-dimensional space into an inside and an outside.

This line of reasoning would be correct in the present case, but it is important to note that this conception of an inside and an outside is in no wise essential to non-Euclidean geometry. Had we adopted the more rational method of presenting three-dimensional non-Euclidean space as due to the peculiar behaviour of our measuring rods, the conception of an inside and an outside would have been meaningless. We must understand that this notion arose merely as a result of our particular choice of a mode of presentation, and in no wise constitutes an intrinsically necessary condition.

If these rather delicate points have been understood, no harm can be done by discussing non-Euclidean geometries as the geometries obtained by applying Euclidean rods on curved surfaces. In many respects, indeed, this method of presentation is helpful; for accustomed as we are in everyday life to effect measurements with Euclidean rods, we are able to visualise more easily a series of abstract investigations when this familiar procedure is followed.

Let us now proceed to a more thorough study of the preceding method. We have said that non-Euclidean results are obtained when we confine ourselves to Euclidean rods while conducting our measurements on curved surfaces. In the case of Riemannian geometry of two dimensions, the required surface is that of a sphere embedded in three-dimensional Euclidean space. In the case of Lobatchewski’s geometry, it is a saddle-shaped surface known as a pseudosphere (subject to the limitations mentioned previously).

We will confine ourselves to the study of Riemann’s geometry for the time being. Such a study in two dimensions permits easy visualisation, owing to the fact that since the earth is spherical in shape it is precisely this type of geometry which we obtain when we conduct measurements on its surface with rigid Euclidean measuring rods (assuming the surface to be perfectly smooth, like that of the ocean on a calm day). In fact, we shall see that the postulates of Riemann’s geometry are immediately verified on the sphere.

To begin with, the shortest distance, or, better still, the most direct distance, between two points on the earth’s surface, when Euclidean rigid rods or inextensible tape measures are employed as a [Pg 65] means of measurement, is an arc of a great circle. Great circles, such as meridians on the surface of our planet, constitute therefore the straight lines or geodesics of Riemann’s geometry; and when discussing Riemann’s geometry we may call them straight lines.

Consider, then, one such straight line or geodesic, and let it be that particular great circle which we call the equator. According to the axioms of Euclid’s geometry two perpendiculars drawn at two different points of the same straight line can never meet; they lie parallel to one another. On the sphere, on the other hand, two perpendiculars to the equator are two meridians and these meridians always meet at the pole. In fact it is impossible to draw any two great circles on the sphere which do not intersect. Hence we see that no parallel geodesics can exist in Riemann’s geometry. Likewise, whereas in Euclid’s geometry the sum of the three angles of a triangle is equal to two right angles, in Riemann’s geometry this sum is always greater than two right angles. This we could easily verify by stretching ropes between three distant points on the earth’s surface so as to form a triangle, and then summing the values of the angles at the three corners of our triangle.

A similar verification would ensue if we were to effect measurements along a circumference and its diameter. Thus, if at a central point on the earth’s surface we fix one end of a rope and cause the other extremity to rotate around the central point, it will describe a circle on the earth’s surface. We shall then find, upon measuring the length of the circumference marked out by the extremity of the rope, that the ratio of the length of this circumference to the length of the diameter is a variable number, always smaller than and depending on the area covered by our circle; whereas in Euclidean geometry this ratio would be the constant number .

There is still another result of importance which must be mentioned. On a plane, the farther we wander along a straight line, the farther we move from our starting point, whereas on our sphere we see that if we follow a straight line or geodesic we shall finally return to our starting point after having circled the earth. We express this fact by saying that in Riemann’s geometry space is finite, yet unbounded. It is finite since we cannot wander away indefinitely from our starting point; it is unbounded because however far we go, we never come to a stone wall or a gap beyond which we can proceed no farther. On the other hand, Euclidean space, such as that of the plane surface, is infinite and unbounded.

And now suppose the earth on which we were conducting our measurements were to swell indefinitely. All the characteristics of Riemann’s geometry which we have discussed would gradually fade away, provided we limited our measurements to the same restricted area of the surface. Our measurements would still yield Riemannian results, of course, since however large our earth had grown, it would still remain spherical. But the results of our measurements over a definite area would approximate [Pg 66] more and more to those of Euclidean geometry. The reason is obvious, since the greater the volume of the sphere, the more nearly would a given area of its surface approximate to an ordinary plane.

In fact a plane surface may be assimilated to the surface of a sphere of infinite radius; so that if our sphere were to grow indefinitely we should find that our measurements lost little by little their Riemannian characteristics, until it would be impossible to distinguish them from Euclidean measurements.

Suppose now that our sphere continued to change in shape after having attained an infinite radius; in particular, suppose that our surface gradually began to assume the shape of a saddle, curved downwards if we intersected it from east to west and curved upwards if we intersected it from north to south. As this double curvature became gradually more pronounced we should find that our measurements lost their Euclidean character and became more and more conspicuously Lobatchewskian.

Thus we see that Euclidean geometry stands at the dividing line of Riemannian and Lobatchewskian geometry. When our surface changes in shape the geometry remains Riemannian so long as there subsists the least trace of sphericity, that is, of positive curvature. Likewise it remains Lobatchewskian so long as there subsists the least trace of saddle-shapedness, or negative curvature. Finally, it is strictly Euclidean only when the surface has become a plane, that is, exhibits zero curvature.

For this reason Euclidean geometry has a uniqueness about it which is denied to the non-Euclidean varieties. These latter constitute a class of geometries; and the precise type of the geometry we may happen to be discussing is determined by the intensity of curvature of the sphere or pseudosphere to which it pertains. We also see why it is so difficult to determine by empirical methods whether the space of the universe is truly Euclidean or not. It is because both Riemannian and Lobatchewskian geometry merge by insensible gradations into Euclidean geometry.

The analysis we have given refers to non-Euclidean geometries as derived from the geometry of surfaces; that is, to the numerical results obtained when we use rigid Euclidean rods to effect measurements on curved surfaces. It must be noted, however, that the surfaces we have discussed are of a very special kind. Both spheres and pseudospheres are known as surfaces of constant curvature. There is no need to go into the mathematical definition of what is meant by this; we will limit ourselves to stating the principal characteristic of such surfaces.

Consider, for example, a net made of inextensible threads (inextensible or undeformable being used in the Euclidean sense). If this net can be applied with perfect contact to any portion of a plane, it can be slid over the entire plane without ever losing its perfect contact. The same is true of the sphere and pseudosphere. Thus, a net which could be fitted with perfect contact to any part of a sphere could be slid over any portion of the surface without ever losing its perfect contact.

[Pg 67]

The only surfaces which possess this property are precisely the sphere, the plane and the pseudosphere (and those derived therefrom without stretching); and these surfaces are the surfaces of constant positive curvature, of constant zero curvature, and of constant negative curvature, respectively. So far, then, as the shape of nets applicable to the surface is concerned, there is no means of distinguishing one part of the surface from any other part, since the net can be slid over the surface, preserving its dimensions and never losing its perfect contact with the surface. In other words, the surface appears to be isotropic and homogeneous when measurements with rigid Euclidean rods are conducted over it, or when the nets we slide over it are Euclideanly inextensible, This is the property called free mobility.

Such would no longer be the case were we to consider surfaces of variable curvature, e.g., the surface of an ellipsoid or of a trumpet. A net which could be applied with perfect contact to one part of the surface would lose its perfect contact when slid over the surface, unless we deformed it by stretching it or causing it to shrink.

It is because all three types of geometry we have mentioned (Euclidean, Riemannian and Lobatchewskian) hold in exactly the same way for all parts of the space in which we are operating, that the surfaces which portray them in two dimensions are necessarily of the constant-curvature type.

We have yet to show the connection between non-Euclidean geometry of two dimensions defined as the geometry obtained with Euclidean rods on a surface of constant curvature, and non-Euclidean geometry obtained with squirming Euclidean rods, i.e., rigid non-Euclidean ones. For this purpose consider a sphere resting on a plane. We may call the south pole that point of the sphere which stands in contact with the plane; and we shall assume that a point-source of light is located at the north pole. If the sphere is transparent to rays of light all figures traced on the surface of the sphere will cast shadows on the plane. But the same will be true of our little Euclidean rods which we place alongside these figures on the sphere for purposes of measurement.

It is easy to see that a circle, for instance, traced on the spherical surface will, generally speaking, have for its shadow an ellipse on the plane. As this circle is displaced on the sphere towards the north pole, its shadow will grow larger and larger, tending finally, when any point of the circle coincides with the north pole, to become a parabola extending to infinity. But inasmuch as the shadows of the little rods with which we measure lengths on the sphere vary in exactly the same way as do the shadows of the figures measured, we shall obtain on the plane exactly the same Riemannian results that endure on the sphere.

In short, if we assume that on the plane our standard measuring rods are given by the shadows of the standard measuring rods used on the sphere, we shall obtain Riemannian geometry on the plane. It is scarcely necessary to state that with our Euclidean ideas of congruence it would be impossible for us to accept the statement that these successive squirming shadows of a Euclidean length displaced on the [Pg 68] surface of the sphere were all congruent to one another; but this is not the point at issue. If we were perfectly flat, two-dimensional beings, living on the plane, and if not only all the flat bodies sliding over the plane behaved as did the shadows in our previous example, but also our measuring rods and our own living bodies behaved likewise, we should have no other alternative than to assume that these shadows represented the displacements of bodies that moved while remaining rigid or without change. We should accordingly regard the geometry of our two-dimensional space as Riemannian.

It is impossible to show more clearly that the geometry we attribute to space is nothing but an expression of the properties of our bodies and measuring rods when displaced, since here we have a plane which is Euclidean or Riemannian in its geometry according to the behaviour of the bodies that glide over its surface. And so we see that two alternative methods of interpreting non-Euclidean geometry are open to us. Either we may regard our rods as Euclidean and applied to a curved surface, or else we may attribute non-Euclideanism to the behaviour of our rods on a flat surface.[20]

Now in this chapter we have discussed only non-Euclidean geometries of two dimensions. Riemann’s geometry and Lobatchewski’s geometry are more particularly these same geometries, extended to the case of three dimensions. We can easily see how Riemann’s geometry of three dimensions would arise.

We have only to suppose that, as contrasted with Euclidean solids, all material bodies expand as did the shadows when displaced from a fixed centre. But the Riemannian observer of course would have no realisation of this expansion, owing to the modified laws of light propagation and owing to his own body’s expanding in company with all other bodies. So he would be of the opinion that all these bodies maintained their same shape and size, that is, remained congruent when displaced. In fact, if he applied the test of practical congruence he would find that these bodies which the Euclidean observer would claim were expanding were, so far as he himself was concerned, always the same both in visual appearance and as regards exploration by the sense of touch. If our Riemannian being were to view bodies which the Euclidean man considered rigid and undeformable, he would assert without hesitation that those bodies were decreasing in size as they approached a certain point. Once again maintenance of shape and size is essentially relative.

All the conclusions which we reached when discussing a Riemannian [Pg 69] world of two dimensions can be extended to the case of three dimensions. Thus, a Riemannian universe or spherical space of three dimensions is finite but unbounded. Travelling for a sufficient length of time along a geodesic or straight line, we should return finally to our starting point, and rays of light which follow geodesics would circle round our space.

Just as in the case of two-dimensional geometry, we may also conceive of a three-dimensional non-Euclidean space as of constant or variable curvature. Prior to Einstein’s discoveries mathematicians concerned themselves more especially with the homogeneous types of geometry, because it was assumed (notwithstanding Riemann’s premonitions) that whatever the practical geometry of real space might turn out to be, it would always remain the same throughout, there being no palpable reason for it to vary here from there. But with the advent of the material or metrical field, conditioned by a more or less capricious distribution of matter, the possibility of a variation in the geometry of space from place to place had to be taken into consideration, and it then became impossible to limit our analysis to spaces of constant curvature. It has been proved that the variable non-Euclideanism of the space surrounding matter in Einstein’s theory—say, over the equatorial plane—would be represented by the geometry of a surface of variable curvature, which turns out to be that of a paraboloid of revolution.

There is one additional aspect of Riemann’s geometry which it may be of interest to mention on account of its possible bearing on the shape of the universe in Einstein’s theory. We refer to elliptical space. The type of Riemann’s geometry which we have discussed so far is known as Riemann’s spherical geometry, because in the case of two dimensions it turns out to be the Euclidean geometry of a spherical surface. But there exists another type of Riemannian geometry discovered by Klein. It is called elliptical space, though it has nothing to do with the surface of an ellipsoid. It corresponds to exactly the same geometry as that of Riemann proper, the only difference consisting in the connectivity of the space. That is to say, the paths of continuous passage from a point to a point are different in the two spaces.

In a similar way the geometry of the cylinder is Euclidean, as is that of the plane (since a plane sheet of paper can be rolled round a cylinder); but its connectivity is different since we can go round the cylinder by following a geodesic and yet return to our starting point.

[Pg 70]

In order to illustrate elliptical space in two dimensions, as we have illustrated spherical space, we must assume that we limit the surface of our sphere to one of its hemispheres. It might appear that in this case the surface would no longer be unbounded since it would stop abruptly at the equator, but the connectivity of elliptical space is such that every point on the equator is identical with its antipodal point on the other side of the equator. It would be as though, in our concrete representation, there were cross-connections of zero length uniting into one sole point, each pair of points situated diametrically opposite one another on the equator.

It is exceedingly difficult to visualise how it can be that a diameter joining two antipodal points of the equator should be vanishing in length, but the difficulty arises from our endeavouring to represent the distance between two points by a line joining these two points. Such a procedure may often be successful, but in the more complex cases of connectivity it breaks down completely. As Eddington remarks, for obscure psychological reasons we find it natural to represent a distance between two points by a line joining them, believing that this line represents a something fundamental in nature. But it is probable that the line we trace does not have the fundamental importance with which we credit it, and that it cannot be regarded as having any intrinsic significance outside of the ease with which it often introduces itself into our graphical description of natural phenomena. Distance would be more akin to a difference of temperature between two points, and it would never enter our minds to represent this difference of temperature by a line joining the two points. Why in the case of distance such a graphical representation appears obligatory is more or less of a mystery. At any rate the graphical representation is not always possible, and our description of elliptical space gives us an illustration to this effect.

Incidentally we see that the problem of space, even when restricted to the particular cases studied by Riemann, is not exhausted by the sole concept of congruence. Connectivity or routes of continuous passage between points must also be taken into consideration. The study of connectivity opens up a new branch of geometry also created by Riemann and known as Analysis Situs.[21]

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CHAPTER VI TIME

OUR awareness of the passage of time constitutes one of the most fundamental facts of consciousness, and our sensations range themselves automatically in this one-dimensional irreversible temporal series. In this respect we must concede a certain difference between space and time, the former being, as we have seen, a concept due in the main to a less inevitable synthetic co-ordination of sense impressions. Yet, on the other hand, there exists a marked similarity between space and duration, in that they both manifest the characteristics of sensory continuity.

Thus, in the case of time, two sensations and may be so close together that it will be impossible for us to determine which of the two was sensed first; the same may happen in the case of two sensations and and yet we might have no difficulty in ascertaining that was prior to . These facts are expressed by the inconsistent relations , , , which, as we have seen, are characteristic of sensory continuity, be it in space, or in time, or in whatever domain we please. In the case of time, as in that of space, only by a process of abstraction can we remedy this inconsistency. We thereby conceive of mathematical duration, a duration in which mathematical continuity is assumed to have replaced sensory continuity.

Now, owing to the ordering relation or path of continuous transfer which appears to be imposed upon us by nature, perceptual time must be regarded as uni-dimensional, and by retaining the natural ordering relation, we will attribute the same uni-dimensionality to mathematical duration. It is, of course, to this mathematical duration that we appeal in all theoretical investigations in mechanics and physics.

Primarily the only time of which we are conscious is an “egoistic” time, the time of our stream of consciousness. But as a result of our perception of change and motion in the exterior universe of space, it becomes necessary to extend the concept of duration throughout space, so that all external events are thought of as having occurred at a definite instant in time as well as at a definite point in space.[22]

Now, when discussing space, we saw that the concept of the same point in space, considered at different times, was ambiguous; it was necessary to specify the frame of reference by which space was defined, and according to the frame selected, an object would continue to occupy the same point or else manifest motion. The analogous problem in the case of time would be to conceive of the same instant in time at two [Pg 72] different places, i.e., of the simultaneity of two spatially separated events.

If we allowed ourselves to be guided by our analysis of position in space, we should be inclined to say that a determination of the simultaneity of spatially separated events must necessitate the choice of a frame of reference. The observer who was to pass an opinion on the simultaneity of the two events would then select that particular frame in which he stood at rest. But from this it would follow that a change in the motion of the observer, hence a change of frame, might bring about a modification in his understanding of simultaneity. Simultaneity would thus manifest itself as relative, just like position.

Although these possibilities cannot be discarded a priori on any rational grounds, classical science and common-sense philosophy refused to entertain them. The fact is that this thesis would have tended to surround with ambiguity the concept of time in physics, since the duration of the same event would vary with the motion of our frame of reference. It had always been felt that although, through an act of our will, we might change our motion through space, yet on the other hand the flowing of time transcended our action. These views were confirmed by the behaviour of physical phenomena known to classical science; hence the uniqueness of time and the absolute character of simultaneity had been accepted. In other words, the simultaneity and the order of succession of two spatially separated events were assumed to constitute facts transcending our choice of a system of reference.

This stand was fully justified in Newton’s day; for inasmuch as a priori arguments could not decide the question, so long as experiment did not compel us to recognise the relativity of simultaneity and the ambiguity of duration there appeared to be no good reason to hamper science with an unnecessary complication. But the point we wish to stress is that if perchance experiment were ever to suggest that simultaneity was not absolute, no rational argument could be advanced to prove the absurdity of this opinion. As we shall see, Einstein’s interpretation of certain refined electromagnetic experiments consists precisely in recognising the relativity of simultaneity and the ambiguity of duration.

Let us now pass to the problem of congruence for time. As is the case with space, mathematical time is an amorphous continuum presenting no definite metrics. The concept of congruence or of the equality of successive durations has no precise meaning; and a definition of time-congruence can be obtained only after we have imposed some conventional measuring standard on an indifferent duration. We might define as equal the durations separating the rising and the setting of the sun, or, again, the durations separating successive rainstorms, etc. Theoretically, all such definitions would be legitimate, since in the final analysis duration, just like space, presents no intrinsic metrics.

In common practice, however, we find that just as in the case of space, a definite species of congruence appears to be imposed on human beings and animals alike. For instance, so long as we are in a conscious state, we all agree that the successive tickings of the [Pg 73] clock correspond to what our psychological sense of duration would qualify as congruent intervals. In other words, we appear to be gifted with a species of psychological time-keeping device which allows us to differentiate between durations which we have come to call “seconds” and those which we have come to call “hours.” What is commonly called the sense of rhythm, which manifests itself in the primitive music of drum beats and in the dances of savages, is nothing but a consequence of this intuitive understanding of time-congruence.

A superficial analysis of the situation might lead us to assume that, contrary to our previous assertions, a definite species of congruence was intrinsic to duration. This opinion, however, is quite untenable unless supplemented by additional explanations. The problem with which we are confronted is in all respects similar to the one we mentioned when discussing space. Of itself, space, or at least mathematical space, is amorphous. In spite of this fact, we have no difficulty in agreeing that a coin is round while an egg is oval. But we saw elsewhere that our unreasoned predilection for a definite type of congruence in space was imposed by the behaviour of material bodies when displaced. Visually these bodies could be made to present us with the same appearances, whether situated here or there, and for that reason they imposed themselves as defining congruent space intervals. Furthermore, our own human limbs, the steps we take, the ground on which we walk, all suggest the same definition of congruence.

Likewise in the case of time-congruence. The human organism is replete with rhythmical motions such as the beating of the heart, the periodic contractions of the arteries, or the respiratory motions of the lungs. Again, an infant feels the pangs of hunger at periodic intervals. Quite independently of these internal processes and changes, certain external mechanical motions affecting our bodies must be considered. Thus, when we walk with our normal gait, our successive steps follow one another in time in a certain definite way, and the same applies to the swinging of our arms. By an effort of our will we might vary the rhythm of our steps, alternating short steps with long ones, now tarrying, now hastening. But we should soon be aware that something was wrong, for we should find that our irregular motions demanded additional effort, so that on relaxing our will we should relapse automatically into our normal gait. It may be that in addition to these periodic motions we have mentioned, cyclic processes occur in our brains, conferring on our consciousness direct norms of time-congruence. This, however, is a point for physiologists to decide. At any rate, when it is realised that these various methods, whether they appeal to our respiratory motions or to our successive steps when walking normally, all issue in the same definition for time-congruence, there is no cause to be surprised at a definite sense of time-congruence imposing itself naturally upon us, and this in spite of the fact that pure duration of itself possesses no metrics.

Thus far we have been considering time-congruence from the standpoint [Pg 74] of the savage and the animal. For the most rudimentary purposes, a definition of equal durations obtained in this crude way would suffice; yet it is obvious that no great precision could thus be arrived at. Our crude psychological clock is subject to variations according to our mental and physical condition; furthermore, though it can differentiate between what we have come to call “one second” and “one minute,” it would be incapable of differentiating between fifty-nine minutes and one hour. In the case of animals this imprecision in their sense of duration is manifested when shepherd dogs lead the flocks back during an eclipse or hens lay eggs under the influence of artificial light, believing that day has come. However, as these last illustrations permit of other interpretations, we need not dwell on them and may confine ourselves to the well-known fact that our human sense of duration varies between one individual and another, and between one psychological state and another.

In addition to these difficulties, our understanding of equal durations is entirely subjective and can give rise to no common knowledge. We must needs objectivise it, hence appeal to some objective phenomenon which all men can observe. Let us assume, for the sake of argument, that the monotonous dripping of water drops falling on a pan may have served as a primitive clock. Still, something would be lacking. We should not be completely satisfied, for we might sense now and then an irregularity in the successive thuds of the drops falling on the pan. Furthermore, if we were to establish two such water clocks we should observe that the drops sometimes fell in unison and sometimes in succession. We should wonder then which of the clocks was most reliable and be in a quandary to know to which to appeal. Just as in the case of space, where we substituted wooden rods for the lengths of our steps, then metallic rods maintained at constant pressure and temperature, so now, in the case of time-keeping devices, we must find some way of selecting clocks which will satisfy all our requirements.

We thus arrive at more scientific methods of time-determination. The psychological clock, water clock, sand clock and burning of candles, which had served men in the earlier days of civilisation, have now been discarded in favour of more precise mechanisms. These, as we shall see, will not differ perceptibly in their indications from our psychological clock, but their measurements will be given with increased refinement. We are thus led to examine what are the conditions that must be satisfied for a time-keeping device to be considered perfect.

In a general way we obtain a definition of time-congruence by appealing to the principle of causality. The following passage, quoted from Weyl, expresses this fact:

“If an absolutely isolated physical system (i.e., one not subject to external influences) reverts once again to exactly the same state as that in which it was at some earlier instant, then the same succession of states will be repeated in time and the whole series of events will constitute a cycle. In general, such a system is called a clock. Each period of the cycle lasts equally long.”

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We shall obtain a more profound understanding of the nature of the problem by considering the various methods of time-determination that have been contemplated by physicists.

We remember that Newton refers to absolute time as “flowing uniformly.” But of course this allusion to time does not lead us very far, for a rate of flow can be recognised as uniform only when measured against some other rate of flow taken as standard. Obviously some further definition will have to be forthcoming. Now both Galileo and Newton recognised as a result of clock measurements that approximately free bodies moving in an approximately Galilean frame described approximately straight lines with approximately constant speeds. Newton then elevates this approximate empirical discovery to the position of a rigorous principle, the principle of inertia, and states that absolutely free bodies will move with absolutely constant speeds along perfectly straight lines, hence will cover equal distances in equal times. When expressed in this way as a rigorous principle, the space and time referred to are the absolute space and time of Newtonian science. In other words, it is the principle of inertia coupled with an understanding of spatial congruence that yields us a definition of congruent stretches of absolute time.

In practice, a definition of this sort entails measurements with rods; that is to say, we measure off equal distances along the straight path of a body and then define as congruent, or equal, the times required by the body to describe these congruent spatial distances. It is obvious that, however perfect our measurements, errors of observation will always creep in. Furthermore, a body moving under ideal conditions of observation can never be contemplated; hence all we can hope to obtain is the greatest possible approximation. But although physical measurements become inevitable as soon as we wish to obtain a concrete definition, an objective criterion of equal durations, we are able to proceed in our mechanical deductions in a purely mathematical way without further appeal to experiment. The principle of inertia, together with the other fundamental principles of mechanics, enables us, therefore, to place mechanics on a rigorous mathematical basis, and rational mechanics is the result.

It will be observed that science, in the case of mechanics, has followed the same course as in geometry. Initially our information is empirical and suffers from all the inaccuracies of human observation and all the various contingencies that characterise physical phenomena. But this empirical information is idealised, then crystallised into axioms, postulates or principles susceptible of direct mathematical treatment. To be sure, as we proceed in our purely theoretical deductions, unless we are to lose all contact with reality, we must check our results by physical experiment; and this necessity is still more apparent in rational mechanics than in geometry. If peradventure further experiment were to prove that our mathematical deductions in mechanics were not borne out in the world of reality, we should have to modify our initial principles and postulates or else agree that nature was irrational. With mechanics, the necessity of modifying the fundamental principles became imperative when it was recognised that [Pg 76] the mass of a body was not the constant magnitude we had thought it to be; hence it was experiment that brought about the revolution. On the other hand, in the case of geometry, it was the mathematicians themselves who foresaw the possibility of various non-Euclidean doctrines, prior to any suggestion of this sort being demanded by experiment.

And now let us revert once again to the problem of time. Theoretically, the law of inertia should permit us to obtain an accurate determination of congruent intervals, but as it was quite impossible to observe freely-moving bodies owing to frictional resistances as also to the gravitational attraction of earth and sun, it was advantageous to discover some other physical method of determination. This was soon obtained. The principles of mechanics enable us to anticipate that if a perfectly rigid sphere, submitted to no external forces, is rotating without friction on an axis fixed in a Galilean frame, its angular rate of rotation will be uniform or constant, as measured in the frame. By “constant,” we mean constant as measured in terms of the standards of time-congruence defined by a free body moving under ideal conditions according to the principle of inertia. Hence we are in possession of a method of measuring time, more convenient than that afforded by the motions of free bodies along straight courses. It may be noted that the definition of congruent durations as given by the rotating sphere is in perfect accord with the definition in terms of causality as formulated in the passage quoted from Weyl.

Unfortunately, here again, ideal conditions can never be established, for we can never obtain perfect isolation, perfect rigidity and complete absence of friction. But a near approach to ideal conditions would appear to be given by the phenomenon of the earth’s rotation. And so the rotation of the earth, defining astronomical time, was regarded as furnishing science with the most reliable objective criterion of congruent time-intervals that it was possible to obtain. Nevertheless, it was well known that a definition of this sort was still far from perfect, since the rate of rotation of the earth could not be absolutely uniform, when “uniform” is defined by the standards of the principle of inertia. This realisation was brought home when the frictional resistance generated by the tides was taken into consideration.[23] The tidal friction would have for effect the slowing down of the earth’s rate of rotation (when measured against the accurate mathematical definition given by a perfectly rigid and isolated body rotating without friction). In addition to this first perturbating influence, our planet is suspected of varying in size, expanding and contracting periodically, a phenomenon known as breathing. According to the laws of mechanics, a change in shape of this sort would entail periodic fluctuations, successive retardations and accelerations in the rate of the earth’s rotation.

It is instructive to understand how this discovery of the earth’s breathing was brought about. It was noticed that the moon’s motion exhibited variations which it seemed impossible to account for under [Pg 77] Newton’s law, even when due consideration was given to the perturbating effects of the sun and other planets. Not only did the moon’s motion appear to be gradually increasing, but in addition, superposed on this first uni-directional effect,[24] periodic accelerations and retardations had been observed. The gradual slowing down of the earth’s rate of rotation under the influence of the tides accounted for the apparent acceleration of the moon’s motion, but the other periodic changes in the velocity of our satellite could not be explained away in so simple a manner. Yet everything would be in order were we to assume that the earth’s rate of rotation was subjected to appropriate periodic changes; and the principles of mechanics then showed that a phenomenon of this kind could only be brought about by periodic variations in the earth’s size.

In all these corrections we are dealing with highly complex phenomena involving Newton’s law of gravitation and the laws of mechanics. Our astronomical time-computations have thus been adjusted so as to satisfy the requirements of these laws, regarded as of absolute validity. We may recall that it was this belief in the accuracy of Newton’s law that led Römer in 1675 to the discovery of the finite velocity of light. He noticed that Jupiter’s satellites appeared to emerge from behind the planet’s disk several minutes later than was required by the law of gravitation. Rather than cast doubts on the accuracy of this law, he ascribed a finite velocity to the propagation of light. Needless to say, his previsions have since been justified by direct terrestrial measurements. The calculations leading to the discovery of the planet Neptune were also based on the assumption that Newton’s law was correct.

And now let us suppose that we had elected to define time by entirely different methods. We might, for instance, appeal to the vibrations of the atoms of some element, such as cadmium. Then the intervals of time marked out by the successive beats of the atom at rest in our Galilean frame would be equal or congruent by definition. These successive congruent intervals of time would be measured by the wave length of the radiation, since a wave length is the distance covered by the luminous perturbation during the interval separating two successive vibrations. We might again proceed in still another way in terms of radioactive processes. It is well known that, when measured by ordinary clocks, the rate of disintegration of radium appears to follow an exponential law. And it is a remarkable fact that this rate of disintegration does not appear to be modified in the slightest degree by the surrounding conditions of temperature or of pressure. Thus the phenomenon may be regarded as being in an ideal state of isolation. Curie had therefore suggested defining time in terms of this rate of disintegration.

We see, then, that a great variety of methods for determining time have been considered. Some were mechanical, others astronomical, [Pg 78] others optical, while still others appealed to radioactive phenomena. In each case our definitions were such as to confer the maximum of simplicity on our co-ordination of a certain type of phenomena. For instance, if in mechanics and astronomy we had selected at random some arbitrary definition of time, if we had defined as congruent the intervals separating the rising and setting of the sun at all seasons of the year, say for the latitude of New York, our understanding of mechanical phenomena would have been beset with grave difficulties. As measured by these new temporal standards, free bodies would no longer move with constant speeds, but would be subjected to periodic accelerations for which it would appear impossible to ascribe any definite cause, and so on. As a result, the law of inertia would have had to be abandoned, and with it the entire doctrine of classical mechanics, together with Newton’s law. Thus a change in our understanding of congruence would entail far-reaching consequences.

Again, in the case of the vibrating atom, had some arbitrary definition of time been accepted, we should have had to assume that the same atom presented the most capricious frequencies. Once more it would have been difficult to ascribe satisfactory causes to these seemingly haphazard fluctuations in frequency; and a simple understanding of the most fundamental optical phenomena would have been well-nigh impossible.

Since we realise that the definitions of time-congruence we ultimately adopt are governed by our desire to obtain the maximum of simplicity in the co-ordination of some particular group of phenomena, considerable divergences in our definitions might be expected when we considered in turn phenomena of a different nature. Why, indeed, should a determination arrived at in terms of the vibrations of atoms agree in any way with the astronomical definition issuing from the earth’s rotation? Even when restricting our attention to the vibrations of the atoms of the same element, it might have been feared that different atoms of cadmium would have beaten out time at different rates, since, after all, even though identical in many respects, yet their life-histories must have differed greatly. Instead of this we find that all the methods of time-congruence mentioned lead to exactly the same determinations so far as experiment can detect. It would appear as though a common understanding existed between the rates of evolution of the various phenomena mentioned. The situation is similar to that presented in the problem of space. There we saw that rods, whether of platinum or of wood, whether situated here or there, appeared to yield the same geometrical results (provided they were maintained at constant temperature and pressure). Again, it seemed as though some common understanding existed among the various rods.

Just as this same mysterious situation in the case of space suggested that the uniformity of our spatial measurements was due to the presence of an all-embracing physical reality, the metrical field of space, regulating the behaviour of material bodies and light rays, so now, as we shall see in later chapters, the general theory of relativity will [Pg 79] compel us to extend the same ideas to time. We shall find that there exists a space-time metrical field which splits up into a temporal and a spatial field. The temporal metrical field will then be responsible for that apparent understanding which seems to exist among the various spatially separated phenomena.

These new ideas modify the position of the problem of time-congruence. While it is still correct to state that pure duration as a mathematical continuum is amorphous and possesses no intrinsic metrics, yet from the standpoint of the physicist we must concede a certain physical reality to this temporal metrical field which controls the processes of all isolated phenomena.

We may summarise the preceding statements by saying that we define time-congruence so as to introduce the maximum of simplicity in our co-ordinations of physical phenomena, regardless of the particular type of phenomenon we may be considering. That a unique definition of time-congruence should be possible, one holding for mechanics, optics, electromagnetics and astronomy, is a fact that could scarcely have been anticipated a priori; hence we must recognise its undeniable existence as establishing the presence of unity in nature.

And here we may mention a type of objection which is often encountered in people who do not trouble to differentiate between physical time and psychological duration. It has been claimed, for instance, that it is absurd to maintain that our sense of time was originated from a desire to retain harmony between the planetary motions and the requirements of Newton’s law, since men possessed an intuitive understanding of equal durations long before Newton’s law or any other natural law was discovered. We trust that those who have taken the trouble to follow the course of this analysis will realise that the criticism is quite beside the mark. No scientist has ever contended that we have derived our sense of time from astronomical observations, for we know that even savages are not lacking in a sense of rhythm. All that is claimed is that the accurate determination of time-congruence must be based on physical processes or laws, since our crude time-sense is too vague to be of any use in the precise problems with which science is concerned. The fact that our crude appreciation of time agrees in a more or less approximate way with the more refined methods of scientific determination offers no great mystery, for, after all, the laws which govern material processes in the outside world would in all probability also govern that which is material in our organisms. Regardless of our ultimate views as to matter and mind, we cannot banish matter from our anatomy.

At any rate, once the existence of a universal type of time-congruence is conceded, the problem is to select some particular phenomenon which can be studied with a minimum of contingency and a maximum of certainty. Now, when we consider the astronomical definition in terms of the earth’s rotation, we realise that were it not for the fact that we had started by assuming the absolute accuracy of Newton’s law, the corrections we made in order to account for the peculiarities of the [Pg 80] lunar motion would have to be discarded or replaced by others. If, therefore, any doubt were to be cast on the accuracy of Newton’s law or on those of mechanics, corresponding modifications would ensue for our definition of time-congruence. But it is obvious that we have no right to assume that Newton’s law or those of mechanics are above all suspicion. The laws of physical science present no a priori inevitableness; and our sole reason for accepting them is because they appear to be borne out to a high degree of approximation when we compare their rational consequences with the results of observation. Approximation is thus all we can aspire to in physical science; and even classical scientists recognised the possibility that more precise observation might prove Newton’s law and those of mechanics to stand in need of correction. All that could be asserted was that the corrections would certainly be very slight, far too slight to entail any perceptible change in our understanding of time-congruence when applied to the processes of commonplace observation.

We now come to Einstein’s definition. As we shall see, it does not differ in spirit from the definitions of classical science; its sole advantage is that it entails a minimum of assumptions, and is susceptible of being realised in a concrete way permitting a high degree of accuracy in our measurements. Einstein’s definition is, then, as follows: If we consider a ray of light passing through a Galilean frame, its velocity in the frame will be the same regardless of the relative motion of the luminous source and frame, and regardless of the direction of the ray. It follows that a definition of equal durations in the frame will be given by measuring equal spatial stretches along the path of the ray, and asserting that the wave front will describe these equal stretches in equal times.

As can be seen, the definition is simple enough, but the major question is to decide on its legitimacy. Classical science would have rejected the definition. Why? Because it would have maintained that Einstein’s definition was equivalent to defining time in terms of a rigid body rotating round an axis, but submitted to frictional forces. It would have claimed that the velocity of the frame through the stagnant ether would have introduced complications entailing anisotropy, that is to say, a variation of the speed of light according to direction. It would then be necessary to take this perturbating effect into account just as it was necessary to take into consideration the frictional effect of the tides on the earth’s angular rate of rotation.

But when it was found that contrary to the anticipations of classical science not the slightest trace of anisotropy could be detected even by ultra-precise experiment, the objections which classical science might have presented against Einstein’s definition lost all force. Henceforth it was not necessary to take into consideration the velocity of the frame through the stagnant ether, since this ether drift appeared to exert no influence one way or the other.

Now, it may not be easy to understand how isotropic conditions could [Pg 81] be demonstrated by experiment, for isotropy signifies that the velocity of light is the same in all directions. And how can we ascertain the equality of a velocity in all directions when we do not yet know how to measure time? Experimenters solved the difficulty by appealing to the observation of coincidences.

Fig. I

The following elementary example will make this point clear. Consider a circle along which two bodies are moving in opposite directions. We shall assume that the two bodies leave a point at the same instant of time, hence in coincidence, and then meet again at a point halfway round the circle, hence diametrically opposite . There is no difficulty in ascertaining that the point is diametrically opposite , since we know how to measure space. All we have to do is to verify the fact that the length of the curve is equal to the length . Neither does the observation of a coincidence, that is, of the simultaneous presence of two objects at the same point of space, entail any knowledge of a measure of duration. Hence, although we know nothing of the measurement of time, we cannot escape the conclusion that the two bodies have taken the same time to pass from to by different routes. But these spatial routes are of equal length; hence we must assume that the speeds of the bodies along their respective paths have been the same. In all rigour, we may only claim that the mean speed has been the same, since one body may have slowed down, then spurted on again, making up for lost time. But if we repeat the experiment with circles of different sizes, and if in every case the bodies meet one another at the opposite point, we are justified in asserting that not only the mean speed, but also the instantaneous speed at every instant, is the same for either body. In short, thanks to spatial measurements coupled with the observation of coincidences, we have been able to establish the equality of two velocities, even though we knew nothing of time-measurement.

A more elaborate presentation of the same problem would be given as follows: Waves of light leaving the centre of a sphere simultaneously are found to return to the centre also in coincidence, after having suffered a reflection against the highly polished inner surface of the sphere. As in the previous example, the light waves have thus covered equal distances in the same time; whence we conclude that [Pg 82] their speed is the same in all directions.

Inasmuch as this experiment has been performed, yielding the results we have just described, even though the ether drift caused by the earth’s motion should have varied in direction and intensity, the isotropy of space to luminous propagations was thus established. (The experiment described constitutes but a schematic form of Michelson’s.) It is to be noted that in this experiment the observation of coincidences is alone appealed to (even spatial measurements can be eliminated).[25] When it is realised that coincidences constitute the most exact form of observation, we understand why it is that Einstein’s definition is justified.

The optical definition presents a marked superiority over those of classical science. Whereas, with the rotation of the earth, we had to assume the correctness of Newton’s law and of those of mechanics in order to determine the compensations necessitated by the earth’s breathing or by the friction of the tides, in the present case we assume practically nothing, and what little we do assume issues from the most highly refined experiments known to science.

Now, the importance of Einstein’s definition lies not so much in its allowing us to obtain an accurate definition of time in our Galilean frame as in its enabling us to co-ordinate time reckonings in various Galilean frames in relative motion. So long as we restrict our attention to space and time computations in our frame, we may, as before, appeal to vibrating atoms for the measurement of congruent time-intervals and to rigid rods for the purpose of measuring space. It is when we seek to correlate space and time measurements as between various Galilean frames in relative motion that astonishing consequences follow. We discover that the concepts of spatial and temporal congruence of classical science must be modified to a very marked degree. They lose those attributes of universality with which we were wont to credit them. It is then found that congruence can only be defined in a universal way when we consider the extension of four-dimensional space-time.

As we have mentioned, any change in our concepts of space and time congruence is not merely a local affair. Owing to the unity of nature, we shall be led to modify our entire understanding of the relatedness of phenomena, and must not be surprised to learn that a totally new science is the necessary outcome of Einstein’s new definitions. Old laws are cast aside, new ones take their place; and Einstein, by following his mathematical deductions with rigorous logic, without introducing any hypotheses ad hoc, proved that if his new definition of space-time congruence was physically correct, a whole series of hitherto unsuspected natural phenomena should ensue and should be revealed by precise experiment. Wonderful as it may appear, Einstein’s previsions have thus far been verified in every minute detail.

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