THE EVOLUTION OF SCIENTIFIC THOUGHT
FROM NEWTON TO EINSTEIN

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

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 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.

CONTENTS

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. 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 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 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 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 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 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 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 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 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 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. 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.

PART I
PRE-RELATIVITY PHYSICS

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.

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 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 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.

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 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 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]

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.

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 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 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 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 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 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 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.

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 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