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
