must be exceedingly minute; for even though finite, the universe is certainly of gigantic proportions, and its curvature therefore exceedingly slight. For the present we shall ignore the
term since in any case its influence would be too insignificant to affect perceptibly such phenomena as those of planetary motions. Only when we come to consider the shape of the universe as a whole and its bearing on the Mach-Einstein principle of the relativity of all motion will the presence of
assume a vast philosophical importance.
Now the methods whereby Einstein was led to the discovery of his law of gravitation might appear rather piecemeal, much like the fitting together of a jig-saw puzzle in which the structural tensors constituted the pieces. But this, in itself, can scarcely be regarded as a drawback to the theory. It is rather the reverse, since it proves that just as the pieces of a puzzle can only be fitted together in certain ways, so now the universe can scarcely be otherwise than it is. In this respect, there is a marked contrast between the present situation and that which endured in classical science, where practically any law of gravitation might have been expected.
Nevertheless, from the standpoint of mathematical elegance, something more was needed. When, therefore, Einstein had cleared the ground and his ideas were more fully understood, the greatest of the pure mathematicians, such as Klein, Hilbert, Weyl and subsequently Einstein himself, undertook to discover the law of gravitation by more general methods. They found that if we accepted the existence of space-time and if we agreed that the gravitational field around matter was due to a curvature of space-time, the Principle of Stationary Action showed that a large number of laws of gravitation could be entertained. If, however, it was specified that by analogy with Newton’s law, the law of curvature should contain no derivatives of the
’s or potentials to an order higher than the second, Einstein’s law of gravitation with or without the