and proceeding initially in a horizontal direction parallel to the earth’s surface. Let us consider, in particular, one of these geodesics, say the one proceeding from east to west. There exists one and only one such geodesic passing through
and starting out horizontally from east to west. According to our tentative theory of gravitation, this would imply that there existed but one path which a body could follow when projected horizontally from
in a westerly direction. But we know that this is not true, since by varying the initial velocity of a body we obtain an infinite variety of parabolas; whence we must conclude that it is impossible to identify the spatial courses of bodies shot out from
with the geodesics of a curved three-dimensional space. But we have seen that unless we are to introduce forces, least action requires that free bodies should describe geodesics. Furthermore, geodesics are the only lines that possess any absolute structural significance. If, then, bodies moving in a gravitational field do not follow them, we cannot attribute the courses of these bodies to the intrinsic geometry of the space. Once more we are compelled to appeal to a force of gravitation, just as Newton did, and our entire theory collapses. In order to succeed in an identification of the courses of bodies in a gravitational field with the geodesics of the continuum, a fourth non-spatial dimension is unavoidable. Only thus can we obtain the infinite number of geodesics necessitated by the infinite number of paths that bodies are known to follow. Here again, space-time solves all our difficulties.
Summarising, we may say that the great merit of Einstein’s general theory resides in the fact that his law of space-time curvature,
(practically the only one possible) yields us not only the motions of bodies, hence the law of gravitation, but also the metrical properties of the fundamental continuum. By this we mean that the law of space-time curvature permits us to anticipate the precise results of measurements with rods and the precise rates of temporal evolution of processes situated in various parts of the gravitational field. Thanks to the law of curvature, we know that an atom must beat faster here than there, hence that its light must be modified in colour. This is, of course, the celebrated Einstein shift-effect since observed by astronomers. The importance Einstein attributes to this phenomenon is plainly illustrated when, writing in 1918, before the shift was observed, he tells us: “If the displacement of spectral lines towards the red by the gravitational potential does not exist, then the general theory of relativity will be untenable.”