depends on the amount of matter in the world—partly by its direct effect on the curvature and partly by its influence on the constant of the law of gravitation. The more matter there is, the more space is created to contain it, and if there were no matter the world would shrink to a point.
In the philosophy of Mach a world without matter is unthinkable. Matter in Mach's philosophy is not merely required as a test body to display properties of something already there, which have no physical meaning except in relation to matter; it is an essential factor in causing those properties which it is able to display. Inertia, for example, would not appear by the insertion of one test body in the world; in some way the presence of other matter is a necessary condition. It will be seen how welcome to such a philosophy is the theory that space and the inertial frame come into being with matter, and grow as it grows. Since the laws of inertia are part of the law of gravitation, Mach's philosophy was summed up—perhaps unconsciously—in the profound saying "If there were no matter in the universe, the law of gravitation would fall to the ground."
No doubt a world without matter, in which nothing could ever happen, would be very uninteresting; and some might deny its claim to be regarded as a world at all. But a world uniformly filled with matter would be equally dull and unprofitable; so there seems to be little object in denying the possibility of the former and leaving the latter possible.
The position can be summed up as follows:—in a space without absolute features, an absolute rotation would be as meaningless as an absolute translation; accordingly, the existence of an experimentally determined quantity generally identified with absolute rotation requires explanation. It was remarked on p. 41 that it would be difficult to devise a plan of the world according to which uniform motion has no significance but non-uniform motion is significant; but such a world has been arrived at—a plenum, of which the absolute features are intervals and geodesics. In a limited region this plenum gives a natural frame with respect to which an acceleration or rotation (but not a velocity) capable of absolute definition can be measured. In the case of rotation the local distortions of the frame are of comparatively little account; and this explains
