Now, if our choice of a frame of reference, and our methods of measurement, were not arbitrary, but were imposed in some inevitable way by the texture or nature of space, we might still claim that position and motion, though relative to our frame, were yet absolute, since they could always be determined without ambiguity. But this inevitability of a particular frame of reference is precisely what the amorphous nature of mathematical space renders impossible, for there is nothing in mathematical space which permits us to single out any particular frame. Hence no absolute unique determination of position and motion can be contemplated. In fine, we must conclude that in mathematical space position and motion, like distance and shape, are entirely relative. Thus far we have been considering mathematical space; and mathematical space is but an abstraction from the space of experience. It remains to be seen, therefore, whether our conclusions will apply without modification when we consider the physical space of experience. Here we may state that so far as absolute position is concerned there is nothing to change in our conclusions. Thus, in everyday life, when we speak of returning to the same point in space, we do not really mean the same point in empty space. We mean rather the same point on the earth’s surface; and we do not stop to consider whether the earth has moved through space in the interval. As a matter of fact it would be totally impossible for us to discover exactly what distance the earth had travelled. Granting that we might ascertain the displacement of the earth along its orbit round the sun, all further progress would be arrested by our ignorance of the velocity of the sun through space. We might even measure the velocity of the sun with respect to the stars, but this again would not yield us its velocity through empty space; for we should still remain in complete ignorance of the velocity of the stars as a whole through space. So we see that when we speak of the same point in physical space we are always referring to the same relative position with respect to some observable frame of reference. It is not absolutely necessary that the frame be material, and we might perfectly well conceive of it as being defined by three perpendicular light rays. The essential requirement is that it should be observable, so that all men should agree on the frame they were talking about. Owing to the important part the earth’s crust occupies in our daily lives, the impossibility of defining the same point at different times without the help of a frame of reference is somewhat obscured. But it is imperative to master this aspect of the elusiveness of space before proceeding to the study of the more complex questions raised by the [Pg 46] theory of relativity.