Finally let us consider how these new views will affect the problem of absolute shape and size. We may say that space-time possesses a definite geometry but that this geometry is subject to local variations both in space and in time, as the masses of the universe modify their positions. Yet, as we have mentioned on several occasions, a geometry, even when fully determined and unchanging, does not imply absolute shape and size. It merely regulates the relationships or defines the laws of the mutual dispositions of bodies, and these laws may remain unmodified even though the absolute shape and size of the bodies vary. Restricting ourselves to the problem of absolute size, we shall find that with the finite universe the metrical field yields a universal standard of length given by the radius of the finite universe. As referred to this natural gauge, an object presents a definite size, but obviously there is nothing absolute about the gauge itself. Not only will its magnitude be governed by the total amount of matter in the universe, an amount determined presumably by accident, but furthermore there would be no means of determining the magnitude of this gauge otherwise than by adopting some other gauge, and so on ad infinitum. [Pg 60] CHAPTER V AN ALTERNATIVE VIEW OF THE NON-EUCLIDEAN GEOMETRIES THE type of geometry we obtain is dependent, as we know, on our definition of congruence. If we define as congruent displacements the displacements of those bodies which in ordinary life we consider rigid and undeformable, we obtain Euclidean geometry. If, on the other hand, we define as congruent the displacements of those bodies which in ordinary life we should regard as of changeable shape, we generally obtain some non-Euclidean type of geometry. In order to simplify what is to follow, we shall first consider the particular case of two-dimensional geometry. Consider, then, a plane surface. If on this plane surface we effect measurements with Euclidean rods, we obtain Euclidean results; if we employ rods which according to the Euclidean point of view squirm in an appropriate way when displaced, we obtain the geometry of Riemann or of Lobatchewski, as the case may be. It is to be noted that the plane is the same in all three cases, and yet the geometry we obtain on its surface may be Euclidean or non-Euclidean. It is not the plane itself but our methods of measurement conducted thereon which have changed. Now it may be mentioned that there exist a number of alternative ways of representing both Euclidean and non-Euclidean geometries. Which way is to be preferred depends largely on the scope of our investigations. One of these alternative procedures has often been appealed to in popular writings (by Helmholtz in particular) because it enables us to visualise the sequence of theorems involved, without abandoning thereby our habitual Euclidean representations.