It is impossible to show more clearly that the geometry we attribute to space is nothing but an expression of the properties of our bodies and measuring rods when displaced, since here we have a plane which is Euclidean or Riemannian in its geometry according to the behaviour of the bodies that glide over its surface. And so we see that two alternative methods of interpreting non-Euclidean geometry are open to us. Either we may regard our rods as Euclidean and applied to a curved surface, or else we may attribute non-Euclideanism to the behaviour of our rods on a flat surface.[20] Now in this chapter we have discussed only non-Euclidean geometries of two dimensions. Riemann’s geometry and Lobatchewski’s geometry are more particularly these same geometries, extended to the case of three dimensions. We can easily see how Riemann’s geometry of three dimensions would arise. We have only to suppose that, as contrasted with Euclidean solids, all material bodies expand as did the shadows when displaced from a fixed centre. But the Riemannian observer of course would have no realisation of this expansion, owing to the modified laws of light propagation and owing to his own body’s expanding in company with all other bodies. So he would be of the opinion that all these bodies maintained their same shape and size, that is, remained congruent when displaced. In fact, if he applied the test of practical congruence he would find that these bodies which the Euclidean observer would claim were expanding were, so far as he himself was concerned, always the same both in visual appearance and as regards exploration by the sense of touch. If our Riemannian being were to view bodies which the Euclidean man considered rigid and undeformable, he would assert without hesitation that those bodies were decreasing in size as they approached a certain point. Once again maintenance of shape and size is essentially relative. All the conclusions which we reached when discussing a Riemannian [Pg 69] world of two dimensions can be extended to the case of three dimensions. Thus, a Riemannian universe or spherical space of three dimensions is finite but unbounded. Travelling for a sufficient length of time along a geodesic or straight line, we should return finally to our starting point, and rays of light which follow geodesics would circle round our space.