Thus three-dimensional Euclidean space is taken as a starting point. It is then proposed to show how, in this three-dimensional Euclidean space, a two-dimensional non-Euclidean geometry can arise. The fundamental space which we are here postulating being Euclidean, distances in this space must be computed with rigid Euclidean rods. Then it is shown that if we apply tiny Euclidean rods to the surface of a sphere and conduct measurements on this surface, we shall obtain a series of numerical results which are representative of Riemann’s geometry. The chief advantage of this method of presentation is that it allows us to foresee at a rapid glance that Riemann’s geometry must be consistent,[17] since in the present case it reduces to Euclidean geometry on a sphere, which is a particular case of Euclidean geometry in three-dimensional space; and Euclidean geometry is known to be [Pg 61] consistent. But aside from this advantage the procedure is to be avoided, for it tends to obscure the philosophical importance of non-Euclidean geometry. In the first place it is not always possible to follow this method. Consider, for example, the case of Lobatchewski’s geometry. Just as Riemann’s geometry was that of the spherical surface, so Lobatchewski’s geometry turns out to be that of a peculiar saddle-shaped surface called the pseudosphere, as was proved by Beltrami. Hilbert, however, has shown that there cannot exist a surface free from singularities which would represent the total spread of Lobatchewski’s plane geometry; hence here is a first reason for being on our guard against a number of unsuspected difficulties. But this is not all. If we adopt the method of representing two-dimensional non-Euclidean geometry as the geometry obtained by taking Euclidean measurements on a curved surface, how are we to conceive of three-dimensional Riemannian geometry? Obviously we must start with a four-dimensional Euclidean space in which a three-dimensional spherical surface is embedded. But as a four-dimensional Euclidean space transcends our immediate experience and is in the nature of a mathematical fiction, we should be led to the erroneous conclusion that three-dimensional non-Euclidean geometry must likewise be a purely conceptual construction having nothing in common with measurements which might be conducted with material rods.