Just as in the case of two-dimensional geometry, we may also conceive of a three-dimensional non-Euclidean space as of constant or variable curvature. Prior to Einstein’s discoveries mathematicians concerned themselves more especially with the homogeneous types of geometry, because it was assumed (notwithstanding Riemann’s premonitions) that whatever the practical geometry of real space might turn out to be, it would always remain the same throughout, there being no palpable reason for it to vary here from there. But with the advent of the material or metrical field, conditioned by a more or less capricious distribution of matter, the possibility of a variation in the geometry of space from place to place had to be taken into consideration, and it then became impossible to limit our analysis to spaces of constant curvature. It has been proved that the variable non-Euclideanism of the space surrounding matter in Einstein’s theory—say, over the equatorial plane—would be represented by the geometry of a surface of variable curvature, which turns out to be that of a paraboloid of revolution. There is one additional aspect of Riemann’s geometry which it may be of interest to mention on account of its possible bearing on the shape of the universe in Einstein’s theory. We refer to elliptical space. The type of Riemann’s geometry which we have discussed so far is known as Riemann’s spherical geometry, because in the case of two dimensions it turns out to be the Euclidean geometry of a spherical surface. But there exists another type of Riemannian geometry discovered by Klein. It is called elliptical space, though it has nothing to do with the surface of an ellipsoid. It corresponds to exactly the same geometry as that of Riemann proper, the only difference consisting in the connectivity of the space. That is to say, the paths of continuous passage from a point to a point are different in the two spaces. In a similar way the geometry of the cylinder is Euclidean, as is that of the plane (since a plane sheet of paper can be rolled round a cylinder); but its connectivity is different since we can go round the cylinder by following a geodesic and yet return to our starting point. [Pg 70] In order to illustrate elliptical space in two dimensions, as we have illustrated spherical space, we must assume that we limit the surface of our sphere to one of its hemispheres. It might appear that in this case the surface would no longer be unbounded since it would stop abruptly at the equator, but the connectivity of elliptical space is such that every point on the equator is identical with its antipodal point on the other side of the equator. It would be as though, in our concrete representation, there were cross-connections of zero length uniting into one sole point, each pair of points situated diametrically opposite one another on the equator.