And now suppose the earth on which we were conducting our measurements were to swell indefinitely. All the characteristics of Riemann’s geometry which we have discussed would gradually fade away, provided we limited our measurements to the same restricted area of the surface. Our measurements would still yield Riemannian results, of course, since however large our earth had grown, it would still remain spherical. But the results of our measurements over a definite area would approximate [Pg 66] more and more to those of Euclidean geometry. The reason is obvious, since the greater the volume of the sphere, the more nearly would a given area of its surface approximate to an ordinary plane. In fact a plane surface may be assimilated to the surface of a sphere of infinite radius; so that if our sphere were to grow indefinitely we should find that our measurements lost little by little their Riemannian characteristics, until it would be impossible to distinguish them from Euclidean measurements. Suppose now that our sphere continued to change in shape after having attained an infinite radius; in particular, suppose that our surface gradually began to assume the shape of a saddle, curved downwards if we intersected it from east to west and curved upwards if we intersected it from north to south. As this double curvature became gradually more pronounced we should find that our measurements lost their Euclidean character and became more and more conspicuously Lobatchewskian. Thus we see that Euclidean geometry stands at the dividing line of Riemannian and Lobatchewskian geometry. When our surface changes in shape the geometry remains Riemannian so long as there subsists the least trace of sphericity, that is, of positive curvature. Likewise it remains Lobatchewskian so long as there subsists the least trace of saddle-shapedness, or negative curvature. Finally, it is strictly Euclidean only when the surface has become a plane, that is, exhibits zero curvature. For this reason Euclidean geometry has a uniqueness about it which is denied to the non-Euclidean varieties. These latter constitute a class of geometries; and the precise type of the geometry we may happen to be discussing is determined by the intensity of curvature of the sphere or pseudosphere to which it pertains. We also see why it is so difficult to determine by empirical methods whether the space of the universe is truly Euclidean or not. It is because both Riemannian and Lobatchewskian geometry merge by insensible gradations into Euclidean geometry.