A similar verification would ensue if we were to effect measurements along a circumference and its diameter. Thus, if at a central point on the earth’s surface we fix one end of a rope and cause the other extremity to rotate around the central point, it will describe a circle on the earth’s surface. We shall then find, upon measuring the length of the circumference marked out by the extremity of the rope, that the ratio of the length of this circumference to the length of the diameter is a variable number, always smaller than
and depending on the area covered by our circle; whereas in Euclidean geometry this ratio would be the constant number
.
There is still another result of importance which must be mentioned. On a plane, the farther we wander along a straight line, the farther we move from our starting point, whereas on our sphere we see that if we follow a straight line or geodesic we shall finally return to our starting point after having circled the earth. We express this fact by saying that in Riemann’s geometry space is finite, yet unbounded. It is finite since we cannot wander away indefinitely from our starting point; it is unbounded because however far we go, we never come to a stone wall or a gap beyond which we can proceed no farther. On the other hand, Euclidean space, such as that of the plane surface, is infinite and unbounded.
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 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.