as the Gaussian curvature, even though the appellation is not strictly accurate.[29]
Before proceeding farther, we must recall that the method we have followed of investigating the geometry of surfaces and using Euclidean rigid rods for the purpose of conducting measurements over the surface, leads to the same geometrical results as would be obtained by an exploration of a two-dimensional space (of a plane, for example) by means of non-Euclidean rods. If, therefore, we conducted measurements with non-Euclidean measurements over a plane, we should of course obtain a non-Euclidean geometry; and the Gaussian curvature of the plane would no longer vanish, as it would were we to make use of the Euclidean measuring rods.[30]
In terms of our non-Euclidean measurements the same plane would be curved. We see, therefore, that the non-vanishing of the Gaussian curvature does not necessarily represent curvature in the usual visualising sense. It represents more truly a relationship between the surface and the behaviour of our measuring rods; in other words, it represents non-Euclideanism, and the word “curvature” is apt to be misleading. In a general way, therefore, we may state that the type of geometry of our two-dimensional space from place to place is defined by the value of the Gaussian curvature from point to point, hence by the law of
-distribution throughout the space; and that when the space is Euclidean, the
-distribution is always such that the Gaussian curvature vanishes at all points, regardless of the particular mesh-system selected.
Now all these discoveries of Gauss relating to a two-dimensional space were extended by Riemann to spaces of any number of dimensions. Riemann found that for spaces of more than two dimensions the results became very much more complex. In the case of a space of
dimensions we of course require