Note 21 — Introductory material

So far, all has been plain sailing—we have been dealing with purely geometrical ideas in a purely geometrical manner—but we have not, as yet, found any sense of the measure of curvature, in which it can be extended to space, still less to an n-dimensional manifold. For this purpose, we must examine Gauss's method, which enables us to determine the measure of curvature of a surface at any point as an inherent property, quite independent of any reference to the third dimension.