And now we come to the main body of Gauss’ discoveries. We have seen that on a given surface the values of the three ’s at any point, or, more correctly, their variations in value from point to point, are defined by our choice of a mesh-system. But we know that a mesh-system, though in large measure arbitrary, is yet not completely independent of the nature of the underlying surface. For instance, a Cartesian mesh-system of equal squares, or again a diamond-shaped one, both of which hold on a plane, cannot be traced on a sphere. Neither can a network of meridians and parallels which holds on a sphere be traced [Pg 92] on a plane. For this reason the representation of the disposition of oceans and continents is necessarily distorted in some way or other when given on a flat map. In short, every species of surface possesses an infinite aggregate of possible mesh-systems, but those systems which are applicable to one type of surface are never applicable to surfaces of any other type. Inasmuch as the Cartesian mesh-system and the diamond-shaped variety are the only ones that entail the constancy of the three ’s throughout the surface, and inasmuch as such co-ordinate systems can be traced only on a plane or on surfaces derived therefrom without stretching (cylinder, cone), we see that the constancy of the three ’s is characteristic of Euclideanism. This does not mean, of course, that all co-ordinate systems traced on a plane yield constant values for the three ’s; it simply means that on a plane it is always possible to trace a Cartesian mesh-system, whereas on all other types of surfaces the task is impossible. All this goes to prove that the curvature of a surface must exert a modifying influence on the -distribution, since when the surface is curved no constant distribution is possible. We must infer, therefore, that the -distribution is governed by two separate influences; first, by the lay of the mesh-system over the surface; secondly, by the intrinsic curvature of the surface from place to place. Gauss realised the importance of separating these two influences and of determining in what measure respectively they affected the -distribution. Obviously, if it were possible to discover some mathematical expression connecting the ’s at one point with the ’s at neighbouring points, and if this mathematical expression remained invariant in value to a change of mesh-system in spite of the variations of the individual ’s which must accompany the change of mesh-system, we should be in the presence of a magnitude which, transcending our choice of a mesh-system, would refer solely to the shape of the surface itself, i.e., to its curvature at the point considered. But before we investigate the nature of Gauss’ discoveries, certain elementary notions must be recalled.