Here again we may calculate any finite area by a process of integration, so we see that the finite geometry of the surface can be studied by concentrating our attention on infinitesimal portions and then extending our results from place to place. In short, the method reduces to an application of the differential calculus to geometrical problems, and for that reason is named differential geometry.
Powerful as this method of differential geometry has proved to be, there are cases in which it cannot be applied. However, as in the problems of physics with which we shall be concerned, difficulties do not arise, we need not dwell on a number of special cases which in the present state of our knowledge are of interest only to the mathematician.[28]
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 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