It follows that a value of u and a value of v belong to every point on the surface of the table. Here the Gaussian coordinates are samply Cartesian ones. So far, these considerations hold for a continuum of two dimensions. But the Gaussian method can be applied also to a continuum of three, four or more dimensions. If, for instance, a continuum of four dimensions be supposed available, we may represent it in the following way.
Adjacent points correspond to adjacent values of the coordinates.
Only when the continuum is a Euclidean one is it possible to associate the co-ordinates x 1 . . However, the Gauss treatment for ds 2 which we have given above is not always possible. It is only possible when sufficiently small regions of the continuum under consideration may be regarded as Euclidean continua. For example, this obviously holds in the case of the marble slab of the table and local variation of temperature. The temperature is practically constant for a small part of the slab, and thus the geometrical behaviour of the rods is almost as it ought to be according to the rules of Euclidean geometry.