[20] Mathematicians have been confronted with our problem in the following form. If we are given a surface (_e.g._ an ellipsoid) in Euclidean three-dimensional space, then there exists for this surface a two-dimensional geometry, just as much as for a plane surface. Gauss undertook the task of treating this two-dimensional geometry from first principles, without making use of the fact that the surface belongs to a Euclidean continuum of three dimensions. If we imagine constructions to be made with rigid rods _in the surface_ (similar to that above with the marble slab), we should find that different laws hold for these from those resulting on the basis of Euclidean plane geometry. The surface is not a Euclidean continuum with respect to the rods, and we cannot define Cartesian co-ordinates _in the surface_. Gauss indicated the principles according to which we can treat the geometrical relationships in the surface, and thus pointed out the way to the method of Riemann of treating multi-dimensional, non-Euclidean _continuum_. Thus it is that mathematicians long ago solved the formal problems to which we are led by the general postulate of relativity. XXV. GAUSSIAN CO-ORDINATES image033 According to Gauss, this combined analytical and geometrical mode of handling the problem can be arrived at in the following way. We imagine a system of arbitrary curves (see Fig. 4) drawn on the surface of the table. These we designate as _u_-curves, and we indicate each of them by means of a number. The Curves _u_ = 1, _u_ = 2 and _u_ = 3 are drawn in the diagram. Between the curves _u_ = 1 and _u_ = 2 we must imagine an infinitely large number to be drawn, all of which correspond to real numbers lying between 1 and 2. We have then a system of _u_-curves, and this “infinitely dense” system covers the whole surface of the table. These _u_-curves must not intersect each other, and through each point of the surface one and only one curve must pass. Thus a perfectly definite value of _u_ belongs to every point on the surface of the marble slab. In like manner we imagine a system of _v_-curves drawn on the surface. These satisfy the same conditions as the _u_-curves, they are provided with numbers in a corresponding manner, and they may likewise be of arbitrary shape. It follows that a value of _u_ and a value of _v_ belong to every point on the surface of the table. We call these two numbers the co-ordinates of the surface of the table (Gaussian co-ordinates). For example, the point _P_ in the diagram has the Gaussian co-ordinates _u_ = 3, _v_ = 1. Two neighbouring points _P_ and _P′_ on the surface then correspond to the co-ordinates _P_: _u, v_ _P′_: _u_ + _du, v_ + _dv_, where _du_ and _dv_ signify very small numbers. In a similar manner we may indicate the distance (line-interval) between _P_ and _P′_, as measured with a little rod, by means of the very small number _ds_. Then according to Gauss we have