Gaussian Co-Ordinates  
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 equals 1, u equals 2 and u equals 3 are drawn in the diagram. Between the curves u equals 1 and u equals 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 equals 3, v equals 1. Two neighbouring points P and upper P 1 on the surface then correspond to the co-ordinates
StartLayout 1st Row 1st Column upper P colon 2nd Column u comma v 2nd Row 1st Column upper P prime colon 2nd Column u plus d u comma v plus d v EndLayout
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
d s squared equals g 11 d u squared plus 2 g 12 d u d v equals g 22 d v squared
where g 11 comma g 12 comma g 22, are magnitudes which depend in a perfectly definite way on u and v. The magnitudes g 11, g 12 and g 22, determine the behaviour of the rods relative to the u-curves and v-curves, and thus also relative to the surface of the table. For the case in which the points of the surface considered form a Euclidean continuum with reference to the measuring-rods, but only in this case, it is possible to draw the u-curves and v-curves and to attach numbers to them, in such a manner, that we simply have:
d s squared equals d u squared plus d v squared
Under these conditions, the u-curves and v-curves are straight lines in the sense of Euclidean geometry, and they are perpendicular to each other. Here the Gaussian coordinates are simply Cartesian ones. It is clear that Gauss co-ordinates are nothing more than an association of two sets of numbers with the points of the surface considered, of such a nature that numerical values differing very slightly from each other are associated with neighbouring............