624 VIII. ON THE STRUCTURE OF MATHEMATICS
a. fixed ratio to it, whence we may put AC*=cy, and so we obtain the important formula s2 =a2x2 +2acxy +b2y2, in which a, b, c are ratios given by fixed numbers. Usually this formula is represented differently, a2 is designated by gu, ac by gn, and b2 by gw, whence our formula becomes s2=gux2+2giixy+gziyi in which the numbers 11,12 and 22 are simply ordering labels without quantitative values, mere subscripts, labels, indices . , which indicate that the different g's have different values. We see that the above formula is the one which was given previously by (2).
The g's with different labels serve just as sides or angles for the determination of the actual sizes of the parallelograms and we call them the factors of the measure determination. They may have different values from mesh to mesh, but if they are known for every mesh, then, by the last formula, the true distance of an arbitrary point P, within an arbitrary mesh from the origin can be calculated.2
The procedure by which we can locate any point on the surface is simple. If our point P is between the two curves x=3 and x=4 we can draw nine curves between these two curves and label them 3,1; 3,2; . . . ; 3,9. If P now lies between curves 3,1 and 3,2 we can draw nine curves between these two curves and label them 3,11; 3,12;. . . j 3,19 , . We could do similarly with the y curves and in this way we would succeed in assigning to any point as accurate a pair of numerical labels as we pleased, and so finally have the gaussian co-ordinates of any point. We used nine curves simply to get the very convenient decimal method of labeling. The cartesian co-ordinate systems which we use in plane geometry obviously represent only special cases of gaussian systems.
As we have already seen, our g's are ratios, and so represent numbers. Such numbers may be regarded as tensors of zero rank for convenience of the mathematical treatment; and the quantities gxx, gxV, gvu, may be treated as components of a tensor. Since this tensor determines the measure relations in any particular region, it is called the metric fundamental tensor. Its value must be given for the region in which we want to make our calculations. It determines the full geometry of the surface in a given region; and, conversely, we can also determine the fundamental tensor in a given region from measurements made in that region, without any previous knowledge of how our curved surface is embedded in 'space' at the place in question. The fundamental tensor in general varies continuously from place to place, and so every geometric manifold may be regarded as the field of its metric fundamental tensor.
Purely mathematical investigations show that the fundamental tensor defines a number called the 'Riemann scalar', which is completely independent of the co-ordinate system and leads to the definition of the curvature tensor, which can be connected with the 'matter tensor'.3
The main importance of the introduction of such arbitrary curves is to produce formulae for the surfaces which remain unaltered for a change of the gaussian co-ordinates - in other words, which remain invariant. This was achieved by the introduction of the relations called ratios which are pure