9 5 8 A P P E N D I X F The distance between two neighboring points, P and P’—he said (and demonstrated)— whose coordinates differ by “du” and “dv”—is given by the expression: (2) The quantities —he added—vary in general, each one individually, on passing over the same surface of one system of coordinates “u,” “v,” to another, but—he stated—there is a mathematical entity, comprised of them, which does not depend on the system of reference. That entity is the so-called fundamental tensor, whose components they form. In that tensor, which is also called a curvature tensor, “a quality inherent to the surface is therefore apparent.” Furthermore, he added, both the g values and the fundamental tensor vary in general in a continuous way from one point to another. The calculation of the distance between two points, with which one can then calculate other geometrical elements, therefore requires not only knowing the coordinates—which was sufficient in Euclidean continua—but also knowing the fundamental tensor. Or, to put it the other way around, if one knows the fun- damental tensor as a function of the coordinates, that is, for various points on the surface, it is possible to calculate the distance between two neighboring points or the geometrical elements pertaining to a figure, a basic triangle, for example by knowing the coordinates of the former or the vertices of the latter, one can solve all the problems of the surface ge- ometry. When the surface is Euclidean, it is possible to draw lines of reference such that on all parts of the surface the expression , given in a general way in (2), takes on the form given in (1). If it is non-Euclidean, he said, that is possible only within a very small area, the part sur- rounding the originating point of the construction. This is the same as saying that a very small portion of a surface comes very close to being flat. But if we move away from that place, it is impossible to give Euclidean form. Further, it is clear that each very small region in itself constitutes a Euclidean continuum but if one wants to establish a geometry that at the same time will account for the behavior of all the regions, for which purpose it is unavoidable to refer it to a single system of reference (which can be chosen arbitrarily), the distance between two points inevitably will acquire the general form given by expression (2). ds2 g11du2 2g12dudv g22dv2 + + = g11, g12 g22 , ds2 ds2