622 VIII. ON THE STRUCTURE OF MATHEMATICS
In the beginning of the nineteenth century the mathematician Gauss formulated the internal theory of surfaces without reference to the plenum in which they are embedded. This theory perhaps is and will remain a model on which all theories should be built. He introduced also a new kind of co-ordinates which have become of paramount importance, and which since Einstein are called gaussian co-ordinates. Gauss investigated the theory of surfaces, which are in general curved, embedded in three-dimensional 'space'. In 1854 the great mathematician Riemann generalized the two-dimensional gaussian theory to a continuous manifold of any number of dimensions. Historically, both Gauss and Riemann can be considered as the precursors of Einstein.
Let us imagine a surveyor to have the task of mapping a thickly wooded hilly region. Because of the conditions of his work, he can not use optical instruments, and he has no 'straight lines' to deal with. So euclidean geometry will, in general, not be applicable to the region as a whole. It can be assumed, however, that euclidean geometry may be applied to very small regions which
can be considered flat. What we know already about the differential and integral calculus shows us that such approximations, when taken on a very small scale, are perfecdy reliable and justifiable.
The surveyor would lay out on his ground a network of smoothly curving lines, in two families, an X family and a Y family. (Fig. 9.) All the curves of the X family would intersect all the curves of the Y family but no X curve Fig. 9 would intersect another X curve, nor a Y curve
another Y curve.