Note 22 — Introductory material

Such a meaning is supplied by Riemann's dissertation, to which, after this long digression, we can now return. We may define a continuous manifold as any continuum of elements, such that a single element is defined by n continuously variable magnitudes. This definition does not really include space, for coordinates in space do not define a point, but its relations to the origin, which is itself arbitrary. It includes, however, the analytical conception of space with which Riemann deals, and may, therefore, be allowed to stand for the moment. Riemann then assumes that the difference—or distance, as it may be loosely called—between any two elements is comparable, as regards magnitude, to the difference between any other two. He assumes further, what it is Helmholtz's merit to have proved, that the difference ds between two consecutive elements can be expressed as the square root of a quadratic function of the differences of the coordinates: i.e.