space, or for any measure of curvature of an _n_-dimensional manifold in general. =22.= 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._ ds^{2} = Σ{1}^{n} Σ{1}^{n} a{ik} dx{i}.dx{k}, where the coefficients _a{ik}_ are, in general, functions of the coordinates _x{1} x{2} ... x{n}_.[28] The question is: How are we to obtain a definition of the measure of curvature out of this formula? It is noticeable, in the first place, that, just as in a surface we found an infinite number of _radii_ of curvature at a point, so in a manifold of three or more dimensions we must find an infinite number of _measures_ of curvature at a point, one for every two-dimensional manifold passing through the point, and contained in the higher manifold. What we have first to do, therefore, is to define such two-dimensional manifolds. They must consist, as we saw on the surface, of a singly infinite series of geodesics through the point. Now a geodesic is completely determined by one point and its direction at that point, or by one point and the next consecutive point. Hence a geodesic through the point considered is determined by the ratios of the increments of coordinates, _dx{1} dx{2} ... dx{n}_. Suppose we have two such geodesics, in which the _i_′th increments are respectively _d′x{i}_ and _d″x{i}_. Then all the geodesics given by dx{i} = λ′d′x{i} + λ″d″x{i} [Illustration]