Gauss, the inventor of this conception[23], proved that, in order that two surfaces may be developable upon each other--_i.e._ may be such that one can be bent into the shape of the other without stretching or tearing--it is necessary that the two surfaces should have equal measures of curvature at corresponding points. When this is the case, every figure which is possible on the one is, in general, possible on the other, and the two have practically the same Geometry[24]. As a corollary, it follows that a necessary condition, for the free mobility of figures on any surface, is the constancy of the measure of curvature[25]. This condition was proved to be sufficient, as well as necessary, by Minding[26]. =21.= So far, all has been plain sailing--we have been dealing with purely geometrical ideas in a purely geometrical manner--but we have not, as yet, found any sense of the measure of curvature, in which it can be extended to space, still less to an _n_-dimensional manifold. For this purpose, we must examine Gauss's method, which enables us to determine the measure of curvature of a surface at any point as an inherent property, quite independent of any reference to the third dimension. The method of determining the measure of curvature from within is, briefly, as follows: If any point on the surface be determined by two coordinates, _u_, _v_, then small arcs of the surface are given by the formula ds^{2} = Edu^{2} + 2Fdu dv + Gdv^{2}, where _E_, _F_, _G_ are, in general, functions of _u_, _v_.[27] From this formula alone, without reference to any space outside the surface, we can determine the measure of curvature at the point _u_, _v_, as a function of _E_, _F_, _G_ and their differentials with respect to _u_ and _v_. Thus we may regard the measure of curvature of a surface as an inherent property, and the above geometrical definition, which involved a reference to the third dimension, may be dropped. But at this point a caution is necessary. It will appear in Chap. III. (§ 176), that it is logically impossible to set up a precise coordinate system, in which the coordinates represent spatial magnitudes, without the axiom of Free Mobility, and this axiom, as we have just seen, holds on surfaces only when the measure of curvature is constant. Hence our definition of the measure of curvature will only be _really_ free from reference to the third dimension, when we are dealing with a surface of constant measure of curvature--a point which Riemann entirely overlooks. This caution, however, applies only in space, and if we take the coordinate system as presupposed in the conception of a manifold, we may neglect the caution altogether--while remembering that the possibility of a coordinate system in space involves axioms to be investigated later. We can thus see how a meaning might be found, without reference to any higher dimension, for a constant measure of curvature of three-dimensional