We then define the total curvature or the Gaussian curvature of the surface at the point by the product . If our surface is a sphere we have , hence the Gaussian curvature becomes or more simply , where is the radius of our sphere; but in the general case, and are unequal. According to [Pg 94] the nature of the surface the two principal curvatures may be of the same or of opposite signs. When of the same sign, the total curvature is obviously positive; hence the surface is said to manifest positive curvature at the point considered. The sphere and ellipsoid are illustrations of surfaces presenting a positive curvature throughout. When, however, the surface is saddle-shaped, the two principal curvatures are of opposite sign; the total Gaussian curvature is then negative, and we have a surface of negative curvature at the point considered. And now let us return to Gauss’ discoveries. We saw that the distribution of the ’s over the surface was affected both by our choice of a mesh-system traced on the surface and by the intrinsic nature or curvature of the surface from point to point. Then we also mentioned that if it were possible to discover some mathematical expression connecting the ’s at a point with the ’s at neighbouring points, and that if this mathematical expression remained invariant in value to a change of mesh-system in spite of the variations of the individual ’s which accompany the change of mesh-system, we should be in the presence of a magnitude which, transcending our choice of mesh-system, would refer solely to the shape of the surface itself, i.e., to its curvature at the point considered. This important mathematical invariant, built up with the ’s, was discovered by Gauss; it is generally designated by the letter and is referred to as the scalar or invariant of curvature at the point . Gauss then proved that this scalar of curvature was none other than minus twice the total curvature defined previously. Hence we may write: [Pg 95] Aside from a constant factor, these two curvatures are thus the same, so that we shall often refer to as the Gaussian curvature, even though the appellation is not strictly accurate.[29] Before proceeding farther, we must recall that the method we have followed of investigating the geometry of surfaces and using Euclidean rigid rods for the purpose of conducting measurements over the surface, leads to the same geometrical results as would be obtained by an exploration of a two-dimensional space (of a plane, for example) by means of non-Euclidean rods. If, therefore, we conducted measurements with non-Euclidean measurements over a plane, we should of course obtain a non-Euclidean geometry; and the Gaussian curvature of the plane would no longer vanish, as it would were we to make use of the Euclidean measuring rods.[30]