is the radius of our sphere; but in the general case,
and
are unequal. According to 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
