"In order that a space of two dimensions should have the geometrical properties with which we are familiar, it is necessary that it should be possible at any place to construct a figure congruent to a given figure; and this is so only if the product of the principle radii of curvature at every point of the space or surface be constant. The product is constant in the case (i) of spherical surfaces, where it is positive; (ii) of plane surfaces (which leads to Euclidean geometry), where it is zero; and (iii) of pseudo-spherical surfaces, where it is negative. A tractroid is an instance of a pseudo-spherical surface; it is saddle-shaped at every point. Hence on spheres, planes, and tractroids we can construct normal systems of geometry. These systems are respectively examples of elliptic, Euclidean, and hyperbolic geometries. Moreover, if any surface be bent without dilation or contraction, the measure of the curvature remains unaltered. Thus these three species of surfaces are types of three kinds on which congruent figures can be constructed. For instance a plane can be rolled into a cone, and the system of geometry on a conical surface is similar to that on a plane."