The analysis we have given refers to non-Euclidean geometries as derived from the geometry of surfaces; that is, to the numerical results obtained when we use rigid Euclidean rods to effect measurements on curved surfaces. It must be noted, however, that the surfaces we have discussed are of a very special kind. Both spheres and pseudospheres are known as surfaces of constant curvature. There is no need to go into the mathematical definition of what is meant by this; we will limit ourselves to stating the principal characteristic of such surfaces. Consider, for example, a net made of inextensible threads (inextensible or undeformable being used in the Euclidean sense). If this net can be applied with perfect contact to any portion of a plane, it can be slid over the entire plane without ever losing its perfect contact. The same is true of the sphere and pseudosphere. Thus, a net which could be fitted with perfect contact to any part of a sphere could be slid over any portion of the surface without ever losing its perfect contact. [Pg 67] The only surfaces which possess this property are precisely the sphere, the plane and the pseudosphere (and those derived therefrom without stretching); and these surfaces are the surfaces of constant positive curvature, of constant zero curvature, and of constant negative curvature, respectively. So far, then, as the shape of nets applicable to the surface is concerned, there is no means of distinguishing one part of the surface from any other part, since the net can be slid over the surface, preserving its dimensions and never losing its perfect contact with the surface. In other words, the surface appears to be isotropic and homogeneous when measurements with rigid Euclidean rods are conducted over it, or when the nets we slide over it are Euclideanly inextensible, This is the property called free mobility. Such would no longer be the case were we to consider surfaces of variable curvature, e.g., the surface of an ellipsoid or of a trumpet. A net which could be applied with perfect contact to one part of the surface would lose its perfect contact when slid over the surface, unless we deformed it by stretching it or causing it to shrink. It is because all three types of geometry we have mentioned (Euclidean, Riemannian and Lobatchewskian) hold in exactly the same way for all parts of the space in which we are operating, that the surfaces which portray them in two dimensions are necessarily of the constant-curvature type.