A strip of a pseudospherical surface may, for example, be represented by the inner surface (turned towards the axis) of a solid anchor-ring. If the plane figure _aabb_ (Fig. 1) is made to revolve on its axis of symmetry AB, the two arcs _ab_ will describe a pseudospherical concave-convex surface like that of the ring. Above and below, towards _aa_ and _bb_, the surface will turn outwards with ever-increasing flexure, till it becomes perpendicular to the axis, and ends at the edge with one curvature infinite. Or, again, half of a pseudospherical surface may be rolled up into the shape of a champagne-glass (Fig. 2), with tapering stem infinitely prolonged. But the surface is always necessarily bounded by a sharp edge beyond which it cannot be directly continued. Only by supposing each single piece of the edge cut loose and drawn along the surface of the ring or glass, can it be brought to places of different flexure, at which farther continuation of the piece is possible. In this way too the straightest lines of the pseudospherical surface may be infinitely produced. They do not, like those on a sphere, return upon themselves, but, as on a plane, only one shortest line is possible between the two given points. The axiom of parallels does not, however, hold good. If a straightest line is given on the surface and a point without it, a whole pencil of straightest lines may pass through the point, no one of which, though infinitely produced, cuts the first line; the pencil itself being limited by two straightest lines, one of which intersects one of the ends of the given line at an infinite distance, the other the other end. [Illustration: FIG. 1.] [Illustration: FIG. 2.] Such a system of geometry, which excluded the axiom of parallels, was devised on Euclid’s synthetic method, as far back as the year 1829, by N. J. Lobatchewsky, professor of mathematics at Kasan,[5] and it was proved that this system could be carried out as consistently as Euclid’s. It agrees exactly with the geometry of the pseudospherical surfaces worked out recently by Beltrami. [Footnote 5: _Principien der Geometrie_, Kasan, 1829-30.] Thus we see that in the geometry of two dimensions a surface is marked out as a plane, or a sphere, or a pseudospherical surface, by the assumption that any figure may be moved about in all directions without change of dimensions. The axiom, that there is only one shortest line between any two points, distinguishes the plane and the pseudospherical surface from the sphere, and the axiom of parallels marks off the plane from the pseudosphere. These three axioms are in fact necessary and sufficient, to define as a plane the surface to which Euclid’s planimetry has reference, as distinguished from all other modes of space in two dimensions.