the surface of constant negative curvature formed by the rotation of a tractrix about its asymptote (see Figure 1). Its name emphasizes both its similarity to and difference from a sphere, which is an example of a surface with constant positive curvature. The pseudosphere is of particular interest because figures drawn on smooth parts of this surface
obey the laws of Lobachevskian non-Euclidean geometry. This fact, established in 1868 by E. Beltrami, was of great importance in the dispute over the reality of Lobachevskian geometry.