GEOMETRICAL AXIOMS. 41 



other piece, and therefore all figures constructed at 

 one place on the surface can be transferred to any 

 other place with perfect congruity of form, and perfect 

 equality of all dimensions lying in the surface itself. 

 The measure of curvature as laid down by Gauss, 

 which is positive for the sphere and zero for the plane, 

 would have a constant negative value for pseudo- 

 spherical surfaces, because the two principal curvatures 

 of a saddle-shaped surface have their concavity turned 

 opposite ways. 



A strip of a pseudospherical surface may, for exam- 

 ple, 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 symme- 

 try AB, the two arcs ab will describe a pseudospherical 

 concave-convex surface like that of the ring. Above 

 and below, towards aa and 66, 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 pseudospheri- 

 cal surface may be rolled up into the shape of a cham- 

 pagne-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 



