40 ORIGIN AND SIGNIFICANCE OF 



the axiom of parallels does not hold good. This is a 

 kind of curved surface which is, as it were, geometri- 

 cally the counterpart of a sphere, and which has there- 

 fore been called the pseudospherical surface by the 

 distinguished Italian mathematician E. Beltrami, who 

 has investigated its properties. 1 It is a saddle-shaped 

 surface of which only limited pieces or strips can be 

 connectedly represented in our space, but which may 

 yet be thought of as infinitely continued in all direc- 

 tions, since each piece lying at the limit of the part 

 constructed can be conceived as drawn back to the 

 middle of it and then continued. The piece displaced 

 must in the process change its flexure but not its 

 dimensions, just as happens with a sheet of paper 

 moved about a cone formed out of a plane rolled up. 

 Such a sheet fits the conical surface in every part, but 

 must be more bent near the vertex and cannot be so 

 moved over the vertex as to be at the same time 

 adapted to the existing cone and to its imaginary 

 continuation beyond. 



Like the plane and the sphere, pseudospherical sur- 

 faces have their measure of curvature constant, so that 

 every piece of them can be exactly applied to every 



1 Saggio di Interpretazio-ne delta Geometric^ Non-Euclidea, Napoli, 

 1 868. Teoria fondamentale degli Spazii di Cwrvativra costante, An- 

 nali di Matematica, Ser. II. Tom. II. pp. 232-55. Both have 

 been translated into French by J. Hoiiel, Annales Scientifiques de 

 VEcole JVormale, Tom V., 1869. 



