42 



ORIGIN AND SIGNIFICANCE OF 



different flexure, at which farther continuation of the 

 piece is possible. 



In this way too the straightest lines of the pseudo- 

 spherical 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 



FIG. l. 



FIG. 2. 



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. 



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. Lo- 



