NON-EUCLIDEAN GEOMETRY 5 



straight line is finite, and is, without any exception, uniquely 

 determined by two distinct points, is due to Klein. 1 



The method inaugurated by Saccheri has now been applied 

 to most of the axioms or fundamental assumptions which lie 

 at the basis of the Euclidean system, and a number of non- 

 Euclidean geometries, many of them of considerable interest, 

 have emerged. We shall be exclusively concerned, however, 

 with the ' classical ' non-Euclidean geometries, Hyperbolic 

 (Lobachevsky-Bolyai) and Elliptic (Riemann-Klein). 



While the development of Hyperbolic geometry in the 

 hands of Lobachevsky and Bolyai led to no apparent internal 

 contradiction, a doubt remained that contradictions might 

 yet be discovered if the investigation were pushed far enough. 

 This doubt was removed by the procedure of Beltrami, 2 who 

 gave a concrete interpretation of non-Euclidean geometry 

 by Euclidean geometry, whereby the straight lines of the former 

 are represented by geodesies upon a surface of constant 

 negative curvature (surface saddle-shaped at every point. 

 The * pseudosphere ' or surface of revolution of the tractrix 

 about its asymptote is a real surface of this description). 

 Any contradiction in non-Euclidean geometry was thus shown 

 to involve a contradiction in Euclidean geometry, and so both 

 geometries must stand or fall together as d priori systems. 



Several other concrete representations have been obtained, 

 and it is proposed to discuss the most important of these. 



1. We shall confine ourselves almost entirely to the 

 representations of plane non-Euclidean geometry, but the 

 extensions to three dimensions will be indicated. We shall 

 also consider for the most part only those representations in 



1 F. Klein, 'tiber die sogenannte Nicht-Euklidische Geometric,' Math. Ann., 

 4 (1871), 6 (1873). French translation in Ann. Fac. sc., Toulouse, 11O (1897). 



2 E. Beltrami, ' Saggio di interpretazione della geometria non-euclidea,' Giorn. 

 Mat., Napoli, 6 (1868). Extended to n dimensions in ' Teoria fondamentale degli 

 spazii di curvatura costante,' Ann. Mat., Milano (2), 2 (1868). Both translated into 

 French by Hoiiel, Ann. fie. Norm., Paris, 6 (1869). 



