4 CONCRETE REPRESENTATIONS OF 



tradictions in the systems of geometry which would be evolved 

 from a contrary assumption. The two geometrical systems 

 which he develops at some length, which are characterised 

 by the sum of the angles of a triangle being greater or less 

 than two right angles, are the well-known non-Euclidean 

 geometries, called by Klein Elliptic and Hyperbolic respec- 

 tively. Saccheri himself, as also Lambert, 1 who struck out 

 the same line independently, believed that the geometry of 

 Euclid was the only logical system, and it was not till Lobach- 

 evsky 2 published the first of his epoch-making works in 1829 

 that non-Euclidean geometry emerged as a system ranking 

 with Euclid's. With the name of Lobachevsky must always 

 be associated that of Bolyai Janos, who arrived independently 

 at the same results by similar methods. His work 3 was 

 published as an appendix to a book of his father, Bolyai Farkas, 

 in 1832. While Saccheri and Lambert both develop the two 

 non-Euclidean geometries, neither Lobachevsky nor Bolyai 

 admitted the possibility of Elliptic geometry, which requires 

 that a straight line be of finite extent. To Riemann 4 is due 

 the conception of finite space, but in his Spherical geometry 

 two straight lines intersect twice like two great circles on a 

 sphere. The conception of Elliptic geometry, in which the 



1 J. H. Lambert, ' Theorie der Parallellinien,' Leipziger Mag. r. ang. Math., 1786. 

 Reprinted in Stackel and Engel's Theorie der Parallellinien. 



2 N. I. Lobachevsky, [On the Foundations of Geometry] (In Russian. German 

 translation by Engel, Leipzig, 1898). Geometrische Unterauchungen zur Theorie der 

 Parallellinien, Berlin, 1840 (2nd ed., 1887), has been translated into English by Halsted 

 (Austin, Texas, 1891). One of the most accessible of his papers is ' GeomStrie imaginaire,' 

 J. Math., Berlin, 17 (1837). Several other papers, originally composed in Russian, 

 have been translated into French, German, or Italian. Lobachevsky's researches 

 first became generally known by means of the translations of Hoiiel in 1866-67. 



3 J. Bolyai, ' Appendix, Scientiam spatii absolute veram exhibens,' Maros-Vasarhely, 

 1832. Translated into English by Halsted (Austin, Texas, 1891 ). 



' B. Riemann, ' Uber die Hypothesen, welche der Geometric zu Grande liegen,' 

 Gottingen, Abh. Oes. Wiss., 13 (1866). The work was written in 1864, but was not 

 published till after the death of the author. English translation by Clifford, Nature, 

 8 (1873). 



