714 SCIENTIFIC THOUGHT. 



ent origin: both made use of the more general con- 

 ception of an extended magnitude, introduced the 

 notion of the curvature of space by analog}' with 

 Gauss's measure of curvature of a surface, and tried 

 to express in algebraical formulae the general and 

 necessary properties of a magnitude which should form 

 the foundation of a geometry. The relation of these 

 algebraical results to those arrived at by the critical 

 and purely geometrical methods of Lobatchevski and 

 Bolyai were set out by Beltrami, who showed clearly 

 that three geometries of two dimensions are possible 

 the Euclidean, that of Lobatchevski, where the three 

 angles of a triangle are less than two right angles, 

 and a third where they are more. He showed the 

 analogy of the third with geometry on the sphere, 

 and suggested the pseudo-sphere as a surface on which 

 the second could be similarly represented. At the 

 same time he indicated the generalisation through the 

 algebraical formula of the conception of dimensions, and 

 introduced the symbolical term geometry of four or 

 more dimensions, as Grassmann and Cayley had done 

 before him. 1 Through all these investigations a habit 



1 The geometry of non-Euclidean i the pioneers in this subject. 



space, as well as the geometry 

 of four or more dimensions (both 

 usually comprised under the term 

 " non - Euclidean geometry "), can 

 now boast of an enormous 

 literature, the enumeration of 

 which alone would fill many 

 pages. A complete bibliography 

 up to the year 1878 is given in 

 vols. L and ii. of the American 

 ' Journal of Mathematics ' by Prof. 

 Bruce Halsted, who has done 

 much to make known to English 

 readers the original writings of 



Later publications are referred to 

 in Dr Victor Schlegel's papers 

 ('Leopoldina,' xxii. , 1886, Nos. 

 9-18) : " Ueber Entwickelung und 

 Stand der n-dimensionalen Ge- 

 ometric," &c., &c. In France 

 Houel published (beginning with the 

 year 1866) translations of memoirs 

 referring to this subject ; in fact, 

 he was almost the first to draw 

 attention to this important modern 

 departure. But it is almost ex- 

 clusively owing to the various 

 writings of Prof. Felix Klein that 



