492 MATHEMATICS 



is distinguished from the infinity of other such multiplicities by 

 certain well-defined characters. Chief of them are (1) the quadratic 

 differential expression which defines the length of an elementary arc, 

 and (2) a property relative to the displacements of this multiplicity 

 about a point. There are an infinity of space multiplicities which 

 satisfy Riemann's axioms. By extending Gauss's definition of a 

 curvature &, of a surface at a point to curvature of space at a point, 

 by considering the geodesic surfaces passing through that point, 

 Riemann finds that all these spaces fall into three classes according 

 as k is equal to, greater, or less than 0. For ?i=3 and fc=0 we have 

 Euclidean space; when fc<0 we have the space found by Gauss, 

 Lobatschewski, and Bolyai; when fc>0 we have the space first 

 considered in the long-forgotten writings of Seccheri and Lambert, 

 in which the right line is finite. 



Helmholtz, like Riemann, considers space as a numerical multiplic- 

 ity. To characterize it further, Helmholtz makes use of the notions 

 of rigid bodies and free mobility. His work has been revised and ma- 

 terially extended by Lie from the standpoint of the theory of groups. 



In the present category also belong important papers by New- 

 comb and Killing. 



The protective direction. We have already noticed the efforts of 

 the synthetic school to express metric properties by means of project- 

 ive relations. In this the circular points at infinity were especially 

 serviceable. An immense step in this direction was taken by Laguerre, 

 who showed, in 1853, that all angles might be expressed as an anhar- 

 monic ratio with reference to these points, that is, with reference to 

 a certain fixed conic. The next advance is made by Cay ley in his 

 famous sixth memoir on quantics, in 1859. Taking any fixed conic 

 (or quadric, for space) which he calls the absolute, Cayley introduces 

 two expressions depending on the anharmonic ratio with reference 

 to the absolute. When this degenerates into the circular points 

 at infinity, these expressions go over into the ordinary expressions 

 for the distance between two points and the angle between two 

 lines. Thus all metric relations may be considered as protective 

 relations with respect to the absolute. Cayley does not seem to be 

 aware of the relation of his work to non-Euclidean geometry. This 

 was discovered by Klein, in 1871. In fact, according to the nature of 

 the absolute, three geometries are possible; these are precisely the 

 three already mentioned. Klein has made many important contri- 

 butions to non-Euclidean geometry. We mention his modification 

 of v. Staudt's definition of anharmonic ratio so as to be independ- 

 ent of the parallel axiom, his discovery of the two forms of Rie- 

 mann's space, and finally his contributions to a class of geometries 

 first noticed by Clifford and which are characterized by the fact that 

 only certain of its motions affect space as a whole. 



