XXXV 



P and Q on the line A,A a can be taken as Xi, + X 2 a 2 , Xj/^-f-X,/?,, 

 Xi7i + Vyi and /iia, + ;< 2 a 2 , fifii+fivfa, nwi+w* respectively. The 

 function (PQ) can then be defined as 



the generalised idea of distance thns finds its definition without any 

 antecedent use of the conception in its ordinary form. Cayley'n view 

 is summed up in his sentence* : " . . . the theory in effect is, that 

 the metrical properties of a figure are not the properties of the figure 

 considered per se apart from everything else, but its properties when 

 considered in connection with another figure, viz. the conic termed 

 the absolute." 



The metrical formulae obtained when the absolute is real are iden- 

 tical with those of Lobatchewsky's and Bolyai's " hyperbolic " 

 geometry: when the absolute is imaginary the formulae are identical 

 with those of Riemann's "elliptic" geometry; the limiting case 

 betweeu the two being that of ordinary Euclidian (" parabolic ") 

 geometry. 



Cayley's memoir leads inevitably to the question, as to how far pro- 

 jective geometry can be defined in terms of space perception without 

 the introduction of distance. This has been discussed by von Staudtf 

 (in 1847, previous to Cayley's memoir), by Klein^ and by Lindemann. 

 The memoir thus points to a division of our space intuitions into two 

 distinct parts : one, the more fundamental as not involving the idea 

 of distance, the other, the more artificial as adding the idea of distance 

 to the former. The consideration of the relation of these ideas to 

 the philosophical account of space has not yet been brought to its 

 ultimate issue. 



It is in analytical geometry, both of curves and of surfaces, that 

 the greatest variety of Cayley's contributions is to be found. There is 

 hardly an important question in the whole range of either subject in 

 the solution of which he has not had some share ; and there are many 

 properties our acquaintance with which is due chiefly, if not entirely, 

 to him. How widely he has advanced the boundaries of knowledge 

 in analytical geometry can be inferred even from the amount of his 

 researches already incorporated in treatises such as those by Salmon, 

 Clebsch and Frost; and yet they represent only a portion of what he 

 has done. In these circumstances only a selection among his con- 



* Loc. cit., 230. 



t ' Geometric der Lage ;' also in his later ' Beitrage zur Geometric der Lage,' 

 1857. 



t ' Math. Ann.,' t. 6 (1873), pp. 112145. 



' Vorlesungen iiber Geometrie ' (Clebech-Lindemann), vol. 2, Part I; tu 

 third section is devoted to the subject. 



