XXX 



substitution transforming the function x* + y* + z* + . . . . into the 

 function x'*+y' z +z'*+ 



How fruitful the subject has proved may be inferred by noting the 

 subsequent investigations of Sylvester, who has developed it on 

 Cayley's lines, and has added to it many new ideas ; of Tait, who 

 developed the theory of quaternions on parallel lines : of the Peirces, 

 father and son, whose researches on linear associate algebra* gave 

 rise to the notion of matrices from a different source ; of Clifford and 

 Buchheim, who connected the theory with Grassmann's methods ; of 

 Laguerre, in whose memoirf the treatment of a " linear system " 

 (the same as a Cayley matrix) is similar to Cayley's ; and of many 

 other writers, among whom Taber should be mentioned. 



Connected with non-commutative algebraical quantities, Cayley's 

 researches on the theory of groups require a passing notice. He 

 devoted several papers to questions in this theory. Some of them 

 relate to those groups of substitutions, the introduction of which by 

 Galois made an epoch in the theory of equations, others of them 

 relate to groups of homographic transformations, particularly those 

 related to the polyhedral functions. But, so far as can be seen, he 

 limited his published investigations to those groups which are finite 

 and discontinuous. 



Abstract geometry the ideal geometry of n dimensions is a sub- 

 ject that he may almost be said to have created ; no other name than 

 his has been associated with its origin. More than anything else, it 

 marks the line of difference between the kinds of homage accorded 

 to him. Experts regard it as an illustration of his imaginative power : 

 the unlearned regard it as an incomprehensible mystery. 



It finds a place among his earliest investigations, J it was steadily 

 present to his mind, illuminating many of his researches ; and occa- 

 sionally it found explicit treatment, e.g., in his ' Memoir on Abstract 

 Geometry,' and in his Presidential Address at Southport. The theory 

 presents itself in two connections : one, as a need in analysis, the other 

 as a generalisation of the ordinary geometries of two dimensions and 

 of three dimensions. 



The former origin can be indicated in a brief statement. When an 

 occasion arises for dealing with a number of variables, connected in 

 any manner and regarded as either variable or determinate (wholly 

 or partially), the nature of the relations among them is frequently 

 indicated, and often is made more easily intelligible, by associating 

 some geometrical interpretation with the given system of relations. 



* ' Amer. Journ. Math.,' vol. 4 (1881), pp. 97229. 



t " Sur le Calcul des Syslemes Lin^aires " (' Journal de 1'lSc. Poly.,' t. 25, 1867, 

 pp. 215264). 



I ' C. M. P.,' Tol. 1, No. 11 ; 'Camb. Math. Journ.,' vol. 4 (1845), pp. 119127. 

 'C. M. P.,' vol. 6, No. 413; ' Phil. Trans.' (1870), pp. 51-63. 



