XX11 



in the background to cause doubt or suggest incompleteness. He 

 has been well described by Glaisher as an unequalled master of 

 analytical processes ; it is especially in algebraical manipulation that 

 his strength and his facility stand out in clear view. His success in 

 this direction was achieved by a skill that cannot be explained by 

 describing it as due to acquired knowledge, or to practice, or to long 

 consideration and patient selection. It was rather an instinct for the 

 management of the most complicated processes, and the way in which 

 he controls the most elaborate calculations is sometimes little short of 

 extraordinary. 



As regards his methods, he does not seem to have cast about so as 

 to choose one rather than another. As soon as he had thought of 

 any method the possible effectiveness of which he could settle almost 

 intuitively (" one's best things are done in five minutes," he once- 

 said to me, in confirmation of the satisfaction I was expressing at the 

 fruitfulness of an idea that had occurred to me unexpectedly), the 

 rest was the exercise of his powers. Among the methods he pre- 

 ferred, especially during the last twenty-five years of his life, was 

 that of verification ; in his hands it proved a weapon of great force. 

 Indeed, only less remarkable than his algebraical skill, was the 

 insight which enabled him to preserve the exact equivalence of all 

 the equations in any particular process, so that he could have reversed 

 each process merely by reversing the steps as they were made, and 

 could have proceeded to the required theorem from the initial expres- 

 sion of an algebraical fact. Numerous instances of this quality in his 

 work could be adduced ; it will be sufficient to refer to some parts of 

 his paper* " On the centre-surface of an ellipsoid." 



But though Cayley was specially happy in the treatment of alge- 

 braical developments, an inadequate estimate of his genius would be 

 obtained by supposing that he was almost entirely an analyst. Much 

 of his thinking, not a little of his writing, is completely geometrical ; 

 and his contributions to line geometry, his introduction of the 

 Absolute into geometry, his continued recurrence to the methods in 

 pure geometry invented by Poncelet and Chasles, should be sufficient 

 to range him among geometricians. 



Moreover, even in strictly analytical work, the synthetic element is 

 often not far away though it does not always appear on the surface. 

 In this connexion an acute suggestion, made by Salmon and perhaps 

 based upon his remembrance of their mathematical correspondence 

 that lasted through many years, is confirmed by one of the Notes 

 Gay ley himself added at the end of the second volume of his 'Collected 

 Mathematical Papers.' An enquiry sometimes begins by a compara- 

 tively easy problem which, when solved, leads to wider inferences ; so 



* ' C. M. P.,' vol. 8, No. 520: ' Camb. Phil. Trans.,' vol. 12 (1873), pp. 319 

 365. 



