226 Memoir of Mr Gregory, 



would remain unchanged. By these considerations many 

 theorems were suggested of which it was thought almost im- 

 possible to obtain direct demonstrations. In this point of 

 view the subject was developed by Lagrange, who left unde- 

 monstrated the results to which he was led, intimating, how- 

 ever, that demonstrations were required. Gradually, how- 

 ever, mathematicians came to perceive that the analogy with 

 which they were dealing involved an essential identity ; and 

 thus results, with respect to which, if the expression may be 

 used, it had only been felt that they must be true, were now 

 actually seen to be so. For, if the algebraical theorems by 

 which these results were suggested, were true, because the 

 symbols they involve represented quantities, and such opera- 

 tions as may be performed on quantities, then indeed the ana- 

 logy would be altogether precarious. But if, as is really the 

 case, these theorems are true, in virtue of certain fundamental 

 laws of combination, which hold both for algebraical symbols, 

 and for those peculiar to the higher branches of mathematics, 

 then each algebraical theorem and its analogue constitute, in 

 fact, only one and the same theorem, except quoad their dis- 

 tinctive interpretations, and therefore a demonstration of 

 either is in reality a demonstration of both.* 



The abstract character of these considerations is doubtless 

 the reason why so long a time elapsed before their truth was 

 distinctly perceived. They would almost seem to require, in 

 order that they may be readily apprehended, a peculiar faculty 

 — a kind of mental disinvoltura which is by no means common. 



Mr Gregory, however, possessed it in a very remarkable 

 degree. He at once perceived the truth and the importance 

 of the principles of which we have been speaking, and pro- 

 ceeded to apply them with singular facility and fearlessness. 



It had occurred to two or three distinguished writers that 



* If, as it has been suggested, the values of certain definite integrals 

 are to be looked upon as merely arithmetical results, then in such cases 

 we are not at liberty to replace the constants involved in the definite 

 integral by symbols of operation. In other cases we are at liberty to do 

 so, and this remarkable application of the principles stated in the text 

 has already led Mr Boole of Lincoln, with whom it seems to have ori- 

 ginated, to several curious conclusions. 



