196 MEMOIR OF MR GREGORY. 



however, 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 

 analogy would be altogether precarious. But if, as is really 

 the case, these theorems are true, in virtue of certain funda- 

 mental 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 distinctive 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 maybe 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 proceeded 

 to apply them with singular facility and fearlessness. 



It had occurred to two or three distinguished writers that the 

 analogy, as it was called, of powers, differentials, &c., might 

 be made available in the solution of differential equations, and 

 of equations in finite differences. 



This idea, however, probably from some degree of doubt as 

 to the legitimacy of the methods which it suggested, had not 

 been fully or clearly developed : it seems to have been chiefly 

 employed as affording a convenient way of expressing solutions 

 already obtained by more familiar considerations. 



To this branch of the subject Mr Gregory directed his 



^ * The values of certain definite integrals are to be looked upon as merely 

 arithmetical results ; in such cases we are not at liberty to replace the constants 

 involved in the definite integrals 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 originated, 

 to several curious conclusions. 



