MEMOIR OF MR GREGORY. 195 



have been effected in this respect, had his activity of mind per- 

 mitted him to devote himself more exclusively to the prescribed 

 course of study. 



From henceforth he felt himself more at liberty to follow 

 original speculations, and, not many months after taking his 

 degree, turned his attention to the general theory of the com- 

 bination of symbols. 



It may be well to say a few words of the history of this part 

 of mathematics. 



One of the first results of the differential notation of Leibnitz, 

 was the recognition of the analogy of differentials and powers. 

 For instance, it was readily perceived that 



d m+n cF d 



or, supposing the y to be understood, that 



dx ~ \dx \dx 

 just as in ordinary algebra we have, a being any quantity, 



This, and one or two other remarks of the same kind, were 

 sufficient to establish an analogy between -j- the symbol of 



differentiation and the ordinary symbols of algebra. And it 

 was not long afterwards remarked that a corresponding analogy 

 existed between the latter class of symbols and that which is 

 peculiar to the calculus of finite differences. It was inferred 

 from hence that theorems proved to be true of combinations of 

 ordinary symbols of quantity, might be applied by analogy to 

 the differential calculus and to that of finite differences. The 

 meaning and interpretation of such theorems would of course 

 be wholly changed by this kind of transfer from one part of 

 mathematics to another, but their form would remain unchanged. 

 By these considerations many theorems were suggested, of which 

 it was thought almost impossible to obtain direct demonstrations. 

 In this point of view the subject was developed by Lagrange, 

 who left undemonstrated the results to which he was led, in- 

 timating, however, that demonstrations were required. Gradually, 



132 



