D'ALEMBEET. 409 



It is here to be observed, that not only Fontaine had, 

 apparently, first of all the geometricians, given the crite- 

 rion of integrability, but he had also given the notation 

 which was afterwards adopted for the calculus of Partial 

 Differences. < being a function of two variables, x and 



v, he makes - stand for the differential coefficient of 

 d x 



d> when x only varies, and ~- for the same differential 



dy 



coefficient when y only varies. Hence he takes - 



CL $ 



X d x, not, as in the ordinary notation it would be, = 

 d 0, the complete differential of ^ ; whereas that differ- 

 ential would, in this solution, be 



d x dy 



Thus if $ = x i/ 2 , its complete dif. d $ = 2 y x d y + 



7/ 2 ^, but d^ = y 2 



It is quite clear, therefore, that Fontaine gave the 

 notation of this calculus. 



But D'Alembert had been anticipated in the method 

 itself, as well as in the notation or algorithm ; for 

 Euler, in a paper entitled ' Investigatio functionum ex 

 data differentialium conditione,' dated 1734,* integrated 

 an equation of partial differences ; and he had after- 

 wards forgotten his own new calculus, so entirely as to 

 believe that it was first applied by D'Alembert in 1744. 

 So great were the intellectual riches of the first of 

 analysts, that he could thus afford to throw away the 

 invention of a new and most powerful calculus ! A 

 germ of the same method is plainly to be traced in 



Petersburgh Memoirs,' Vol. VII. 



