D'ALEMBERT. 421 



Thus if </> =xy\ its complete dif. d <#> = 2 y JL dy + y*dx, 

 but * ' 



It is quite clear, therefore, that Fontaine gave the nota- 

 tion 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 differ- 

 entialium conditioned dated 1 734,"'' integrated an equation 

 of partial differences; and he had afterwards 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 Nicolas Bernouilli's paper f in 

 the ' Acta Eruditorum' for 1720, on Orthogonal Trajec- 

 tories.;]; 



* 'Petersburg!! Memoirs,' Vol. VII. 



t See, too, the paper in John Bernouilli's Works, Vol. II., p. 442, 

 where he investigates the transformation of the differential equation 

 d x = P d y (P being a function of a, x, and y) into one, in which a 

 also is variable. 



J While upon the subject of Partial Differences, we must natu- 

 rally feel some disappointment that this important subject has not 

 been treated more systematically, especially by later analysts. 

 Some of these, indeed, seem to have formed an extremely vague 

 notion of its nature. Thus Professor Leslie, in his declamatory and 

 inaccurate Dissertation on the progress of mathematical and physical 

 science, (' Encyc. Brit.,' I., 600,) gives a definition of this calculus, 

 which is really that of the fluxional or differential calculus in 

 general, and which, though authorized by an inaccurate passage in 

 Bossut's excellent work, (' Integ. and Dif. Cal.,' II., 351,) could never 

 have been adopted by any one who did more than copy after another. 



