48 DYNAMICAL PBINCIPLE. 



ences ; and he had afterwards forgotten his own new calculus, 

 so entirely as to believe that it was first applied by D'Aleru- 

 bert in 1744. So great were the intellectual riches of the first 

 of analysts, that he could thus afford to throw away the inven- 

 tion of a new and most powerful calculus ! A germ of the 

 same method is plainly to be traced in Nicolas Bernoulli's 

 paper* in the ' Acta Eruditorum ' for 1720, on Orthogonal 

 Trajectories. 



While mentioning Fontaine's great and original genius for 

 analytical investigations, we must not overlook his having 

 apparently come very near the Calculus of Variations. In a 



testable, though he laid down no general method ; which, indeed, 

 D'Alembert himself never did, nor any geometrician before the publication 

 of Euler's third vol. of the ' Institutions of the Integral and Differential 

 Calculus.' The problem, as given in the ' Mem. Acad. Petersb.' vol. vii. 

 was this : We have the equation dz ~Pdx + Qda, z being a function 

 of x and a ; and the problem is to find the most general value of P and Q, 

 which will satisfy the equation. Q = Fz + PK, F being a function of a, 

 and R a function of a and x, Euler seeks for the factor which will make 

 d x + E d a integrable. Call this factor S, and make 

 and make f F d a = log. B. 



He finds for the values required 



P = BS/':T, Q = - 



joda 

 and from thence he deduces 



+ z^? and 



. 

 consequently z - B/ : T. 



It is thus clear, that Euler had, in or before 1734, integrated an equa- 

 tion of Partial Differences ; and it must further be remarked, that 

 D'Alembert, in his paper on the Winds, the first application of the cal- 

 culus, quotes Euler's paper of 1734. D'Alembert always differed with 

 Euler respecting the extent to which this calculus can be applied, hold- 

 ing, contrary to Euler's opinion, that it does not include irregular and 

 discontinuous arbitrary functions. 1 



* See, too, the paper in John Bernoulli's Works, vol. ii. p. 442, where 

 he investigates the transformation of the differential equation dx = Pdy 

 (P being a function of a, x, and y} into one, in which a also is variable. 



1 Cousin lias mentioned the anticipation of Euler. ' Astronomic, Disc. Prelim." 



