410 D'ALEMBEET. 



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 over- 

 look his having apparently come very near the Calculus 

 of Variations. In a paper read at the Academy, 17th 

 February, 1734, we find a passage that certainly looks 

 towards that calculus, and shows that he used a new 

 algorithm as requisite for conducting his operation : 

 " J'ai etc oblige," he says, " de faire varier les memes 

 lignes en deux manieres differentes. 11 a fallu designer 

 leurs variations differemment." " J'ai marque les unes 

 commes les geometres Anglais par des fluxions (points) ; 

 les autres par des differences (d x) a notre maniere ; 

 de sorte qu'ici d x ne sera pas la meme chose que #, d x 

 que #," (p. 18.) "II peut y avoir," he afterwards adds, 

 " des problemes qui dependroient de cette methode 

 fluxio-differentielle." 



* See, too, the paper in John Bernoulli's Works, Vol. II., p. 442, 

 where he investigates the transformation of the differential equation dx = 

 P d y (P heing a function of a, or, and y) into one, in which a also is 

 variable. 



f While upon the subject of Partial Differences, we must naturally 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 defi- 

 nition of this calculus, which is really that of the fluxional or differential 

 calculus in general, and which, thouph authorized by an inaccurate pas- 

 sage in Bossut's excellent work, ('Cal. Dif. et Int,' II., 351,) could 

 never have been adopted by any one who did more than copy after another. 

 He afterwards (p. 606) supposes Clairaut's addition to the 'inverse square 



of the distance ( 4 J to have been adding what he calls " a small 



portion of the inverse cube joined to the ordinary term of the inverse 

 square ;" and he considers, most unaccountably, that this is not a function 

 of the distance at all. His account of the calculus of variations is equally 

 vague ; and the example unhappily chosen is one in which the relations of 

 the co-ordinates do not change, but only the amount of the parameter 

 (Ib., p. 600.) I must also most respectfully enter my protest here, once 

 more, against mathematicians writing metaphorically" and poetically, as 

 this learned Professor does in almost every sentence. 



