405 



the earliest intimation on this important subject ; for 

 the function of one or both variables which is multi- 

 plied by d x being called M, and that function of one or 

 both which is multiplied \>y d y being called N, the 

 canon or criterion of integrability is that 



= 



d y d x 



and we certainly find this clearly given in a paper of 

 Fontaine's read before the Academy, 19th Nov., 1738. 

 It is the third theorem of that paper. Clairaut laid 

 down the same rule in a Memoir which he presented 

 in 1739 ; but he admits in that Memoir his having seen 

 Fontaine's paper. He expounds the subject more 

 largely in his far fuller and far abler paper of 1740 ; 

 and there he says that Fontaine showed his theorem to 



lande, "If any unexpected discovery shall be made, I believe it will be 

 Fontaine that will make it." (Montucla, iv., 77, note by Lalande.) His 

 name is not even mentioned in the scientific Encyclopaedias; nor does 

 Professor Leslie, in his Dissertation to the ' Encyc. Brit.,' show that he had 

 ever heard of it. The delay of the Academy in publishing his papers is 

 apparently suspected by Montucla as having resulted from some unfair 

 feeling towards him. He was a person of the most philosophic habits, 

 living always in the country, where he cultivated a small estate ; and 

 having had the misfortune to be involved in an oppressive litigation he 

 appears to have abandoned scientific pursuits during the latter years of his 

 life. (Mem., 1771.) We find him mentioned in some of the contemporary 

 Memoirs, among the very first geometricians. Grimm always treats him 

 as such, and he gives some anecdotes of him. " Fontaine vit a la campagne, 

 et ne vient a Paris que rarement. II passe aupres des connaisseurs 

 pour le premier geometre du royaume. II met du genie dans ses ouvrages, 

 et quand on le connait on n'est pas difficile a persuader sur ce point. C'est 

 un homme d'un tour d' esprit tres-piquant. II r6unit une finesse extreme 

 a je ne sais quoi de niais." (Corr. ii., 287.) It must, however, be con- 

 fessed, that Grimm writes on a subject he knew nothing of, having mixed 

 error with truth. Thus he says of D'Alembert, " Sans avoir rien invente, 

 il passe pour mettre beaucoup d'elegance et de clarte dans ses ouvrages 

 geometriques," p. 215 ; thus praising him for exactly that in which he is 

 most deficient, and denying him the originality which was his great merit. 

 Of Clairaut he elsewhere says : " Un tres-grand geometre, presque sur la 

 ligne des Euler, des Fontaine, des Bernouilli, et des D'Alembert. II avait 

 moins de genie que Fontaine, plus de justesse et de surete et moins de 

 penetration que D'Alembert. Ce dernier a perdu a son mort un rival 

 qui le tenait sans cesse en haleine, et c'est une grande perte." (Corr. iv., 

 456.) This latter passage is very just in all respects; but the word 

 "presque" at the beginning is altogether absurd. 



