1843.] Treatment of Geometry as a branch oj Analysis 111 



2. In the 2nd volume of the Memoirs of Turin, there is a demonstra- 

 tion, purporting to be by M. Daviet de Foncenex, of the parallelogram 

 of forces. Assuming two forces, each equal to a, acting at an angle 9 and 

 denoting their resultant by z t he states z to be a determinate func- 

 tion of a and 0, and that this expression must by the principle of homo- 

 geneity be of the form 



It follows from thence, that the angle remaining constant, z is always 

 proportional to a. " On pourrait," continues the author, " de meme 

 demontrer par cette methode d'une maniere directe et fort naturelle 

 plusieurs theoremes sur la proportionality des cotes des figures et un 

 grand nombre d'autres propositions de geometrie et de mecanique." 



This essay, I have said, bears the name of Foncenex, but I am induc- 

 ed to attribute it to Lagrange, on the foundation of some curious facts 

 revealed by Delambre in his eloge on that mathematician, (Annals of 

 Philosophy, vol. in). It is there stated that Foncenex received the 

 analytical part of his memoirs from Lagrange, and only performed the 

 task of developing the reasoning on which the formulas depend.* 

 Parts of this very memoir were afterwards reclaimed and re-written by 

 Lagrange, and the beauty and boldness of the portion we are consider- 

 ing, betray I think undeniable traces of being ex ungue leonem, even 

 without the collateral evidence. The conclusion of this historiette 

 is amusing. In recompense for the science displayed in these iden- 

 tical memoirs, Foncenex was appointed Minister of Marine by the 

 Sardinian monarch, an honour which separated him from Lagrange, 

 and he ceased in a short time to take interest in mathematical pursuits. 

 Too simple minded to discern between cause and effect, Montucla 

 laments the unaccountable apathy which Foncenex thenceforward dis- 

 played towards researches which had given him profit, and might have 

 added honour. Certain it is, the Minister died and made no sign anent 

 the "plusieurs theoremes de geometrie et de mecanique." 



The essay, which we may therefore attribute to Lagrange, is quoted 

 by Legendre at the foot of his celebrated second note, as doing for me- 



* This, by the way, is the manner in which Gothe is said to have accounted for 

 the fertility and variety of Scott's pen. Sir Walter was supposed to have sketched 

 the plot and skeletoned the chief characters, the whole being then worked up by 

 younger artists at the foot of this Gamaliel ! A delicious theory on fertility and variety 

 by one of the most fertile and varied intellects of the age ! 



