1SI4.] M, Lagrange. 323 



life, is astonished that Foncenex interrupted those researches which 

 migiit liave given him a great reputfuion. 



M. Lagrange, while he abandoned to his friend insulated theo- 

 rems, published at the same time, under his own name, tlieories 

 which he promised to develope furtlier. Thus after l)aving givea 

 new formulas of the mnx'nuum and minimum in all cases, after 

 having shown the insufficiency of the known method^, he announces 

 tliat he will treat this subject, which he considered as important, 

 in a work whi<.h he was preparing, in which would be deduqed 

 from the same principles all the mechanical properties of bodies 

 whether solid or fluid. Thus at the age of 23 he had laid the 

 foundation of the great works, which constitute the admiration of 

 philosophers. 



In the same volume he reduces under the differential calculus the 

 theory of recurrent series and the doctrine of ch mces; which be- 

 fore that time had only been treated by indirect methods. He 

 established them upon more natural and general principles. 



Newton had undertaken to submit the motions of Auids to calcu- 

 lation. He had made researches on the propagation of sound ; but 

 his principles were insufficient and even faulty, and his suppositions 

 inconsistent with each other. Lagrange demonstrates this. He 

 founds his new researches on the known laws of dynamics, and by 

 considering only in the air the particles which are in a straight 

 line, he reduces the problem to that of vibrating cords, respecting 

 which the greatest mathematicians differed in opinion. He shows 

 that their calculations are insufficient to decide the question. Ke 

 undertakes a general solution by an analysis equally new and inter- 

 esting, \yhich enables him to resolve at once an indefinite number 

 of equations, and which embraces even discontinued functions. 

 He establishes on more solid grounds the theory of the mixture of 

 simple and regular vibrations of Daniel Bernoulli. He sliows the 

 limits within which this theory is exact, and beyond which it be- 

 comes faulty. Then he comes to the construction given by Euler, 

 a construction true in itself, although its first author had arrived at 

 it by calculations which were not quite rigorous. He answers the 

 objections of D'Alembert He demonstrates that whatever figure 

 is given to the cord, the duration of the oscillations is always the 

 «ame: a truth derived from experiment which D'Alembert consi- 

 dered as very dilficiilt if not impossible to demonstrate. He passes 

 to the propagation of sound, treats of simple and compound echos, 

 of the mixture of sound<, of the possibility of their spreading in 

 the same space without interfering with each osher. He demon- 

 strates rigoiouily the generation of harmonious sounds. Finally, he 

 announces that his intention is to destroy the prejudices of those 

 who still doui)t whether the mathematics eta ever throw a real light 

 upon phyBics. 



We have given this long account of that memoir, because it 19 

 the first by which M, Lagnnige became known. If the analytical 

 reasoning in it be of the n»osi transcendent kind, the object at least 



X 2 



