Lagrange^s Memoirs. 105 



lators, and the labor that the use of their formula ought to impose 

 upon them, even after it has been suitably reduced. We would not 

 dare to affirm that Lagrange was more often of this opinion. More 

 than once he openly expressed his wish to see researches purely 

 analytical encouraged ; and evefn when he seemed to propose to 

 himself the utmost facility of common calculations, he still chiefly 

 perfected analysis. 



The general solution of algebraic equations is subject to obstacles 

 thought insurmountable : but. in practice, every determinate prob- 

 lem leads to an equation, of which all the coefficients are given in 

 numbers. It will suffice then to have a sure method of finding all 

 the roots of that equation, which we call numerical. This is the 

 object which Lagrange proposes to himself: he analyzes known 

 methods, demonstrates their uncertainty and insufficiency : he redu- 

 ces the problem to the determination of a quantity smaller than the 

 smallest difference between the roots. Here is much. We cannot 

 too much admire the analytical science that shines every where in 

 this work ; but, notwithstanding all the resources of the genius of 

 Lagrange, we cannot conceal that the work is still too long, and cal- 

 culators will doubtless continue to give the preference to means less 

 direct and more expeditious. Four times the author has returned 

 upon this subject. It must be believed that a convenient and gene- 

 ral solution will be always refused, or that at least it will be by other 

 means than will be worth while to seek one. The author seems to 

 have so recognized himself, by recommending that of M. Budau as 

 the most easy and elegant for resolving all equations : all the roots 

 are real. 



The desire of multiplying useful applications made him undertake 

 a new edition of his Mecanique Analytique. His intent was to un- 

 fold its more common parts. He therein labored with all the ardor 

 and force of head that he had put to it in his better times. But 

 this application left him a fatigue which sometimes made him fall 

 into a swoon. He was found in this state by Madame Lagrange. 

 His head, as he fell, struck upon the corner of a piece of furniture, 

 and the shock deprived him of the use of his senses. This was a 

 warning to take more care of himself. He thought so too ; but he 

 held too much at heart the final digesting of this work. The edition 

 long in suspense, was not finished till 1815. The first volume ap- 

 peared some time before his death : it had been followed by a new 

 edition of his fonctions analytiques. So many labors exhausted 



Vol. XXXI.— No. 1. 14 



