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 even when he seemed to propose to 
himself the utmost facility of common colon 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 
Laatenne, we cannot conceal that the work is still too long, wid 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 
Vou. XXXI.—No. 1. 14 
