40S Biographical Account of [June^ 



been suitably reduced. M. Lagrange had more than once attempted 

 to abridge the usual calculations. 



The general resolution of algebraic equations is subject to diffi- 

 culties which are considered as insurmountable ; but in practice 

 every dctenninate problem brings us to an eqv'.ation, all the coeffi- 

 cients of which are given in numbers. It would be sufficient 

 therefore to have a sure method of finding all the roots of such an 

 equation, which is called numerical. This was the object which 

 M. Lagrange proposed to himself. He analyses all the known 

 methods, and shows their uncertainty and insufficiency. He re- 

 duces tlie problem to the determination of a quantity smaller than 

 the smallest difference between the roots. This is sometliing. We 

 cannot too much admire the analytical skill displayed throughout 

 the whole work. But notwithstanding all the resources of the 

 genius of M. Lagrange, we cannot conceal that the labour of his 

 method is exceedingly great, and calculators will doubtless continue 

 to prefer methods less direct indeed, but more expeditious. The 

 author resumed this subject no less than four times. It is to be 

 feared tliat a commodious and general solution will never be disco- 

 vered, or at least it must be sought for by other means. The author 

 seems to have acknowledged this himself, as he recommends the 

 method of M. Budan as the most convenient and elegant for re- 

 solving equations whose roots are all real. 



'J he desire of midtiplying useful ;ipplications induced him to 

 undertake a new edition of the Mccanique Anaiytique. His project 

 was to develope the most useful parts of it. He lal)oured at it with 

 all the ardour and intcllectunl power which lie could have applied at 

 any period of his life. But this application occasioned a degree of 

 fatigue which threw him into a fainting fit. He was found in that 

 state by Madame Lagrange. His head in falling had struck against 

 the corner of a table, and this shock had not restored him to his 

 senses. This was a warning to take more care of himself. He 

 thought so at first; but he was too anxious to finish his work, the 

 printing of which is at present at the 26th sheet of the second 

 volume. The first volume had appeared some time before his death. 

 It bad been followed by a new edition of his Fonctions Analytiques. 

 So mucli labour exhausted him. Towards the end of March a fever 

 came on, he lost his appetite, his sleep was uneasy, and his waking 

 M'as accompanied by alarming swoonings. He perceived his 

 danger; but, jjrcserving his undisturbable serenity, he studied what 

 passed within him, and, as if he were assisting at a great and un- 

 comrnon' experiment, he bestowed all his attention on it. His 

 remarks have not been lost. Friendship conducted to his house on 

 the Sth of April, in the morning, MM. Lacepede, Morge, and* 

 Chapi;d, who took care to write down the principal ])oiuts of a 

 conversation which v.as his last. (We have scrupulously followed 

 these notes, and the passages under inverted commas are faithfully 

 copied from the manuscript of M, Chaptal.) 



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