The Normal School, of which he was named professor, but which had only an ephemeral existence, scarcely gave him time to explain his ideas respecting the foundation of arithmetic and algebra, and their application to geometry.
The Polytechnic School, the result of a happier idea, had likewise a more durable success : and among the best effects which it produced, we may place that of having restored La Grange to Analysis. It was there that he had an opportunity of developing those ideas, the germ of which was to be found in two memoirs that he had published in 1772, and the object of which was to explain the true me taphysics of the differential and integral calculus. To render these happy developements more easily understood, the professor associated himself with his pupils. It was then that he composed his Analytical Functions, and his Lectures on that Calculus, of which he published several editions.
It was then.likewise that he published his treatise on the numerical solution of equations, with notes on several points of the theory of algebraic equations.
It is said that Archimedes, whose great reputation, at least with the historians, is founded upon the machines of all kinds, by means of which he retarded the taking of Syra cuse, despised these mechanical inventions, on which he wrote nothing, and placed importance only in his works of pure theory. We may sometimes conceive, that the great mathematicians of our age entertained the same sentiments with Archimedes. They consider a problem as solved when it presents no analytical difficulty, when nothing re mains to be done but differentiations, substitutions and re ductions, operations which require merely patience, and a certain dexterity derived from practice. Satisfied with having removed all the real difficulties, they concern them selves perhaps too little with the embarrassments which they leave to the calculator, and with the long labour ne cesary in order to make use of their formula, even after it has been suitably reduced. M. la Grange had more than once attempted to abridge the usual calculations.
The general resolution of algebraic equations is subject to difficulties which are considered as insurmountable ; but in practice every determinate problem brings us to an equation, all the co-efficients of which are given in num bers. It would be sufficient, therefore, to have a sure me thod of finding all the roots of such an equation which is called numerical. This was the object which M. la Grange proposed to himself. Ile analyses all the known methods, and shews their uncertainty and insufficiency. He reduces the problem to the determination of a quantity smaller than the smallest difference between the roots. This is something. We cannot too much admire the analytical skill displayed throughout the whole work. But notwith standing all the resources of the genius of M. la Grange, we cannot conceal that the labour of his method is exceed ingly great, and calculators will doubtless continue to pre fer methods less direct indeed, but more expeditious. The author resumed this subject no less than four times. It is to be feared, that a commodious and general solution will never be discovered, 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 resolving equations whose roots are all real.
The desire of multiplying useful applications induced him to undertake a new edition of the fifecanique Analy ague . His project was to develope the most useful parts of it. He laboured at it with all the ardour and intellec tual power which he could have applied at any period of his life. But this application occasioned a degree of fa tigue, which threw him into a fainting fit. He was found in that state by Madame La Grange. 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 first volume had appeared some time before his death. It had been follow ed by a new edition of his Fonctions ?lnalytiques. So much labour exhausted him. Towards the end of March a fever came on, he lost his appetite, his sleep was uneasy, and his waking was accompanied by alarming swoonings. He perceived his danger ; but, preserving his undisturb able serenity, he studied what passed within him, and, as if he were assisting at a great and uncommon experiment, he bestowed all his attention on it. His remarks have not been lost. Friendship conducted to his house on the 8th of April, in the morning, MM. Lacepede, Monge. and Chaptal, who took care to write down the principal points of a conversation which was his last. (We have scrupu lously followed these notes, and the passages under invert ed commas are faithfully copied from the manuscript of M. Chaptal.) " He received them with tenderness and cordiality. I was very ill, my friends, (said he,) the day before yester day; I perceived myself dying, my body became weaker, my moral and physical powers were gradually declining; I observed with pleasure the gradual diminution of my strength, and I arrived at the point without pain, without regret, and by a very gentle declivity. Death is not to be feared, and when it comes without violence, it is a last function, which is neither painful nor disagreeable." Then he explained to them his ideas respecting life, the seat of which he considered as spread over the whole body, in every organ and all parts of the machine, which in his case became equally feebler in every part by the same degrees. " A little longer, and there would have been no functions, death would have overspread the whole body, for death is merely the absolute repose of the body ; I wished to die," added he with greater force, " I found a pleasure in it ; but my wife did not wish it. I should have preferred at that time a wife less kind, less eager to restore my strength, and who would have allowed me gently to have finished my career. I have performed my task. I have acquired some celebrity in the mathematics, I have hated nobody, I have clone no ill ; it is now proper to finish." As he was very animated, especially at these last words, his friends, notwithstanding the interest with which they lis tened to him, proposed to retire. He retained them, began to relate to them the history of his life, of his labours, of his success, of his residence at Berlin, where he had often told us what he had seen near a king ; of his arrival at Paris, the tranquillity he had enjoyed at first, the anxiety occasioned to him by the Revolution, and how he had been finally rewarded by a powerful monarch, capable of appre ciating his worth.