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Although Lagrange had made numerous major contributions to mechanics, he had not produced a comprehensive work. He decided to write a definitive work incorporating his contributions and wrote to  Laplace on 15 September 1782:-

I have almost completed a Traité de mécanique analytique, based uniquely on the principle of virtual velocities; but, as I do not yet know when or where I shall be able to have it printed, I am not rushing to put the finishing touches to it.

Caraccioli, who was by now in Sicily, would have liked to see Lagrange return to Italy and he arranged for an offer to be made to him by the court of Naples in 1781. Offered the post of Director of Philosophy of the Naples Academy, Lagrange turned it down for he only wanted peace to do mathematics and the position in Berlin offered him the ideal conditions. During his years in Berlin his health was rather poor on many occasions, and that of his wife was even worse. She died in 1783 after years of illness and Lagrange was very depressed. Three years later Frederick II died and Lagrange's position in Berlin became a less happy one. Many Italian States saw their chance and attempts were made to entice him back to Italy.

The offer which was most attractive to Lagrange, however, came not from Italy but from Paris and included a clause which meant that Lagrange had no teaching. On 18 May 1787 he left Berlin to become a member of the Académie des Sciences in Paris, where he remained for the rest of his career. Lagrange survived the French Revolution while others did not and this may to some extent be due to his attitude which he had expressed many years before when he wrote:-

I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable.

The Mécanique analytique which Lagrange had written in Berlin, was published in 1788. It had been approved for publication by a committee of the Académie des Sciences comprising of  Laplace, Cousin,  Legendre and  Condorcet.  Legendre acted as an editor for the work doing proof reading and other tasks. The Mécanique analytique summarised all the work done in the field of mechanics since the time of  Newton and is notable for its use of the theory of differential equations. With this work Lagrange transformed mechanics into a branch of mathematical analysis. He wrote in the Preface:-

One will not find figures in this work. The methods that I expound require neither constructions, nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform course.

Lagrange was made a member of the committee of the Académie des Sciences to standardise weights and measures in May 1790. They worked on the metric system and advocated a decimal base. Lagrange married for a second time in 1792, his wife being Renée-Françoise-Adélaide Le Monnier the daughter of one of his astronomer colleagues at the Académie des Sciences. He was certainly not unaffected by the political events. In 1793 the Reign of Terror commenced and the Académie des Sciences, along with the other learned societies, was suppressed on 8 August. The weights and measures commission was the only one allowed to continue and Lagrange became its chairman when others such as the chemist Lavoisier,  Borda,  Laplace,  Coulomb,  Brisson and  Delambre were thrown off the commission.

In September 1793 a law was passed ordering the arrest of all foreigners born in enemy countries and all their property to be confiscated. Lavoisier intervened on behalf of Lagrange, who certainly fell under the terms of the law, and he was granted an exception. On 8 May 1794, after a trial that lasted less than a day, a revolutionary tribunal condemned Lavoisier, who had saved Lagrange from arrest, and 27 others to death. Lagrange said on the death of Lavoisier, who was guillotined on the afternoon of the day of his trial:-

It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like.

The École Polytechnique was founded on 11 March 1794 and opened in December 1794 (although it was called the École Centrale des Travaux Publics for the first year of its existence). Lagrange was its first professor of analysis, appointed for the opening in 1794. In 1795 the École Normale was founded with the aim of training school teachers. Lagrange taught courses on elementary mathematics there. We mentioned above that Lagrange had a 'no teaching' clause written into his contract but the Revolution changed things and Lagrange was required to teach. However, he was not a good lecturer as  Fourier, who attended his lectures at the École Normale in 1795 wrote:-

His voice is very feeble, at least in that he does not become heated; he has a very pronounced Italian accent and pronounces the s like z ... The students, of whom the majority are incapable of appreciating him, give him little welcome, but the professors make amends for it.

Similarly Bugge who attended his lectures at the École Polytechnique in 1799 wrote:-

... whatever this great man says, deserves the highest degree of consideration, but he is too abstract for youth.

Lagrange published two volumes of his calculus lectures. In 1797 he published the first theory of functions of a real variable with Théorie des fonctions analytique although he failed to give enough attention to matters of convergence. He states that the aim of the work is to give:-

... the principles of the differential calculus, freed from all consideration of the infinitely small or vanishing quantities, of limits or fluxions, and reduced to the algebraic analysis of finite quantities.

Also he states:-

The ordinary operations of algebra suffice to resolve problems in the theory of curves.

Not everyone found Lagrange's approach to the calculus the best however, for example  de Prony wrote in 1835:-

Lagrange's foundations of the calculus is assuredly a very interesting part of what one might call purely philosophical study: but when it is a case of making transcendental analysis an instrument of exploration for questions presented by astronomy, marine, geodesy, and the different branches of science of the engineer, the consideration of the infinitely small leads to the aim in a manner which is more felicitous, more prompt, and more immediately adapted to the nature of the questions, and that is why the Leibnizian method has, in general, prevailed in French schools.

The second work of Lagrange on this topic Leçons sur le calcul des fonctions appeared in 1800.

Napoleon named Lagrange to the Legion of Honour and Count of the Empire in 1808. On 3 April 1813 he was named grand croix of the Ordre Impérial de la Réunion. He died a week later.

J J O'Connor and E F Robertson

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