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astrous crisis of the Revolution roused him once more to activity and cheerfulness. Curiosity impelled him to remain and watch the progress of such a novel phenomenon; but curiosity was changed into dismay as the terrific character of the phenomenon unfolded itself. He now bitterly regretted his temerity in braving the danger. "Tu l'as voulu," he would repeat self-reproachfully. Even from revolutionary tribunals, however, the name of Lagrange uniformly commanded respect. His pension was continued by the National Assembly, and he was partially indemnified for the depreciation of the currency by remunerative appointments. Nominated president of the Academical commission for the reform of weights and measures, his services were retained when its "purification" by the Jacobins removed his most distinguished colleagues. He again sat on the commission of 1799 for the actual construction of the metrical system, arid by his zealous advocacy of the decimal principle of subdivision largely contributed to its adoption.
The interval had, however, been marked by some of the most considerable events in the placid life of our mathematician. On the 31st of May 1792 he married Mademoiselle Lemonnier, daughter of the astronomer of that name, a young and beautiful girl, whose devotion ignored disparity of years, and formed the one tie with life which Lagrange found it hard to break. He had no children by either marriage, and never regretted their absence. Although specially exempted from the operation of the decree of October 1793, imposing banishment on foreign residents, he took alarm at the fate of Bailly and Lavoisier, and prepared to resume his former situation in Berlin. His design was frustrated by the establishment of and his official connexion with the École Normale, and then the École Polytechnique. The former institution had an ephemeral existence, and his lectures there were consequently few and elementary; but amongst the benefits derived from the foundation of the École Polytechnique one of the greatest, it has been observed,[1] was the restoration of Lagrange to mathematics. The remembrance of his teachings was long treasured by such of his auditors – amongst whom were Delambre and Lacroix – as were capable of appreciating them. In expounding the principles of the differential calculus, he started, as it were, from the level of his pupils, and ascended with them by almost insensible gradations from elementary to abstruse conceptions. He seemed, not a professor amongst students, but a learner amongst learners; pauses for thought alternated with luminous exposition; invention accompanied demonstration; and thus originated his Théorie des fonctions analytiques (Paris, 1797). The leading idea of this remarkable work was contained in a paper published in the Berlin Memoirs for 1772.[2] Its object was the elimination of the to some minds unsatisfactory conception of the infinite from the metaphysics of the higher mathematics, and the substitution for the differential and integral calculus of an analogous method depending wholly on the serial development of algebraical functions. By means of this "calculus of derived functions" Lagrange hoped to give to the solution of all analytical problems the utmost "rigour of the demonstrations of the ancients";[3] but it cannot be said that the attempt was successful. The validity of his fundamental position was impaired by the absence of a well-constituted theory of series; the notation employed was inconvenient, and was abandoned by its inventor in the second edition of his Mécanique; while his scruples as to the admission into analytical investigations of the idea of limits or vanishing ratios have long since been laid aside as idle. Nowhere, however, were the keenness and clearness of his intellect more conspicuous than in this brilliant effort, which, if it failed in its immediate object, was highly effective in secondary results. His purely abstract mode of regarding functions, apart from any mechanical or geometrical considerations, led the way to a new and sharply characterized development of the higher analysis in the hands of Cauchy, Jacobi, and others.[4] The Théorie des Fonctions is divided into three parts, of which the first explains the general doctrine of functions, the second deals with its application to geometry, and the third with its bearings on mechanics.
On the establishment of the Institute, Lagrange was placed at the head of the section of geometry; he was one of the first members of the Bureau des Longitudes; and his name appeared in 1791 on the list of foreign members of the Royal Society. On the annexation of Piedmont to France in 1796, a touching compliment was paid to him in the person of his aged father. By direction of Talleyrand, then minister for foreign affairs, the French commissary repaired in state to the old man's residence in Turin, to congratulate him on the merits of his son, whom they declared "to have done honour to mankind by his genius, and whom Piedmont was proud to have produced, and France to possess." Bonaparte, who styled him "la haute pyramide des sciences mathématiques," loaded him with personal favours and official distinctions. He became a senator, a count of the empire, a grand officer of the legion of honour, and just before his death received the grand cross of the order of reunion.
The preparation of a new edition of his Mécanique, to which he devoted himself with extraordinary zeal, exhausted his already failing powers. Frequent fainting fits gave presage of a speedy end, and on the 8th of April 1813 he had a final interview with his friends Lacépède, Monge, and Chaptal. He spoke with the utmost calm of his approaching death; "c'est une dernière fonction," he said, "qui n'est ni pénible ni désagréable." He, however, looked forward to a future meeting, when he promised to complete the autobiographical details which weakness obliged him to interrupt. They remained untold, for he died two days later, April 10, at the age of seventy-seven, and was buried in the Pantheon, the funeral oration being pronounced by Laplace and Lacépède.
Lagrange would never allow his portrait to be painted, holding that a man's works, not his features, deserve remembrance. From a sketch, however, obtained by stealth at a meeting of the Institute, coupled with the descriptions of those who knew him, we can, in some sort, construct an image of his mild and venerable aspect. He was of the middle height, with a slight, well-proportioned figure. His head was finely formed, though not massive; his features strongly marked, with a stamp of grave and noble beauty; eyes ashy blue, habitually cast down in meditation, but when raised, clear and penetrating; complexion pale and faded. The whole physiognomy was more expressive of benignity than of strength, and his social attitude was one of deprecation rather than of self-assertion. He was timid and affable in conversation, slow to give his opinion, though frequently betraying, by his remarks even on subjects alien to his habitual studies, unexpected stores of information and depths of thought. The phrase "Je ne sais pas" became habitual with him, serving to express his sense of failure in the search for words to fit accurately with ideas always precise. Of music he used to say "Je l'aime, parce qu'elle m'isole"; and his most abstruse reasonings were frequently pursued under its soothing influence. The sight of suffering was intolerable to him; he abhorred controversy, tolerated –
