Editing Joseph-Louis Lagrange (section)

==Biography==
{{Quote box |align=right |width=20% |quote=

In appearance he was of medium height, and slightly formed, with pale blue eyes and a colourless complexion. In character he was nervous and timid, he detested controversy, and to avoid it willingly allowed others to take the credit for what he had himself done.

He always thought out the subject of his papers before he began to compose them, and usually wrote them straight off without a single erasure or correction.

[[W.W. Rouse Ball]]<ref name="RouseBall">[[W. W. Rouse Ball]], 1908, [http://www.maths.tcd.ie/pub/HistMath/People/Lagrange/RouseBall/RB_Lagrange.html Joseph Louis Lagrange (1736–1813)]," ''[[Rouse History of Mathematics|A Short Account of the History of Mathematics]]'', 4th ed. pp. 401–412. Complete article online, p.338 and 333: [http://www.gutenberg.org/files/31246/31246-pdf.pdf]</ref>
}}
[[File:Joseph Louis Lagrange2.jpg|thumb|Portrait of Joseph-Louis Lagrange (18th-century)]]

=== Early years ===
Firstborn of eleven children as ''Giuseppe Lodovico Lagrangia'', Lagrange was of Italian and French descent.<ref name=laei/> His paternal great-grandfather was a [[Kingdom of France (1498-1791)|French]] captain of cavalry, whose family originated from the French region of [[Tours]].<ref name=laei/> After serving under [[Louis XIV]], he had entered the service of [[Charles Emmanuel II]], [[Duchy of Savoy|Duke of Savoy]], and married a [[Conti di Segni|Conti]] from the noble Roman family.<ref name=laei/> Lagrange's father, Giuseppe Francesco Lodovico, was a doctor in Law at the [[University of Torino]], while his mother was the only child of a rich doctor of [[Cambiano]], in the countryside of [[Turin]].<ref name=laei/><ref name="St Andrew">[http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Lagrange.html Lagrange] {{webarchive|url=https://web.archive.org/web/20070325121637/http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Lagrange.html |date=25 March 2007 }} St. Andrew University</ref> He was raised as a Roman Catholic (but later on became an [[agnostic]]).<ref>{{cite book|title=Mathematics and the Search for Knowledge|date=1986|publisher=Oxford University Press|isbn=978-0-19-504230-6|author=Morris Kline|page=214|quote=Lagrange and Laplace, though of Catholic parentage, were agnostics.}}</ref>

His father, who had charge of [[Charles Emmanuel III|the King]]'s military chest and was Treasurer of the Office of Public Works and Fortifications in Turin, should have maintained a good social position and wealth, but before his son grew up he had lost most of his property in speculations. A career as a lawyer was planned out for Lagrange by his father,<ref name=laei/> and certainly Lagrange seems to have accepted this willingly. He studied at the [[University of Turin]] and his favourite subject was classical Latin. At first, he had no great enthusiasm for mathematics, finding Greek geometry rather dull.

It was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by [[Edmond Halley]] from 1693<ref>{{cite journal | author = Halley, E. | title = IV. An Instance of the Excellence of the Modern ALGEBRA, in the Resolution of the Problem of finding the Foci of Optick Glasses universally| journal = [[Philosophical Transactions of the Royal Society of London]] | year = 1693 | volume = 17 | pages = 960–969 | url = https://doi.org/10.1098/rstl.1693.0074| issue = 205| doi = 10.1098/rstl.1693.0074| s2cid = 186212029}}</ref>
which he came across by accident. Alone and unaided he threw himself into mathematical studies; at the end of a year's incessant toil he was already an accomplished mathematician. [[Charles Emmanuel III]] appointed Lagrange to serve as the "Sostituto del Maestro di Matematica" (mathematics assistant professor) at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army's early adoption of the ballistics theories of [[Benjamin Robins]] and [[Leonhard Euler]].  In that capacity, Lagrange was the first to teach calculus in an engineering school.  According to [[:it:Alessandro Vittorio Papacino D'Antoni|Alessandro Papacino D'Antoni]], the academy's military commander and famous artillery theorist, Lagrange unfortunately proved to be a problematic professor with his oblivious teaching style, abstract reasoning, and impatience with artillery and fortification-engineering applications.<ref>{{cite book|last=Steele|first=Brett|title=The Heirs of Archimedes: Science and the Art of War through the Age of Enlightenment|date=2005|publisher=MIT Press|location=Cambridge|isbn=0-262-19516-X|pages=368, 375|author-link=Military 'Progress' and Newtonian Science|editor=Brett Steele |editor2=Tamera Dorland|chapter=13}}</ref> In this academy one of his students was [[François Daviet de Foncenex|François Daviet]].<ref>{{cite book | last = de Andrade Martins | first = Roberto | chapter = A busca da Ciência ''a priori'' no final do Seculo XVIII e a origem da Análise dimensional | editor= Roberto de Andrade Martins |editor2=Lilian Al-Chueyr Pereira Martins |editor3=Cibelle Celestino Silva |editor4=Juliana Mesquita Hidalgo Ferreira| title = Filosofia E Historia Da Ciência No Cone Sul. 3 Encontro | url = https://books.google.com/books?id=yiuwanC2NlYC | date = 2008 | publisher = AFHIC | page = 406 | isbn = 978-1-4357-1633-9|language=pt}}</ref>

==== Variational calculus ====
Lagrange is one of the founders of the [[calculus of variations]]. Starting in 1754, he worked on the problem of the [[tautochrone]], discovering a method of maximizing and minimizing functionals in a way similar to finding extrema of functions. Lagrange wrote several letters to [[Leonhard Euler]] between 1754 and 1756 describing his results. He outlined his "δ-algorithm", leading to the [[Euler–Lagrange equation]]s of variational calculus and considerably simplifying Euler's earlier analysis.<ref>Although some authors speak of a general method of solving "[[isoperimetric]] problems", the eighteenth-century meaning of this expression amounts to "problems in variational calculus", reserving the adjective "relative" for problems with isoperimetric-type constraints. The celebrated method of [[Lagrange multipliers]], which applies to the optimization of functions of several variables subject to constraints, did not appear until much later. See {{cite journal | last = Fraser | first = Craig | title = Isoperimetric Problems in the Variational Calculus of Euler and Lagrange | journal = Historia Mathematica | volume = 19 | pages = 4–23 |year = 1992 | doi = 10.1016/0315-0860(92)90052-D | doi-access = free }}</ref> Lagrange also applied his ideas to problems of classical mechanics, generalising the results of Euler and [[Pierre Louis Maupertuis|Maupertuis]].

Euler was very impressed with Lagrange's results. It has been stated that "with characteristic courtesy he withheld a paper he had previously written, which covered some of the same ground, in order that the young Italian might have time to complete his work, and claim the undisputed invention of the new calculus"; however, this chivalric view has been disputed.<ref>Galletto, D., ''The genesis of Mécanique analytique'', La Mécanique analytique de Lagrange et son héritage, II (Turin, 1989). Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 126 (1992), suppl. 2, 277–370, {{MathSciNet|id=1264671}}.</ref> Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773.

==== ''Miscellanea Taurinensia'' ====
In 1758, with the aid of his pupils (mainly with Daviet), Lagrange established a society, which was subsequently incorporated as the [[Turin Academy of Sciences]], and most of his early writings are to be found in the five volumes of its transactions, usually known as the ''Miscellanea Taurinensia''. Many of these are elaborate papers. The first volume contains a paper on the theory of the propagation of sound; in this he indicates a mistake made by [[Isaac Newton|Newton]], obtains the general [[differential equation]] for the motion, and integrates it for motion in a straight line. This volume also contains the complete solution of the problem of a [[vibrating string|string vibrating transversely]]; in this paper, he points out a lack of generality in the solutions previously given by [[Brook Taylor]], [[Jean le Rond d'Alembert|D'Alembert]], and Euler, and arrives at the conclusion that the form of the curve at any time ''t'' is given by the equation <math>y = a \sin (mx) \sin (nt)\,</math>. The article concludes with a masterly discussion of [[echo (phenomenon)|echo]]es, [[beat (acoustics)|beat]]s, and compound sounds. Other articles in this volume are on [[recurrence relation|recurring]] [[series (mathematics)|series]], [[probability|probabilities]], and the [[calculus of variations]].

The second volume contains a long paper embodying the results of several papers in the first volume on the theory and notation of the calculus of variations, and he illustrates its use by deducing the [[principle of least action]], and by solutions of various problems in [[dynamics (mechanics)|dynamics]].

The third volume includes the solution of several dynamical problems by means of the calculus of variations; some papers on the [[integral calculus]]; a solution of a [[Pierre de Fermat|Fermat]]'s problem: given an integer {{math|''n''}} which is not a [[square number|perfect square]], to find a number {{math|''x''}} such that {{math|''nx''<sup>2</sup> + 1}}{{verify source|reason=Not sure that this is the correct formula|date=January 2022}} is a perfect square; and the general differential equations of [[N-body problem|motion for three bodies]] moving under their mutual attractions.

The next work he produced was in 1764 on the [[libration]] of the Moon, and an explanation as to why the same face was always turned to the earth, a problem which he treated by the aid of [[virtual work]]. His solution is especially interesting as containing the germ of the idea of generalised equations of motion, equations which he first formally proved in 1780.

=== Berlin ===
Already by 1756, [[Euler]] and [[Maupertuis]], seeing Lagrange's mathematical talent, tried to persuade Lagrange to come to Berlin, but he shyly refused the offer. In 1765, [[Jean le Rond d'Alembert|d'Alembert]] interceded on Lagrange's behalf with [[Frederick the Great|Frederick of Prussia]] and by letter, asked him to leave Turin for a considerably more prestigious position in Berlin. He again turned down the offer, responding that<ref name="Vinter2000">{{cite book|author=Richard B. Vinter|title=Optimal Control|url=https://books.google.com/books?id=PY9xscfbfiYC|date=2000|publisher=Springer|isbn=978-0-8176-4075-0}}</ref>{{rp|361}}
: ''It seems to me that Berlin would not be at all suitable for me while M.Euler is there''.

In 1766, after Euler left Berlin for [[Saint Petersburg]], Frederick himself wrote to Lagrange expressing the wish of "the greatest king in Europe" to have "the greatest mathematician in Europe" resident at his court. Lagrange was finally persuaded. He spent the next twenty years in [[Prussia]], where he produced a long series of papers published in the Berlin and Turin transactions, and composed his monumental work, the ''Mécanique analytique''. In 1767, he married his cousin Vittoria Conti.

Lagrange was a favourite of the king, who frequently lectured him on the advantages of perfect regularity of life. The lesson was accepted, and Lagrange studied his mind and body as though they were machines, and experimented to find the exact amount of work which he could do before exhaustion. Every night he set himself a definite task for the next day, and on completing any branch of a subject he wrote a short analysis to see what points in the demonstrations or the subject-matter were capable of improvement. He carefully planned his papers before writing them, usually without a single erasure or correction.

Nonetheless, during his years in Berlin, Lagrange's health was rather poor, and that of his wife Vittoria was even worse. She died in 1783 after years of illness and Lagrange was very depressed. In 1786, Frederick II died, and the climate of Berlin became difficult for Lagrange.<ref name="St Andrew" />

=== Paris ===
In 1786, following Frederick's death, Lagrange received similar invitations from states including Spain and [[Naples]], and he accepted the offer of [[Louis XVI of France|Louis XVI]] to move to Paris. In France he was received with every mark of distinction and special apartments in the Louvre were prepared for his reception, and he became a member of the [[French Academy of Sciences]], which later became part of the [[French Institute|Institut de France]] (1795). At the beginning of his residence in Paris, he was seized with an attack of melancholy, and even the printed copy of his ''Mécanique'' on which he had worked for a quarter of a century lay for more than two years unopened on his desk. Curiosity as to the results of the [[French Revolution]] first stirred him out of his lethargy, a curiosity which soon turned to alarm as the revolution developed.

It was about the same time, 1792, that the unaccountable sadness of his life and his timidity moved the compassion of 24-year-old Renée-Françoise-Adélaïde Le Monnier, daughter of his friend, the astronomer [[Pierre Charles Le Monnier]]. She insisted on marrying him and proved a devoted wife to whom he became warmly attached.

In September 1793, the [[Reign of Terror]] began. Under the intervention of [[Antoine Lavoisier]], who himself was by then already thrown out of the academy along with many other scholars, Lagrange was specifically exempted by name in the decree of October 1793 that ordered all foreigners to leave France. On 4 May 1794, Lavoisier and 27 other [[Tax farming|tax farmers]] were arrested and sentenced to death and guillotined on the afternoon after the trial. Lagrange said on the death of Lavoisier:

: ''It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like.''<ref name="St Andrew" />

Though Lagrange had been preparing to escape from France while there was yet time, he was never in any danger; different revolutionary governments (and at a later time, [[Napoleon I of France|Napoleon]]) gave him honours and distinctions. This luckiness or safety may to some extent be due to his life attitude he expressed many years before: "''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''".<ref name="St Andrew" /> A striking testimony to the respect in which he was held was shown in 1796 when the French commissary in Italy was ordered to attend in the full state on Lagrange's father and tender the congratulations of the republic on the achievements of his son, who "had done honour to all mankind by his genius, and whom it was the special glory of [[Piedmont]] to have produced". It may be added that Napoleon, when he attained power, warmly encouraged scientific studies in France, and was a liberal benefactor of them. Appointed [[Sénat conservateur|senator]] in 1799, he was the first signer of the [[Sénatus-consulte]] which in 1802 annexed his fatherland Piedmont to France.<ref name=laei/> He acquired French citizenship in consequence.<ref name=laei/> The French claimed he was a French mathematician, but the Italians continued to claim him as Italian.''<ref name="St Andrew" />

==== Units of measurement ====
Lagrange was involved in the development of the [[metric system]] of measurement in the 1790s. He was offered the presidency of the Commission for the reform of weights and measures (''[[General Conference on Weights and Measures|la Commission des Poids et Mesures]]'') when he was preparing to escape. After Lavoisier's death in 1794, it was largely Lagrange who influenced the choice of the [[metre]] and [[kilogram]] units with [[decimal]] subdivision, by the commission of 1799.<ref name="Delambre1816">{{cite book |last1=Delambre |first1=Jean Baptiste Joseph |title=Mémoires de la classe des Sciences mathématiques et physiques de l'Institut de France, Année 1812, Seconde Partie |location=Paris |publisher=Firmin Didot |date=1816 |pages=xxvii–lxxx |chapter=Notice sur la vie et les ouvrages de M. Malus, et de M. le Comte Lagrange |chapter-url=https://books.google.com/books?id=BbxeAAAAcAAJ&pg=PR1 }}</ref> Lagrange was also one of the founding members of the [[Bureau des Longitudes]] in 1795.

==== École Normale ====
In 1795, Lagrange was appointed to a mathematical chair at the newly established [[École Normale Supérieure|École Normale]], which enjoyed only a short existence of four months. His lectures there were elementary; they contain nothing of any mathematical importance, though they do provide a brief historical insight into his reason for proposing [[undecimal#Undecimal in the history of measurement|undecimal]] or Base 11 as the base number for the reformed system of weights and measures.<ref>{{cite book |last1=Lagrange |first1=Joseph-Louis |last2=Laplace |first2=Pierre-Simon |chapter=Mathématiques |title=Séances des écoles normales, recueillies par des sténographes, et revues par les professeurs. Seconde partie. Débats. Tome premier |location=Paris |publisher=L. Reynier |date=1795 |pages=3–23 |oclc=780161317}}</ref>{{rp|23}} The lectures were published because the professors had to "pledge themselves to the representatives of the people and to each other neither to read nor to repeat from memory" ["Les professeurs aux Écoles Normales ont pris, avec les Représentants du Peuple, et entr'eux l'engagement de ne point lire ou débiter de mémoire des discours écrits"<ref name="Book1">{{cite book |title=Séances des Écoles normales, recueillies par des sténographes, et revues par les professeurs. Nouvelle édition. Leçons. Tome premier |chapter=Avertissement |pages=iii–viii |location=Paris |publisher=Cercle-Social |date=1795 |oclc=490193660 }}</ref>{{rp|iii}}]. The discourses were ordered and taken down in shorthand to enable the deputies to see how the professors acquitted themselves. It was also thought the published lectures would interest a significant portion of the citizenry ["Quoique des feuilles sténographiques soient essentiellement destinées aux élèves de l'École Normale, on doit prévoir quיelles seront lues par une grande partie de la Nation"<ref name="Book1"/>{{rp|v}}].

==== École Polytechnique ====
In 1794, Lagrange was appointed professor of the [[École Polytechnique]]; and his lectures there, described by mathematicians who had the good fortune to be able to attend them, were almost perfect both in form and matter.{{Citation needed|date=February 2008}} Beginning with the merest elements, he led his hearers on until, almost unknown to themselves, they were themselves extending the bounds of the subject: above all he impressed on his pupils the advantage of always using general methods expressed in a symmetrical notation.

However, Lagrange does not seem to have been a successful teacher. [[Joseph Fourier|Fourier]], who attended his lectures in 1795, wrote:
:his voice is very feeble, at least in that he does not become heated; he has a very marked 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 ''professeurs'' make amends for it.<ref>Ivor Grattan-Guinness. Convolutions in French Mathematics, 1800–1840. Birkhäuser 1990. Vol. I, p.108. [https://books.google.com/books?id=dHe00X4MDKMC&q=%22His+voice+is+very+feeble%22&pg=PA108]</ref>

==== Late years ====
[[File:Lagrange's tomb at the Pantheon.jpg|thumb|right|250px|Lagrange's tomb in the crypt of the [[Panthéon, Paris|Panthéon]]]]
In 1810, Lagrange started a thorough revision of the ''Mécanique analytique'', but he was able to complete only about two-thirds of it before his death in Paris in 1813, in 128 [[rue du Faubourg Saint-Honoré]]. Napoleon honoured him with the Grand Croix of the Ordre Impérial de la Réunion just two days before he died. He was buried that same year in the [[Panthéon]] in Paris. The inscription on his tomb reads in translation:<blockquote>JOSEPH LOUIS LAGRANGE. Senator. Count of the Empire. Grand Officer of the Legion of Honour. Grand Cross of the Imperial [[Order of the Reunion]]. Member of the Institute and the Bureau of Longitude. Born in Turin on 25&nbsp;January 1736. Died in Paris on 10&nbsp;April 1813.</blockquote>