Editing Joseph-Louis Lagrange (section)

=== 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.