Editing Adrien-Marie Legendre

{{Short description|French mathematician (1752–1833)}}
{{other uses|Legendre (disambiguation){{!}}Legendre}}
{{Use dmy dates|date=July 2017}}
{{Infobox scientist
| name              = Adrien-Marie Legendre
| birth_date        = {{birth date|df=yes|1752|9|18}}
| birth_place       = [[Paris]], France
| death_date        = {{death date and age|df=yes|1833|1|9|1752|9|18}}
| death_place       = Paris, France
| nationality       = French
| field             = Mathematician
| work_institution  = [[École Militaire]]<br>[[École Normale Supérieure|École Normale]]<br>[[École Polytechnique]]
| alma_mater        = [[Collège Mazarin]]
| doctoral_advisor  = <!--there were no PhDs in France before 1808-->
| doctoral_students = <!--there were no PhDs in France before 1808-->
| known_for         = [[Associated Legendre polynomials]]<br>[[Legendre transformation]]<br>[[Legendre polynomial]]s<br>[[Elliptic function]]s<br>Introducing the character [[∂]]<ref>{{cite web|url=http://jeff560.tripod.com/calculus.html|author=Aldrich, John|title=Earliest Uses of Symbols of Calculus|access-date=20 April 2017}}</ref>
| image             = Legendre Adrien-Marie (croquis Barral).png
| caption           = Portrait made from an 1823 sketch
}}
[[File:Blason Adrien-Marie Legendre.svg|thumb|Coat of Arms of Adrien-Marie Legendre, as he was knighted in 1811]]'''Adrien-Marie Legendre''' ({{IPAc-en|l|ə|ˈ|ʒ|ɑː|n|d|ər|,_|-|ˈ|ʒ|ɑː|n|d}};<ref>[http://dictionary.reference.com/browse/legendre "Legendre"]. ''[[Random House Webster's Unabridged Dictionary]]''.</ref> {{IPA|fr|adʁiɛ̃ maʁi ləʒɑ̃dʁ|lang}}; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the [[Legendre polynomials]] and [[Legendre transformation]] are named after him. He is also known for his contributions to the [[Least squares|method of least squares]], and was the first to officially publish on it, though [[Carl Friedrich Gauss]] had discovered it before him.<ref name=":3">{{Cite journal |last=Plackett |first=R.L. |author-link=R. L. Plackett |date=1972 |title=The discovery of the method of least squares |url=https://hedibert.org/wp-content/uploads/2016/08/plackett1972-thediscoveryofthemethodofleastsquares.pdf |journal=Biometrika |volume=59 |issue=2 |pages=239–251}}</ref><ref name=":4">{{Cite journal |last=Stigler |first=Stephen M. |date=1981 |title=Gauss and the Invention of Least Squares |journal=The Annals of Statistics |volume=9 |issue=3 |pages=465–474 |doi=10.1214/aos/1176345451 |jstor=2240811 |issn=0090-5364|doi-access=free }}</ref>

== Life ==
Adrien-Marie Legendre was born in [[Paris]] on 18 September 1752 to a wealthy family. He received his education at the [[Collège Mazarin]] in Paris, and defended his thesis in physics and mathematics in 1770. He taught at the [[École Militaire]] in Paris from 1775 to 1780 and at the [[École Normale Supérieure|École Normale]] from 1795. At the same time, he was associated with the [[Bureau des Longitudes]]. In 1782, the [[Prussian Academy of Sciences|Berlin Academy]] awarded Legendre a prize for his treatise on projectiles in resistant media. This treatise also brought him to the attention of [[Lagrange]].<ref name=":0">{{MacTutor Biography|id=Legendre}}</ref>

The ''[[Académie des sciences]]'' made Legendre an adjoint member in 1783 and an associate in 1785. In 1789, he was elected a [[Fellow of the Royal Society]].<ref>{{cite web| url = http://royalsociety.org/dserve/dserve.exe?dsqIni=Dserve.ini&dsqApp=Archive&dsqCmd=Show.tcl&dsqDb=Persons&dsqPos=0&dsqSearch=%28Surname%3D%27le%20gendre%27%29|title = Library and Archive|publisher= Royal Society|access-date = 6 August 2012}}</ref>

He assisted with the [[Anglo-French Survey (1784–1790)]] to calculate the precise distance between the [[Paris Observatory]] and the [[Royal Greenwich Observatory]] by means of [[trigonometry]]. To this end in 1787 he visited Dover and London together with [[Dominique, comte de Cassini]] and [[Pierre Méchain]]. The three also visited [[William Herschel]], the discoverer of the planet [[Uranus]].

Legendre lost his private fortune in 1793 during the [[French Revolution]]. That year, he also married Marguerite-Claudine Couhin, who helped him put his affairs in order. In 1795, Legendre became one of six members of the mathematics section of the reconstituted Académie des Sciences, renamed the Institut National des Sciences et des Arts. Later, in 1803, Napoleon reorganized the Institut National, and Legendre became a member of the Geometry section. From 1799 to 1812, Legendre served as mathematics examiner for graduating artillery students at the École Militaire and from 1799 to 1815 he served as permanent mathematics examiner for the [[École Polytechnique]].<ref>[[André Weil]], ''Number Theory: An approach through history From Hammurapi to Legendre'', Springer Science & Business Media2006, p. 325.</ref> In 1824, Legendre's pension from the École Militaire was stopped because he refused to vote for the government candidate at the Institut National. In 1831, he was made an officer of the [[Légion d'Honneur]].<ref name=":0"/>

Legendre died in Paris on 9 January 1833, after a long and painful illness, and Legendre's widow carefully preserved his belongings to memorialize him. Upon her death in 1856, she was buried next to her husband in the village of [[Auteuil, Paris|Auteuil]], where the couple had lived, and left their last country house to the village. Legendre's name is one of the [[List of the 72 names on the Eiffel Tower|72 names inscribed on the Eiffel Tower]].
[[File:Tombe Adrien-Marie Legendre, Cimetière d'Auteuil, Paris.jpg|thumb|Legendre's grave at the Auteuil cemetery]]

== Mathematical work==
[[Niels Abel|Abel]]'s work on [[elliptic function]]s was built on Legendre's, and some of [[Carl Friedrich Gauss|Gauss]]'s work in statistics and [[number theory]] completed that of Legendre. He developed, and first communicated to his contemporaries before Gauss, the [[least squares]] method <ref>{{cite journal |author= Stephen M. Stigler|year=1981 |title=Gauss and the Invention of Least Squares |url=http://projecteuclid.org/euclid.aos/1176345451 |journal=Ann. Stat. |volume=9 |issue=3 |pages=465–474 |doi=10.1214/aos/1176345451 |doi-access=free }}</ref> which has broad application in [[linear regression]], [[signal processing]], statistics, and [[curve fitting]]; this was published in 1806 as an appendix to his book on the paths of comets. Today, the term "least squares method" is used as a direct translation from the French "méthode des moindres carrés".

His major work is ''Exercices de Calcul Intégral'', published in three volumes in 1811, 1817 and 1819. In the first volume he introduced the basic properties of elliptic integrals, [[beta function]]s and [[gamma function]]s, introducing the symbol Γ and normalizing it to Γ(n+1) = n!. Further results on the beta and gamma functions along with their applications to mechanics – such as the rotation of the earth, and the attraction of ellipsoids – appeared in the second volume.<ref name=":1">{{Cite book|url=https://books.google.com/books?id=bENTBQAAQBAJ&pg=PA218|title=Creators of mathematical and computational sciences|last1=Agarwal|first1=Ravi P.|last2=Sen|first2=Syamal K.|publisher=Springer|year=2014|isbn=9783319108704|pages=218–19|oclc=895161901}}</ref> In 1830, he gave a proof of [[Fermat's Last Theorem]] for exponent ''n''&nbsp;=&nbsp;5, which was also proven by [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]] in 1828.<ref name=":1" />

In [[number theory]], he conjectured the [[quadratic reciprocity]] law, subsequently proved by Gauss; in connection to this, the [[Legendre symbol]] is named after him. He also did pioneering work on the distribution of [[prime number|primes]], and on the application of analysis to number theory. His 1798 conjecture of the [[prime number theorem]] was rigorously proved by [[Jacques Hadamard|Hadamard]] and [[Charles Jean de la Vallée-Poussin|de la Vallée-Poussin]] in 1896.

Legendre did an impressive amount of work on [[elliptic function]]s, including the classification of [[elliptic integral]]s, but it took [[Niels Henrik Abel|Abel]]'s study of the inverses of [[Carl Gustav Jakob Jacobi|Jacobi]]'s [[Jacobi's elliptic functions|functions]] to solve the problem completely.

He is known for the [[Legendre transformation]], which is used to go from the [[Lagrangian mechanics|Lagrangian]] to the [[Hamiltonian mechanics|Hamiltonian]] formulation of [[classical mechanics]]. In [[thermodynamics]] it is also used to obtain the [[enthalpy]] and the [[Helmholtz free energy|Helmholtz]] and [[Gibbs free energy|Gibbs]] [[Thermodynamic free energy|(free) energies]] from the [[internal energy]]. He is also the namesake of the [[Legendre polynomials]], solutions to Legendre's differential equation, which occur frequently in physics and engineering applications, such as [[electrostatics]].

Legendre is best known as the author of ''Éléments de géométrie'', which was published in 1794 and was the leading elementary text on the topic for around 100 years. This text greatly rearranged and simplified many of the propositions from [[Euclid's Elements|Euclid's ''Elements'']] to create a more effective textbook.

==Honors==
*Foreign Honorary Member of the [[American Academy of Arts and Sciences]] (1832)<ref name=AAAS>{{cite web|title=Book of Members, 1780–2010: Chapter L|url=http://www.amacad.org/publications/BookofMembers/ChapterL.pdf|publisher=American Academy of Arts and Sciences|access-date=28 July 2014}}</ref>
*The [[Moon]] crater [[Legendre (crater)|Legendre]] is named after him.
*Main-belt asteroid [[26950 Legendre]] is named after him.
*Legendre is one of the [[List of the 72 names on the Eiffel Tower|72 prominent French scientists]] who were commemorated on plaques at the first stage of the [[Eiffel Tower]] when it first opened.
*A [[:fr:Rue_Legendre_(Paris)|street]] in [[17th arrondissement of Paris|Paris' 17th Arrondissement]] is named after him.

==Publications==

;Essays
* 1782 ''Recherches sur la trajectoire des projectiles dans les milieux résistants'' (prize on projectiles offered by the Berlin Academy)

;Books
* ''Eléments de géométrie'', textbook 1794
* ''Essai sur la Théorie des Nombres'' 1797-8 ("An VI"), 2nd ed. 1808, 3rd ed. in 2 vol. 1830
* ''Nouvelles Méthodes pour la Détermination des Orbites des Comètes'', 1805
* ''Exercices de Calcul Intégral'', book in three volumes 1811, 1817, and 1819
* ''Traité des Fonctions Elliptiques'', book in three volumes 1825, 1826, and 1830

;Memoires in ''Histoire de l'Académie Royale des Sciences''
* 1783 ''Sur l'attraction des Sphéroïdes homogènes'' (work on Legendre polynomials)
* 1784 ''Recherches sur la figure des Planètes'' p.&nbsp;370
* 1785 ''Recherches d'analyse indéterminée'' p.&nbsp;465 (work on number theory)
* 1786 ''Mémoire sur la manière de distinguer les Maxima des Minima dans le Calcul des Variations'' p.&nbsp;7 (as Legendre)
* 1786 ''Mémoire sur les Intégrations par arcs d'ellipse'' p.&nbsp;616 (as le Gendre)
* 1786 ''Second Mémoire sur les Intégrations par arcs d'ellipse'' p.&nbsp;644
* 1787 ''L'intégration de quelques équations aux différences Partielles'' (Legendre transform)

;In ''Memoires présentés par divers Savants à la l'Académie des Sciences de l'Institut de France''
* 1806 ''Nouvelle formula pour réduire en distances vraies les distances apparentes de la Lune au Soleil ou à une étoile'' (30–54)
* 1807 ''Analyse des triangles tracés sur la surface d'un sphéroide'' (130–161)
* Tome 10 ''Recherches sur diverses sortes d'intégrales défines'' (416–509)
* 1819 ''Méthode des moindres carrés pour trouver le milieu le plus probable entre les résultats de différentes observations'' (149–154), ''Mémoire sur l'attraction des ellipsoïdes homogènes'' (155–183)
* 1823 ''Recherches sur quelques objets d'Analyse indéterminée et particulièrement sur le théorème de Fermat'' (1–60)
* 1828 ''Mémoire sur la détermination des fonctions Y et Z que satisfont à l'équation 4(X^n-1) = (X-1)(Y^2+-nZ^2), n étant un nombre premier 4i-+1'' (81–100)
* 1833 ''Réflexions sur différentes manières de démontrer la théorie des parallèles ou le théorème sur la somme des trois angles du triangle, avec 1 planche'' (367–412)

==Mistaken portrait==
For two centuries, until the recent discovery of the error in 2005, books, paintings and articles have incorrectly shown a profile portrait of the obscure French politician [[Louis Legendre]] (1752–1797) as a portrait of the mathematician. The error arose from the fact that the sketch was labelled simply "Legendre" and appeared in a book along with contemporary mathematicians such as Lagrange. One of only two known portraits of Legendre, rediscovered in 2008, is found in the 1820 book ''Album de 73 portraits-charge aquarellés des membres de I'Institut'', a book of caricatures of seventy-three members of the Institut de France in Paris by the French artist [[Julien-Léopold Boilly]] as shown below.<ref name=Boilly>Boilly, Julien-Léopold. (1820). ''Album de 73 portraits-charge aquarellés des membres de I'Institut'' ([https://www.photo.rmn.fr/C.aspx?VP3=SearchResult&IID=2C6NU00YI4TE watercolor portrait] {{Webarchive|url=https://web.archive.org/web/20220827170003/https://www.photo.rmn.fr/C.aspx?VP3=SearchResult&IID=2C6NU00YI4TE |date=27 August 2022 }} #29). Biliotheque de l'Institut de France.</ref><ref name="Duren2009">{{cite journal | last=Duren | first=Peter | title=Changing Faces: The Mistaken Portrait of Legendre | journal=[[Notices of the American Mathematical Society|Notices of the AMS]] | volume=56 | issue=11 |date=December 2009 | pages=1440–1443, 1455 | url=https://www.ams.org/notices/200911/rtx091101440p.pdf}}</ref> The other portrait is from the book ''Le Panthéon scientifique de la tour Eiffel''.<ref>{{Cite web |last=Nuttle |first=William |date=2022-09-17 |title=Mathematics of Choice — Adrien-Marie Legendre |url=https://medium.com/eiffels-paris-an-engineers-guide/mathematics-of-choice-adrien-marie-legendre-287cdd8052c2 |access-date=2025-05-18 |website=Eiffel’s Paris — an Engineer’s Guide |language=en}}</ref>

{{multiple image
| align = center
| width = 250
| image_gap = 10
| image1 = Legendre and Fourier (1820).jpg
| caption1 = 1820 watercolor caricatures of the French mathematicians Adrien-Marie Legendre (left) and [[Joseph Fourier]] (right) by French artist [[Julien-Léopold Boilly]], watercolor portrait numbers 29 and 30 of ''Album de 73 portraits-charge aquarellés des membres de I'Institut''.<ref name=Boilly/>
| image2 = Louis Legendre.jpg
| caption2 = Side view sketching of French politician [[Louis Legendre]] (1752–1797), whose portrait had been mistakenly used, for nearly 200 years, to represent French mathematician Adrien-Marie Legendre, i.e. up until 2005 when the mistake was discovered.<ref name="Duren2009"/>
}}

== See also ==
{{Portal|Mathematics|Physics}}
* [[List of things named after Adrien-Marie Legendre]]
* [[Associated Legendre polynomials]]
* [[Gauss–Legendre algorithm]]
* [[Legendre's constant]]
* [[Legendre's equation]] in number theory
* [[elliptic integral#Functional relations|Legendre's functional relation]] for elliptic integrals
* [[Legendre's conjecture]]
* [[Legendre sieve]]
* [[Legendre symbol]]
* [[Legendre's theorem on spherical triangles]]
* [[Saccheri–Legendre theorem]]
* [[Least squares]]
* [[Least-squares spectral analysis]]
* [[Seconds pendulum]]

==Notes==
{{reflist|30em}}

== External links ==
*{{Commons category-inline}}
{{EB1911 poster|Legendre, Adrien Marie}}
* {{PlanetMath|urlname=adrienmarielegendre}}
* [http://www.numericana.com/answer/record.htm#legendre The True Face of Adrien-Marie Legendre] (Portrait of Legendre)
* [http://fermatslasttheorem.blogspot.com/2005/10/adrien-marie-legendre.html Biography] at [http://fermatslasttheorem.blogspot.com Fermat's Last Theorem Blog]
* [http://www-history.mcs.st-andrews.ac.uk/References/Legendre.html References for Adrien-Marie Legendre]
* {{in lang|fr}} [http://catalogue.bnf.fr/ark:/12148/cb39315327d Eléments de géométrie] (Paris  : F. Didot, 1817)
* [http://name.umdl.umich.edu/ABN7066.0001.001 Elements of geometry and trigonometry, from the works of A. M. Legendre. Revised and adapted to the course of mathematical instruction in the United States, by Charles Davies.] (New York: A. S. Barnes & co., 1858) : English translation of the above text
* [http://gallica.bnf.fr/ark:/12148/bpt6k62641x/f2.image Mémoires sur la méthode des moindres quarrés, et sur l'attraction des ellipsoïdes homogènes] (1830)
* [http://gallica.bnf.fr/notice?N=FRBNF30775984 Théorie des nombres] (Paris : Firmin-Didot, 1830)
* [http://gallica.bnf.fr/ark:/12148/bpt6k110147r Traité des fonctions elliptiques et des intégrales eulériennes] (Paris : Huzard-Courcier, 1825–1828)
* [http://echo.mpiwg-berlin.mpg.de/MPIWG:PMG5H8WQ Nouvelles Méthodes pour la Détermination des Orbites des Comètes] (Paris  : Courcier, 1806)
* [http://catalogue.bnf.fr/ark:/12148/cb30775982w Essai sur la Théorie des Nombres] (Paris  : Duprat, 1798)
* [https://archive.org/details/exercicesdecalc00legegoog Exercices de Calcul Intégral V.3] (Paris : Courcier, 1816)
* [http://gallica.bnf.fr/ark:/12148/bpt6k90209g/f398.image Correspondance mathématique avec Legendre] in C. G. J. Jacobis gesammelte Werke (Berlin: 1852)

{{Authority control}}

{{DEFAULTSORT:Legendre, Adrien Marie}}
[[Category:1752 births]]
[[Category:1833 deaths]]
[[Category:University of Paris alumni]]
[[Category:18th-century French mathematicians]]
[[Category:19th-century French mathematicians]]
[[Category:French number theorists]]
[[Category:French textbook writers]]
[[Category:Officers of the Legion of Honour]]
[[Category:Fellows of the American Academy of Arts and Sciences]]
[[Category:Members of the French Academy of Sciences]]
[[Category:Fellows of the Royal Society]]
[[Category:Fellows of the Royal Society of Edinburgh]]
[[Category:Scientists from Paris]]