Adrien-Marie Legendre [ adʁiɛ̃ maʁi ləʒɑ̃ːdʁ ] (born September 18, 1752 in Paris , † January 9, 1833 there ) was a French mathematician .

Caricature Legendres by the French artist Julien Léopold Boilly

## Life

Legendre attended the Collège Mazarin , where he received his doctorate in 1770 (Thèse). Since he came from a wealthy family, he then lived as a private scholar until the French Revolution , and only out of interest did he accept an apprenticeship at the Paris Military Academy ( École Militaire ) from 1775 to 1780, recommended by d'Alembert . In 1782 he won the award of the Berlin Academy of Sciences for determining the trajectory of a projectile taking air resistance into account, which attracted the attention of Lagrange , who was then director of the academy in Berlin. A work submitted to the Paris Academy in January 1783 on the attraction of ellipsoids , in which he also introduced the Legendre polynomials , earned him recognition from the leading French astronomer and mathematician Pierre Simon de Laplace , who ensured that he was a corresponding member and 1785 became an associate member of the Académie des Sciences . In 1785 he dealt with elliptic integrals and in 1786 with number theory - he formulated the Quadratic Reciprocity Law , which Leonhard Euler was already familiar with.

In 1787 he was commissioned by Delambre and Méchain (another member was Cassini ) to measure the longitude between Dunkirk and Barcelona  - the longitudes of the two places differ by only 13 angular minutes - using geodetic triangulation , also with the aim of establishing the basis for determining the Meters to win. They worked together with the observatory in Greenwich and also carried out a triangulation from Greenwich to Paris . At this time he visited William Herschel in England with Cassini and in 1789 became a member of the Royal Society . The publication Exposé des operations, faites en France en 1787 (Paris 1792) reports on the results . In 1791 he became a member of the Commission for the Reorganization of Weights and Measures (Metric Commission). From 1792 he was involved with Gaspard de Prony and other mathematicians such as Lazare Carnot in an extensive project to create mathematical tables ( logarithm tables ).

During the French Revolution he lost his property and had to look for a job. In the time of terror he even had to hide for a while. In 1793 he married Marguerite-Claudine Cohin. In 1794 the first edition of his textbook of geometry appeared, which was used not only in France but also in B. was very influential in the USA in the 19th century and had many editions. From 1795 he taught at the École normal supérieure . In 1808 he was appointed lifelong head of the university, in 1815 an honorary member of the commission for public education and in 1816 an examiner at the École polytechnique as successor to Laplace. In 1812 he replaced Lagrange in the Bureau des Longitudes. In 1820 he was elected a Fellow of the Royal Society of Edinburgh and in 1832 the American Academy of Arts and Sciences .

After he had fallen out with the government - in 1824 he refused to approve a candidate it had proposed for the Institut de France - his pension was canceled. He became impoverished and died in Paris in 1833.

Legendre's tombstone in Auteuil

## plant

Legendre made important contributions in the most diverse areas of mathematics, but was already overshadowed during his lifetime by those of Carl Friedrich Gauß , who was 25 years his junior , who worked in almost all areas on the same subjects as Legendre in strange parallelism, but always penetrated deeper. Legendre discovered the method of least squares before Gauss (and was the first to publish it in 1805), which he also used in astronomy (for determining the orbits of comets from three observations), and before Gauss he also found the law of square reciprocity (which, however, was already Euler in works from 1751 and 1783), whose first evidence comes from Gauss. The term Legendre symbol in number theory is still today a reminder of Legendre's achievements in its formulation. Legendre recognized the contributions of Gauss and took them into account in the heavily revised second edition of his number theory from 1808, but at the same time complained bitterly that Gauss, conversely, claimed all priorities for himself. The asymptotic formula for the prime number distribution can also be found in Legendre's number theory of 1798. It is at the beginning of the use of analytical methods in number theory.

Legendre gave the proof (1825) of Fermat's Great Theorem for the special case n = 5 . He also found a new pair of friendly numbers in 1830 , conjectured the theorem later proved by Dirichlet that there are infinitely many prime numbers in arithmetic progressions in which the first term is relatively prime to the difference of successive terms, and established the Legendre's conjecture that for n> 0 lies between and at least one prime number. Legendre also wrote the three-squares theorem in number theory . ${\ displaystyle n ^ {2}}$${\ displaystyle (n + 1) ^ {2}}$

In analysis , Legendre is not only known for his Legendre polynomials in potential theory, but also for his work on elliptic integrals , in which his division into three “genera” is named after him. He treated it together with other functions defined by integrals such as the gamma function and the beta function in his Exercises du calcul integral, which appeared in three volumes in 1811, 1817, and 1819. It also contains applications of elliptic integrals and extensive tables. Legendre was later no longer satisfied with the presentation and instead of a new edition published the three volumes of the Traite des fonctions elliptiques (1825, 1826, 1830). At that time, however, his book was already outdated by the groundbreaking work of Niels Henrik Abel and Carl Gustav Jacobi .

The geometry textbook by Legendre, first published in 1794, in which he simplified and modernized the elements of Euclid, was of lasting influence . It had 15 editions during its lifetime, was translated into many languages ​​and was widely used in schools in the 19th century, sometimes in an abbreviated form (Blanchet, 1854, 1862). In the appendix there are also simplifications of the proofs of the irrationality of (first proven by Johann Heinrich Lambert ) and of . In contrast to Gauss, he was convinced of the validity of Euclid's postulate of parallels and tried to prove it in vain for 30 years. He published the "proofs" in several editions of his elements of geometry, each time after refutation by other mathematicians he published a new proof until he thought he had given a correct proof in the 12th edition in 1823, which he then no longer replaced. In 1787 he found Legendre's theorem , an approximation formula for approximating spherical triangles . ${\ displaystyle \ pi}$${\ displaystyle \ pi ^ {2}}$

In mechanics, Legendre is also known for the Legendre transformation .

## Others

An engraving by François-Seraphin Delpech (1778–1825), which is often reproduced as a portrait of Legendre, does not show him but the politician Louis Legendre . However, his portrait is among the 73 watercolor caricatures by members of the Institut de France by the artist Julien Léopold Boilly .

Legendre is immortalized by name on the Eiffel Tower, see: The 72 names on the Eiffel Tower .

The Legendre lunar crater and the asteroid (26950) Legendre are named after him.

## Fonts

• Sur la figure des planetes. 1784. Here the Legendre polynomials are mentioned for the first time.
• Éléments de géométrie. Paris 1794. This work was published many times, re-edited by Girard in 1881 and translated into German by Crelle in 1858 (Berlin).
• Memoire sur les transcendantes elliptiques. Paris 1794.
• Essai sur la theory des nombres. Paris 1797/1798. 2nd edition, two volumes; Paris 1808. 3rd edition 1830, two volumes; German Leipzig 1886.
• Nouvelle théorie des paralleles. Paris 1803.
• Nouvelles methodes pour la détermination des orbites des comètes, etc. Paris 1807. New edition 1819, three volumes.
• Exercises du calcul intégral. Paris 1811/1817, three volumes.
• Traité des fonctions elliptiques et integral Euleriennes. Paris 1826–1829, three volumes.
• The elements of geometry, and plane and spherical trigonometry. Rücker, Berlin 1833. Digitized