**Adrien-Marie Legendre**'s major work on elliptic integrals provided basic analytical tools for mathematical physics. He gave a simple proof that $\pi$ is irrational as well as the first proof that $\pi^2$ is irrational.

- It is not surprising that, given these views of Legendre, there are few details of his early life.
- In 1770, at the age of 18, Legendre defended his thesis in mathematics and physics at the Collège Mazarin but this was not quite as grand an achievement as it sounds to us today, for this consisted more of a plan of research rather than a completed thesis.
- With no need for employment to support himself, Legendre lived in Paris and concentrated on research.
- His essay Recherches sur la trajectoire des projectiles dans les milieux résistants Ⓣ(Research on the trajectories of projectiles in a resistant medium) won the prize and launched Legendre on his research career.
- In 1782 Lagrange was Director of Mathematics at the Academy in Berlin and this brought Legendre to his attention.
- Legendre next studied the attraction of ellipsoids.
- He then introduced what we call today the Legendre functions and used these to determine, using power series, the attraction of an ellipsoid at any exterior point.
- Legendre submitted his results to the Académie des Sciences in Paris in January 1783 and these were highly praised by Laplace in his report delivered to the Académie in March.
- Within a few days, on 30 March, Legendre was appointed an adjoint in the Académie des Sciences filling the place which had become vacant when Laplace was promoted from adjoint to associé earlier that year.
- Over the next few years Legendre published work in a number of areas.
- This is fair since Legendre's proof of quadratic reciprocity was unsatisfactory, while he offered no proof of the theorem on primes in an arithmetic progression.
- Legendre's career in the Académie des Sciences progressed in a satisfactory manner.
- This work resulted in his election to the Royal Society of London in 1787 and also to an important publication Mémoire sur les opérations trigonométriques dont les résultats dépendent de la figure de la terre which contains Legendre's theorem on spherical triangles.
- On 13 May 1791 Legendre became a member of the committee of the Académie des Sciences with the task to standardise weights and measures.
- At this time Legendre was also working on his major text Eléments de géométrie which he had been encouraged to write by Condorcet.
- However the Académie des Sciences was closed due to the Revolution in 1793 and Legendre had special difficulties since he lost the capital which provided him with a comfortable income.
- Following the work of the committee on the decimal system on which Legendre had served, de Prony in 1792 began a major task of producing logarithmic and trigonometric tables, the Cadastre.
- Legendre and de Prony headed the mathematical section of this project along with Carnot and other mathematicians.
- In 1794 Legendre published Eléments de géométrie Ⓣ(Elements of geometry) which was the leading elementary text on the topic for around 100 years.
- Legendre's work replaced Euclid's "Elements" as a textbook in most of Europe and, in succeeding translations, in the United States and became the prototype of later geometry texts.
- Each section of the Institut contained six places, and Legendre was one of the six in the mathematics section.
- In 1803 Napoleon reorganised the Institut and a geometry section was created and Legendre was put into this section.
- Legendre published a book on determining the orbits of comets in 1806.
- In an appendix Legendre gave the least squares method of fitting a curve to the data available.
- However, Gauss published his version of the least squares method in 1809 and, while acknowledging that it appeared in Legendre's book, Gauss still claimed priority for himself.
- This greatly hurt Legendre who fought for many years to have his priority recognised.
- In 1808 Legendre published a second edition of his Théorie des nombres Ⓣ(Number theory) which was a considerable improvement on the first edition of 1798.
- For example Gauss had proved the law of quadratic reciprocity in 1801 after making critical remarks about Legendre's proof of 1785 and Legendre's much improved proof of 1798 in the first edition of Théorie des nombres Ⓣ(Number theory).
- Gauss was correct, but one could understand how hurtful Legendre must have found an attack on the rigour of his results by such a young man.
- Of course Gauss did not state that he was improving Legendre's result but rather claimed the result for himself since his was the first completely rigorous proof.
- To his credit Legendre used Gauss's proof of quadratic reciprocity in the 1808 edition of Théorie des nombres Ⓣ(Number theory) giving proper credit to Gauss.
- Again Gauss would claim that he had obtained the law for the asymptotic distribution of primes before Legendre, but certainly it was Legendre who first brought these ideas to the attention of mathematicians.
- Legendre's major work on elliptic functions in Exercices du Calcul Intégral Ⓣ(Exercises in integral calculus) appeared in three volumes in 1811, 1817, and 1819.
- In the first volume Legendre introduced basic properties of elliptic integrals and also of beta and gamma functions.
- However, despite spending 40 years working on elliptic functions, Legendre never gained the insight of Jacobi and Abel and the independent work of these two mathematicians was making Legendre's new three volume work obsolete almost as soon as it was published.
- Legendre's attempt to prove the parallel postulate extended over 30 years.
- In 1824 Legendre refused to vote for the government's candidate for the Institut National.
- As a result of Legendre's refusal to vote for the government's candidate in 1824 his pension was stopped and he died in poverty.

Born 18 September 1752, Paris, France. Died 10 January 1833, Paris, France.

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Algebra, Analysis, Astronomy, Geometry, Number Theory, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive