Legendre

Adrien-Marie Legendre

Adrien-Marie Legendre was a wealthy Parisian mathematician who lived from 1752-1833. In Paris he studied physics and math at Collège Mazarin and then taught for five years at Ecole Militaire. In 1782 he entered an essay to the Berlin Academy’s contest to “determine the curve described by cannonballs and bombs, taking into consideration the resistance of the air; give rules for obtaining the ranges corresponding to different initial velocities and to different angles of projection¹. Legendre won the projectile contest with his essay Recherches sur la trajectoire des projectiles dans les milieux resistants. The knowledge of projectiles impacted weaponry in the French Revolution (1789-1795), France’s war on Austria (1792) and France’s war against Prussia (1792)².

Legendre contributed a great deal of knowledge to field of mathematics. His study of the attraction of ellipsoids led to the formation of “Legendre functions” or “Legendre Polynomials”⁴, which are used today in differential equations⁸ He used these polynomials when writing on celestial mechanics. He also published papers on number theory and the theory of elliptic functions⁶. His 1785 paper contains the beginnings of the law of quadratic reciprocity for residues—he didn’t have the correct proof that Gauss came up with later. He worked extensively on trying to solve Fermat’s Last Theorem (It is impossible to find any two cubes, whose sum, or difference, is a cube) but ended up only restating Euler’s work on the problem in a different form in 1785 and 1798³. In 1830, he gave a proof of the problem up to n=5⁵ but as we all know the problem wasn’t fully proven until the mid-1990’s.

Legendre was a member of a group that set out to make measurements of the Earth involving a triangulation survey between the Paris and Greenwich observatories. His work here resulted in his election to the Royal Society of London in 1787⁷ and in publishing Mémoire sur les opérations trigonométriques dont les résultats dépendent de la figure de la terre, which contains Legendre's theorem on spherical triangles¹. In 1791 Legendre became a member of the committee of the Académie des Sciences that worked to standardize weights and measures using the metric system. From 1792-1801, he supervised 70-80 assistants who produced logarithmic and trigonometric tables. Legendre also invented spherical harmonics⁹ and discovered the mean-square approximation, which Gauss also discovered. Later on in 1806 he published a book on determining the orbits of comets. He is also known for the method of least squares¹⁰.

Legendre Legendre published Elélments de géométrie in 1794, which became the leading textbook on geometry for the next 100 years. In his "Eléments" Legendre greatly rearranged and simplified many of the propositions from Euclid's "Elements" to create a more effective textbook. 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. In "Eléments" Legendre gave a simple proof that p is irrational, as well as the first proof that p² is irrational, and conjectured that p is not the root of any algebraic equation of finite degree with rational coefficients¹. Supposedly Legendre worked on the parallel postulate for 29 years but never gave a true proof because he used statements that cannot be proved from Euclid’s first four postulates⁸. Historians credit both theorems 4.4 (p.125) and 4.7 (p.131) to Saccheri and Legendre⁸. He attempted to prove that the sum of every triangle is 180 degrees but failed to prove that the defect of every triangle is zero⁸. Legendre has proved theorem 5.1 (p158) that states: For any acute angle A and any point D in the interior of angle A, there exists a line through D and not through A which intersects both sides of angle A, which proves that the angle sum of every triangle is 180 degrees.