Mathematician:Leonhard Paul Euler

Swiss mathematician and physicist who pioneered much of the foundation of modern mathematics.
 * Introduced much of the notation which is used today, including e and the modern notation for trigonometric functions.
 * Proved Fermat's Little Theorem.
 * In 1783, on the basis of considerable numerical evidence, conjectured the Law of Quadratic Reciprocity, which was eventually proven by Gauss in 1798.


 * Proved the converse of the result known to Euclid, that if $2^p - 1$ is prime, then $2^{p-1} \left({2^p - 1}\right)$ is perfect. That is, Euler proved that if $n$ is an even perfect number, then $n$ is of the form $2^{p-1} \left({2^p - 1}\right)$ where $p$ is prime. The results together are known as the Theorem of Even Perfect Numbers.

Nationality
Swiss

History

 * Born: 15 April 1707, Basel, Switzerland
 * Died: 18 Sept 1783, St Petersburg, Russia

Geometry

 * Euler Triangle Formula

Analysis and Calculus

 * Euler-Maclaurin Summation Formula (with Colin Maclaurin)
 * Euler Formula for Sine Function
 * Often credited with solving the Basel Problem, but it is believed that this was in fact solved by Nicolaus I Bernoulli.
 * Euler's Reflection Formula

Complex Analysis

 * Euler's Formula
 * Euler's Identity

Number Theory

 * Euler's Criterion
 * Theorem of Even Perfect Numbers
 * Euler's Theorem
 * Euler-Binet Formula (with Jacques Philippe Marie Binet) (also known as Binet's Formula)
 * Eulerian Integer (also known as Eisenstein Integer for Ferdinand Eisenstein)

Graph Theory

 * Handshake Lemma
 * The Bridges of Königsberg Problem
 * Euler Polyhedron Formula

Analysis and Calculus

 * Euler's Number
 * Euler-Mascheroni Constant (with Lorenzo Mascheroni)
 * Cauchy-Euler Equation (with Augustin Louis Cauchy)

Number Theory

 * Euler Phi Function

Graph Theory
... and the list goes on.
 * Euler Characteristic
 * Eulerian Circuit
 * Eulerian Graph
 * Semi-Eulerian Graph
 * Eulerian Path

Books and Papers

 * 1736: Solutio problematis ad geometriam situs pertinentis (The solution of a problem relating to the geometry of position) in which was given the Handshake Lemma and solution to the Bridges of Königsberg problem, possibly the first ever paper in graph theory.
 * 1736-37: Mechanica
 * 1739: Tentamen Novae Theoriae Musicae
 * 1740: Methodus Inveniendi Lineas Curvas
 * 1748: Introductio in Analysin Infinitorum
 * 1755: Institutiones Calculi Differentialis
 * 1765: Theoria Motus Corporum Solidorum
 * 1768-70: Institutionum Calculi Integralis

Linguistic Note
The correct pronunciation of Euler is Oi-ler, consistent with convention in Germanic languages.

Uninitiated English native speakers may be tempted to pronounce You-ler, but this is definitely wrong.

Also see

 * : Chapter $\text{IX}$
 * : Chapter $\text{A}.21$