Mathematician:Leonhard Paul Euler

Mathematician
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.

A student of who outstripped his teacher early on.

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 in $1798$.

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

According to anecdote (source to be ascertained), learning of new techniques for calculating approximations to $\pi$ (pi), demonstrated their power by calculating $\pi$ to $10$ decimal places (possibly $20$) in the space of $1$ hour.

Possibly the most prolific writer of all time, in any field.

Was blind during the last $17$ years of his life, but did not let that slow down his output.

Nationality
Swiss

History

 * Born: 15 April 1707, Basel, Switzerland
 * 1724: Took Masters' Degree at University of Basel
 * 1727: Received honourable mention for memoir on the masting of ships
 * 1727: Took up position in St. Petersburg Academy amid political confusion
 * 1733: Took over 's position
 * 1733: Resigned himself to settling in St. Petersburg, married Catharina Gsell and started a legendarily large family
 * 1740: Accepted invitation from Frederick the Great to join Berlin Academy
 * 1766: Left Berlin for St. Petersburg at invitation of Catherine the Great
 * Died: 18 Sept 1783, St Petersburg, Russia

Geometry

 * Euler Triangle Formula

Analysis and Calculus

 * Euler-Maclaurin Summation Formula (with ) (also known as Euler-Maclaurin Summation Formula)
 * Euler Formula for Sine Function
 * Often credited with solving the Basel Problem, but it is believed that this was in fact solved by.
 * Euler-Darboux Equation (with )
 * Euler-Poisson-Darboux Equation (with and )


 * Euler's Reflection Formula
 * Euler's Product form of Riemann Zeta Function
 * Euler's Integral Representation of Hypergeometric Function‎
 * Euler's Transformation‎

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 ) (also known as Binet's Formula)
 * Euler's Pentagonal Numbers Theorem

Numerical Analysis

 * Euler Method
 * Improved Euler Method

Graph Theory

 * Handshake Lemma
 * The Bridges of Königsberg Problem
 * Euler's Theorem for Planar Graphs
 * Euler Polyhedron Formula

Combinatorics

 * Euler's Conjecture on Orthogonal Latin Squares (refuted $1959$ by, and )

Mechanics

 * Euler's Equations of Motion for Rotation of Rigid Body
 * Euler's Hydrodynamical Equation for Flow of Ideal Incompressible Fluid
 * Euler-Bernoulli Beam Equation (with ) (also known as the Euler-Bernoulli Law)
 * Euler Buckling Formula (also known as the Euler Column Formula)

Linear Algebra

 * Euler-Rodrigues Formula (with )

Geometry

 * Euler Line
 * Euler Spiral (also known as Cornu Spiral for )
 * Euler Triangle

Analysis and Calculus

 * Euler's Equation
 * Euler's Equation for Vanishing Variation
 * Eulerian Integral of the First Kind (also known as the Beta Function)
 * Eulerian Logarithmic Integral
 * Euler Multiplier
 * Euler's Number (also known as Napier's Constant for )
 * Euler Numbers


 * Euler Substitution:
 * Euler's First Substitution
 * Euler's Second Substitution
 * Euler's Third Substitution


 * Cauchy-Euler Equation (with )
 * Euler-Gompertz Constant (with ) (also known as the Gompertz Constant)
 * Euler-Mascheroni Constant (with ) (also known as Euler's Constant)

Number Theory

 * Euler Phi Function
 * Euler Lucky Number
 * Eulerian Integer (also known as Eisenstein Integer for )

Graph Theory

 * Euler Characteristic
 * Eulerian Circuit
 * Eulerian Graph
 * Semi-Eulerian Graph
 * Eulerian Trail
 * Eulerian Walk

Set Theory

 * Euler Diagram

Mechanics

 * Euler-Lagrange Equation (with )
 * Eulerian Strain Rate

... and the list goes on.

Conjectures later proved false

 * The Euler Quartic Conjecture
 * Euler's Sum of Powers Conjecture (refuted by and  in $1966$)
 * Euler's Conjecture on Orthogonal Latin Squares

Publications

 * 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:
 * 1739: Tentamen Novae Theoriae Musicae
 * 1744:
 * 1748:
 * 1755:
 * 1765: Theoria Motus Corporum Solidorum
 * 1768 -- 94:
 * 1768:
 * 1769:
 * 1770:
 * 1768:
 * 1769:
 * 1770:



Notable Quotes

 * Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
 * -- $1751$


 * Sir, $\dfrac {a + b^n} n = x$, hence God exists; reply!
 * -- To, who had been stating the case for Atheism


 * I die.
 * -- Reportedly his last words.

Critical View

 * Read Euler: he is our master in everything.


 * He calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind.


 * One of the most remarkable features of Euler's mathematical genius was its equal strength in both of the main currents of mathematics, the continuous and the discrete.

Also known as
Some sources render his name as Léonard.

Also see

 * Euler Archive