Mathematician:Leonhard Paul Euler

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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 Johann Bernoulli 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 Gauss in $1798$.

Proved the converse of the result known to Euclid, that if $2^p - 1$ is prime, then $2^{p - 1} \paren {2^p - 1}$ is perfect. That is, Euler 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.




  • 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 Daniel Bernoulli'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

Theorems and Problems

Results named for Leonhard Paul Euler can be found here.


Analysis and Calculus

Complex Analysis

Number Theory

Numerical Analysis

Graph Theory



Linear Algebra


Definitions of concepts named for Leonhard Paul Euler can be found here.


Analysis and Calculus

Number Theory

Graph Theory

Set Theory


... and the list goes on.

Conjectures later proved false

Results named for Leonhard Paul Euler can be found here.

Definitions of concepts named for Leonhard Paul Euler can be found here.


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 Denis Diderot, who had been stating the case for Atheism

I die.
-- Reportedly his last words.

Critical View

Read Euler: he is our master in everything.
-- Pierre-Simon de Laplace

He calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind.
-- Dominique François Jean Arago

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.
-- Eric Temple Bell

Also known as

Some sources render his name as Léonard.

Also see

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 Yoo-ler, but this is definitely wrong.

Consequently, noun phrases which begin with Euler's name would be preceded by "an" rather than "a", for example an Eulerian graph.