# Mathematician:Leonhard Paul Euler

## Contents

## 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 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 $2^{p-1} \left({2^p - 1}\right)$ 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.

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 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

### Geometry

### 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-Darboux Equation (with Jean-Gaston Darboux)
- Euler-Poisson-Darboux Equation (with Siméon-Denis Poisson and Jean-Gaston Darboux)

### Complex Analysis

- Euler's Formula
- Euler's Identity
- Eulerian Integer, also known as Eisenstein Integer for Ferdinand Gotthold Max Eisenstein.

### 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**) - Euler's Pentagonal Numbers Theorem
- Euler's Sum of Powers Conjecture (refuted by Leon J. Lander and Thomas R. Parkin in $1966$)

### Numerical Analysis

### Graph Theory

### Combinatorics

- Euler's Conjecture on Orthogonal Latin Squares (refuted $1959$ by Raj Chandra Bose, Sharadchandra Shankar Shrikhande and Ernest Tilden Parker)

### 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 Daniel Bernoulli)
- Euler Buckling Formula

### Linear Algebra

## Definitions

### Geometry

### Analysis and Calculus

- Euler's Number (also known as Napier's Constant for John Napier)
- Euler-Mascheroni Constant (with Lorenzo Mascheroni)
- Cauchy-Euler Equation (with Augustin Louis Cauchy)
- Eulerian Logarithmic Integral
- Euler Multiplier
- Euler's Equation for Vanishing Variation

### Number Theory

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

### Graph Theory

### Set Theory

### Mechanics

... 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.

## 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:
*Mechanica* - 1738:
*De Progressionibus Harmonicis Obseruationes*(*Commentarii Acad. Sci. Imp. Pet.***Vol. 7**: 150 – 161) - 1739:
*Tentamen Novae Theoriae Musicae* - 1741:
*Observationes Analyticae Variae de Combinationibus*(*Commentarii Acad. Sci. Imp. Pet.***Vol. 13**: 64 – 93) - 1744:
*Methodus Inveniendi Lineas Curvas* - 1748:
*Introductio in Analysin Infinitorum* - 1750:
*De Partitione Numerorum*(*Novi Comment. Acad. Sci. Imp. Petropol.***Vol. 3**: 125 – 169) - 1755:
*Institutiones Calculi Differentialis* - 1765:
*Theoria Motus Corporum Solidorum* - 1768 -- 94:
*Institutiones 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.

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

## 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.*

*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**.

## Sources

- John J. O'Connor and Edmund F. Robertson: "Leonhard Paul Euler": MacTutor History of Mathematics archive

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{IX}$: Analysis Incarnate - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($1707$ – $1783$) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): A List of Mathematicians in Chronological Sequence - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$