# 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 $2^{p - 1} \paren {2^p - 1}$ 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.

## 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

### 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

### Numerical Analysis

### 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 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

- Euler Line
- Euler Spiral (also known as Cornu Spiral for Marie Alfred Cornu)
- Euler Triangle

### 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
- Euler-Gompertz Constant (with Benjamin Gompertz) (also known as the Gompertz Constant)

### Number Theory

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

### Graph Theory

### Set Theory

### Mechanics

... and the list goes on.

## Conjectures later proved false

- Euler's Quartic Conjecture
- Euler's Sum of Powers Conjecture (refuted by Leon J. Lander and Thomas R. Parkin in $1966$)
- Euler's Conjecture on Orthogonal Latin Squares

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

## 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**.

## 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**.

## 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 - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): A List of Mathematicians in Chronological Sequence - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Euler, Leonhard**(1707-83) - 1991: David Wells:
*Curious and Interesting Geometry*... (previous) ... (next): A Chronological List Of Mathematicians - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$) - 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): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Euler, Leonhard**(1707-83) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Euler, Leonhard**(1707-83) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Euler, Leonhard**(1707-83)