Leonhard Euler: His Life and Work

Michael P. Saclolo, Ph.D.

St. Edward’s University

Austin, Texas

Pronunciation

Euler = “Oiler”

Leonhard Euler

Lisez Euler, lisez Euler, c'est notre maître à tous.”

-- Pierre-Simon Laplace

Read Euler, read Euler, he’s the master (teacher) of us all.

Images of Euler

Euler’s Life in Bullets

• Born: April 15, 1707, Basel, Switzerland

• Died: 1783, St. Petersburg, Russia

• Father: Paul Euler, Calvinist pastor

• Mother: Marguerite Brucker, daughter of a pastor

• Married-Twice: 1)Katharina Gsell, 2)her half sister

• Children-Thirteen (three outlived him)

Academic Biography

• Enrolled at University of Basel at age 14– Mentored by Johann Bernoulli– Studied mathematics, history, philosophy

(master’s degree)

• Entered divinity school, but left to pursue more mathematics

Academic Biography

• Joined Johann Bernoulli’s sons in St. Russia (St. Petersburg Academy-1727)

• Lured into Berlin Academy (1741)

• Went back to St. Petersburg in 1766 where he remained until his death

Other facts about Euler’s life

• Loss of vision in his right eye 1738

• By 1771 virtually blind in both eyes– (productivity did not suffer-still averaged 1

mathematical publication per week)

• Religious

Mathematical Predecessors

• Isaac Newton

• Pierre de Fermat

• René Descartes

• Blaise Pascal

• Gottfried Wilhelm Leibniz

Mathematical Successors

• Pierre-Simon Laplace

• Johann Carl Friedrich Gauss

• Augustin Louis Cauchy

• Bernhard Riemann

Mathematical Contemporaries

• Bernoullis-Johann, Jakob, Daniel

• Alexis Clairaut

• Jean le Rond D’Alembert

• Joseph-Louis Lagrange

• Christian Goldbach

Contemporaries: Non-mathematical

• Voltaire– Candide– Academy of Sciences, Berlin

• Benjamin Franklin

• George Washington

Great Volume of Works

• 856 publications—550 before his death

• Works catalogued by Enestrom in 1904 (E-numbers)

• Thousands of letters to friends and colleagues

• 12 major books– Precalculus, Algebra, Calculus, Popular

Science

Contributions to Mathematics

• Calculus (Analysis)

• Number Theory—properties of the natural numbers, primes.

• Logarithms

• Infinite Series—infinite sums of numbers

• Analytic Number Theory—using infinite series, “limits”, “calculus, to study properties of numbers (such as primes)

Contributions to Mathematics

• Complex Numbers

• Algebra—roots of polynomials, factorizations of polynomials

• Geometry—properties of circles, triangles, circles inscribed in triangles.

• Combinatorics—counting methods

• Graph Theory—networks

Other Contributions--Some highlights

• Mechanics

• Motion of celestial bodies

• Motion of rigid bodies

• Propulsion of Ships

• Optics

• Fluid mechanics

• Theory of Machines

Named after Euler

• Over 50 mathematically related items (own estimate)

Euler Polyhedral Formula (Euler Characteristic)

• Applies to convex polyhedra

Euler Polyhedral Formula (Euler Characteristic)

• Vertex (plural Vertices)—corner points

• Face—flat outside surface of the polyhedron

• Edge—where two faces meet

• V-E+F=Euler characteristic

• Descartes showed something similar (earlier)

Euler Polyhedral Formula (Euler Characteristic)

• Five Platonic Solids– Tetrahedron

– Hexahedron (Cube)

– Octahedron

– Dodecahedron

– Icosahedron

• #Vertices - #Edges+ #Faces = 2

Euler Polyhedral Formula (Euler Characteristic)

• What would be the Euler characteristic of– a triangular prism?

– a square pyramid?

The Bridges of Königsberg—The Birth of Graph Theory

• Present day Kaliningrad (part of but not physically connected to mainland Russia)

• Königsberg was the name of the city when it belonged to Prussia

The Bridges of Königsberg—The Birth of Graph Theory

The Bridges of Königsberg—The Birth of Graph Theory

• Question 1—Is there a way to visit each land mass using a bridge only once? (Eulerian path)

• Question 2—Is there a way to visit each land mass using a bridge only once and beginning and arriving at the same point? (Eulerian circuit)

The Bridges of Königsberg—The Birth of Graph Theory

The Bridges of Königsberg—The Birth of Graph Theory

• One can go from A to B via b (AaB).

• Using sequences of these letters to indicate a path, Euler counts how many times a A (or B…) occurs in the sequence

The Bridges of Königsberg—The Birth of Graph Theory

• If there are an odd number of bridges connected to A, then A must appear n times where n is half of 1 more than number of bridges connected to A

The Bridges of Königsberg—The Birth of Graph Theory

• Determined that the sequence of bridges (small letters) necessary was bigger than the current seven bridges (keeping their locations)

The Bridges of Königsberg—The Birth of Graph Theory

• Nowadays we use graph theory to solve problem (see ACTIVITIES)

Knight’s Tour (on a Chessboard)

Knight’s Tour (on a Chessboard)

• Problem proposed to Euler during a chess game

Knight’s Tour (on a Chessboard)

Knight’s Tour (on a Chessboard)

• Euler proposed ways to complete a knight’s tour

• Showed ways to close an open tour

• Showed ways to make new tours out of old

Knight’s Tour (on a Chessboard)

Basel Problem

• First posed in 1644 (Mengoli)

• An example of an INFINITE SERIES (infinite sum) that CONVERGES (has a particular sum)

6...

1...

3

1

2

1

1

1 2

2222

k

Euler and Primes

• If

• Then

• In a unique way• Example

22 bap

14 np

22 121)1(45

Euler and Primes

• This infinite series has no sum

• Infinitely many primes

...1

...11

1

7

1

5

1

3

1

2

11

p

Euler and Complex Numbers

• Recall

1i

Euler and Complex Numbers

p

Euler’s Formula:

Euler and Complex Numbers

• Euler offered several proofs

• Cotes proved a similar result earlier

• One of Euler’s proofs uses infinite series

Euler and Complex Numbers

...54321432132121

15432

xxxx

xex

...54321

)(

4321

)(

321

)(

21

)(1

5432

ixixixix

ixeix

...54321432132121

15432

ixxixx

ixeix

Euler and Complex Numbers

...432121

1cos42

xx

...54321321

sin53

xx

xx

...54321321

sin53

ixix

ixxi

Euler and Complex Numbers

...54321432132121

15432

ixxixx

ixeix

...54321321

...432121

15342

ixix

ixxx

eix

Euler and Complex Numbers

Euler’s Identity:

01ie1)sin(cos1 iei

1011 iei

01ie

How to learn more about Euler

• “How Euler did it.” by Ed Sandifer– http://www.maa.org/news/howeulerdidit.html– Monthly online column

• Euler Archive– http://www.math.dartmouth.edu/~euler/– Euler’s works in the original language (and

some translations)

• The Euler Society– http://www.eulersociety.org/

How to learn more about Euler

• Books– Dunhamm, W., Euler: the Master of Us All, Dolciani

Mathematical Expositions, the Mathematical Association of America, 1999

– Dunhamm, W (Ed.), The Genius of Euler: Reflections on His Life and Work, Spectrum, the Mathematical Association of America, 2007

– Sandifer, C. E., The Early Mathematics of Leonhard Euler, Spectrum, the Mathematical Associatin of America, 2007