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Leonhard Euler His Life and Work

- Michael P. Saclolo, Ph.D.
- St. Edwards 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, hes the master (teacher)

of us all.

Images of Euler

Eulers 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

(masters degree) - Entered divinity school, but left to pursue more

mathematics

Academic Biography

- Joined Johann Bernoullis 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 Eulers 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 DAlembert
- Joseph-Louis Lagrange
- Christian Goldbach

Contemporaries Non-mathematical

- Voltaire
- Candide
- Academy of Sciences, Berlin
- Benjamin Franklin
- George Washington

Great Volume of Works

- 856 publications550 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 Theoryproperties of the natural numbers,

primes. - Logarithms
- Infinite Seriesinfinite sums of numbers
- Analytic Number Theoryusing infinite series,

limits, calculus, to study properties of

numbers (such as primes)

Contributions to Mathematics

- Complex Numbers
- Algebraroots of polynomials, factorizations of

polynomials - Geometryproperties of circles, triangles,

circles inscribed in triangles. - Combinatoricscounting methods
- Graph Theorynetworks

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
- Faceflat outside surface of the polyhedron
- Edgewhere two faces meet
- V-EFEuler 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 KönigsbergThe 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 KönigsbergThe Birth of Graph

Theory

The Bridges of KönigsbergThe Birth of Graph

Theory

- Question 1Is there a way to visit each land mass

using a bridge only once? (Eulerian path) - Question 2Is 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 KönigsbergThe Birth of Graph

Theory

The Bridges of KönigsbergThe 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 KönigsbergThe 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 KönigsbergThe 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 KönigsbergThe Birth of Graph

Theory

- Nowadays we use graph theory to solve problem

(see ACTIVITIES)

Knights Tour (on a Chessboard)

Knights Tour (on a Chessboard)

- Problem proposed to Euler during a chess game

Knights Tour (on a Chessboard)

Knights Tour (on a Chessboard)

- Euler proposed ways to complete a knights tour
- Showed ways to close an open tour
- Showed ways to make new tours out of old

Knights 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)

Euler and Primes

- If
- Then
- In a unique way
- Example

Euler and Primes

- This infinite series has no sum
- Infinitely many primes

Euler and Complex Numbers

- Recall

Euler and Complex Numbers

Eulers Formula

Euler and Complex Numbers

- Euler offered several proofs
- Cotes proved a similar result earlier
- One of Eulers proofs uses infinite series

Euler and Complex Numbers

Euler and Complex Numbers

Euler and Complex Numbers

Euler and Complex Numbers

- Eulers Identity

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/
- Eulers 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