1. Leonhard Euler: His Life and Work
Michael P. Saclolo, Ph.D.
St. Edward’s University
Austin, Texas

2. Pronunciation
Euler = “Oiler”

3. 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.

4. Images of Euler

5. 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)

6. 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

7. 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

8. 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

9. Mathematical Predecessors
Isaac Newton
Pierre de Fermat
René Descartes
Blaise Pascal
Gottfried Wilhelm Leibniz

10. Mathematical Successors
Pierre-Simon Laplace
Johann Carl Friedrich Gauss
Augustin Louis Cauchy
Bernhard Riemann

11. Mathematical Contemporaries
Bernoullis-Johann, Jakob, Daniel
Alexis Clairaut
Jean le Rond D’Alembert
Joseph-Louis Lagrange
Christian Goldbach

12. Contemporaries: Non-mathematical
Voltaire
Candide
Academy of Sciences, Berlin
Benjamin Franklin
George Washington

13. 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

14. 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)

15. 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

16. Other Contributions--Some highlights
Mechanics
Motion of celestial bodies
Motion of rigid bodies
Propulsion of Ships
Optics
Fluid mechanics
Theory of Machines

17. Named after Euler
Over 50 mathematically related items (own estimate)

18. Euler Polyhedral Formula (Euler Characteristic)
Applies to convex polyhedra

19. 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)

20. Euler Polyhedral Formula (Euler Characteristic)
Five Platonic Solids
Tetrahedron
Hexahedron (Cube)
Octahedron
Dodecahedron
Icosahedron
#Vertices - #Edges+ #Faces = 2

21. Euler Polyhedral Formula (Euler Characteristic)
What would be the Euler characteristic of
a triangular prism?
a square pyramid?

22. 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

23. The Bridges of Königsberg—The Birth of Graph Theory

24. 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)

25. The Bridges of Königsberg—The Birth of Graph Theory

26. 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

27. 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

28. 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)

29. The Bridges of Königsberg—The Birth of Graph Theory
Nowadays we use graph theory to solve problem (see ACTIVITIES)

30. Knight’s Tour (on a Chessboard)

31. Knight’s Tour (on a Chessboard)
Problem proposed to Euler during a chess game

32. Knight’s Tour (on a Chessboard)

33. 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

34. Knight’s Tour (on a Chessboard)

35. Basel Problem First posed in 1644 (Mengoli)
An example of an INFINITE SERIES (infinite sum) that CONVERGES (has a particular sum)

36. Euler and Primes If
Then
In a unique way
Example

37. Euler and Primes This infinite series has no sum
Infinitely many primes

38. Euler and Complex Numbers
Recall

39. Euler and Complex Numbers
Euler’s Formula:

40. Euler and Complex Numbers
Euler offered several proofs
Cotes proved a similar result earlier
One of Euler’s proofs uses infinite series

41. Euler and Complex Numbers

42. Euler and Complex Numbers

43. Euler and Complex Numbers

44. Euler and Complex Numbers
Euler’s Identity:

45. How to learn more about Euler
“How Euler did it.” by Ed Sandifer
Monthly online column
Euler Archive
Euler’s works in the original language (and some translations)
The Euler Society

46. 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