Title: Leonhard Euler: His Life and Work
1Leonhard Euler His Life and Work
- Michael P. Saclolo, Ph.D.
- St. Edwards University
- Austin, Texas
2Pronunciation
3Leonhard 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.
4Images of Euler
5Eulers 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)
6Academic 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
7Academic 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
8Other 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
9Mathematical Predecessors
- Isaac Newton
- Pierre de Fermat
- René Descartes
- Blaise Pascal
- Gottfried Wilhelm Leibniz
10Mathematical Successors
- Pierre-Simon Laplace
- Johann Carl Friedrich Gauss
- Augustin Louis Cauchy
- Bernhard Riemann
11Mathematical Contemporaries
- Bernoullis-Johann, Jakob, Daniel
- Alexis Clairaut
- Jean le Rond DAlembert
- Joseph-Louis Lagrange
- Christian Goldbach
12Contemporaries Non-mathematical
- Voltaire
- Candide
- Academy of Sciences, Berlin
- Benjamin Franklin
- George Washington
13Great 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
14Contributions 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)
15Contributions to Mathematics
- Complex Numbers
- Algebraroots of polynomials, factorizations of
polynomials - Geometryproperties of circles, triangles,
circles inscribed in triangles. - Combinatoricscounting methods
- Graph Theorynetworks
16Other Contributions--Some highlights
- Mechanics
- Motion of celestial bodies
- Motion of rigid bodies
- Propulsion of Ships
- Optics
- Fluid mechanics
- Theory of Machines
17Named after Euler
- Over 50 mathematically related items (own
estimate)
18Euler Polyhedral Formula (Euler Characteristic)
- Applies to convex polyhedra
19Euler 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)
20Euler Polyhedral Formula (Euler Characteristic)
- Five Platonic Solids
- Tetrahedron
- Hexahedron (Cube)
- Octahedron
- Dodecahedron
- Icosahedron
- Vertices - Edges Faces 2
21Euler Polyhedral Formula (Euler Characteristic)
- What would be the Euler characteristic of
- a triangular prism?
- a square pyramid?
22The 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
23The Bridges of KönigsbergThe Birth of Graph
Theory
24The 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)
25The Bridges of KönigsbergThe Birth of Graph
Theory
26The 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
27The 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
28The 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)
29The Bridges of KönigsbergThe Birth of Graph
Theory
- Nowadays we use graph theory to solve problem
(see ACTIVITIES)
30Knights Tour (on a Chessboard)
31Knights Tour (on a Chessboard)
- Problem proposed to Euler during a chess game
32Knights Tour (on a Chessboard)
33Knights 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
34Knights Tour (on a Chessboard)
35Basel Problem
- First posed in 1644 (Mengoli)
- An example of an INFINITE SERIES (infinite sum)
that CONVERGES (has a particular sum)
36Euler and Primes
- If
- Then
- In a unique way
- Example
37Euler and Primes
- This infinite series has no sum
- Infinitely many primes
38Euler and Complex Numbers
39Euler and Complex Numbers
Eulers Formula
40Euler and Complex Numbers
- Euler offered several proofs
- Cotes proved a similar result earlier
- One of Eulers proofs uses infinite series
41Euler and Complex Numbers
42Euler and Complex Numbers
43Euler and Complex Numbers
44Euler and Complex Numbers
45How 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/
46How 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