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Leonhard Euler: His Life and Work

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Leonhard Euler: His Life and Work Michael P. Saclolo, Ph.D. St. Edward s University Austin, Texas Pronunciation Euler = Oiler Leonhard Euler Lisez Euler ... – PowerPoint PPT presentation

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Title: Leonhard Euler: His Life and Work


1
Leonhard Euler His Life and Work
  • Michael P. Saclolo, Ph.D.
  • St. Edwards 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, hes the master (teacher)
    of us all.

4
Images of Euler
5
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)

6
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

7
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

8
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

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 DAlembert
  • 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 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

14
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)

15
Contributions to Mathematics
  • Complex Numbers
  • Algebraroots of polynomials, factorizations of
    polynomials
  • Geometryproperties of circles, triangles,
    circles inscribed in triangles.
  • Combinatoricscounting methods
  • Graph Theorynetworks

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
  • Faceflat outside surface of the polyhedron
  • Edgewhere two faces meet
  • V-EFEuler 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 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

23
The Bridges of KönigsbergThe Birth of Graph
Theory
24
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)

25
The Bridges of KönigsbergThe Birth of Graph
Theory
26
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

27
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

28
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)

29
The Bridges of KönigsbergThe Birth of Graph
Theory
  • Nowadays we use graph theory to solve problem
    (see ACTIVITIES)

30
Knights Tour (on a Chessboard)
31
Knights Tour (on a Chessboard)
  • Problem proposed to Euler during a chess game

32
Knights Tour (on a Chessboard)
33
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

34
Knights 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

Eulers Formula
40
Euler and Complex Numbers
  • Euler offered several proofs
  • Cotes proved a similar result earlier
  • One of Eulers proofs uses infinite series

41
Euler and Complex Numbers
42
Euler and Complex Numbers
43
Euler and Complex Numbers
44
Euler and Complex Numbers
  • Eulers Identity

45
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/

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
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