leonhard euler: his life and work michael p. saclolo, ph.d. st. edward’s university austin, texas
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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, 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
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