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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, lisez Euler, c'est notre maître à tous.” -- Pierre-Simon Laplace Read Euler, read Euler, he’s the master (teacher) of us all.**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**• 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**• 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)**• Problem proposed to Euler during a chess game**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**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**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**Euler’s 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/ • 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