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Leonhard Euler (pronounced Oiler) Analysis Incarnate “Read Euler, read Euler. He is the master of us all” - Laplace “Euler calculated without apparent effort, as men breathe, or eagles sustain themselves in the wind” - Arago

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Biography Born Leonhard Euler, in Switzerland (April 15, 1707 - September 18, 1783 at age 76) His early education was given by his father. Entered the University of Basel at 14, received Masters in Philosophy at age 17. Studied Hebrew & Theology, but soon focused on mathematics. Moved to Russia and found a position at the St. Petersburg Academy of Sciences. His efforts there helped make Russia a naval power. Married Katharina Gsell, a Swiss girl, in 1733. He had thirteen children with her, all of whom he loved dearly. Accepted invitation to move to Prussia, escaping political unrest in Russia. Frederick the Great, the leader of Prussia, was an atheist, and constantly ridiculed Euler’s faith. Euler lost the sight in one eye in 1735, and lost the sight in the other in 1766. He had an operation to repair them, but both became infected. He later said that only his faith in God allowed him to bear that torment. Produced works almost until the day of his death in 1783, working on the “black slate of his mind”. In an astonishing feat, his works became more clear after his blindness set in.

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Euler’s Worldview Raised in a Calvinist home, son of a Protestant minister Held to the Reformed Worldview all his life Held family worship & prayer daily in his home; often preached; read Scripture to his children every night Faced biting criticism from Frederick and Voltaire, an atheist and a deist, respectively Spent much time writing apologetics to respond to these two thinkers

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Euler’s Accomplishments Wrote a total of 886 works His collected works total 74 volumes Made first rate discoveries in –Analysis –Functions –Calculus –Summations –Combinatorics –Number Theory –Higher Algebra –Convergent series –Hydromechanics –Physical Mechanics –Astronomy –Topology

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Königsberg Bridge Problem

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Euler’s Accomplishments (continued) Analyzed –mechanics –planetary motion –ballistics, projectile trajectories –lunar orbit theory (tides) –design & sailing of ships –construction & architecture –acoustics, theory of musical harmony –investment theory insurance, annuities, pensions Other topics of interest –chemistry –medicine –geography –cartography –languages –philosophy –apologetics –religion –family he taught his 13 children and many grandchildren

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(more) Euler’s Accomplishments (more) This work greatly influenced Riemann and Maxwell He wrote textbooks that remained standards for hundreds of years Wrote research papers at the rate of 800 per year The epitome of his mathematical analysis is summed up in his formula : e i +1 = 0 Promoted partial solutions to: –Gravitational Problems –Optic Problems –Etheric Problems –Electromagnetic Problems

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Euler’s Textbooks The first was the Introductio in Analysin Infinitorum (Introduction to the Analysis of the Infinite). This is considered by mathematics historians to be one of the most influential textbooks in history. This was Euler’s ‘Pre-Calculus’ textbook, which introduced topics that were “absolutely required for analysis” so that the reader “almost imperceptibly becomes acquainted with the idea of the infinite” He was the first to devise the ingenious teaching art of skillfully letting mathematical formulae “speak for themselves.” Euler wrote three Latin textbooks on the topics of Calculus and Pre-Calculus

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Euler’s “Introduction” (Did you know this…?) The most important part of this book dealt with exponential, logarithmic, and trigonometric functions. It was there that Euler first introduced important notations such as: –Functional notation; f(x) –The base of natural logarithms; e –The sides of a triangle ABC; a, b, c –The semiperimeter of triangle ABC; s –The summation sign; –The imaginary unit - 1; i

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More Eulerian Textbooks Euler’s remaining books in the series were Institutiones Calculi Differentialis (Methods of the Differential Calculus) and Institutiones Calculi Integralis (Methods of the Integral Calculus) Euler’s Differential Calculus contains: –Introduction to differential equations –Discussed various methods for converting functions to power series –Extensive chapters on finding sums of various series –A pair of chapters on finding maxima and minima This is especially impressive, because his text contains no graphs or charts. All discussion given to maxima and minima is done purely analytically. Euler’s Integral Calculus contains: –Integrals of various functions –Solutions of differential equations –Integration by infinite series, integration by parts, formulas for integration of powers of trigonometric functions All three books are an exercise in analysis, so much so that they contain no applications to geometry. The integral is not even used to calculate area under a curve.

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“Euler” functions & formulae Discovered “Euler’s identity” e i x = cos(x) + i sin(x) for any simple closed polyhedron with vertices V, edges E, and faces F V – E + F = 2 Euler curvature formula = 1 cos 2 + 2 sin 2

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“Euler” functions & formulae Number Theory Euler’s function (or phi-function), (n), is defined as the number of integers less than n and relatively prime to n, i.e. sharing no common factor with n. Here are the first 10 values of (n): n1234 5 6 7 8 9 10 (n)1122 4 2 6 4 6 4 (10)=4 because of all the integers between 1 and 10 only 1,3,7, & 9 share no common factor with 10. So when n is prime (n)=n-1 since all integers less than n are relatively prime to n.

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Rigid body motion Euler angles Hydrodynamics the Euler equation Dynamics of rigid bodies Euler’s equation of motion Theory of elasticity Bernoulli-Euler law Trigonometric series Euler-Fourier formulas Infinite Series Euler’s constant Euler numbers Euler’s transformations DEs & Partial Diff Eqs Euler’s polygonal curves Euler’s theorem on homogeneous functions Calculus of variations Euler-Lagrange equation Numerical Methods Euler-Maclaurin formula “Euler” functions & formulae

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Rigid body motion Euler angles Hydrodynamics the Euler equation Dynamics of rigid bodies Euler’s equation of motion Theory of elasticity Bernoulli-Euler law Trigonometric series Euler-Fourier formulas Infinite Series Euler’s constant Euler numbers Euler’s transformations DEs & Partial Diff Eqs Euler’s polygonal curves Euler’s theorem on homogeneous functions Calculus of variations Euler-Lagrange equation Numerical Methods Euler-Maclaurin formula “Euler” functions & formulae Euler’s Infinite ProductEuler’s Formula

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Euler line The most famous line in the subject of triangle geometry is named in honor of Leonhard Euler, who penned more pages of original mathematics than any other human being.

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G=centroid O=circumcenter H=orthocenter N=nine-point center L=DeLongchamps point } } O to H } N is 1/2 way from O to H } G is 1/3 of the way from O to H O always lies 1/2 way from H to L

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Euler stops calculating Mathematics was used by Euler as God’s ally. He wrote Letters to a German Princess to give lessons in mechanics, physical optics, astronomy, sound, etc. In it he combined piety and the sciences. Their extreme popularity resulted in their translation into seven languages. Euler remained virile and powerful of mind to the very second of his death [despite his total blindness], which occurred in his seventy seventh year, on September 18, 1783. That day he had amused himself by calculating the laws of ascent of balloons, dined with his family and friends. Uranus being a recent discovery, Euler outlined the calculation of its orbit. A little latter he asked his grandson to be brought in. While playing he suffered a stroke. “Euler ceased to live and calculate.”

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