Leonhard Euler (1707 - 1783) (Mathematician, Physicts , Engineer, Astronomer, Philosopher )

1. Leonhard Euler(1707 - 1783)(Mathematician, Physicts, Engineer, Astronomer, Philosopher ) By: Vivek Joseph Department of Mathematics BBDNIIT, Lucknow

2. Leonhard Euler, 1753

3. References • “Euler: The Master of Us All” by William Dunham, Mathematical Association of America • “Leonhard Euler: His Life & His Faith” by Dr. George W Benthein, 2008. • “Leonhard Euler: His Life, The Man and his work” by Walter Gautschi, SIAM Review, Vol 50 No. 1, pp 3 – 33, 2008.

4. “Read Euler, read Euler, he is a master in Every thing” By: Pierre Simon Laplace Early Life • On April 15th, 1707, Leonhard Euler was born in Basel, Switzerland. • His father named Paul Euler, who was pastor of the reformed church and mother named Marguerite Brucker. • Leonhard Euler had two younger sister Anna Maria and Maria Magdalena.

5. Home of Leonhard Euler

6. Early Life • Euler’s early formal education started in Basel where he lived with his maternal grandmother. • Euler’s father wanted his son to follow him into the church and sent him to the University of Basel to prepare for the ministry. He entered University in 1720 at the age of 13. Euler was studying theology, Greek and Hebrew languages in order to become a pastor. • In meantime Leonhard Euler encountered its most famous Professor, Johann Bernoulli (1667 -1748) the greatest active mathematician at that time.

7. Johann Bernoulli

8. University of Basel

9. Academic Life • At University, Euler completed a master’s degree in Philosophy. Then fulfilling his apparent destiny . Euler entered divinity school to study for the ministry. But the call of mathematics was too strong. He left to others and become a mathematician. • His progress was rapid. At the age of 20, he earned recognition in an international scientific competition for the analysis of the placement of masts on a sailing ship.

10. Life in St. Petersburg’s Academy • In 1725, Johann’s son Daniel Bernoulli (1700 – 1782) arrived to assume a position in mathematics at the New St. Petersburg Academy. • In 1727 Euler wrote his dissertation in Physics, presenting a theory of sound on the basis of which he applied for an open physics professorship in Basel. Since this application was unsuccessful, the young Euler decided an invitation to the St. Petersburg Academy in the same year.

11. St. Petersburg Academy

12. Daniel Bernoulli

13. Life in St. Petersburg Academy • In 1733 Daniel Bernoulli left for an academic post in Switzerland. The position was occupied by Euler. • In 1733 he married his Swiss compatriot Katharina Gsell, the daughter of a well known painter. Euler fathered 13 children, only 5 of which reached adolescence. His children provided him with 38 grand children. • Intellectual life at the St. Petersburg Academy suited Euler perfectly. He became a scientific consultant to the Government in which capacity he prepared maps, advised the Russian navy and even design for the fire engines.

14. Only Picture of Leonhard Euler before he lost sight

15. Life in St. Petersburg Academy • Meanwhile his frame was growing. One of his triumphs was a solution of the so called “Basel Problem”. • The issue was to determine the exact value of the infinite series. • The answer was given by Euler in 1735. • With the Basel problem behind him and a promise of good things ahead Euler pursued his research at a breath taking pace in St. Petersburg Academy.

16. Euler lost his vision in One Eye • In 1735, he fell seriously ill and almost lost his life. To the great relief of all, he recovered, but suffered a repeat attack three years later of (probably) the same infectious disease. This time it cost him his right eye. • The political turmoil in Russia that followed the death of the czarina Anna Ivanovna induced Euler to seriously consider, and eventually decide, to leave St. Petersberg

17. Life At Royal Academy at Berlin • Political unrest in Russia led to a tense working environment at St. Petersburg Academy, prompting Euler to accept an invitation from Frederick the Great of Prussia to join his Royal Academy at Berlin in 1741. • Frederick asked Euler to tutor one of his niece in the physical sciences and Euler agreed. • This resulted in a series of over 200 letters that have been collected under the title “Letter of Euler to a German Princess on a Different subject in a Natural Philosophy”. • These letters explained in Laymen’s language the basic concepts of physics as well as Euler’s view on philosophy and theology. These letters were later published by the St. Petersburg Academy in two large illustrated volumes.

18. Berlin Academy

19. Frederick II

20. Letters written by Euler to Princesses

21. Contribution of Euler in Field of Mathematics During his stay in Berlin • In the last 25 years that Euler spent in Berlin he made important discoveries in the Calculus of Variation. • Established Euler Identity for complex numbers. • Produced two treatises in analysis (Introduction to the Analysis of the Infinite in 1748) and Foundations of Differential Calculus with Applications to Finite Analysis and series in 1755.

22. List of Publication During his stay in Berlin

23. Return of St. Petersburg Academy

24. Return of St. Petersburg Academy • In 1762 , the politics in Russia changed. Empress Catherine II later named changed Catherine the Great, came to the thorne. The atmosphere in Russian society improved dramatically. • Catherine II aimed to create in Russia a regime of Educated Absolutism. She invited many progressive people to Russia and increased the budget of the St. Petersburgh Academy to 60000 rubles per year, which was much motre than the budget of the Berlin Academy.

25. Return of St. Petersburg Academy • Catherine II offered Euler an important post in the mathematics department, conference-secretary of the Academy, with a big salary. She instructed her representative in Berlin to agree to Euler's terms if he does not like her first offer. • In 1766 he accepted the invitation of Catherine the Great to return to the St. Petersburg Academy.

26. Lost of Eye Sight • In 1771 Euler's home was destroyed by fire and he was able to save only himself and his mathematical manuscripts. • In September 1771, Euler had surgery to remove his cataract. The surgery was very successful - the mathematicians vision was restored. • Unfortunately, Euler didn’t take care of his eyes; he continued to work and after a few days lost his vision again, this time without any hope of recovery. • He was virtually blind for the last seventeen years of his life. After losing the ability to see he commented “Now I will have less distraction”.

27. Death of Euler • On September 18, 1783, Euler passed away in St. Petersburg after suffering a stroke, and was buried with his wife in the Smolensk Lutheran • Cemetery on Vasilievsky Island.

28. Tomb of Leonhard Euler Tomb of Leonhard Euler in St. Petersburg Academy

29. List of Field in which Leonhard Euler worked • Differential and integral calculus • Logarithmic, exponential, and trigonometric functions • Differential equations, ordinary and partial • Elliptic functions and integrals • Hypergeometric integrals • Classical geometry • Number theory • Algebra • Continued fractions • Zeta and other (Euler) products • Infinite series and products • Divergent series • Mechanics of particles • Mechanics of solid bodies • Calculus of variations • Optics (theory and practice) • Hydrostatics • Hydrodynamics • Astronomy • Lunar and planetary motion • Topology • Graph theory

30. Bibliography • Euler has an extensive bibliography but his best known books are • Elements of Algebra: This elementary algebra text starts with a discussion of the nature of numbers and gives a comprehensive introduction to algebra, including formulae for solutions of polynomial equations. • Introductio in analysininfinitorum (1748). English translation Introduction to Analysis of the Infinite by John Blanton (Book I, ISBN 0-387-96824-5, Springer-Verlag 1988; Book II, ISBN 0-387-97132-7, Springer-Verlag 1989).

31. Bibliography • Two influential textbooks on calculus: Institutiones calculi differentialis (1755) and Institutiones calculi integralis (1768-1770). • Lettres a une Princesse d'Allemagne (Letters to a GermanPrincess)(1768-1772). English translation, with notes, and a life of Euler, available online from Google Books: Volume 1, Volume 2 • Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti (1744). (Method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense.)

32. Book of Euler on Differential Calculus

33. Books written by Euler on Mechanics

34. Book written by Euler on Navigation

35. Book written by Euler on curves

36. Book on Leonhard Euler

37. Character of Leonhard Euler • Clifford Truesdell wrote “He was exceptionally generous, never once making a claim of priority and in some cases actually giving away discoveries that were his own. He was the first to cite the work of others in what is now regarded as the just way, that is, so as to acknowledge their worth” • Euler was a committed Christian and frequently expressed awe at the work of the creator. Euler was particularly impressed by the design of eye.

38. Overview of some work of Leonhard Euler Euler established the use of • e for the base of Natural lograthim • for the ratio of circumference to the diameter of the circle. • f(x) for function value • sinx and cosx for values of sine and cosine function • i for Imaginary unit • Σ for summation • Δ for finite difference

39. Few works of Euler • In December 1729, Goldbach wrote a letter to Euler in regarding with the Fermat’s statement that all numbers of the form were primes. Euler became very much interested in number theory and afterwards disproved the assertion about Fermat’s prime. • Euler discovered that on this point Fermat was wrong for = 4,294,697,297 is evenly divisible by 641

40. Few works of Euler • In 1730 Euler established Gamma Function and Beta Function. • In 1735 Euler solved the famous problem known as Basel Problem. Euler found that the sum of infinite series

41. Periodical work of Euler • Konigsberg Bridge • The river Pregel, which flows through the Prussian city of Konigsberg, divides the city into an island and three distinct land masses, one in the north, one in the east, and one in the south. There are altogether seven bridges

42. Periodical work of Euler • Problem: Is it possible to follow a path that crosses each bridge exactly once and returns to the starting point? • Euler solved the problem in 1735, published as E53 in 1741, by showing that such paths cannot exist. • This solution is considered to be the first theorem of graph theory and planar graph theory. Euler also introduced the notion now known as the Euler characteristic of a space and a formula relating the number of edges, vertices, and faces of a convex polyhedron with this constant.

43. Few works of Euler • Prime Numbers and the Zeta Function. Let P = {2, 3, 5, 7,11,13,17......} be a set of prime numbers, i.e., the integers > 1 that is only divisible by 1 or by themselves.Euler’s fascination with prime numbers started quite early and continued throughout his life. An example of his profound insight into the theory of numbers is the discovery in 1737 (E72) of the fabulous product formula , s > 1, connecting prime numbers with zeta function.

44. Few works of Euler • Euler in his Element of Algebra introduced as imaginary unit. • Euler stated the general result what we known as De- Moivre Theorem for n ≥ 1 • Euler with the help of De Moivre theorem developed an well known Euler identity • In 1728 Euler wrote a paper entitled On finding the equation of geodesic curves. Later in 1744, he published a more general work entitled “A method of discovering curved lines that enjoy the maximum and minimum property

45. Few works of Euler One thing he considered in his paper the minimization of integral of the form He showed that the necessary condition for the minimum was Euler equation

46. Few works of Euler • Euler’s Polyhedral Formula: In a 1752 study of polyhedra, Euler observed that V + F = E + 2, where V is the number of vertices, F is the number of faces, and E is the number of edges of a solid figure. Because of the utter simplicity of this relationship, Euler confessed that, “I find it surprising that these general results in solid geometry have not previously been noticed by anyone, so far as I am aware. ” Of course, no previous mathematician had had Euler’s penetrating insight.

47. Physics and Astronomy • Euler helped develop the Euler-Bernoulli beam equation, which became a cornerstone of engineering. • He applied his analytic tools to problems in classical mechanics and to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes. • His accomplishments in astronomy include determining the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the sun. • In addition, Euler made important contributions in optics.

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