1. In Celebration of Euler's 300th Birthday V. Frederick Rickey West Point

2. Euler’s Life Basel1707-172720 Petersburg I1727-174114 Berlin1741-176625 Petersburg II1766-178317 ____ 76

3. Euler born 15 April 1707 in Basel Son of a Lutheran minister, Paul Euler Paul Euler studied mathematics with Jakob Bernoulli

4. LEONHARD EULER 1707 - 1783 mathematician, physicist, engineer, astronomer, and philosopher spent his early years in Riehen. He was a great savant and a kind-hearted man.

5. Paul Euler taught algebra to his son using Michael Stifel’s 1553 edition of Christoff Rudolff’s 1525 Coss, the first German book entirely devoted to algebra.

6. At age 7 Euler returned to Basel to attend a Latin school No mathematics taught Tutored by Johannes Burckhardt, “magni Euleri praeceptor in mathematicis”

7. Entered the University of Basel (founded 1460) in 1720, age 13. 19 professors, about 100 students Johann I Bernoulli was one professor Johann II Bernoulli was a fellow student Euler became friends with his brothers Daniel I and Nicolaus II

9. I soon found the opportunity to become acquainted with the famous professor Johann Bernoulli [who advised me] to look at more difficult mathematics books and work through them with great diligence [and visit him every Saturday]. When he resolved one of my objections, ten others disappeared, which is certainly the best method of making auspicious progress in the mathematical sciences.

10. Research in Basel by Euler The brachistochrone problem in resisting media The masting of ships won an accessit in the 1728 Paris Prize competition An essay on the physics of sound, written in an unsuccessful attempt to become professor of physics

11. St. Petersburg I 1727 – 1741 Nicolaus II and Daniel I Bernoulli got Euler invited Also there: –Jakob Hermann –Christian Goldbach –F. C. Meyer

12. Vue sur le quai de l'Université et l'Académie, depuis le quai opposé Saint-Petersbourg n'est-elle pas la Venise du Nord?

13. Euler about 1737, age 30 Painting by J. Brucker 1737 mezzotint by Sokolov Black below and above right eye Fluid around eye is infected “Eye will shrink and become a raisin” Ask your ophthalmologist Thanks to Florence Fasanelli

14. Research in Petersburg 1727 - 41 Published 55 papers; wrote 35 more –Started research in number theory Wrote 4 books: –Mechanica, 1736 –Rechen-Kunst, 1738 –Tentamen novae theoriae musicae, 1739 –Scientia navalis, 2 vols, 1749

15. An arithmetic book Assumes the students are highly literate Covers surprisingly little material E17, E35 1738, 1740

16. June 7th 1742, Christian Goldbach makes a conjecture: "Es scheinet wenigstens, daß eine jede Zahl, die größer ist als 2, ein aggregatum trium numerorum primorum sey."

17. “Mediation on experiments made recently on the firing of cannon.” Euler’s first paper on cannon, E853, written 1727, published 1862.

18. A polemic against Euler by Robins, 1739 Too algebraical and uses infinitesimals

19. From Teacher to Professor ? Robins hoped to be the first professor of mathematics at Woolwich Planned a course on fortifications and gunnery Politics intervened Mr. Derham served 1741 -1743.

20. Mathematics at Woolwich, 1741 That the second Master shall teach the Science of Arithmetic, together with the principles of Algebra and the Elements of Geometry, under the direction of the Chief Master. That the chief Master shall further instruct the hearers in Trigonometry and the Elements of the Conick Sections. To which he shall add the Principles of Practical Geometry and Mechanics, applied to raising and transporting great Burthens; With the Knowledge of Mensuration, and Levelling, and its Application to the bringing of water and the draining of Morasses; And lastly, shall teach Fortification in all its parts. But no calculus

21. 1742 Preface –55 pages Ch I: Internal ballistics –65 pages Ch2: External ballistics –30 pages Total: 150 pp.

22. Euler 1745 Frederick the Great asks about the best book on gunnery Euler magnanimously recognizes Robins Robins sets a research program for Euler Euler adds Annotations 2400

23. Euler returns to gunnery in E217 Presented 1752 Published 1755 Translated in 1777

24. Euler translated into English, 1777

25. From Euler’s preface: Some are of the opinion that fluxions are applicable only in such subtile speculations as can be of no practical use... But what has been just now said of artillery is sufficient to remove this prejudice...

26. It may be affirmed, that things which depend on mathematics cannot be explained in all their circumstances without the help of fluxions, and even this sublime part of mathematics has met with difficulties which it has not fully mastered.

28. Translations of Euler’s Observations upon the new principles of gunnery translation by Hugh Brown – p. 276 and p. 303 – 28 pages

29. 7 Postulates for motion of a projectile in a vacuum Postulate 2: If the Parabola, in which the Body moves, be terminated on a horizontal Plain, then the Vertex of the Parabola will be equally distant from its two Extremities. Postulate 4: If a Body be projected in different Angles, but with the same Velocity, then its greatest horizontal Range will be, when it is projected in an Angle of 45º with the horizon.

30. Euler’s Annotations Analytically derives the equations of motion from the fundamental principles of motion in a vacuum Confirms each of the 7 postulates for the trajectory of a projectile in a vacuum

31. Robins gives experimental evidence to confute the postulates posed in Proposition V. For example, according to postulate 5 in Prop V: –A musket ball ¾ of an inch in diameter that has an initial velocity of 1700 feet per second at an angle of 45º should have a horizontal range of about 17 miles according to the fifth postulate. –Actual range: Less than half a mile.

32. DatePages Robins1742150 Robins / Euler 1745720 Euler1777423 Robins / Euler 1922409 Editions of New Principles of Gunnery

33. –Differentials to fluxions –German to English –Leibnizian to Newtonian notation 1745 to 1777 is a triple translation:

34. Mathematics at Woolwich, 1772 1.The Elements of Euclid 2.Trigonometry applied to Fortification, and the Mensuration of Superficies and Solids 3.Conic Sections. 4.Mechanics applied to the raising and transporting heavy bodies, together with the use of the lever pulley, wheel, wedge and screw, &c. 5.The Laws of Motion and Resistance, Projectiles, and Fluxions. Now some calculus!

35. Question: –What was the impact of ballistics on mathematics? Answer: –Calculus was taught to engineers and artillerists.

36. Euler’s Berlin House Behrenstraße 21/22

37. Research in Berlin 1741 - 66 275 papers published; 105 more written Wrote 14 books, including –Methodus inveniendi, 1744 –Theoria motuum planetarum et cometarum, 1744 –Rescue of divine revelation from objections of freethinkers, 1747 –Théorie général de la dioptrique, 1765 –Lettres à une Princesse d’Allemagne, 1768

38. In addition, Euler supervised the Library Observatory Botanical garden Scientific publications Calendars and maps Various financial matters

39. Euler’s Calculus Books 1748Introductio in analysin infinitorum 399 402 1755Institutiones calculi differentialis 676 1768Institutiones calculi integralis 462 542 508 _____ 2982

40. Euler creates trig functions in 1739

43. Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From the preface of the Introductio

44. Chapter 1: Functions A change of Ontology: Study functions not curves

45. VIII. Trig Functions

46. He showed a new algorithm which he found for circular quantities, for which its introduction provided for an entire revolution in the science of calculations, and after having found the utility in the calculus of sine, for which he is truly the author... Eulogy by Nicolas Fuss, 1783

47. Sinus totus = 1 π is “clearly” irrational Value of π from de Lagny Note error in 113 th decimal place “scribam π” W. W. Rouse Ball discovered (1894) the use of π in W m Jones 1706. Arcs not angles Notation: sin. A. z

49. Research in Petersburg 1766 - 83 More than 400 papers –And, yes, he was blind ! 4 more books: –Dioptrica, 1768 –Vollstandige Anleitung zur Algebra, 1770 –Theoria motuum lunaea, 1772 –Théorie Complete de la Construction et de la Manoeuvre des Vaissaux, 1773

50. Euler's method, invented in 1768 in THIS house The address is Naberezhnaya Leitenanta Shmidta No. 15, St. Petersburg.

51. Euler is buried in the Lavra Cemetery in St. Petersburg. "Leonhardo Eulero Academia Metropolitana MDCCLXXXVII."

52. Read Euler, read Euler, he is our teacher in everything. Laplace as quoted by Libri, 1846

53. To be issued on the 300th anniversary of Euler’s birth, April 15, 1707