1. Historical Notes on Archimedes, Trigonometry, and Algebra Fred Rickey USMA USMAPS, 29 October 2008

2. October 29, 1998 A 10th century codex is sold for $2,000,000 to an anonymous buyer It contains the unique copies of two works by Archimedes: –The Stomachion –The Method The Walters Art Museum in Baltimore is conserving the manuscript

4. http://www.math.ucsd.edu/~fan/stomach/ How many ways can you rearrange the pieces of the stomachion?

5. The balancing act involves thin layers of the cone, sphere and the cylinder. x x x 2a http://mthwww.uwc.edu/uwmc-math/pmartin/

6. Archimedes: The Palimpsest Project http://www.archimedespalimpsest.org Many great pictures here of the restoration of the palimpsest.

7. At 2pm on October 29th, 2008, ten years after the Archimedes Palimpsest was purchased by the present owner, the core data generated by the project to conserve, image and study the manuscript, will be released on the web.

9. What is a sine ? The Greeks used chords The Arabs used half-chords NB: These are line segments, not numbers!

10. Almagest Katz paper

11. Calculus Differentialis 1727 1.The Calculus of Finite Differences 2.The differential Calculus in General 3.Differentiation of Algebraic Functions 4.Differentiation of Logarithmic and Exponential Quantities

12. Draft on Differential Calculus, 1827 Euler defines functions and then divides them into two classes: –Algebraic –Transcendental The only transcendental functions are logarithms and exponentials Euler gives a differential calculus of these functions NB: no trigonometry

13. Daniel Bernoulli to Euler, May 4, 1735 The DE arises in a problem about vibrations on an elastic band. “This matter is very slippery.”

14. Euler to Johann Bernoulli September 15, 1739 after treating this problem in many ways, I happened on my solution entirely unexpectedly; before that I had no suspicion that the solution of algebraic equations had so much importance in this matter.

15. Euler creates trig functions in 1739

18. 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

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

20. VIII. Trig Functions

21. 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

22. 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

23. Linear Differential Equations with constant coefficients De Integratione Aequationum Differentialium altiorum graduum 1743 E62

24. Editor’s introduction in 1754 there occurs in analysis a very important type of transcendental quantity, namely the sine... which demands a special calculus, which the celebrated author of this dissertation is able rightly to claim all for himself.

26. Abu Ja'far Muhammad ibn Mūsā al-Khwārizmī Lived c. 780 to c. 850 Stamp issued September 6, 1983 in the Soviet Union to commemorate the 1200 th anniversary of al-Khwārizmī's birth. September 61983Soviet Union

27. Kitab al-jabr wa l-muqabala The book of restoration and balancing “what is easiest and most useful in arithmetic” Origin of our word algebra

28. Latin translation, beginning with "Dixit algorizmi" His name is the origin of our word “algorithm”

29. Six Types of Quadratics 1.Squares equal to roots x² = 5x 2.Squares equal to numbers x² = 9 3.Roots equal to numbers 4x = 20 4.Squares and roots equal to numbers x² + 10x = 39 5.Squares and numbers equal to roots x² + 21 = 10x 6.Roots and numbers equal to squares 3x + 4 = x²

31. Abstraction makes mathematics easier ! The introduction of zero The coefficients include negative reals a x 2 + b x + c = 0

32. Questions On what I have just said On any topic