1 / 32

Historical Notes on Archimedes, Trigonometry, and Algebra

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

xiang
Download Presentation

Historical Notes on Archimedes, Trigonometry, and Algebra

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

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

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

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

  6. 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.

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

  8. Almagest Katz paper

  9. Calculus Differentialis1727 • The Calculus of Finite Differences • The differential Calculus in General • Differentiation of Algebraic Functions • Differentiation of Logarithmic and Exponential Quantities

  10. 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

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

  12. Euler to Johann BernoulliSeptember 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.

  13. Euler creates trig functions in 1739

  14. 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

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

  16. VIII. Trig Functions

  17. 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

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

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

  20. 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.

  21. 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 1200th anniversary of al-Khwārizmī's birth.

  22. 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

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

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

  25. Abstraction makes mathematics easier ! • The introduction of zero • The coefficients include negative reals • a x2 + b x + c = 0

  26. Questions • On what I have just said • On any topic

More Related