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Some History of the Calculus of the Trigonometric Functions

Some History of the Calculus of the Trigonometric Functions. V. Frederick Rickey West Point. The 13 th is more likely to occur on Friday than on any other day of the week. The Gregorian calendar has a 400 year cycle. 7 does not divide 12 ∙ 400. So the days are not equally likely.

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Some History of the Calculus of the Trigonometric Functions

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  1. Some History of the Calculus of the Trigonometric Functions V. Frederick Rickey West Point

  2. The 13th is more likely to occur on Friday than on any other day of the week. The Gregorian calendar has a 400 year cycle. 7 does not divide 12∙400. So the days are not equally likely. A Theorem for Triskaidekaphobics

  3. The 13th is more likely to occur on Friday than on any other day of the week. Saturday 684 Sunday 687 Monday 685 Tuesday 685 Wednesday 687 Thursday 684 Friday 688 A Theorem for Triskaidekaphobics

  4. Reviel Netz • Professor of Classics at Stanford • The Works of Archimedes: Translation and Commentary • An editor of The Archimedes Palimpsest

  5. Archimedes (died 212 BCE)

  6. Sphere and Cylinder, Prop 21 If in an even-sided and equilateral polygon is inscribed inside a circle, and the lines are draw through, joining the sides of the polygon (so that they are parallel to one – whichever – of the lines subtended by two sides of the polygon), all the joined lines have to the same diameter of the circle that ratio, which the line (subtending the sides, whose number is smaller by one, than half the sides) has to the side of the polygon.

  7. Problem • Mesopotamians created trig, 3rd BCE • Hipparchus constructed a table, 150 BCE • Archimedes was killed in 212 BCE • So who did this? Cardano, Kepler, Roberval

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

  9. Etymology • Chord in Arabic: • Jya • Half-chord in Arabic: • jiba • Arabic abbreviation: • jb • Latin mistranslation: • Jaib • Sinus

  10. Etymology • Chord in Arabic: • Jya • Half-chord in Arabic: • jiba • Arabic abbreviation: • jb • Latin mistranslation: • Jaib • Sinus

  11. Isaac Newton 1642 - 1727 • Series for arcsine and sine in De analysi, 1669 • Portrait: Kneller 1689

  12. Newton: 1664, 1676 (Epistola prior)

  13. Gottfried Wilhelm von Leibniz1646 - 1716 • The sine series could be derived from the cosine series by term-by-term integration

  14. The derivatives of the trigonometric functions are rather amazing when one thinks about it. Of all the possible outcomes, D sin x = cos x. Simply cos x, not Is it just luck on the part of mathematicians who derived trig and calculus? I assume trig was developed before calculus, why or how could the solution prove to be so simple? Luck. A Student Fl. 1988

  15. Roger Cotes Sir Isaac Newton, speaking of Mr. Cotes, said “If he had lived we might have known something.”

  16. The small variation of any arc of a circle is to the small variation of the sine of that arc, as the radius to the sine of the complement.

  17. The small variation of any arc of a circle is to the small variation of the sine of that arc, as the radius to the sine of the complement.

  18. 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

  19. Euler’s Life • Basel 1707-1727 20 • Petersburg I 1727-1741 14 • Berlin 1741-1766 25 • Petersburg II 1766-1783 17 ____ 76

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

  21. Euler was prolific I Mathematics 29 volumes II Mechanics, astronomy 31 III Physics, misc. 12 IVa Correspondence 8 IVb Manuscripts 7 87 One paper per fortnight, 1736-1783 Half of all math-sci work, 1725-1800

  22. Euler creates trig functions in 1739

  23. 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

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

  25. VIII. Trig Functions

  26. 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

  27. 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

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

  29. Joseph Fourier 1768 - 1830

  30. Georg Cantor, 1845 - 1918

  31. Euler, age 71 • 1778 painting by Darbes • In Geneva • Used glass pane, á la Leonardo

  32. Power Point • http://www.dean.usma.edu/departments/math/people/rickey/talks-future.html • Full text to follow

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