Some History of the Calculus of the Trigonometric Functions

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Some History of the Calculus of the Trigonometric
Functions

• V. Frederick Rickey
• West Point

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A Theorem for Triskaidekaphobics

• 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 12400.
• So the days are not equally likely.

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A Theorem for Triskaidekaphobics

• 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

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Reviel Netz

• Professor of Classics at Stanford
• The Works of Archimedes Translation and
  Commentary
• An editor of The Archimedes Palimpsest

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Archimedes (died 212 BCE)
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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.

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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

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What is a sine ?

• The Greeks used chords
• The Arabs used half-chords
• NB These are line segments, not numbers!

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Etymology

• Chord in Arabic
• Jya
• Half-chord in Arabic
• jiba
• Arabic abbreviation
• jb
• Latin mistranslation
• Jaib
• Sinus

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Etymology

• Chord in Arabic
• Jya
• Half-chord in Arabic
• jiba
• Arabic abbreviation
• jb
• Latin mistranslation
• Jaib
• Sinus

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Isaac Newton 1642 - 1727

• Series for arcsine and sine in De analysi, 1669
• Portrait Kneller 1689

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Newton 1664, 1676 (Epistola prior)
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Gottfried Wilhelm von Leibniz1646 - 1716

• The sine series could be derived from the cosine
  series by term-by-term integration

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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

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Roger Cotes

• Sir Isaac Newton, speaking of Mr. Cotes, said
  If he had lived we might have known something.

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• 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.

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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.
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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

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Eulers Life

• Basel 1707-1727 20
• Petersburg I 1727-1741 14
• Berlin 1741-1766 25
• Petersburg II 1766-1783 17
• ____
• 76

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Eulers Calculus Books

• 1748 Introductio in analysin infinitorum
• 399
• 402
• 1755 Institutiones calculi differentialis
• 676
• 1768 Institutiones calculi integralis
• 462
• 542
• 508
• _____
• 2982

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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

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Euler creates trig functions in 1739
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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
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Chapter 1 Functions

• A change of Ontology
• Study functions
• not curves

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VIII. Trig Functions
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• 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

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

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Read Euler, read Euler, he is our teacher in
everything.

• Laplace
• as quoted by Libri, 1846

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Joseph Fourier 1768 - 1830
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Georg Cantor, 1845 - 1918
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Euler, age 71

• 1778 painting by Darbes
• In Geneva
• Used glass pane, á la Leonardo

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Power Point

• http//www.dean.usma.edu/departments/math/people/r
  ickey/talks-future.html
• Full text to follow