Some History of the Calculus of the Trigonometric Functions

1
Some History of the Calculus of the Trigonometric
Functions

• V. Frederick Rickey
• West Point

2
Archimedes (died 212 BCE)
3
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.

4
(No Transcript)
5
(No Transcript)
6
(No Transcript)
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
Isaac Newton 1642 - 1727

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

10
Newton 1664, 1676 (Epistola prior)
11
Gottfried Wilhelm von Leibniz1646 - 1716

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

12
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

13
Roger Cotes

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

14
(No Transcript)
15

• 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.

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
Euler creates trig functions in 1739
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
(No Transcript)
20
(No Transcript)
21
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
22
Chapter 1 Functions

• A change of Ontology
• Study functions
• not curves

23
VIII. Trig Functions
24

• 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

25
Euler creates trig functions in 1739
26
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

27
(No Transcript)
28
(No Transcript)
29

• 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

30

• 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

31

• 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

32
Institutionum calculi integralis, 1769
E366
33
(No Transcript)
34
Read Euler, read Euler, he is our teacher in
everything.

• Laplace
• as quoted by Libri, 1846

35
Euler, age 71

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

36
Power Point

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