Some History of the Calculus of the Trigonometric Functions

V. Frederick Rickey

West Point

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 12∙400.

• So the days are not equally likely.

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

Reviel Netz

• Professor of Classics at Stanford

• The Works of Archimedes: Translation and Commentary

• An editor of The Archimedes Palimpsest

Archimedes (died 212 BCE)

Sphere and Cylinder, Prop 21If 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.

EK Z B HN M

A

E

EA

EK Z B HN M

A

E

EA

Let angle EA n and r 1.So EK 2 sin n,

Z 2 sin 2n,B 2 sin 3n, etc.

Also E 2 cos n and EA 2 sin n2 sin n 2 sin 2n . . . 2sinn 1n2 cot n

2 sin n 2 sin 2n . . . 2 sinn 1n2 cot nThis is not a Riemann sum,so add one more term and divide by n

nj1

n

2 sin jn

ncot

n

n2 sin

n

n

The limit yields0

sinxx

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

What is a sine ?

• The Greeks used chords

• The Arabs used half-chords

• NB: These are line segments, not numbers!

Etymology

• Chord in Arabic:– Jya

• Half-chord in Arabic:– jiba

• Arabic abbreviation:– jb

• Latin mistranslation:– Jaib– Sinus

Etymology

• Chord in Arabic:– Jya

• Half-chord in Arabic:– jiba

• Arabic abbreviation:– jb

• Latin mistranslation:– Jaib– Sinus

Isaac Newton 1642 - 1727

• Series for arcsine and sine in De analysi, 1669

• Portrait: Kneller 1689

Newton: 1664, 1676 (Epistola prior)

If from a given right sine,or the versed sine, the arc is required,let r be the radius and x the right sine,and the arc will be

x x3

6r2

3x5

40r4

5x7

112r6 etc.

Gottfried Wilhelm von Leibniz1646 - 1716

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

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

1

542cos x1

2x.

Roger Cotes

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

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.

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.

CE

EG

AC

AD

drdsin r

cos

d

dsin cos

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

Euler’s Life

• Basel 1707-1727 20

• Petersburg I 1727-1741 14

• Berlin 1741-1766 25

• Petersburg II 1766-1783 17____

76

Euler’s Calculus Books

• 1748 Introductio in analysin infinitorum399

402

• 1755 Institutiones calculi differentialis676

• 1768 Institutiones calculi integralis462

542

508

_____

2982

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

Euler creates trig functions in 1739

Solve y k4d4y

dx4 0.

Factor 1 k4p4 0 :1 k p1 kp1 k2p2The solution is :

y xk C

xk D E Cosx

k F Sinx

k

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

Chapter 1: Functions

A change of Ontology:

Study functions

not curves

VIII. Trig Functions

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

• 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

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

Laplace

as quoted by Libri, 1846

Joseph Fourier 1768 - 1830

Georg Cantor, 1845 - 1918

Euler, age 71

• 1778 painting by Darbes

• In Geneva

• Used glass pane, á la Leonardo

Power Point

• http://www.dean.usma.edu/departments/math/people/rickey/talks-future.html

• Full text to follow