Some History of the Calculus of the Trigonometric Functions
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Transcript of Some History of the Calculus of the Trigonometric Functions
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