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
Sunday
Monday
Tuesday
Wednesday
Thursday
Friday
684
687
685
685
687
684
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 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.
EK
Z
B
A
HN
M
E
EA
EK
Z
B
A
HN
M
E
EA
Let angle E A
n and r
1.
So EK
2 sin
n,
Z
2 sin 2
n,
B
2 sin 3
n, etc.
Also E
2 cos
n and EA
2 sin
2 sin
n
2 sin 2
n
.. . 2
sin n 1
n
2
cot
n
n
2 sin
n
n 1
2 sin 2
n
2
n
.. .
cot
n
This is not a Riemann sum,
so add one more term and divide by
2 sin
n
n
n
2 sin j
n
n
j 1
cot
The limit yields
sin x
0
x
n
n
2 sin
n
n
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
x3
3 x5
5 x7
x
etc.
6 r2
40 r4
112 r6
Gottfried Wilhelm von Leibniz
1646 - 1716
• The sine series could
be derived from the
cosine series by termby-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
1
542
cos x
1
2 x.
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
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
AC
EG
AD
d r
d sin
d
d
sin
r
cos
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 infinitorum
399
402
• 1755
Institutiones calculi differentialis
676
• 1768
Institutiones calculi integralis
462
542
508
_____
2982
Euler was prolific
I
Mathematics
II
Mechanics, astronomy
III
Physics, misc.
IVa Correspondence
IVb Manuscripts
29 volumes
31
12
8
7
87
One paper per fortnight, 1736-1783
Half of all math-sci work, 1725-1800
Euler creates trig functions in 1739
k4
Solve y
d4 y
0.
dx4
Factor 1 k4 p4
0:
1 k p 1 kp 1
2
k p
The solution is :
y
x
k
C
x
k
D
E Cos
2
x
k
F Sin
x
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/m
ath/people/rickey/talks-future.html
• Full text to follow