Half Is Better
Sine and Cosine
Hipparchus of Rhodes (190 – 120 B.C.)
• Planetary motion
– Celestial sphere
– Position of stars were specified by angles
– Relate angles to a straight line segment
• Chords
– Future positions of stars and planets
Hipparchus of Rhodes
• Table of chords
• Worked with a circle with radius 3438
– Why 3438?
• Didn’t survive
• Referenced by other mathematicians
Claudius Ptolemy (85 – 165 A.D.)
• The greatest ancient Greek
astronomer
• Almagest – theory of chords
– “Spherical triangles”
• Explained how to construct a table of
chords
• Devised a method to compute
o
o
approx. to chords from 1/2 to 180
Going to India!
• Table of “half-chords” – 5th century
• Many situations require one to use
half the chord of twice the angle
• Indian astronomers understood this
– Called them jyā-ardha – “half-chords”
– Shortened to jyā
Still in India
• Earliest tables used circles with radius
3438 (Hipparchus of Rhodes)
• No way to exactly find the length of a
chord of an arbitrary angle
• Many Indian mathematicians found
approximations through the 12th
century and beyond
• Rediscovered by European
mathematicians
Arabs
• Indian mathematics came to Europe
by way of Arabic mathematicians
• Arabs learned astronomy from jyā
tables
• Instead of translating jyā, they
invented the word jiba
• Discovered connections between
trigonometry and algebra
“Trigonometry”
• Computing sines of arbitrary angles
and solving cubic equations
• Expanded understanding of spherical
triangles
• Added a “shadow” function (tangent)
• Improved methods for computing
“half-chord” and “shadow” tables
The Mistake
• Europeans discovered Arabic material
• Translating jiba
– jb → jaib – “cove” or “bay”
– Chose sinus – “Something is sinuous if it has
lots of coves and hollows.”
• This turned into our modern word sine
16th Century
• Our “trigonometry” was a part of
astronomy until this time
• Began to break apart as a topic of
interest itself
• Johannes Müller (1436 – 1476)
– On All Sorts of Triangles (1463)
• Not published for several decades
• Knows of tangent but only uses sine
• Applications of both plane and spherical
triangles
Cosine?
• Needed to use the sine of the
complementary angle
o

– sin(90 - )
– No special name yet
– By the 17th century, sinus complementi
had become co. sinus, then cosinus.
The Next Few Decades…
• Works influenced by On All Sorts of
Triangles by Müller
– Re-workings of On All Sorts of Triangles
– George Joachim Rheticus (1514 – 1574)
• Sines and other functions of right triangles
• No reference to circles
– Thomas Finche (1561 – 1656)
• Invented the words tangent and secant
– Bartholomeo Pitiscus (1561 – 1613)
• Invented the word trigonometry for his book title (1595)
After Calculus
• Leonhard Euler (1707 – 1783)
– Thought of sine as a ratio instead of a line
segment
– Used sine as a function, the way we now
use functions
– Sine is a function of the arc in a unit circle
Sine Curve
• Gilles de Roberval (1602 – 1675)
– Sketched the
sine curve
– He was
computing the
area under a
cycloid
– Not clear if he
understood what he did
Timeline
• Hipparchus of Rhodes...............190 – 120 B.C.
• Claudius Ptolemy……………......85 – 165 A.D.
– Almagest – theory of chords
• Table of “Half Chords”…………....5th century
• Indian mathematics came
to Europe….........…..……......… ~12th century
• Mistranslation of jiba….... ~12 – 16th centuries
Timeline Continued
• Johannes Müller……………………1436 – 1476
– On All Sorts of Triangles………………………….1463
• Cosine…………………………..……17th century
• Bartholomeo Pitiscus………………1561 – 1613
– Invented “trigonometry”………………………..1595
• Gilles de Roberval…………….……1602 – 1675
– Sketched sine curve
• Leonhard Euler………………….…..1707 – 1783
References
• Berlinghoff, William and Gouvea, Fernando.
Math through the Ages. Maine: Oxton House
Publishers, 2002.
• Sine curve - http://edgrenweb.se/math/.
• Half the chord of twice the angle example http://aleph0.clarku.edu/~djoyce/java/trig/
sines.html.