Featured essay/article of this month
Title: The bows and arrows of trigonometry
By Randy Schwartz (Schoolcraft College) and Rheta Rubenstein (University of Michigan-Dearborn)
... Excerpt from Math Horizons, September 2006, pp26-27.
[The Sinuous Route to Sine]
: To steer by the stars or to predict tides and eclipses, ancient astronomers
had to figure out many things about angles. For measuring an agle and its
associated arcs and line segments, they imagined placing the vertex of the
angle at the center of a circle. Studying angles in this way was one of the
main ancient sources of trigonometry: a word that literally means
"measurement of three-angled figures."
In Figure 1, we've drawn twin copies of an angle A in "standard position"
inside a circle. The archer's bow and bowstring represent an ancient way
of visualizing angles. They help explain several mathematical terms that
we still use today. For example, we call the "bowstring" a chord,
based on the Latin chorda, meaning a musical string or other piece
of cord, such as might be used to string a bow. Similarly, our term for
the circular arc is derived from the Latin arcus, meaning a
bow, also seen in the words arch and archer.
The bow-and-arrow image also explains the origin of our term sine,
abbreviated sin. In this case, the naming process was a surprising
one, involving several different cultures and a resultant misunderstanding.
Among astronomers in India during the early Middle Ages, the term for a
circular arc was chapa (Sanskrit for "bow"), and a chord was called
samasta-jya ("bowstring"). Each half of the chord was called
jya-ardha ("half-string"), a term frequently abbreviated to
jya or its variant jiva. Certain Medieval Arab mathematicians,
borrowing these concepts, wrote the term as jiba, but this was
routinely confused with a term that had already existed in Arabic and
looked similar, but had a completely different meaning: jaib,
a pocket or bay. Later, when a European scholar translating the Arabic
into Latin sought a word with meaning most similar to jaib, he
came up with sinus, designating a bend, fold, or pocket. The same
root is seen in words like sinuous, insinuate, and sinuses ("pockets in
Our term "sine," then, traces back in a very sinuous and roundabout way
to "half-chord," designating a line segment opposite to an angle placed
in standard position. Notice that originally, the sine was thought of
as a certain line segment (or its length), not as a ratio of two lengths.
This was standard practice from ancient times all the way until the
1700s, when Leonhard Euler redefined the sine function (and its sisters)
in such a way as to depend purely on the angle involved, not on the
size of the circle. Of course, in a unit circle (a circle whose radius
is one unit long), the two types of definitions give the same results.
If the arc is bow, and the chord is a bowstring, where is the arrow
itself? The Indians visualized an arrow with its head positioned at
the bow and its tail resting on the straight bowstring (see Figure 1).
This line segment therefore came to be called "arrow": sara in
Sanskrit, sahm in Arabic and sagitta in Latin, a root
also seen in Sagittarius, the archer constellation in the zodiac.
Another Latin term for the same line segment is versus sinus,
which literally means "turned sine," since it can be visualized as
the half-chord (sine) turned through 90° but still
bounded by the arc of the circle. In English we call this the
versed sine or versine, abbreviated versin
(see Figure 2).
Astronomers and mahtematicians needed to understand the many relationships
involving bows, chords, and arrows. For example, Muhammad al-Khuwarizmi,
writing in Baghdad around 825, indicated how to use the chord and the
arrow to reconstruct the diameter of the circle: "If you want to ascertain
the circle to which it belongs, multiply the half-chord by itself, divide
it by the arrow, and add the quotient to the arrow; the sum is the
diameter of the circle to which this bow belongs". See if you can
verify that formula. Then provide, as did al-Khuwarizmi, an algorithm
for computing the area between the bow and bowstring, based purely on
the lengths of the bow, chord, and arrow, and without reference to pi.
... (For the full text of the article, see Sep 2006 issue of Math Horizons.) ...
... (See also References cited in the article) ...