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]

Figure 1
Figure 1.
: 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 your head").

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.

Figure 2
Figure 2.
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) ...