# Chords

#### What is a chord?

As used in mathematics, the word chord refers to a straight line drawn between two points on a circle (or more generally, on any curve). The known first trigonometric table was a table of chords. In modern times, the sine is used instead (sines and chords are closely related), but, perhaps, chords are more intuitive.

For example, the angle AOB in the diagram shows the curve connecting A to B to be an arc of a circle. The straight line AB is the chord. Of course, the length of the chord depends on the radius of the circle, in fact, it is proportional to the radius of the circle.

#### Trigonometry began with chords

Hipparchus (190–120 B.C.E.) produced the first trigonometric table for use in astronomy. It was a table of chords for angles in a circle of large fixed radius. Incidentally, his table was not in terms of degrees, but “steps”, each step being 1/24 of a circle. Later, Ptolemy (100–178 C.E.) constructed a more complete table of chords. His table had chords for angles increasing from 1/2 degree to 180 degrees by steps of 1/2 degree. It also included aids for interpolating chords for minutes of angle. Ptolemy used a different large fixed radius than Hipparchus. The advantage of a large radius is that fractions can be avoided. In contrast, our present-day trigonometric functions are based on a unit circle, that is, a circle of radius 1. Of course using a unit circle doesn’t avoid fractions, but we have decimal fractions which are easy to work with.

Claudius Ptolemy (ca. 90 ca. 168)

Portrait of Claudius Ptolemy (c. 1475)
Louvre Museum
Date ca. 1475, by Justus van Gent
Source: The Mathematical Tourist

Although trigonometry was, and still could be, based on chords as the primary trigonometric function, a slight modification of chords, called “sines,” turns out to be more convenient. Sines were first used in India a few centuries after chords were first used in ancient Greece. Sines are described on the next page.