- INDIA 259
FIGURE 12. For a quadrilateral inscribed in a circle, the product
of the diagonals equals the sum of the products of the two pairs of
opposite sides.To construct his table of chords, Ptolemy had to make use of some subtle
geometry developed earlier: in particular, the fact that for a quadrilateral inscribed
in a circle the product of the diagonals is the sum of the products of the two pairs of
opposite sides.^14 Ptolemy proved this result by drawing AE (see Fig. 12) so that
ÄΒΑΕ = ZD AC, thus obtaining two pairs of similar triangles: ABAE ~ ACAD
and AADE ~ AACB. (Angles ABD and ACD are equal, both being inscribed in
the same arc AD; and similarly Æ AC Β — ZADB.) Ptolemy used this relation to
compute the chord of the difference of two arcs and the chord of half an arc.
Since Ptolemy knew the construction of the regular dodecagon and the regular
decagon, he was easily able to compute the chords of 36° and 30°, expressed in
units of one-sixtieth of the radius. His difference theorem then gave the chord of
6°. Then by repeated bisection he got the chord of 3°, then 1° 30', and finally, 45'.
Using these two values and certain inequalities, he was able to set upper and lower
bounds on the chord of 1° with sufficient precision for his purposes. He then set
out a table with 360 entries, giving the chords of arcs at half-degree increments up
to 180°.
Although this table fulfilled its purpose in astronomy, the chord is a cumber-
some tool to use in studying plane geometry. For example, it was well known that
in any triangle, the angle opposite the larger of two sides will be larger than the an-
gle opposite the smaller side. But what is the exact, quantitative relation between
the two sides and the two angles? The ratio of the sides has no simple relationship
to the ratio of the angles or to the chords of those angles. There is, however, a
very simple relation between the sides and the chords of twice the opposite angles,
that is, the chords these angles cut off on the circumscribed circle. One might have
thought that the constant comparison of a chord with the diameter would have
inspired someone to associate the arc with the angle inscribed in it rather than the
central angle it subtends. After all, a side of any triangle is the chord of a central
angle in the circumscribed circle equal to the double of the opposite angle. Hindu
astronomers discovered that trigonometry is simpler if you express the relations
between circular arcs and chords in terms of half-chords, what are now called sines.
(^14) When the quadrilateral is a rectangle, this fact is the form of the Pythagorean theorem given
in the Sulva Sutras. Gow (1884, P- 194) describes this result as "now appended to Euclid VI."