FIGURE 13. The "bowstring" diagram. The sine of the arc AR is
the line AB.In Fig. 13 the arc AR can be measured by either line AB or AR. Ptolemy chose
AR and was led to the complications already mentioned. The Hindus preferred
AB, which is succinctly described as half the chord of twice the arc. We mentioned
above that the Chinese word (xian) for the hypotenuse of a right triangle means
bowstring. The Hindus used the Sanskrit term for a bowstring (jya or jiva) to mean
the sine. The reason for the colorful language is obvious from the figure.
To all appearances, then, trigonometry began to assume its modern form among
the Hindus some 1500 years ago. A few reservations are needed, however. First, for
the Hindu mathematicians the sine was not, as it is to us, a ratio. It was a length,
and that physical dimension had to be taken into account in all computations.
Second, the only Hindu concept corresponding approximately to our trigonometric
functions was the sine. The tangent, secant, cosine, cotangent, and cosecant were
not used. Third, the use of trigonometry was restricted to astronomy. Surveying,
which is the other natural place to use trigonometry, did not depend on angle
measurement.
Aryabhata used the sine function developed in the Surya Siddhanta, giving a
table for computing its values at intervals of 225' (3° 45') of arc from 0° to 90°
degrees and expressing these values in units of 1' of arc, rounded to the nearest
integer, so that the sine of 90° is 3438 = 360 · 60 • ð. This interval suggests that the
tables were computed independently of Ptolemy's work. If the Hindu astronomers
had read Ptolemy, their tables of sines could easily have been constructed from his
table of chords, and with more precision than is actually found. Almost certainly,
this interval was reached by starting with an angle of 30°, whose sine was known to
be half of the radius, then applying the formula for a half-angle to get successively
15°, 7° 30', and finally 3° 45'. Arybhata's table is actually a list of the differences
of 24 successive sines at intervals of 225 minutes. Since one minute of arc is a
very small quantity relative to the radius, the 24 values provide sufficient precision
for the observational technology available at the time. Notice, however, that to
calculate the sine of half of an angle è one would have to apply the cumbersome