1 The Birth of Science - MSRI

2.8 Trigonometry and Spherical Geometry 47
for use in astronomy. The only difference between ancient and modern
trigonometry is in the choice of the fundamental function, which was then page 78
the chord, rather than the sine. The two choices are clearly equivalent, as
one can pass from one function to the other using the obvious formula
chord a = 2 sin a
2 ,
since the sine of an angle is simply half the chord subtended by twice the
angle.
It is not possible to compute the chord associated to an arc of a given
length using the methods typical of geometric algebra; that is, one cannot
compute the chord function using ruler and compass (this task being
one of the possible formulations of the famous problem of squaring
the circle). 63 That this impossibility had not blocked the development of
trigonometry, but had instead channeled it toward methods other than
geometric algebra, such as numerical tables written in positional notation,
shows that the use of ruler and compass was a matter of convenience
rather than an intellectual prejudice.
In the fourth century A.D., trigonometry was imported into India together
with astronomy, to which it had become an indispensable technical
adjunct. 64 Indeed, at various times during the century, Alexandrian astronomers
and mathematicians decided to emigrate to India, pressed by
their ever more precarious situation in Alexandria. 65 It seems, for example,
that the Paulisa who authored the Indian astronomical work Paulisa page 79
siddhanta was the astronomer Paulus, a refugee from Alexandria.
Indian mathematicians, having to use half-chords often, decided to pick
the half-chord as the main variable. (They eventually even transferred to
it the Sanskrit term for the chord or bowstring itself, jiva. The Arabs, instead
of translating this term, transliterated it with consonants that could
also be interpreted as jaib, meaning “bosom of dress, cavity”; this was subsequently
translated into Latin as “sinus”, with the same meaning. 66 ) The
novelty consisted of a trivial change in variables, which eliminated factors
of 2 in some formulas but did not alter in any way the theorems of Hellenistic
trigonometry; the latter were recovered intact on the other side of
63 Obviously one can draw, with ruler alone, the chord corresponding to a given arc, but computing
the chord function with ruler and compass would require constructing with these instruments
the chord of an arc whose length is the same as that of a given segment. The construction of such
an arc is effectively equivalent to the inverse operation of rectifying a circumference.
64 It seems certain that the astronomical methods developed in the West were first introduced in
India in the second century A.D., but these were Mesopotamian arithmetic methods used in Greek
astrological texts; see [Neugebauer: HAMA], vol. 1, p. 6. The use in Indian astronomy of geometric
methods, which required trigonometric functions, started in the fourth century.
65 Hypatia’s end early in the next century (see page 9) shows that this was not at all a bad idea.
66 [Rosenfeld], p. 11.
Revision: 1.12 Date: 2003/01/10 06:11:21