A history of Greek mathematics - Wilbourhall.org

HIPPARCHUS 259
in his Commentary, he used the formulae of spherical trigonometry
to get his results. In the particular case where it is
required to find the time in which a star of 27-§° northern
declination describes, in the latitude of
Rhodes, the portion of
its arc above the horizon, Hipparchus must have used the
equivalent of the formula in the solution of a right-angled
spherical triangle, tan b = cos A tan c, where C is the right
angle. Whether, like Ptolemy, Hipparchus obtained the
formulae, such as this one, which he used from different
applications of
the one general theorem (Menelaus's theorem)
it is not possible to say. There was of course no difficulty
in calculating the tangent or other trigonometrical function
of an angle if only a table of sines was given ; for Hipparchus
and Ptolemy were both aware of the fact expressed by
sin 2 a + cos 2 a = 1
or, as they would have written it,
(crd. 2a) 2 + {crd. (180°-2a)} 2 = 4r 2 ,
where (crd. 2 a) means the chord subtending an arc 2 a, and r
is the radius, of the circle of reference.
Table of Chords.
We have no details of Hipparchus's Table of Chords sufficient
to enable us to compare it with Ptolemy's, which goes
by half-degrees, beginning with angles of |°, 1°, l-§°, and so
on. But Heron 1 in his Metrica says that 'it is proved in the
books about chords in a circle ' that, if « 9
and a n are the sides
of a regular enneagon (9 -sided figure) and hendecagon (1 1 -sided
figure) inscribed in a circle of diameter d, then (1) a 9
= ^d,
(2) a u = £gd very nearly, which means that sin 20° was
taken as equal to 0-3333 ... (Ptolemy's table makes it
Rn( 20 "** fin + fin?)' S0 ^a^ ^e ^rs^ a PP roxmiati°n is §), and
sin T X T
.
180° or sin 16° 21' 49" was made equal to 0-28 (this corresponds
to the chord subtending an angle of 32° 43' 38",nearly
half-way between 32J° and 33°, and the mean between the two
1 /lt> 54 55 \
chords subtending the latter angles gives —
(
+ — H |
as
the required sine, while eV (
16A) = Iff, which only differs
1
Heron, Metrica, i. 22, 24, pp. 58. 19 and 62. 17.
s2