A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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
260 TRIGONOMETRY
by ¥Jo from §§§ or , -^T
Heron's figure). There is little doubt
that it is to Hipparchus's work that Heron refers, though the
author is not mentioned.
While for our knowledge of Hipparchus's trigonometry we
have to rely for the most part upon what we can infer from
Ptolemy, we fortunately possess an original source of information
about Greek trigonometry in its highest development
in the Sphaerica of Menelaus.
The date of Menelaus of Alexandria is roughly indicated
by the fact that Ptolemy quotes an observation of
his made in the first year of Trajan's reign (a.d. 98). He
was therefore- a contemporary of Plutarch, who in fact
represents him as being present at the dialogue De facie in
orbe lunae, where (chap. 17) Lucius apologizes to Menelaus 'the
mathematician ' for questioning the fundamental proposition
in optics that the angles of incidence and reflection are equal.
He wrote a variety of treatises other than the Sphaerica.
We have seen that Theon mentions his work on Chords in a
Circle in six Books. Pappus says that he wrote a treatise
(Trpayixareia) on the setting (or perhaps only rising) of
different arcs of the zodiac. 1 Proclus quotes an alternative
proof by him of Eucl. I. 25, which is direct instead of by
reductio ad absurdum, and he would seem 2, to have avoided
the latter kind of proof throughout. Again, Pappus, speaking
of the many complicated curves discovered by Demetrius of
'
Alexandria (in his " Linear considerations ") and by Philon
of Tyana as the result of interweaving plectoids and other
surfaces of all kinds ', says that one curve in particular was
investigated by Menelaus and called by him ' paradoxical
(irapd8o£os) 3 ;
the nature of this curve can only be conjectured
(see below).
But Arabian tradition refers to other works by Menelaus,
(l) Elements of Geometry, edited by Thabit b. Qurra, in three
Books, (2) a Book on triangles, and (3) a work the title of
which is translated by Wenrich de cognitione quantitatis
discretae corporum permixtomm. Light is thrown on this
last title by one al-Chazini who (about A.D. 1121) wrote a
1
Pappus, vi, pp. 600-2.
1
Heron, Metrica, i. 22, 24, pp. 58. 19 and 62. 17.
s2
2
Proclus on Eucl. I, pp. 345. 14-346. 11.
3
Pappus, iv, p. 270. 25.