A history of Greek mathematics - Wilbourhall.org

284 TRIGONOMETRY
the sines obtained from Ptolemy's Table are correct to 5
places.
(l) Plane trigonometry in effect used.
There are other cases in Ptolemy in which plane trigonometry
is in effect used, e.g. in the determination of the
eccentricity of the sun's orbit. 1 Suppose that AGBD is
the eccentric circle with centre 0,
and A B, GD are chords at right
angles through E, the centre of the
earth. To find OE. The arc BG
is known (= a, say) as also the arc
GA (=P). If BF be the chord
parallel to CD, and GG the chord
parallel to A B, and if iV, P be the
middle points of the arcs BF, GG,
Ptolemy finds (1) the arc BF
(= oc + /3- 180°), then the chord BF,
crd. (a +/3-180 ), then the half of it, (2) the arc GG
— arc (a + /3— 2/3) or arc (a — /?), then the chord GG, and
lastly half of it. He then adds the squares on the halfchords,
i.e.
he obtains
0# 2 = i{crd. (
a + 0-18O)} 2 + f{crd.(a-/3)} 2 ,
that is, 0E*/r* = cos 2 J (oc + /3) + sin 2 § (a - 0).
He proceeds to obtain the angle OEG from its sine OR/ OE,
which he expresses as a chord of double the angle in the
circle on OE as diameter in relation to that diameter.
Spherical trigonometry : formulae in solution of
spherical triangles.
In spherical trigonometry, as already stated, Ptolemy
obtains everything that he wants by using the one fundamental
proposition known as Menelaus's theorem ' ' applied
to the sphere (Menelaus III. 1), of which he gives a proof
following that given by Menelaus of the first case taken in
his proposition. Where Ptolemy has occasion for other propositions
of Menelaus's Sphaerica, e.g. III. 2 and 3, he does
1
Ptolemy, Syntaxis, iii. 4, vol. i, pp. 234-7.