284 
TRIGONOMETRY 
the sines obtained from Ptolemy’s Table are correct to 5 
places. 
(i) Plane trigonometry in effect used. 
There are other cases in Ptolemy in which plane trigono 
metry is in effect used, e.g. in the determination of the 
eccentricity of the sun’s orbit. 1 Suppose that ACBD is 
the eccentric circle with centre 0, 
and AB, Cl) are chords at right 
angles through E, the centre of the 
earth. To find OE. The arc BO 
is known (= a, say) as also the arc G 
CA (= (3). If BE be the chord p 
parallel to CP, and CG the chord 
parallel to AB, and if JP, P be the 
middle points of the arcs BE, GC, 
Ptolemy finds (1) the arc BE 
{= oc + f3 — 180°), then the chord BE, 
crd. (a + /3 —180°), then the half of it, (2) the arc GC 
= arc (a + (3 — 2(3) or arc (a —/3), then the chord GC, and 
lastly half of it. He then adds the squares on the half 
chords, i.e. he obtains 
OE 2 = \ {crd. (a + /3— 180)} 2 + |{crd, (a — (3) ] 2 , 
that is, OE 2 /r 2 = cos 2 \ (a + /3) + sin 2 |(a — (3). 
He proceeds to obtain the angle OEC from its sine OB / 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 funda 
mental 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 pro 
positions of Menelaus’s Sphderica, e.g. III. 2 and 3, he does 
1 Ptolemy, Syntaxis, iii, 4, vol. i, pp. 284-7.