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

PTOLEMY'S SYNTAXIti 283
284 TRIGONOMETRY
(77) Table of Chords.
From this Ptolemy deduces that (crd. §°) is very nearly
0/' 31' 25", and by the aid of the above propositions he is in
a position to complete his Table of Chords for arcs subtending
angles increasing from J
° to 180° by steps of -|°; in other
words, a Table of Sines for angles from -|° to 90° by steps
of* .
(6) Further use of proportional increase.
Ptolemy carries further the principle of proportional increase
as a method of finding approximately the chords of
arcs containing an odd number of minutes between r and 30'.
Opposite each chord in the Table he enters in a third column
3 X oth of the excess of that chord over the one before, i.e. the
chord of the arc containing 30' less than the chord in question.
For example (crd. 2-|°) is stated in the second column of the
Table as 2P 37' 4". The excess of (crd. 2|°) over (crd. 2°) in the
Table is OP 31' 24"; ^th of this is OP Y 2" 48'", which is
therefore the amount entered in the third column opposite
(crd. 2^°). Accordingly, if we want (crd. 2° 25'), we take
(crd. 2°) or 2P 5' 40" and add 25 times OP l'2"48'"; or we
take (crd. 2|°) or 2P 37' 4" and subtract 5 times OP Y 2" 48'".
Ptolemy adds that if, by using the approximation for 1° and
J°, we gradually accumulate an error, we can check the calculation
by comparing the chord with that of other related arcs,
e.g. the double, or the supplement (the difference between the
arc and the semicircle).
Some particular results obtained from the Table may be
mentioned.
Since (crd. 1°) = 1 P 2' 50", the whole circumference
= 360 (IP 2' 50"), nearly, and, the length of the diameter
being 120*>, the value of n is 3 (1 +^_ + _5o_ _
) 3 + ^8_ + _|o_
which is the value used later by Ptolemy and is equivalent to
3-14166... Again, a/3 = 2 sin 60° and, 2 (crd. 120°) being
equal to 2 (103? 55' 23"), we have V3 = ^ (103 + f£ + dnta)
= 1 +— .]
43 55 23
+ _ = 1-7320509,
60 60 2 60 3
which is correct to 6 places of decimals.
Speaking generally,
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.