A history of Greek mathematics - Wilbourhall.org

PTOLEMY'S SYNTAXIti 283
(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
+
60 60 2 60 3
_ = 1-7320509,
which is correct to 6 places of decimals.
Speaking generally,