PTOLEMY’S SYNTAXW 
283 
(rj) Table of Chords. 
Fipm this Ptolemy deduces that (crd. i°) is very nearly 
OP 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 |° to 180° by steps of -|°; in other 
words, a Table of Sines for angles from -|° to 90° by steps 
of i°. 
(6) Further use of proportional increase. 
Ptolemy carries further the principle of proportional in 
crease as a method of finding approximately the chords of 
arcs containing an odd number of minutes between 0' and 30'. 
Opposite each chord in the Table he enters in a third column 
Yo'th 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. 2i°) over (crd. 2°) in the 
Table is OP 31' 24"; ^th of this is OP V 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 1'2"48'"; or we 
take (crd. 2^°) or 2P 37' 4" and subtract 5 times OP V 2" 48'". 
Ptolemy adds that if, by using the approximation for 1° and 
|°, we gradually accumulate an error, we can check the calcu 
lation 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°) = IP 2' 50", the whole circumference 
= 360 (IP 2' 50"), nearly, and, the length of the. diameter 
being 12OP, the value of tt is 3 (1 +#o+xl§o) = 3 + tht + al&o> 
which is the value used later by Ptolemy and is equivalent to 
3-14166... Again, V3 = 2 sin 60° and, 2 (crd. 120°) being 
equal to 2 (103P 55' 23"), we have Vs = (103 + |£ + T f|o) 
43 55 23 
= 1 + — + —- + —5 = 1-7320509, 
60 60 2 60 d 
which is correct to 6 places of decimals. Speaking generally,