A history of Greek mathematics - Wilbourhall.org

PTOLEMY'S SYNTAXIS 277
set out correctly, but may be in possession of a ready proof of
our method of obtaining them based on geometrical consideration^'
1
He explains that he will use the division (1) of the circle into
360 equal parts or degrees and (2) of the diameter into 120
equal parts, and will express fractions of these parts on the
sexagesimal system. Then come the geometrical propositions,
as follows.
(a) Lemma for finding sin 18° and sin 36°.
To find the side of a pentagon and decagon inscribed in
a circle or, in other words, the chords subtending arcs of 72°
and 36° respectively.
Let AB be the diameter of a circle, the centre, OC the
radius perpendicular to AB.
Bisect OB at D, join DC, and measure
DE along DA equal to DC.
Join EC.
Then shall OE be the side of the inscribed
regular decagon, and EC the side
of the inscribed regular pentagon.
For, since OB is bisected at D,
BE.E0 + 0D 2 = DE 2
= DC 2 =D0 2 + 0C 2 .
Therefore BE. E0 = OC 2 = OB 2 ,
and BE is divided in extreme and mean ratio.
But (Eucl. XIII. 9) the sides of the regular hexagon and the
regular decagon inscribed in a circle when placed in a straight
line with one another form a straight line divided in extreme
and mean ratio at the point of division.
Therefore, BO being the side of the hexagon, EO is the side
of the decagon.
Also (by Eucl. XIII. 10)
(side of pentagon) 2 = (side of hexagon) 2 + (side of decagon) 2
= CO 2 + OE 2 = EC 2 ;
therefore EC is the side of the regular pentagon inscribed
in the circle.
*
1
Ptolemy, Syntaxis, i. 10, pp. 31 2.