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 con 
siderations.’ 1 
He explains that he will u^e 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, 0 the centre, OG the 
radius perpendicular to AB. 
Bisect OB at 1). join 1)G, and measure 
DE along DA equal to DC. Join EG. 
Then shall OE be the side of the in 
scribed regular decagon, and EG the side 
of the inscribed regular pentagon. 
For, since OB is bisected at D, 
BE.EO + OD 2 = DE % 
= DC 2 = D0 2 + 0G 2 
Therefore BE. EO — OG 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 
= (70 2 + OE 2 = EG 2 ; 
therefore EG is the side of the regular pentagon inscribed 
in the circle. 
1 Ptolemy, Syntaxis, i. 10, pp. 31 2.