438 
PAPPUS OF ALEXANDRIA 
Take points P, R on EG, EF such that 
EP' 2 = GE. EM, and ER 2 = FE.EN. 
Then EP is half the major axis, and ER half the minor axis. 
Pappus omits the proof. 
Problem of seven hexagons in a circle. 
Prop. 19 (chap. 23) is a curious problem. To inscribe seven 
equal regular hexagons in a circle in such a way that one 
is about the centre of the circle, while six others stand on its 
sides and have the opposite sides in each case placed as chords 
in the circle. 
Suppose GHKLXM to be the hexagon so described on UK, 
a side of the inner hexagon; OKL will then be a straight line. 
Produce OL to meet the circle in P. 
Then OK — KL — LX. Therefore, in the triangle OLX, 
OL — 2LX, while the included angle OLX (=120°) is also 
given. Therefore the triangle is given in species; therefore 
the ratio OX: XL is given, and, since OX is given, the side XL 
of each of the hexagons is given. 
Pappus gives the auxiliary construction thus. Let AF be 
taken equal to the radius OP. Let AC = ^AF, and on AC as 
base describe a segment of a circle containing an angle of 60°. 
Take GE equal to *AC, and draw EB to touch the circle at B.