A history of Greek mathematics - Wilbourhall.org

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 GHKLNM to be the hexagon so described on HK,
a side of the inner hexagon ;
OKL will then be a straight line.
Produce OL to meet the circle in P.
Then OK = KL = LN. Therefore, in the triangle OLN,
OL - 2LN, while the included angle OLN (— 120°) is also
given. Therefore the triangle is given in species; therefore
the ratio ON : NL is given, and, since ON is given, the side NL
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 A 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.