A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

so that
or
THE COLLECTION. BOOK VIII 437
PQ:QV=Q'W:PQ'
i
PQ.PQ'=QV.Q'W.
Thus P can be found, and similarly P'.
The conjugate diameter is found by virtue of the relation
(conjugate diam.) 2 : PP'
where p is the latus rectum to PP f
of the curve
f
,
2 = p
:
PF = AV'-.PV.VF
PP'.
determined by the property
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
Problem, Given tivo conjugate diameters of an ellipse,
to find the
axes.
Lastly, Pappus shows (Prop. 14, chap. 17) how, when we are
given two conjugate diameters, we can find the axes. The
construction is as follows.
(CD being the greater), E the centre.
Produce EA to Hso that
EA.AH=DE 2 .
Let A B, CD be conjugate diameters
Through A draw FG parallel to CD. Bisect EH in K, and
draw KL at right angles to EH meeting FG in L.
r*-^^ P\
M"\\V
,* ~y
^ -
l\\/> / N J/0
^^.a
/K F
With L as centre, and LE as radius, describe a circle cutting
GF in G, F.
Join EF, EG, and from A draw AM, AN parallel to EF, EG
respectively.
B
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.