412
PAPPUS OF ALEXANDPJA
Therefore
AP. PD : BP . PC = AD 2 : { V{AC . BP) - V{AB . CD) } 2 .
The proofs of Props. 62 and 64 are different, the former
being long and involved. The results are:
Prop. 62. If P is between C and D, and
AD . DB : AC. GB = DP 2 : PC 2 ,
then the ratio AP . PB : CP . PD is singular and a minimum
and is equal to { V{AC. BP) + V{AD . BC)} 2 : PC 2 .
Prop. 64. If P is on AD produced, and
AB . BP : AG. QD = BP 2 : CP 2 ,
then the ratio AP . PD : BP .PC is singular and a maximum,
and is equal to AD 2 : { V{AG. BP) + V{AB . CD)] 2 .
(y) Lemmas on the Neva-eis of Apollonius.
After a few easy propositions (e.g. the equivalent of the
proposition that, if ax + x 2 = by + y 2 , then, according as a >
or < b, a + x > or < 6 + 2/), Pappus gives (Prop. 70) the
lemma leading to the solution of the vevens with regard to
the rhombus (see pp. 190-2, above), and after that the solu
tion by one Heraclitus of the same problem with respect to
a square (Props. 71, 72, pp. 780-4). The problem is, Given a
square A BCD, to draw th rough B a straight line, meeting CP
in H and AD produced in E, such that HE is equal to a given
length.
The solution depends on a lemma to the effect that, if any
straight line BHE through B meets CD in H and AP pro
duced in E, and if EF be drawn perpendicular to BE meeting
BC produced in F, then
. GF 2 — BC 2 + HE 2 .