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 .