A history of Greek mathematics - Wilbourhall.org

412 PAPPUS OF ALEXANDRIA
Therefore
AP.PD.BP .PC = AD 2 :
{ V(AC .
BD)- V{AB .
CD)}*.
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.CB = DP 2 : PG\
then the ratio AP . PB : CP . PD is singular and a minimum
2
and is equal to { V(AC . BD) + V(AD . BC) :
}
DC
Prop. 64.
If P is on AD produced, and
2 .
then the ratio AP . Pi)
AB.BD:AC.CD = BP 2 : CP
.
: 5P PC
and is equal to AD 2 :
{ V^C. BD) + V(AB .
2 ,
is singular and a maximum,
CD)}
2 .
,
(y) Lemmas on the Nevveis 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 < b + y), Pappus gives (Prop. 70) the
lemma leading to the solution of the vevais with regard to
the rhombus (see pp. 190-2, above), and after that the solution
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 through B a straight line, meeting CD
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 AD produced
in E, and if EF be drawn perpendicular to
BC produced in F, then
CF 2 = BC 2 +HE 2 .
BE meeting