A history of Greek mathematics - Wilbourhall.org

i
4?2 PAPPUS OF ALEXANDRIA
Props. 149, 151.
If AB .
then
BC = BD 2 ,
(AD±DC)BD = AD.DG,
(AD±DC)BC= DC 2 ,
A C B
A
i
1
— C
B
and (AD±DC)BA = AD 2 .
Props. 152, 153.
If AB:BC=AD 2 : DC
2 , then
AB .
BG
= BD 2 .
DC
-4 1
1
B
Prop. 160.
If AB : BC=AD
:
DC,
then, if ^be the middle point of AC,
BE. ED = EC 2 ,
BD.DE= AD. DC,
EB.BD = AB. BC.
A £ D C B
1 1 1 —
The Lemmas about the circle include the harmonic properties
of the pole and polar, whether the pole is external to the
circle (Prop. 154) or internal (Prop. 161). Prop. 155 is a
problem, Given a segment of a circle on AB as base, to inflect
straight lines AC, BC to the segment in a given ratio to one
another.
Prop. 156 is one which Pappus has already used earlier
in the Collection. It proves that the straight lines drawn
from the extremities of a chord (DE) to any point (F) of the
circumference divide harmonically the diameter (AB) perpendicular
to the chord. Or, if ED, FK be parallel chords, and
EF, DK meet in G, and EK, DF in H, then
AH:BB = AG:GB.