408
PAPPUS OF ALEXANDRIA
VI. Props. 37, 38.
If AB: BG — AD 2 : DC 2 , whether AB be greater or less
than AD, then
AB. BC = BD 2 .
[E in the figure is a point such that ED = CD.]
(?)
C B
if)
The algebraical equivalent is: If ^ ^» then ac —
These lemmas are subsidiary to the next (Props. 39, 40),
being used in the first proofs of them.
Props. 39, 40 prove the following;
If ACDEB be a straight line, and if
BA . AE-. BD.DE = AC 2 : CD 2 ,
then AB.BD-.AE.ED = BG 2 : GE 2 ;
if, again, AG.CB-.AE.EB = CD 2 :DE 2 ,
then EA . AC :CB.BE= AD 2 : DB 2 .
If AB — a, BG — h, BD — c, BE = d, the algebraic equiva
lents are the following.
If
a{a — d) (a —by
and if
c{c — d) (6 — cf ’
■ (a — h)h _ (b — c) 2
{a—d)d (c — d) 2
then
ac
then
{a — d) (c — d)
(a — d) {a — b)
bd
b 2
{b-df
_ (a—c) 2
~ A '
VII. Props. 41, 42, 43.
If AD . DC = BD . DE, suppose that in Figures (1) and (2)
0) 0___A_
(2) A_
(3) A
E B
D C
f 6
k = AE+GB, and in Figure (3) k = AE—BC, then
k.AD = BA.AE, k.CD = BG.GE, k.BD = AB.BG,
k.DE = AE.EG.