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.