A history of Greek mathematics - Wilbourhall.org

408 PAPPUS OF ALEXANDRIA
VI. Props. 37, 38.
If AB:BC = AD 2 : DC
than AD, then
2 ,- whether AB be "greater or less
AB.BG = BD 2 .
\E in the figure is a point such that ED = CD.]
A (E) D C B
i 1 1 1
A C B P (f)
i
i
The algebraical equivalent is: If - = .,"" L t
then ac — b 2 .
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 AGDEB be a straight line, and if
BA . AE: BD.DE = AC 2 : CD
2 ,
2 ;
then AB.BD:AE.ED = BG 2 : GE
if, again, AG .CB.AE . EB = CD 2 :DE 2 ,
then EA .AC:CB.BE = AD 2 : DB
2 .
If AB = a, BG = b, BD = c, BE— d, the algebraic equivalents
are the following.
a (a-d) (a-b) 2 ac b 2
c(c-d)
'
'
(b-cf '
(a-d) (c-d) " (6-d) 2 '
, .„ (a — b)b (b —
2 c) .. (a —
2
c£)( then
(a — d)d (c — df bd c 2
VII. Props. 41, 42, 43.
If AD.DC=BD.DE, suppose that in Figures (1) and (2)
0) O A $ D E B
(2) A E D C B°
" : 1 1
(3) A E B C ,p
k = AE+GB, and in Figure (3) k = AE-BG, then
k.AD = k.GD = BA.AEt BG.GE, k.BD = AB.BC,
k.DE=AE.EC.