A history of Greek mathematics - Wilbourhall.org

406 PAPPUS OF ALEXANDRIA
Props. 32,
34 are alternative proofs based on other lemmas
The algebraic equivalent may be
(Props. 31, 33 respectively).
stated thus : if ax = by, then - =
,
III. Props. 35, 36.
V
~ —-
•
(x + y)(a-y)
If AB.BE = CB.BD, then AB : BE = DA . AC :CE . ED,
and CB:BD = AC .CE-.AD.DE, results equivalent to the
following : if ax = by, then
a __
(ct — y) (a-b) b_ (a — b)(b-x)
x" (b-x) (y-x)
IV. Props. 23, 24, 31, 57, 58.
n
y" (a-y)(y-x)'
A §
1 '
i
C E D
1 1
If
AB = CD, and E is any point in CD,
AC.CD = AE.ED + BE.EC,
and similar formulae hold for other positions of E. If E is
between B and C, AC . CD = AE . ED-BE .EC; and if E
EC ED DC.
is on AD produced, BE .
= AE .
+ BD .
V. A small group of propositions relate to a triangle ABC
with two straight lines AD, AE drawn from the vertex A to
points on the base BC in
conditions (a)
(b)
accordance with one or other of the
that the angles BAC, DAE are supplementary,
that the angles BAE, DAC are both right angles or, as we
may add from Book VI, Prop. 12, (c) that the angles BAD,
EAC are equal. The theorems are :
In case (a) BC.CD:BE.ED = CA 2 : AE\
(b) BC.CE:BD.DE= CA 2 : AD\
(c) DC.CE:EB.BD = AC-.AB 2 .