A history of Greek mathematics - Wilbourhall.org

A QPE B
'
THE COLLECTION. BOOK VII 405
17-20 deal with three straight lines a, b, c in geometrical
progression, showing how to mark on a straight line containing
a, b, c as segments (including the whole among 'segments'),
lengths equal to a + c ± 2 V(ac) ;
the lengths are of course equal
to a 4- c + 2 b respectively. These lemmas are preliminary to
the problem (Prop. 21), Given two straight lines AB, BC
(C lying between A and B), to find a point D on BA produced
such that BD:DA=CD: (AB + BC-2 VABTBC). This is,
of course, equivalent to the quadratic equation (a + x):x
= (a — c + x):(a + c — 2 Vac), and, after marking off AE along
AD equal to the fourth term of this proportion, Pappus solves
the equation in the usual way by application of areas.
(fi) Lemmas to the ' Determinate Section ' of Apollonius.
The next set of Lemmas (Props. 22-64, pp. 704-70) belongs
As we have seen
to the Determinate Section of Apollonius.
(pp. 180-1, above), this work seems to have amounted to
a Theory of Involution. Whether the application of certain
of Pappus's lemmas corresponded to the conjecture of Zeuthen
or not, we have at all events in this set of lemmas some
remarkable applications of ' geometrical algebra '. They may
be divided into groups as follows
I. Props. 22, 25, 29
If in the figure AD. DC = BD .
DE\ then
BD:DE = AB.BC:AE. EC.
The proofs by proportions are not difficult. Prop. 29 is an
alternative proof by means of Prop. 26 (see below). The
algebraic equivalent may be expressed thus : if ax = by, then
II. Props. 30, 32, 34.
If in the same figure
BD :
b (a + b)(b + x)
y " (*+y){x+y)
AD.DE= BD. DC, then
DC = AB .
BE
:
EC
.
CA
.