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 + c±2b respectively. These lemmas are preliminary to 
the problem (Prop. 21), Given two straight lines AB, BC 
(G lying between A and B), to find a point D on BA produced 
such that BD:DA = CD: (AB + BG-2 NAB. BC). This is, 
of course, equivalent to the quadratic equation (a + x): x 
= {a — c + x): (a+ c — 2 N ac), 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. 
(fS) Lemmas to the ‘ Determinate Section ’ of Apollonius. 
The next set of Lemmas (Props. 22-64, pp. 704-70) belongs 
to the Determinate Section of Apollonius. As we have seen 
(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.BG.AE. EC. 
A CPE B 
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 
b _ (a + b) {b + x) 
y ~ (a + y)(x + y) 
II. Props. 30, 32, 34. 
If in the same figure AD . DE = BD. DC, then 
BD : DC = AB. BE: EC. GA.