THE COLLECTION. BOOK YII 
409 
The algebraical equivalents for Figures (1) and (2) re 
spectively may be written (if a — AD, h = DC, c = BD, 
d = DE): 
If ah = cd, then (a + d + c + h) a = (a + c) (a± d), 
{a + d + c + 6) h — (c + 6) (b -J- d), 
(a + c + 6) c = (c + u) (c + 6), 
(ci + d + c + i) d — (a + d) (d + 6). 
Figure (3) gives other varieties of sign. Troubles about 
sign can be avoided by measuring all lengths in one direction 
from an origin 0 outside the line. Thus, if 0A = a, OB — b, 
&c., the proposition may be as follows: 
If (d — a){d — c) = {h — d}{e — d) and k = e — a + h — c, 
then k(d — a) = (b — a){e — a), k{d — c) = {b — c)(e—c), 
k{b — d) = (b—a) (b — c) and k (e — d) = (e — a) (e — c). 
VIII. Props. 45-56. 
More generally, if AD . DC = BD .DE and k = AE+BC, 
then, if F be any point on the line, we have, according to the 
position of F in relation to A, B, C, D, E, 
+ AF. FG±EF. FB = k. DF. 
Algebraically, if 0A = a, OB = b ... OF = x, the equivalent 
is: If (d — a){d — c) = {b — d){e — d), and k = (e — a) + (b — c), 
then (x — a) [x — c) + (ic — e) (b — x) — k{x — d). 
By making x — a, b, c, e successively in this equation, we 
obtain the results of Props. 41-3 above. 
IX. Props. 59-64. 
In this group Props. 59, 60, 63 are lemmas required for the 
remarkable propositions (61, 62, 64) in which Pappus investi 
gates ‘ singular and minimum ’ values of the ratio 
AP.FD-.BP .PC, 
where {A, D), {B, G) are point-pairs on a straight line and P 
is another point on the straight line. He finds, not only when 
the ratio has the ‘ singular and minimum (or maximum) ’ value,