A history of Greek mathematics - Wilbourhall.org

:
:
THE COLLECTION. BOOK VII 409
The algebraical equivalents for Figures (1) and (2) respectively
may be written (if a — AD, b — DC, c = BD,
d = DE)
If ab — cd, then (a±d + c±b) a = (a + c) (a±d),
(a±d + c±b)
b = (c±b) (b + d),
(a±d + c±b) c = (c + a)(c±b),
(a±d + c±b) d— (a±d)(d + b).
Figure (3) gives other varieties of sign. Troubles about
sign can be avoided by measuring all lengths in one direction
from an origin outside the line. Thus, if A = a, OB = b,
&c, the proposition may be as follows
If (d — a) (d — c) = (6 — d) (e-d) and k = e — a + b — c,
then k(d — a) — (b — a)(e — a), k(d — c) = (b — c)(e—c),
Jc(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,
F be any point on the line, we have, according to the
then, if
position of i^in relation to A, B, (7, D, E,
±AF. FC±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) + (x — 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 investigates
singular and minimum ' ' values of the ratio
AP.FDiBP.FC,
where (A, D), (B, C) are point-pairs on a straight line and F
is another point on the straight line. He finds, not only when
the ratio has the singular and minimum ' '
(or maximum) value,