Again, since a = b, dividing c by each, we have
therefore c : a :: c : b.
Observation.—2 follows at once from 1 by Proposition B.
PROP. VIII.—Theorem.
1. Of two unequal magnitudes, the greater has a greater ratio to any third magnitude than the less has; 2. any third magnitude has a greater ratio to the less of two unequal magnitudes than it has to the greater.
1. Let a be greater than b, and let c be any other magnitude of the same kind, then the ratio a : c is greater than the ratio b : c.
Dem.—Since a is greater than b, dividing each by c,
therefore the ratio a : c is greater than the ratio b : c.