The First Six Books of the Elements of Euclid

by Euclid; John Casey

	Again, since	a	: b :: c : d,
			= ;

therefore

is greater than .

or the ratio of a : b is greater than the ratio of e : f.

PROP. XIV.—Theorem.

If two ratios be equal, then, according as the antecedent of the first ratio is greater than, equal to, or less than the antecedent of the second, the consequent of the first is greater than, equal to, or less than the consequent of the second.

Let a : b :: c : d; then if a be greater than c, b is greater than d; if equal, equal; if less, less.

	Dem.—Since	a : b	:: c : d.
	we have		= ,

and multiplying each by

we get