The First Six Books of the Elements of Euclid

by Euclid; John Casey

	Hence	1 ÷	is less than 1 ÷,
	that is,		is less than .

Hence, multiplying each by c, we get

PROP. XI.—Theorem.
Ratios that are equal to the same ratio are equal to one another.

Let a : b :: e : f, and c : d :: e : f, to prove a : b :: c : d.

Dem.—Since a : b :: e : f,

			= .
	In like manner,		= .
	Hence		= [I., Axiom i.],
	and	a :	b :: c : d.

PROP. XII.—Theorem.

If any number of ratios be equal to one another, any one of these equal ratios is equal to the ratio of the sum of all the antecedents to the sum of all the consequents.

Let the ratios a : b, c : d, e : f, be all equal to one another; it is required to prove that any of these ratios is equal to the ratio a + c + e : b + d + f.