The First Six Books of the Elements of Euclid

by Euclid; John Casey

PROP. XVI—Theorem.
If four magnitudes of the same kind be proportionals they are also proportionals by alternation (alternando).

Let a : b :: c : d, then a : c :: b : d.

Dem.—Since a : b :: c : d,

and multiplying each by

, we get

		.	= .,
	or		= ;
	therefore	a : c	:: b : d.

PROP. XVII.—Theorem.

If four magnitudes be proportional, the difference between the first and second : the second :: the difference between the third and fourth : the fourth (dividendo).