The First Six Books of the Elements of Euclid

by Euclid; John Casey

=

;

	therefore		= ,
	Hence	c	= .

PROP. VII.—Theorem.
1. Equal magnitudes have equal ratios to the same magnitude.
2. The same magnitude has equal ratios to equal magnitudes.

Let a and b be equal magnitudes, and c any other magnitude.

Then 1.		a : c	:: b : c,
2.		c : a	:: c : b.

Dem.—Since a = b, dividing each by c, we have

therefore a : c :: b : c.