The First Six Books of the Elements of Euclid

by Euclid; John Casey

Dem.—Let m denote the multiple which the magnitudes of the first system are of those of the second system.

	Then we have		a = ma′ (hyp.),
			b = mb′,
			c = mc′.
			&c., &c.

Hence, by addition,

PROP. II.—Theorem.

If two magnitudes of the same kind (a, b) be the same multiples of another (c) which two corresponding magnitudes (a′, b′) are of another (c′), then the sum of the two first is the same multiple of their submultiple which the sum of their corresponding magnitudes is of their submultiple.

Dem.—Let m and n be the multiples which a and b are of c.

Then we have

Therefore