The First Six Books of the Elements of Euclid

by Euclid; John Casey

Hence a + b is the same multiple of c that a′ + b′ is of c′.

This Proposition is evidently true for any number of multiples.

PROP. III.—Theorem.

If two magnitudes (a, b) be equimultiples of two others (a′, b′); then any equimultiples of the first magnitudes (a, b) will be also equimultiples of the second magnitudes (a′, b′).

Dem.—Let m denote the multiples which a, b are of a′, b′; then we have

Hence, multiplying each equation by n, we get

Hence, na, nb are equimultiples of a′, b′.