The First Six Books of the Elements of Euclid

by Euclid; John Casey

PROP. IV.—Theorem.

If four magnitudes be proportional, and if any equimultiples of the first and third be taken, and any other equimultiples of the second and fourth; then the multiple of the first : the multiple of the second :: the multiple of the third : the multiple of the fourth.

Let a : b :: c : d; then ma : nb :: mc : nd.

Dem.—We have a : b :: c : d (hyp.);

	therefore		= .
Hence, multiplying each fraction by , we get
			= ;

therefore ma : nb :: mc : nd.

PROP. V.—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 difference of the two first is the same multiple of their submultiple (c), which the difference of their corresponding magnitudes is of their submultiple (c′) (compare Proposition ii.).

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

	Then we have		a = mc, and a′ = mc′,
			b = nc, and b′ = nc′.