Euclid's Elements, Book V, Proposition 3

Proposition 3

If a first magnitude is the same multiple of a second that a third is of a fourth, and if equimultiples are taken of the first and third, then the magnitudes taken also are equimultiples respectively, the one of the second and the other of the fourth.
Let a first magnitude A be the same multiple of a second B that a third C is of a fourth D, and let equimultiples EF and GH be taken of A and C.		V.Def.2
I say that EF is the same multiple of B that GH is of D. Since EF is the same multiple of A that GH is of C, therefore there are as many magnitudes as in EF equal to A as there are in GH equal to C.
Divide EF into the magnitudes EK and KF equal to A, and divide GH into the magnitudes GL and LH equal to C. Then the number of the magnitudes EK and KF equals the number of the magnitudes GL and LH.
And, since A is the same multiple of B that C is of D, while EK equals A, and GL equals C, therefore EK is the same multiple of B that GL is of D. For the same reason KF is the same multiple of B that LH is of D.
Since a first magnitude EK is the same multiple of a second B that a third GL is of a fourth D, and a fifth KF is the same multiple of the second B that a sixth LH is of the fourth D, therefore the sum EF of the first and fifth is the same multiple of the second B that the sum GH of the third and sixth is of the fourth D.		V.2
Therefore, if a first magnitude is the same multiple of a second that a third is of a fourth, and if equimultiples are taken of the first and third, then the magnitudes taken also are equimultiples respectively, the one of the second and the other of the fourth.
Q.E.D.

This proposition says that equimultiples of equimultiples are equimultiples, that is, if w and x are equimultiples of y and z, and u and v are equimultiples of w and x, then u and v are equimultiples of y and z. The proof depends on an associativity of multiplication: m (ny) = (mn) y. In Euclid's proof, n is 3 and m is 2.

As in the last proposition, the magnitudes need not all be of the same kind.

Although this proposition is not actually a statement about ratios, it can be interpreted as one. The hypotheses that A and C are equimultiples of B and D can be interpreted as a proportion A:B = C:D, and the conclusion that mA and mC are equimultiples of B and D can be interpreted as a proportion mA:B = mC:D. Under these interpretations this proposition becomes a special case of the next, and it is the special case that is used to prove the general case in the next proposition.

Book V Introduction - Proposition V.2 - Proposition V.4.