If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken will be 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 / E F /, / G H / be taken of A, C;
I say that / E F / is the same multiple of B that / G H / is of D.
For, since / E F / is the same multiple of A that / G H / is of C,
therefore, as many magnitudes as there are in / E F / equal to A, so many also are there in / G H / equal to C.
Let / E F / be divided into the magnitudes / E K /, / K F /equal to A, and / G H / into the magnitudes / G L /, / L H /equal to C; then the multitude of the magnitudes / E K /, / K F / will be equal to the multitude of the magnitudes / G L /, / L H /.
And, since A is the same multiple of B that C is of D,
while / E K / is equal to A, and / G L / to C,
therefore / E K / is the same multiple of B that / G L / is of D.
For the same reason
/ K F / is the same multiple of B that / L H / is of D.
Since, then, a first magnitude / E K / is the same multiple of a second B that a third / G L / is of a fourth D, and a fifth / K F / is also the same multiple of the second B that a sixth / L H / is of the fourth D,
therefore the sum of the first and fifth, / E F /, is also the same multiple of the second B that the sum of the third and sixth, / G H /, is of the fourth D. [G.172]
