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 EF, GH be taken of A, C;
I say that EF is the same multiple of B that GH is of D.
For, since EF is the same multiple of A that GH is of C,
therefore, as many magnitudes as there are in EF equal to A, so many also are there in GH equal to C.
Let EF be divided into the magnitudes EK, KFequal to A, and GH into the magnitudes GL, LHequal to C; then the multitude of the magnitudes EK, KF will be equal to the multitude of the magnitudes GL, LH.
And, since A is the same multiple of B that C is of D,
while EK is equal to A, and GL to 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, then, 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 also the same multiple of the second B that a sixth LH is of the fourth D,
therefore the sum of the first and fifth, EF, is also the same multiple of the second B that the sum of the third and sixth, GH, is of the fourth D. [V. 2]
