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.*

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.*

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.

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.