The least numbers of those which have the same ratio with them measure those which have the same ratio with them the same number of times; the greater the greater; and the less the less.

Let *CD* and *EF* be the least numbers of those which have the same ratio with *A* and *B.*

I say that *CD* measures *A* the same number of times that *EF* measures *B.*

Now *CD* is not parts of *A.* If possible, let it be so. Therefore *EF* is also the same parts of *B* that *CD* is of *A.*

Therefore there are as many parts of *B* in *EF* are there are parts of *A* in *CD.*

Divide *CD* into the parts of *A,* namely *CG* and *GD,* and divide *EF* into the parts of *B,* namely *EH* and *HF.* Thus the multitude of *CG* and *GD* equals the multitude of *EH* and *HF.*

Now, since the numbers *CG* and *GD* equal one another, and the numbers *EH* and *HF* also equal one another, while the multitude of *CG* and *GD* equals the multitude of *EH* and *HF,* therefore *CG* is to *EH* as *GD* is to *HF.*

Since one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents, therefore *CG* is to *EH* as *CD* is to *EF.*

Therefore *CG* and *EH* are in the same ratio with *CD* and *EF,* being less than they, which is impossible, for by hypothesis *CD* and *EF* are the least numbers of those which have the same ratio with them. Therefore *CD* is not parts of *A,* therefore it is a part of it.

And *EF* is the same part of *B* that *CD* is of *A,* therefore *CD* measures *A* the same number of times that *EF* measures *B.*

Therefore, *the least numbers of those which have the same ratio with them measure those which have the same ratio with them the same number of times; the greater the greater; and the less the less.*

Q.E.D.

The proof goes along like this. Suppose *a* : *b* reduces to *c* : *e* in lowest terms. In order to show that *c* divides *a,* assume that it doesn’t, assume that *c* = (*m/n*)*a.* Since *a* : *b* is the same ratio as *c* : *e,* therefore *e* = (*m/n*)*d.* But then *c/m* = (1/*n*)*a,* and *e/m* = (1/*n*)*b.* Therefore *c/m* : *e/m* is the same ratio as *a* : *b,* which shows that *c* : *e* is not in lowest terms, a contradiction. Therefore *c* does divide *a,* and *e* divides *b* the same number of times.