If a magnitude is the same multiple of a magnitude that a subtracted part is of a subtracted part, then the remainder also is the same multiple of the remainder that the whole is of the whole.

Let the magnitude *AB* be the same multiple of the magnitude *CD* that the subtracted part *AE* is of the subtracted part *CF.*

I say that the remainder *EB* is also the same multiple of the remainder *FD* that the whole *AB* is of the whole *CD.*

Make *CG* so that *EB* is the same multiple of *CG* that *AE* is of *CF.*

Then, since *AE* is the same multiple of *CF* that *EB* is of *GC,* therefore *AE* is the same multiple of *CF* that *AB* is of *GF.*

But, by the assumption, *AE* is the same multiple of *CF* that *AB* is of *CD.*

Therefore *AB* is the same multiple of each of the magnitudes *GF* and *CD.* Therefore *GF* equals *CD.*

Subtract *CF* from each. Then the remainder *GC* equals the remainder *FD.*

And, since *AE* is the same multiple of *CF* that *EB* is of *GC,* and *GC* equals *DF,* therefore *AE* is the same multiple of *CF* that *EB* is of *FD.*

But, by hypothesis, *AE* is the same multiple of *CF* that *AB* is of *CD,* therefore *EB* is the same multiple of *FD* that *AB* is of *CD.*

That is, the remainder *EB* is the same multiple of the remainder *FD* that the whole *AB* is of the whole *CD.*

Therefore, *If a magnitude is the same multiple of a magnitude that a subtracted part is of a subtracted part, then the remainder also is the same multiple of the remainder that the whole is of the whole.*

Q.E.D.

Note that all the magnitudes in this proposition are of the same kind.

- Make

Alternative proofs that don’t require constructions of parts are relatively easy to find. A more interesting problem of general constructions for magnitudes is discussed in the Guide for proposition V.18.

This proposition is not used in the rest of the *Elements.*