If magnitudes are proportional taken jointly, then they are also proportional taken separately.

Let *AB, BE, CD,* and *DF* be magnitudes proportional taken jointly, so that *AB* is to *BE* as *CD* is to *DF.*

I say that they are also proportional taken separately, that is, *AE* is to *EB* as *CF* is to *DF.*

Take equimultiples *GH, HK, LM,* and *MN* of *AE, EB, CF,* and *FD,* and take other, arbitrary, equimultiples, *KO* and *NP* of *EB* and *FD.*

Then, since *GH* is the same multiple of *AE* that *HK* is of *EB,* therefore *GH* is the same multiple of *AE* that *GK* is of *AB.*

But *GH* is the same multiple of *AE* that *LM* is of *CF,* therefore *GK* is the same multiple of *AB* that *LM* is of *CF.*

Again, since *LM* is the same multiple of *CF* that *MN* is of *FD,* therefore *LM* is the same multiple of *CF* that *LN* is of *CD.*

But *LM* was the same multiple of *CF* that *GK* is of *AB,* therefore *GK* is the same multiple of *AB* that *LN* is of *CD.*

Therefore *GK* and *LN* are equimultiples of *AB* and *CD.*

Again, since *HK* is the same multiple of *EB* that *MN* is of *FD,* and *KO* is also the same multiple of *EB* that *NP* is of *FD,* therefore the sum *HO* is also the same multiple of *EB* that *MP* is of *FD.*

And, since *AB* is to *BE* as *CD* is to *DF,* and of *AB* and *CD* equimultiples *GK* and *LN* have been taken, and of *EB* and *FD* equimultiples *HO* and *MP,* therefore, if *GK* is in excess of *HO,* and *LN* is also in excess of *MP*; if equal, equal; and if less, less.

Let *GK* be in excess of *HO.* Subtract *HK* from each. Therefore *GH* is also in excess of *KO.*

But we saw that, if *GK* was in excess of *HO,* then *LN* was also in excess of *MP,* therefore *LN* is also in excess of *MP,* and, if *MN* is subtracted from each, then *LM* is also in excess of *NP,* so that, if *GH* is in excess of *KO,* then *LM* is also in excess of *NP.*

Similarly we can prove that, if *GH* equals *KO,* then *LM* also equals *NP*; and if less, less.

And *GH* and *LM* are equimultiples of *AE* and *CF,* while *KO* and *NP* are other, arbitrary, equimultiples of *EB* and *FD,* therefore *AE* is to *EB* as *CF* is to *FD.*

Therefore, *if magnitudes are proportional taken jointly, then they are also proportional taken separately.*

Q.E.D.

The converse is given in the next proposition.

This proposition is used in the next two propositions and a couple in Book X.