If four magnitudes are proportional, then they are also proportional alternately.

Let *A, B, C,* and *D* be four proportional magnitudes, so that *A* is to *B* as *C* is to *D.*

I say that they are also so alternately, that is *A* is to *C* as *B* is to *D.*

Take equimultiples *E* and *F* of *A* and *B,* and take other, arbitrary, equimultiples *G* and *H* of *C* and *D.*

Then, since *E* is the same multiple of *A* that *F* is of *B,* and parts have the same ratio as their equimultiples, therefore *A* is to *B* as *E* is to *F.*

But *A* is to *B* as *C* is to *D,* therefore *C* is to *D* also as *E* is to *F.*

Again, since *G* and *H* are equimultiples of *C* and *D,* therefore *C* is to *D* as *G* is to *H.*

But *C* is to *D* as *E* is to *F,* therefore as *E* is to *F* also as *G* is to *H.*

But, if four magnitudes are proportional, and the first is greater than the third, then the second is also greater than the fourth; if equal, equal; and if less, less.

Therefore, if *E* is in excess of *G, F* is also in excess of *H*; if equal, equal; and if less, less.

Now *E* and *F* are equimultiples of *A* and *B,* and *G* and *H* other, arbitrary, equimultiples of *C* and *D,* therefore *A* is to *C* as *B* is to *D.*

Therefore, *if four magnitudes are proportional, then they are also proportional alternately.*

Q.E.D.

so by the definition of proportion V.Def.5, *a* : *c* = *b* : *d*. Q.E.D.

This proposition requires using V.Def.4 as an axiom of comparability.