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

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

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

Since *A* is to *B* as *C* is to *D,* therefore, *A* is the same part or parts of *B* as *C* is of *D.*

Therefore, alternately, *A* is the same part or parts of *C* as *B* is of *D.*

Therefore *A* is to *C* as *B* is to *D.*

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

Q.E.D.

This proposition is used frequently in Books VII through IX starting with the next proposition.