If a number multiplied by two numbers makes certain numbers, then the numbers so produced have the same ratio as the numbers multiplied.

Let the number *A* multiplied by the two numbers *B* and *C* make *D* and *E.*

I say that *B* is to *C* as *D* is to *E.*

Since *A* multiplied by *B* makes *D,* therefore *B* measures *D* according to the units in *A.*

But the unit *F* also measures the number *A* according to the units in it, therefore the unit *F* measures the number *A* the same number of times that *B* measures *D.* Therefore the unit *F* is to the number *A* as *B* is to *D.*

For the same reason the unit *F* is to the number *A* as *C* is to *E,* therefore *B* is to *D* as *C* is to *E.*

Therefore, alternately *B* is to *C* as *D* is to *E.*

Therefore, *if a number multiplied by two numbers makes certain numbers, then the numbers so produced have the same ratio as the numbers multiplied.*

Q.E.D.

Symbolically,
*b* : *c* = *ab* : *ac.*

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