If two numbers multiplied by one another make certain numbers, then the numbers so produced equal one another.

Let *A* and *B* be two numbers, and let *A* multiplied by *B* make *C,* and *B* multiplied by *A* make *D.*

I say that *C* equals *D.*

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

But the unit *E* also measures the number *A* according to the units in it, therefore the unit *E* measures *A* the same number of times that *B* measures *C.*

Therefore, alternately, the unit *E* measures the number *B* the same number of times that *A* measures *C.*

Again, since *B* multiplied by *A* makes *D,* therefore *A* measures *D* according to the units in *B.* But the unit *E* also measures *B* according to the units in it, therefore the unit *E* measures the number *B* the same number of times that *A* measures *D.*

But the unit *E* measures the number *B* the same number of times that *A* measures *C,* therefore *A* measures each of the numbers *C* and *D* the same number of times.

Therefore *C* equals *D.*

Therefore, *if two numbers multiplied by one another make certain numbers, then the numbers so produced equal one another.*

Q.E.D.

This proposition is used in VII.18 and a few others in Book VII.