If a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number.

Let the number *A* be measured by any number *B.*

I say that *A* has a part called by the same name as *B.*

Let there be as many units in *C* as the times that *B* measures *A.*

Since *B* measures *A* according to the units in *C,* and the unit *D* also measures the number *C* according to the units in it, therefore the unit *D* measures the number *C* the same number of times as *B* measures *A.*

Therefore, alternately, the unit *D* measures the number *B* the same number of times as *C* measures *A.* Therefore, whatever part the unit *D* is of the number *B,* the same part is *C* of *A* also. But the unit *D* is a part of the number *B* called by the same name as it, therefore *C* is also a part of *A* called by the same name as *B,* so that *A* has a part *C* which is called by the same name as *B.*

Therefore, *if a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number.*

Q.E.D.