If a composite number multiplied by any number makes some number, then the product is solid.

Let the composite number *A* multiplied by any number *B* make *C.*

I say that *C* is solid.

Since *A* is composite, it is measured by some number *D.* Let there be as many units in *E* as times that *D* measures *A*

Since *D* measures *A* according to the units in *E,* therefore *E* multiplied by *D* makes *A.* And, since *A* multiplied by *B* makes *C,* and *A* is the product of *D* and *E,* therefore the product of *D* and *E* multiplied by *B* makes *C.*

Therefore *C* is solid, and *D, E,* and *B* are its sides.

Therefore, *if a composite number multiplied by any number makes some number, then the product is solid.*

Q.E.D.

Euclid takes extra steps because he sees “*d* measures *a* a number *n* times” as saying something different from
“the product of *d* and *n* equals *a.*” See the discussion in the Guide for VII.Def.2.