To find two medial straight lines commensurable in square only, containing a medial rectangle, such that the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater.

Set out three rational straight lines *A*, *B*, and *C* commensurable in square only, such that the square on *A* is greater than the square on *C* by the square on a straight line commensurable with *A*. Let the square on *D* equal the rectangle *A* by *B*.

Then the square on *D* is medial. Therefore *D* is also medial.

Let the rectangle *D* by *E* equal the rectangle *B* by *C*.

Then since as the rectangle *A* by is is to the rectangle *B* by *C* as *A* is to *C*, while the square on *D* equals the rectangle *A* by *B*, and the rectangle *D* by *E* equals the rectangle *B* by *C*, therefore *A* is to *C* as the square on *D* is to the rectangle *D* by *E*.

But the square on *D* is to the rectangle *D* by *E* as *D* is to *E*, therefore *A* is to *C* as *D* is to *E*. But *A* is commensurable with *C* in square only, therefore *D* is also commensurable with *E* in square only.

But *D* is medial, therefore *E* is also medial.

And, since *A* is to *C* as *D* is to *E*, while the square on *A* is greater than the square on *C* by the square on a straight line commensurable with *A*, therefore the square on *D* is greater than the square on *E* by the square on a straight line commensurable with *D*.

I say next that the rectangle *D* by *E* is also medial.

Since the rectangle *B* by *C* equals the rectangle *D* by *E*, while the rectangle *B* by *C* is medial, therefore the rectangle *D* by *E* is also medial.

Therefore two medial straight lines *D* and *E*, commensurable in square only, and containing a medial rectangle, have been found such that the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater.

Similarly again it can be proved that the square on *D* is greater than the square on *E* by the square on a straight line incommensurable with *D* when the square on *A* is greater than the square on *C* by the square on a straight line incommensurable with *A*.

Q.E.D.