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

Set out two rational straight lines *A* and *B* commensurable in square only such that the square on *A*, being the greater, is greater than the square on *B* the less by the square on a straight line commensurable in length with *A*.

Let the square on *C* equal the rectangle *A* by *B*.

Now the rectangle *A* by *B* is medial, therefore the square on *C* is also medial. Therefore *C* is also medial.

Let the rectangle *C* by *D* equal the square on *B*.

Now the square on *B* is rational, therefore the rectangle *C* by *D* is also rational. And since *A* is to *B* as the rectangle *A* by *B* is to the square on *B*, while the square on *C* equals the rectangle *A* by *B*, and the rectangle *C* by *D* equals the square on *B*, therefore *A* is to *B* as the square on *C* is to the rectangle *C* by *D*.

But the square on *C* is to the rectangle *C* by *D* as *C* is to *D*, therefore *A* is to *B* as *C* is to *D*.

But *A* is commensurable with *B* in square only, therefore *C* is also commensurable with *D* in square only.

And *C* is medial, therefore *D* is also medial.

Since *A* is to *B* as *C* is to *D*, and the square on *A* is greater than the square on *B* by the square on a straight line commensurable with *A*, therefore the square on *C* is greater than the square on *D* by the square on a straight line commensurable with *C*.

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

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

Q.E.D.