| 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. | X.29 | |
| Then the square on D is medial. Therefore D is also medial. | X.21 | |
| 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. | X.11 | |
| But D is medial, therefore E is also medial. | X.23,Note | |
| 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. | X.14 | |
| 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. | X.21 | |
| 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. | X.30 | |
| Q.E.D. | ||
