A straight line commensurable with that which produces with a rational area a medial whole is a straight line which produces with a rational area a medial whole.

Let *AB* be a straight line which produces with a rational area a medial whole, and *CD* commensurable with *AB*.

I say that *CD* is also a straight line which produces with a rational area a medial whole.

Let *BE* be the annex to *AB*, therefore *AE* and *EB* are straight lines incommensurable in square which make the sum of the squares on *AE* and *EB* medial but the rectangle contained by them rational.

Make the same construction.

Then we can prove, in manner similar to the foregoing, that *CF* and *FD* are in the same ratio as *AE* and *EB*, the sum of the squares on *AE* and *EB* is commensurable with the sum of the squares on *CF* and *FD*, and the rectangle *AE* by *EB* is commensurable with the rectangle *CF* by *FD*, so that *CF* and *FD* are also straight lines incommensurable in square which make the sum of the squares on *CF* and *FD* medial but the rectangle contained by them rational.

Therefore *CD* is a straight line which produces with a rational area a medial whole.

Therefore, *a straight line commensurable with that which produces with a rational area a medial whole is a straight line which produces with a rational area a medial whole.*

Q.E.D.