The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial.
Proof
Commensurable-in-square-only means the square on each is rational
but the lengths are not in integer ratio. The rectangle is then in
a non-rational ratio to a rational area; its square root is the
medial straight line (Definition XIII.3).