From the straight line AB let there be subtracted the straight line BC which is incommensurable in square with AB and fulfills the given conditions.
I say that the remainder AC is the irrational straight line aforesaid.
Since the sum of the squares on AB and BC is medial, while twice the rectangle AB by BC is rational, therefore the sum of the squares on AB and BC is incommensurable with twice the rectangle AB by BC. Therefore the remainder, the square on AC, is also incommensurable with twice the rectangle AB by BC.
And twice the rectangle AB by BC is rational, therefore the square on AC is irrational. Therefore AC is irrational. Let it be called that which produces with a rational area a medial whole.