For, since AB, BC are medial, the squares on AB, BC are also medial.
But twice the rectangle AB, BC is rational;
therefore the squares on AB, BC are incommensurable with twice the rectangle AB, BC;
therefore twice the rectangle AB, BC is also incommensurable with the remainder, the square on AC, [Cf. II. 7]
since, if the whole is incommensurable with one of the magnitudes, the original magnitudes will also be incommensurable. [X. 16]
But twice the rectangle AB, BC is rational;
therefore the square on AC is irrational;
therefore AC is irrational. [X. Def. 4]
And let it be called a first apotome of a medial straight line.
Complete text
Book I