For, since AG is incommensurable with GE, [X. 18] therefore AH is also commensurable with HE, [VI. 1, X. 11] that is, the square on MN with the square on NO;
therefore MN, NO are incommensurable in square.
And, since AD is a fifth binomial straight line, and ED the lesser segment, therefore ED is commensurable in length with AB. [X. Deff. II. 5]
But AE is incommensurable with ED;
therefore AB is also incommensurable in length with AE. [X. 13]
Therefore AK, that is, the sum of the squares on MN, NO, is medial. [X. 21]
And, since DE is commensurable in length with AB, that is, with EK, while DE is commensurable with EF, therefore EF is also commensurable with EK. [X. 12]
And EK is rational;
therefore EL, that is, MR, that is, the rectangle MN, NO, is also rational. [X. 19]
Therefore MN, NO are straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational.
Complete text
Book I