For since, as D is to AB, so is the square on E to the square on FG,
and, as BA is to AC, so is the square on FG to the square on GH,
therefore, ex aequali, as D is to AC, so is the square on E to the square on GH. [V. 22]
But D has not to AC the ratio which a square number has to a square number;
therefore neither has the square on E to the square on GH the ratio which a square number has to a square number;
therefore E is incommensurable in length with GH. [X. 9]
And since, as BA is to AC, so is the square on FG to the square on GH,
therefore the square on FG is greater than the square on GH.
Let then the squares on GH, K be equal to the square on FG;
therefore, convertendo, as AB is to BC, so is the square on FG to the square on K. [V. 19, Por.]
But AB has to BC the ratio which a square number has to a square number;
therefore the square on FG also has to the square on K the ratio which a square number has to a square number;
therefore FG is commensurable in length with K. [X. 9]
Therefore the square on FG is greater than the square on GH by the square on a straight line commensurable with FG.
And FG, GH are rational straight lines commensurable in square only, and neither of them is commensurable in length with E.
Complete text
Book I