For since, as D is to AB, so is the square on E to the square on FG, and also, 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]
But it was also proved incommensurable with FG;
therefore each of the straight lines FG, GH is incommensurable in length with E.
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 not to BC the ratio which a square number has to a square number;
so that neither has the square on FG to the square on K the ratio which a square number has to a square number.
Therefore FG is incommensurable 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 incommensurable with FG.
And FG, GH are rational straight lines commensurable in square only, and neither of them is commensurable in length with the rational straight line E set out.