For since, as the number BA is to AC, so is the square on EF to the square on FG, while BA is greater than AC, therefore the square on EF is also greater than the square on FG.
Let then the squares on FG, H be equal to the square on EF.
Now since, as BA is to AC, so is the square on EF to the square on FG,
therefore, convertendo, as AB is to BC, so is the square on EF to the square on H. [V. 19, Por.]
But AB has to BC the ratio which a square number has to a square number;
therefore the square on EF also has to the square on H the ratio which a square number has to a square number.
Therefore EF is commensurable in length with H; [X. 9]
therefore the square on EF is greater than the square on FG by the square on a straight line commensurable with EF.
And EF, FG are rational, and EF is commensurable in length with D.