For let the square on H be that by which the square on BG is greater than the square on GC.
Since then, as the square on BG is to the square on GC, so is the number ED to the number DF,
therefore, convertendo, as the square on BG is to the square on H, so is DE to EF. [V. 19, Por.]
And each of the numbers DE, EF is square;
therefore the square on BG has to the square on H the ratio which a square number has to a square number;
therefore BG is commensurable in length with H. [X. 9]
And the square on BG is greater than the square on GC by the square on H;
therefore the square on BG is greater than the square on GC by the square on a straight line commensurable in length with BG.
And CG, the annex, is commensurable with the rational straight line A set out.
Therefore BC is a second apotome. [X. Deff. III. 2]