Let a rational straight line A be set out, and BG commensurable in length with it;
therefore BG is also rational.
Let two numbers DF, FE be set out such that the whole DE has not to either of the numbers DF, EF the ratio which a square number has to a square number.
Let it be contrived that, as DE is to EF, so is the square on BG to the square on GC; [X. 6, Por.]
therefore the square on BG is commensurable with the square on GC. [X. 6]
But the square on BG is rational;
therefore the square on GC is also rational;
therefore GC is rational.
Now, since DE has not to EF the ratio which a square number has to a square number,
therefore neither has the square on BG to the square on GC the ratio which a square number has to a square number;
therefore BG is incommensurable in length with GC. [X. 9]
And both are rational;
therefore BG, GC are rational straight lines commensurable in square only;
therefore BC is an apotome. [X. 73]