Let a rational straight line A be set out, and let BG be commensurable in length with A;
therefore BG is also rational.
Let two square numbers DE, EF be set out, and let their difference FD not be square;
therefore neither has ED to DF the ratio which a square number has to a square number.
Let it be contrived that, as ED is to DF, 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 also rational.
And, since ED has not to DF 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]
I say next that it is also a first apotome.