Let a rational straight line A be set out, and let CG be commensurable in length with A;
therefore CG is rational.
Let two numbers DF, FE be set out such that DE again has not to either of the numbers DF, FE the ratio which a square number has to a square number;
and let it be contrived that, as FE is to ED, so is the square on CG to the square on GB.
Therefore the square on GB is also rational; [X. 6]
therefore BG is also rational.
Now since, as DE is to EF, so is the square on BG to the square on GC, while 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]
I say next that it is also a fifth apotome.