Let two numbers AC, CB be set out such that AB neither has to BC, nor yet to AC, the ratio which a square number has to a square number.
Let a rational straight line D be set out, and let EF be commensurable in length with D;
therefore EF is also rational.
Let it be contrived that, as the number BA is to AC, so is the square on EF to the square on FG; [X. 6, Por.]
therefore the square on EF is commensurable with the square on FG; [X. 6]
therefore FG is also rational.
Now, since BA has not to AC the ratio which a square number has to a square number, neither has the square on EF to the square on FG the ratio which a square number has to a square number;
therefore EF is incommensurable in length with FG. [X. 9]
Therefore EF, FG are rational straight lines commensurable in square only;
so that EG is binomial.
I say next that it is also a fourth binomial straight line.