Let a rational straight line A be set out, let three numbers E, BC, CD be set out which have not to one another the ratio which a square number has to a square number,
but let CB have to BD the ratio which a square number has to a square number.
Let it be contrived that, as E is to BC, so is the square on A to the square on FG,
and, as BC is to CD, so is the square on FG to the square on GH. [X. 6, Por.]
Since then, as E is to BC, so is the square on A to the square on FG,
therefore the square on A is commensurable with the square on FG. [X. 6]
But the square on A is rational;
therefore the square on FG is also rational;
therefore FG is rational.
And, since E has not to BC the ratio which a square number has to a square number,
therefore neither has the square on A to the square on FG the ratio which a square number has to a square number;
therefore A is incommensurable in length with FG. [X. 9]
Again, since, as BC is to CD, so is the square on FG to the square on GH,
therefore the square on FG is commensurable with the square on GH. [X. 6]
But the square on FG is rational;
therefore the square on GH is also rational;
therefore GH is rational.
And, since BC has not to CD the ratio which a square number has to a square number,
therefore neither has the square on FG to the square on GH the ratio which a square number has to a square number;
therefore FG is incommensurable in length with GH. [X. 9]
And both are rational;
therefore FG, GH are rational straight lines commensurable in square only;
therefore FH is an apotome. [X. 73]
I say next that it is also a third apotome.
Complete text
Book I