Book 10
53
Let any rational straight line E be set out, and let it be contrived that, as D is to AB, so is the square on E to the square on FG; [X. 6, Por.] therefore the square on E is commensurable with the square on FG. [X. 6]
53
And E is rational; therefore FG is also rational.
53
Now, since D has not to AB the ratio which a square number has to a square number, neither has the square on E to the square on FG the ratio which a square number has to a square number; therefore E is incommensurable in length with FG. [X. 9]
53
Again, let it be contrived that, as BA is to AC, so is the square on FG to the square on GH. [X. 6, Por.]
53
Therefore the square on FG is commensurable with the square on HG. [X. 6]
53
Therefore the square on HG is rational; therefore HG is rational.
53
And, since BA has not to AC the ratio which a square number has to a square number, 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]
53
Therefore FG, GH are rational straight lines commensurable in square only; therefore FH is binomial. [X. 36]
53
It is next to be proved that it is also a sixth binomial straight line.
53
For since, as D is to AB, so is the square on E to the square on FG, and also, as BA is to AC, so is the square on FG to the square on GH, therefore, ex aequali, as D is to AC, so is the square on E to the square on GH. [V. 22]
