Kitap 10
90
But E has not to CD the ratio which a square number has to a square number; therefore neither has the square on A to the square on GH the ratio which a square number has to a square number; therefore A is incommensurable in length with GH; [X. 9] therefore neither of the straight lines FG, GH is commensurable in length with the rational straight line A.
90
Now let the square on K be that by which the square on FG is greater than the square on GH.
90
Since then, as BC is to CD, so is the square on FG to the square on GH, therefore, convertendo, as CB is to BD, so is the square on FG to the square on K. [v. 19, Por.]
90
But CB has not to BD the ratio which a square number has to a square number; therefore neither has the square on FG to the square on K the ratio which a square number has to a square number; therefore FG is incommensurable in length with K. [X. 9]
90
And the square on FG is greater than the square on GH by the square on K; therefore the square on FG is greater than the square on GH by the square on a straight line incommensurable in length with FG.
90
And neither of the straight lines FG, GH is commensurable with the rational straight line A set out.
90
Therefore FH is a sixth apotome. [X. Deff. III. 6]
90
Therefore the sixth apotome FH has been found. Q. E. D.
PROPOSITION 91.
91
If an area be contained by a rational straight line and a first apotome, the side of the area is an apotome.
91
For let the area AB be contained by the rational straight line AC and the first apotome AD;
