For let BG be the annex to AB;
therefore AG, GB are medial straight lines commensurable in square only which contain a rational rectangle. [X. 74]
To CD let there be applied CH equal to the square on AG, producing CK as breadth, and KL equal to the square on GB, producing KM as breadth;
therefore the whole CL is equal to the squares on AG, GB; therefore CL is also medial. [X. 15 and 23, Por.]
And it is applied to the rational straight line CD, producing CM as breadth;
therefore CM is rational and incommensurable in length with CD. [X. 22]
Now, since CL is equal to the squares on AG, GB, and, in these, the square on AB is equal to CE, therefore the remainder, twice the rectangle AG, GB, is equal to FL. [II. 7]
But twice the rectangle AG, GB is rational; therefore FL is rational.
And it is applied to the rational straight line FE, producing FM as breadth;
therefore FM is also rational and commensurable in length with CD. [X. 20]
Now, since the sum of the squares on AG, GB, that is, CL, is medial, while twice the rectangle AG, GB, that is, FL, is rational, therefore CL is incommensurable with FL.
But, as CL is to FL, so is CM to FM; [VI. 1]
therefore CM is incommensurable in length with FM. [X. 11]
And both are rational;
therefore CM, MF are rational straight lines commensurable in square only;
therefore CF is an apotome. [X. 73]
I say next that it is also a second apotome.