For let BG be the annex to AB;
therefore AG, GB are straight lines incommensurable in square which make the sum of the squares on AG, GB rational, but twice the rectangle AG, GB medial. [X. 76]
To CD let there be applied CH equal to the square on AG and producing CK as breadth, and KL equal to the square on BG, producing KM as breadth;
therefore the whole CL is equal to the squares on AG, GB.
And the sum of the squares on AG, GB is rational;
therefore CL is also rational.
And it is applied to the rational straight line CD, producing CM as breadth;
therefore CM is also rational and commensurable in length with CD. [X. 20]
And, since the whole CL is equal to the squares on AG, GB, and, in these, CE is equal to the square on AB,
therefore the remainder FL is equal to twice the rectangle AG, GB. [II. 7]
Let then FM be bisected at the point N, and let NO be drawn through N parallel to either of the straight lines CD, ML;
therefore each of the rectangles FO, NL is equal to the rectangle AG, GB.
And, since twice the rectangle AG, GB is medial and is equal to FL, therefore FL is also medial.
And it is applied to the rational straight line FE, producing FM as breadth;
therefore FM is rational and incommensurable in length with CD. [X. 22]
And, since the sum of the squares on AG, GB is rational, while twice the rectangle AG, GB is medial,
the squares on AG, GB are incommensurable with twice the rectangle AG, GB.
But CL is equal to the squares on AG, GB, and FL equal to twice the rectangle AG, GB;
therefore CL is incommensurable with FL.
But, as CL is to FL, so is CM to MF; [VI. 1]
therefore CM is incommensurable in length with MF. [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 that it is also a fourth apotome.