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 them medial but twice the rectangle contained by them rational. [X. 77]
To CD let there be applied CH equal to the square on AG, and KL equal to the square on GB;
therefore the whole CL is equal to the squares on AG, GB.
But the sum of the squares on AG, GB together is medial;
therefore CL is medial.
And it is applied to the rational straight line CD, producing CM as breadth;
therefore CM is rational and incommensurable with CD. [X. 22]
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 N, and through N let NO be drawn 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 rational and equal to FL, therefore FL is rational.
And it is applied to the rational straight line EF, producing FM as breadth;
therefore FM is rational and commensurable in length with CD. [X. 20]
Now, since CL is medial, and FL rational, 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 next that it is also a fifth apotome.