Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

But, if neither of the straight lines AE, EB is commensurable with the rational straight line set out, neither of the straight lines CF, FD will be commensurable with it, [X. 13] and each of the straight lines AB, CD is a third binomial. [X. Deff. II. 3]

But, if the square on AE is greater than the square on EB by the square on a straight line incommensurable with AE, the square on CF is also greater than the square on FD by the square on a straight line incommensurable with CF. [X. 14]

And, if AE is commensurable with the rational straight line set out, CF is also commensurable with it, and each of the straight lines AB, CD is a fourth binomial. [X. Deff. II. 4]

But, if EB is so commensurable, so is FD also, and each of the straight lines AB, CD will be a fifth binomial. [X. Deff. II. 5]

But, if neither of the straight lines AE, EB is so commensurable, neither of the straight lines CF, FD is commensurable with the rational straight line set out, and each of the straight lines AB, CD will be a sixth binomial. [X. Deff. II. 6]

Hence a straight line commensurable in length with a binomial straight line is binomial and the same in order. Q. E. D.

PROPOSITION 67.

Let AB be bimedial, and let CD be commensurable in length with AB; I say that CD is bimedial and the same in order with AB.

For, since AB is bimedial, let it be divided into its medials at E;