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

Let it be contrived that, as AB is to CD, so is AE to CF; [VI. 12] therefore also the remainder EB is to the remainder FD as AB is to CD. [V. 19]

But AB is commensurable in length with CD; therefore AE is also commensurable with CF, and EB with FD. [X. 11]

And AE, EB are rational; therefore CF, FD are also rational.

And, as AE is to CF, so is EB to FD. [V. 11]

Therefore, alternately, as AE is to EB, so is CF to FD. [V. 16]

But AE, EB are commensurable in square only; therefore CF, FD are also commensurable in square only. [X. 11]

And they are rational; therefore CD is binomial. [X. 36]

I say next that it is the same in order with AB.

For the square on AE is greater than the square on EB either by the square on a straight line commensurable with AE or by the square on a straight line incommensurable with it.

If then the square on AE is greater than the square on EB by the square on a straight line commensurable with AE, the square on CF will also be greater than the square on FD by the square on a straight line commensurable with CF. [X. 14]

And, if AE is commensurable with the rational straight line set out, CF will also be commensurable with it, [X. 12] and for this reason each of the straight lines AB, CD is a first binomial, that is, the same in order. [X. Deff. II. 1]

But, if EB is commensurable with the rational straight line set out, FD is also commensurable with it, [X. 12] and for this reason again CD will be the same in order with AB, for each of them will be a second binomial. [X. Deff. II. 2]