Book 10
66
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]
66
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]
66
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]
66
Hence a straight line commensurable in length with a binomial straight line is binomial and the same in order. Q. E. D.
PROPOSITION 67.
67
A straight line commensurable in length with a bimedial straight line is itself also bimedial and the same in order.
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.
67
For, since AB is bimedial, let it be divided into its medials at E; therefore AE, EB are medial straight lines commensurable in square only. [X. 37, 38]
67
And let it be contrived that, as AB is to CD, so is AE to CF; therefore also the remainder EB is to the remainder FD as AB is to CD. [V. 19]
67
But AB is commensurable in length with CD; therefore AE, EB are also commensurable with CF, FD respectively. [X. 11]
67
But AE, EB are medial; therefore CF, FD are also medial. [X. 23]
