Classification of Irrational Straight Lines derived from Binomial Straight Line

Proof
From :
 * the square on a medial, applied to a rational straight line, produces as breadth a rational straight line which is incommensurable in length with that to which it is applied.

From :
 * the square on a binomial, applied to a rational straight line, produces as breadth the first binomial.

From :
 * the square on a first bimedial, applied to a rational straight line, produces as breadth the second binomial.

From :
 * the square on a second bimedial, applied to a rational straight line, produces as breadth the third binomial.

From :
 * the square on the major, applied to a rational straight line, produces as breadth the fourth binomial.

From :
 * the square on the side of a rational plus medial area, applied to a rational straight line, produces as breadth the fifth binomial.

From :
 * the square on the side of the sum of two medial areas, applied to a rational straight line, produces as breadth the sixth binomial.

All of these breadths so produced differ from the first and from each other:
 * from the first because it is rational

and:
 * from each other because they are different in order.