# Classification of Irrational Straight Lines derived from Binomial Straight Line

## Theorem

In the words of Euclid:

The binomial straight line and the irrational straight lines after it are neither the same with the medial nor with one another.

## Proof

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.
the square on a binomial, applied to a rational straight line, produces as breadth the first binomial.
the square on a first bimedial, applied to a rational straight line, produces as breadth the second binomial.
the square on a second bimedial, applied to a rational straight line, produces as breadth the third binomial.
the square on the major, applied to a rational straight line, produces as breadth the fourth binomial.
the square on the side of a rational plus medial area, applied to a rational straight line, produces as breadth the fifth binomial.
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.

$\blacksquare$

## Historical Note

This proof is Proposition $72$ of Book $\text{X}$ of Euclid's The Elements.