# Classification of Irrational Straight Lines derived from Binomial Straight Line

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## 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.*

(*The Elements*: Book $\text{X}$: Proposition $72$ : Summary)

## Proof

From Proposition $22$ of Book $\text{X} $: Square on Medial Straight Line:

- 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*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions