# Classification of Irrational Straight Lines derived from Apotome

## Theorem

In the words of Euclid:

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

(*The Elements*: Book $\text{X}$: Proposition $111$ : Summary)

## Proof

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

- the square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied.

From Proposition $97$ of Book $\text{X} $: Square on Apotome applied to Rational Straight Line:

- the square on an apotome applied to a rational straight line produces as breadth a first apotome.

- the square on a first apotome of a medial straight line applied to a rational straight line produces as breadth a second apotome.

- the square on a second apotome of a medial straight line applied to a rational straight line produces as breadth a third apotome.

- the square on a minor straight line applied to a rational straight line produces as breadth a fourth apotome.

- the square on the straight line which produces with a rational area a medial whole, if applied to a rational straight line, produces as breadth a fifth apotome.

- the square on the straight line which produces with a medial area a medial whole, if applied to a rational straight line, produces as breadth a sixth apotome.

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.

Thus it follows that the irrational straight lines themselves are different from one another.

From Proposition $111$ of Book $\text{X} $: Apotome not same with Binomial Straight Line:

- an apotome is not the same as a binomial straight line.

Combining the above analysis with Classification of Irrational Straight Lines derived from Binomial Straight Line, it can be seen that the following $13$ types of irrational straight line are all different in nature:

- Binomial
- First bimedial
- Second bimedial
- Major
- "Side" of a rational plus medial area
- "Side" of the sum of two medial areas

- Apotome
- First apotome of a medial straight line
- Second apotome of a medial straight line
- Minor
- Straight line which produces with a rational area a medial whole
- Straight line which produces with a medial area a medial whole.

$\blacksquare$

## Historical Note

This proof is Proposition $111$ 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