# 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.

## Proof

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.
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.

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:

Medial
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.