Classification of Irrational Straight Lines derived from Apotome

Proof
From :
 * 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 :
 * the square on an apotome applied to a rational straight line produces as breadth a first apotome.

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

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

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

From :
 * 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.

From :
 * 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 :
 * 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.