From Medial Straight Line arises Infinite Number of Irrational Straight Lines

Proof

 * Euclid-X-115.png

Let $A$ be a medial straight line.

It is to be shown that there is an infinite number of irrational straight lines, and none of them are:
 * Binomial
 * First bimedial
 * Second bimedial
 * Major
 * The side of a rational plus a medial area
 * The side of the sum of two medial areas
 * Apotome
 * First apotome of a medial
 * Second apotome of a medial
 * Minor
 * That which produces a medial whole with a rational area
 * That which produces a medial whole with a medial area

Let the above set be called $\mathcal K$.

Let $B$ be a rational straight line.

Let $C^2$ be equal to $A \cdot B$.

Therefore by:

and from:

it follows that:
 * $C$ is irrational.

Consider a square on any of $\mathcal K$.

None of them, when applied to a rational straight line, produces as breadth a medial straight line.

Let $D^2$ be equal to $B \cdot C$.

Therefore by:

and from:

it follows that:
 * $D$ is irrational.

Consider a square on any of $\mathcal K$.

None of them, when applied to a rational straight line, produces $C$ as breadth.

This process can be continued ad infinitum.