From Medial Straight Line arises Infinite Number of Irrational Straight Lines

Theorem

In the words of Euclid:

From a medial straight line there arise irrational straight lines infinite in number, and none of them is the same as any of the preceding.

Proof

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 $\KK$.

Let $B$ be a rational straight line.

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

Therefore by:

Book $\text{X}$ Definition $4$: Rational Area

and from:

Proposition $20$ of Book $\text{X}$: Quotient of Rationally Expressible Numbers is Rational

it follows that:

$C$ is irrational.

Consider a square on any of $\KK$.

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:

Book $\text{X}$ Definition $4$: Rational Area

and from:

Proposition $20$ of Book $\text{X}$: Quotient of Rationally Expressible Numbers is Rational

it follows that:

$D$ is irrational.

Consider a square on any of $\KK$.

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

This process can be continued ad infinitum.

$\blacksquare$

Historical Note

This proof is Proposition $115$ of Book $\text{X}$ of Euclid's The Elements.