# 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.*

(*The Elements*: Book $\text{X}$: Proposition $115$)

## 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:

and from:

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:

and from:

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*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions