# Construction of First Apotome of Medial is Unique

## Theorem

In the words of Euclid:

*To a first apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle.*

(*The Elements*: Book $\text{X}$: Proposition $80$)

## Proof

Let $AB$ be the first apotome of a medial straight line.

Let $BC$ be added to $AB$ such that:

- $AC$ and $CB$ are medial straight lines
- $AC$ and $CB$ are commensurable in square only
- $AC \cdot CB$ is a rational rectangle.

It is to be proved that no other medial straight line can be added to $AB$ which is commensurable in square only with the whole and which contains with the whole a rational rectangle.

Suppose $BD$ can be added to $AB$ so as to fulfil the conditions stated.

Then by definition of the first apotome of a medial straight line, $AD$ and $DB$ are such that:

- $AD$ and $DB$ are medial straight lines
- $AD$ and $DB$ are commensurable in square only
- $AD \cdot DB$ is a rational rectangle.

From Proposition $7$ of Book $\text{II} $: Square of Difference:

- $AD^2 + DB^2 - 2 \cdot AD \cdot DB = AC^2 + CB^2 - 2 \cdot AC \cdot CB = AB^2$

Therefore:

- $AD^2 + DB^2 - AC^2 + CB^2 = 2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$

But $AD \cdot DB$ and $AC \cdot CB$ are both rational.

Therefore $2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$ is rational.

Therefore $AD^2 + DB^2 - AC^2 + CB^2$ is rational.

But from:

and:

both $AD^2 + DB^2$ and $AC^2 + CB^2$ is medial.

By Proposition $26$ of Book $\text{X} $: Medial Area not greater than Medial Area by Rational Area this cannot happen.

Therefore only one medial straight line can be added to $AB$ which is commensurable in square only with the whole and which contains with the whole a rational rectangle.

$\blacksquare$

## Historical Note

This proof is Proposition $80$ 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