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:

Proposition $15$ of Book $\text{X} $: Commensurability of Sum of Commensurable Magnitudes

and:

Porism to Proposition $23$ of Book $\text{X} $: Straight Line Commensurable with Medial Straight Line is Medial:

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