Construction of Apotome is Unique
Theorem
Let $D$ be the domain $\left\{{x \in \R_{>0} : x^2 \in \Q}\right\}$, the rationally expressible numbers.
Let $a, b \in D$ be two rationally expressible numbers such that $a - b$ is an apotome.
Then, there exists only one $x \in D$ such that $a - b + x$ and $a$ are commensurable in square only.
In the words of Euclid:
- To an apotome only one rational line segment can be annexed which is commensurable with the whole in square only.
(The Elements: Book $\text{X}$: Proposition $79$)
Proof
Let $AB$ be an apotome.
Let $BC$ be added to $AB$ so that $AC$ and $CB$ are rational straight lines which are commensurable in square only.
It is to be proved that no other rational straight line can be added to $AB$ which is commensurable in square only with the whole.
Suppose $BD$ can be added to $AB$ so as to fulfil the conditions stated.
Then $AD$ and $DB$ are rational straight lines which are commensurable in square only.
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 $AD^2 + DB^2$ and $AC^2 + CB^2$ are both rational.
Therefore $AD^2 + DB^2 - AC^2 + CB^2$ is rational.
Therefore $2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$ is rational.
But from Proposition $21$ of Book $\text{X} $: Medial is Irrational:
- $2 \cdot AD \cdot DB$ and $2 \cdot AC \cdot CB$ are both medial.
From Proposition $26$ of Book $\text{X} $: Medial Area not greater than Medial Area by Rational Area:
- a medial area does not exceed a medial area by a rational area.
Therefore no other rational straight line can be added to $AB$ which is commensurable in square only with the whole.
Therefore only one rational straight line can be added to $AB$ which is commensurable in square only with the whole.
$\blacksquare$
Historical Note
This proof is Proposition $79$ 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