First Apotome of Medial is Irrational
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Theorem
In the words of Euclid:
- If from a medial straight line there be subtracted a medial straight line commensurable with the whole in square only, and which contains with the whole a rational rectangle, the remainder is irrational. And let it be called a first apotome of a medial straight line.
(The Elements: Book $\text{X}$: Proposition $74$)
Proof
Let $AB$ be a medial straight line.
Let a medial straight line $BC$ such that:
- $BC$ is commensurable in square only with $AB$
- the rectangle contained by $AB$ and $BC$ is rational
be cut off from $AB$.
We have that $AB$ and $BC$ are medial.
So by definition $AB^2$ and $BC^2$ are both medial.
But $2 \cdot AB \cdot BC$ is rational.
Therefore:
- $AB^2$ and $BC^2$ are incommensurable with $2 \cdot AB \cdot BC$.
From Proposition $16$ of Book $\text{X} $: Incommensurability of Sum of Incommensurable Magnitudes:
- if $AB$ is incommensurable with either $AC$ or $CB$, $AC$ and $CB$ are incommensurable with each other.
Therefore by Proposition $7$ of Book $\text{II} $: Square of Difference:
- $2 \cdot AB \cdot BC$ is incommensurable with $AC^2$.
Therefore $AC^2$ is irrational.
Therefore by definition $AC$ is irrational.
Such a straight line is known as a first apotome of a medial.
$\blacksquare$
Historical Note
This proof is Proposition $74$ 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