First Bimedial is Irrational
Theorem
In the words of Euclid:
- If two medial straight lines commensurable in square only and containing a rational rectangle be added together, the whole is irrational; and let it be called a first bimedial straight line.
(The Elements: Book $\text{X}$: Proposition $37$)
Proof
Let $AB$ and $BC$ be medial straight lines which are commensurable in square only.
Let $AB$ and $BC$ contain a rational rectangle.
By definition, $AB$ and $BC$ are incommensurable in length.
We have:
- $AB : BC = AB \cdot BC : BC^2$
- $AB : BC = AB^2 : AB \cdot BC$
Therefore from Proposition $11$ of Book $\text{X} $: Commensurability of Elements of Proportional Magnitudes:
- $AB \cdot BC$ is incommensurable with $AB^2$
and
- $AB \cdot BC$ is incommensurable with $BC^2$.
But by Proposition $6$ of Book $\text{X} $: Magnitudes with Rational Ratio are Commensurable:
- $2 AB \cdot BC$ is commensurable with $AB \cdot BC$.
We have that $AB$ and $BC$ are commensurable in square.
So from Proposition $15$ of Book $\text{X} $: Commensurability of Sum of Commensurable Magnitudes:
- $AB^2 + BC^2$ is commensurable with $BC^2$.
- $2 AB \cdot BC$ is incommensurable with $AB^2 + BC^2$.
Thus from Proposition $16$ of Book $\text{X} $: Incommensurability of Sum of Incommensurable Magnitudes:
- $2 AB \cdot BC + AB^2 + BC^2$ is incommensurable with $AB \cdot BC$.
We have that $AB$ and $BC$ contain a rational rectangle.
Thus $AB \cdot BC$ is rational.
From Proposition $4$ of Book $\text{II} $: Square of Sum:
- $AC^2 = \left({AB + BC}\right)^2 = 2 AB \cdot BC + AB^2 + BC^2$
Thus from Book $\text{X}$ Definition $4$: Rational Area:
- $AC$ is irrational.
Such a straight line is called first bimedial.
$\blacksquare$
Historical Note
This proof is Proposition $37$ 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