Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome
Theorem
In the words of Euclid:
- If a rational straight line be cut in extreme and mean ratio, each of the segments is the irrational straight line called apotome.
(The Elements: Book $\text{XIII}$: Proposition $6$)
Proof
Let $AB$ be a rational straight line.
Let $AB$ be cut in extreme and mean ratio at the point $C$.
Let $AC$ be the greater segment.
It is to be demonstrated that each of $AC$ and $CB$ is the irrational straight line known as apotome.
Let $BA$ be produced.
Let $AD = \dfrac {BA} 2$.
We have that $AB$ is cut in extreme and mean ratio at $C$.
- $CD^2 = 5 \cdot DA^2$
Therefore $CD^2 : DA^2$ is the ratio that a number has to a number.
Therefore by Proposition $6$ of Book $\text{X} $: Magnitudes with Rational Ratio are Commensurable:
- $CD^2$ is commensurable with $DA^2$.
But by Book $\text{X}$ Definition $4$: Rational Area:
- $DA^2$ is rational.
Therefore by Book $\text{X}$ Definition $4$: Rational Area:
- $CD^2$ is also rational.
Therefore $CD$ is rational.
But $CD^2 : DA^2$ is not the ratio that a square number has to a square number.
Therefore by Proposition $9$ of Book $\text{X} $: Commensurability of Squares:
- $CD$ and $DA$ are incommensurable in length.
Thus $CD$ and $DA$ are commensurable in square only.
Therefore by definition, $AC$ is an apotome.
$\Box$
We have that $AB$ is cut in extreme and mean ratio at $C$.
Therefore from:
and:
it follows that:
- $AB \cdot BC = AC^2$
The square on the apotome $AC$, applied to the rational straight line $AB$, produces $BC$ as breadth.
Therefore from Proposition $97$ of Book $\text{X} $: Square on Apotome applied to Rational Straight Line:
- $CB$ is a first apotome.
$\blacksquare$
Historical Note
This proof is Proposition $6$ of Book $\text{XIII}$ 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 $\text{XIII}$. Propositions