Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome

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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

Euclid-XIII-6.png

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 $C$.

From Proposition $1$ of Book $\text{XIII} $: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio:

$CD^2 = 5 \cdot DA^2$

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:

Proposition $17$ of Book $\text{VI} $: Rectangles Contained by Three Proportional Straight Lines

and:

Book $\text{VI}$ Definition $3$: Extreme and Mean Ratio

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 theorem is Proposition $6$ of Book $\text{XIII}$ of Euclid's The Elements.


Sources