Major Straight Line is Divisible Uniquely

Theorem

In the words of Euclid:

A major straight line is divided at one and the same point only.

(The Elements: Book $\text{X}$: Proposition $45$)

Proof

Let $AB$ be a major straight line.

Let $AB$ be divided at $C$ to create $AC$ and $CB$ such that:

$AC$ and $CB$ are incommensurable in square

$AC^2 + CB^2$ is rational

$AC$ and $CB$ contain a medial rectangle.

Let $AB$ be divided at $D$ such that $AD$ and $DB$ have the same properties as $AB$ and $CB$.

From Proposition $4$ of Book $\text{II} $: Square of Sum:

$AB^2 = \left({AC + CB}\right)^2 = AC^2 + CB^2 + 2 \cdot AC \cdot CB$

and:

$AB^2 = \left({AD + DB}\right)^2 = AD^2 + DB^2 + 2 \cdot AD \cdot DB$

and so:

$\left({AC^2 + CB^2}\right) - \left({AD^2 + DB^2}\right) = 2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$

But $AB^2 + CB^2$ and $AD^2 + DB^2$ are rational.

So $\left({AC^2 + CB^2}\right) - \left({AD^2 + DB^2}\right)$ is rational.

Therefore $2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$ is rational.

But $2 \cdot AD \cdot DB$ and $2 \cdot AC \cdot CB$ are medial.

From Proposition $26$ of Book $\text{X} $: Medial Area not greater than Medial Area by Rational Area this cannot be the case.

So there can be no such $D$.

$\blacksquare$

Historical Note

This proof is Proposition $45$ 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