Minor is Irrational

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In the words of Euclid:

If from a straight line there be subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the squares on them added together rational, but the rectangle contained by them medial, the remainder is irrational; and let it be called a minor.

(The Elements: Book $\text{X}$: Proposition $76$)



Let $AB$ be a straight line.

Let a straight line $BC$ such that:

$BC$ is incommensurable in square with $AB$
$AB^2 + BC^2$ is rational
the rectangle contained by $AB$ and $BC$ is medial

be cut off from $AB$.

We have that:

$AB^2 + BC^2$ is rational


$2 \cdot AB \cdot BC$ is medial.

Therefore $AB^2 + BC^2$ is incommensurable with $2 \cdot AB \cdot BC$.


Proposition $7$ of Book $\text{II} $: Square of Difference


Proposition $16$ of Book $\text{X} $: Incommensurability of Sum of Incommensurable Magnitudes

it follows that:

$AB^2 + BC^2$ is incommensurable with $AC^2$.

But $AB^2 + BC^2$ is rational.

Therefore $AC^2$ is irrational.

Therefore $AC$ is irrational.

Such a straight line is known as a minor.


Historical Note

This proof is Proposition $76$ of Book $\text{X}$ of Euclid's The Elements.