# Second Apotome of Medial is Irrational

## Theorem

In the words of Euclid:

*If from a medial straight line there be subtracted a medial straight line commensurable with the whole in square only, and which contains with the whole a medial rectangle, the remainder is irrational; and let it be called a***second apotome of a medial***straight line.*

(*The Elements*: Book $\text{X}$: Proposition $75$)

## Proof

Let $AB$ be a medial straight line.

Let a medial straight line $CB$ such that:

- $CB$ is commensurable in square only with $AB$
- the rectangle contained by $AB$ and $BC$ is medial

be cut off from $AB$.

Let $DI$ be a rational straight line.

- Let $DE$ be a parallelogram set out on $DI$ equal to $AB^2 + BC^2$.

Let its breadth be $DG$.

Similarly:

- Let $DH$ be a parallelogram set out on $DI$ equal to $2 \cdot AB \cdot BC$.

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

- $FE = AC^2$

We have that $AB^2$ and $BC^2$ are medial areas which are commensurable.

Therefore from:

and:

it follows that:

- $DE$ is medial.

We have that $DE$ has been applied to the rational straight line $DI$ producing $DG$ as breadth.

So from Proposition $22$ of Book $\text{X} $: Square on Medial Straight Line:

- $DG$ is rational and incommensurable in length with $DI$.

We have that $AB \cdot BC$ is medial.

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

But $2 \cdot AB \cdot BC = DH$.

Therefore $DH$ is medial.

We have that $DH$ has been applied to the rational straight line $DI$ producing $DF$ as breadth.

So from Proposition $22$ of Book $\text{X} $: Square on Medial Straight Line:

- $DF$ is rational and incommensurable in length with $DI$.

We have that $AB$ and $BC$ are commensurable in square only.

Therefore from Proposition $11$ of Book $\text{X} $: Commensurability of Elements of Proportional Magnitudes:

- $AB$ is incommensurable in length with $BC$.

But from Proposition $15$ of Book $\text{X} $: Commensurability of Sum of Commensurable Magnitudes:

- $AB^2 + BC^2$ are commensurable with $AB^2$

and from Proposition $6$ of Book $\text{X} $: Commensurability of Sum of Commensurable Magnitudes:

- $2 \cdot AB \cdot BC$ is commensurable with $AB \cdot BC$

therefore from Proposition $13$ of Book $\text{X} $: Commensurable Magnitudes are Incommensurable with Same Magnitude:

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

But:

- $DE = AB^2 + BC^2$

and:

- $DH = 2 \cdot AB \cdot BC$

and so $DE$ is incommensurable with $DH$.

But from Proposition $1$ of Book $\text{VI} $: Areas of Triangles and Parallelograms Proportional to Base:

- $DE : DH = GD : DF$

Therefore from Proposition $11$ of Book $\text{X} $: Commensurability of Elements of Proportional Magnitudes:

- $GD$ is incommensurable with $DF$.

But both $GD$ and $DF$ are rational.

Therefore $GD$ and $DF$ are rational straight lines which are commensurable in square only.

Therefore, by definition, $FG$ is an apotome.

But $DI$ is rational.

From Proposition $20$ of Book $\text{X} $: Quotient of Rationally Expressible Numbers is Rational:

- a rectangle contained by a rational and an irrational straight line is irrational.

Hence its "side" is irrational.

But $AC$ is the "side" of $FE$.

Therefore $AC$ is irrational.

Such a straight line is known as a second apotome of a medial.

$\blacksquare$

## Historical Note

This proof is Proposition $75$ 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