# Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted

## Theorem

In the words of Euclid:

*If, from a medial area a rational area be subtracted, there arise two other irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole.*

(*The Elements*: Book $\text{X}$: Proposition $109$)

## Proof

Let $BC$ be a medial area.

Let the rational area $BD$ be subtracted from $BC$.

It needs to be demonstrated that the "side" of the remainder $EC$ is either:

or:

Let $FG$ be a rational straight line.

Let the rectangle $GH$ be applied to $FG$ equal to $BC$ producing $FH$ as breadth.

Let the area $GK$ equal to $BC$ be subtracted from $GH$.

Then the remainder $EC$ is equal to $LH$.

We have that $BC$ is medial and $BD$ is rational.

We also have:

- $BC = GH$

and:

- $BD = GK$

Therefore $GH$ is medial and $GK$ is rational.

But $GH$ and $GK$ are applied to a rational straight line $FG$.

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

- $FK$ is rational and commensurable in length with $FG$

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

- $FH$ is rational and incommensurable in length with $FG$.

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

- $FH$ is incommensurable in length with $FK$.

Therefore $FH$ and $FK$ are rational straight lines which are commensurable in square only.

Therefore $KH$ is an apotome and $KF$ is the annex to $KH$.

We have that:

- $HF^2 = FK^2 + \lambda^2$

where either:

- $\lambda$ is commensurable in length with $HF$

or:

- $\lambda$ is incommensurable in length with $HF$.

First suppose $\lambda$ is commensurable in length with $HF$.

Then $FK$ is commensurable in length with the rational straight line $FG$.

Therefore $KH$ is a second apotome.

- the "side" of $LH$ is a first apotome of a medial straight line.

Next suppose $\lambda$ is incommensurable in length with $HF$.

Then $FK$ is incommensurable in length with the rational straight line $FG$.

Therefore $KH$ is a fifth apotome.

- the "side" of $LH$ is a straight line which produces with a rational area a medial whole.

$\blacksquare$

## Historical Note

This proof is Proposition $109$ 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