Side of Remaining Area from Rational Area from which Medial Area Subtracted

Theorem

In the words of Euclid:

If, from a rational area a medial area be subtracted, the "side" of the remaining area becomes one of two irrational straight lines, either an apotome or a minor straight line.

(The Elements: Book $\text{X}$: Proposition $108$)

Proof

Let $BC$ be a rational area.

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

It needs to be demonstrated that the "side" of the remainder $EC$ is either an apotome or a minor straight line.

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 rational and $BD$ is medial.

We also have:

$BC = GH$

and:

$BD = GK$

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

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:

$FH$ is rational and commensurable in length with $FG$

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

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

Therefore $KH$ is a first apotome.

But from Proposition $91$ of Book $\text{X} $: Side of Area Contained by Rational Straight Line and First Apotome:

the "side" of $LH$ is an apotome.

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

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

Therefore $KH$ is a fourth apotome.

But from Proposition $94$ of Book $\text{X} $: Side of Area Contained by Rational Straight Line and Fourth Apotome:

the "side" of $LH$ is minor.

$\blacksquare$

Historical Note

This proof is Proposition $108$ 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