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