Construction of First Apotome
Theorem
In the words of Euclid:
- To find the first apotome.
(The Elements: Book $\text{X}$: Proposition $85$)
Proof
Let $A$ be a rational straight line.
Let $BG$ be commensurable in length with $A$.
Therefore, by definition, $BG$ is also rational.
Let $DE$ and $EF$ be square numbers such that $FD = DE - EF$ is not square.
Then $ED : FD$ is not the ratio that a square number has to another square number.
Using Porism to Proposition $6$ of Book $\text{X} $: Magnitudes with Rational Ratio are Commensurable, let it be contrived that:
- $ED : DF = BG^2 : CG^2$
Therefore by Proposition $6$ of Book $\text{X} $: Magnitudes with Rational Ratio are Commensurable:
- $BG^2$ is commensurable with $CG^2$.
But $BG^2$ is rational.
Therefore $GC^2$ is rational.
Therefore $GC$ is rational.
We have that $ED : FD$ is not the ratio that a square number has to another square number.
Therefore $BG^2 : GC^2$ is not the ratio that a square number has to another square number.
Therefore from Proposition $9$ of Book $\text{X} $: Commensurability of Squares:
- $BG$ is incommensurable in length with $GC$.
Both $BG$ and $GC$ are rational.
Therefore $BG$ and $GC$ rational straight lines which are commensurable in square only.
Therefore $BC$ is an apotome.
It remains to be shown that $BC$ is a first apotome.
Let $H$ be a straight line such that $H^2 = BG^2 - GC^2$.
We have that:
- $ED : FD = BG^2 : GC^2$
So by Porism to Proposition $19$ of Book $\text{V} $: Proportional Magnitudes have Proportional Remainders:
- $DE : EF = GB^2 : H^2$
But $DE$ and $EF$ are both square numbers.
So $DE$ has to $EF$ the ratio that a square number has to another square number.
Therefore $GB^2$ has to $H^2$ the ratio that a square number has to another square number.
Therefore from Proposition $9$ of Book $\text{X} $: Commensurability of Squares:
- $BG$ is commensurable in length with $H$.
We have that:
- $BG^2 = GC^2 + H^2$
Therefore $BG^2$ is greater than $GC^2$ by the square on a straight line which is commensurable in length with $BG$.
Also, the whole $BG$ is commensurable in length with the rational straight line $A$.
Therefore, by definition, $BC$ is a first apotome.
$\blacksquare$
Historical Note
This proof is Proposition $85$ 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