Construction of First Binomial Straight Line
Theorem
In the words of Euclid:
- To find the first binomial straight line.
(The Elements: Book $\text{X}$: Proposition $48$)
Proof
Let $AC$ and $CB$ be straight lines constructed such that $AB = AC + CB$ is itself a straight line.
- $AB : BC = m^2 : n^2$
where $m$ and $n$ are numbers such that $m^2 - n^2$ is not square.
Let $D$ be a rational straight line.
Let $EF$ be constructed commensurable in length with $D$.
Then $EF$ is also a rational straight line.
Using Porism to Proposition $6$ of Book $\text{X} $: Magnitudes with Rational Ratio are Commensurable, let:
- $BA : AC = EF^2 : FG^2$
where $FG$ is a straight line constructed such that $EG = EF + FG$ is itself a straight line.
But:
- $AB : AC = m^2 : \left({m^2 - n^2}\right)$
where $\left({m^2 - n^2}\right)$ is not square.
So from Proposition $6$ of Book $\text{X} $: Magnitudes with Rational Ratio are Commensurable:
- $EF^2 : FG^2 = m^2 : \left({m^2 - n^2}\right)$
$EF$ and $FG$ are commensurable in square.
But from Proposition $9$ of Book $\text{X} $: Commensurability of Squares: $EF$ and $FG$ are incommensurable in length.
Therefore $EF$ and $FG$ are rational straight lines which are commensurable in square only.
Therefore by definition $EG$ is a binomial.
We have that:
- $BA : AC = EF^2 : FG^2$
while:
- $BA > AC$
Therefore:
- $EF^2 > FG^2$
Let:
- $FG^2 + H^2 = EF^2$
for some $H$.
From Porism to Proposition $19$ of Book $\text{V} $: Proportional Magnitudes have Proportional Remainders:
- $AB : BC = EF^2 : H$
But $AB : BC$ has the ratio that a square number has to a square number.
Therefore by Proposition $9$ of Book $\text{X} $: Commensurability of Squares:
- $EF$ is commensurable in length with $H$.
Therefore $EF^2$ is greater than $FG^2$ by the square on a straight line commensurable with $EF$.
We have that:
- $EF$ and $FG$ are rational straight lines
and:
- $EF$ is commensurable in length with $D$.
Therefore $EF$ is a first binomial straight line.
$\blacksquare$
Historical Note
This proof is Proposition $48$ 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
- in which a mistake appears.