Square on Binomial Straight Line applied to Rational Straight Line
Theorem
In the words of Euclid:
- The square on the binomial straight line applied to a rational straight line produces as breadth the first binomial.
(The Elements: Book $\text{X}$: Proposition $60$)
Lemma
In the words of Euclid:
- If a straight line be cut into unequal parts, the squares on the unequal parts are greater than twice the rectangle contained by the unequal parts.
(The Elements: Book $\text{X}$: Proposition $60$ : Lemma)
Proof
Let $AB$ be a binomial straight line divided into its terms at $C$.
Let $AC > CB$.
Let $DE$ be a rational straight line.
Let $DEFG$ equal to $AB^2$ be applied to $DE$ producing $DG$ as its breadth.
It is to be demonstrated that $DG$ is a first binomial straight line.
From Proposition $4$ of Book $\text{II} $: Square of Sum:
- $AB^2 = AC^2 + CB^2 + 2 \cdot AC \cdot CB$
Let the rectangle $DH$ be applied to $DE$ such that $DH = AC^2$.
Let the rectangle $KL$ be applied to $DE$ such that $KL = BC^2$.
Then the rectangle $MF$ is equal to $2 \cdot AC \cdot CB$.
Let $MG$ be bisected at $N$.
Let $NO$ be drawn parallel to $ML$ (or $GF$, which is the same thing).
Therefore each of the rectangles $MO$ and $NF$ equals $AC \cdot CB$.
We have that $AB$ is a binomial which has been divided into its terms at $C$.
Therefore, by definition, $AC$ and $CB$ are rational straight lines which are commensurable in square only.
Therefore $AC^2$ and $CB^2$ are commensurable and rational.
Thus by Proposition $15$ of Book $\text{X} $: Commensurability of Sum of Commensurable Magnitudes:
- $AC^2 + CB^2$ is a rational area.
But $AC^2 + CB^2 = DL$.
Therefore $DL$ is a rational area.
We have that $DL$ is applied to the rational straight line $DE$.
Therefore from Proposition $20$ of Book $\text{X} $: Quotient of Rationally Expressible Numbers is Rational:
- $DM$ is rational and commensurable in length with $DE$.
We have that $AC$ and $CB$ are rational straight lines which are commensurable in square only.
So from Proposition $21$ of Book $\text{X} $: Medial is Irrational:
- $2 \cdot AC \cdot CB = MF$ is medial.
We have that $MF$ is applied to the rational straight line $ML$.
Therefore from Proposition $22$ of Book $\text{X} $: Square on Medial Straight Line:
- $MG$ is also rational and incommensurable in length with $DE$.
But $MD$ is also rational and commensurable in length with $DE$.
Therefore from Proposition $13$ of Book $\text{X} $: Commensurable Magnitudes are Incommensurable with Same Magnitude:
- $DM$ is incommensurable in length with $MG$.
But $DM$ and $MG$ are both rational straight lines which are commensurable in square only.
Therefore, by definition, $DG$ is binomial.
It remains to be proved that $DG$ is first binomial.
- $AC \cdot CB$ is a mean proportional between $AC^2$ and $CB^2$.
Therefore $MO$ is a mean proportional between $DH$ and $KL$.
Therefore by Proposition $1$ of Book $\text{VI} $: Areas of Triangles and Parallelograms Proportional to Base:
- $DK : MN = MN : MK$
Therefore from Proposition $17$ of Book $\text{VI} $: Rectangles Contained by Three Proportional Straight Lines:
- $DK \cdot KM = MN^2$
We have that $AC^2$ is commensurable with $CB^2$.
Therefore $DH$ is commensurable with $KL$.
So from:
and:
it follows that:
- $DK$ is commensurable in length with $KM$.
- $AC^2 + CB^2 > 2 \cdot AC \cdot CB$
Therefore:
- $DL > MF$
and so from Proposition $1$ of Book $\text{VI} $: Areas of Triangles and Parallelograms Proportional to Base:
- $DM > MG$
And:
- $DK \cdot KM = MN^2 = \dfrac {MG^2} 4$
and:
- $DK$ is commensurable in length with $KM$.
- $DM^2$ is greater than $MG^2$ by the square on a straight line commensurable in length with $DM$.
We also have that:
- $DM$ and $MG$ are rational
and:
- the greater term $DM$ is commensurable in length with the rational straight line $DE$.
Therefore by definition $DG$ is a first binomial straight line.
$\blacksquare$
Historical Note
This proof is Proposition $60$ of Book $\text{X}$ of Euclid's The Elements.
It is the converse of Proposition $54$: Root of Area contained by Rational Straight Line and First Binomial.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions