Straight Line Commensurable with Major Straight Line is Major
Theorem
In the words of Euclid:
- A straight line commensurable with a major straight line is itself also major.
(The Elements: Book $\text{X}$: Proposition $68$)
Proof
Let $AB$ be a major.
Let $CD$ be commensurable in length with $AB$.
It is to be shown that $CD$ is also a major.
Let $AB$ be divided into its terms by $E$.
Let $AE$ be the greater term.
By definition, $AE$ and $EB$ are straight lines such that:
- $AE$ and $EB$ are incommensurable in square
- $AE^2 + EB^2$ is rational
- $AE \cdot EB$ is medial.
Using Proposition $12$ of Book $\text{VI} $: Construction of Fourth Proportional Straight Line, let it be contrived that:
- $AB : CD = AE : CF$
Therefore by Proposition $19$ of Book $\text{V} $: Proportional Magnitudes have Proportional Remainders:
- $EB : FD = AB : CD$
Therefore by Proposition $11$ of Book $\text{V} $: Equality of Ratios is Transitive:
- $AE : CF = EB : FD$
But $AB$ is commensurable in length with $CD$.
Therefore from Proposition $11$ of Book $\text{X} $: Commensurability of Elements of Proportional Magnitudes:
- $AE$ is commensurable in length with $CF$
and:
- $EB$ is commensurable in length with $FD$.
We have that:
- $AE : CF = EB : FD$
Therefore by Proposition $16$ of Book $\text{V} $: Proportional Magnitudes are Proportional Alternately:
- $AE : EB = CF : FD$
and by by Proposition $18$ of Book $\text{V} $: Magnitudes Proportional Separated are Proportional Compounded:
- $AB : BE = CD : DF$
Therefore by Proposition $20$ of Book $\text{VI} $: Similar Polygons are composed of Similar Triangles:
- $AB^2 : BE^2 = CD^2 : DF^2$
Using a similar line of reasoning to the above:
- $AB^2 : AE^2 = CD^2 : CF^2$
and:
- $AB^2 : AE^2 + EB^2 = CD^2 : CF^2 + FD^2$
Therefore by Proposition $16$ of Book $\text{V} $: Proportional Magnitudes are Proportional Alternately:
- $AB^2 : CD^2 = AE^2 + EB^2 : CF^2 + FD^2$
But $AB^2$ is commensurable with $CD^2$.
Therefore $AE^2 + EB^2$ is commensurable with $CF^2 + FD^2$.
We have that $AE^2 + EB^2$ is rational.
Therefore $CF^2 + FD^2$ is rational.
Similarly, $2 \cdot AE \cdot EB$ is commensurable with $2 \cdot CF \cdot FD$.
- $2 \cdot AE \cdot EB$ is medial.
Therefore $2 \cdot CF \cdot FD$ is medial.
Thus it has been demonstrated that $CF$ and $FD$ are straight lines such that:
- $CF$ and $FD$ are incommensurable in square
- $CF^2 + FD2$ is rational
- $CF \cdot FD$ is medial.
Thus by definition $CD$ is a major.
$\blacksquare$
Historical Note
This proof is Proposition $68$ 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