Straight Line Commensurable with Minor Straight Line

Theorem

In the words of Euclid:

A straight line commensurable with a minor straight line is minor.

(The Elements: Book $\text{X}$: Proposition $105$)

Proof

Let $AB$ be a minor straight line.

Let $CD$ be commensurable in length with $AB$.

It is to be demonstrated that $CD$ is a minor straight line.

Let $BE$ be the annex of $CD$.

Therefore by definition of minor straight line:

$AE$ and $EB$ are incommensurable in square

$AE^2 + EB^2$ is rational

$2 \cdot AE \cdot EB$ is a medial rectangle.

From Proposition $12$ of Book $\text{VI} $: Construction of Fourth Proportional Straight Line, let it be contrived that:

$BE : DF = AB : CD$

From Proposition $12$ of Book $\text{V} $: Sum of Components of Equal Ratios:

$AE : CF = AB : CD$

and so from Proposition $11$ of Book $\text{V} $: Equality of Ratios is Transitive:

$AE : CF = BE : DF$

We have that $AE$ and $EB$ are incommensurable in square.

Therefore from Proposition $13$ of Book $\text{X} $: Commensurable Magnitudes are Incommensurable with Same Magnitude:

$CF$ and $FD$ are incommensurable in square.

We have that:

$AE : CF = BE : DF$

So from Proposition $16$ of Book $\text{V} $: Proportional Magnitudes are Proportional Alternately:

$AE : EB = CF : FD$

So from Proposition $22$ of Book $\text{VI} $: Similar Figures on Proportional Straight Lines:

$AE^2 : EB^2 = CF^2 : FD^2$

Therefore from Proposition $18$ of Book $\text{V} $: Magnitudes Proportional Separated are Proportional Compounded:

$AE^2 + EB^2 : EB^2 = CF^2 + FD^2 : FD^2$

But $EB^2$ is commensurable with $FD^2$.

So from:

Proposition $16$ of Book $\text{V} $: Proportional Magnitudes are Proportional Alternately:

and:

Proposition $11$ of Book $\text{X} $: Commensurability of Elements of Proportional Magnitudes:

we have that:

$AE^2 + EB^2$ is commensurable with $CF^2 + FD^2$.

But by definition $AE^2 + EB^2$ is rational.

Therefore by Book $\text{X}$ Definition $4$: Rational Area:

$CF^2 + FD^2$ is rational.

We have that:

$AE : EB = CF : FD$

Therefore:

$AE^2 : AE \cdot EB = CF^2 : CF \cdot FD$

Therefore $AE \cdot EB$ is commensurable with $CF \cdot FD$.

Therefore $CF \cdot FD$ is medial.

Thus $CF$ and $FD$ are such that:

$CF$ and $FD$ are incommensurable in square

$CF^2 + FD^2$ is rational

$2 \cdot CF \cdot FD$ is a medial rectangle.

Hence the result.

$\blacksquare$

Historical Note

This proof is Proposition $105$ 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