# Straight Line Commensurable with Apotome of Medial Straight Line

## Theorem

In the words of Euclid:

*A straight line commensurable with an apotome of a medial straight line is an apotome of a medial straight line and the same in order.*

(*The Elements*: Book $\text{X}$: Proposition $104$)

## Proof

Let $AB$ be an apotome of a medial straight line.

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

It is to be demonstrated that:

- $CD$ is an apotome of a medial straight line

and:

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

Therefore by definition of apotome of a medial straight line:

- $AE$ and $EB$ are medial straight lines which are commensurable in square only.

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:

- $AE : CF = BE : DF$

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:

- $BE$ is commensurable in length with $DF$.

We have that $AE$ and $EB$ are medial straight lines which are commensurable in square only.

- $CF$ and $FD$ are medial straight lines.

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

- $CF$ and $FD$ are medial straight lines which are commensurable in square only.

Thus $CD$ is an apotome of a medial straight line.

$\Box$

It remains to be shown that the order of $CD$ is the same as the order of $AB$.

We have that:

- $AE : CF = BE : DF$

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

- $AE : EB = CF : FD$

Therefore:

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

But $AE^2$ is commensurable with $CF^2$.

Therefore from:

and:

we have that:

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

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

Thuis $CD$ is an apotome of a medial straight line of the same as the order as $AB$.

$\blacksquare$

## Historical Note

This proof is Proposition $104$ 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