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:

the order of $CD$ is the same as the order of $AB$.

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.

From Proposition $23$ of Book $\text{X} $: Straight Line Commensurable with Medial Straight Line is Medial:

$CF$ 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:

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:

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

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

if $AE \cdot EB$ is rational, then $CF \cdot FD$ is rational.

From Porism to Proposition $23$ of Book $\text{X} $: Straight Line Commensurable with Medial Straight Line is Medial:

if $AE \cdot EB$ is medial, then $CF \cdot FD$ is medial.

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