# Construction of Incommensurable Lines/Lemma

## Lemma to Construction of Incommensurable Lines

In the words of Euclid:

It has been proved in the arithmetical books that similar plane numbers have to one another the ratio which a square number has to a square number,
and that, if two numbers have to one another the ratio which a square number has to a square number, they are similar plane numbers.

And it is manifest from these propositions that numbers which have not to one another the ratio which a square number has to a square number, that is, those which have not their sides proportional, have not to one another the ratio which a square number has to a square number.

## Proof

The propositions referred to are:

Proposition $26$ of Book $\text{VIII}$: Similar Plane Numbers have Same Ratio as between Two Squares

and its converse, which Euclid does not explicitly prove.

In the words of Euclid:

For, if they have, they will be similar plane numbers: which is contrary to the hypothesis.
Therefore numbers which are not similar plane numbers have not to one another the ratio which a square number has to a square number.

$\blacksquare$

## Historical Note

This proof is Proposition $10$ of Book $\text{X}$ of Euclid's The Elements.
It was suggested by Heiberg that this lemma is a later interpolation, and not part of the original work by Euclid. The theorem Construction of Incommensurable Lines that it supports is itself also questionable.

Moreover, there is no reason why numbers which are not similar plane numbers should be introduced here.