Product of Rationally Expressible Numbers is Rational/Lemma

Theorem

In the words of Euclid:

Since it has been proved that straight lines are commensurable in length are always commensurable in square also, while those commensurable in square are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line be commensurable in length with a given rational straight line, it is called rational and commensurable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in square also.

But, if any straight line be commensurable in square with a given rational straight line, then, if it is also commensurable in length with it, it is called in this case also rational and commensurable with it in both in length and in square; but, if again any straight line, being commensurable in square with a given rational straight line, be incommensurable in length with it, it is called in this case also rational but commensurable in square only.

(The Elements: Book $\text{X}$: Proposition $19$ : Lemma)

Proof

This lemma does little more than restate (in a considerably more wordy form) Proposition $9$ of Book $\text{X} $: Commensurability of Squares.

$\blacksquare$

Historical Note

This proof is Proposition $19$ 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. In his own edition of this work, he relegated it to the appendix.

Heath similarly separates it out from the canon in his case by enclosing it in brackets $[\,]$.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions