# Euclid:Proposition/X/10/Lemma

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## Proposition

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.

(*The Elements*: Book $\text{X}$: Proposition $10$ : Lemma)

## Sources

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