# Definition:Euclid's Definitions - Book X (II)

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## Euclid's Definitions: Book $\text{X (II)}$

These definitions appear between Propositions $47$ and $48$ of Book $\text{X}$ of Euclid's *The Elements*.

- Given a rational straight line and a binomial, divided into its terms, such that the square on the greater term is greater than the square on the lesser by the square on a straight line commensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a
**first binomial**straight line; - but if the lesser term be commensurable in length with the rational straight line set out, let the whole be called a
**second binomial**; - and if neither of the terms be commensurable in length with the rational straight line set out, let the whole be called a
**third binomial**. - Again, if the square on the greater term be greater than the square on the lesser by the square on a straight line incommensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a
**fourth binomial**; - if the lesser, a
**fifth binomial**; - and if neither, a
**sixth binomial**.

## Sources

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