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

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

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

1. 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;
2. but if the lesser term be commensurable in length with the rational straight line set out, let the whole be called a second binomial;
3. 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.
4. 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;
5. if the lesser, a fifth binomial;
6. 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 $\text{II}$