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

Definitions II

These definitions appear between Propositions 47 and 48 of Book X of The Elements by Euclid.

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 term 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.