"The beauty of [ Eudoxus' ] theory of proportions [ expounded in Book V of Euclid's Elements ] was its adaptability to this new climate. ...The length \sqrt2 is determined by the two sets of positive rationalsL_\sqrt2 = \{r: r^2 < 2\}, \qquad U_\sqrt2 = \{r: r^2 > 2\}Dedekind... decided to let \sqrt2 be this pair of sets! In general, let any partition of the positive rationals into sets L, U such that any member of L is less than any member of U be a positive real number. This idea, now known as the Dedekind cut, is more than just a twist of Eudoxus; it gives a complete and uniform construction of all real numbers, or points on a line, using just the discrete, finally resolving the fundamental conflict in Greek mathematics."