Euclid Book V

David Fowler and C. Zeeman have made many important contributions to understanding this book of Euclid's which is assumed to be reorganization of the work of Eudoxus.

The theory is quite abstract and can be framed in terms of an action of the natural numbers $\mathbb{N}$ on any linearly ordered structure $\mathcal{A}$. We write $a:b=c:d$ for $a,b,c,d\in \mathcal{A}$ iff the following three conditions are satisfied:$$\mathrm{\forall }n,m\in \mathbb{N}((n.a=m.b)\iff (n.c=m.d))$$$$\mathrm{\forall }n,m\in \mathbb{N}((n.a>m.b)\iff (n.c>m.d))$$$$\mathrm{\forall }n,m\in \mathbb{N}((n.a<m.b)\iff (n.c<m.d))$$ 

In the case of $\mathbb{Q}$ and the action is the usual multiplication the first condition is sufficient. It just says that $\mathrm{\forall }q\in \mathbb{Q}(\frac{a}{b}=q\iff \frac{c}{d}=q)$.  What is amazing is that Eudoxus' definition works for the real numbers too ! Consider the particular case in which $b=d=1$. Then $a:1=c:1$ is just equality in $\mathbb{R}$. Eudoxus' definition is saying in this case that $a$ and $b$ determine the same Dedekind cut.