Archimedes' Axiom

An Axiom actually attributed to Eudoxus (Boyer 1968) which states that

$${a / b} = {c / d}$$

Iff the appropriate one of following conditions is satisfied for Integers $m$ and $n$:

1. If $ma < nb$, then $mc < md$.

2. If $ma = nd$, then $mc = nd$.

3. If $ma > nd$, then $mc > nd$.

Archimedes' Lemma is sometimes also known as Archimedes' axiom.

References

Boyer, C. B. A History of Mathematics. New York: Wiley, p. 99, 1968.