Archimedes' axiom, also known as the continuity axiom or Archimedes' lemma, survives in the writings of Eudoxus 
(Boyer and Merzbach 1991), but the term was first coined by the Austrian mathematician Otto Stolz (1883). It states that, given two magnitudes having a ratio, one can find a multiple of either which will exceed the other. This principle was the basis for the method of exhaustion, which Archimedes invented to solve problems of area and volume.
Symbolically, the axiom states that
iff the appropriate one of following conditions is satisfied for integers m and n:
- 1. If ma < nb, then mc < nd.
- 2. If ma = nb, then mc = nd.
- 3. If ma > nb, then mc > nd.
Formally, Archimedes' axiom states that if AB and CD are two line segments, then there exist a finite number of points 
,
,
on 
such that
and B is between A and (Itô 1986, p. 611). A geometry in which Archimedes' lemma does not hold is called a non-Archimedean Geometry.
Continuity Axioms, Fraction, Inequality, Method of Exhaustion, Non-Archimedean Geometry
Boyer, C. B. and Merzbach, U. C. "The Abacus and Decimal Fractions." A History of Mathematics, 2nd ed. New York: Wiley, p. 100, 1991.
Itô, K. (Ed.). §155B and 155D in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 611, 1986.
Stolz, O. "Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes." Math. Ann. 22, 504-520, 1883.
Stolz, O. "Über das Axiom des Archimedes." Math. Ann. 39, 107-112, 1891.
Veronese, G. "Il continuo rettilineo e l'assioma cinque d'Archimede." Atti della Reale Accademia dei Lincei Ser. 4, No. 6, 603-624, 1890.