Archimedes' axiom

Archimedes' axiom

[¦är·kə¦mēd‚ēz ′ak·sē·əm]
(mathematics)
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Archimedes' axiom - the fifth in his The Sphere and the Cylinder - states (in modern terminology) that if AB and CD are any rectilinear segments, then there exists a positive integer n and a point B[prime], collinear with AB, such that AB goes into AB[prime] exactly n times and A C goes into AB[prime] at least once.
Second, he pointed out that non-Archimedean continua cannot be excluded on empirical or logical grounds: non-Archimedean geometry, in which Archimedes' axiom is denied, is just as legitimate as non-Euclidean geometry, in which Euclid's fifth postulate is denied.