An Archimedean field (or ordered ring, were someone to use such a beast) is an ordered field with the following property:
(Underlined letters are elements of the field, which, being ordered, contains all natural numbers.)

Any complete ordered field which is Archimedean is (isomorphic to) the real numbers.

Archimedes was the first mathematician to NOTICE this axiom; he did so in the context of geometry, but the name stuck to the general case.

Ar`chi*me*de"an (#), a. [L. Archimedeus.]

Of or pertaining to Archimedes, a celebrated Greek philosopher; constructed on the principle of Archimedes' screw; as, Archimedean drill, propeller, etc.

Archimedean screw, or Archimedes' screw, an instrument, said to have been invented by Archimedes, for raising water, formed by winding a flexible tube round a cylinder in the form of a screw. When the screw is placed in an inclined position, and the lower end immersed in water, by causing the screw to revolve, the water is raised to the upper end.

Francis.