Eudoxus employed 27 spheres: three each for the Sun and Moon, four each for the five planets, and one for the fixed stars.
Callipus (370?-300?) improved the Eudoxian system by adding spheres. Aristotle (384-322 B.C.) further modified it, but, unlike Eudoxus, he maintained that the spheres were material bodies. Accordingly, certain presuppositions of Aristotelian physics needed to be satisfied. This required 22 additional spheres. Unfortunately, these models failed to explain certain phenomena.
In the second century A.D. Ptolemy proposed a more satisfactory system in the Almagest. He endorsed Aristotelian physics and its conclusions regarding Earth being at rest and the center of a spherical universe. He also acknowledged that planetary motions could only be explained kinematically by uniform circular motion or combinations thereof. However, instead of concentric spheres, Ptolemy employed eccentric orbits, epicycles, and equants.
In its simplest form, a planet's motion might be represented by an eccentric orbit—a circle about Earth, known as the deferent, with Earth offset from the center. In addition, a planet could be made to travel on an epicycle—a smaller circular orbit whose center moves along the circumference of the deferent. These geometrical devices had been exploited to great effect by Hipparchus (170?-120? B.C.).
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