Figure 7.13. The rolling motion of one planetary sphere on another: In this illustration, the labeled axis emerging from the sphere at point B gives rise to motion along the arc B1 to B5. The comparison of this motion to that on the arc A1 to A5 (the celestial equator in this example) gives rise to the relative motion shown to the left. This is the hippopede. Drawn by E.F. Milone, after Pannekoek (1961/1989, Fig. 8, p. 109).

The rolling motion of one planetary sphere on another resulted in a curious motion on the sky (somewhat like the solar analemma cf. ยง4.1.2), known as the hippopede, after a hobble used on the front legs of horses. The hippopede motion is illustrated in Figure 7.13. In all, Eudoxos required an elaborate series of 26 nested spheres, with some rotating counter to the adjacent one in an attempt to decouple one planet's motion from another.

The concentric sphere cosmology was further elaborated by Kallippos (or Callippus in Latin) [fl. 4th century b.c.] of

Cyzicus, on the Sea of Marmora, who studied with an associate of Eudoxos in that city, Polemarchus. Simplicius26 states that Polemarchus was aware that Eudoxos's model could not account for planetary brightness variations, brought on by their varying distances from Earth. Dicks (1970, p. 191) speculates that Kallippos introduced seven additional spheres (two each to the Sun and Moon, one each to Mercury, Venus, and Mars) to provide retrograde motion for Mars and Venus and that these motions were not in Eudoxos's original model. If true, this would have been a more glaring omission than the brightness variation, the true cause of which could scarcely have been known to Eudoxos. Aristotle [Athens, 384-322 b.c.] elaborated the model still further: He added 22 celestial spheres to Calippos's 33, making 55, in order to counteract the motions of the driving spheres for each planet.

The Hellenic period marked a highpoint for the concentric spheres; observations compelled astronomers of the Hellenistic and Roman periods to displace both the physical centers of these spheres from the Earth, and their centers of uniform motions (the equants). Elaborate systems of epicycles were further required to "save the phenomena." In a subsequent period, schools of Islamic astronomers strove to recapture the perfection of the uniform circular motion that the spheres represent. Visions of the celestial spheres dominated European cosmological thinking through the Middle Ages, and into the Reformation period, when Tycho's geo-heliocentric model broke through the crystalline spheres with overlapping orbits, and the Copernican system as elaborated by Kepler destroyed the perfect circular motion once and for all.

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