Main astronomical systems:

Eudoxus of Cnidus (1st half 4th C. B.C.E.), one of the greatest mathematicians of antiquity, developed the first geometrical theory of celestial motions, hitherto described by purely arithmetical models such as those used by the Babylonians. Eudoxus's system, like almost all ancient cosmological systems, is geocentric-geostatic, in other words, it posits an immobile Earth at the center of the universe. The system assumes the circularity and uniformity of celestial motions. Although the latter axiom was preserved—in name only—even by Ptolemy (mid-2nd C. C.E.), both axioms characterized all of ancient and medieval cosmology until Copernicus (1473-1543). But Eudoxus also based his system on the concentricity of celestial motions around the Earth. For each planet, the system postulated a certain number of concentric spheres "nested" one inside the other. As a result, it became known, many centuries later, as the system of homocentric spheres. The treatise On velocities in which Eudoxus expounded it was lost, like all his other writings; however, we know its broad outlines thanks to descriptions by later authors.

To the two luminaries, i.e., the Sun and the Moon, whose motions never display stations or regressions, Eudoxus assigned three spheres each. Of these, the outermost, rotating in 24 hours from east to west on the polar axis, explains the daily rising and setting of the celestial body. The innermost sphere, in whose equator the celestial body is "embedded," completes one revolution in one year for the Sun, and in one synodic month (period between two new moons) for the Moon. It thus represents the motion of the celestial body along the zodiac. In other words, Eudoxus believed that not only the Moon but also the Sun had uniform zodiacal motions. This was despite the fact that, in about 430 B.C.E., the Athenian astronomers Meton and Euctemon had demonstrated their non-uniformity when they had discovered that the seasons had different durations. The middle sphere explains the recession of the Moon's nodes and, probably, a presumed but non-existent phenomenon of latitudinal motion of the Sun.

For the five remaining planets, characterized by relatively brief periods of retrograde—i.e., westward—motion (called the "second anomaly"), the model provides four spheres each. The two outer ones explain, as with the Sun and Moon, the diurnal motion and the zodiacal motion (even here regarded as uniform). The two inner spheres, instead, move with equal velocity but in opposite directions. The result is that the planet, which lies on the equator of the innermost sphere, follows a to-and-fro motion along a closed, eight-shaped curve that Eudoxus named hippòpedon, i.e., "horse foot," from the name of an equestrian exercise (today, this type of curve is called lemniscate, i.e., "ribbon-shaped"; more precisely, in the present case, it is a spherical lemniscate lying on the surface of a sphere). The planet's velocity along the hippòpedon is not uniform, but increases as it approaches its central node. Thus the planet moves in the same direction (west to east) on the sphere of zodiacal motion when it travels along one half of the curve, and in the opposite direction when its travels along the other half. On the latter half lies a point in which, for the terrestrial observer, the plane's velocity is equal to that of the sphere of the mean motion, and the planet will seem to be moving westward (i.e., in retrograde motion) for a while. For Mercury and Venus, the hippòpedon was always centered on the Sun, a construct that explained why these two planets never diverged from the Sun by more than a given angular value.

A generation later, Callippus of Cyzicus perfected Eudoxus's system by introducing seven additional spheres: two for the Sun and two for the Moon to account for the elliptical anomaly, and one each for Mercury, Venus, and Mars, to allow their regression without assuming values for the rotation velocity of the spheres that would be incompatible with the observation data.

It is not known whether Eudoxus regarded the spheres of his system as physical realities or merely as a mathematical expedient to compute planetary positions. What is certain is that Aristotle (384-322 B.C.E.) adapted the Eudoxus-Callippus system by introducing 22 other spheres (for a total of 55!), whose velocities and rotation directions were selected so as to cancel the motion of the overlying planet when necessary. This approach unified the individual planetary models into a single great mechanism in which motion was propagated from the periphery to the center—in accordance with the principles of Aristotelian physics.

Notwithstanding the authoritative endorsement by Aristotle, and although the crystal-sphere concept would not be abandoned until the late sixteenth century, Eudoxus's elegant system soon gave way to the epicyclical theory. His construct had been unable to account for some obvious observational phenomena such as the increase in the planets' brightness when nearing opposition, or the fact that for the inferior planets the time elapsed between the western elongation and the eastern elongation is considerably longer than the time between the eastern elongation and the western one.

The geocentric system was outlined by Eudoxus (1st half 4th C. B.C.E.), who saw the universe as a set of concentric solid spheres (each carrying one planet) revolving one over the other in uniform motion. Aristotle (384-322 B.C.E.) facilitated its definitive formulation. He established that the universe was divided into two clearly separate zones: the superlunar world, characterized by crystal spheres of absolute perfection, and the sublunar world, theater of the perpetual changes in the four elements (earth, water, air, and fire). In the second century C.E., Ptolemy elaborated this cosmology with the aid of a complex mathematical structure. The Ptolemaic system was almost universally accepted until the late sixteenth century. To reconcile the astronomical hypotheses with observational data, Ptolemy resorted to ingenious geometrical solutions such as epicycles and equants, which made his system's structure extremely complicated. His basic assumption was that the Earth is at rest at the center of the universe. Around it, in ever wider circular orbits, the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn revolve at a constant, uniform pace. The planetary spheres are surrounded by the heaven of fixed stars, whose rotation is driven by the Primum Mobile (the starless, swift-moving ninth heaven).

Contrary to the Aristotelian-Ptolemaic system, the Copernican system places the Sun at the heart of the universe, making it the center of the revolutions of the planets, all moving at constant velocity. Promoted by authoritative ancient astronomers, such as Aristarchus and Hipparchus (3rd-2nd C. B.C.E.), the heliocentric system was revived by Copernicus (1473-1543) in the mid-sixteenth century. But it did not win final acceptance until after a long, strenuous confrontation with the die-hard supporters of the geocentric hypothesis. The Copernican system upset the structure of the universe envisaged by the Ptolemaic system, without, however, simplifying its complex geometrical arrangements. Copernicus actually preserved the solid crystal spheres and the division of the universe into two distinct parts: the superlunar world (world of perfection) and the sublunar world, theater of perpetual change. The Copernican system places the Sun in the central position; around it orbit Mercury, Venus, the Earth (around which orbits the Moon), Mars, Jupiter, and Saturn. For Copernicus, the spheres of all the planets then known were enclosed by the immobile sphere of the fixed stars.

Tycho Brahe built the great astronomical observatory of Uraniborg in Hveen (Denmark), where he observed the heavens with very large instruments for more than 20 years. Tycho was convinced of the Earth's absolute immobility, as demonstrated by the fact that a stone dropped from the top of a tower falls at its base. If the Earth revolved on its axis, Tycho argued, the stone should fall to the west of the tower. However, he rejected the Ptolemaic system, in which phenomena such as comets were deemed to occur inside the Earth's atmosphere. In 1572, a nova (new star) appeared in the constellation of Cassiopeia. Tycho sought to measure its distance by means of triangulation. The narrowness of the angle he found proved that the nova lay far beyond the Moon. With the same method, Tycho demonstrated that even the comet of 1577 moved beyond the sphere of the Moon, on a circular orbit around the Sun, with a non-uniform motion. Tycho examined the Sun's annual path and concluded that it moved uniformly on a circumference eccentric to the Earth. For the monthly path of the Moon, instead, Tycho devised a model comprising five circles in uniform rotation; using the model, he defined the Moon's position relative to the Sun and the stars with unprecedented precision. In 1582, the results of new triangulations led Tycho to assert that Mars in opposition to the Sun was closer to the Earth than the Sun itself. He concluded that Mars, too, revolved around the Sun. But since the Martian orbit intersected the solar orbit, Tycho deduced that Mars could not lie on a solid crystal sphere. He accordingly eliminated the celestial spheres of the Aristotelian-Ptolemaic tradition, proclaiming the fluidity of the heavens. Tycho extended the circumsolar motion of Mars to the other planets. Under the sphere of the fixed stars, even Mercury, Venus, Jupiter, and Saturn revolved around the Sun, which, at the same time, carried them with it around the immobile Earth.