Eudoxus of Cnidos, who lived in the 4th century B.C.E., was the first to develop a geometrical model of celestial movements.

Convinced that the Earth was immobile at the center of the universe, he devised a system of four concentric spheres for each planet. For this reason, his concept was later known as the system of homocentric spheres.

The axis of each sphere was attached to the sphere immediately above. This ensured that the inner spheres were driven by the outer ones.

The first sphere, whose axis passed through the celestial poles, revolves in 24 hours from east to west, simulating the planet's daily rising and setting.

The axis of the sphere immediately below is inclined in such a way that its equator lies on the plane of the ecliptic.

This sphere revolves from west to east in the same period in which the planet travels across the Zodiac. It therefore reproduces the planet's mean movement.

The combination of these two spheres describes the planet's diurnal movement and its movement along the Zodiac, but not its retrograde movement. To explain the latter, Eudoxus used the two other spheres.

The planet—in this case Jupiter—is fixed on the equator of the innermost sphere, which completes a full rotation in the time interval between two successive retrogradations.

The upper sphere revolves with equal velocity, but in the opposite direction, thus offsetting the movement of the other sphere.

The two movements would cancel each other out if the axes of the spheres coincided; but the reciprocal inclination, set at an appropriate value for each planet, causes the planet to move to and fro along the ecliptic.

The path generated in this manner is actually an eight-shaped curve that Eudoxus called hippòpedon, i.e., "horse foot," from the name of an equestrian exercise.

The hippòpedon is the core of the mechanism: the planet effectively travels from west to east along one-half of the curve, and from east to west along the other half.

In other words, half of its travel is in the same direction as the sphere immediately above, the other half in the opposite direction.

Since the planet moves faster near the center of the hippòpedon than at its extremes, it will reach a point where its longitudinal velocity will exceed that of the mean movement, and the planet will briefly retrograde.

In keeping with the cosmological precepts of his time—i.e., using only circular and uniform movements centered on the immobile Earth—Eudoxus was thus able to account for the complicated paths that each planet displays for the terrestrial observer.