## Greek Cosmology The First Stage

The fourth and third centuries b.c. marked the emergence and flowering of Greek mathematical science. The three greatest figures of Greek mathematicsâ€” Euclid, Apollonius, and Archimedesâ€”worked during the period from 320 to 200 b.c. Their efforts were preceded in the fourth century by the seminal contributions of Thaetetus and Eudoxus of Rhodes. Eudoxus studied at Plato's Academy and went on to establish a school of mathematics in Cnidus in Asia Minor. Eudoxus created the proportion theory at the foundation of Greek mathematics, and he was also the one who founded geometric cosmology.

The basis of the Greek geometrical view of the universe is what is known as the two-sphere model, a conception suggested by Plato in the Timaeus and developed more formally by Eudoxus. The Earth is a very small sphere at the center of the universe, surrounded at an immense distance by a celestial sphere, on which lie the fixed stars. The celestial sphere rotates once every 24 hours, taking with it the fixed stars, the planets, the Moon, and the Sun on their daily circuits through the sky.

The sphericity of the Earth was a fact that was supported by several pieces of evidence. The mast of a ship sailing off in the distance is the last part of the ship to disappear from view, just as we would expect if it moved on a curved arc on the spherical Earth. An eclipse of the Moon occurs when the Sun, Earth, and Moon are aligned, and the zone of darkness as it passes across the Moon possesses a circular shape, apparently the result of the Moon passing into the shadow of the spherical Earth. It is possible to travel within the Mediterranean region a considerable distance from south to north. As one does so, changes are observed in the altitude of the Sun at noon and in the total length of day at different times of the year, observational facts that seem explicable only by assuming that the Earth is a sphere.

The celestial sphere was both a conceptual object that facilitated the measurement of the position of objects in the sky and a material body to which the stars were attached and that rotated daily. Today, in surveying and navigation the celestial sphere endures as a mathematical idealization useful in organizing line-of-sight observations. The Greek conception of it as a material body seems to have derived primarily from the fact of its daily rotation: the sphere moved as one would expect a rigid body to move, with the relative distances of the different parts remaining unchanged during the motion. Thus it was the diurnal motion of the heavens which led to the reification of what otherwise would have been a purely mathematical concept.

The two-sphere model of the universe was well established in Greek astronomical thinking by the beginning of the fourth century b.c. and is believed to have been the inspiration for a system of planetary models created by Eudoxus. Although Eudoxus was also responsible for fundamental contributions to mathematics, none of his original writings have survived. The basic idea of his planetary system was adopted by Aristotle, who wrote about it in his Metaphysics, and there is also an account of the system by the Aristotelian commentator Simplicius in the fifth century a.d. Modern interest in the Eudoxan spheres stems from the writings of the nineteenth-century Italian astronomer Giovanni Schiaparelli (1835-1910), who reconstructed the Eudoxan explanation of planetary motion.

The Eudoxan conception is known as the system of homocentric or concentric spheres. In a slightly simplified form it works as follows. Each celestial body is assumed to be moved by a set of spheres concentric with the Earth and all at the same distance from the Earth. In the case of the Sun, there are two motions to be modeled: the daily motion of the Sun westward in the sky and the much-slower annual motion of the Sun eastward on the ecliptic, that is, on the great circle it traces annually on the celestial sphere. The two motions are understood to result from the action of two rotating spheres, to which the Sun is affixed in some manner. One of the spheres produces the daily motion of the Sun westward in the sky; this motion coincides with the daily rotation of the celestial sphere. A second sphere produces the slower motion eastward of the Sun along the ecliptic, with a period of rotation equal to the sidereal period of the Sun, that is, the time it takes for the Sun to complete a 360-degree circuit of the ecliptic with respect to the fixed stars. The axes of rotation of the two spheres are inclined to each other at an angle of approximately 23 degrees. (Yet a third sphere was added to model the motion of the Sun, although its purpose is not clear.)

Similarly, two spheres produce the motion of the Moon. The first coincides with the daily rotation of the celestial sphere and produces the Moon's daily westward circuit of the sky, and the second carries the Moon eastward along the ecliptic, completing one rotation in 27 1/2 days, the Moon's sidereal period. A third sphere was added, apparently, to account for some variations in the Moon's motion with respect to the ecliptic.

The main difference between the planets on the one hand and the Moon and the Sun on the other is that the planets exhibit periodic retrograde motions in their passage eastward along the ecliptic. For the sake of simplicity we consider the case of the superior planets. Figure 3.1 depicts the path of Saturn along the ecliptic over a two-year period. The backward motion occurs around the time when Saturn is in opposition, that is to say, when it is 180 degrees opposite the Sun in the sky. Today, we are aware that retrogradation is an optical effect that results as the faster-moving Earth in its orbit about the Sun passes the slower-moving Saturn in its orbit. As Saturn is sighted against the distant stellar background, it appears to move backward for a while, with the midpoint of the retrogradation occurring at opposition.

ecliptic ecliptic

Eudoxus was able to come up with a geocentric model that at least qualitatively produced the retrograde motion of a planet. Consider the case of Saturn. He first introduced two spheres to produce the daily rotation and the eastward circuit of Saturn around the ecliptic; the first sphere has a rotational period of 24 hours, and the second sphere has a period of 29 1/2 years, the sidereal period of Saturn. A third and fourth sphere were introduced to account for the retrograde motion. These spheres rotate with equal and opposite angular velocities about axes that are tilted with respect to each other (see Figure 3.2). Consider a point that is initially on the intersection of the equators of the two spheres. It will be carried by the two motions in a figure eight-shaped curve, whose axis of symmetry lies perpendicular to one of the equators. If the two spheres are positioned so that this axis of symmetry lies along the ecliptic path of the planet, the effect of the combined action of the third and fourth spheres will be to superimpose the figure-eight-shaped motion on the steady eastward motion of the planet, producing the retrogradations of the planet that are periodically observed. The motion of the planet is thereby successfully modeled using a set of four spheres.

The Eudoxan system was adopted by Callipus (370-300 b.c.) and by Aristotle, both of whom further developed the conception and added some embellishments of their own. Aristotle supposed that the motion of the spheres for a given planet occurs as the result of the transference of motion from the outermost sphere inward. In order to make this work mechanically, it was believed necessary to introduce additional "counterturning" spheres to counteract the westward rotational motion of the outer sphere. This modification introduced some complications into the original Eudoxan conception. In the Eudoxan system, there were a total of 25 spheres, while in the final Aristotelian scheme, 64 spheres were required to make the system work.

The Eudoxus-Aristotelian system was the first geometrical attempt to model the motions of the planets, and it continued to be upheld by some writers well into the Middle Ages. Nevertheless, it possessed some serious defects that led to its abandonment by virtually all later astronomers of note. First, in this system the distance of each planet from the Earth always remains the same, a fact that seems to contradict the periodic and substantial variations in the observed brightness of the planets. Whatever the cause of a planet's brightness, it is difficult to account for its changes other than to suppose that they result from changes in distance of the planet from the Earth. Second, the mechanism to produce retrograde motion succeeds only in a crude qualitative way in accounting for this phenomenon; in the case of certain of the planets it does not succeed at all. The basic problem is that there is only one degree of

freedom, given by the angle of inclination between the axes of rotation of the two spheres that produce the retrogradation. This angle determines the width in latitude of the retrograde loops, and the latter is fixed once this width is given. There is then no flexibility in the model to produce the correct period relations for the planet's phases, its stationary points, opposition, and so on.

Despite its limitations, the Eudoxan system of homocentric spheres was a significant step forward in theorizing about the cosmos. It extended the geometric method of modeling evident in the two-sphere model to the motions of the Sun, Moon, and planets. There was now a complete geometric system of the heavens, consonant with the fundamental geometric outlook of Greek mathematical thought, which described a cosmos with a stationary Earth at the center of the universe. A crucial new intellectual element had entered into astronomy: each motion was revealed to result from a definite cause. For example, retrograde motion was (at least in principle) a consequence of the combined motion of two planetary spheres rotating in a specified way. The notion of causality was completely absent in Babylonian mathematical astronomy. Its emergence in Greek astronomy was connected to an interest in spatial geometrical modeling and was reflective, at a very general level, of the concern for the notion of cause in Greek philosophy and for deductive proof in Greek mathematics.

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