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Kepler learned about heliocentric astronomy at the University of Tubingen, where he studied first as a student in the faculty of arts and later as a clergyman in training in the faculty of theology. Tubingen was a leading center of Lutheran theology, and Kepler planned to pursue a career in the church. One of his professors in Tubingen was Michael Maestlin, a supporter of the Copernican system and a prominent astronomer in the last decades of the sixteenth century. From his first encounter with the Copernican system Kepler became an advocate for the new cosmology, about whose correctness he seems never to have had any doubts.

At the recommendation of the Tubingen authorities, Kepler, in 1594, took up a position teaching mathematics in Gradz, Austria, a post that also required him to dispense astrological advice and prepare an annual almanac. In the course of a lecture in mathematics he was struck by what he took to be a deep connection between the heliocentric orbits of the six planets in the Copernican system and the mathematical properties of geometrical solids. He developed this idea in his book Cosmographic Mystery (1596), a treatise important for being one of the first astronomical works written from an avowedly Copernican viewpoint. Consider a geometrical solid or polyhedron whose surface is composed of plane polygonal faces. If the polygonal faces are all congruent to each other, then the solid is said to be a regular polyhedron. (We consider only convex polyhedra, that is, those without indentations.) For example, the cube is a regular polyhedron with six faces, each face being a square. It turns out that there exist only five regular solids: the tetrahedron (4 triangular faces), the cube (6 square faces), the octahedron (8 triangular faces), the dodecahedron

(12 pentagonal faces), and the icosahedron (20 triangular faces). The last part of Euclid's great book on geometry, the Elements, was devoted to a demonstration of this remarkable fact. Because the philosopher Plato discussed the regular polyhedra, the five solids are sometimes called the Platonic solids.

Consider now the Copernican planetary system, consisting of the six planets revolving about the Sun in circular orbits. For Kepler the fact there were five Platonic solids and six planets was no coincidence, as he attempted to show in his planetary cosmology. In the sphere corresponding to the orbit of Saturn, inscribe a cube. In this cube, inscribe another sphere. It turns out that the latter is a very close fit to the orbital sphere of Jupiter. Within Jupiter's sphere, inscribe a tetrahedron, and within the tetrahedron, inscribe a sphere; doing so, we obtain the orbital sphere of Mars. Within the sphere of Mars, inscribe a dodecahedron, and inside it, inscribe a sphere, thereby obtaining the sphere of Earth. Within Earth's sphere we place an icosahedron, and the sphere inscribed in it contains the orbit of Venus. Finally, within the sphere of Venus, inscribe an octahedron, and within it, inscribe a sphere, obtaining in this final step the sphere of Mercury. Figure 5.3 is taken from Kepler's book and depicts the resulting cosmo-graphic system. For Kepler the nesting of the planetary spheres in terms of the Platonic solids was the key to the mystery of the planetary system.

Kepler spent a good deal of time and effort configuring various possible nestings of solids with spheres until he obtained one that worked. He was very proud of his geometric cosmology and arranged to have actual paper models built of the solids and nested planetary spheres. Although his subsequent research went off in different directions, he continued to believe throughout his career that he had discovered something important in his first book. Today, we are aware of the unsound and even fanciful character of the theory. In fact, there are nine planets and a host of minor planets or asteroids as well as more distant objects ranging beyond Pluto. There is no connection between the five Platonic solids and the planetary orbits, and the fitting that Kepler derived is fortuitous. Even for his own purposes and times, the cosmology could not serve as the basis for an empirical project to produce planetary tables. Johannes Praetorius, an older contemporary of Kepler and a believer in traditional geocentric cosmology, was critical of the Cosmographic Mystery:

But that speculation of the regular solids, what, I beg, does it offer to Astronomy? It can (he says) be useful for marking the limits or defining the order or magnitude of the celestial orbs, yet clearly the distance of the orbs are derived from another source, i.e., a posteriori, from the observations. And, having defined

these [distances] and shown that they agree with the regular solids, what does it matter? (Westman 1975, 303)

Despite its speculative character, Kepler's research on geometric cosmology possessed definite scientific value. Through it he gained valuable experience in the numerical analysis of the planetary orbits and dimensions of the Copernican system, experience that would serve him well in his later investigations. The Cosmographic Mystery also established Kepler as a young astronomer of promise and served to promote Copernicanism as an astronomical theory.

Kepler's most important book was the New Astronomy (1609), one of the great scientific classics of the seventeenth century. In it Kepler set about analyzing Tycho's observations of Mars in order to determine the exact geometric shape of its orbit. The book marked a clear break with medieval thinking about cosmology and helped to launch modern physical theories of the universe. It is written in an autobiographical style in which the various turns and false steps in the investigation are chronicled in lengthy detail. The book offers an unusually revealing study of the process of discovery in mathematical science, something that normally has to be reconstructed from draft notes or various pieces of circumstantial evidence. It is believed that Kepler wrote in this way so that a skeptical reader, critical of the new cosmology and celestial physics, would see for himself how one would be led to the remarkable conclusions of his investigation.

The eccentricity of Mars (the deviation of its orbit from a circle) is, with the exception of Mercury, larger than that of the other planets, and its motion is more difficult to reduce to a combination of circular motions. Kepler had gained access to Tycho's observations of Mars at a time when Tycho had restricted access to his full set of planetary observations. Kepler was fortunate to have found a problem that would lend itself to a solution by new methods and for which there was ample reliable data from which to work.

The New Astronomy marked a radical new approach to the analysis of planetary motion. In order to understand the nature of Kepler's innovation, it is helpful to consider earlier planetary models. Take as an example the Copernican model of the Earth's motion. The annual motion of the Sun occurs along a great circle on the celestial sphere, and the speed of the motion varies in a regular pattern. Assuming that the Earth moves on a circle with constant angular velocity, and assuming that the Sun is located at rest at a point offset from the center of the circle by a small distance, the observed solar motion will result. All we have to do is determine two parameters: an angle that gives the orientation of the line joining the center of the circle and the Sun with respect to the celestial sphere and the distance of this line as a fraction of the distance from the Earth to the Sun.

The eccentric-circle model successfully accounts for the observations, in the sense of showing that if the planet moves in the way specified in the model, then we will indeed see what we see: the model has excellent predictive value. The model also provides what would appear to be a very reasonable physical representation of how the planet actually moves in the heavens. However, there is no explanation of why or how the planet moves as it does in the model. There was the older concept of a planetary soul, according to which the planet possesses a sort of intelligent soul that powers and directs the different spheres involved with its motion. Copernicus was silent on this part of traditional cosmology, perhaps because he felt that the underlying conception of a soul contributed little to the primary goal of predictive astronomy. For him the planetary system resembled a rather complicated clockwork mechanism, in which the spring, or weight, and regulator were hidden from view; given that the various spheres were positioned in definite ways and rotated with specified speeds and periods, the observed motions of the parts would follow.

Unlike Copernicus, Kepler was working from a perspective in which the planetary spheres had been shown not to exist. He needed to investigate how it comes about that the planets moved in the way that they did. He seriously considered the possibility of a planetary soul or intelligence as the efficient cause of planetary motion. Each planet would possess both an innate power of motion and an intelligence that would direct this power. In the case of the Earth moving around the Sun the Earth's soul would use the apparent size of the Sun as an indicator, enabling it to maintain the proper distance to the Sun in its eccentric-circle orbit. (For Copernicus, this was unnecessary since the Earth was carried around on the eccentric sphere.) In order to account for Tycho's very accurate observations, Kepler found it necessary to use the equant to model the Earth's motion. This was a novel step since Ptolemy, in the equivalent solar model of the Almagest, had not used the equant, the latter being employed only for the five planets. Although Kepler found that a model with an equant was in fact not good enough to account for the observations, it served as a useful working hypothesis in his investigation. As more complicated motions were posited, it became very difficult to understand how a planetary intelligence would be able to perform the difficult calculations required in order to direct the motion.

To model the motion of Mars, Copernicus had replaced Ptolemy's equant by a secondary epicycle that produced a motion about as good in agreement with observation as Ptolemy's had been. In general, Kepler found the use of epicycles objectionable since it was unclear how the planetary intelligence could direct the motion of the center of the epicycle, which was only a mathematical point. He therefore reintroduced the equant, this time for the heliocentric orbit of Mars, and attempted to determine the exact relation between the Sun, the center of the orbit, and the equant. Even here, it was not easy to understand how the planetary intelligence could direct the motion of Mars in such a model.

To supplement the action of the soul, Kepler advanced the idea of a causal physical connection between the Sun and the Earth: the motion of the Earth about the Sun is seen to result from the physical action of the Sun on the Earth. There is a direct causal link between a central physical agent and the orbit described by the planet. As his investigation progressed, Kepler came to dispense with the intelligent soul altogether and cast his analysis solely in terms of the Sun's physical action on the planet. Historian E. J. Dijksterhuis

(1961, 310) sees Kepler's shift from the conception of an intelligent soul animating planetary motion to an inanimate process involving solar force to be one of the most profound moments in the mechanization of the world view that occurred during the Scientific Revolution. Like other scientists of the period, Kepler was influenced by William Gilbert's (1544-1603) On the Magnet (1600), a work that was notable for its methodical and experimental investigation of magnetic phenomena. Kepler conjectured that the Sun acts on the planets in a way analogous to the magnetic action of a lodestone on iron. The Sun rotates on an axis perpendicular to the plane of the ecliptic, setting in play a series of concentric, rotating filaments that propel the planets in their orbital trajectories. Through a process of reasoning involving some very questionable steps Kepler arrived at the conclusion that the line joining the Sun to the planet sweeps out equal areas in equal times. Using the area law, he deduced, after a series of attempts, that an ellipse with the Sun at one focus was the best fit to the orbit of Mars. His analysis of this planet contained two facts that he would later formulate as general laws of planetary motion: the area swept out by the radius from the Sun to the planet is a linear function of time; and the orbit of each planet is an ellipse with the Sun at one focus.

By a complicated route involving a mixture of geometric and physical reasoning Kepler had managed to derive facts of the utmost importance about the planetary orbits. Although he did not succeed in developing this idea into a coherent theory of celestial dynamics, he had at least initiated a line of investigation that would eventually lead to a satisfactory dynamical theory of planetary motion.

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