Both Tycho Brahe and Galileo Galilei used a relatively modern approach to scientific problems. Brahe insisted on the importance of systematic, extensive, and accurate collection of data. Galileo was not content simply to make observations, but recognized the necessity to alter experimental conditions in order to eliminate extraneous factors. Both men were very rational and logical in their approach to their work. Copernicus, on the other hand, and Kepler (of whom more will be said later) even more so, were concerned with the philosophical implications of their work. Kepler was well aware that the ancient Pythagoreans had set great store on finding simple numerical relationships among phenomena. It was the Pythagoreans who recognized that musical chords are much more pleasing if the pitches of their component notes are harmonics, that is, simple multiples of each other. The Pythagoreans also discovered that the pitch of notes from a lyre string, for example, are simply related to the length of the string, the

Platonic solids

Figure 2.8. The Platonic solids. From left to right: tetrahedron, cube, octahedron, dodecahedron, icosahedron.

Platonic solids

Figure 2.8. The Platonic solids. From left to right: tetrahedron, cube, octahedron, dodecahedron, icosahedron.

higher harmonics being obtained by changing the length of the string in simple fractions. Turning to the heavens, they asserted that each planetary sphere, as well as the sphere of the stars, emits its own characteristic musical sound. Most humans never notice such sounds, if only because they have heard them continuously from birth. The spatial intervals between the spheres were also supposed to be proportional to musical intervals, according to some of the legends about the Pythagoreans.

Johannes Kepler was born some 28 years after Copernicus's great work was published. He loved to indulge in neo-Pythagorean speculations on the nature of the heavens. His personality was undoubtedly affected by an unhappy childhood, relatively poor health, and a lack of friends in his youth. He showed mathematical genius as an adolescent, and as a young man succeeded in obtaining positions as a teacher of mathematics. His mathematical ability was soon recognized, and as a result his deficiencies as a teacher and his personality problems were often overlooked. In fact, he was such a poor teacher that he did not have many students, and this allowed him more time for astronomical studies and speculations.

Kepler was convinced of the essential validity of the heliocentric theory, and looked upon the Earth as one of the six planets that orbited the Sun. Because of his fascination with numbers, he wondered why there were six planets rather than some other number, why they were spaced as they were, and why they moved at their particular speeds. Seeking a connection to geometry, he recalled that the Greek mathematicians had proved there were only five ' 'perfect'' solids, known as the "Pythagorean" or "Platonic" solids, aside from a sphere.

A perfect solid is a multifaced figure, each of whose faces is identical to the other faces. Each of the faces is itself a perfect figure (see Fig. 2.8). A cube has six faces, all squares; a tetrahedron has four faces, all equilateral triangles; the eight faces of an octahedron are also equilateral triangles; the 12 faces of a dodecahedron are pentagons; and the 20 faces of the icosahedron are all equilateral triangles. Kepler proposed that the six spheres carrying the planets were part of a nested set of hollow solids, in which the spheres alternated with the Platonic solids. If the innermost and outermost solids are spheres, then there could be only six such spheres, corresponding to the number of known planets. The size of the five spaces between the spheres might be determined by carefully choosing the particular order of the Platonic solids, as shown in Fig. 2.9.

The largest sphere is the sphere of Saturn. Fitted to the inner surface of that

Figure 2.9. Kepler's nesting of spheres and Platonic solids to explain spacings and order of the planters. The lower drawing shows the spheres of Mars, Earth, Venus, and Mercury in more detail. Note the Sun at the center. (Top: Burndy Library; bottom: From illustrations in Kepler's Mysterium Cosmographicum.)

Figure 2.9. Kepler's nesting of spheres and Platonic solids to explain spacings and order of the planters. The lower drawing shows the spheres of Mars, Earth, Venus, and Mercury in more detail. Note the Sun at the center. (Top: Burndy Library; bottom: From illustrations in Kepler's Mysterium Cosmographicum.)

sphere is a cube with its comers touching the sphere. Inside the cube is the sphere of Jupiter, just large enough to be touching the cube at the middle of its faces. Inside the sphere of Jupiter is a tetrahedron, and inside that the sphere of Mars. Inside the sphere of Mars is the dodecahedron, then the sphere of the Earth, then the icosahedron, the sphere of Venus, then the octahedron, and finally the smallest sphere, the sphere of Mercury. With this order of nesting, the ratios of the diameters of the spheres are fairly close to the ratios of the average distances of the planets from the Sun.10 Of course, Kepler knew that the spheres of the planets were not homocentric on the Sun; Copernicus had found it necessary to introduce eccentrics in his calculations. Kepler therefore made the spheres into spherical shells thick enough to accommodate the meanderings of the planets from perfect circular motion about the Sun. The results did not quite fit the data, but came close enough that Kepler wanted to pursue the idea further. He published this scheme in a book called the Mysterium Cosmographicum (the Cosmic Mystery), which became very well known and established his reputation as an able and creative mathematician and astronomer.11

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