To Explain the World: The Discovery of Modern Science (24 page)

This scheme dictated the relative sizes of the orbits of all the planets, with no freedom to adjust the results, except by choosing the order of the five regular polyhedrons that fit into the spaces between the planets. There are 30 different ways of choosing the order of the regular polyhedrons,
*
so it is not surprising that
Kepler could find one way of choosing their order so that the predicted sizes of planetary orbits would roughly fit the results of Copernicus.

In fact, Kepler’s original scheme worked badly for Mercury, requiring Kepler to do some fudging, and only moderately well for the other planets.
*
But like many others at the time of the Renaissance, Kepler was deeply influenced by Platonic philosophy, and like Plato he was intrigued by the theorem that regular polyhedrons exist in only five possible shapes, leaving room for only six planets, including the Earth. He proudly proclaimed, “Now you have the reason for the number of planets!”

No one today would take seriously a scheme like Kepler’s, even if it had worked better. This is not because we have gotten over the old Platonic fascination with short lists of mathematically possible objects, like regular polyhedrons. There are other such short lists that continue to intrigue physicists. For instance, it is known that there are just four kinds of “numbers” for which a version of arithmetic including division is possible: the real numbers, complex numbers (involving the square root of −1), and more exotic quantities known as quaternions and octonions. Some physicists have expended much effort trying to incorporate quaternions and octonions as well as real and complex numbers in the fundamental laws of physics. What makes Kepler’s scheme
so foreign to us today is not his attempt to find some fundamental physical significance for the regular polyhedrons, but that he did this in the context of planetary orbits, which are just historical accidents. Whatever the fundamental laws of nature may be, we can be pretty sure now that they do not refer to the radii of planetary orbits.

This was not just stupidity on Kepler’s part. In his time no one knew (and Kepler did not believe) that the stars were suns with their own systems of planets, rather than just lights on a sphere somewhere outside the sphere of Saturn. The solar system was generally thought to be pretty much the whole universe, and to have been created at the beginning of time. It was perfectly natural then to suppose that the detailed structure of the solar system is as fundamental as anything else in nature.

We may be in a similar position in today’s theoretical physics. It is generally supposed that what we call the expanding universe, the enormous cloud of galaxies that we observe rushing apart uniformly in all directions, is the whole universe. We think that the constants of nature we measure, such as the masses of the various elementary particles, will eventually all be deduced from the yet unknown fundamental laws of nature. But it may be that what we call the expanding universe is just a small part of a much larger “multiverse,” containing many expanding parts like the one we observe, and that the constants of nature take different values in different parts of the multiverse. In this case, these constants are environmental parameters that will never be deduced from fundamental principles any more than we can deduce the distances of the planets from the Sun from fundamental principles. The best we could hope for would be an anthropic estimate. Of the billions of planets in our own galaxy, only a tiny minority have the right temperature and chemical composition to be suitable for life, but it is obvious that when life does begin and evolves into astronomers, they will find themselves on a planet belonging to this minority. So it is not really surprising that the planet on which we live is not twice or half as far from the Sun as the Earth actually is. In the same way, it seems likely that only
a tiny minority of the subuniverses in the multiverse would have physical constants that allow the evolution of life, but of course any scientists will find themselves in a subuniverse belonging to this minority. This had been offered as an explanation of the order of magnitude of the dark energy mentioned in
Chapter 8
, before dark energy was discovered.
¹⁴
All this, of course, is highly speculative, but it serves as a warning that in trying to understand the constants of nature we may face the same sort of disappointment Kepler faced in trying to explain the dimensions of the solar system.

Some distinguished physicists deplore the idea of a multiverse, because they cannot reconcile themselves to the possibility that there are constants of nature that can never be calculated. It is true that the multiverse idea may be all wrong, and so it would certainly be premature to give up the effort to calculate all the physical constants we know about. But it is no argument against the multiverse idea that it would make us unhappy not to be able to do these calculations. Whatever the final laws of nature may be, there is no reason to suppose that they are designed to make physicists happy.

At Graz Kepler began a correspondence with Tycho Brahe, who had read the
Mysterium Cosmographicum.
Tycho invited Kepler to visit him in Uraniborg, but Kepler thought it would be too far to go. Then in February 1600 Kepler accepted Tycho’s invitation to visit him in Prague, the capital since 1583 of the Holy Roman Empire. There Kepler began to study Tycho’s data, especially on the motions of Mars, and found a discrepancy of 0.13° between these data and the theory of Ptolemy.
*

Kepler and Tycho did not get along well, and Kepler returned to Graz. At just that time Protestants were being expelled from Graz, and in August 1600 Kepler and his family were forced to leave. Back in Prague, Kepler began a collaboration with Tycho, working on the Rudolphine Tables, the new set of astronomical tables intended to replace Reinhold’s Prutenic Tables. After Tycho died in 1601, Kepler’s career problems were solved for a while by his appointment as Tycho’s successor as court mathematician to the emperor Rudolph II.

The emperor was enthusiastic about astrology, so Kepler’s duties as court mathematician included the casting of horoscopes. This was an activity in which he had been employed since his student days at Tübingen, despite his own skepticism about astrological prediction. Fortunately, he also had time to pursue real science. In 1604 he observed a new star in the constellation Ophiuchus, the last supernova seen in or near our galaxy until 1987. In the same year he published
Astronomiae Pars Optica
(
The Optical Part of Astronomy
), a work on optical theory and its applications to astronomy, including the effect of refraction in the atmosphere on observations of the planets.

Kepler continued work on the motions of planets, trying and failing to reconcile Tycho’s precise data with Copernican theory by adding eccentrics, epicycles, and equants. Kepler had finished this work by 1605, but publication was held up by a squabble with the heirs of Tycho. Finally in 1609 Kepler published his results in
Astronomia Nova
(
New Astronomy Founded on Causes, or Celestial Physics Expounded in a Commentary on the Movements of Mars
).

Part III of
Astronomia Nova
made a major improvement in the Copernican theory by introducing an equant and eccentric for the Earth, so that there is a point on the other side of the center of the Earth’s orbit from the Sun around which the line to the Earth rotates at a constant rate. This removed most of the discrepancies that had bedeviled planetary theories since the time of Ptolemy, but Tycho’s data were good enough so that Kepler
could see that there were still some conflicts between theory and observation.

At some point Kepler became convinced that the task was hopeless, and that he had to abandon the assumption, common to Plato, Aristotle, Ptolemy, Copernicus, and Tycho, that planets move on orbits composed of circles. Instead, he concluded that planetary orbits have an oval shape. Finally, in Chapter 58 (of 70 chapters) of
Astronomia Nova
, Kepler made this precise. In what later became known as Kepler’s first law, he concluded that planets (including the Earth) move on ellipses, with the Sun at a focus, not at the center. Just as a circle is completely described (apart from its location) by a single number, its radius, any ellipse can be completely described (aside from its location and orientation) by two numbers, which can be taken as the lengths of its longer and shorter axes, or equivalently as the length of the longer axis and a number known as the “eccentricity,” which tells us how different the major and minor axes are. (See
Technical Note 18
.) The two foci of an ellipse are points on the longer axis, evenly spaced around the center, with a separation from each other equal to the eccentricity times the length of the longer axis of the ellipse. For zero eccentricity, the two axes of the ellipse have equal length, the two foci merge to a single central point, and the ellipse degenerates into a circle.

In fact, the orbits of all the planets known to Kepler have small eccentricities, as shown in the following table of modern values (projected back to the year 1900):

This is why simplified versions of the Copernican and Ptolemaic theories (with no epicycles in the Copernican theory and only one epicyle for each of the five planets in the Ptolemaic theory) would have worked pretty well.
*

The replacement of circles with ellipses had another far-reaching implication. Circles can be generated by the rotation of spheres, but there is no solid body whose rotation can produce an ellipse. This, together with Tycho’s conclusions from the comet of 1577, went far to discredit the old idea that planets are carried on revolving spheres, an idea that Kepler himself had assumed in the
Mysterium Cosmographicum.
Instead, Kepler and his successors now conceived of planets as traveling on freestanding orbits in empty space.

The calculations reported in
Astronomia Nova
also used what later became known as Kepler’s second law, though this law was not clearly stated until 1621, in his
Epitome of Copernican Astronomy.
The second law tells how the speed of a planet changes as the planet moves around its orbit. It states that as the planet moves, the line between the Sun and the planet sweeps out equal areas in equal times. A planet has to move farther along its orbit to sweep out a given area when it is near the Sun than when it is far from the Sun, so Kepler’s second law has the consequence that each planet must move faster the closer it comes to the Sun. Aside from tiny corrections proportional to the square of the eccentricity, Kepler’s second law is the same as the statement that the line to the planet from the
other
focus (the one where the Sun isn’t) turns at a constant rate—that is, it turns by the same angle
in every second. (See
Technical Note 21
.) Thus to a good approximation, Kepler’s second law gives the same planetary velocities as the old idea of an equant, a point on the opposite side of the center of the circle from the Sun (or, for Ptolemy, from the Earth), and at the same distance from the center, around which the line to the planet turns at a constant rate. The equant was thus revealed as nothing but the empty focus of the ellipse. Only Tycho’s superb data for Mars allowed Kepler to conclude that eccentrics and equants are not enough; circular orbits had to be replaced with ellipses.
¹⁵

The second law also had profound applications, at least for Kepler. In
Mysterium Cosmographicum
Kepler had conceived of the planets as being moved by a “motive soul.” But now, with the speed of each planet found to decrease as its distance from the Sun increases, Kepler instead concluded that the planets are impelled in their orbits by some sort of force radiating from the Sun:

If you substitute the word “force” [
vis
] for the word “soul” [
anima
], you have the very principle on which the celestial physics in the
Commentary on Mars
[
Astronomia Nova
] is based. For I formerly believed completely that the cause moving the planets is a soul, having indeed been imbued with the teaching of J. C. Scaliger
*
on motive intelligences. But when I recognized that this motive cause grows weaker as the distance from the Sun increases, just as the light of the Sun is attenuated, I concluded that this force must be as it were corporeal.
¹⁶

Of course, the planets continue in their motion not because of a force radiating from the Sun, but rather because there is nothing to drain their momentum. But they are held in their orbits rather than flying off into interstellar space by a force radiating from the Sun, the force of gravitation, so Kepler was not entirely wrong. The idea of force at a distance was becoming popular at this
time, partly owing to the work on magnetism by the president of the Royal College of Surgeons and court physician to Elizabeth I, William Gilbert, to whom Kepler referred. If by “soul” Kepler had meant anything like its usual meaning, then the transition from a “physics” based on souls to one based on forces was an essential step in ending the ancient mingling of religion with natural science.

Astronomia Nova
was not written with the aim of avoiding controversy. By using the word “physics” in the full title, Kepler was throwing out a challenge to the old idea, popular among followers of Aristotle, that astronomy should concern itself only with the mathematical description of appearances, while for true understanding one must turn to physics—that is, to the physics of Aristotle. Kepler was staking out a claim that it is astronomers like himself who do true physics. In fact, much of Kepler’s thinking was inspired by a mistaken physical idea, that the Sun drives the planets around their orbits, by a force similar to magnetism.