One mathematician tried to reconcile this belief with the Copernican theory. In 1568, Melanchthon’s son-in-law Caspar Peucer, professor of mathematics at Wittenberg, argued in Hypotyposes orbium coelestium that it should be possible by a mathematical transformation to rewrite the theory of Copernicus in a form in which the Earth rather than the Sun is stationary. This is precisely the result achieved later by one of Peucer’s students, Tycho Brahe.
Tycho Brahe was the most proficient astronomical observer in history before the introduction of the telescope, and the author of the most plausible alternative to the theory of Copernicus. Born in 1546 in the province of Skåne, now in southern Sweden but until 1658 part of Denmark, Tycho was a son of a Danish nobleman. He was educated at the University of Copenhagen, where in 1560 he became excited by the successful prediction of a partial solar eclipse. He moved on to universities in Germany and Switzerland, at Leipzig, Wittenberg, Rostock, Basel, and Augsburg. During these years he studied the Prutenic Tables and was impressed by the fact that these tables predicted the date of the 1563 conjunction of Saturn and Jupiter to within a few days, while the older Alfonsine Tables were off by several months.
Back in Denmark, Tycho settled for a while in his uncle’s house at Herrevad in Skåne. There in 1572 he observed in the constellation Cassiopeia what he called a “new star.” (It is now recognized as the thermonuclear explosion, known as a type Ia supernova, of a preexisting star. The remnant of this explosion was discovered by radio astronomers in 1952 and found to be at a distance of about 9,000 light-years, too far for the star to have been seen without a telescope before the explosion.) Tycho observed the new star for months, using a sextant of his own construction, and found that it did not exhibit any diurnal parallax, the daily shift in position among the stars of the sort that would be expected to be caused by the rotation of the Earth (or the daily revolution around the Earth of everything else) if the new star were as close as the Moon, or closer. (See Technical Note 20.) He concluded, “This new star is not located in the upper regions of the air just under the lunar orb, nor in any place closer to Earth . . . but far above the sphere of the Moon in the very heavens.”10 This was a direct contradiction of the principle of Aristotle that the heavens beyond the orbit of the Moon can undergo no change, and it made Tycho famous.
In 1576 the Danish king Frederick II gave Tycho the lordship of the small island of Hven, in the strait between Skåne and the large Danish island of Zealand, along with a pension to support the building and maintenance of a residence and scientific establishment on Hven. There Tycho built Uraniborg, which included an observatory, library, chemical laboratory, and printing press. It was decorated with portraits of past astronomers—Hipparchus, Ptolemy, al-Battani, and Copernicus—and of a patron of the sciences: William IV, landgrave of Hesse-Cassel. On Hven Tycho trained assistants, and immediately began observations.
Already in 1577 Tycho observed a comet, and found that it had no observable diurnal parallax. Not only did this show, again contra Aristotle, that there was change in the heavens beyond the orbit of the Moon. Now Tycho could also conclude that the path of the comet would have taken it right through either Aristotle’s supposed homocentric spheres or the spheres of the Ptolemaic theory. (This, of course, would be a problem only if the spheres were conceived as hard solids. This was the teaching of Aristotle, which we saw in Chapter 8 had been carried over to the Ptolemaic theory by the Hellenistic astronomers Adrastus and Theon. The idea of hard spheres was revived in early modern times,11 not long before Tycho ruled it out.) Comets occur more frequently than supernovas, and Tycho was able to repeat these observations on other comets in the following years.
From 1583 on, Tycho worked on a new theory of the planets, based on the idea that the Earth is at rest, the Sun and Moon go around the Earth, and the five known planets go around the Sun. It was published in 1588 as Chapter 8 of Tycho’s book on the comet of 1577. In this theory the Earth is not supposed to be moving or rotating, so in addition to having slower motions, the Sun, Moon, planets, and stars all revolve around the Earth from east to west once a day. Some astronomers adopted instead a “semi-Tychonic” theory, in which the planets revolve around the Sun, the Sun revolves around the Earth, but the Earth rotates and the stars are at rest. (The first advocate of a semi-Tychonic theory was Nicolas Reymers Bär, although he would not have called it a semi-Tychonic system, for he claimed Tycho had stolen the original Tychonic system from him.)12
As mentioned several times above, the Tychonic theory is identical to the version of Ptolemy’s theory (never considered by Ptolemy) in which the deferents of the inner planets are taken to coincide with the orbit of the Sun around the Earth, and the epicycles of the outer planets have the same radius as the Sun’s orbit around the Earth. As far as the relative separations and velocities of the heavenly bodies are concerned, it is also equivalent to the theory of Copernicus, differing only in the point of view: a stationary Sun for Copernicus, or a stationary and nonrotating Earth for Tycho. Regarding observations, Tycho’s theory had the advantage that it automatically predicted no annual stellar parallax, without having to assume that the stars are very much farther from Earth than the Sun or planets (which, of course, we now know they are). It also made unnecessary Oresme’s answer to the classic problem that had misled Ptolemy and Buridan: that objects thrown upward would seemingly be left behind by a rotating or moving Earth.
For the future of astronomy, the most important contribution of Tycho was not his theory, but the unprecedented accuracy of his observations. When I visited Hven in the 1970s, I found no sign of Tycho’s buildings, but there, still in the ground, were the massive stone foundations to which Tycho had anchored his instruments. (A museum and formal gardens have been put in place since my visit.) With these instruments, Tycho was able to locate objects in the sky with an uncertainty of only 1/15°. Also at the site of Uraniborg stands a granite statue, carved in 1936 by Ivar Johnsson, showing Tycho in a posture appropriate for an astronomer, facing up into the sky.13
Tycho’s patron Frederick II died in 1588. He was succeeded by Christian IV, whom Danes today regard as one of their greatest kings, but who unfortunately had little interest in supporting work on astronomy. Tycho’s last observations from Hven were made in 1597; he then left on a journey that took him to Hamburg, Dresden, Wittenberg, and Prague. In Prague, he became the imperial mathematician to the Holy Roman Emperor Rudolph II and started work on a new set of astronomical tables, the Rudolphine Tables. After Tycho’s death in 1601, this work was continued by Kepler.
Johannes Kepler was the first to understand the nature of the departures from uniform circular motion that had puzzled astronomers since the time of Plato. As a five-year-old he was inspired by the sight of the comet of 1577, the first comet that Tycho had studied from his new observatory on Hven. Kepler attended the University of Tübingen, which under the leadership of Melanchthon had become eminent in theology and mathematics. At Tübingen Kepler studied both of these subjects, but became more interested in mathematics. He learned about the theory of Copernicus from the Tübingen mathematics professor Michael Mästlin and became convinced of its truth.
In 1594 Kepler was hired to teach mathematics at a Lutheran school in Graz, in southern Austria. It was there that he published his first original work, the Mysterium Cosmographicum (Mystery of the Description of the Cosmos). As we have seen, one advantage of the theory of Copernicus was that it allowed astronomical observations to be used to find unique results for the order of planets outward from the Sun and for the sizes of their orbits. As was still common at the time, Kepler in this work conceived these orbits to be the circles traced out by planets carried on transparent spheres, revolving in the Copernican theory around the Sun. These spheres were not strictly two-dimensional surfaces, but thin shells whose inner and outer radii were taken to be the minimum and maximum distance of the planet from the Sun. Kepler conjectured that the radii of these spheres are constrained by an a priori condition, that
each sphere (other than the outermost sphere, of Saturn) just fits inside one of the five regular polyhedrons, and each sphere (other than the innermost sphere, of Mercury) just fits outside one of these regular polyhedrons. Specifically, in order outward from the Sun, Kepler placed (1) the sphere of Mercury, (2) then an octahedron, (3) the sphere of Venus, (4) an icosahedron, (5) the sphere of Earth, (6) a dodecahedron, (7) the sphere of Mars, (8) a tetrahedron, (9) the sphere of Jupiter, (10) a cube, and finally (11) the sphere of Saturn, all fitting together tightly.
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):
Planet
Eccentricity
Mercury
0.205615
To Explain the World: The Discovery of Modern Science Page 17