The Music of Pythagoras (39 page)

For complicated reasons involving Danish politics and personal issues, Tycho Brahe eventually abandoned this remarkable, beloved palace, and Denmark, and went into exile—the exile that made it possible for him to meet Johannes Kepler.

Sixteenth and Seventeenth Centuries

A
S THE LAST DECADE OF THE
sixteenth century began, the two-thousand-year-old Pythagorean dream of rationality, unity, and the power of numbers was about to be given a serious test. Pythagoras and his followers had been sure they had caught a glimpse, as through a crack or a keyhole, of truth based on numbers that lay beyond the façade of nature. Johannes Kepler would force the door wide open, once and for all. After him, ironically, and though Kepler did not intend it to be so, the Pythagorean concept of the music of the spheres would survive only in poetic imagery. Yet in a profound and magnificent way, the faith embodied in that concept—faith in a wondrously rational and ordered universe—tempered by Kepler’s imaginative genius and rigorous mathematics, would finally place real examples of that music under the feet of science.

The higher seminary at Maulbronn, which Kepler attended in the 1580s as a troubled but exuberantly intellectual and religious teenager, taught “spherics” and arithmetic, but it was not until he enrolled at the University of Tübingen that he encountered astronomy. The mission of the Stift at the university where Kepler studied and had his lodgings was to prepare young men for careers of service to the Duke of Württemberg or for the Lutheran clergy, but the course of study was broadly
focused. The conviction that there was a unity to all knowledge lived on in the “Philippist” curriculum at the great Lutheran universities after the Reformation, as it had in the classical and medieval quadrivium and trivium and in humanist thinking. “Philippist” referred to the educational philosophy of Martin Luther’s disciple and friend Philipp Melanchthon, who had insisted that one could not truly comprehend and master any part of knowledge unless one comprehended and mastered the whole of it—a sentiment the Pythagorean Archytas would have applauded. Melanchthon felt the church could not succeed in teaching the path to salvation unless it produced a well-read scholarly clergy thoroughly grounded in the liberal arts. Reading the Scriptures, the church fathers, and the classical philosophers required facility in Hebrew, Latin, and Greek. Arithmetic and geometry were necessary for comprehension of both the secular and the sacred aspects of the world, and astronomy was the most heavenly of the sciences. Philippist philosophy also held that since the cosmos was orderly and harmonious, one could, and should, not only observe and record things but also hypothesize about them.

Early in his university career, Kepler realized that theology, mathematics, and astronomy would all be essential in his personal search for truth. He never ceased to be a devoutly religious man, but, as he later wrote, he believed that “God also wants to be known through the Book of Nature.” Perhaps it was in that interest (Kepler would have thought so) that God had placed a superb professor of mathematics and astronomy at the University of Tübingen: Michael Mästlin.

When Kepler first arrived there in 1589, forty-six years had passed since the death of Copernicus and the publication of his
De revolutionibus
in 1543. Many scholars were finding Copernicus’ grasp of celestial mechanics and his mathematics invaluable, while choosing to ignore his rearrangement of the cosmos. The University of Tübingen still officially taught Ptolemaic astronomy, and Michael Mästlin made sure his pupils had a good grounding in that, for which Kepler would later be grateful when he sought to overturn it. But Mästlin believed that Copernicus’ system had to be taken literally and that the planets and the Earth do, indeed, orbit the Sun. Kepler also read Nicholas of Cusa and was soon writing: “I have by degrees—partly out of Mästlin’s lectures, partly out of myself—collected all the mathematical advantages which Copernicus has over Ptolemy.” In a letter he wrote later to Mästlin, Kepler called Pythagoras the “grandfather of all Copernicans.”
¹

Johannes Kepler

During his university years, Kepler rapidly became well-read in the classics and also encountered neo-Platonic/Pythagorean thinkers of his own era. He gave all of it a religious and Pythagorean spin of his own: A universe created by God must surely be the perfect expression of a profound hidden order, harmony, simplicity, and symmetry, no matter how complicated and confusing it might appear to people who, like himself, were only beginning to understand it. This was the conviction that set fire to his spiritual and scientific imagination, and that flame would last him a lifetime. He was about to pin this idea to the wall using more precise observations of the heavens and his innate genius for rigorous mathematics: a potent combination.

While still a student at Tübingen, Kepler openly defended Copernican astronomy in two formal debates, arguing that the planets’ periods and their distances from the Sun made far better sense in the Copernican system; and that if the Sun was indeed (like the Creator) the source of all change and motion, then it might follow that the closer a planet was to the Sun, the faster it would travel. He worked busily and happily
on astronomical questions and wrote a piece about how the movements of the heavens would appear to someone on the Moon. Despite all that, it seems not to have occurred to him that he might pursue any career other than as a clergyman.

Near the end of his fifth university year, Kepler learned that his time at Tübingen was to end immediately, and not in the way he had planned. A Protestant school in southern Austria appealed to the university for a teacher, mainly for mathematics but also with knowledge of history and Greek. Tübingen had decided to send Kepler. Sorely discouraged and frustrated, he made the move to Graz. It was there that, about a year after his arrival, while drawing a diagram on the board for his pupils, he made the startling discovery that a triangle seemed somehow to be dictating the distance between the orbits of Jupiter and Saturn. The triangle was the Pythagorean
tetractus
.

The date was January 19, 1595, and Kepler was lecturing about the Great Conjunctions that occur when Jupiter and Saturn, as viewed from the Earth, appear to pass each other. This does not happen often in anyone’s lifetime, for Jupiter overtakes Saturn only approximately every twenty years. Imagine the two planets moving on a great circular belt around the Earth. During the twenty-year interval between two Great Conjunctions, Saturn moves about two thirds of the way around the belt, while Jupiter makes one complete revolution and two thirds of another. The locations of the Great Conjunctions leap forward on the belt by two thirds of the circle every twenty years.

Kepler had drawn a circle on the chalkboard to represent the great circle of the zodiac belt, and then marked the points in the zodiac where the successive Great Conjunctions occurred, viewed from Earth. If one plotted only three Great Conjunctions, those points were very near to being the corners of an equilateral triangle, but not quite. Beginning another triangle where the first ended (plotting the next conjunctions), the new triangle did not precisely retrace the first one. For example, the fourth conjunction in Kepler’s drawing (the conjunction that occurred in the year 1643) happened at almost the same point as the first (in 1583), and the fifth at almost the same point as the second. Draw lines connecting them and you almost have an equilateral triangle . . . but, again, not quite, and you have not retraced the first triangle. So the triangle “rotates,” as Kepler’s diagram shows. The result is two circles, outer and inner, with the distance between them set by the rotating triangle. Thus, Kepler’s triangle seemed to be mysteriously dictating the distance between the orbits of the first two planets. Interestingly, the radius of the inner circle looked as though it were half that of the outer circle, and observations of the heavens showed that the radius of Jupiter’s orbit was approximately half the radius of Saturn’s.

Drawing from Kepler’s
Mysterium cosmographicum
depicting the pattern of Jupiter-Saturn conjunctions and where they happened in the zodiac. The conjunction in 1583 (right) occurred when the two planets were in Aries/Pisces. The conjunction in 1603 (lower left) was in Sagittarius, in 1623 in Leo, in 1643 in Aries, in 1663 in Sagittarius, and so on. If the conjunctions occurred repeatedly in the same positions in the zodiac, Kepler’s drawing would have looked like the insert (upper right). Instead they “progress,” as represented in the central figure.

An amazed Kepler decided immediately to try the next regular polygon—the square (the triangle has three sides, the square four)—to
see whether it would serve similarly for the separation between the orbits of Jupiter and Mars.
*
If it did, he planned to try a pentagon (five sides) for the separation between the orbits of Mars and Earth, a hexagon for Earth and Venus, and so forth. He hoped the arrangement of the cosmos would resemble this diagram, with the triangle, then the square, then the pentagon, then the hexagon, and so forth, all nested between the separate planetary orbits. The idea failed on the first try, when the square would not work for the known separation between the orbits of Jupiter and Mars.

Kepler experimented with other regular polygons, searching for a fit, but he realized that given the infinite number of polygons available, success was assured. To the early Pythagoreans, this might have seemed adequate. Not to Kepler, for the question remained, why—among all the possibilities—
these
particular polygons worked and not others. Why had God chosen to construct the universe in this way and not in some other?

Though many of his contemporaries considered questions like these naive, they bothered Kepler, who had already been focusing his thinking along two lines of investigation: what reasoning God was using when he made things the way they are; and the physical reasons why the universe operates as it does. Clearly, for Kepler, shuffling through all the polygons and finding five that fit neatly between the six planetary orbits was not satisfactory. Since there were regular polygons to fit any planetary distances one might find, he felt there had to be a scheme that would
limit
the
actual, possible ratios (Saturn to Jupiter, Jupiter to Mars, Mars to Earth, Earth to Venus, Venus to Mercury), accounting for why some ratios, not others, existed in the heavens and there were only six planets.

It occurred to Kepler that he was making a mistake in trying to apply two-dimensional, flat figures (polygons) to a three-dimensional universe, and he decided to experiment instead with solid figures, the regular polyhedra.
*
That thought was a Pythagorean knockout. There were, after all, only five regular polyhedra (the Pythagorean or Platonic solids) not an infinite number of possibilities. To Kepler’s immense satisfaction, he found he could fit the five polyhedra into a nested arrangement that quite nicely coincided with the known separations between the six “spheres” in which the planets orbit.
^†