Tycho and Kepler

by Kitty Ferguson

  Tycho’s hook dangled enticingly before Kepler’s eyes. He was so overjoyed that he penned in the margin, “One can see his high opinion of my method.”

  There was, however, more to Tycho’s letter, a lengthy postscript containing what was, under the circumstances, a remarkably restrained complaint about Kepler’s unctuous praise of Ursus. Tycho wrote that he could not imagine that Kepler had been aware that Ursus would use his letter in his “defamatory and criminal publication.” He suggested that Kepler might give him a statement of his opinion of Ursus’s behavior, which document Tycho could use in legal proceedings against Ursus.

  Tycho had given Kepler a glimpse of the reward that might be his and had named the price of his forgiveness. Kepler also knew that in a separate letter to Mästlin, Tycho had criticized Mysterium much more severely and complained about the letter reprinted in Ursus’s book. Tycho surely anticipated that Kepler’s influential mentor would communicate with his errant pupil, and he was treading a delicate line between flattery and encouragement on the one hand and criticism and reprimand on the other.

  Even before he had read the copy of Tycho’s actual words to him, Kepler had taken Tycho’s bait on the strength of Mästlin’s correspondence. He had also not seen Ursus’s book and did not know whether Ursus had quoted him correctly or distorted what he had written. Kepler could not even remember precisely what he had written, nor did he know which letter Ursus had quoted, for there had been more than one, and he had not kept copies. In a blind panic about the appalling situation he had got himself into with one of the great men of the age, who actually was inclined to like his work, Kepler threw himself at Tycho’s feet in a long reply, written with characteristic eloquence and dramatic flair:

  Why does [Ursus]4 set such value on my flatteries? . . . If he were a man he would despise them, if he were wise he would not display them in the market place. The nobody, that I was then, searched for a famous man who would praise my new discovery. I begged him for a gift and behold, it was he who extorted a gift from the beggar. . . . My spirit was soaring and melting away with joy over the discovery I had just made. If, in the selfish desire to flatter him I blurted out words which exceeded my opinion of him, this is to be explained by the impulsiveness of youth.

  Tycho’s response this time5 was gracious, but he kept Kepler in his place by saying that he had not required such an elaborate apology.

  THAT WINTER, Tycho was sumptuously lodged in a famous old house that had once belonged to Melanchthon, with space and time for his studies and chemical work and for attending to his publications. Kirsten and their children joined him. He was surrounded by intellectual friends who set enormous value on his company, and he was temporarily not required to fawn over kings or emperors. Tycho knew that the success of his pilgrimage to seek imperial patronage depended on his continuing to appear frequently and splendidly in public, maintaining a reputation in his own field but also as someone with an extensive knowledge of natural philosophy and the arts. No situation could have suited him better. He was in his element.

  Even though Rudolph’s patronage seemed almost assured, Tycho had experienced the mercurial nature of kings and knew he could not take it for granted that the emperor would offer to fund him and his research permanently and in as generous a manner as he required. It might still happen that he would be greeted as a distinguished visitor, converse with the emperor, but only be given gold chains and medallions and finally have to return to Wandsburg. Worse yet, the emperor might grant him no audience at all. Power brokerage was such that the word of influential friends and informants at court was almost as good as a contract, but it was nothing signed, sealed, and delivered. Tycho, for all his assurances and letters, would still be approaching Rudolph as a supplicant. Hence Tycho continued to use every means possible to make his position in Prague stronger, not neglecting to curry favor in other courts as well. He dispatched Tengnagel across the Alps to present copies of Mechanica and the star catalog to rulers in Venice, Florence, and Parma. Again, Tengnagel did well for himself and for Tycho. The Order of San Marco in Venice gave him a knighthood, and he secured for Tycho the favor of the doge of Venice, Grand Duke Ferdinand de Medici in Florence, and the Farnese family of Parma, all of whom had close ties with the court in Prague.

  There had been a belated flicker of hope for Tycho in Denmark the previous autumn. King Christian had married Princess Anne Catherine of Brandenburg, and Viceroy Rantzau had attended the wedding. On their journey home in mid-December, the bride’s parents, Margrave Joachim Frederick and Margravine Catherine of Brandenburg-Küstrin, passed through Schleswig-Holstein, and Rantzau arranged for Tycho to meet them. So convincingly did Tycho and Rantzau make their case that the new royal in-laws agreed to take up Tycho’s cause with their son-in-law Christian. In January, the margrave became elector of Brandenburg, which meant he was even better positioned to exert influence. He and his wife wrote to Christian. They even took care to disguise their letters as personal family correspondence so that they would not be intercepted, for Tycho knew he had enemies at the Danish court and even suspected that the reply to his earlier appeal had not been written by King Christian himself. The margrave’s and margravine’s letters nevertheless fell into the wrong hands and were too much delayed to be of any use.

  By this time Tycho, with his eyes set on Prague, was less discouraged by this setback than by another having to do with his astronomy. He heard from Longomontanus, who had returned to his boyhood home in Jutland, that he had observed a lunar eclipse from there in late January, and it had occurred half an hour earlier than Tycho had predicted. Tycho’s plan to present a book about his lunar theory to the emperor when he had his first audience had to be scrapped in favor of another manuscript about daily solar and lunar positions for 1599. Nevertheless, Tycho’s hopes were finally reaching fruition. By March he had his summons: His Imperial Majesty desired to receive him in Prague.

  Tycho had bookbinders create a presentation copy of the book of solar and lunar positions. He wrote to Longomontanus, then in Rostock, asking him to come quickly and travel with him. He bought new, superb carriage horses. In his chemical laboratory he cooked up medicines against plague and other diseases to take with him as gifts and for his own use. The journey to Prague was delayed again when Magdalene became seriously ill, and then delayed yet again because the spring thaw and heavy rains made the roads impassable.

  At last, on June 14, 1599, Tycho and his entourage left Wittenberg, overnighted with lavish entertainment at Castle Pretzsch along the way, and arrived at Dresden, where Kirsten, their children, and the servants would wait in the household of a high Saxon official. Tycho, in his finest carriage with the new horses, accompanied only by his eldest son and servants—for Longomontanus had not yet arrived—proceeded to Prague. Early in July, as they approached the city, Ursus slipped away.

  IN GRAZ, Kepler heard about Tycho’s triumphal entry into Prague. He also heard about Ursus’s ignominious departure. He had no idea whether these events had any relevance to his own prospects. The only thing that could be said was that Tycho had come nearer. It was no longer completely out of the question that Kepler might someday travel to meet him. The successful continuation of Kepler’s work seemed to him increasingly to depend on being able to consult observations that only Tycho could supply.

  The summer of 1599 was a tragic one for the Keplers. Their second child, Susanna, born in June, lived only thirty-five days. Kepler refused a Catholic burial for his infant daughter and was fined for his stubbornness. When he appealed, the fine was lessened, but he still had to pay it and was forbidden to bury the tiny body until he did.

  As he grieved for his child, tried to comfort his wife, and worried about how long they could hold on in Graz, Kepler’s mind was also busy in a new and happier direction. He wrote to Edmund Bruce,6 an English acquaintance living in Padua, about a new theory he thought might interest Bruce’s friend Galileo, and he described the theory in letters to Mästlin and Herwart von Hohenbu
rg. In his search for answers to the questions that interested him, Kepler had begun to look to music.7 He was not being illogical, mystical, or, for that matter, original.

  One of the most profound and far-reaching intellectual advances in the history of human knowledge occurred in the sixth century B.C. when the Pythagoreans recognized that there are fundamental mathematical relationships hidden in nature. It was in music that they made this discovery, and it seemed to them, with ample reason, a glimpse into the mind of God. Plato also recognized that musical harmony is a manifestation of deep mathematical relationships, and he, as Kepler would later, speculated about the possibility that the arrangement of the cosmos might be another of those manifestations.

  Tycho had been thinking along the same lines when he designed Uraniborg: Plucking the string of a harp produces a musical tone—the “ground note” or “fundamental.” If one presses the string down at its center—creating, in effect, two strings each half as long as the original one—and plucks again, the tone is an octave higher than the fundamental. A ratio of string length one to two produces the octave; two to three produces a fifth; three to four, a fourth; four to five, a major third; five to six, a minor third; three to five, a major sixth; and five to eight, a minor sixth. The ancients preferred only the ratios 1:2:3:4, but Kepler learned of more modern music that used the other ratios as well. All these ratios produce musical intervals that human ears find “harmonious.” There is some deep connection between these lengths of string, these ratios, and the human mind.

  For Kepler this was not at root a matter of mathematics or numbers. He was no numerologist. For him, the most fundamental attribute of nature was geometry, a clearer and simpler demonstration than mathematics that some things are possible and others are not—that some things “fit,” and others do not. It was the link between geometry and musical harmony that intrigued him.

  Kepler’s polyhedral theory failed to explain two things about the solar system about which Kepler was particularly curious. The first was the size of the planetary “eccentricities.” In Ptolemy, when a planet’s orbit was not precisely centered on Earth, but rather on a point near Earth, the orbit was said to be “eccentric.” Similarly, a Copernican like Kepler called a planetary orbit eccentric if it seemed to be centered not precisely on the Sun but near it. Kepler wanted to know how eccentric the orbits of the planets were, but he also wanted to know the physical reasons why a planet had a certain degree of eccentricity and not another.

  The second question his polyhedral theory could not answer was how a planet’s distance from the Sun was related to the length of time it took to complete one orbit, known as its period. Kepler had discussed that relationship in Mysterium, but not to his satisfaction. Again, he wanted to know not just how they were related but also the physical causes why this must be so. Kepler continued to adhere to his belief that God’s creativity lay deeper than merely setting things up and moving them around. God, he was certain, had established an underlying logic and perfect harmony from which all things had to proceed. Kepler had already found that discovering pieces of this logic was pure delight, which he felt must echo God’s delight in creating. Since musical harmony was a manifestation of that same divine logic, it seemed a highly promising area in which to find links with the cosmos.

  His first results seemed to indicate that he was right. In his letters to Bruce, and to Mästlin in July 1599, and a little later to Herwart von Hohenburg, Kepler pointed out that his new “harmonic theory” gave more accurate predictions than a theory he had earlier included in Mysterium. He was suggesting that the planets, moving through something similar to air, made sounds the way the strings of a lyre do if the lyre is hung up so that the breeze moves across them. The velocities (the “vigor,” as he put it) of the six planets might be related to each other in the same relationships that would produce a harmonious chord if one translated them into lengths of strings on a stringed instrument. For example, say the relationship between the velocities of Saturn and Jupiter was 3:4. On a stringed instrument a 3:4 ratio between string lengths produced the musical interval of a fourth. It followed, said Kepler, that the “interval” between Saturn and Jupiter could be thought of as a musical fourth. Kepler worked out this scheme with all the planets’ velocities. He calculated the proportions of their velocities as follows: Saturn to Jupiter, 3:4—a fourth; Jupiter to Mars, 4:8 (or 1:2—an octave); Mars to Earth, 8:10 (4:5—a major third); Earth to Venus, 10:12 (5:6—a minor third); Venus to Mercury, 12:16 (3:4—a fourth). From all these intervals, Kepler built up a chord, a C major chord in what musicians call its “second inversion (see figure 15.1).” Kepler was not entirely pleased with this second inversion. He would have preferred the chord to be in root position—a C major chord with C as the lowest note. However, he wrote with a shrug, “thus it is in the heavens.”

  Figure 15.1: Kepler’s 1599 planetary chord.

  Having chosen the velocities with the goal of creating a harmonious chord, Kepler was encouraged when he found that the musical intervals were not far off from the spatial intervals between the planets in his polyhedral theory. The planetary periods—the amount of time it took each planet to complete its orbit—were well known. He set out to calculate how large the different planetary orbits had to be in relation to one another if the planets, with these periods, were traveling at the velocities his musical intervals predicted. Then he compared his results with the orbital sizes calculated from Copernican principles. Though still far from perfect agreement, his harmonic theory was in somewhat better agreement than his polyhedral theory.

  Kepler proceeded to sort out what he had learned from each theory. The harmonic theory allowed him to figure out the planets’ distances from the Sun, relative to one another. The polyhedral theory gave him the thickness of the empty spaces between the spheres in which the planets moved. He thought that the space left over might be the space required by the planetary eccentricities, for a sphere had to be thick enough to accommodate the eccentricity of the planet’s orbit (see figure 15.2). The task then was to figure out how to distribute that leftover space among the different spheres. It seemed possible that the answer to the question of why the eccentricities of the planets’ orbits were what they were was that any other eccentricities would spoil the harmony.

  Figure 15.2: In this drawing, the borders of the planet’s sphere are shown by broken lines, the orbit by a continuous line. The sphere of a planet had to be roomy enough to accommodate the planet’s eccentricity. In other words, the planet had to be able to travel in the parts of the orbit where it was nearest to and farthest from the Sun without breaking out beyond the borders of its sphere. The more eccentric the orbit, the thicker the sphere needed to be.

  Kepler soon wrote to Mästlin about a clever way he had found to relate his arrangement of the Platonic solids to three of the five intervals in his chord. It was the sort of connection Kepler loved. The polyhedron that separated the spheres of Saturn and Jupiter in his theory was a cube. Each corner of a cube is the meeting place of three flat squares, and the corner of each of the three squares is a ninety-degree angle. Add those three angles together, and the sum is 270 degrees. The ratio between 270 and 360 (the number of degrees in a complete circle) is 3:4. Thus it seemed appropriate to Kepler that the musical interval that requires a ratio of string lengths 3:4 should define the space interval between Saturn and Jupiter. Similar relationships worked for the interval between Jupiter and Mars and between Earth and Venus.

  Kepler confided to Mästlin and Herwart von Hohenburg, later in the summer, that having come thus far with his harmonic theory, he felt as though he had “a bird under a bucket.”8 His sense was that the harmony he was suggesting reflected the mind of the Creator and therefore surely had to be carried out in the cosmos. Kepler thought he was close to an answer. In fact, it would be twenty years before he found it, and when he did, he would also have discovered his third law of planetary motion.

  In the meantime, Kepler’s great frustration was t
hat being so near, he believed, to resolving the discrepancies in his theories, he lacked precise, accurate values for the observed eccentricities of the planets and the dimensions of the solar system. Kepler had no astronomical instruments of sufficient quality to make the necessary observations himself. The only man in the world who did was Tycho.

  Kepler also longed to know whether Tycho’s observations revealed any tiny differences in the altitudes of the Pole Star at the winter solstice and the equinoxes. The absence of any stellar parallax shift was still a difficulty for a Copernican, because the shift ought to be visible as Earth orbited the Sun. Kepler remained troubled by the fact that Copernican astronomy, combined with the absence of stellar parallax, required the stars to be so inconceivably far away.

  At the same time, Kepler had little respect for the “patchwork” Tychonic system, no matter who had thought of it, Tycho or Ursus. To Mästlin he wrote of Tycho’s ideas about the arrangement of the planetary system as “little paper houses” and wondered whether his own theory of celestial harmony might not easily blow them over. However, Kepler also wrote that he was reluctant to contradict a man like Tycho, especially since he needed Tycho’s observations. If only, he complained, Tycho would publish them so they would be widely available, or, failing that, be persuaded to send Kepler the information he needed. “I did not wish to be discouraged, but to be taught,” Kepler complained to Mästlin, rankled by the reservations about Mysterium that Tycho had expressed. Kepler could hardly contain his impatience. “My opinion about Tycho9 is this: he has abundant wealth. Only, like most rich men, he does not know how to make proper use of his riches. Therefore, one must take pains to wring his treasures from him, to get from him, by begging, the decision to publish all his observations without reservation.”