The school where Kepler taught was thick in the middle of this uneasy truce and by no means neutral. It had been established in 1574 as a deliberate countermove to the founding of a Catholic Jesuit college the previous year. Its four preachers and twelve teachers were influential members of the Protestant community, and the school was, to all intents and purposes, the rallying center for Protestants in Graz.
At first none of this controversy touched Kepler. His concerns were almost entirely academic. His teaching duties were in the upper school, where he taught advanced mathematics, including astronomy. Either he was not a particularly exciting teacher or the subjects were not popular, because few students attended his classes in the first year and none in the second. School officials looked for other classes Kepler could teach and assigned him rhetoric, Virgil, less advanced arithmetic, ethics, and history.
Graz, in an early seventeenth-century engraving attributed to Matthäus Merian Sr.
Kepler’s duties extended beyond the classroom. He was also district mathematician, a public office that had considerable responsibilities connected with it. One of them was the compilation of an annual calendar with astrological predictions for the coming year. Kepler could approach this task with confidence. Already at Tübingen he had been known for his astrological skills. The district mathematician’s calendar told what to expect about weather, harvests, war, disease, the most auspicious periods during which a physician might bleed a patient or perform surgery, when farmers should sow seed, when the weather would be most benign or most inclement or dangerous, when the Turks would attack, and when one should anticipate religious or political troubles. What Tycho had to produce for the princes of Denmark, Kepler had to produce for the entire citizenry of Graz and the surrounding countryside—albeit for only a year at a time instead of a lifetime.
Kepler’s attitude toward astrology fell far short of the confidence most of his contemporaries had in it. He was already calling it the “foolish little daughter”1 of respectable astronomy. Later he would write that he abhorred “nourishing the superstition2 of fatheads” and that “if astrologers do sometimes tell the truth, it ought to be attributed to luck.” However, he did not fully reject the idea that there were links between the cosmos and human beings. He, like Tycho, thought that the movements of the planets must in some way influence what happened on Earth, but probably far more subtly and far less deterministically than was commonly supposed.
Whatever his reservations, Kepler produced the calendars; they were, after all, a part of his job description. Before long, Graz found it had a very able district mathematician indeed, though he was not exactly a bearer of good news. For 1595, he had predicted an exceptionally cold winter, an attack by the Turks from the south, and a peasant uprising. All of those prophecies came true.
Michael Mästlin, Kepler’s mentor at Tübingen, was particularly scornful of Kepler’s astrological activities. Kepler took exception with him in a letter: “If God gave each animal3 tools for sustaining life, what harm is there if for the same purpose he joined astrology and astronomy?”
However, Kepler, the astrologer, failed to predict the most momentous event in his own life that year, a discovery he made in his classroom. Until then, Kepler had been an obscure teacher with some mathematical skills and little to set him apart from hundreds like him in Europe. Now emerged the Kepler who would transform astronomy, and also the Kepler whom some would think quite mad, for the discovery he made that day sounds to the twenty-first century almost as outlandish as the astrology for which he was admired.
On July 19, 1595—he kept careful record of the date, so significant did it seem to him—Kepler drew a diagram for his students on the chalkboard. The drawing demonstrated the progression of the Great Conjunctions of the planets Jupiter and Saturn; that is, when Jupiter passes Saturn in the zodiac. Jupiter and Saturn are the slowest moving of the planets that were known in Kepler’s time, and since Tycho Brahe had recorded the Great Conjunction of 1563 as the second entry in his logbook, there had been only one other.
To understand what Kepler drew, picture the celestial sphere with Earth at the center. The two planets, Jupiter and Saturn, travel in enormous circles around Earth. Every twenty years Jupiter catches up with Saturn, the more distant of the two planets, and passes it. Kepler’s drawing showed that these passings, or conjunctions, do not happen every time at the same points in the zodiac. For example, the fourth conjunction in the drawing (1643) occurred at almost the same point as the first (1583), but not quite; the fifth at almost the same point as the second, but again not quite, and so on. As Kepler drew the lines, each went just beyond joining a former line to make a closed triangle. Instead, the quasi-triangle “rotated,” creating the pattern on the chalkboard. In Kepler’s words, “I inscribed4 within a circle many triangles, or quasi-triangles, such that the end of one was the beginning of the next. In this manner a smaller circle was outlined by the points where the lines of the triangles crossed each other.”
That second circle was visible in his drawing, half the size of the outer circle. While the points of the triangle “drew” the outer circle as the triangle rotated, the middles of the sides of the triangle “drew” the inner circle. The triangle’s lines never trespassed into the inner circle, and its points never went outside the outer circle. The triangle dictated how far apart the two circles had to be.
Figure 12.1:7 The pattern of Jupiter-Saturn conjunctions, showing where they happened in the zodiac. The conjunction in 1583 (right side of drawing) 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. The drawing is from Kepler’s Mysterium, with the zodiac names and the dates added around the rim. The reason for the names of the elements—earth, air, fire, water—will be explained later in the discussion of the appearance of “Kepler’s Star.”
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.
Looking at what he had drawn, Kepler was struck by an insight that made him feel as though he had suddenly and unexpectedly opened a book and found inscribed there the secret of creation. As he wrote later: “The delight that I took5 in my discovery, I shall never be able to describe in words.”
It was in Kepler’s nature that as soon as he had resigned himself a year earlier to teaching mathematics and astronomy, he had put his whole heart and mind into their study. “I pondered on this subject6 with the whole energy of my mind,” wrote Kepler, “and there were three things above all for which I sought the causes as to why it was this way and not another—the number, the dimensions, and the motions of the orbs.” There were six planets: Why not more or fewer? The planets orbited at certain relative distances from the Sun: Why those distances and not others? Each planet moved at a certain speed and seemed to change its speed in a certain way: Why this particular speed and this particular change? Like many great scientific discoverers, Kepler asked simple, naive questions that most scholars of his time thought not worth asking and to which they would have responded at best with a tolerant smile for a poor schoolteacher. Part of Kepler’s genius was that these questions nagged him.
Twentieth- and twenty-first-century scientists regard it as their mandate to try to discover why things are as they are, rather than simply to describe how they are, but that was not the case for astronomers prior to the late sixteenth century. Although it would be incorrect to say that scholars such as Ptolemy and Copernicus never pondered such causal questions, their primary concern was to describe and predict where heavenly bodies were positioned and the patterns of their movements, not to answer what caused them to be where they were and to move in certain patterns and at certain speeds and distances.
There was good philosophical precedent for concentrating on the one and not the other. In the fourth century B.C., Aristotle h
ad defined a difference between mathematics (including astronomy) on the one hand and “physics” on the other. His definition could be interpreted to mean that those who studied physics were obligated to think in terms of Aristotelian “causes,” while mathematicians and astronomers could ignore these concerns. Being let off that particular philosophical hook proved a great boon to astronomy in eras when looking for causes could have been no more than guesswork. Ignoring causal questions became a pleasant habit. Medieval astronomers and philosophers thought that if one had to look for causes, the simple “naturalness”8 of the cosmos was reason enough for things being as they found them.
Two thousand years after Aristotle, Kepler bucked this tradition, thinking about such questions as, What lies behind this? According to what larger plan is this so? Why has God chosen to construct the solar system in this way and not another? Kepler knew that many of these questions might turn out to be unanswerable, but by the time he drew his fateful diagram on the board, he had begun to focus them in two questions that he thought he could answer: What line of reasoning was God using when he made things this way? and, What are the physical reasons why the universe operates as it does? He had begun to focus that second question in a way that would prove enormously important to him, asking whether one body in the solar system influences the way the others move. Maybe, for instance, the Sun did more than simply sit in the center of a neat arrangement. Kepler was not the first to wonder whether there were physical explanations for celestial phenomena, but he was the first to insist there must be and to insist on seeking them out.
When he plotted the Great Conjunctions for his students in July 1595, Kepler had already tried out and discarded some possible answers to his question about God’s line of reasoning. “Almost the whole summer9 was lost with this agonizing labor,” he reported. He had speculated, for example, that the orderly progression that underlay the relative distances of the planets from the Sun was a scheme in which Venus’s orbit was twice the size of Mercury’s, Earth’s twice the size of Venus’s, Mars’s twice the size of Earth’s, and so on. But neither that nor any similar set of relationships had fit. Kepler had speculated that there might be another planet, too small for us to see, between Mercury and Venus, and another between Venus and Earth, and so on, but that had not fit either.
The exercise Kepler had set himself was like some problems on modern standardized tests: Given a list of numbers, discern what mathematical regularity generates the sequence. Find the pattern that lies behind it. Break the code. On a sophisticated standardized test, one of the possible choices of answers is likely to be, “There is no pattern to this sequence, no code.” In his attempt to decipher the solar system, Kepler rejected entirely the possibility of that answer. His Philippist education and his own natural inclinations caused him to believe that a universe created by God could not be random and meaningless or subject to arbitrary whim. Underlying all the seemingly disconnected aspects of nature, the complexity and the confusion, there had to be pattern, logic, and harmony. That conviction implied also that there must be hidden connections between things that seemed unrelated. Geometry, music, medicine, and astronomy had to be linked at some deep level. Kepler thought this must surely be the way God created the universe; therefore, a man created in the image of God could comprehend the logic10 and discover the links, with effort.
There are those who argue that Kepler’s preoccupation with the harmony of the universe made him a medieval mystical throwback. He was not. His assumption of underlying harmony has become one of the pillars of the scientific method. There are indeed many connections of the sort Kepler was seeking, and these are understood now as he could not understand them. Some of the connections he experimented with turn out with hindsight to look ludicrous today, but the marvel of the man was that he thought to put them to rigorous testing. His error, and it cannot be called an error in the context of what he knew and could know in the sixteenth and seventeenth centuries, was that he had no idea how deeply such harmony lies hidden.
There are also those who would say Kepler naively contradicted himself by believing both in divine providence and in a universe not subject to the arbitrary decisions of God. But Kepler was not a naive man. He could not dismiss either side of that “contradiction” without being intellectually dishonest. Over the years, as his understanding increased, he continued, perhaps aided by his strong conviction that hidden, deeper resolutions lay behind apparent contradictions, but also out of the simple need to live with what his science and his life experience told him was true, to be exuberantly enthusiastic about both beliefs.
It was with an outburst of this exuberance that Kepler reported the Great Conjunction insight that finally did look as though it might break the code of the planetary system: “Finally I came close11 to the true facts on a quite unimportant occasion. I believe Divine Providence arranged matters in such a way that what I could not obtain with all my efforts was given to me. I believe all the more that this is so as I have always prayed to God that He should make my plan succeed, if what Copernicus had said was the truth.” Kepler stepped back from the diagram and saw that the smaller circle was half as large as the larger circle, and that this relationship was dictated by the triangle. Was it coincidence, he wondered, that the orbit of Jupiter was about half as large as the orbit of Saturn? Could a triangle have something to do with that relationship? Saturn and Jupiter were the two planets in conjunction and the two outermost planets, and the triangle was the first figure in geometry. (The smallest number of lines from which one can create a closed geometric figure is three.) Kepler immediately began experimenting to see whether a square (the second figure of geometry) would similarly fit between the orbits of Jupiter and Mars, a pentagon between the orbits of Mars and Earth, a hexagon between the orbits of Earth and Venus, and so forth. Unfortunately this scheme did not match the observed distances between the planetary orbits. Kepler wondered whether those “known” distances were really correct.
Something else troubled Kepler about his scheme: It was too loose, leaving too much room for arbitrary choice. Beginning with a triangle and adding sides of equal length produces a square, a pentagon, a hexagon, a heptagon, an octagon, and so forth. One can go on forever adding yet another side of equal length and produce an infinite number of these so-called polygons. Certainly it would not be surprising if among those infinite polygons it were possible to find five that fit snugly between the orbits of the planets. However, to Kepler’s mind, this achieved nothing, because one still had to ask why these polygons and not others had been chosen for the design, and why there were only six planets.
Kepler nevertheless felt he was breathing down the neck of the answer. If only he could discover why certain polygons and not others—five of them and not more—had been chosen to dictate the distances between the orbits. It occurred to him that while drawings on the chalkboard were of necessity flat (two-dimensional drawings), the real universe was three-dimensional. Perhaps it was not appropriate to apply polygons, which are two-dimensional figures, to a three-dimensional system. He considered using solid figures—three-dimensional forms—instead. “And behold, dear reader,”12 he wrote, “you have my discovery in your hands.” Kepler knew that although it is possible to create an infinite number of two-dimensional shapes in which all the edges have the same length—triangle, square, pentagon, hexagon, etc.—there is no such extensive a collection with solid, three-dimensional shapes. Experimenting with polyhedrons—solids in which all the edges are the same length, in which all the sides are the same shape, and that have other characteristics that appealed to Kepler—reveals that only five have all the defining characteristics of “perfect solids,” also known as Pythagorean or Platonic solids. Nature, God, Creation, mathematical logic—they allow these five and no others.
It seemed significant to Kepler that each Platonic solid can be nested inside a sphere so that every corner of the solid touches the inside surface of the sphere. And a sphere can be nested inside any Platonic so
lid so that the sphere touches the center of every face of the solid. To Kepler’s mind this meant there was something deeply “sphere-like” about these solids. Only of these five polyhedrons could it be said that each can be “inscribed into a sphere” and “circumscribed around a sphere.”
Figure 12.2: The five Platonic solids: The tetrahedron has four faces, all of them identical equilateral triangles. The cube has six identical square faces. The octahedron has eight faces, all identical equilateral triangles. The dodecahedron has twelve identical pentagonal faces. The icosahedron has twenty faces, all identical equilateral triangles.
Five figures thus stood apart from all other possible solid figures because of their simplicity, their mathematical beauty and perfection. Here, thought Kepler, was what God must have been thinking when he set the Sun and planets in their places. The reason there were six planets—no more, no less—was because there were five perfect solids to dictate their relative distances. As Kepler would write in the introduction to his book on the subject:
Behold, reader, the invention13 and whole substance of this little book! In memory of the event, I am writing down for you the sentence in the words from that moment of conception: The Earth’s orbit is the measure of all things; circumscribe around it a dodecahedron [twelve-sided regular solid], and the circle containing this will be Mars [Mars’s sphere]; circumscribe around Mars a tetrahedron [four-sided solid], and the circle containing this will be Jupiter; circumscribe around Jupiter a cube, and the circle containing this will be Saturn. Now inscribe within the earth [within Earth’s sphere] an icosahedron [twenty-sided solid], and the circle contained in it will be Venus; inscribe within Venus an octahedron [eight-sided solid], and the circle contained in it will be Mercury. You now have the reason for the number of planets.
Tycho and Kepler Page 18