Tycho and Kepler (27 page)

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 subject
⁶
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 had 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 summer
⁹
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 logic
¹⁰
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 close
¹¹
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 solid 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 invention
¹³
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.