He envisioned Tycho’s domain as a refuge from the evils of the time, as the place where his Cosmic Mystery would be confirmed. He aspired to become a colleague of the great Tycho Brahe, who for thirty-five years had devoted himself, before the invention of the telescope, to the measurement of a clockwork universe, ordered and precise. Kepler’s expectations were to be unfulfilled. Tycho himself was a flamboyant figure, festooned with a golden nose, the original having been lost in a student duel fought over who was the superior mathematician. Around him was a raucous entourage of assistants, sycophants, distant relatives and assorted hangers-on. Their endless revelry, their innuendoes and intrigues, their cruel mockery of the pious and scholarly country bumpkin depressed and saddened Kepler: ‘Tycho . . . is superlatively rich but knows not how to make use of it. Any single instrument of his costs more than my and my whole family’s fortunes put together.’
Impatient to see Tycho’s astronomical data, Kepler would be thrown only a few scraps at a time: ‘Tycho gave me no opportunity to share in his experiences. He would only, in the course of a meal and, in between other matters, mention, as if in passing, today the figure of the apogee of one planet, tomorrow the nodes of another . . . Tycho posesses the best observations . . . He also has collaborators. He lacks only the architect who would put all this to use.’ Tycho was the greatest observational genius of the age, and Kepler the greatest theoretician. Each knew that, alone, he would be unable to achieve the synthesis of an accurate and coherent world system, which they both felt to be imminent. But Tycho was not about to make a gift of his life’s work to a much younger potential rival. Joint authorship of the results, if any, of the collaboration was for some reason unacceptable. The birth of modern science – the offspring of theory and observation – teetered on the precipice of their mutual mistrust. In the remaining eighteen months that Tycho was to live, the two quarreled and were reconciled repeatedly. At a dinner given by the Baron of Rosenberg, Tycho, having robustly drunk much wine, ‘placed civility ahead of health,’ and resisted his body’s urgings to leave, even if briefly, before the baron. The consequent urinary infection worsened when Tycho resolutely rejected advice to temper his eating and drinking. On his deathbed, Tycho bequeathed his observations to Kepler, and ‘on the last night of his gentle delirium, he repeated over and over again these words, like someone composing a poem: “Let me not seem to have lived in vain . . . Let me not seem to have lived in vain.” ’
After Tycho’s death, Kepler, now the new Imperial Mathematician, managed to extract the observations from Tycho’s recalcitrant family. His conjecture that the orbits of the planets are circumscribed by the five platonic solids was no more supported by Tycho’s data than by Copernicus’. His ‘Cosmic Mystery’ was disproved entirely by the much later discoveries of the planets Uranus, Neptune and Pluto – there are no additional platonic solids* that would determine their distances from the Sun. The nested Pythagorean solids also made no allowance for the existence of the Earth’s moon, and Galileo’s discovery of the four large moons of Jupiter was also discomfiting. But far from becoming morose, Kepler wished to find additional satellites and wondered how many satellites each planet should have. He wrote to Galileo: ‘I immediately began to think how there could be any addition to the number of the planets without overturning my Mysterium Cosmographicum, according to which Euclid’s five regular solids do not allow more than six planets around the Sun . . . I am so far from disbelieving the existence of the four circumjovial planets that I long for a telescope, to anticipate you, if possible, in discovering two around Mars, as the proportion seems to require, six or eight round Saturn, and perhaps one each round Mercury and Venus.’ Mars does have two small moons, and a major geological feature on the larger of them is today called the Kepler Ridge in honor of this guess. But he was entirely mistaken about Saturn, Mercury and Venus, and Jupiter has many more moons than Galileo discovered. We still do not really know why there are only nine planets, more or less, and why they have the relative distances from the Sun that they do. (See Chapter 8.)
* The proof of this statement can be found in Appendix 2.
Tycho’s observations of the apparent motion of Mars and other planets through the constellations were made over a period of many years. These data, from the last few decades before the telescope was invented, were the most accurate that had yet been obtained. Kepler worked with a passionate intensity to understand them: What real motion of the Earth and Mars about the Sun could explain, to the precision of measurement, the apparent motion of Mars in the sky, including its retrograde loops through the background constellations? Tycho had commended Mars to Kepler because its apparent motion seemed most anomalous, most difficult to reconcile with an orbit made of circles. (To the reader who might be bored by his many calculations, he later wrote: ‘If you are wearied by this tedious procedure, take pity on me who carried out at least seventy trials.’)
Pythagoras, in the sixth century B.C., Plato, Ptolemy and all the Christian astronomers before Kepler had assumed that the planets moved in circular paths. The circle was thought to be a ‘perfect’ geometrical shape and the planets, placed high in the heavens, away from earthly ‘corruption,’ were also thought to be in some mystical sense ‘perfect’ Galileo, Tycho and Copernicus were all committed to uniform circular planetary motion, the latter asserting that ‘the mind shudders’ at the alternative, because ‘it would be unworthy to suppose such a thing in a Creation constituted in the best possible way.’ So at first Kepler tried to explain the observations by imagining that the Earth and Mars moved in circular orbits about the Sun.
After three years of calculation, he believed he had found the correct values for a Martian circular orbit, which matched ten of Tycho’s observations within two minutes of arc. Now, there are 60 minutes of arc in an angular degree, and 90 degrees, a right angle, from the horizon to the zenith. So a few minutes of arc is a very small quantity to measure – especially without a telescope. It is one-fifteenth the angular diameter of the full Moon as seen from Earth. But Kepler’s replenishable ecstasy soon crumbled into gloom – because two of Tycho’s further observations were inconsistent with Kepler’s orbit, by as much as eight minutes of arc:
Divine Providence granted us such a diligent observer in Tycho Brahe that his observations convicted this . . . calculation of an error of eight minutes; it is only right that we should accept God’s gift with a grateful mind . . . If I had believed that we could ignore these eight minutes, I would have patched up my hypothesis accordingly. But, since it was not permissible to ignore, those eight minutes pointed the road to a complete reformation in astronomy.
The difference between a circular orbit and the true orbit could be distinguished only by precise measurement and a courageous acceptance of the facts: ‘The universe is stamped with the adornment of harmonic proportions, but harmonies must accommodate experience.’ Kepler was shaken at being compelled to abandon a circular orbit and to question his faith in the Divine Geometer. Having cleared the stable of astronomy of circles and spirals, he was left, he said, with ‘only a single cartful of dung,’ a stretched-out circle something like an oval.
Eventually, Kepler came to feel that his fascination with the circle had been a delusion. The Earth was a planet, as Copernicus had said, and it was entirely obvious to Kepler that the Earth, wracked by wars, pestilence, famine and unhappiness, fell short of perfection. Kepler was one of the first people since antiquity to propose that the planets were material objects made of imperfect stuff like the Earth. And if planets were ‘imperfect,’ why not their orbits as well? He tried various oval-like curves, calculated away, made some arithmetical mistakes (which caused him at first to reject the correct answer) and months later in some desperation tried the formula for an ellipse, first codified in the Alexandrian Library by Apollonius of Perga. He found that it matched Tycho’s observations beautifully: ‘The truth of nature, which I had rejected and chased away, returned by stealth through the back door, disguising itself to be accepted . . . Ah, what a foolish bird I have been!’
Kepler had found that Mars moves about the Sun not in a circle, but in an ellipse. The other planets have orbits much less elliptical than that of Mars, and if Tycho had urged him to study the motion of, say, Venus, Kepler might never have discovered the true orbits of the planets. In such an orbit the Sun is not at the center but is offset, at the focus of the ellipse. When a given planet is at its nearest to the Sun, it speeds up. When it is at its farthest, it slows down. Such motion is why we describe the planets as forever falling toward, but never reaching, the Sun. Kepler’s first law of planetary motion is simply this: A planet moves in an ellipse with the Sun at one focus.