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Born into the Danish nobility in 1546, Tycho Brahe would earn lasting fame among astronomers for two particular reasons. First, in 1566, Tycho became embroiled in a disagreement with his cousin Manderup Parsberg, possibly because Parsberg had insulted and mocked Tycho over a recent astrological prediction that had fallen flat. Tycho had foretold the death of Suleiman the Great, and even embedded his prophecy within a Latin poem, apparently unaware that the Ottoman leader had already been dead for six months. The dispute culminated in an infamous duel. During the sword fight, a slash from Parsberg cut Tycho’s forehead and hacked through the bridge of his nose. An inch deeper and Tycho would have died. Thereafter he glued into place a false metal nose, so cleverly composed of a gold-silver–copper alloy that it blended in with his skin tone.

The second and more important reason for Tycho’s fame was that he took observational astronomy to an entirely new level of accuracy. He earned such a high reputation that King Frederick II of Denmark gave him the island of Hven, 10 km off the Danish coast, and paid for him to build an observatory there. Uraniborg (Castle of the Heavens) would grow over the years into a vast ornate citadel that consumed more than 5% of Denmark’s gross national product, an all-time world record for research centre funding.

Uraniborg housed a library, a paper mill, a printing press, an alchemist’s laboratory, a furnace and a prison for unruly servants. The observation turrets contained giant instruments, such as sextants, quadrants and armillary spheres (all naked-eye instruments, as astronomers had not yet learned to exploit the potential of lenses). There were four sets of every instrument for simultaneous and independent measurements, thereby minimising errors in assessing the angular positions of stars and planets. Tycho’s observations were generally accurate to ¹/₃₀°, five times better than the best previous measurements. Perhaps Tycho’s measurements were aided by his ability to remove his nose and align his eye more perfectly.

Figure 11 Uraniborg, on the island of Hven, the best funded and most hedonistic astronomical observatory in history.

Tycho’s reputation was such that a stream of VIPs visited his observatory. As well as being interested in his research, these visitors were also attracted by Uraniborg’s wild parties, which were famous all over Europe. Tycho provided alcohol in excess and entertainment in the shape of mechanical statues and a story-telling dwarf called Jepp, who was said to be a gifted clairvoyant. To add to the spectacle, Tycho’s pet elk was allowed to freely wander the castle, but tragically it died after stumbling down a staircase after drinking too much alcohol. Uraniborg was more like the setting for a Peter Greenaway film than a research institute.

While Tycho had been raised in the traditions of Ptolemaic astronomy, his painstaking observations forced him to reconsider his confidence in the ancient view of the universe. In fact, we know that he had a copy of De revolutionibus in his study and that he was sympathetic to Copernicus’s ideas, but, instead of adopting them unreservedly, he developed his own model of the universe, which was a faint-hearted halfway house between Ptolemy and Copernicus. In 1588, almost fifty years after Copernicus’s death, Tycho published De mundi ætherei recentioribus phænomenis (‘Concerning the New Phenomena in the Ethereal World’), in which he argued that all the planets orbited the Sun, but that the Sun orbited the Earth, as shown in Figure 12. His liberalism stretched as far as allowing the Sun to be the hub for the planets, but his conservatism obliged him to retain the Earth at the centre of the universe. He was reluctant to dislodge the Earth, because its supposed centrality was the only way to explain why objects fall towards the centre of the Earth.

Figure 12 Tycho’s model makes the same error as Ptolemy’s and places the Earth at the centre of the universe, being orbited by the Moon and the Sun. His main breakthrough was to realise that the planets (and the fiery comet) orbit the Sun. This illustration is from Tycho’s De mundi ætherei.

Before Tycho could continue to the next stage of his programme of astronomical observation and theorising, his research suffered a severe blow. His patron, King Frederick, died after a session of binge drinking in the same year that Tycho published De mundi ætherei, and the new king, Christian IV, was no longer prepared to fund Tycho’s lavish observatory or tolerate his hedonistic lifestyle. Tycho had no option but to abandon Uraniborg and leave Denmark with his family, assistants, Jepp the dwarf and cartloads of astronomical equipment. Fortunately, Tycho’s instruments had been designed to be transportable, because he had shrewdly realised: ‘An astronomer must be cosmopolitan, because ignorant statesmen cannot be expected to value their services.’

Tycho Brahe migrated to Prague, where Emperor Rudolph II appointed him Imperial Mathematician and allowed him to establish a new observatory in Benatky Castle. The move turned out to have a silver lining, because it was in Prague that Tycho teamed up with a new assistant, Johannes Kepler, who would arrive in the city a few months later. The Lutheran Kepler had been forced to flee his previous home in Graz when the fiercely Catholic Archduke Ferdinand had threatened to execute him, in keeping with his stated declaration that he would rather ‘make a desert of the country than rule over heretics’.

Fittingly, Kepler set out on his journey to Prague on 1 January 1600. The start of a new century would mark the start of a new collaboration that would lead to a reinvention of the universe. Together, Tycho and Kepler made the perfect double act. Scientific advance requires both observation and theory. Tycho had accumulated the best collection of observations in the history of astronomy, and Kepler would prove to be an excellent interpreter of those observations. Although Kepler suffered from myopia and multiple vision from birth, he would ultimately see farther than Tycho.

It was a partnership that was formed in the nick of time. Within a few months of Kepler’s arrival, Tycho attended a dinner hosted by the Baron of Rosenberg and drank to his usual excess, refusing nonetheless to break etiquette by leaving the table before the Baron. Kepler recorded: ‘When he drank more, he felt the tension in his bladder increase, but he put politeness before his health. When he got home, he was scarcely able to urinate.’ That night he developed a fever, and from then on he alternated between bouts of unconsciousness and delirium. Ten days later he was dead.

On his deathbed, Tycho repeatedly uttered the phrase: ‘May I not have lived in vain.’ There was no need to fear, because Kepler would guarantee that Tycho’s meticulous observations bore fruit. In fact, it is quite possible that Tycho had to die in order for his work to flourish, because while he was alive he carefully guarded all his notebooks and never shared his observations, always dreaming of publishing a solo masterwork. Tycho certainly never considered embracing Kepler as an equal partner – he was, after all, a Danish aristocrat, whereas Kepler was a mere peasant. However, seeing the deeper meaning of his own observations was beyond Tycho, and required the skills of a trained mathematician such as Kepler.

Kepler was born into a lowly family that struggled to survive the upheavals caused by war, religious strife, a wayward criminal father and a mother who had been exiled after accusations of witchcraft. Not surprisingly, he grew up as an insecure hypochondriac with little self-esteem. In his own self-deprecating horoscope, written in the third person, he described himself as a little dog:

He likes gnawing bones and dry crusts of bread, and is so greedy that whatever his eyes chance on he grabs; yet, like a dog, he drinks little and is content with the simplest food… He continually seeks the goodwill of others, is dependent on others for everything, ministers to their wishes, never gets angry when they berate him and is anxious to get back into their favour… He has a dog-like horror of baths, tinctures and lotions. His recklessness knows no limits, which is surely due to Mars in quadrature with Mercury and in trine with the Moon.

His passion for astronomy seems to have been his only respite from self-loathing. At the age of twenty-five he wrote Mysterium cosmographicum, the first book to defend Copernicus’s De revolutionibus. Thereafter, convinced of the veracity of the Sun-centred model, he dedicated himself to identifying just what it was that made it inaccurate. The greatest error was in predicting the exact path of Mars, a problem that had plagued Copernicus’s assistant, Rheticus. According to Kepler, Rheticus had been so frustrated with his failure to solve the Mars problem that ‘he appealed as a last resort to his guardian angel as an Oracle. The ungracious spirit thereupon seized Rheticus by the hair and alternately banged his head against the ceiling, then let his body down and crashed it against the floor.’

With access at last to Tycho’s observations, Kepler was confident that he could solve the problem of Mars and remove the inaccuracies in the Sun-centred model within eight days; in fact, it took him eight years. It is worth stressing the amount of time that Kepler spent perfecting the Sun-centred model– eight years!– because the brief summary that follows could easily underplay his immense achievement. Kepler’s eventual solution was the result of arduous and tortuous calculations that filled nine hundred folio pages.

Kepler made his great breakthrough by jettisoning one of the ancient tenets, namely that the planets all move in paths that are circles or combinations of circles. Even Copernicus had clung loyally to this circular dogma, and Kepler pointed out that this was just one of Copernicus’s flawed assumptions. In fact, Kepler claimed that his predecessor had wrongly assumed the following three points:

1. the planets move in perfect circles,

2. the planets move at constant speeds,

3. the Sun is at the centre of these orbits.

Although Copernicus was right in stating that the planets orbit the Sun and not the Earth, his belief in these three false assumptions sabotaged his hopes of ever predicting the movements of Mars and the other planets with a high degree of accuracy. However, Kepler would succeed where Copernicus had failed because he discarded these assumptions, believing that the truth emerges only when all ideology, prejudice and dogma are set aside. He opened his eyes and mind, took Tycho’s observations as his rock and built his model upon Tycho’s data. Gradually an unbiased model of the universe began to emerge. Sure enough, Kepler’s new equations for the orbits matched the observations, and the Solar System took shape at last. Kepler exposed Copernicus’s errors, and showed that:

1. the planets move in ellipses, not perfect circles,

2. the planets continuously vary their speed,

3. the Sun is not quite at the centre of these orbits.

When he knew he had the solution to the mystery of planetary orbits, Kepler shouted out: ‘O, Almighty God, I am thinking Thy thoughts after Thee.’

In fact, the second and third points in Kepler’s new model of the Solar System emerge out of the first, which states that planetary orbits are elliptical. A quick guide to ellipses and how they are constructed reveals why this is so. One way to draw an ellipse is to pin a length of string to a board, as shown in Figure 13, and then use a pencil to extend the string. If the pencil is moved around the board, keeping the string taut, it will trace out half an ellipse. Switch to the other side of the string, and make it taut again, and the other half of the ellipse can be traced out. The length of the string is constant and the pins are fixed, so a possible definition of the ellipse is the set of points whose combined distance to the two pins has a specific value.

Figure 13 A simple way to draw an ellipse is to use a piece of string attached to two pins, as shown in diagram (a). If the pins are 8 cm apart and the string is 10 cm long, then each point on the ellipse has a combined distance of 10 cm from the two pins. For example, in diagram (b), the 10 cm of string forms two sides of a triangle, both 5 cm long. From Pythagoras’ theorem, the distance from the centre of the ellipse to the top must be 3 cm. This means that the total height (or minor axis) of the ellipse is 6 cm. In diagram (c), the 10 cm of string is pulled to one side. This indicates that the total width (or major axis) of the ellipse is 10 cm, because it is 8 cm from pin to pin plus 1 cm at both ends.

The ellipse is quite squashed, because the minor axis is 6 cm compared with the major axis of 10 cm. As the two pins are brought closer together, the major and minor axes of the ellipse become more equal and the ellipse becomes less squashed. If the pins merge into a single point, then the string would form a constant radius of 5 cm and the resulting shape would be a circle.

The positions of the pins are called the foci of the ellipse. The elliptical paths followed by the planets are such that the Sun sits at one of the foci, and not at the centre of the planetary orbits. Therefore there will be times when a planet will be closer to the Sun than at other times, as if the planet has fallen towards the Sun. This process of falling would cause the planet to speed up and, conversely, the planet would slow down as it moved away from the Sun.

Kepler showed that, as a planet follows its elliptical path around the Sun, speeding up and slowing down along the way, an imaginary line joining the planet to the Sun will sweep out equal areas in equal times. This somewhat abstract statement is illustrated in Figure 14, and it is important because it precisely defines how a planet’s speed changes over the course of its orbit, contrary to Copernicus’s belief in constant planetary speeds.

The geometry of the ellipse had been studied since ancient Greek times, so why had nobody ever before suggested ellipses as the shape of the planetary orbits? One reason, as we have seen, was the enduring belief in the sacred perfection of circles, which seemed to blinker astronomers to all other possibilities. But another reason was that most of the planetary ellipses are only very slightly elliptical, so under all but the closest scrutiny they appear to be circular. For example, the length of the minor axis divided by the length of the major axis (see Figure 13) is a good indication of how close an ellipse is to a circle. The ratio equals 1.0 for a circle, but the Earth’s orbit has a ratio of 0.99986. Mars, the planet that had given Rheticus nightmares, was so problematic because its orbit is more squashed, but the ratio of the two axes is still very close to 1, at 0.99566. In short, the Martian orbit was only slightly elliptical, so it duped astronomers into thinking it was circular, but the orbit was elliptical enough to cause real problems for anybody who tried to model it in terms of circles.

Figure 14 The diagram shows a highly exaggerated planetary orbit. The height of the ellipse is roughly 75% of its width, whereas for most planetary orbits in the Solar System this proportion is typically between 99% and 100%. Similarly, the focus occupied by the Sun is far off-centre, whereas it is only slightly off-centre for actual planetary orbits. The diagram demonstrates Kepler’s second law of planetary motion. He explained that the imaginary line joining a planet to the Sun (the radius vector) sweeps out equal areas in equal times, which is a consequence of a planet’s increase in speed as it approaches the Sun. The three shaded sectors all have equal areas. When the planet is closer to the Sun the radius vector is short, but this is compensated by its greater speed, which means that it covers more of the ellipse’s circumference in a fixed time. When the planet is far from the Sun the radius vector is much longer, but it has a slower speed so it covers a smaller section of the circumference in the same time.

Kepler’s ellipses provided a complete and accurate vision of our Solar System. His conclusions were a triumph for science and the scientific method, the result of combining observation, theory and mathematics. He first published his breakthrough in 1609 in a huge treatise entitled Astronomia nova, which detailed eight years of meticulous work, including numerous lines of investigation that led only to dead ends. He asked the reader to bear with him: ‘If thou art bored with this wearisome method of calculation, take pity on me who had to go through with at least seventy repetitions of it, at a very great loss of time.’

Kepler’s model of the Solar System was simple, elegant and undoubtedly accurate in terms of predicting the paths of the planets, yet almost nobody believed that it represented reality. The vast majority of philosophers, astronomers and Church leaders accepted that it was a good model for making calculations, but they were adamant that the Earth remained at the centre of the universe. Their preference for an Earth-centred universe was based largely on Kepler’s failure to address some of the issues in Table 2 (pp. 34—5), such as gravity – how can the Earth and the other planets be held in orbit around the Sun, when everything that we see around us is attracted to the Earth?

Also, Kepler’s reliance on ellipses, which was contrary to the doctrine of circles, was considered laughable. The Dutch clergyman and astronomer David Fabricius had this to say in a letter to Kepler: ‘With your ellipse you abolish the circularity and uniformity of the motions, which appears to me increasingly absurd the more profoundly I think about it… If you could only preserve the perfect circular orbit, and justify your elliptic orbit by another little epicycle, it would be much better.’ But an ellipse cannot be built from circles and epicycles, so a compromise was impossible.

Disappointed by the poor reception given to Astronomia nova, Kepler moved on and began to apply his skills elsewhere. He was forever curious about the world around him, and justified his relentless scientific explorations when he wrote: ‘We do not ask for what useful purpose the birds do sing, for song is their pleasure since they were created for singing. Similarly, we ought not to ask why the human mind troubles to fathom the secrets of the heavens… The diversity of the phenomena of Nature is so great, and the treasures hidden in the heavens so rich, precisely in order that the human mind shall never be lacking in fresh nourishment.’

Beyond his research into elliptical planetary orbits, Kepler indulged in work of varying quality. He misguidedly revived the Pythagorean theory that the planets resonated with a ‘music of the spheres’. According to Kepler, the speed of each planet generated particular notes (e.g. doh, ray, me, fah, soh, lah and te). The Earth emitted the notes fah and me, which gave the Latin word fames, meaning ‘famine’, apparently indicating the true nature of our planet. A better use of his time was his authorship of Somnium, one of the precursors of the science fiction genre, recounting how a team of adventurers journey to the Moon. And a couple of years after Astronomia nova, Kepler wrote one of his most original research papers, ‘On the Six-Cornered Snowflake’, in which he pondered the symmetry of snowflakes and put forward an atomistic view of matter.

‘On the Six-Cornered Snowflake’ was dedicated to Kepler’s patron, Johannes Matthaeus Wackher von Wackenfels, who was also responsible for delivering to Kepler the most exciting news that he would ever receive: an account of a technological breakthrough that would transform astronomy in general and the status of the Sun-centred model in particular. The news was so astonishing that Kepler made a special note of Herr Wackher’s visit in March 1610: ‘I experienced a wonderful emotion while I listened to this curious tale. I felt moved in my deepest being.’

Kepler had just heard for the first time about the telescope, which was being used by Galileo to explore the heavens and reveal completely new features of the night sky. Thanks to this new invention, Galileo would discover the evidence that would prove that Aristarchus, Copernicus and Kepler were all correct.