In the late 16th and early 17th centuries two astronomers strove to calculate the positions of the planets. Tycho Brahe spent many years measuring the paths of the planets across the sky from his observatory on the island of Hven, while Johannes Kepler was able to prove from this data that the orbits of the planets were ellipses.
It was Christmas 1566, and in the town of Rostock in Germany a university professor was holding a festive gathering for staff and students. At some point during the proceedings, perhaps fueled by wine, a furious row broke out between two of the guests, both of them Danish noblemen. Before anyone could intervene, one of the participants threw down the gauntlet and challenged his antagonist to a duel. A week later tempers had still not cooled off, and so the duel took place at an appointed place in the nearby countryside. One of the noblemen was called Manderup Parbsjerg, and the other was called Tycho Brahe (1546–1601). When the swords were drawn and the blades were flashing in anger it was Parbsjerg who got the better of the encounter. Tycho Brahe’s nose received a severe cut, his face was covered in blood and he was forced to concede. Brahe had his nose patched up with a concoction of gold, silver and wax. The doctor did a good job and, apart from a rather startling appearance that sometimes gave him an advantage, Brahe was hardly troubled by his injury. His relationship with Manderup Parbsjerg was also patched up and the two became good friends.
The “man with the golden nose” was a very successful astrologer. When he witnessed an eclipse of the Moon just two months before his duel with Parbsjerg, he forecast that the sultan of Turkey was about to die. Soon afterward, news arrived that the sultan had indeed died. This enhanced Brahe’s reputation and it also convinced him of the truth of astrology. Then news arrived that the sultan had died before the eclipse of the Moon and the eclipse was therefore not a valid prediction of the event. This was disturbing, and Brahe soon discovered that there were other problems with his astrological predictions. He had to admit that many of them did not seem to work. He was very confident that his methods were correct, so he concluded that the problem lay with the fact that he was unable to calculate the positions of the planets with sufficient accuracy when he was casting the horoscopes. He had two sets of tables at his disposal, one was based on the well-tried system of Ptolemy, the other was a new set of tables based on the heretical ideas of an upstart Pole called Nicolaus Copernicus, who thought that the Sun was at the center of the universe. Brahe discovered that neither of the two systems, Ptolemaic nor Copernican, gave him accurate positions for the planets. It was bad news for his horoscopes, but he was undaunted and he made a decision to dedicate his life to calculating the positions of the planets as accurately as he could.
Brahe continued with his astronomical studies. Then he made a discovery that caused him great concern. In 1572 he witnessed a very rare event, in the constellation of Cassiopeia. A very bright new star had appeared. Brahe knew that all the world’s astronomers agreed that the sphere of the stars was fixed; it had been made by God at the creation and it never changed. New stars simply did not appear. He first saw the star in November 1572 when it was brighter than Jupiter. He knew he was not in error, for it was unmistakable. For several months it was the brightest star in the sky. By December its brightness had faded to equal Jupiter. By March it had faded again, but it still ranked with the first magnitude stars. It dimmed steadily through the magnitudes, until by April 1574 it was no longer visible. What Brahe was observing was a supernova—an event so rare that it has happened only three times in our galaxy during the past 1000 years. However, Brahe was more interested in the astrological significance of the new star. It was not good news:
The star was at first like Venus and Jupiter, giving pleasing effects; but as it then became like Mars, there will next come a period of wars, seditions, captivity and death of princes, and destruction of cities, together with dryness and fiery meteors in the air, pestilence, and venomous snakes. Lastly, the star became like Saturn, and there will finally come a time of want, death, imprisonment and all sorts of sad things.
In time, Brahe decided to move from Denmark to Germany, but when King Frederick II heard about this he became alarmed to think that he might lose the services of such a wise man. So the king made Brahe a generous offer. He was to have his own private observatory. It would be built on the island of Hven, an isolated but inhabited piece of land in the straits between Denmark and Sweden. Brahe would become landlord of Hven, and by collecting the rents from the local farmers he would have financial independence over and above his royal patronage as well as his own small kingdom. Tycho Brahe could not refuse such an offer. He began to build the most magnificent observatory the world had seen.
Brahe’s observatory looked like a magical fairy-tale palace. Built in the Flemish style, it rose to 12 meters (40 ft) in height, and was surmounted by domes, spires and pinnacles sufficient to grace a cathedral. It had two semicircular observing bays on the north and south walls. It was also a luxurious home with running water in the bedrooms. It even had a jail—a useful facility for tenants who could not, or would not, pay their rent! The observatory became known as the Palace of Uraniborg. Brahe kept a dwarf called Jep to enhance his importance, and he also acquired two large dogs, presented to him on a royal visit by King James VI of Scotland.
Brahe was an artist as well as a scientist and craftsman, and everything he undertook or surrounded himself with was innovative and beautiful. He imported Augsburg craftsmen to construct the finest astronomical instruments. He established a printing shop to produce and bind his manuscripts in his own individual way. He induced Italian and Dutch artists and architects to design and decorate his observatory, and he invented a hydraulic pressure system to provide one of the great luxuries of the time—sanitary lavatory facilities. Uraniborg fulfilled the hopes of Brahe’s king and friend, Frederick II of Denmark, that it would become the center of astronomical study and discovery in northern Europe.
The greatest and most unusual feature of the building was the many astronomical instruments it contained. There were quadrants, sextants, armillary spheres, parallactic rules, astrolabes and clocks. On the island of Hven there was every astronomical instrument known to mankind, all made by skilled craftsmen and fashioned to the highest quality. The largest instrument was the great mural quadrant that could measure the positions of the planets to within a few minutes of arc.
Night after night Brahe and his assistants searched for the planets and carefully measured their positions in the night sky. Month after month the positions of more and more stars were added to a great catalog of 777 stars—all located with greater accuracy than ever before. Night after night and year after year for 20 years the lonely vigil was kept on the island of Hven. The data were collected for what would turn out to be the last, and greatest, of the catalogs created using observations made with the naked eye, and everything was carefully recorded. But where was it all leading? What was the purpose of this great enterprise? The main object was to create a set of tables to record the positions of the stars more accurately than ever before, but it also involved plotting the positions of the planets whenever they were visible—this was in some ways a greater task, for they changed their positions nightly. But Brahe wanted to do more than simply record the nightly positions of the planets. He wanted to predict what would happen in the future as well as what had happened in the past. What he really needed more than anything was a mathematician to study the data and to formulate a new theory that could predict the planetary positions in the future. Such knowledge would be a great bonus to the science of astrology.
Then Brahe received a great setback to his ambitions. In 1588 King Frederick II died, and it became clear that his son and successor Christian IV was not prepared to patronize the eccentric astronomer any longer. Brahe was forced to continue his work funded by his own resources. He soldiered on for nearly a decade in this way, but in 1597 he finally left Hven and by 1599 he had moved to Prague. It was there that he was fortunate enough to obtain a second royal patron in the person of Rudolf II, the Holy Roman Emperor, who was prepared to pay for the publication of his works. It was also in Prague that Brahe met a much younger man called Johannes Kepler (1571–1630). As we shall see shortly, the meeting between Brahe and Kepler was a famous and significant one, for Brahe found in Kepler someone who was capable of formulating a mathematical theory that would fit his data to the motion of the planets.
Our story moves to Germany, where Johannes Kepler was born in 1571 in the town of Weil de Stadt. Kepler was a small, frail man. He was near-sighted, and he was a hypochondriac. He was always plagued by fevers and stomach ailments. He was also a strange and mystical character who was very interested in astrology, and at least 800 of his horoscopes are still preserved. When he was casting horoscopes for his family he described his grandfather as “quick tempered and obstinate,” his grandmother was “clever, deceitful, blazing with hatred, the queen of busybodies,” his father Heinrich was “criminally inclined, quarrelsome, liable to a bad end” and his mother was “thin, garrulous and bad-tempered.” In later life he spent many months trying to clear his meddlesome mother of a charge of witchcraft.
In 1597 Kepler married Barbara Muller. She had been twice married before and she was twice widowed. When he first met her she “set his heart on fire,” but unfortunately when they tied the matrimonial knot he did not consult his horoscopes or he would have realized that the planets were in the wrong signs of the zodiac for such an event to take place. The marriage produced two children, but both died very young. Kepler became disillusioned with his wife when she told him in no uncertain terms that his precious astrology was nothing more than nonsense. He responded by accusing her of being “fat, confused and simple-minded.”
In spite of his personal problems and his attachment to astrology, Kepler was an excellent mathematician and he was convinced that God had created the universe with a mathematical pattern. When Kepler came to study the planets he tried to fit regular plane polygons between their orbits. But despite his efforts he was unable to find any geometric pattern that fitted.
Then, to his great delight, he found that when he modeled the problem in three dimensions instead of two, using the crystal spheres to carry the planets instead of plane circles, he could fit the five regular solids between the spheres. It seemed so perfect; the regular solids formed the framework that supported the spheres. The sphere was the perfect figure, but in terms of beauty and symmetry the regular solids ranked next. It seemed to explain why God had created the five regular solids and how they fitted into the universe. He published this finding with great enthusiasm in his Mysterium Cosmographicum of 1596.
And then again it struck me, why have plane figures among three-dimensional orbits? Behold, reader, the invention 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, and the sphere containing this will be Mars; circumscribe around Mars a tetrahedron and the sphere containing this will be Jupiter; circumscribe around Jupiter a cube and the sphere containing this will be Saturn. Now inscribe within the Earth an icosahedron, and the sphere contained in it will be Venus; inscribe within Venus an octahedron, and the sphere contained in it will be Mercury. You now have the reason for the number of planets.
Johannes Kepler first met Tycho Brahe early in the 17th century when Brahe was living in Prague. Brahe realized that Kepler had a good grasp of mathematics and would be the right person to help him complete and publish the Rudolphine Tables, named after his patron Rudolf II. On his deathbed Brahe pleaded with Kepler to finish his work on the planets and to publish his findings after his death. Kepler did his utmost to oblige. He worked very hard at studying the data, but try as he might he could not get the observations to fit the cycles and epicycles described by Ptolemy. The heretical work of Nicolaus Copernicus was of little help to him, for Copernicus had moved the Sun to the center of the universe and thereby he had simplified the calculations, but the system devised by Copernicus was no better than that devised by Ptolemy for predicting the future positions of the planets.
In the year 1604 Kepler observed a new star in the sky. He was not the first person to observe it, however, that honor goes to a court official called Johann Brunowski, a keen amateur astronomer who told Kepler about the new star. At first Kepler did not believe the report, but when the clouds cleared from the Prague sky he could not miss it. It was October 17 and the star, in the constellation of Ophiuchus, was shining as brightly as the planet Jupiter. Kepler continued to observe the star—which became known, perhaps wrongly, as Kepler’s Star—for about two years, during which time it slowly faded away. Thus it was that both Brahe and Kepler had been fortunate enough to witness a supernova. It is a surprising coincidence that these two contemporaries should both find a new supernova in the sky, when we consider that it is such a rare event.
Kepler was very moved by his astronomy and he left behind a record of his feelings:
It is true that a divine voice, which enjoins humans to study astronomy, is expressed in the world itself, not in words or syllables, but in things themselves and in the conformity of the human intellect and senses with the sequence of celestial bodies and of their dispositions. Nevertheless, there is a kind of fate, by whose invisible agency various individuals are driven to take up various arts, which makes them certain that, just as they are part of the work of creation, they, likewise also partake to some extent in divine providence. When, in my early years, I was able to taste the sweetness of philosophy, I embraced the whole of it with an overwhelming desire, and with no special interest whatever in astronomy. I certainly had enough knowledge, and I had no difficulty understanding the geometrical and astronomical topics included in the normal curriculum, aided as I was by figures, numbers and proportions …
Kepler used Tycho Brahe’s observations when he constructed his famous laws of planetary movement. Kepler tried to fit the data for the planet Mars into an elliptical orbit instead of a circle. Eventually he found to his great joy that the data for an elliptical orbit fitted well, and it explained perfectly the errors of a few minutes of arc. The ellipse can be defined as a slanted section through a cone. Kepler knew much about the properties of the conic sections, for he had studied the work of the Greek mathematician Apollonius of Perga.
There is some irony in the fact that Ptolemy would accept nothing but the perfect circle to describe the motions in the heavens, when all the time he had the work of Apollonius available to him. To the uninitiated, Kepler’s ellipses seem just as ridiculous as his idea of the five regular polyhedrons. Why should the planets, in their orbits around the Sun, be forever following a path represented by the section through a cone? What had slanted sections of cones to do with the system of the world? But as Kepler developed his theory he discovered that this really was the case. He was absolutely right and he had made a major step forward. He formulated his three laws of planetary motion as follows:
Law 1. The orbits of the planets are ellipses with the Sun at one focus. The focus is not at the center of the ellipse. The ellipse has two symmetrically placed foci on its longer axis. The circle is a special case of the ellipse where both the foci coincide with the center. The foci are so called because if a ray of light from one focus is reflected by the surface of the ellipse, then whatever its direction it will always be reflected through the second focus.
Law 2. The radius vector sweeps out equal areas in equal times. As the planet moves around the Sun, the line joining Sun and planet sweeps out equal areas in equal time intervals. Once the constants of the orbit are known this law can be used to predict the position of the planet at any time in the future. It is a special case of the law of conservation of angular momentum.
Law 3. The cubes of the planets’ mean distances from the Sun are proportional to the squares of their periods. There is some uncertainty about what is meant by the “mean distance” from the Sun, but it can be taken as the geometric mean of the maximum and minimum distances. Using this law, if we know the period of a planet (time to orbit the Sun) then we can calculate its mean distance from the Sun, and vice versa.
Laws 1 and 2 enable the position of any planet to be predicted once the orbital plane—in other words, the period and the position of the perihelion—are known. Kepler was particularly pleased with his third law for he liked discovering numerical relationships. Kepler did not produce a popular scientific work like that of Copernicus before him or Galileo after him. Instead he edited and published the Rudolphine Tables. The Holy Roman Emperor Rudolf II was, like Kepler himself, more interested in casting horoscopes than in astronomy. The tables brought together the most accurate set of astronomical observations ever made (those of Tycho Brahe) and the most perfect theory for the motions of the planets (the elliptical theory of Johannes Kepler).
The tables first became available to the world at large in 1628, but scientific knowledge traveled only very slowly in the early 17th century. The first people in England to make use of the Rudolphine Tables were the young astronomer Jeremiah Horrocks and his friend William Crabtree, in 1639, after they had spent two years trying to make sense of earlier tables.