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Kepler continued eking out a living in Graz as an ineffective teacher, went on with his astronomical work, wrote some books about astrology, and married – a marriage that lasted, though there is some indication it was not an entirely happy one. Then in 1598, the Archduke Ferdinand began to make life miserable for Lutheran leaders and teachers. Things went from bad to worse and, eventually, given a day’s notice to leave Graz altogether or be sentenced to death, Kepler departed. It soon became evident that the Archduke’s attitude was intransigent. Kepler would no longer be able to live and work in Graz.
Figure 3.1
(a) The five regular or ‘cosmic’ solids. In every case, all of the sides are identical and only equilateral figures are used for them. These are the only possible regular or ‘cosmic’ solids.
(b) Kepler’s arrangement of them in relation to the spheres of the planets. Saturn’s sphere is outside the cube. Jupiter’s sphere is between the cube and the tetrahedron. Mars’s sphere is between the tetrahedron and the dodecahedron. Earth’s sphere is between the dodecahedron and the icosahedron. Venus’s sphere is between the icosahedron and the octahedron. Mercury’s sphere is within the octahedron.
However, Kepler was not completely out in the cold, for the opportunity had arisen to join Tycho Brahe at Prague. Kepler had sent the older man a copy of Mysterium, and Tycho had recognized a serious talent. Kepler was understandably apprehensive about how well the two of them would get along, for Tycho had a reputation as a proud, imperious eccentric. His nose had been partly sliced off in a duel, and he had restored the missing bit himself with gold, silver and wax. But Tycho was also unarguably the greatest astronomer of his generation . . . and Kepler needed a job. So Kepler, now 30 years old, moved to Prague in 1601. Whatever the difficulties working with Tycho, he didn’t have to cope with them for long, for within two years Tycho died. Kepler rose to the title of as Imperial Mathematician.
This new position was a considerable improvement over teaching at Graz, but there was a negative side. Although he had an impressive title, Kepler’s salary was much lower than Tycho’s had been and often it wasn’t paid. Kepler was obliged to waste a great deal of time trying to collect what was due him. The job as Imperial Mathematician did, however, have other compensations, for, over the objections of Tycho’s relatives and fortunately for the future of astronomy, it was Kepler who fell heir to Tycho’s magnificent set of astronomical observations, the best the world had ever known. Tycho had found that circular orbits were difficult to reconcile with the actual paths of the planets, and he had undertaken an exhaustive series of observations that he hoped would throw more light on the problem. No man was better suited than Kepler to put this precious inheritance to optimum use.
Kepler brought to this work both a firm belief in Copernican astronomy and an unwillingness to accept that the skilled and meticulous Tycho’s data could be faulty. The two must somehow fit, even though Tycho himself had rejected Copernican astronomy in favour of the scheme that had the Sun orbiting the Earth and all the other planets orbiting the Sun. Kepler also brought to his task a concept of his own that the Sun moves the planets by a ‘whirling force’ centred in itself. Orbits centred elsewhere than on the Sun would not do.
Tycho Brahe had paid particular attention to the planet Mars, and it was in trying to make sense of these observations that Kepler finally found himself obliged to abandon circular orbits, removing the stumbling block that had hobbled Copernican astronomy since its inception. He realized that by using elliptical orbits he could explain Tycho’s observations. Kepler signalled his overwhelming joy and astonishment at his insight by falling to his knees and exclaiming, ‘My God, I am thinking Thy thoughts after Thee.’ Those who describe Kepler as dry and passionless and his life as uniformly drab and sad simply fail to appreciate the sorts of things that moved him.
Kepler’s discovery and Tycho’s earlier observations (without a telescope) that made it possible were achievements that still inspire awe in modern astronomers. The ellipse in which the Earth orbits is so nearly circular that any attempt to make it obviously an ellipse in a scale drawing ends up being a distortion. It is only slightly more obvious that the orbit of Mars is an ellipse. Yet with this seemingly trivial geometric alteration, the Copernican system fell into place. Kepler, aware of the power of his discovery and also of the reaction it would inevitably receive, commented wryly that with this introduction of elliptical orbits, he had ‘laid an enormous egg’. He had indeed, and even many who admired his work found this egg difficult to digest. Galileo, for one, never accepted it.
Saying an ellipse is an ‘egg shape’ or an ‘oval’ isn’t precise enough. It’s best to think of an ellipse in one of two ways. One is as a slice out of a cone. Not just any oval or egg shape can be produced by slicing a cone, and so not all ovals and egg shapes are ellipses. A circle, however, is an ellipse. It’s a slice directly across the cone. See Figure 3.2.
Figure 3.2
A second way of thinking about an ellipse is a little more difficult to describe, but it gets the same result: imagine a piece of thick cardboard lying on the table before you. Stick two drawing pins into it, a distance apart. Then take a piece of string that is longer than the distance between the two pins and attach its ends to the pins – something like the string shown in See Figure 3.3.
Figure 3.3
Next take a pencil and pull the string taut with its point (Figure 3.4), still keeping the string flat on the surface of the cardboard.
Figure 3.4
If you move the pencil, allowing it to slide along the string while the string remains taut, the point of the pencil will draw an ellipse. Changing the length of the string or moving the drawing pins closer together or further apart changes the shape of the ellipse, just as changing the tilt of the slices in the cone does. The closer together the two drawing pins are, the nearer the ellipse comes to being a circle. If they are in precisely the same location, it is a circle.
Each drawing pin represents a focus of the ellipse, so an ellipse has two foci. If a planet’s orbit is near to circular, that means that the ellipse’s foci must be very close together. By contrast, comets, which also orbit the Sun in elliptical orbits, have foci that are very far apart, producing an extremely elongated ellipse.
The first of Kepler’s two laws of planetary motion that appeared in his book titled Astronomia Nova (New Astronomy) states that a planet moves in an elliptical orbit, and the Sun is located at one of the two foci of that ellipse (see Figure 3.5).
Kepler’s second law has to do with the variation in the speed of a planet as it travels in its orbit. The law states that an imaginary straight line (called the ‘radius vector’) joining the centre of the planet to the centre of the Sun ‘sweeps out’ equal areas in equal intervals of time.
Picture an elliptical orbit, the Sun (one of its foci), and a planet travelling along in the orbit (see Figure 3.6). Picture the planet at A and draw an imaginary straight line from A to the centre of the Sun. Set a stopwatch and time the planet as it orbits to B (which for purposes of this demonstration could be located anywhere on the part of the orbit nearer the Sun). Think of the imaginary line moving with it like the hand of a clock, shading the area it sweeps across. At B, stop the shading and the timing. You have now created a shaded area that we say is ‘swept out’ as the planet travels from A to B – looking like the left side of Figure 3.6. You know how long it took the planet to travel that distance and sweep out that area. Wait until the planet has travelled somewhat further. Decide on a point C somewhere on the side of the ellipse further from the Sun. When the planet reaches that point, again draw a line from the planet to the centre of the Sun and start the shading and the stopwatch. The planet travels along and so does the sweeping line. When the planet has travelled for the same amount of time it took to go from A to B, stop the shading and call that final point D.
Figure 3.5
Figure 3.6
The resulting drawing resembles Figure 3.6. What does it mean? Notice that on the left side of the picture, from A to B, the planet travelled a much greater distance, in the same interval of time, than it did from C to D over on the right side. Clearly it couldn’t have maintained a constant speed. It had to travel much faster from A to B, to cover all that distance, than it did to cover the smaller distance from C to D in the same interval of time. The two shaded ‘swept out’ portions, however, are equal in area. Kepler’s law thus predicts that a planet will always travel at a faster rate the nearer its path comes to the Sun. Kepler thought the variation in speed occurred because the distance from the Sun affected how much or how little the planet felt the Sun’s ‘whirling force’.
Kepler’s second law, then, accounted for the way a planet varies in speed and made it possible to predict those variations. While one can’t make sense of the system by saying ‘this planet always travels thus-and-so-many miles per hour’, one can make sense of it by saying ‘this planet’s imaginary line to the Sun always sweeps out an area of thus-and-so-many square miles per hour’. This was another of Kepler’s crazy ideas, a ‘connection’ he found between astronomy and geometry, arising from his belief in the intrinsic harmony of a universe created by God. We still today think he got this one right.
Kepler’s Astronomia Nova contained more than these two laws. He was well on track towards our modern understanding of gravity, previewing Newton’s later discoveries about the tides and understanding them a great deal better than Galileo did. Kepler had discerned that the more massive a body, the stronger its attractive effect. He also wrote that it couldn’t be the Sun’s light that provided the ‘whirling force’, for the Earth didn’t screech to a halt during a solar eclipse.
Kepler finished Astronomia Nova in 1606 but didn’t publish it until 1609. That was the same year Galileo first looked through a telescope.
In 1611, after the death of Kepler’s wife and son, and with difficulties for Protestants mounting in Prague with the counter-reformation as they had done earlier in Graz, Kepler moved to Linz. He lived there for 14 years, marrying a second time. It was in Linz that he published, in 1619, a book called Harmonices Mundi (Harmonies of the World), relating the orbital speeds of the planets to melodic lines. He speculated that it might be possible to discover the moment of Creation by calculating backwards in time and determining the moment when the orbits of the planets would have produced the equivalent of the most perfect musical harmony.
Much more significant, Harmonices Mundi also contained Kepler’s third law of planetary motion, which establishes a relationship between the lengths of time the planets take to complete their orbits and their distances from the Sun. Those not conversant in the language of mathematical equations may skip the following paragraph.
The equation is (TA /TB)2 = (RA /RB)3. TA is the time it takes planet A to complete its orbit (its ‘orbital period’). TB is the time it takes planet B to complete its orbit. RA is the average distance from planet A to the Sun. RB is the average distance from planet B to the Sun. The ratio of the squares of the orbital periods of the two planets is equal to the ratio of the cubes of their average distances from the Sun.
This equation makes it possible to build a scale model of the solar system. The question remains, however: What is the scale? For even though the orbital periods could be timed and were known, Kepler’s law didn’t provide the means to calculate any actual measurement for the distance from any planet to the Sun. The third law was an equation waiting for just one absolute, known distance to plug in.
Kepler’s discovery of his first and second laws allowed him to put together far more accurate tables for calculating the planets’ positions for any time in the past or future. When he published his third law, he had already begun the laborious task of producing these, but the completed tables were still slow in coming. Kepler’s preface explains that this was partly because of difficulties finding financial backing (he finally paid for the publication out of his own pocket) and also because of ‘the novelty of my discoveries and the unexpected transfer of the whole of astronomy from fictitious circles to natural causes’. He went on to point out that no one had ever attempted anything of this kind before. It required intense study of Tycho’s astronomical observations. Kepler tried orbit after orbit, leaving one after another behind when, after prodigious computation, they didn’t match his data. He was not particularly happy doing this work. It must have been very pedantic for so imaginative and creative a mind. It wasn’t until 1627 that he finally published his Rudolphine Tables, and they were a triumph for Copernican astronomy.
Kepler spent his last years at Sagan, in Silesia. He died at Regensburg on a trip seeking a new job and trying to collect back salary, in November 1630, just a year short of seeing that his prediction, in accordance with the Rudolphine tables, that Mercury would cross the disc of the Sun on 7 November 1631 was correct. He had composed his own epitaph:
I measured the skies, now the shadows I measure
Skybound was the mind, Earthbound the body rests.
Compared with Kepler’s bleak childhood, Galileo’s apparently was a pleasant one in a family that valued intellectual pursuits. He was born seven years before Kepler, in 1564, in the north Italian city of Pisa, and attended school both there and in Florence. At the age of 17, honouring his father’s wishes, he entered the University of Pisa as a medical student. Two years later, when it was clear that mathematics and mechanics, not medicine, were Galileo’s forte, his father allowed him to change his course of study.
One of Galileo’s important discoveries took place during his student years. Good Catholic he may have been, but evidently he was not always completely attentive at services. A lamp swinging on a long cord in the cathedral caught his eye. He noticed that regardless of whether the length (in space) of each swing was long or short, it seemed that the time it took to complete the swing remained the same. Curious, he experimented on his own and found that the length of time it takes a pendulum to complete one swing depends not on how big the swing is but only on the length of the cord by which it hangs.
In 1585, Galileo’s formal education ended when his father ran short of money to pay university costs. Galileo came home to Florence, where his family was now living. Undeterred, he studied on his own and with scholars among his father’s acquaintances and was soon making a local reputation with several inventions and discoveries, circulating his own short book on measuring the specific gravities of bodies, and brashly voicing suspicions about Aristotle’s mental capacities. One bizarre undertaking during these years was a series of public lectures he delivered about the shape, size and location of Dante’s hell – a topic that might draw a considerable audience even today.
In 1589, when Galileo was 25, four years after he had left the University of Pisa without a degree, he returned there as a lecturer. Pisa, like Tübingen, still taught Ptolemaic astronomy. Whatever Galileo may have been thinking privately, that was what he taught. It isn’t clear just when he became a convinced Copernican.
A discovery Galileo made while he was a lecturer at Pisa caused his already low opinion of Aristotle to sink even further. Though Aristotle had in truth been rather ambiguous on the subject, at least in those writings that survive, most scholars of Galileo’s time, including Galileo, thought that Aristotle had said that if two objects were dropped simultaneously from the same height, the heavier would strike the ground first. Galileo either dropped weights on numerous occasions from the Tower of Pisa, as his earliest biographer reported, or rolled them down a ramp, or perhaps he tried both. But he established to his satisfaction that the heavier and the lighter hit the ground at the same time. This was, of course, an experiment that he couldn’t perform in the absence of air resistance, and science historian Thomas Kuhn has quipped that it probably wasn’t Galileo who carried out the legendary public demonstration from the Leaning Tower but a defender of Aristotle, who thereby proved quite decisively for all present that Galileo was wrong and Aristotle was right. In the 20th century, astronauts performed the experiment in the airless environment of the Moon. Galileo was right.
Never a tactful man, Galileo somewhat incautiously proceeded to debunk the still highly venerated Aristotle. Aristotle ‘wrote the opposite of truth’ . . . was ‘ignorant’. Galileo also showed a lack of judgement when he bluntly advised the Grand Duke of Tuscany, Ferdinand I (who had granted him his professorship), that a dredging machine designed by the Grand Duke’s brother-in-law wasn’t going to work. The machine was built nevertheless and didn’t work, which made Galileo even less popular. When his father died in 1591, leaving him as eldest son responsible for the family, Galileo’s salary at Pisa was far from sufficient and, thanks to the dredging machine incident, not likely to rise. He looked for a new job and found one at the University of Padua, where Copernicus had studied medicine 90 years earlier. The Venetian Republic, the source of funding for Padua, was liberal and tolerant compared with Tuscany, and Padua and Venice were much more sympathetic venues for someone with ideas as radical and speech as imprudent as Galileo’s.
Though his financial difficulties didn’t end, Galileo was happy and influential in the intellectual milieu of Padua and Venice. On the personal front, he fathered three children by Marina Gamba, though he never married her. Their relationship seems to have ended amicably when he finally moved back to Florence, for he remained good friends with her and with the man she later married. Galileo’s two daughters both became nuns, and one of them was a source of great comfort and help to him when he was elderly.
