Tycho and Kepler

by Kitty Ferguson

  Kepler was exhausted when he rode out of Sagan on October 8. He had pushed the printer and himself unmercifully. He had needed to ship ahead of him, for the autumn book fair in Leipzig, a large stock of books: fifty-seven copies of ephemerides, sixteen copies of the Tables, and seventy-three other books. It had also taken time and effort to gather together all the documentation he had collected through the years about everything that was owed him.

  After Leipzig, Kepler rode to Regensburg and arrived there on November 2 after a cold autumn journey on a nearly worthless old horse that he sold in the city for a few florins. He felt ill but, determined not to neglect anything he had set out to do, shrugged off his illness as no more than a nuisance. For a man his age, it was more than that. He grew worse, his fever soared, and he lapsed into delirium5. A doctor came and bled him, which did not help.

  At last a Protestant pastor was summoned. Kepler drifted in and out of consciousness for several days and in his few lucid moments tried to explain to the pastor that he had done his utmost to reconcile Catholics and Protestants. The pastor admonished the dying Kepler that this was like expecting to bring together Satan and Christ. Kepler, as usual, was not dissuaded from his own beliefs. When asked on what basis he hoped for salvation, he answered, “Solely on the merit6 of our Savior Jesus Christ, in whom is found all refuge, solace, and salvation.” Those words were the very essence of Protestantism, and no pastor could take exception to them.

  On November 15, 1630, Kepler died. Though his grave has been lost, he was not buried in obscurity. Some of the most powerful and illustrious men of the empire were in Regensburg for the meeting Kepler had planned to attend, and many of them walked in the funeral procession for the imperial mathematician they had so celebrated but also so poorly supported. That evening there was a meteor shower. As it was reported at the time, fiery balls fell from heaven.

  Kepler’s gravestone in the Protestant cemetery bore an epitaph that he had written himself:

  I measured7 the heavens, Now the earth’s shadows I measure, Skybound, my mind. Earthbound, my body rests.

  Tycho Brahe in the portrait from his book Astronomiae Instauratae Mechanica, 1598, printed shortly before he went to Prague.

  Johannes Kepler in a portrait painted in 1610 when he was at the height of his career as Imperial Mathematician in Prague.

  Knutstorps Borg, Tycho Brahe’s birthplace and ancestral manor house, as it appears today.

  Knutstorps Borg, in a seventeenth-century drawing by Gerhard von Burman.

  King Frederick II of Denmark, Tycho’s patron, who granted him the fiefdom of Hven and whose support enabled him to build Uraniborg, in a portrait by an unknown artist.

  Tycho’s elevation drawing for Uraniborg.

  Tycho’s plan of the gardens of Uraniborg, with the mansion at the center.

  The great globe, begun in Augsburg in 1570 and completed at Uraniborg a decade later. It became the centerpiece of Uraniborg’s library.

  The great mural quadrant, one of Tycho’s most splendid instruments, built into the structure of Uraniborg in 1582. Tycho considered this portrait of himself an excellent likeness. The two men shown in the foreground were not part of the mural; they were part of this particular drawing of the mural.

  Stjerneborg (“Star Castle”), the partly subterranean observatory Tycho built in the 1580s outside the perimeter wall of Uraniborg to accommodate some of his largest and most powerful instruments.

  The Chapel of the Magi at Roskilde Cathedral, with tombs of Frederick II and his father, as it appears today. It was this chapel that Tycho neglected to keep in good repair, thus incurring the wrath of the teenage Christian IV.

  King Christian IV of Denmark, whose birth horoscope Tycho drew up, and who later became Tycho’s nemesis, in a portrait by Pieter Isaacsz.

  Wedding medallion portraits (1597) of Johannes Kepler and his first wife, Barbara.

  Emperor Rudolph II of the Holy Roman Empire, the eccentric, reclusive patron of both Tycho and Kepler, in a portrait by Hans von Aachen.

  The cliff-top Benatky Castle, northeast of Prague, that Emperor Rudolph gave to Tycho to create a second Uraniborg. The mural depicting hunting scenes and featuring the emperor is visible on its walls.

  Drawing by an unknown artist depicting riots between Archduke Leopold’s troops and Protestant vigilantes in the streets of Prague, near the Keplers’ home, in February 1611.

  APPENDIX 1

  ANGULAR DISTANCE

  A simple way to approach the definitions of the terms angular distance, angle of separation, and degree of arc is to imagine oneself at the center of a giant clock face, where the two hands meet. From that point of view, the angle of separation or angular distance between an object at “twelve o’clock” and an object at “one o’clock” is thirty degrees of arc. Likewise the angular distance between “one o’clock” and “two o’clock,” and so forth. The angular distance between an object at “twelve o’clock” and another at “two o’clock” is sixty degrees of arc, and so forth. The entire circle has 360 degrees. To understand the concept roughly with regard to the sky, draw an imaginary line between two stars directly overhead whose angle of separation you want to measure and let that line continue all the way around you and the earth beneath your feet until it comes up the other side of the sky and joins its other end, so that the line has drawn a huge circle all the way around the celestial sphere. That huge circle is the equivalent of the clock face, and you are in the center where the two hands meet. If the two stars look to be at, let us say, one and two o’clock, then their angular separation is thirty degrees.

  Because celestial objects that interest astronomers are often closer together than one degree of arc, degrees are divided into smaller segments. There are sixty minutes of arc in one degree of arc; sixty seconds of arc or arcseconds in one minute of arc.

  Two objects whose angle of separation is, let us say, thirty degrees of arc (from twelve to one on the clock face as viewed from the center of the clock) can actually be either quite close to one another or very far apart. For example, looking from the window of my study, I see two trees whose distance from one another (if you go out and measure it) is about twelve feet. Their angular separation from where I stand is about thirty degrees. Beyond them is the sky. Lining up each tree with a star, those two stars also have an angular separation from one another of about thirty degrees when viewed from my study. However, those two stars are definitely not just twelve feet apart. Knowing what angle separates two objects does not tell us the distance between them.

  APPENDIX 2

  VOCABULARY OF ASTRONOMY

  Much of the vocabulary that is essential to understanding this book is explained in the relevant chapters, but here are a few more useful terms:

  Meridian circle: Starting at the north celestial pole, draw an imaginary line to the zenith above where you are standing, then continue the line around the celestial sphere until you have brought it all the way around the celestial sphere, through the south celestial pole, to meet its tail again at the north celestial pole. What you have drawn is a line of longitude or a meridian, the celestial equivalent of the lines of longitude or meridians to be found on a globe of Earth. This meridian is perpendicular to the horizon.

  Altitude is the distance of a star or planet above the horizon, measured in degrees. A complete circle is 360 degrees, so the altitude of a star at the zenith is 90 degrees. No star can ever have an altitude greater than 90 degrees.

  Azimuth is the distance of an object from the meridian, also measured in degrees. Imagine again drawing the meridian hoop. Stand facing north and imagine that line. If you see a star off to the left or right of that line, that star is not on your meridian. Its azimuth is the measurement of how far it is from the meridian.

  Meridian, altitude, and azimuth—like horizon and zenith—are dependent on where an observer is standing.

  Astronomers need measurements that will not change with the position of the observer—measurements that stay p
ut, as do the celestial equator and the celestial poles. An astronomer in Denmark must be able to tell an astronomer in Italy what the position of a star or planet is without using Denmark as a reference point. Hence another set of terms:

  The prescribed meridian is the meridian line established not by the position of the observer but by the position of the Sun at the vernal equinox.

  The declination of a celestial object is its distance in degrees above the celestial equator.

  Right ascension is its distance in degrees east of the prescribed meridian.

  Declination and right ascension are hence independent of the observer’s position on Earth. Whether you are in New York or Arizona or Turkey, the declination and right ascension of a particular star will be the same.

  Two more measurements are related not to the horizon or the celestial equator but to the ecliptic:

  The latitude of a celestial body is how many degrees it is above or below the ecliptic.

  Longitude is a body’s position along the ecliptic, measured in degrees eastward from the vernal equinox.

  It is useful to remember all these terms in groups of four:

  Altitude and azimuth are measurements related to the horizon and the meridian.

  Declination and right ascension are measurements related to the celestial equator and the prescribed meridian.

  Latitude and longitude are measurements related to the ecliptic and the vernal equinox.

  APPENDIX 3

  KEPLER’S USE OF TYCHO’S OBSERVATIONS OF MARS TO FIND THE ORBIT OF EARTH

  To discover what Earth’s motion was like, Kepler put himself and the readers of Astronomia Nova in the position of a Martian astronomer observing Earth.

  The Martian begins the series of observations when Mars is on Earth’s apsidal line (that is, the line running through the Sun, the center of Earth’s orbit, Earth, and Earth’s positions at aphelion and perihelion—see figure 19.1). Every 687 Earth-days after that (687 Earth-days is one Martian year), the Martian takes another observation of Earth. Each time, Mars has completed an orbit and returned to Earth’s apsidal line. To put himself and his readers in the place of that Martian observer, Kepler reversed the direction along which Tycho had observed Mars from Earth, in effect allowing himself to watch Earth from a stationary Mars.

  Appendix 3, Figure 1 shows the Sun and Earth and Mars arranged so that Mars is at opposition, but the diagram gives no indication of distances. It shows only that at this moment an observer on Earth (were it possible to see the stars in the sky at the same time as the Sun) would find the Sun at the point called Z-1 in the zodiac belt and Mars at Z-2; an observer on the Sun would find Earth and Mars at Z-2; an observer on Mars would find Earth and the Sun both at Z-1. At the moment captured here, Earth, Sun, Mars, and those points in the zodiac are all located on the same straight line; if Earth is on its apsidal line, then obviously so is Mars.

  Using Tycho’s collection of observations as his background data, Kepler found instances 687 Earth-days (a Martian year) apart, beginning when Mars was on Earth’s apsidal line. Though Mars would have arrived back on that line at the end of every 687-day period, Earth—which takes only 365 days to complete an orbit—would not have. Earth would be in a different place each rime. Appendix 3, Figure 2 imagines Earth showing up at positions designated Earth1, Earth2, and Earth3, while Mars’s position is always on Earth’s apsidal line. Mars, Sun, and Earth1 are points of a triangle; Mars, Sun, and Earth2 are points of another triangle; Mars, Sun, and Earth3 are points of a third triangle. All the triangles have a side in common, the Mars-to-Sun line. The length of that line (which coincides with Earth’s apsidal line) is the same in all three triangles.

  Appendix 3, Figure 1: The drawing represents a large circular room, with the stars of the zodiac “belt” painted on its walls. These stars, whose positions and angular distances from one another were well known to early astronomers and astrologers, are the fixed background against which an inhabitant of the Solar System observes the planets. According to Copernicus, the Sun is in the center, and Earth is near it. The stars are, of course, much farther away from the Sun and Earth than the dimensions of a room can possibly simulate. But their great distance means that whether viewed from Earth, the Sun, or Mars, they occupy the same positions in the zodiac, just as though they really were on the walls of a huge room.

  Appendix 3, Figure 2: At the end of each 687-day interval, Mars has completed an orbit and made its way back to Earth’s apsidal line, while Earth (requiring only 365 days to complete an orbit) shows up at different positions. In each case, Mars, the Sun, and Earth are points of a triangle. The three triangles have one line in common, the line that coincides with Earth’s apsidal line—the Sun-to-Mars line.

  From this “Martian astronomer” exercise, Tycho’s solar theory, and his own Vicarious Hypothesis, Kepler had the information needed to find out where observers on any of the three bodies (Earth, Mars, Sun) would see the other two bodies against the background stars of the zodiac, when Earth was at Earth1, Earth2, and Earth3 (see Appendix 3, Figure 2).

  Knowing the angular distances between the positions where the three lines illustrated in Appendix 3, Figure 3 ended in the zodiac gave Kepler the angles of the triangle when Earth was at Earth1, which in turn told him how the lengths of the sides were related to one another. He made this calculation for the triangles when Earth was at Earth1, Earth2, and Earth3. All three triangles had one side in common—the Sun-Mars line (line C in Appendix 3, Figure 3), which coincided with Earth’s apsidal line. Comparing the lengths of the other sides with the length of that common side told him how all the sides of all three triangles compared with one another. And that knowledge, in principle, allowed Kepler to find the position of Earth in its orbit at the time each of the observations were made—Earth1, Earth2, and Earth3—in other words, to draw a diagram placing the three Earth positions where they really do occur, not just in imaginary places. Using those three points, he could draw a circle through them to represent Earth’s orbit and find out where the center of the circle was. That should give him the position of the center of the orbit relative to the Sun and the radius of the orbit. Kepler double-checked his findings using several different sets of triangles based on different observations.

  Appendix 3, Figure 3: The triangle when Earth was at Earth1: Tycho’s observation, made from Earth, of Mars’s position in the zodiac told Kepler where the Earth-Mars line (line A) ended in the zodiac. Tycho’s solar theory (the theory in which the Sun orbits the Earth and the planets orbit the Sun) gave what Kepler judged to be accurate positions for the Earth in the zodiac as viewed from the Sun, which means he knew where the Earth-Sun line (line B) ended in the zodiac. Kepler’s Vicarious Hypothesis’s heliocentric logitudes told him where Mars appeared in the zodiac when viewed from the Sun; i.e., where the Mars-Sun line (line C) ended in the zodiac.

  Drawing all those points in turn might seem certain to have described the shape of Earth’s orbit and to have revealed that it was an ellipse. To think so is to underestimate the pitfalls and uncertainties involved, and the subtlety of the answer Kepler was seeking. The ellipse he would later discover is so close to being a circle that it was impossible to find it by this method, even had there been no error at all in his calculations. For this particular procedure, each piece of data provided an opportunity for error, and in a process that used three triangles with one common side, that meant seven chances to go wrong. The trigonometry used in the computation, especially when it involved small angles, magnified any error. Even if the true orbit had been circular, Kepler might easily have got different results for different sets of triangles. To his chagrin, he did.

  However, this outcome was by no means a complete disappointment, because Kepler’s results showed that contrary to what Ptolemy, Copernicus, and Tycho had thought, the center of Earth’s orbit lay somewhere in the middle between the equant point and the Sun, and that was where astronomers had traditionally put the center of the orbit of a planet. Also, K
epler had found that Earth was moving like a planet, speeding up when it came closer to the Sun and slowing down as it moved away.

  NOTES

  JKGW refers to the twentieth-century compendium of Kepler’s works and letters: Max Caspar, Walther von Dyck, Franz Hammer, and Volker Bialas, eds. Johannes Kepler Gesammelte Werke. 22 vols. Munich: Deutsche Forschungsgemeinschaft, and the Bavarian Academy of Sciences, 1937–.

  TBDOO refers to Tycho Brahe’s collected works: John Lewis E. Dreyer, ed. Tychonis Brahe Dani Opera Omnia. 15 vols. Copenhagen: Libraria Gyldendaliana, 1913–29.

  Mechanica refers to Tycho Brahe, Astronomiae Instauratae Mechanica, Raeder et al. translation.

  I have used the following form for quotations where the original is in Danish and I have taken the translation from an English source: Original source/English translation source. For example: Gassendi 3:20/Thoren 20 means that the original is Gassendi, volume 3, page 20; and I have used the translation on page 20 of Thoren.

  PROLOGUE

  1 “You will come”: Brahe to Kepler, Jan. 26, 1600, JKGW, vol. 14, letter 154.