Tycho and Kepler (53 page)

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 Earth
¹
, Earth
²
, and Earth
³
, while Mars’s position is always on Earth’s apsidal line. Mars, Sun, and Earth
¹
are points of a triangle; Mars, Sun, and Earth
²
are points of another triangle; Mars, Sun, and Earth
³
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 Earth
¹
, Earth
²
, and Earth
³
(
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 Earth
¹
, 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 Earth
¹
, Earth
²
, and Earth
³
. 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—Earth
¹
, Earth
²
, and Earth
³
—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 Earth
¹
:
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, Kepler 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.

N
OTES

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. LEGACIES

For much of the information in the early chapters of this book I am indebted to Victor Thoren, whose
Lord of Uraniborg
(1990) superseded Dreyer’s 1890 book
Tycho
Brahe: A Picture of Scientific
Life and Work in the Sixteenth Century
as the definitive biography of Tycho Brahe. Tycho provided an autobiographical summary in his
Mechanica
, 106 ff; and Christianson has done a splendid job of recreating Tycho’s youth in
Cloister and Observatory: Herrevad Abbey and Tycho Brahe’s Uraniborg
.

2. ARISTOCRAT BY BIRTH, ASTRONOMER BY NATURE

3. BEHAVIOR UNBECOMING A NOBLEMAN