Tycho and Kepler (43 page)

Common experience indicates that objects
seem
to move much more slowly the farther away they are. A bird flying close overhead can easily appear
to win a race with a plane far higher above it in the sky. Closer
looks
faster. Ptolemaic astronomers, Copernicus, and Tycho had all considered the possibility that what appears to be a variation in the speed of a planet, as viewed from the center of the system (Earth or Sun, depending on whether one followed Ptolemy or Copernicus), is only an illusion created by the fact that the planet, traveling
on an eccentric orbit, is sometimes closer than at other times. But they had realized that this explanation was not sufficient to account for the extent of speeding up and slowing down revealed by observations. Ptolemy experimented with an equalizing point or “equant” that was not the center of the system (Earth in his case), and not the center of the eccentric orbit. It was a third point along
the apsidal line, a point from which an observer would find that the planet appeared to be moving with constant speed.

Figure 19.1 (Apsidal line): An off-center orbit was said to be “eccentric.” The distance between the center of the orbit and the center of the system (Earth or Sun) was known as the “eccentricity” of the orbit. A straight line drawn through both of those two points is the apsidal line. Extended farther, the apsidal line passes through the point where the orbiting planet is farthest from the
center of the system (at aphelion) and the point where it is closest (at perihelion).

The “problem of Mars” was that it was the most difficult of the three outer planets to accommodate in this manner. Mars has by far the greatest eccentricity. Tycho must have been well aware, when he decided to train the instruments of Uraniborg and Stjerneborg on Mars, that that planet provided not only the
stickiest problems but also the best opportunity to define what the problems were and, one might hope, to solve them. As Kepler put it, “I consider it a divine decree
¹⁴
that I came at exactly the time when Longomontanus was busy with Mars. Because assuredly either through it we arrive at the
knowledge
of the secrets of astronomy or else they remain forever concealed from us.”

Although scholars
had been studying the heavens for centuries, no one had discovered that Mars’s orbit was not a circle, but ancient and medieval astronomers cannot be accused of ignoring data to adhere to an erroneous assumption. The amount by which a planet’s orbit, even the orbit of Mars, departs from a circle is extremely small. The errors in the observations available prior to Tycho were at least as great
as ten minutes of arc, and this margin of error—of which astronomers were well aware—made it impossible to discern that the orbit was other than circular. Tycho’s observations were trustworthy to within two or three minutes of arc, a great improvement on the ten-minute error tolerated before, but the discovery that Mars’s orbit was elliptical would not come—as one might naively suspect—from Kepler’s
simply plotting a number of points where Tycho had found Mars and then failing in the attempt to draw a circle whose rim would pass through all of them. To discover the true orbit of Mars from Tycho’s observations required a level of subtlety, insight, and inventiveness from Kepler that arguably has not been surpassed in the history of science.

The attack on the problem of Mars as it was originally
assigned by Tycho to Longomontanus and Kepler was not intended to address the question whether or not Mars’s orbit was circular. It involved two basic calculations. The first was the position of Mars’s apsidal line—the straight line passing through its aphelion, equant, eccentric, Sun (for in the Tychonic system Mars orbited the Sun), and perihelion. The second was the extent of Mars’s eccentricity
(how far the center of Mars’s orbit was from the Sun). If Ptolemy was right about the position of the equant in relation to the center of the system and the eccentric, they could expect to find that Mars’s equant was twice as far from the Sun as the eccentric was.

From the time Kepler first began working with Tycho’s data at Benatky, he chose to let the tight constraints of mathematical/geometrical
logic
and precise observations be his primary guides and to give them, for a while, precedence over the ideals of symmetry and harmony. However, he had by no means abandoned those ideals. He would continue to measure his theories against them and be uncomfortable when his results did not survive the test. Kepler was setting a precedent still followed in science, where symmetry, harmony, and logical
beauty are not the most important criteria for judging whether a theory is correct, but where there is suspicion if those hallmarks are absent. Kepler had also not given up on his former theories: He hoped with Tycho’s observations to be able to find out whether his polyhedral and harmonic theories were correct.

In other ways as well, Kepler did not begin his assault on the orbit of Mars with
his mind a tabula rasa. Though he described his efforts as a voyage of exploration, he, like most explorers, knew the direction he thought he was heading, if not exactly what he would find there. He had already come to believe that understanding planetary motion required knowing the physical explanation for the motion, and he had already reached the conclusion that an essential part of the physical
explanation was a force residing in the Sun that caused the planets to move in their orbits.

Kepler did not think that mathematical rigor, ideals of symmetry and harmony, and the search for a physical explanation were incompatible. He dove into a body of data that he trusted implicitly, though it was not his own, placing his bets that if his math was good enough and his instincts correct,
he would come out the other side with his convictions about a physical explanation confirmed, and also clutching the trophies of symmetry and harmony. No one had traveled this particular route before. Kepler was not merely using science to find answers; he was working out what “science” was and would be, for himself and future generations.

Unlike Galileo when he wrote his famous
Dialogo
, Kepler
did not intend the book he first referred to as “Commentaries on the Theory of Mars,” which later evolved into
Astronomia Nova
, for the popular
market
. His target audience were his fellow early-seventeenth-century astronomers, men well versed in mathematics and planetary astronomy, including Copernicus’s astronomy, although most of them did not think Copernicus had been suggesting anything “real”
when he put the Sun in the center. Kepler realized that a battle to discredit the ancient models of astronomy had best take place on familiar ground, with familiar weapons, and not look like a battle. Hence his book would have to spend some time meandering benignly through intellectual landscapes where his contemporaries felt comfortable, not to mention where they were capable of recognizing and
coming to trust his own knowledge and skill.

Kepler’s mind was well suited to this kind of discourse. At age twenty-six he had written, “There was nothing I could state
¹⁵
that I could not also contradict.” He had no problem setting more traditional models in a fair and serious manner against Tycho’s data as a way of discovering for himself and leading his readers to see that no astronomer,
even with the utmost mathematical and geometrical maneuvering, could rest his case with these theories. He hoped to convince his readers that a theory had to be able to survive the commonsense questions of what might actually be going on in the heavens among these huge, real bodies . . . and
why
. Thus Kepler thought he would set the stage for his new astronomy, and it would prove to be accurate
to the limits of Tycho’s observations. At the outset, a confident Kepler who had promised to finish this book by Easter of 1602 had no way of knowing how long it would take him, how doubtful its outcome, how much ingenuity it would require of him, how many times he would fail, and how new his new astronomy would have to be.

To understand Kepler’s achievement in writing
Astronomia Nova
, one
must bear in mind that for most of Kepler’s contemporaries there was no reason to wonder what the orbit of Earth was like. In the Ptolemaic system and even in the Tychonic system, Earth had no orbit. It sat still. Copernicus had implied that Earth orbited the Sun like the other planets, but he had not carried through with this in his
mathematical
analysis. It was a matter of extreme interest to
Kepler whether or not Earth did indeed orbit
like the other planets
, speeding up when closer to the Sun (at perihelion) and slowing down when farther away (at aphelion). He believed it must and that if he used the true, visible Sun as his reference point he would find he was right. It had been a significant move when, at Benatky in the spring of 1600, he asked for and obtained Tycho’s permission
to use the true Sun when calculating Mars’s orbit. He again used the true Sun when he determined that Earth does move more quickly when it is closer to the Sun and more slowly when it is farther away. Earth, at least in this regard, is nothing unique. It is just a planet. Kepler would spend many pages persuading his readers that it was better to use the true Sun. A physical explanation demanded
it.

Such an approach also made it ridiculous to use devices consisting of invisible circles centered on invisible points on invisible circles on invisible points. However, it was not only to keep his readers with him that Kepler continued to use traditional Ptolemaic tools such as epicycles and equants, for Kepler needed all the mathematical and geometrical help they could provide to find
his way to an astronomy that could do without them.

While Kepler had a superb mathematical mind, and improved his skills continually as he wrote
Astronomia Nova
, much of modern mathematics, including calculus, had not been invented yet. He also lacked the concept of inertia, though his contemporary Galileo understood it. Kepler did not visualize a universe in which an object keeps moving in
a straight line at the same speed unless something comes along to affect that movement. Instead he thought an object sat still unless something moved it, and if that something ceased to move it, it reverted to stillness. In looking for the causes of motion he had to ask not only, “Why does it move as it does?” but also, “Why does it move at all, and why does it keep moving?”

Perhaps most significant
of all, as Kepler set out on his quest, he was, as he put it, “armed with incredulity”—and
¹⁶
hence ready to
question
all assumptions from the past as well as the theories and discoveries he made himself along the way.

Kepler’s work with the Mars data had begun at Benatky in the unhappy spring of 1600. If the hand of God was in this enterprise, as Kepler believed—and Tycho Brahe would not have
disagreed—then God’s work on it had begun considerably earlier, and
Astronomia Nova
, which would be Kepler’s masterpiece, was the result toward which divine purpose had been moving Tycho and Kepler with the most intricate and unlikely maneuvering for at least fifty years.

No one could possibly have obeyed Tycho’s dying plea—“Let me not seem to have lived in vain”—more magnificently than Kepler
did, but as Tycho had feared, he did it his own way, not the way that Tycho had intended, and history would celebrate the Copernican revolution, not the Tychonic revolution.

20

A
STRONOMIA
N
OVA

1600–1605

AFTER BEGINNING
ASTRONOMIA
Nova
with a strong demonstration of how important the role of the Sun is, something he intended to hammer home even more emphatically later, Kepler turned his and his readers’ attention immediately to Ptolemy. His strategy was to improve and generalize Ptolemy’s theories as Ptolemy himself might have done had he had Tycho’s
data—an exercise of which no Ptolemaic astronomer could disapprove. Kepler appropriately titled this section of his book “In Imitation of the Ancients.”

Kepler felt obliged to preface his “imitation” with six chapters describing and justifying the rigorous reworking he had given Tycho’s observations in order to use them effectively. One problem was the necessity of undoing some choices and
corrections made by Tycho and his assistants. For example, Kepler made extensive use of observations of Mars at opposition. Opposition is normally loosely defined as being when a planet is on the opposite side of Earth from the Sun. However, at opposition, Mars is rarely
directly
opposite the Sun, because of its latitude north or south of the ecliptic (review
figure 7.6
). Tycho and his assistants
knew they had to compensate for this,
but
they had failed to do so consistently, nor was it always clear in the logs whether they had or had not.