Tycho and Kepler (45 page)

Kepler was still left with a stubborn problem. He had an explanation for why a planet more distant
from the Sun would move more slowly than one closer, and why a planet’s speed would vary as its distance from the Sun changed. But he lacked an explanation for why a planet’s distance from the Sun
should ever change at all
. If the only thing at work were a planet-moving force residing in the Sun, then the planets would be carried around in circles centered on the Sun, never speeding up or slowing
down. Something as yet unknown was alternately moving the planets toward and away from the Sun, to distances where the Sun’s planet-moving force was stronger or weaker. He speculated that a planet might have a mind, or some other non-mental mechanism, yet if this were so, there was still a question how that mind or mechanism would know how distant it was from the Sun. Yet a planet seemed mysteriously
able to decide how far it should move away before coming back, and when it was close enough to move away again. Kepler could think of only one way in which distance from the Sun shows up clearly, and that is in the apparent size of the Sun’s disk. He conceded that there might be ways of “perceiving” that humans knew nothing about.

An explanation for planetary movement that endows a planet
with a “mind” to help steer it through the heavens does not sound to modern ears like a “physical explanation.” Kepler was not fond of the idea himself, but he could not dismiss it.

At this point in his book, Kepler decided that he had, for the moment, done all he could in the way of pursuing physical explanations and needed to return to the problem of describing the planets’ true motions
mathematically and geometrically.

How much time it took a planet to travel a given distance along its orbit depended on how far it was from the Sun, so much was clear. The planet was slightly changing its distance from the Sun, and
hence
its speed, continually all the way around the orbit. To get a handle on these changes, Kepler needed integral calculus, but that would not be invented for
at least three-quarters of a century. He nonetheless found another way.

A circle has 360 degrees, so Kepler divided the circumference of a circular orbit into 360 equal arcs. Then he laboriously proceeded to calculate the distance from the Sun (off-center in that circle) to each of these separate arcs, as if measuring the length of every spoke of a wheel that has an off-center hub. Of course
he had to calculate only 180 of the 360 “spokes,” since planet-Sun distances on the other side of the orbit would be the same. Having done that, he could, for instance, imagine a planet starting at aphelion and passing through the first 30 of the 360 degrees of its orbit. The sum of those 30 orbit-to-Sun distances was to the sum of all 360 distances, as the time it took the planet to move those
30 degrees was to the time it took the planet to complete an entire orbit of 360 degrees. Even for Kepler this procedure became unendurably tedious and complicated. He decided to look for a shortcut.

Kepler recalled a method that the ancient mathematician Archimedes had used to calculate the area of a circle. Archimedes reasoned that a circle was made up of an infinite number of isosceles
triangles
^fn3
with their bases on the rim of the circle and their apexes at the center. Knowing he must use something more manageable than an infinite number, Archimedes, like someone who has baked a pie for a few that many show up to eat, divided the circle instead into very fine, equal isosceles triangles, again with their bases on the rim of the circle and their peaks at the center.

It went
without saying that when Archimedes combined a few adjacent triangles of equal size, he was doing two things. One, he was combining their bases along the rim of the circle. The more triangles
he
combined, the longer the portion of the rim taken up by those bases. If he combined all of the triangles, he would have taken up the complete circle. Two, he was combining the areas of the triangles. If
he doubled the number of triangles in the group, the area doubled; triple the number, and the area tripled; and so forth. Clearly there was a relationship between the amount of the rim that the combined bases covered and the area of the combined triangles. For example, for an arc twice as long, the area would be twice as big.

Kepler suggested that combining the triangles does more than combine
their areas and combine the lengths their bases take up on the rim of the circle. He thought of the circle as a spoked wheel with an
infinite number
of spokes representing center-to-rim distances. No matter what size one made the triangles,
each
triangle would contain an infinite number of these spokes. Kepler, like Archimedes, could not compute with infinite numbers. However, though it was not
possible to say how many spokes were contained in any one triangle or combination of triangles, it was reasonable to conclude that the more triangles you combined, the more of these spokes there were in the combined area. Even more precise than that: Combine two triangles of equal area, and the number of spokes doubles, and so forth. With this idea, Kepler had found a way to think about the relationship
between orbit-to-Sun distances, the time that passed as the planet moved along its orbit, and areas within the circle. Whether this line of reasoning could be applied to an off-center orbit, where the triangles were no longer isosceles triangles, was problematic. And of course the whole point of the exercise was to illuminate the workings of an off-center orbit.

Kepler persevered, and he reached
a tentative conclusion: A straight line drawn from a planet to the Sun, as the planet orbits, would sweep out equal areas of the circle in equal times. When he tried this rule with Earth’s orbit, it worked. Though he was not yet nearly confident enough about it to declare that it was correct, and never even clearly stated it in
Astronomia Nova
, Kepler had arrived at what has come down through
history as his “area rule,” his “second law of planetary motion.” Confusingly, he did not discover his “first law” until somewhat later.

Figure 20.3: If the Sun were in the precise center of the orbit (a), the triangles would be isosceles triangles. But with the Sun off-center (b), the triangles Kepler was considering were no longer isosceles triangles.

Figure 20.4: Kepler’s area rule, shown here in its final form, as his second law of planetary motion, with an elliptical orbit. When he first arrived at the area rule, he was still trying to apply it to a circular orbit.

Imagine the planet moving around the orbit with a straight line drawn from it to the Sun. As the planet moves, so does the straight line. Watch the line move for, let
us say, two minutes, then measure the area of the pie wedge it has “swept out.” For every two-minute interval, the wedge will have that same
area
, but it will not always be the same shape, nor will the edge of it that touches the orbit always be the same length. Near the Sun the wedge will be fat and cover a long portion of the orbital path. Far from the Sun it will be thin and cover a much shorter
portion, meaning that far from the Sun the planet is moving a shorter distance in the two-minute interval.

Kepler realized immediately that his old distance rule and this new area rule were not necessarily the same. The limits of observational accuracy made it impossible to judge which was correct for Earth. He knew he must look at the orbit of another planet.

At this critical moment in
October 1602, Kepler was rudely interrupted by Tengnagel’s return to Prague and his conclusion that Kepler had made no progress in the use of Tycho’s observations. It is not difficult to understand why Kepler secretly kept the Mars data, judging, correctly, that Tengnagel would not notice at least for a while.

The old “problem of Mars” now offered a splendid opportunity. Because Mars’s orbit
was farther from being centered on the Sun than Earth’s orbit was, a flaw in the area rule was more likely to reveal itself. Kepler used the area rule to compute where Mars should be in its orbit at given times during the 687 days the planet takes to complete the orbit, and then he checked these predictions against the heliocentric longitudes of his Vicarious Hypothesis. He found agreement when
it came to certain parts of the orbit but not others. In fact, he was back to an eight-minute discrepancy! Again, it had come to a showdown: Either the circular orbit was wrong, or the area rule was wrong. Kepler could not rule out even the possibility that
both
might be wrong.

Though still far from completely trusting his area rule, Kepler decided to take the plunge and try a noncircular
orbit. A triangulation like the one he had used earlier for Earth’s orbit indicated that Mars’s path was indeed not a circle but bowed in at the sides. Mars was like a racer who cheats by coming within the circle of a circular racetrack. Doing that while still having to make it around two goalposts (aphelion and perihelion) would change a circular race into an oval race.

The precise amount
by which Mars was “cheating” in this race was fiendishly difficult to establish. Circles, except for size, are identical. “Oval,” on the other hand, is a much less precise term. An ellipse is one
kind
of oval, the best defined geometrically and the one a man obsessed with harmony and symmetry in nature might be expected to assume was correct. Kepler did not. Not only did movement in an elliptical
orbit appear to defy a physical explanation, but it also seemed too easy an answer. Kepler wrote to his friend David Fabricius
⁶
that surely if the orbit were a perfect ellipse the problem he had been struggling with would have been solved long ago by Archimedes or Apollonius.

At the time he wrote that letter, in July 1603, Kepler had been forced to abandon the struggle with Mars, because Tengnagel
had finally noticed that the Mars observations were missing and confiscated them. Kepler was working on
Astronomiae Pars Optica
instead. It was not until a year later that he had the Mars data again.

As Kepler resumed juggling ovals, which he would continue to do for the rest of 1604 and in the beginning of 1605, his frustration grew intense. His math was inadequate. He was suspicious of his
area rule. He even had some doubts whether Mars’s orbit made sense mathematically
at all
. An attempt he made to calculate Mars’s positions degree by degree gave him unsatisfactory results and was the sort of procedure he despised. This was not geometry, and Kepler took issue with God on the matter, in words he might have used to comment about a human colleague: “Heretofore we have not
⁷
found such
an ungeometrical conception in his other works.” Kepler had not changed in his intolerance of procedures or results that insulted his geometrical sensibilities.

Kepler resorted to working with an ellipse that he called the “approximating ellipse,” to see what he might learn from the exercise. That presented a new problem. He had earlier (as he described it) been obliged to “squeeze in” his
circular orbit as though he were holding a “fat-bellied sausage” in his hand and squeezing it in the middle so that the meat was forced out into the ends. With his approximating ellipse, he had squeezed the sausage too much. The correct orbit had to be something in between.

Kepler’s desire for a physical explanation made his efforts more
difficult
. He had begun to think that the force resembling
a magnetic force might account not only for the motion of the planets around the Sun but also for their motion toward and away from it. That was out of the question with an elliptical orbit. One of Kepler’s uses of an epicycle as a computational device had led him to have rather high hopes for the magnetic hypothesis, so he brought yet another epicycle out of storage. That resulted in a “puffy-cheeked”
orbit
(via buccosa)
. It is one of the ironies of scientific history that it was an error in his calculations that caused Kepler to reject this orbit. Kepler had reached chapter 58 when he wrote, “I was almost driven to madness
⁸
considering and calculating this matter. I could not find out why the planet would rather go on an elliptical orbit!”