The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)
Part II. Hope and Monsters
Chapter 27. Cracking the Cosmic Safe
For nearly twenty years Kepler would stare at Tycho’s figures, certain that they concealed a hidden message but for months and years at a time unable to make any headway in deciphering it. He had long since abandoned his geometric models, on the grounds that they simply did not fit the data. The problem was that nothing else did, either.
Kepler knew, for example, how long it took each planet to orbit the sun—Mercury, 3 months; Venus, 7 months; Earth, 1 year; Mars, 2 years; Jupiter, 12 years; Saturn, 30 years—but try as he might he could not find a rule to connect those numbers. This was a task with some resemblance to making sense of the numbers on a football scoreboard if you had never heard of football. The number 3 appears sometimes and so do 7 and 14, but never 4 or 5. What could be going on?
Even armed with Tycho’s astronomical data, Kepler took six years to find the first two of the three laws now named for him. The story of Kepler’s discovery of his laws is a saga of false starts and dead ends piled excruciatingly one upon another, while poor Kepler despaired of ever finding his way.
Kepler’s first law has to do with the paths the planets travel as they orbit the sun. Kepler shocked his fellow astronomers—he shocked himself—by banishing astronomy’s ancient emblem of perfection, the circle. But Tycho’s data were twice as accurate as any that had been known before him, and Kepler, who had indulged himself in endless speculative daydreams, now turned the world upside down because of a barely discernible difference between theory and reality. “For us, who by divine kindness were given an accurate observer such as Tycho Brahe,” Kepler wrote, “for us it is fitting that we should acknowledge this divine gift and put it to use.” To take Tycho’s measurements seriously meant to acknowledge, albeit slowly and reluctantly, that the planets simply did not travel in circles (or in circles attached to circles or any such variant).
Worn down by endless gruesome calculations, Kepler nearly despaired of ever finding the patterns hidden inside the astronomical records. (He referred wearily to his hundreds of pages of calculations as his “warfare” with the unyielding data). Finally he found that each planet orbits the sun not in a circle but in an ellipse, a kind of squeezed-in circle. This meant, among other things, that the distance from the sun to a planet was not constant, as it would be if the planet traveled in a circle, but rather always changing.
All circles are identical except in size—this was part of what made them perfect—but ellipses come in infinite variety, some barely distinguishable from circles and others long and skinny. An ellipse is not just an oval but an oval of a specific sort. (To draw an ellipse push two tacks into a piece of cardboard and drop a loop of string around them. Pull the string taut with a pencil and move the pencil along. Each tack is called a focus. The defining property of an ellipse is that, for every point on the curve, if you measure the distance from one focus to the pencil tip and then add to that number the distance from the pencil tip to the other focus, the sum is always the same.37)
For every point on the ellipse, the distance from F1 to the pencil’s tip plus the distance from F2 to the pencil tip is the same.
In the case of the planets, Kepler found, the sun sits at one focus of an ellipse. (The other focus does not correspond to a physical object.) This was Kepler’s first law—the planets travel in an ellipse with the sun at one focus. This was truly radical. Even Galileo, revolutionary though he was, never abandoned the belief that the planets move in circles.
According to Kepler’s first law, the planets orbit the sun not in a circle but in an ellipse.
* * *
Kepler’s second law was heretical, too. It had to do with the planets’ speed as they travel, and it involved another assault on uniformity. The planets didn’t travel in perfect circles, Kepler claimed, and they didn’t travel at a steady pace, either. The spur was Kepler’s belief that the sun somehow pushed the planets on their way. If so, it stood to reason that the force pushed harder when a planet was near the sun and more weakly when it was farther away. When a planet neared the sun, it would race along; when far away, it would dawdle.
It took Kepler two years of false starts to find his second law. (He earned his living, in the meantime, as imperial mathematician to Rudolph II, the Habsburg emperor whose court was in Prague. Kepler’s official duties largely centered on such tasks as preparing horoscopes and making astrology-based forecasts of next season’s weather or a stalemated war’s outcome.) His great insight was finding a way to capture the planets’ uneven motion in a precise, quantitative rule. The natural way to describe a planet’s motion was to chart its position every ten days, say, and then compute the distance between one point and the next. But that procedure turned out not to reveal any general rule. In a moment of inspiration Kepler saw a better way. The key was to think not of distance, which seemed natural, but of area, which seemed irrelevant.
According to Kepler’s second law, the triangles (in gray) have different shapes but are identical in area.
Kepler’s second law: the line from a planet to the sun sweeps out equal areas in equal times.
Proud of his discoveries though he was, Kepler had no great fondness for these laws, because he had no idea where they’d come from. Why had God not employed circles? Circles were perfect; ovals and ellipses were, Kepler lamented, a “cartload of dung.” And if for some reason He had chosen ovals, why choose ellipses in particular rather than egg shapes or a thousand other possibilities?
Kepler’s third law seemed the most arbitary of all, and proved the hardest to find. Kepler’s first two laws had to do with the planets considered one at a time. His third law dealt with all the planets at once. Kepler’s quest was once again, just as it had been with his Platonic solids model, to find what the various orbits had to do with one another. God surely had not set the planets in the sky arbitrarily. What was His plan?
Kepler had two sets of numbers to work with—the size of each planet’s orbit and the length of each planet’s “year.” Neither set of figures on its own revealed any pattern. The size of Earth’s orbit, for instance, revealed nothing about the size of Mars’s orbit, nor did the length of one planet’s “year” (the time it took to complete one circuit of the sun) provide any clue to the length of a year on a different planet. Kepler turned his attention to looking at both numbers together, in the hope of finding a magic formula.
The general trend was clear—the farther a planet was from the sun, the longer its year. That stood to reason, because near-to-the-sun planets had small orbits to trace and distant planets big ones. But it wasn’t a matter of a simple proportion. Planets farther from the sun had more distance to cover than closer-in planets and they traveled more slowly. It would be as if ships crossing the ocean traveled more slowly than ships hopping along the coast from port to nearby port.
Since he had no idea of the forces that moved the planets, Kepler took on the code-breaking challenge as if it were purely a task in numerology. Like a safecracker armed with nothing but patience, he tried every combination he could think of. If there was no pattern in the lengths of the different planets’ years, for instance, perhaps there was a pattern if you took the lengths of the years and squared them. Or cubed them. Or perhaps you could see a pattern if you computed each planet’s maximum speed and compared those. Or the minimum speeds. For more than a dozen years, Kepler tried one combination after another. He failed every time.
Then, out of the blue, “On March 8 of this present year 1618, if precise dates are wanted, the solution turned up in my head.” The discovery itself was complicated. Characteristically, so was Kepler’s response, which combined gratitude to God, immense pride in his own achievement, and his customary willingness to paint himself unflatteringly. “I have consummated the work to which I pledged myself, using all the abilities that You gave to me; I have shown the glory of Your works to men,” he wrote, “but if I have pursued my own glory among men while engaged in a work intended for Your glory, be merciful, be compassionate, and forgive.”
What Kepler had found was a way—a mysterious, complicated way—to tie the orbits of the various planets together. It required that you perform a messy calculation. Choose a planet, Kepler said, and then take its orbit and cube it (multiply it by itself three times). Next, take the planet’s year and square it (multiply it by itself ). Divide the first answer by the second answer. For every planet, the result of that calculation will be the same. Kepler’s third law is the assertion that if you follow that unappetizing recipe the answer always comes out the same.
Kepler knew, for instance, that Mars’s distance from the sun is 1.53 times Earth’s distance, and Mars’s year is 1.88 times Earth’s year. He saw—somehow—that 1.53 × 1.53 × 1.53 = 1.88 × 1.88. The other planets all told the same story. (Put another way, the length of a planet’s year depends not on its distance from the sun, or on that distance squared, but on something in between—the distance raised to the 3/2power.)
But why? What did it mean?
The numbers worked out, which seemed beyond coincidence, but it all sounded like mumbo jumbo. Of all the ways that God might have arrayed the planets and set them orbiting on their way, why had He picked one built around this curious business of squaring and cubing?
The safe door had swung open, but Kepler had no idea why.