The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)

Part II. Hope and Monsters

Chapter 24. The Secret Plan

When Newton declared that he stood on the shoulders of giants, he was at least partly sincere. He did genuinely admire some of his fellow scientists, particularly those who’d had the good judgment to die before he came along. One of the great predecessors he had in mind was the astronomer Johannes Kepler. A contemporary of Galileo, Kepler was a genius and a mystic whose faith in God and faith in mathematics had fused into an inseparable unit.

Kepler was both astronomer and astrologer, though he never sorted out just how much the heavens influenced human affairs. “In what manner does the countenance of the sky at the moment of a man’s birth determine his character?” he wrote once, and then he answered his own question. “It acts on the person during his life in the manner of the loops which a peasant ties at random around the pumpkins in his field: they do not cause the pumpkin to grow, but they determine its shape. The same applies to the sky: it does not endow man with his habits, history, happiness, children, riches or a wife, but it molds his condition.”

For many years the sky seemed set against Kepler. He grew up poor, sick, and lonely. His childhood, according to an account he compiled later, was a long series of afflictions (“I was born premature . . . I almost died of smallpox . . . I suffered continually from skin ailments, often severe sores, often from the scabs of chronic putrid wounds in my feet”). He remained adrift into his twenties, cut off from others not only by his intelligence but also by his quarrelsome, touchy, defensive manner. “That man has in every way a dog-like nature,” he wrote, for some reason describing himself in the third person. “His appearance is that of a little lap-dog. . . . He liked gnawing bones and dry crusts of bread, and was so greedy that whatever his eyes chanced on he grabbed.”

Kepler was brilliant but restless, hopping from obsession to obsession. Astrology, astronomy, theology, mathematics all captivated him. They related to each other in some way that he could sense but not articulate. After his own university days, he managed to find work as a high school teacher, but his students found him disorganized and hard to follow, and soon his classroom was nearly deserted. And then, on a summer day, while teaching a class on astronomy, Kepler had his Eureka! moment. To the end of his life, he would remember the instant when he glimpsed God’s blueprint.

It was July 9, 1595. Kepler was twenty-four years old, and he believed fervently in Copernicus’s doctrine of a sun-centered universe. For weeks he had been laboring to find some pattern in the planets’ orbits. If you knew the size of one planet’s orbit, what did that tell you about the others? There had to be a rule. Kepler tried ever more complicated numerical manipulations. Each one failed. Now, standing at the front of his classroom, he began drawing a diagram having to do with the positions of Jupiter and Saturn, the two most distant planets then known. Kepler knew the size of both orbits, but he couldn’t see any connection between the two.

Jupiter and Saturn were important astrologically—our words jovial and saturnine are fossils of bygone doctrines—and what was especially important were the times the two planets were “in conjunction,” near one another in the sky. If they met at a certain point today, astronomers knew, they would meet next (in twenty years) at a point 117 degrees away, just under one-third of the way around the zodiac. The conjunction point after that one would be another 117 degrees along, and so on. Kepler drew a circle showing the first conjunction point, the second, and the third.


The diagram shows where Saturn and Jupiter appear in the sky together. If today they can be seen at point 1, in twenty years they will appear at 2, in twenty more at 3, and so on.

He filled in more conjunction points, each one 117 degrees along from its predecessor. (If the points had been 120 degrees apart, exactly one-third of the way around a circle, there would have been a total of only three conjunction points, because all the points after the first three would have overlapped.)


Continuing in the same way, Kepler soon had a circle with evenly spaced, numbered dots marked all the way around it. (Look at the diagram below, in which points 1 through 5 are labeled.) Each dot represented a point where Saturn and Jupiter met.

For no especially clear reason, Kepler drew a line from the first conjunction to the second, from the second to the third, and on and on. From that series of straight lines emerged, mysteriously and unexpectedly, not some straight-sided shape but a new circle. To Kepler it seemed as if his original circle had conjured up a new, smaller counterpart inside itself.


Staring at that circle within a circle, Kepler found himself almost staggering. (Kepler would have loved The Da Vinci Code.) At the moment that the new, inner circle swam into his view, he saw the secret plan behind the universe’s design. “The delight that I took in my discovery,” he wrote, “I shall never be able to describe in words.”

Only a skilled geometer with a bone-deep faith that God himself delighted in geometric riddles would have seen anything noteworthy in Kepler’s drawing. But Kepler, who knew that nothing in nature is mere coincidence, looked at his two circles and thought of his two planets, and marveled. What could it mean except that the outer circle represented the orbit of the outmost planet, Saturn, and the inner circle the orbit of the inner planet, Jupiter? And the inner circle was half the size of the outer circle, just as Jupiter’s orbit was half the size of Saturn’s!

But that was only the start. Kepler’s full discovery had an even more mystical, more geometric flavor. Saturn and Jupiter were the first two planets (counting from farthest from the sun to nearest). What connected their orbits? What else was “first”?

The answer struck Kepler like a hammerblow. This was the eureka insight. “The triangle is the first figure in geometry,” Kepler exclaimed—“first” in this case meaning “simplest”—and that first, simplest geometric figure was the key to the mystery of the first two orbits. Kepler had known all along that Saturn’s orbit and Jupiter’s orbit could be depicted as a circle inside a circle, but there are countless ways to draw one circle inside another. The mystery Kepler yearned to solve was why God had chosen these two circles in particular. The triangle gave him the answer.

Feverishly, Kepler put his brainstorm to the test. He drew a circle and inside it he drew the one triangle that stood out from all the other possibilities—the simplest triangle of all, the only one that fit perfectly inside the circle and was completely symmetrical, with all three sides identical. Inside the triangle he drew another circle. Again, he could have chosen any of countless circles; again, he made the only “natural” choice, the one circle that fit the triangle perfectly. He looked again at his drawing. The inner circle nestled snugly inside the triangle, and the triangle fit neatly and naturally into the outer circle. In Kepler’s mind, the outer circle represented Saturn’s orbit, the inner circle Jupiter’s. The triangle that tied the two together was the first shape in geometry. Kepler stared at that geometric emblem.


He performed a quick calculation—the outer circle in his diagram was twice the circumference of the inner circle. And Saturn’s orbit was twice Jupiter’s. He had broken God’s code. Now Kepler set to work in a frenzy. If the orbits of the first two planets depended on the simplest geometric shape, a triangle, then the orbits of the next two planets must depend on the next simplest shape, a square.

Kepler drew a circle, representing Jupiter’s orbit. The question was what circle would represent the orbit of the next planet toward the sun, Mars. In Kepler’s mind, the answer nearly shouted aloud. Inside the Jupiter circle, he drew a square. Inside that square he drew the one, special, God-designated circle that fit perfectly. That inner circle depicted Mars’s orbit.


And Kepler could continue in this way for all the planets, working his way in toward the sun, arraying the planets just as Copernicus had shown them. The orbits were nested one within the other, and the size of one automatically dictated the size of the next. The first two orbits were built around a triangle, which has three sides; the next two around a square, with four sides; the next two around a pentagon, with five sides; and so on. Kepler set to work drawing squares, pentagons, hexagons, septagons, with circles in between.


Kepler believed that God had arranged the planets’ orbits according to this geometric scheme. (For clarity, the diagram shows only the four outermost planets, not all six planets known in Kepler’s day.)

Johannes Kepler had discovered the architecture of the solar system. Or so he believed, and in his fever dream he filled sheet after sheet with ever more elaborate geometric diagrams. For the young, unknown astronomer, this was dizzyingly exciting. Without looking out the window, he had not only assigned each planet to its proper place but shown why it had to occupy that place.

It was perfect, it was elegant, and it was wrong. As Kepler took more time to compare the actual sizes of the planets’ orbits with the sizes his model predicted, he found mismatches he couldn’t explain away. He tried endless fixes. Nothing. How could God have led him astray?