Two right triangles, two sets of numbers, like two words from a coded message. First set: 3, 4, 5. Second set: 5, 12, 13. What do the two triplets have in common?

For two thousand years, we have called the answer the Pythagorean theorem-the length of one short side, squared, plus the other short side, squared, equals the long side, squared. 3 + 4 + 4 = 5 = 5. For the second triangle, 5 + 12 + 12 = 13 = 13.30 More to the point, the relationship holds for More to the point, the relationship holds for every every right triangle whatsoever, whether it is scratched in the sand or splayed across the heavens. right triangle whatsoever, whether it is scratched in the sand or splayed across the heavens.31 In modern terms, the theorem is usually written as a + b + b = c = c. In the pictorial terms the Greeks preferred, the theorem is about squares, not numbers, and Pythagoras's claim is that the area of one small square added to the area of the other small square is exactly the same as the area of the large square. (See drawing below.) The two approaches, numerical and pictorial, are exactly equivalent. The choice between them is purely a matter of taste, like the choice between an architectural drawing and a scale model.

[image]

Pythagoras's theorem says that the area of one small square plus the area of the other small square is exactly equal to the area of the large square.

Chapter Twenty-Three.

G.o.d's Strange Cryptography If you had somehow happened to guess the Pythagorean theorem, how would you prove it? It's not enough to draw a right triangle, measure the sides, and do the arithmetic. That would only serve to verify the theorem for one example, not for all all right triangles. Moreover, not even the most careful measuring could confirm that the sum worked out precisely, to the millionth decimal point and beyond, as it must. But even a dozen successful examples, or a hundred, or a thousand, would still fall short of proof. "True beyond a reasonable doubt" applies in law and in ordinary life-who doubts that the sun will rise tomorrow?-but the Greeks demanded more. right triangles. Moreover, not even the most careful measuring could confirm that the sum worked out precisely, to the millionth decimal point and beyond, as it must. But even a dozen successful examples, or a hundred, or a thousand, would still fall short of proof. "True beyond a reasonable doubt" applies in law and in ordinary life-who doubts that the sun will rise tomorrow?-but the Greeks demanded more.

Here is one cla.s.sic proof, which proceeds in jigsaw-puzzle fashion and almost wordlessly. In math as in chess, crucial moves often look mysterious at the time. Why move a knight there there when all the action is when all the action is here here? In this case, the unexpected move that brings the answer in reach is this: take the original triangle and make three identical copies of it, so that you have four triangles all exactly the same.

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What's the gain in dealing with four triangles when we weren't sure what to do with one? The gain comes in imagining the triangles as cardboard cutouts and then sliding them around on a table in different arrangements. Look at Figure X and Figure Y below. Two different arrangements, both with the same four triangles and some white s.p.a.ce. In both cases, the outlines (in bold) look like squares. How do we know they really are squares and not just four-sided, squarish shapes?

Stare at Figure X and Figure Y for a few seconds. All the bold sides are the same length (because each bold side is made up of a long side of the original triangle and a short side). And all the corners are right angles. So the bold shape in Figure X is a square, and so is the bold shape in Figure Y, and both squares are precisely the same size.

Almost done. Each bold square encloses the same area. Each bold square is made up of four identical triangles and some white s.p.a.ce. Stare at the pictures again. The large white square in Figure X has to be has to be exactly the same in area as the two smaller white squares in Figure Y. Voila, we have Pythagoras! exactly the same in area as the two smaller white squares in Figure Y. Voila, we have Pythagoras!

[image]

Figure X [image]

Figure Y Why did the Greeks find that discovery so astonishing? Not for its utility. No Greek would have asked, "What good is it?" What good is a poem or a play? Would a sculpture be more admirable if it could also serve as a doorstop? Mathematics was true and it was beautiful, and that was more than enough. The point was not to find the length of a diagonal across a rectangular field without having to measure it, although the Pythagorean theorem lets you do that. The Greeks had loftier goals.

The Pythagorean theorem thrilled the Greeks for two reasons. First, simply by thinking-without using any tools whatsoever-they had discovered one of nature's secrets, an eternal and never-before-suspected truth about the structure of the world. Second, they could prove it. Unlike nearly any other valid observation-vinegar is tart, Athens is dusty, Socrates is short-this particular observation was not only true but necessarily necessarily true. One of G.o.d's thoughts, finally grasped by man. true. One of G.o.d's thoughts, finally grasped by man.

Like all the best insights, it is simultaneously inevitable and surprising. But it may also be surprising that the Greeks took for granted that their mathematical theorems were facts about the world about the world rather than man-made creations like houses or songs. Is mathematics invented or discovered? The Greeks came down emphatically in favor of "discovered," but the question is ancient, like rather than man-made creations like houses or songs. Is mathematics invented or discovered? The Greeks came down emphatically in favor of "discovered," but the question is ancient, like what is justice? what is justice? and apparently just as difficult to resolve. and apparently just as difficult to resolve.

On the one hand, what could more plainly be human inventions than the concepts of geometry and algebra? Even the simplest mathematical notion has no tangible existence in the everyday world. Who goes for a walk and trips over a 3? On the other hand, what could be more obvious than that the truths of mathematics are facts about the world, whether or not any humans catch on to them? If two dinosaurs were at a watering hole and two more dinosaurs came along to join them, the science writer Martin Gardner once asked, weren't there four dinosaurs altogether? Didn't three stars in the sky form a triangle before the first humans came along to define triangles?32 Newton and the other scientists of the seventeenth century shared the Greek view, and they coupled it with their own fundamental belief that the world was a cosmic code, a riddle designed by G.o.d. Their mission, in the words of one prominent writer of the day, was to decode that "strange Cryptography." The Greeks had harbored similar ambitions, but the new scientists had advantages their predecessors had lacked. First, they had no taboos about studying motion mathematically. Second, they had calculus, a gleaming new weapon in the mathematical a.r.s.enal, to study it with.

Just as important, they had complete, unbreakable faith that the riddle had an answer. That was vital. No one would stick with a crossword puzzle if they feared that the areas of order might be mixed with patches of gibberish. Nature presented a greater challenge than any crossword, and only the certain knowledge that G.o.d had played fair kept scientists struggling year after year to catch on to His game.

Even so, the task was enormously difficult. Nonmathematicians underestimated the challenge. When Francis Bacon spoke of the mysteries of science, for instance, he made it sound as if G.o.d had set up an Easter egg hunt to entertain a pack of toddlers. G.o.d "took delight to hide his works, to the end to have them found out."

Why would G.o.d operate in such a roundabout way? If his intent was to proclaim His majesty, why not arrange the stars to spell out BEHOLD BEHOLD in blazing letters? To seventeenth-century thinkers, this was no mystery. G.o.d in blazing letters? To seventeenth-century thinkers, this was no mystery. G.o.d could could have put on a display of cosmic fireworks, but that would have been to win us over by shock and fear. When it came to intellectual questions, coercion was the wrong tool. Having created human beings and endowed us with the power of reason, G.o.d surely meant for us to exercise our gifts. have put on a display of cosmic fireworks, but that would have been to win us over by shock and fear. When it came to intellectual questions, coercion was the wrong tool. Having created human beings and endowed us with the power of reason, G.o.d surely meant for us to exercise our gifts.

The mission of science was to honor G.o.d, and the best way to pay Him homage was to discover and proclaim the perfection of His plans.

Chapter Twenty-Four.

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 G.o.d 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 cla.s.sroom was nearly deserted. And then, on a summer day, while teaching a cla.s.s on astronomy, Kepler had his Eureka! Eureka! moment. To the end of his life, he would remember the instant when he glimpsed G.o.d's blueprint. moment. To the end of his life, he would remember the instant when he glimpsed G.o.d'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...o...b..t, 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 cla.s.sroom, 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 jovial and and saturnine 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. 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.

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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.) [image]

Continuing in the same way, Kepler soon had a circle with evenly s.p.a.ced, 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.

[image]

Staring at that circle within a circle, Kepler found himself almost staggering. (Kepler would have loved The Da Vinci Code. 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 G.o.d 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...o...b..t 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 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...o...b..t and Jupiter's...o...b..t 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 G.o.d had chosen 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...o...b..t and Jupiter's...o...b..t 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 G.o.d had chosen these these two circles in particular. The triangle gave him the answer. 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...o...b..t, the inner circle Jupiter's. The triangle that tied the two together was the first shape in geometry. Kepler stared at that geometric emblem.

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He performed a quick calculation-the outer circle in his diagram was twice the circ.u.mference of the inner circle. And Saturn's...o...b..t was twice Jupiter's. He had broken G.o.d'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 next two planets must depend on the next simplest shape, a square. two planets must depend on the next simplest shape, a square.

Kepler drew a circle, representing Jupiter's...o...b..t. 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, G.o.d-designated circle that fit perfectly. That inner circle depicted Mars's...o...b..t.

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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.

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Kepler believed that G.o.d 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 a.s.signed each planet to its proper place but shown why it had had to occupy that place. 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 G.o.d have led him astray?

Chapter Twenty-Five.

Tears of Joy At last the light dawned. Kepler had been thinking in two dimensions, in the flat world of circles and triangles and squares. But the universe has three dimensions. How much time had he wasted? "And now I pressed forward again. Why look for two-dimensional forms to fit orbits in s.p.a.ce? One has to look for three-dimensional forms-and, behold dear reader, now you have my discovery in your hands!"

The switch to three dimensions represented far more than a chance to salvage a pet theory. One of the riddles that tormented Kepler had to do with the number of planets-there were exactly six. (Ura.n.u.s, Neptune, and Pluto were not yet known.33) Why had G.o.d picked six, Kepler asked, "instead of twenty or one hundred"? He had no idea, and all his fussing with squares and pentagons and hexagons had brought him no nearer to an answer.

But now he realized that he had overlooked a glaring clue. Euclid had proved, two thousand years before, that in three dimensions the story of symmetrical shapes has an extraordinary twist. Working in two dimensions, you can draw an endless succession of perfectly symmetrical, many-sided figures-triangles, squares, pentagons, hexagons, and so on, forever. If you had enough patience, you could draw a hundred-sided polygon or a thousand-sided one. (All you'd have to do is draw a circle, mark equally s.p.a.ced dots on it, and then connect each one to its next-door neighbors.) In three dimensions, where there is more room, you might expect the same story-a handful of simple shapes like pyramids and cubes and then a cascade of increasingly complicated ones. Just as a pyramid is made of triangles pasted together, and a cube is made of squares, so you might guess that you could glue fifty-sided shapes together, or thousand-sided ones, and make infinitely many new objects.

But you can't. Euclid proved that there are exactly five "Platonic solids"-three-dimensional objects where each face is symmetrical and all the faces are identical. (If you needed dice to play a game, the mathematician Marcus du Sautoy points out, these five shapes are the only possible ones.) Here is the complete array. There are no others: [image]

Only five. And there are six planets. Now Now Kepler had it. He still had to work out the details, but at last he'd seen the big picture. Each planet traveled around the sun, its...o...b..t confined to a particular sphere. The spheres sat one inside the other. But what determined the sizes of the spheres? G.o.d, the greatest of all geometers, surely had a plan. After a false start, Kepler had seen it. Each sphere fit snugly and symmetrically inside a Platonic solid. Each Platonic solid, in turn, fit snugly and symmetrically inside a larger sphere. In a flash, Kepler saw why G.o.d had designed the cosmos to have six planets and why those orbits have the sizes they do. He burst into tears of joy. Kepler had it. He still had to work out the details, but at last he'd seen the big picture. Each planet traveled around the sun, its...o...b..t confined to a particular sphere. The spheres sat one inside the other. But what determined the sizes of the spheres? G.o.d, the greatest of all geometers, surely had a plan. After a false start, Kepler had seen it. Each sphere fit snugly and symmetrically inside a Platonic solid. Each Platonic solid, in turn, fit snugly and symmetrically inside a larger sphere. In a flash, Kepler saw why G.o.d had designed the cosmos to have six planets and why those orbits have the sizes they do. He burst into tears of joy.

"Now I no longer regretted the lost time," he cried. "I no longer tired of my work; I shied from no computation, however difficult." On and on he calculated, computing orbits, contemplating octahedrons and dodecahedrons, working without rest in the hope that at last he had it right but always terrified that once again his "joy would be carried away by the winds."

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Kepler devised a new, more elaborate scheme to explain the planets' orbits. G.o.d had built the solar system around the five "Platonic solids." The diagram at right, with the sun at its center, is a detail of the drawing at left. The sun sits inside a nested cage; the inmost shape is an octahedron.

But it wasn't. "Within a few days everything fell into its place. I saw one symmetrical solid after the other fit in so precisely between the appropriate orbits, that if a peasant were to ask you on what kind of hook the heavens are fastened so that they don't fall down, it will be easy for you to answer him."

Kepler rejoiced in his success. "For a long time I wanted to become a theologian," he told an old mentor. "For a long time I was restless. Now, however, behold how through my effort G.o.d is being celebrated through astronomy."

In 1596 he presented his theory to the world in a book called The Mystery of the Universe The Mystery of the Universe. Even with his book completed, Kepler fretted about whether his model fit the actual data about the planets' orbits quite well enough. For the time being, he managed to fight down his doubts. He happily devoted long hours to constructing models of his solar system from colored paper and drawing plans for a version made of silver and adorned with diamonds and pearls. "No one," he boasted, "ever produced a first work more deserving of admiration, more auspicious and, as far as its subject is concerned, more worthy."

In the decades to come Kepler would make colossal discoveries, but his pride in his elaborate geometric model never faded. Centuries later the biologist James Watson would proclaim his double helix model of DNA "too pretty not to be true." Kepler had felt the same joy and the same certainty, but eventually the data left him no choice but to acknowledge that he had gone wrong, again.

His perfect theory was only a fantasy, but it proved enormously fruitful even so. For one thing, Mystery of the Universe Mystery of the Universe transformed Kepler's career. He sent a copy of the book to Tycho Brahe, the leading astronomer of the day, who found it impressive. In time Kepler would gain access to Tycho's immense and meticulous trove of astronomical data. He would pore over those figures incessantly, over the course of decades, trying to make his model work and uncovering other patterns concealed in the night sky. Later scientists would rummage through Kepler's collection of numerical discoveries and find genuine treasure among the dross. transformed Kepler's career. He sent a copy of the book to Tycho Brahe, the leading astronomer of the day, who found it impressive. In time Kepler would gain access to Tycho's immense and meticulous trove of astronomical data. He would pore over those figures incessantly, over the course of decades, trying to make his model work and uncovering other patterns concealed in the night sky. Later scientists would rummage through Kepler's collection of numerical discoveries and find genuine treasure among the dross.

Kepler valued Mystery of the Universe Mystery of the Universe so highly because it was there that he had unveiled his great breakthrough. But in the course of discussing his model of the heavens, he had scored another history-making coup. Kepler had followed Copernicus in placing the sun at the center of his model, but then Kepler had moved a crucial step beyond all his predecessors. Not only did all the planets circle the sun, he noted, but the farther a planet was from the sun, the slower it traveled in its...o...b..t. Somehow the sun must so highly because it was there that he had unveiled his great breakthrough. But in the course of discussing his model of the heavens, he had scored another history-making coup. Kepler had followed Copernicus in placing the sun at the center of his model, but then Kepler had moved a crucial step beyond all his predecessors. Not only did all the planets circle the sun, he noted, but the farther a planet was from the sun, the slower it traveled in its...o...b..t. Somehow the sun must propel propel the planets, and whatever force it employed plainly grew weaker with distance. the planets, and whatever force it employed plainly grew weaker with distance.

Kepler had not yet found the law that described that force-that would take him another seventeen grueling years-but this was a breakthrough even so. Astrologers and astronomers had always focused their attention on mapping the stars and charting the planets' journeys across the sky. The goal had been description and prediction, not explanation. No one before Kepler had ever focused on asking what it was that moved the planets on their way. From now on, scientists looking at the heavens would picture the stars and planets as actual, physical objects pushed and tugged by some cosmic engine and not simply as dots on a chart.

"Never in history," marvels the historian of science Owen Gingerich, "has a book so wrong been so seminal in directing the future course of science."

Chapter Twenty-Six.

Walrus with a Golden Nose From the start Kepler's faith that G.o.d was a mathematician both impeded him and spurred him. First his faith lured him into devoting years to his Platonic pipe dream; when that dream dissolved it motivated him to search elsewhere, in the certain knowledge that there had to be some some mathematical pattern that explained the solar system. Through all his years of searching, Kepler's fascination was less with the objects in the sky-the sun, stars, and planets-than with the relationships among them. Not the things but the patterns. "Would that G.o.d deliver me from astronomy," Kepler once wrote, "so I can devote all my time to my work on harmonies." mathematical pattern that explained the solar system. Through all his years of searching, Kepler's fascination was less with the objects in the sky-the sun, stars, and planets-than with the relationships among them. Not the things but the patterns. "Would that G.o.d deliver me from astronomy," Kepler once wrote, "so I can devote all my time to my work on harmonies."

In time that work would yield many patterns, a few of them among the highest achievements of human thought but most of them nearly inexplicable to modern readers. When Kepler finally abandoned his elaborate geometric model of the planets, for instance, he replaced it with an equally arcane model based on music. This new search for "harmonies" built on Pythagoras's age-old insight about strings of different lengths producing notes of different pitch. Kepler's notion was that the planets in their various...o...b..ts, traveling at different speeds, corresponded to different musical notes, and "the heavenly motions are nothing but a continuous song for several voices (perceived by the intellect, not by the ear)."34 Kepler's new system, with its sopranos and tenors and ba.s.ses, was as farfetched as its predecessor, with its cubes and pyramids and dodecahedrons. As it turned out, neither model had anything to do with reality. But in the course of his obsessive, misguided quest to prove the truth of his theories, Kepler did make genuine, epochal discoveries. Scientists would eventually dub three of these "Kepler's laws," though Kepler never gave them that name nor deemed them any more praiseworthy than his other finds.35 Late in his life, when he looked back over his career, Kepler himself could scarcely pick out his breakthroughs from the mathematical fantasies that surrounded them. "My brain gets tired when I try to understand what I wrote," he said later, "and I find it hard to rediscover the connection between the figures and the text, that I established myself."

Kepler was one of the most daring, insightful thinkers who ever lived, but his career only took off when he joined forces with an astronomer who was his opposite in almost every respect. Kepler was poor and lean, a creature of ribs and patches. Tycho Brahe was rich beyond measure. Kepler was shy and ascetic, Tycho hard-drinking and rowdy. Kepler was imaginative and creative, sometimes alarmingly so, Tycho a brilliant observer but a run-of-the-mill theoretician. But the two great astronomers needed one another.

Tycho36 was a Danish n.o.bleman with a private observatory on a private island. The most eminent astronomer of the generation before Kepler and Galileo, it was Tycho who had startled the world in 1572 by proving that the new star that had somehow materialized in the sky truly was a star. Nothing about Tycho was run-of-the-mill. Round, bald, sumptuously dressed, he looked like Humpty Dumpty with a walrus mustache and a velvet cloak. He ruled his mini-kingdom like a mini-king, presiding over lavish banquets and cackling over the antics of his court jester, a dwarf named Jepp. was a Danish n.o.bleman with a private observatory on a private island. The most eminent astronomer of the generation before Kepler and Galileo, it was Tycho who had startled the world in 1572 by proving that the new star that had somehow materialized in the sky truly was a star. Nothing about Tycho was run-of-the-mill. Round, bald, sumptuously dressed, he looked like Humpty Dumpty with a walrus mustache and a velvet cloak. He ruled his mini-kingdom like a mini-king, presiding over lavish banquets and cackling over the antics of his court jester, a dwarf named Jepp.

In his student days, Tycho had lost part of his nose in a swordfight. In one version of the story, the trouble started at a wedding celebration when another wealthy young Dane reminded everyone of some odd events from a few months before. Tycho had announced with great fanfare, in a poem written in elegant Latin flourishes, that a recent eclipse of the moon foretold the death of the Turkish sultan. But, it turned out, the sultan had died six months before before the eclipse. Tycho's rival told the story with gusto, and nearly all his listeners enjoyed it. Not Tycho. The retelling of the story led to bad blood and, soon after, a duel. Tycho nearly lost his life and did lose a chunk of his nose. For the rest of his life he sported a replacement made of gold and silver. the eclipse. Tycho's rival told the story with gusto, and nearly all his listeners enjoyed it. Not Tycho. The retelling of the story led to bad blood and, soon after, a duel. Tycho nearly lost his life and did lose a chunk of his nose. For the rest of his life he sported a replacement made of gold and silver.

Despite the bl.u.s.ter and showmanship, Tycho was a genuine scholar. His observatory was the best in Europe, outfitted with a dazzling array of precision-made s.e.xtants, quadrants, and other devices for pinpointing the positions of stars. The observatory stood in a grand, turreted castle that boasted fourteen fireplaces and an astonishing luxury, running water. In Tycho's library stood a celestial globe five feet in diameter and made of bra.s.s; when a star's position was established beyond a doubt, a new dot was carefully added to the globe. Tycho boasted that his observatory had cost a ton of gold, and Kepler complained that "any single instrument cost more than my and my whole family's fortune put together."

Kepler had sent his Mystery of the Universe Mystery of the Universe to Tycho and all the other eminent scientists he could think of. Many could not fathom what he was up to. Tycho, more mystically minded than Galileo and some of the other skeptics, replied enthusiastically and soon took Kepler on as his a.s.sistant. It was an arrangement with obvious benefits for both men. Tycho had devised a hybrid model of the solar system, partway between the ancient Earth-centered model and Copernicus' sun-centered version. In this picture, the sun and moon orbited the Earth, and the five other planets...o...b..ted the sun. Tycho had compiled reams of scrupulously accurate observations, but without Kepler's mathematical help he could not demonstrate the truth of his hybrid model. Kepler had no interest in Tycho's model, but in order to make progress on his own theories he desperately needed Tycho's records. to Tycho and all the other eminent scientists he could think of. Many could not fathom what he was up to. Tycho, more mystically minded than Galileo and some of the other skeptics, replied enthusiastically and soon took Kepler on as his a.s.sistant. It was an arrangement with obvious benefits for both men. Tycho had devised a hybrid model of the solar system, partway between the ancient Earth-centered model and Copernicus' sun-centered version. In this picture, the sun and moon orbited the Earth, and the five other planets...o...b..ted the sun. Tycho had compiled reams of scrupulously accurate observations, but without Kepler's mathematical help he could not demonstrate the truth of his hybrid model. Kepler had no interest in Tycho's model, but in order to make progress on his own theories he desperately needed Tycho's records.

But Tycho h.o.a.rded them. Torn between hope that the younger man could find patterns hidden within twenty years' worth of figures and fear that he was giving away his treasure, Tycho clung to his numbers with a miser's grip. Kepler snarled helplessly. And then, out of the blue, Tycho died. (He died of a bladder infection brought on, according to Kepler, by drinking too much at a banquet and refusing to leave the table to pee.) Kepler had been with Tycho only eighteen months, but now he had what he needed. "I was in possession of the observations," Kepler noted contentedly, "and refused to hand them over to the heirs."

Chapter Twenty-Seven.

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) [image]

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.

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According to Kepler's first law, the planets...o...b..t the sun not in a circle but in an ellipse.