The Clockwork Universe Part 6

Kepler's second law was heretical, too. It had to do with the planets' speed as they travel, and it involved another a.s.sault 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, quant.i.tative 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.

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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 G.o.d 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...o...b..ts had to do with one another. G.o.d 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...o...b..t and the length of each planet's "year." Neither set of figures on its own revealed any pattern. The size of Earth's...o...b..t, for instance, revealed nothing about the size of Mars's...o...b..t, 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 and they traveled more slowly. It would be as if s.h.i.+ps crossing the ocean traveled more slowly than s.h.i.+ps hopping along the coast from port to nearby port. they traveled more slowly. It would be as if s.h.i.+ps crossing the ocean traveled more slowly than s.h.i.+ps 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 grat.i.tude to G.o.d, 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 compa.s.sionate, 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...o...b..t 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 a.s.sertion 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 3/2 power.) 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 G.o.d 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.

Chapter Twenty-Eight.

The View from the Crow's Nest Kepler's laws represented a giant advance in decoding G.o.d's cryptography, even if he did not know why they were true or what they meant. The next advance came from his fellow astronomer, Galileo, who was almost exactly Kepler's contemporary.

Galileo was born in the same year as Shakespeare, and Galileo's stature in science nearly matches Shakespeare's in literature. "I believe that if a hundred of the men of the seventeenth century had been killed in infancy, the modern world would not exist," wrote Bertrand Russell. "And of these hundred, Galileo is the chief." In truth, that seems unlikely. Galileo's genius is beyond dispute, but every great scientist, from Galileo to Darwin to Einstein, had rivals at his heels. If Shakespeare had not lived, we would not have "to be or not to be." If Einstein had not lived, we might have had to wait a few years for e = mc e = mc.

The same holds for Galileo, great as he was. But Galileo did shake the scientific world out of its doldrums, and perhaps Russell was right that no one could have been better suited to that task, either temperamentally or intellectually. Galileo was brilliant, cantankerous, and expert in wielding all the weapons of intellectual combat. (Even his hair bristled, as if it, too, were poised for battle.) He spoke wittily and wrote vividly, with a knack for metaphors and homey a.n.a.logies; he had a flair for mockery, name-calling, and sarcastic put-downs; he had, when he wanted, a honeyed tongue and, as an acquaintance noted, "a way of bewitching people."

Curiously, Galileo did not not build on Kepler's work. Indeed, he seems not to have known that Kepler's laws existed at all, even though Kepler had sent him his build on Kepler's work. Indeed, he seems not to have known that Kepler's laws existed at all, even though Kepler had sent him his New Astronomy New Astronomy, which included the first two laws and endless astronomical speculation besides. (Galileo put the book away unread.) What Galileo did instead was turn to a completely different part of the cosmic riddle.

Galileo focused on solving a mystery far older than Kepler's. Nearly a century after Copernicus, the question does the Earth move? does the Earth move? still struck nearly everyone as absurd. Armed with the fame he had earned by turning his telescope to the heavens, Galileo set out to do what Copernicus and Kepler had never done, and what Kepler's laws did not do, either-find a response to the claim that the Earth could not possibly be in motion. Then he set out to tell the world. still struck nearly everyone as absurd. Armed with the fame he had earned by turning his telescope to the heavens, Galileo set out to do what Copernicus and Kepler had never done, and what Kepler's laws did not do, either-find a response to the claim that the Earth could not possibly be in motion. Then he set out to tell the world.

Galileo wrote his most important scientific works not in formal, impenetrable prose but in pugnacious dialogues, like miniature plays. He put the arguments of his rivals into the mouth of a character he named Simplicio, the embodiment of intellectual mediocrity. The name perhaps referred to an actual figure, an Aristotelian named Simplicius who lived around a thousand years before Galileo. More likely, Galileo made up "Simplicio" because it was so close to simpleton simpleton ( (sempliciotto in Italian). Certainly his readers jumped to that conclusion. And one of them, Pope Urban VIII, was notably unamused to find a pet argument of his own put in Simplicio's mouth. in Italian). Certainly his readers jumped to that conclusion. And one of them, Pope Urban VIII, was notably unamused to find a pet argument of his own put in Simplicio's mouth.

That proved to be a disastrous miscalculation, but a telltale one. Galileo's Italy was a flamboyant place. Showmans.h.i.+p was far more common than bashfulness, and Galileo was never much inclined to hide his talent in any case. Still, he dangerously overestimated his own powers of persuasion. He liked holding forth, about wine and cheese and literature to be sure, but especially about the excellence of the new picture of the heavens and the foolishness of the old view. "He discourses often amid fifteen or twenty guests who make hot a.s.saults upon him, now in one house, now in another," one friend recalled, after a dinner party, "but he is so well b.u.t.tressed that he laughs them off."

To be outnumbered was part of the fun. "If reasoning were like hauling," Galileo proclaimed, "I should agree that several reasoners would be worth more than one, just as several horses can haul more sacks of grain than one can. But reasoning is like racing and not like hauling, and a single Arabian steed can outrun a hundred plowhorses."

Galileo not only defended Copernicus against his critics but, in the course of making his argument, devised a theory of relativity. Three centuries before Einstein's version, Galileo's theory proved nearly as hard for common sense to grasp. Inside a room with the curtains pulled, Galileo showed, there is no way to tell if you're standing still or traveling in a straight line at a steady speed. Inside the compartment of a smooth-running new train, to take a modern example, no experiment you can do (short of peeking out the window) will reveal whether you're sitting motionless or racing down the track. You might think that dropping your keys would give the game away-if the train is moving to the east, wouldn't the keys fall a bit toward the west?-but in fact they fall straight down, as usual.

More to the point, what is true of a s.h.i.+p or a train is true of the Earth itself-there is no way to tell if the Earth is moving or standing still, short of carrying out sophisticated astronomical measurements. No ordinary actions we can carry out reveal whether we're moving. The same holds true for any any motion that is smooth, steady, and straight, no matter how fast it might be. (The Earth's...o...b..t is nearly circular, not straight, but the circle is so huge, compared to our speed, that any short stretch is effectively a straight line.) motion that is smooth, steady, and straight, no matter how fast it might be. (The Earth's...o...b..t is nearly circular, not straight, but the circle is so huge, compared to our speed, that any short stretch is effectively a straight line.) This was a direct attack on Aristotle and all his followers. We can be sure that the Earth does not move, Aristotle had insisted, because we see proof everywhere we look. Rocks fall straight down, not on some curved or slanted path. Buildings don't shake or topple, as they would if the ground beneath them was on the move. A moving world would be chaotic, Aristotle taught, and the most routine task would be as difficult as trying to paint a room while standing on a ladder mounted on wheels.

Galileo showed this was false. Nothing is special about a motionless world. Smooth, steady motion looks and feels exactly the same as utter stillness. The strongest argument against Copernicus-that he began by a.s.suming something that was plainly ridiculous-was invalid.

Galileo reached these far-ranging conclusions by means of the humblest experiments imaginable. He began with a metal ball and a wooden ramp. (In time he would add a bucket of water with a hole poked in it.) Galileo's pet subject was motion, in particular the motion of falling objects. For Aristotle, as we have seen, to be in motion meant to change-from one position to another, perhaps, but also from one "quality" to another, as from "foolishness" to "wisdom." Galileo was after what looked like simpler game. He wanted to know the rules that govern inanimate objects in free fall. But how could he look closely enough to tell precisely how rocks plummet?

The answer, he decided, was to slow things down. Rather than drop a rock through the air, he would roll a ball down a ramp and hope that what held for the ramp would hold for free fall. This was a nervy move. Arguments by a.n.a.logy are always risky, and here the a.n.a.logy seemed far from ironclad. But Galileo, a brilliant teacher and debater, presented this leap of faith as if it were but another step on a casual walk, and his audience leaped with him.

He began by seeing what happened when he let a ball roll down one ramp, across a table, and then up a second ramp. If the two ramps were identical, it turned out, the ball ended up at virtually the same height it had started at. (In the same way, if you let go of a marble in a circular bowl it will roll to the bottom and then up the other side to very, very near its starting height.) Then came the crucial observation. Galileo chose a second ramp that was less steep than the first one. Once again the ball ended up at the height it had started at, though this time it had to roll farther to get there. Then still another repet.i.tion, this time with a second ramp that was only tilted ever so slightly. Again, the ball eventually reached its starting height but it had to roll and roll to get there.

And suppose the second ramp was perfectly flat, not tilted at all? Then, said Galileo, the ball would roll horizontally forever forever. The flat ramp was a thought experiment, not a real one, but Galileo proclaimed a new law of nature-any object moving horizontally will continue moving horizontally forever, at the same speed, unless something comes along to intervene. (Newton's first law of motion is a generalization of the same principle.) Aristotle had decreed exactly the opposite, as we saw earlier. In Aristotle's world, motion was unnatural and always called for explanation; unless a force kept pus.h.i.+ng or pulling it, a moving object would always slow and then stop.

We should not downplay Galileo's boldness. In rejecting Aristotle, he was also dismissing what everyone has seen for themselves countless times-moving objects do do always stop. Ignore what all your experience and your common sense have taught you, Galileo said. More important than the world you actually see, more true to the essential nature of things, is an idealized, abstract, mathematical world that you can only see with the mind's eye. always stop. Ignore what all your experience and your common sense have taught you, Galileo said. More important than the world you actually see, more true to the essential nature of things, is an idealized, abstract, mathematical world that you can only see with the mind's eye.

In Galileo's hands, the simple statement that motion was natural had enormous consequences. Here was the key to his theory of relativity and the reb.u.t.tal to the Aristotelians' guffawing about a moving Earth. In his day, when roads were rutted and coaches horse-drawn, the most familiar example of smooth travel was on s.h.i.+pboard. What would happen, Galileo asked, if a sailor climbed to the top of the mast and dropped a rock? For Aristotle, this would have been an easy question. If the s.h.i.+p was motionless, at rest in a quiet harbor, the rock would fall straight down and hit the deck at the base of the mast. If the s.h.i.+p was gliding along on a gla.s.sy sea, the rock would crash to the deck several inches from the mast. Galileo disagreed. In both both cases, he insisted, the rock would fall straight down to the base of the mast. cases, he insisted, the rock would fall straight down to the base of the mast.

The reason was his first law. The s.h.i.+p, the sailors, the pa.s.sengers, the rock falling from the mast, are all in horizontal motion, all of them moving together. The rock lands at the base of the mast because mast and rock are both moving horizontally, in unison, at the same time as the rock is hurtling downward.

"Shut yourself up with some friend in the main cabin below decks on some large s.h.i.+p," Galileo wrote. Bring in some b.u.t.terflies, a fishbowl with some fish swimming around, a leaky jug dripping water into a pan on the floor. No matter how closely you looked for something out of the ordinary (the fish cl.u.s.tered against one side of their bowl, for instance, or the drops of water missing the pan), Galileo went on, "you could not tell from any of them whether the s.h.i.+p was moving or standing still."

The same holds for the Earth itself, and all its pa.s.sengers, as it speeds along on its voyage. The speeding Earth, which races in its...o...b..t at about eighteen miles per second per second, is as safe and solid a home as a s.h.i.+p safely moored at anchor in a mirror-smooth sea.

Even today, Galileo's insight doesn't come naturally. We believe him, though, because we've all carried out countless tests of our own. It sometimes happens, for instance, that we're speeding down the highway, with the car windows rolled up, when we notice a fly buzzing around. The car might be traveling at 70 miles per hour, much faster than any fly can manage, and yet the fly continues unperturbed. Why doesn't the back window slam into it at 70 miles per hour?

Or think about traveling by plane. Drop your phone in a jet and during the fraction of a second it takes to hit the floor, the plane will have traveled perhaps a hundred yards. How is it that it falls at your feet and not a football field behind you? For that matter, how can the flight attendants dare to pour coffee? While the coffee is in midair, on its way to the cup but not yet there, the cup itself will have moved hundreds of feet. How can the crew serve first-cla.s.s without scalding everyone in economy?

"A company of chessmen standing on the same squares of the chessboard where we left them, we say are all in the same place or unmoved: though perhaps the chessboard has been in the meantime carried out of one room into another." So wrote the philosopher John Locke in 1690, in one of the earliest discussions of relativity. Whether the board sits on a table or is carried from here to there makes no difference to how the game is played. As for the chess pieces, so for us. Whether the Earth sits immobile at the center of the cosmos or speeds around the sun, all our activities go on in their customary ways.

Chapter Twenty-Nine.

Sputnik in Orbit, 1687 In a story called "The Red-Headed League," Dr. Watson looks hard at Sherlock Holmes's latest visitor, but nothing strikes him as noteworthy. He turns toward the great detective. Perhaps Holmes has seen more? "Beyond the obvious facts that he has at some time done manual labour, that he takes snuff, that he is a Freemason, that he has been in China, and that he has done a considerable amount of writing lately, I can deduce nothing else," says Holmes.

Galileo and his fellow scientists favored a similar technique. By paying close attention to what others had overlooked, they could find their way to utterly unexpected conclusions. Galileo's a.n.a.lysis of life on s.h.i.+pboard showed, for instance, that a marble that rolled off a table would take precisely the same time to reach the floor whether the s.h.i.+p was moving at a steady speed or standing still. The s.h.i.+p's horizontal motion has no effect on the rock's vertical fall. In Galileo's hands, that seemingly small observation had momentous consequences.

Picture any projectile moving through the air-a baseball soaring toward the outfield, a penny flipped into the air, a dancer leaping across the stage. In all such cases, the moving object's horizontal motion and its vertical motion take place independently and can be examined separately. The horizontal movement is steady and unchanging, in line with Galileo's law of motion. Ball and coin and dancer travel a certain distance horizontally in the first second, the same distance in the next second, and so on, moving at a constant speed from liftoff until touchdown.38 At the same time, the projectile's At the same time, the projectile's vertical vertical progress-its height above the ground-changes according to a different rule. At the moment of launch, the projectile rises quickly but then it rises slower and slower, stops rising altogether, and sits poised for an instant neither rising nor falling, and then plummets earthward faster and faster. The change in speed follows a simple, precise rule, and the upward part of the flight and the downward part are exactly symmetrical. progress-its height above the ground-changes according to a different rule. At the moment of launch, the projectile rises quickly but then it rises slower and slower, stops rising altogether, and sits poised for an instant neither rising nor falling, and then plummets earthward faster and faster. The change in speed follows a simple, precise rule, and the upward part of the flight and the downward part are exactly symmetrical.

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Any object launched into the air-arrow, bullet, cannonball-travels in a curved path like this one. The moving object covers the same horizontal horizontal distance during each second of its flight. distance during each second of its flight.

Mathematically, it's easy to show that the combination of steady horizontal motion and steadily changing vertical motion makes for a parabolic path. (A parabola is an arch-shaped curve, but it is not just a generic arch; it is one that satisfies specific technical conditions, just as an ellipse is not a generic oval but one of a specific sort.) Parabolas had been painted against the sky ever since the first caveman threw a rock, but no one before Galileo had ever recognized them, and he was immensely proud of his discovery. "It has been observed that missiles and projectiles describe a curved path of some sort," he wrote. "However no one has pointed out the fact that this path is a parabola. But this and other facts, not few in number or less worth knowing, I have succeeded in proving."

G.o.d had once again shown his taste for geometry. The planets in the heavens traveled not in haphazard curves but in perfect ellipses, and objects here on Earth traced exact parabolas.

Concealed within the same observation about the independence of horizontal motion and vertical motion was a further surprise. Galileo might have found it, but he didn't. Isaac Newton did. Imagine someone firing a gun horizontally, and at the same instant someone standing next to the shooter and dropping a bullet from the same height as the gun. When the two bullets reach the ground, they will be far apart. The one shot from the gun will have traveled hundreds of yards; the other will rest in the gra.s.s directly below the spot where it was dropped. Which bullet will hit the ground first?

Surprisingly, both reach the ground at exactly the same moment. That's what it means for the bullet's vertical motion-its fall-to be independent of its horizontal motion. For Newton, that was enough to draw a remarkable conclusion.

Suppose it takes one second for a bullet dropped from a certain height to hit the ground. That means that a bullet shot horizontally from the same height would also hit the ground in one second. A more powerful gun would send the bullet faster and farther, but-if the ground was perfectly flat-that bullet, too, would fall to the ground in one second.

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Bullets shot horizontally with different force travel different distances before they come to rest, but they all fall fall at the same rate. Each second a bullet is in the air it falls 16 feet toward the ground. at the same rate. Each second a bullet is in the air it falls 16 feet toward the ground.

Newton preferred to imagine a cannon blasting away horizontally. He imagined faster and faster cannonb.a.l.l.s, covering greater and greater distances in their one-second journey. But the Earth is round, not flat.

That makes all the difference. Since the Earth isn't flat, it curves away beneath the speeding cannonball. In the meantime, the cannonball is falling toward the ground. Suppose you fired a cannonball from high above the atmosphere, horizontally. With nothing to slow it down, it would continue at the same speed forever, falling all the while. If you launched it at just the right speed, then by the time the cannonball had fallen, say, four feet, the ground itself would have fallen four feet below horizontal.

And then what? The cannonball would continue on its journey forever, always falling but never coming any closer to the ground. Why? Because the cannonball always falls at the same rate, and the ground always curves beneath it at the same rate, so the cannonball falls and falls, and the Earth curves and curves, and the picture never changes. We've launched a satellite.

Newton pictured it all in 1687.

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Chapter Thirty.

Hidden in Plain Sight Kepler had taken the first giant steps toward showing that mathematics governed the heavens. Galileo showed that mathematics reigned here on Earth. Newton's great achievement, to peek ahead for a moment, was to demonstrate that Kepler's discoveries and Galileo's fit seamlessly together, and to explain why.

It was Kepler who spelled out explicitly the credo that all the great seventeenth-century scientists endorsed. When he began studying astronomy he had talked of planets as if they had souls. He soon recanted. The planets surely moved, but their motion had nothing in common with that of galloping horses or leaping porpoises. "My aim is to show that the machine of the universe is not similar to a divine animated being," Kepler declared, "but similar to a clock."

Galileo was the first to grasp, in detail, the workings of the cogs and gears of that cosmic clock. He liked to tell a story, perhaps invented, about how he had made his first great discovery. He had been young and bored, in church, daydreaming. An attendant had lit the candles on a giant chandelier and inadvertently set it swinging. Rather than listen to the service, Galileo watched the chandelier. It swung widely at first and then gradually in smaller and smaller arcs. Using his pulse beat to measure the time (in his day no one had yet built a clock with a second hand), Galileo discovered what has ever since been known as the law of pendulums-a pendulum takes the same time to swing through a small arc as through a large one.

Perhaps it was because Galileo had been raised in a musical household-his father was a renowned composer and musician-that counting time came naturally to him.39 Eventually his counting would lead to one of history's profound discoveries. What Galileo did, and what no one before him had ever done, was find a new way to think about time. It was an accomplishment akin to a fish's finding a new way to think about water. "Galileo spent twenty years wrestling with the problem before he got free of man's natural biological instinct for time as that in which he lives and grows old," wrote the historian Charles C. Gillispie. "Time eluded science until Galileo." Eventually his counting would lead to one of history's profound discoveries. What Galileo did, and what no one before him had ever done, was find a new way to think about time. It was an accomplishment akin to a fish's finding a new way to think about water. "Galileo spent twenty years wrestling with the problem before he got free of man's natural biological instinct for time as that in which he lives and grows old," wrote the historian Charles C. Gillispie. "Time eluded science until Galileo."

Galileo's solution was so successful and so radical that everyone today-even those without the slightest knowledge of physics-takes his insight for granted. The breakthrough was to identify time-not distance or temperature or color or any of a thousand other possibilities-as the essential variable that governs the world. For years Galileo had tried to find a relations.h.i.+p between the speed of a falling object and the distance it had fallen. All his efforts failed. Finally he turned away from distance and focused on time. Suddenly everything fell into place. Galileo had found a way to pin numbers to the world.

The crucial experiments might have occurred only to a musician. Once again they involved rolling a ball down a ramp. The setup was bare-bones: a wooden ramp with a thin groove down the middle, a bronze ball to roll down the groove, and a series of movable catgut strings. The strings lay on the surface of the ramp, at a right angle to the groove, like frets on the neck of a guitar. When the ball crossed a string, it made an audible click but its speed continued almost unchanged.

Galileo may actually have dropped rocks from the Leaning Tower of Pisa, as legend has it, but if he did they fell too quickly to study. So he picked up a ball, released it at the top of the ramp, and c.o.c.ked his ears.

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Now the strings came into play. Galileo could hear the ball cross each string in turn, and he painstakingly rolled the ball again and again, each time trying to position the strings so that the travel time between each pair of strings was the same. He needed to arrange the strings, in other words, so that the time it took the ball to move from the top of the ramp to string A was the same as the time it took to move from string A to string B, which was the same as the time from B to C, and C to D, and so on. (He measured time intervals by weighing the water that leaked through a hole in the bottom of a jug. Twice as much water meant twice as much time.) It was finicky, tedious work.

Finally satisfied, Galileo measured the distance between strings. That yielded this little table.

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The pattern in the right-hand column was easy to spot, but Galileo looked at the numbers again and recast the same data into a new table. Instead of looking at the ball as it traveled from one string to the next, he focused on the total distance the ball had traveled from the starting line. (All he had to do was add up the distances in the right-hand column.) This time he saw something more tantalizing.

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Each number in the right column of this new table represented the distance the ball had traveled in a certain amount of time-in one second, in two seconds, in three seconds, and so on. That distance, Galileo saw, could be expressed as a function of time. In t t seconds, a ball rolling down a ramp at gravity's command traveled precisely seconds, a ball rolling down a ramp at gravity's command traveled precisely t t inches. inches.40 In 1 second, a ball rolled 1 In 1 second, a ball rolled 1 inches, in 2 seconds 2 inches, in 2 seconds 2 inches, in 5 seconds 5 inches, in 5 seconds 5 inches, and so on. inches, and so on.

What was just as surprising was what the law didn't didn't say-it didn't say anything about how much the ball weighed. Roll a cannonball and a BB down a ramp in side-by-side grooves, and they would travel alongside one another all the way and reach the bottom at precisely the same moment. For a given ramp, the same tidy law always held-the distance the ball traveled was proportional to time squared. All that counted was the height above the ground of the point where the ball was released. say-it didn't say anything about how much the ball weighed. Roll a cannonball and a BB down a ramp in side-by-side grooves, and they would travel alongside one another all the way and reach the bottom at precisely the same moment. For a given ramp, the same tidy law always held-the distance the ball traveled was proportional to time squared. All that counted was the height above the ground of the point where the ball was released.

Repeat the experiment on a steeper ramp, and the cannonball and the BB would both travel faster, but they would still travel side by side every inch of the way. That was enough. Galileo made a daring leap: what held for a steep ramp and for an even steeper ramp would also hold for the steepest "ramp" of all, a free fall through the air. All objects, regardless of their weight, fall at exactly the same rate.

Chapter Thirty-One.

Two Rocks and a Rope Tradition has it that Galileo discovered how objects fall by dropping weights from the top of the Leaning Tower of Pisa. Unlike most legends-Archimedes and his bathtub, Columbus and the flat Earth, George Was.h.i.+ngton and the cherry tree-historians believe this one might possibly be true. A tower drop would have disproved Aristotle's claim that heavy objects fall faster than light ones. But it took the ramp experiments, which Galileo indisputably carried out, to yield the quant.i.tative law about distance and time.

Whether he really climbed a tower or not, Galileo did propose a thought experiment to test Aristotle's claim. Imagine for a moment, said Galileo, that it was true that the heavier the object, the faster its fall. What would happen, he asked, if you tied a small rock and a big rock together, with some slack in the rope that joined them? On the one hand, the tied-together rocks would fall slower slower than the big rock alone, because the small rock would lag behind the big one and bog it down, just as a toddler tied to a sprinter would slow him down. (That was where the slack in the rope came into play.) On the other hand, the tied-together rocks would fall than the big rock alone, because the small rock would lag behind the big one and bog it down, just as a toddler tied to a sprinter would slow him down. (That was where the slack in the rope came into play.) On the other hand, the tied-together rocks would fall faster faster than the big rock alone, because they const.i.tuted a new, heavier "object." than the big rock alone, because they const.i.tuted a new, heavier "object."

Which meant, Galileo concluded triumphantly, that Aristotle's a.s.sumption led to an absurd conclusion and had to be abandoned. Regardless of what Aristotle had decreed, logic forced us to conclude that all objects fall at the same rate, regardless of their weight. This is a story with a curious twist. Galileo, the great pioneer of experimental science, may never have bothered to perform his most famous experiment. No one is sure. What we know with certainty is that, like the Aristotelians he scorned, Galileo sat in a chair and deduced the workings of the world with no tool but the power of logic.

Since Galileo's day, countless tests have confirmed his Leaning Tower principle (including some at the Leaning Tower itself ). In ordinary circ.u.mstances, air resistance complicates the picture-feathers flutter to the ground and arrive long after cannonb.a.l.l.s. Not until the invention of the air pump, which came after Galileo's death, could you drop objects in a vacuum. A century after Galileo, the demonstration retained its power to surprise. King George III demanded that his instrument makers arrange a test for him, featuring a feather and a one-guinea coin falling in a vacuum. "In performing the experiment," one observer wrote, "the young optician provided the feather, the King supplied the guinea and at the conclusion the King complimented the young man on his skill as an experimenter but frugally returned the guinea to his waistcoat pocket."

Today we've all seen the experiment put to the test, at every Olympic games. When television shows a diver leaping from the ten-meter board, thirty feet above the pool, how does the camera stick with her as she plummets toward the water? Galileo could have solved the riddle-just as a small stone falls at exactly the same rate as a heavy one, a camera falls at exactly the same rate as a diver. The trick is to set up a camera near the diver, at exactly the same height above the water. Attach the camera to a vertical pole and release the camera at the instant the diver starts her fall poolward. Gravity will do the rest.

Galileo exulted in his discovery that "distance is proportional to time squared." The point was not merely that nature could be described in numbers but that a single, simple law-in operation since the dawn of time but unnoticed until this moment (just as the Pythagorean theorem had been true but unknown before its its discovery)-applied to the infinite variety of falling objects in the world. A geranium knocked off a windowsill, a painter tumbling off his ladder, a bird shot by a hunter, all fell according to the same mathematical law. discovery)-applied to the infinite variety of falling objects in the world. A geranium knocked off a windowsill, a painter tumbling off his ladder, a bird shot by a hunter, all fell according to the same mathematical law.

The difference between Galileo's world and Aristotle's leaps out, as we have seen. Galileo had stripped away the details that fascinated Aristotle-the color of the bird's plumage, the motives behind the painter's absentmindedness-and replaced the sensuous, everyday world with an abstract, geometric one in which both a bird and a painter were simply moving dots tracing a trajectory against the sky. Ever since, we have been torn between celebrating the bounty that science and technology provide and lamenting the cost of those innovations.

Chapter Thirty-Two.

A Fly on the Wall The mathematical patterns that Kepler had found in the heavens looked different from those Galileo had found on Earth. Perhaps that was to be expected. What did falling rocks have to do with endlessly circling planets, which plainly were not falling at all?