The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)
Part II. Hope and Monsters
Chapter 30. 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.”
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 relationship 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 cocked his ears.
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.
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.
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 seconds, a ball rolling down a ramp at gravity’s command traveled precisely t2 inches.40 In 1 second, a ball rolled 12 inches, in 2 seconds 22 inches, in 5 seconds 52 inches, and so on.
What was just as surprising was what the law 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.
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.