The Clockwork Universe (25 page)

This was an enormous breakthrough, and perhaps a recap is in order to make sure we see just what Newton and Leibniz had
done. They had found a way to define a moving object's speed
at a given instant.
Instantaneous speed was the number that average speeds approached
,
as you looked at shorter and shorter time intervals.

The slope of the tangent line (short dashes) represents the speed of a falling rock at the instant t = 1 second.

Instantaneous speed wasn't a paradoxical idea or an arcane one. You could grab it and examine it at your leisure. The speed of a moving object at a given instant was just an ordinary number, the slope of the tangent line at that point. How did you compute that slope? By looking at the slopes of the straight lines that approached the tangent line and seeing if those numbers approached a limit. That limit was the number we were after, the grail in this long quest.

In the case of Galileo's rock, Newton and Leibniz found that the rock's speed at the instant when it had been falling for 1 second was precisely 32 feet per second. They discovered an array of tricks that made such calculations easy whenever you had an equation to work with. Nearly always, you did. (I will skip the procedure, but it is a hint at how neatly things work out that the number they ended up with in the Galileo example—32—can be written as 16 × 2, and there were both a 16 and a 2 lurking in the equation of the curve they started with,
d
= 16
t
²
.)

Better yet, the same calculation that revealed the rock's speed at a single instant also told its speed at
every
instant. Without bothering to lift a finger or draw another straight line (let alone an infinite sequence of straight lines homing in on a target line), this once-and-for-all calculation showed that the rock's speed at any time
t
was precisely 32
t
. The speed was always changing, but a single formula captured all the changes. When the rock had been falling for 2 seconds, its speed was 64 feet per second (32 × 2). At 2
½
seconds, its speed was 80 feet per second (32 ×
2
½
)
; at three seconds, 96 feet per second, and so on.

This new tool for describing the moving, changing world was called calculus. With its discovery, every scientist in the world suddenly held in his hands a magical machine. Pose a question that asked
how far? how fast? how high?
and then press a button, and the machine spit out the answer. Calculus made it easy to take a snapshot—to freeze the action at any given instant—and then to examine, at leisure, an arrow momentarily motionless against the sky or an athlete hovering in midleap.

Questions that had been out of reach forever now took only a moment. How fast is a high diver traveling when she hits the water? If you shoot a rifle with the barrel at a given angle, how far will the bullet travel? What will its speed be when it reaches its target? If a drunken reveler shoots a pistol in the air to celebrate, how high will the bullet rise? More to the point, how fast will it be traveling when it returns to the ground?

Calculus was “the philosopher's stone that changed everything it touched to gold,” one historian wrote, and he seemed almost resentful of the new tool's power. “Difficulties that would have baffled Archimedes were easily overcome by men not worthy to strew the sand in which he traced his diagrams.”

Chapter Forty-Two
When the Cable Snaps

If infinity had not always inspired terror, like some mythological dragon blocking access to a castle, someone would have discovered calculus long before Newton and Leibniz. They did not slay the dragon—the crucial concepts in calculus all hinge on infinity—but they did manage to capture and tame it. Their successors harnessed it to a plow and set it to work. For the next two centuries, science would consist largely of finding ways to exploit the new power that calculus provided. Patterns once invisible to the naked eye now showed up in vivid color. Galileo had expended huge amounts of effort to come up with the law of falling bodies, for example, but his
d
= 16
t
²
equation contained far more information than he ever knew. Without calculus he could not see it. With calculus, there was no missing it.

Galileo knew that his law described position; he didn't know that it contained within itself a hidden law that described speed. Better yet, the law describing position was complicated, the law describing speed far simpler. In words, Galileo's position law says that after
t
seconds have passed, an object's distance from its
starting point is 16
t
²
feet. It is the
t
²
in that equation, rather
than a simple
t
, that makes life complicated. As we have seen,
calculus takes that law and, with only the briefest of calculations,
extracts from it a new law, this one for the speed of a falling object. In words, when an object has been falling for
t
seconds, its speed is exactly 32
t
feet per second. In symbols (using
v
for velocity),
v
= 32
t
.

That tidy speed equation contains three surprises. First, it's simple. There's no longer any need to worry about messy numbers like
t
²
. Plain old
t
will do. Second, it holds for
every
falling object, pebbles and meteorites alike. Third, this single equation tells you a falling object's speed at
every
instant
t
, whether
t
represents 1 second or 5.3 seconds or 50. There's never any need to switch to a new equation or to modify this one. For a complete description of falling objects, this is the only equation you'll ever need.

We began with a law that described the position of a falling body and saw that it concealed within itself a simpler law describing speed. Scientists looked at that speed law and saw that it, too, concealed within itself a simpler law. And
that
one, that gem inside a gem, is truly a fundamental insight into the way the world works.

What is speed? It's a measure of how fast you're changing position. It is, to put it in a slightly more general way, a rate of change. (To be barreling down the highway at 80 miles per hour means that you're changing position at a rate of 80 miles every hour.) If we repeat the same process, by starting with speed and looking at
its
rate of change—in other words, if we compute the falling rock's acceleration—what do we find?

We find good news. Calculus tells us, literally at a glance, that a falling rock's acceleration never changes. Unlike position, that is, which depends on time in a complicated way, and unlike speed, which depends on time in a simpler way, acceleration doesn't depend on time at all. Whether a rock has been falling for one second or ten, its acceleration is always the same. It is always 32 feet per second per second. Every second that a rock continues to fall, in other words, its speed increases by another 32 feet per second.
This
is nature's doubly concealed secret.

When a rock falls, its position changes in a complicated way, its velocity in a simpler way, and its acceleration in the simplest possible way.

There is a pattern in the position column, but it hardly blazes forth. The pattern in the speed column is less obscure. The pattern in the acceleration column is transparent. What do all falling objects have in common? Not their weight or color or size. Not the height they fall from or the time they take to reach the ground or their speed on impact or their greatest speed. What is
true of all falling objects—an elevator snapping its cables, an
egg slipping through a cook's fingers, Icarus with the wax melting from his wings—is that they all accelerate at precisely the same rate.

Acceleration
is a familiar word (“the acceleration in my old car was just pitiful”), but it is a remarkably abstract notion. “It is not a fundamental quantity, such as length or mass,” writes the mathematician Ian Stewart. “It is a rate of change. In fact, it is a ‘second order' rate of change—that is, a rate of change of a rate of change.”

Acceleration is a measure of how fast velocity is changing, in other words, and that's tricky, because velocity is a measure of how fast position is changing. “You can work out distances with a tape measure,” Stewart goes on, “but it is far harder to work out a rate of change of a rate of change of distance. This is why it took humanity a long time, and the genius of a Newton, to discover the law of motion. If the pattern had been an obvious feature of distances, we would have pinned motion down a lot earlier in our history.”

Acceleration turns out to be a fundamental feature of the world—unless we understand it, whole areas are off-limits to us—but it does not correspond to anything tangible. We can run a finger across a pineapple's prickly surface or heft a brick or feel the heat of a cup of coffee even through gloved hands. We could put the brick on a scale or take a ruler and measure it. Acceleration seems different from such things as the pineapple's texture and the brick's weight. We
can
measure it, but only in an indirect and cumbersome way, and we cannot quite touch it.

But it is this elusive, abstract property, Newton and Leibniz showed, that tells us how objects fall. Once again, seeing nature's secrets required looking through a mathematical lens.

Calculus had still more riches to offer. It not only revealed that distance, speed, and acceleration were all closely linked, for instance, but also showed how to move from any one of them to any of the others. That was important practically—if you wanted to know about speed, say, but you only had tools to measure time and distance, you could still find all the information you wanted, and you could do it easily—and it was important conceptually. Galileo invested endless hours in showing that if you shoot an arrow or throw a ball it travels in a parabola. Newton and Leibniz reached the same conclusion with hardly any work. All they had to know was that falling objects accelerate at 32 feet per second per second. That single number, decoded with calculus's aid, tells you almost at once that cannonballs and arrows and leaping kangaroos all travel in parabolas.

Again and again, simple observations or commonplace equations transformed themselves into wondrous insights, the mathematical counterpart of Proust's “little pieces of paper” that “the moment they are immersed in [water] stretch and shape themselves, color and differentiate, become flowers, houses, human figures, firm and recognizable.”

Calculus was a device for analyzing how things change as time passes. Just what those things were made no difference. How long will it take the world's population to double? How many thousands of years ago was this mummy sealed in his tomb? How soon will the oyster harvest in the Chesapeake Bay fall to zero?

Questions about bests and worsts, when this quantity was at a maximum or that one at a minimum, could also be readily answered. Of all the roller-coaster tracks that start at a peak here and end at a valley there, which is fastest? Of all the ways to fire a cannon at a fortress high above it on a mountain, which would inflict the heaviest damage? (This was Halley's contribution, written almost as soon as he had heard of calculus. It turns out that he had also found the best angle to shoot a basketball so that it swishes through the hoop.) Of all the shapes of soap bubble one can imagine, which encloses the greatest volume with the least surface? (Nature chooses the ideal solution, a spherical bubble.) Of all the ticket prices a theater could charge, which would bring in the most money?

Not every situation could be analyzed using the techniques of calculus. If in a tiny stretch of time a picture changed only a tiny bit, then calculus worked perfectly. From one millisecond to the next, for instance, a rocket or a sprinter advanced only a tiny distance, and calculus could tell you everything about their paths. But in the strange circumstances where something shifts abruptly, where the world jumps from one state to a different one entirely without passing through any stages in between, then calculus is helpless. (If you're counting the change in your pocket, for example, no coin is smaller than a penny, and so you jump from “twelve cents” to “thirteen cents” to “fourteen cents,” with nothing in between.) One of the startling discoveries of twentieth-century science was that the subatomic world works in this herky-jerky way. Electrons jump from here to there, for instance, and in between they are . . . nowhere. Calculus throws up its hands.

But in the world we can see, most change is smooth and continuous. And whenever anything changes smoothly—when a boat cuts through the water or a bullet slices through the air or a comet speeds across the heavens, when electricity flows or a cup of coffee cools or a river meanders or the high, quavering notes of a violin waft across a room—calculus provides the tools to probe that change.

Scientists wielding the new techniques talked as if they had witnessed sorcery. The old methods compared to the new, one dazed astronomer exclaimed, “as dawn compares to the bright light of noon.”