The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)
Part III. Into the Light
Chapter 41. The World in Close-Up
Newton and Leibniz framed their discoveries in different vocabulary, but both had found the same thing. The challenge confronting both men was finding a way to stop time in its tracks. Their solution, hundreds of years before the birth of photography, was essentially to imagine the movie camera. They pictured the world not as the continuous, flowing panorama we see but as a series of still photos, each one barely different from those before and after it, and all the frames flashing before the eye too quickly to register as static images.
But how could you be sure that, no matter what moment you wanted to scrutinize, there happened to be a sharply focused image on hand? It seemed clear that the briefer the interval between successive still photos, the better. The problem was finding a stopping place—if sixty-four frames a second was good, wouldn’t 128 be better? Or 1,000, or 100,000?
Think of Galileo near the top of the Leaning Tower, a bit winded from the long climb. He extends an arm out into space, opens his fingers, and releases the rock he has lugged up all this way. It falls faster and faster—in each successive second, that is, it covers more distance than it did the second before—as the figures in this table show. (As we have seen, it was no easy matter to make such measurements without clocks or cameras, which was why Galileo ended up working with ramps rather than towers.)
Galileo found that rocks fall according to a precise rule that can be expressed in symbols. Scientists write the rule as d = 16 t2, where d stands for distance and t stands for time. In one second, a rock falls a distance of 16 × 1 feet, or 16 feet. In two seconds, it falls a distance of 16 × 4 feet, or 64 feet; in three seconds, 16 × 9 feet, or 144 feet.
The graph shows how far a rock dropped from a height falls in t seconds. The rock obeys the rule d = 16 t2.
The table can be converted to a graph, and, as usual, a picture helps reveal what the numbers only imply. (And a picture, unlike a table, shows the rock’s position at every moment rather than at a select few.) The horizontal axis depicts time, the vertical axis distance. The drawing shows the distance the rock has fallen at a given time. At the moment that Galileo uncurled his fingers and released the rock (in other words, at t = 0), the rock has fallen 0 feet. At 1 second it has fallen 16 feet; in 2 seconds, 64 feet; and so on.
The transition from rock to table to graph is one of increasing abstraction, and early mathematicians had a hard time keeping their bearings. Few things could be more tangible than a rock. When Galileo let it go, anyone who happened to be passing by could see it. The table took that ordinary event—a stone whooshing toward the ground—and transformed it into a list of numbers. The graph represented still another move away from everyday reality. It shows a curve that represents the rock’s distance from Galileo’s hand, but the curve in the graph only matches the real-life descent of the falling rock in a subtle way. The actual rock dropped in a straight line. The graph shows a curve, a parabola. Worse yet, the rock fell down, while the parabola headed up. To “see” the rock falling in the way that the graph depicts required a far more laborious and roundabout process than merely looking at a rock.
And yet it was this second way of looking at a rock’s fall, this unnatural way, that held the key to nature’s secrets. For it was this curve that showed Newton and Leibniz how to seize time in their fists and hold it still. We saw in chapter 36 that in a graph that shows distance measured against time, a straight line corresponds to a steady speed. (The greater the speed, the steeper the slope of that line, because a steeper slope indicates more distance covered in a given amount of time.) But in the graph of a falling rock, we have a curve, not a straight line. How can we talk about the rock’s speed? In particular, how can we find its speed at a specific instant, at, for instance, precisely one second into its fall?
We could do it, Newton and Leibniz explained, if we could find the slope of the curve at precisely the one-second point. Which they proceeded to do. The idea was to look at the curve in extraordinary close-up. If you look closely enough, a curve looks like a straight line. (A jogger on a huge circular track would feel as if she were running in a straight line. Only a bird’s-eye view would reveal the track’s true shape.) And although curves are hard to work with, straight lines are easy.
First they froze time by selecting a single frame from nature’s ongoing movie. (Newton and Leibniz worked in parallel, unaware of one another, as we have seen, but they independently hit on the same strategy.) Second, they tunneled into that frame, as if it were a slide under a microscope.
In the case of a falling rock, they began by freezing the picture at the instant t = 1 second. They wanted to know the rock’s speed at that moment, but the only information they had to work with was a graph depicting time and distance. Even so, they were nearly done.
All they had to do was focus their conceptual microscope. Speed is a measure of distance traveled in a given time. Sixty miles per hour. Three inches per second. To solve the problem they cared about, they began by solving an easier problem, in the hope that the solution to the easy problem would point to the solution they truly wanted.
The rock’s speed at a given instant was hard to find because that speed was constantly changing. But the rock’s average speed over any particular span of time was easy to find. ( Just divide the distance the rock fell by the length of the time span.) With that in mind, Newton and Leibniz did something clever. They put the actual rock to one side for a moment and concentrated instead on an easier-to-deal-with imaginary rock. The great virtue of this imaginary rock was that, in contrast with a real rock, it fell at a constant speed. What speed to pick?
The answer, Newton and Leibniz decided, was that the imaginary rock should fall at a steady speed that exactly matched the average speed of the actual rock in the interval between t = 1 and t = 2. This roundabout procedure seems like a detour, but in fact it brought them closer to their goal.
Look at the graph below. The dotted line depicts the imaginary rock, the curve the real rock. At the one-second mark (in other words, at t = 1) the imaginary rock and the real one have both fallen 16 feet. At t = 2, both the imaginary rock and the real one have fallen 64 feet.
The dotted line represents the fall of an imaginary rock traveling at constant speed. The slope of the dotted line gives the imaginary rock’s speed in the one-second interval between t = 1 and t = 2.
The dotted line is straight. That’s crucial. Why? Because it means we can talk about its slope, which is a number—a regular, run-of-the-mill number, not an infinitesimal or any other colorful beast. That number is the speed of the imaginary rock. (It is easy to compute. Slope is a measure of steepness, which means that it is a ratio of vertical change to horizontal change. In this case, the vertical change was from 16 feet to 64 feet, and the horizontal change was from 1 second to 2 seconds, so the slope was [64–16] feet ÷ [2–1] seconds, or 48 feet per second.)
Now Newton and Leibniz made their big move. Forty-eight feet per second was the imaginary rock’s speed over a one-second span. That gave a fair approximation to what they really wanted to know about, an actual rock’s speed at the precise instant t = 1.
How could you get a better approximation? By zooming in for a closer look at the graph. And the way to do that was once again to focus your attention at t = 1 but this time to look at a shorter time interval than one second. As usual, pictures came to the rescue.
Look at the diagram below. The new, dashed line represents the path of a new imaginary rock. This imaginary rock, too, is falling at a constant speed. What speed? Not the same speed as the first imaginary rock. This new imaginary rock is falling at a speed exactly equal to the actual rock’s average speed in a newer, shorter time interval, the interval between t = 1 and t = 1 ½. The point is that the speed of this new imaginary rock gives us a better estimate of the actual rock’s speed at the instant t = 1.
* * *
The dashed line represents the fall of a new imaginary rock. The slope of the line gives the imaginary rock’s speed, which is constant, in the one-half second interval between t = 1 and t = 1 ½.
If we zoomed in on an even shorter interval starting at t = 1, we could draw still another straight line. We might, for instance, focus on the interval between t = 1 second and t = 1 ¼ seconds. The new line, too, would have a slope that we could compute. We could repeat the procedure still another time, this time focusing on a yet-shorter interval, say between t = 1 second and t = 1 ⅛ seconds. And so on.
Newton and Leibniz saw that you could continue drawing new straight lines forever. Pictorially, you would be drawing straight lines that passed through two dots on the curve. One dot was fixed in place at t = 1, and the other moved down the curve, like a bead on a wire, approaching ever nearer to the fixed dot.
Those lines would approach ever nearer to one particular straight line. That “target” line was unique, in a natural way—it was the line that just grazed the curve at a single place, the point corresponding to t = 1. The target line—in mathematical jargon the tangent line—was the prize that all the fuss was about. (In the diagram below, the tangent line is the straight line made up of short dashes.) Until this moment, mathematicians had never managed to close their fingers around the notion of instantaneous speed. Now they had it.
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 t2.)
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 32t. 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.”