The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)
Part II. Hope and Monsters
Chapter 36. Out of the Whirlpool
The answer began with Descartes’ graphs. Since steady motion was far easier to deal with than uneven motion, scientists started there. Imagine a man trudging home from work at the end of a long day, dragging himself along at 2 miles per hour. A younger colleague might scoot along at 4 miles per hour. A runner might whiz by at 8 miles per hour.
We could chart their journeys in a table that shows how much distance they covered.
But a graph, à la Descartes, makes matters clearer. A steady pace corresponds to a straight line, as we see in the drawing below, and the faster the pace the steeper the line’s slope. Slope, in other words, is a measure of speed. (Slope is a textbook term with a symbol-laden definition, but the technical meaning is the same as the everyday one. A line’s slope is simply a measure of how quickly a situation is changing. A flat slope means no change at all; a steep slope, like a spike in blood pressure, means a fast change.)
So we can say, with the aid of our picture, precisely what it means to travel at a steady speed of 2 miles per hour (or 4, or 8). It means that, if we were to make a graph of the traveler’s path, the result would be a straight line with a certain slope.
This seems simple enough, and so it is, but there is a subtle point hidden inside our tidy graph. The picture has let us dodge a vital but tricky question: What does it mean to travel at two miles per hour if you’re not traveling for a full hour? Before Descartes came along, such questions had spawned endless confusion. But we have no need to vanish into the philosophical fog. We can nearly do without words and debates and definitions altogether. At least in the case of a traveler moving at a steady speed, we can blithely make statements like, “At this precise instant she’s traveling at a rate of two miles per hour.” All this with the aid of a graph.
But suppose our task was to look at a more complicated journey than a steady march down the street. What does a graph of a cannonball’s flight look like? Galileo knew that. It looks like this, as we have seen before.
Now our trick seems to have let us down. So long as we were dealing with graphs of straight lines, we’d found a way to talk about instantaneous speed. It was easy to talk about the slope of a straight line, because the slope was always the same. But what does it mean to talk about the slope of a curve, which by definition is never straight?
The question was important for two reasons. First, few changes in real life are as simple as the steady plink plink plink of drops from a leaky faucet. Second, if thinkers could devise a way to deal with complicated change of one sort, then presumably they could deal with complicated changes of many sorts. Mathematics is so powerful—and so difficult for us to learn—because it is a universal tool. We balk at algebra, for instance, because those inscrutable x’s are so off-putting. But algebra is useful precisely because it allows us to fill in the blanks in countless different ways.
A mathematics of change dangled the same promise. Planets and comets speeding across the heavens, populations growing and shrinking, bank accounts swelling, divers plummeting, snowbanks melting, all would yield up their mysteries. Questions that asked when a given change would reach its high point or its low—what angle should a cannon be tilted at to shoot the farthest? when will a growing population level off? what is the ideal shape for the arch of a bridge?—could be answered quickly and definitively.
This was a gleaming prize. But how to win it?
The riddle at the heart of the mystery of motion was the question of speed at a given instant. What did it mean? How could you keep from drowning in the whirlpool of “zero distance in zero seconds”?
Answering that question meant learning how to focus on infinitesimally brief stretches of time. The first step was to see that Zeno was not so unnerving as he had seemed. Take his argument that it would take forever to cross a room because it would take a certain amount of time to cross to the halfway point, and then more time to cross half the remaining distance, and so on.
Zeno’s paradox. If it takes 1 second to walk to the middle of a room, and ½ second more to walk half the rest of the way, and ¼ second more to walk halfway again, and so on, then it takes an infinitely long time to cross the room.
In essence Zeno’s argument is a claim about infinity. It seems common sense to say that if you add up numbers forever, and if each number you add is bigger than zero, then eventually the sum is infinite. If you piled up blocks forever, wouldn’t the stack eventually reach the ceiling, no matter how vast a room you started in?
Well, no, actually, not necessarily.
It all depends on the size of each new block that you added to the stack. If all the blocks were the same size, then the tower would eventually reach the ceiling, the moon, the stars. And even if each new block were thinner than its predecessor, a tower might still grow forever.43 But it might not, if you picked the sizes of the blocks just so.
In modern terms, Zeno’s paradox amounts to saying that if you add up 1 + ½ + ¼ + ⅛ + 1/16 + . . . the total is infinite. Zeno never framed it that way. He focused not on a specific chain of fractions like this one, but on a general argument about what had to be true for any endless list of numbers whatsoever.
But Zeno was wrong. If the sum were infinite, as he believed, then it would be bigger than any number you can think of—bigger than 100, bigger than 100,000, and so on. But Zeno’s sum does not exceed any number you can name. On the contrary, the sum is the perfectly ordinary number 2.
In a minute, we’ll see why that is. But think how surprising this result is. Suppose you took a block one inch high, and put a ½-inch-thick block on top of it, and then a ¼-inch-thick one on top of that, and so on. If you added new blocks forever, one a second, through your lifetime and your children’s lifetimes and the universe’s lifetime, even so the tower would never reach above a two-year-old’s ankles.