Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

To time the ball’s descent he used a water clock. It worked like a stopwatch. To start the clock he would open a valve. Water would then flow steadily, at a constant rate, straight down through a thin pipe and into a container. To stop the clock, he would close the valve. By weighing how much water had accumulated during the ball’s descent, Galileo could quantify how much time had elapsed to within “one-tenth of a pulse-beat.”

To spell out this law of odd numbers more explicitly, let’s suppose the ball rolls a certain distance in the first unit of time. Then, in the next unit of time, it will roll three times as far. And in the next unit of time after that, it will roll five times as far as it did originally. It’s amazing; the odd numbers 1, 3, 5, and so on are somehow inherent in the way things roll downhill. And if falling is just the limit of rolling as the tilt approaches vertical, the same rule must hold for falling.

To see the most important implication of Galileo’s rule, let’s look at what happens if we add consecutive odd numbers. After one unit of time, the ball has traveled one unit of distance. After the next unit of time the ball has traveled another three units of distance, for a total of 1 + 3 = 4 units traveled since the motion started. After the third unit of time, the total becomes 1 + 3 + 5 = 9 units of distance. Notice the pattern: the numbers 1, 4, and 9 are the squares of consecutive integers—12 = 1, 22 = 4, 32 = 9. So Galileo’s odd-number rule seems to be implying that the total distance fallen is proportional to the square of the time elapsed.

In our own era, the challenge of navigating on Earth still relies on the precise measurement of time. Consider the global positioning system. Just as mechanical clocks were the key to the longitude problem, atomic clocks are the key to pinpointing the location of anything on Earth to within a few meters. An atomic clock is a modern-day version of Galileo’s pendulum clock. Like its forebear, it keeps time by counting oscillations, but instead of tracking the movements of a pendulum bob swinging back and forth, an atomic clock counts the oscillations of cesium atoms as they switch back and forth between two of their energy states, something they do 9,192,631,770 times per second. Though the mechanism is different, the principle is the same. Repetitive motion, back and forth, can be used to keep time.

The calculation relies on triangulation, an ancient geolocation technique based on geometry. For GPS, it works like this: When the signals from the four satellites arrive at the receiver, your GPS gadget compares the time they were received to the time they were transmitted. Those four times are all slightly different, because the satellites are at four different distances away from you. Your GPS device multiplies those four tiny time differences by the speed of light to calculate how far away you are from the four satellites overhead. Because the positions of the satellites are known and controlled extremely accurately, your GPS receiver can then triangulate those four distances to determine where it is on the surface of the Earth. It can also figure out its elevation and speed. In essence, GPS converts very precise measurements of time into very precise measurements of distance and thereby into very precise measurements of location and motion.

The global positioning system was developed by the US military during the Cold War. The original intent was to keep track of US submarines carrying nuclear missiles and give them precise estimates of their current locations so that if they needed to launch a nuclear strike, they could target their intercontinental ballistic missiles very accurately. Peacetime applications of GPS nowadays include precision farming, blind landings of airplanes in heavy fog, and enhanced 911 systems that automatically calculate the fastest routes for ambulances and fire trucks.

To grasp how different a million is from a billion, think about it like this: A million seconds is a little under two weeks; a billion seconds is about thirty-two years. The first is the length of a vacation; the second is a significant fraction of a lifetime. The lesson here is that we need to use powers of ten with care. They are dangerously strong compressors, capable of shrinking enormous numbers down to sizes we can fathom more easily.