The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)
Part II. Hope and Monsters
Chapter 29. Sputnik in Orbit, 1687
In a story called “The Red-Headed League,” Dr. Watson looks hard at Sherlock Holmes’s latest visitor, but nothing strikes him as noteworthy. He turns toward the great detective. Perhaps Holmes has seen more? “Beyond the obvious facts that he has at some time done manual labour, that he takes snuff, that he is a Freemason, that he has been in China, and that he has done a considerable amount of writing lately, I can deduce nothing else,” says Holmes.
Galileo and his fellow scientists favored a similar technique. By paying close attention to what others had overlooked, they could find their way to utterly unexpected conclusions. Galileo’s analysis of life on shipboard showed, for instance, that a marble that rolled off a table would take precisely the same time to reach the floor whether the ship was moving at a steady speed or standing still. The ship’s horizontal motion has no effect on the rock’s vertical fall. In Galileo’s hands, that seemingly small observation had momentous consequences.
Picture any projectile moving through the air—a baseball soaring toward the outfield, a penny flipped into the air, a dancer leaping across the stage. In all such cases, the moving object’s horizontal motion and its vertical motion take place independently and can be examined separately. The horizontal movement is steady and unchanging, in line with Galileo’s law of motion. Ball and coin and dancer travel a certain distance horizontally in the first second, the same distance in the next second, and so on, moving at a constant speed from liftoff until touchdown.38 At the same time, the projectile’s vertical progress—its height above the ground—changes according to a different rule. At the moment of launch, the projectile rises quickly but then it rises slower and slower, stops rising altogether, and sits poised for an instant neither rising nor falling, and then plummets earthward faster and faster. The change in speed follows a simple, precise rule, and the upward part of the flight and the downward part are exactly symmetrical.
Any object launched into the air—arrow, bullet, cannonball—travels in a curved path like this one. The moving object covers the same horizontal distance during each second of its flight.
Mathematically, it’s easy to show that the combination of steady horizontal motion and steadily changing vertical motion makes for a parabolic path. (A parabola is an arch-shaped curve, but it is not just a generic arch; it is one that satisfies specific technical conditions, just as an ellipse is not a generic oval but one of a specific sort.) Parabolas had been painted against the sky ever since the first caveman threw a rock, but no one before Galileo had ever recognized them, and he was immensely proud of his discovery. “It has been observed that missiles and projectiles describe a curved path of some sort,” he wrote. “However no one has pointed out the fact that this path is a parabola. But this and other facts, not few in number or less worth knowing, I have succeeded in proving.”
God had once again shown his taste for geometry. The planets in the heavens traveled not in haphazard curves but in perfect ellipses, and objects here on Earth traced exact parabolas.
Concealed within the same observation about the independence of horizontal motion and vertical motion was a further surprise. Galileo might have found it, but he didn’t. Isaac Newton did. Imagine someone firing a gun horizontally, and at the same instant someone standing next to the shooter and dropping a bullet from the same height as the gun. When the two bullets reach the ground, they will be far apart. The one shot from the gun will have traveled hundreds of yards; the other will rest in the grass directly below the spot where it was dropped. Which bullet will hit the ground first?
Surprisingly, both reach the ground at exactly the same moment. That’s what it means for the bullet’s vertical motion—its fall—to be independent of its horizontal motion. For Newton, that was enough to draw a remarkable conclusion.
Suppose it takes one second for a bullet dropped from a certain height to hit the ground. That means that a bullet shot horizontally from the same height would also hit the ground in one second. A more powerful gun would send the bullet faster and farther, but—if the ground was perfectly flat—that bullet, too, would fall to the ground in one second.
Bullets shot horizontally with different force travel different distances before they come to rest, but they all fall at the same rate. Each second a bullet is in the air it falls 16 feet toward the ground.
Newton preferred to imagine a cannon blasting away horizontally. He imagined faster and faster cannonballs, covering greater and greater distances in their one-second journey. But the Earth is round, not flat.
That makes all the difference. Since the Earth isn’t flat, it curves away beneath the speeding cannonball. In the meantime, the cannonball is falling toward the ground. Suppose you fired a cannonball from high above the atmosphere, horizontally. With nothing to slow it down, it would continue at the same speed forever, falling all the while. If you launched it at just the right speed, then by the time the cannonball had fallen, say, four feet, the ground itself would have fallen four feet below horizontal.
And then what? The cannonball would continue on its journey forever, always falling but never coming any closer to the ground. Why? Because the cannonball always falls at the same rate, and the ground always curves beneath it at the same rate, so the cannonball falls and falls, and the Earth curves and curves, and the picture never changes. We’ve launched a satellite.
Newton pictured it all in 1687.