But what happens if one raises the plane a bit, making the inclination steeper? Galileo worked many angles and found this same relationship, this sequence of squares, at each inclination, from gentle to less gentle, until the motion proceeded too swiftly for his "clock" to record distances accurately enough. The crucial thing is that Galileo demonstrated that a falling object doesn't just drop, but drops faster and faster and faster over time. It accelerates, and the acceleration is constant.
Being a mathematician, he came up with a formula to describe this motion. The distance's that a falling body covers is equal to a number A times the square of the time t it takes to cover the distance. In the ancient language of algebra, we abbreviate this by: s = At2. The constant A changed with each inclination of the plane. A represents the crucial concept of acceleration, that is, the increase of speed as the object continues to fall. Galileo was able to deduce that speed changes with time in a simpler way than distance, increasing simply with the time (rather than with the square of the time).
The inclined plane, the trained ear's ability to measure times to a sixty-fourth of a second, and the ability to measure distances to somewhat better than a tenth of an inch gave Galileo the precision he needed to make his measurements. Galileo later invented a clock based upon the regular period of the pendulum. Today the Bureau of Standards' atomic cesium clocks keep time to a precision better than one millionth of a second per year! These clocks are rivaled by nature's own timepieces: astronomical pulsars, which are whirling neutron stars that sweep beams of radio waves across the cosmos with a regularity you can set your watch to. They may in fact be more precise than the atomic pulse in the cesium atom. Galileo would have been entranced by this deep connection between astronomy and atomism.
Well, what is so important about s = At2?
It was the first time, as far as we know, that motion was correctly described mathematically. The crucial concepts of acceleration and velocity were sharply defined. Physics is a study of matter and motion. The movement of projectiles, the motion of atoms, the whirl of planets and comets must all be described quantitatively. Galileo's mathematics, confirmed by experiment, provided the starting point.
Lest all of this sound too easy, we should note that Galileo's obsession with the law of free fall lasted for decades. He even got the law wrong in one publication. Most of us, being basically Aristotelians (did you know that you, dear reader, are a basic Aristotelian?), would guess that the speed of the fall would depend on the weight of the ball. Galileo, because he was smart, reasoned otherwise. But is it so crazy to think that heavy things should fall faster than light things? We do so because nature misleads us. Smart as Galileo was, he had to do careful experiments to show that the apparent dependence of a body's time of fall on its weight comes from the friction of the ball on the plane. So he polished and polished to decrease the effect of friction.
THE FEATHER AND THE PENNY
Extracting a simple law of physics from a set of measurements is not so simple. Nature hides the simplicity in a thicket of complicating circumstances, and the experimenter's job is to prune away these complications. The law of free fall is a splendid example. In freshman physics we hold a feather and a penny at the top of a tall glass tube and drop them simultaneously. The penny falls rapidly and clinks to the bottom in less than a second. The feather floats gently down, arriving in five or six seconds. Such observations led Aristotle to postulate his law that heavier objects fall faster than light ones. Now we pump the air out of the tube and repeat the experiment. Feather and penny drop with equal times. Air resistance obscures the law of free fall. To make progress, we must remove this complicating feature to get the simple law. Later, if it is important, we can learn how to add this effect back in to arrive at a more complex but more applicable law.
The Aristotelians believed that an object's "natural" state was to be at rest. Push a ball along a plane and it conies to rest, no? Galileo knew all about imperfect conditions, and that understanding led to one of the great discoveries. He read physics in inclined planes as Michelangelo saw magnificent bodies in slabs of marble. He realized, however, that because of friction, air pressure, and other imperfect conditions, his inclined plane was not ideal for studying the forces on various objects. What happens, he pondered, if you have an ideal plane? Like Democritus mentally sharpening his knife, you mentally polish the plane until it attains the ultimate smoothness, completely free of friction. Then you stick it in an evacuated chamber to get rid of air resistance. And you extend the plane to infinity. You make sure the plane is absolutely horizontal. Now when you give a tiny nudge to the perfectly polished ball sitting on your smooth, smooth plane, how far will it roll? For how long will it roll? (As long as all of this is in the mind, the experiment is possible and cheap.)
The answer is forever. Galileo reasoned thus: when a plane, even an earthly imperfect plane, is tilted up, a ball, started by a push from the bottom, will go slower and slower. If the plane is tilted down, a ball released at the top will go faster and faster. Therefore, using the intuitive sense of continuity of action, he concluded that a ball on a flat plane will neither slow down nor speed up but continue forever. Galileo had made an intuitive jump to what we now call Newton's first law of motion: a body in motion tends to remain in motion. Forces are not necessary for motion, only for changes in motion. In contrast to the Aristotelian view, a body's natural state is motion with constant velocity. Rest is the special case of zero velocity, but in the new view that is no more natural than any other constant velocity. For anyone who has driven a car or a chariot, this is a counterintuitive idea. Unless you keep your foot on the pedal or keep whipping the horse, the vehicle will halt. Galileo saw that to find the truth you must mentally attribute ideal conditions to your instrument. (Or drive your car on an ice-slicked road.) It was Galileo's genius to see how to remove natural obfuscations such as friction and air resistance to establish a set of fundamental relations about the world.
As we shall see, the God Particle itself is a complication imposed upon a simple, beautiful universe, perhaps in order to hide this dazzling symmetry from an as yet undeserving humanity.
THE TRUTH OF THE TOWER
The most famous example of Galileo's ability to strip complications away from simplicity is the Leaning Tower of Pisa story. Many experts doubt that this fabled event ever took place. Stephen Hawking, for one, writes that the story is "almost certainly untrue." Why, Hawking asks, would Galileo bother dropping weights from a tower with no accurate way of timing their descent when he already had his inclined plane to work with? Shades of the Greeks! Hawking, the theorist, is using Pure Reason here. That doesn't cut it with a guy like Galileo, an experimenter's experimenter.
Stillman Drake, the biographer of choice of Galileo, believes the Leaning Tower story is true for a number of sound historical reasons. But it also fits Galileo's personality. The Tower experiment was not really an experiment at all but a demonstration, a media happening, the first great scientific publicity stunt. Galileo was showing off, and showing up his critics.
Galileo was an irascible sort of guy—not really contentious, but quick of temper and a fierce competitor when challenged. He could be a pain in the ass when annoyed, and he was annoyed by foolishness in all its forms. An informal man, he ridiculed the doctoral gowns that were required attire at the University of Pisa and wrote a humorous poem called "Against the Toga" that was appreciated most by the younger and poorer lecturers, who could ill afford the robes. (Democritus, who loves togas, didn't enjoy the poem at all.) The older professors were less than amused. Galileo also wrote attacks on his rivals using various pseudonyms. His style was distinct, and not too many people were fooled. No wonder he had enemies.
His worst intellectual rivals were the Aristotelians, who believed that a body moves only if driven by some force and that a heavy body falls faster than a light one because it has a greater pull toward the earth. The thought of testing these ideas never occurred to them. Aristotelian sch
olars pretty much ruled the University of Pisa and, for that matter, most universities in Italy. As you can imagine, Galileo wasn't a big favorite of theirs.
The stunt at the Leaning Tower of Pisa was directed at this group. Hawking is right that it wouldn't have been an ideal experiment. But it was an event. And as in any staged event, Galileo knew in advance how it was going to come out. I can see him climbing the tower in total darkness at three in the morning and tossing a couple of lead balls down at his postdoc assistants. "You should feel both balls hitting you in the head simultaneously," he yells at his assistant. "Holler if the big one hits you first." But he didn't really have to do that, because he had already reasoned that both balls should strike the ground at the same instant.
Here's how his mind worked: let us suppose, he said, that Aristotle was right. The heavy ball will land first, meaning that it will accelerate faster. Let us then tie the heavy ball to the light ball. If the light ball is indeed slower, it will hold back the heavy ball, making it fall more slowly. However by tying them both together, we have created an even heavier object, and this combination object should fall faster than each ball individually. How do we solve this dilemma? Only one solution satisfies all conditions: both balls must fall at the same rate of speed. That is the only conclusion that gets around the slower/ faster conundrum.
According to the story, Galileo spent a good part of the morning dropping lead balls from the tower, proving his point to interested observers and scaring the heck out of everybody else. He was wise enough to not use a penny and a feather but instead unequal weights of very similar shapes (such as a wooden ball and a hollow lead sphere of the same radius) to roughly equalize the air resistance. The rest is history, or it should be. Galileo had demonstrated that free fall is utterly independent of mass (though he didn't know why, and it would take Einstein, in 1915, to really understand it). The Aristotelians were taught a lesson they never forgot—or forgave.
Is this science or show biz? A little of both. It's not only experimenters who are so inclined. Richard Feynman, the great theorist (but one who always showed a passionate interest in experiment), thrust himself into the public eye when he was on the commission investigating the Challenger space shuttle disaster. As you may recall, there was a controversy over the ability of the shuttle's O-rings to withstand low temperatures. Feynman ended the controversy with one simple act: when the TV cameras were on him, he tossed a bit of O-ring into a glass of ice water and let the audience view its loss of elasticity. Now, don't you suspect that Feynman, like Galileo, knew in advance what was going to happen?
In fact, in the 1990s, Galileo's Tower experiment has emerged with a brand-new intensity. The issue involves the possibility that there is a "fifth force," a hypothetical addition to Newton's law of gravitation that would produce an extremely small difference when a copper ball and, say, a lead ball are dropped. The difference in time of fall through, say, one hundred feet might be less than a billionth of a second, unthinkable in Galileo's time but merely a respectable challenge with today's technology. So far, evidence for the fifth force, which appeared in the late 1980s, has all but vanished, but keep watching your newspaper for updates.
GALILEO'S ATOMS
What did Galileo think about atoms? Influenced by Archimedes, Democritus, and Lucretius, Galileo was intuitively an atomist. He taught and wrote about the nature of matter and light over many decades, especially in his book The Assayer of 1622 and his last work, the great Dialogues Concerning the Two New Sciences. He seemed to believe that light consisted of pointlike corpuscles and that matter was similarly constructed.
Galileo called atoms "the smallest quanta." Later he pictured an "infinite number of atoms separated by an infinite number of voids." The mechanistic view is closely tied to the mathematics of infinitesimals, a precursor to the calculus that would be invented sixty years later by Newton. Here we have a rich lode of paradox. Take a simple circular cone—a dunce cap?—and think of slicing it horizontally, parallel to its base. Let's examine two contiguous slices. The top of the lower piece is a circle, the bottom of the upper piece is a circle. Since they were previously in direct contact, point to point, they have the same radius. Yet the cone is continuously getting smaller, so how can the circles be the same? However, if each circle is composed of an infinite number of atoms and voids, one can imagine that the upper circle contains a smaller though still infinite number of atoms. No? Let's remember that we are in 1630 or so and dealing with exceedingly abstract ideas—ideas that were almost two hundred years from experimental test. (One way around this paradox is to ask how thick the knife is that slices the cone. I think I hear Democritus giggling again.)
In Dialogues Concerning the Two New Sciences, Galileo presents his last thoughts on atom structure. In this hypothesis, according to recent historical scholars, atoms are reduced to the mathematical abstraction of points, lacking any dimension, clearly indivisible and uncuttable, but devoid of the shapes that Democritus had envisioned.
Here Galileo moves the idea closer to its most modern version, the pointlike quarks and leptons.
ACCELERATORS AND TELESCOPES
Quarks are even more abstract and difficult to visualize than atoms. No one has ever "seen" one, so how can they exist? Our proof is indirect. Particles collide in an accelerator. Sophisticated electronics receive and process electrical pulses generated by particles in a variety of sensors in the detector. A computer interprets the electronic impulses from the detector, reducing them to a bunch of zeroes and ones. It sends these results to a monitor in our control room. We look at the representation of ones and zeroes and say, "Holy cow, a quark!" It seems so far-fetched to the layman. How can we be so sure? Couldn't the accelerator or the detector or the computer or the wire from the computer to the monitor have "manufactured" the quark? After all, we never see the quark with our own God-given eyes. Oh, for a time when science was simpler! Wouldn't it be great to be back in the sixteenth century? Or would it? Ask Galileo.
Galileo built, according to his records, a huge number of telescopes. He tested his telescope, in his own words, "a hundred thousand times on a hundred thousand stars and other objects." He trusted the thing. Now I have this little mental picture. Here's Galileo with all his graduate students. He's looking out the window with his telescope and describing what he sees, and they're all scribbling it down: "Here's a tree. It's got a branch this way and a leaf that way." After he tells them what he sees through the telescope, they all get on their horses or wagons—maybe a bus—and go across the field to look at the tree close up. They compare it to Galileo's description. That's how you calibrate an instrument. You do that ten thousand times. So a critic of Galileo describes the meticulous nature of the testing and says, "If I follow these experiments on terrestrial objects, the telescope is superb. I trust it, even though it interposes something between the God-given eye and the God-given object. Nevertheless, it does not fool you. On the other hand, if you look up at the sky, there's a star. And if you look through the telescope, there are two stars. It's totally cracked!"
Okay, those weren't his exact words. But one critic did use words to this effect to dispute Galileo's claim that Jupiter has four moons. Since the telescope allowed him to see more than could be seen with the naked eye, it must be lying. A math professor also dismissed Galileo, saying he, too, could find four moons of Jupiter if given enough time "to build them into some glasses."
Anyone who uses an instrument runs into this problem. Is the instrument "manufacturing" the results? Galileo's critics seem foolish today, but were they off the wall or just scientific conservatives? Some of both, no doubt;. In 1600 people believed that the eye had an active role in vision; the eyeball, given to us by God, interpreted the visual world for us. Today we know the eye is no more than a lens with a bunch of receptors in it that passes visual information along to our brain's visual cortex, where we actually "see." The eye is in fact a mediator between the object and the brain, just as the telescope is. Do you wear eyegl
asses? You're already modifying. In fact, among devout Christians and philosophers in sixteenth-century Europe, wearing spectacles was considered almost sacrilegious, even though they had been around for three hundred years. One notable exception was Johannes Kepler, who was very religious but who nonetheless wore specs because they helped him see; this was fortunate, given that he became the greatest astronomer of his time.
Let's accept that a well-calibrated instrument can provide a good approximation of reality. As good perhaps as the ultimate instrument, our brain. Even the brain must be calibrated at times, and safeguards and fudge factors applied to compensate for distortion. For example, even if you have 20/20 vision, a few glasses of wine can double the number of friends around you.
THE CARLSAGAN OF 1600
Galileo helped pioneer the acceptance of instruments, an accomplishment whose importance to science and experimentation cannot be overemphasized. What sort of person was he? He comes across as a deep thinker with a subtle mind, capable of intuitive insights that would be the envy of any theoretical physicist today, but with energy and technical skills that included lens polishing and the construction of many instruments, including telescopes, the compound microscope, and the pendulum clock. Politically he alternated from docile conservatism to bold, slashing attacks on his opponents. He must have been a dynamo of activity, constantly engaged, for he left behind an enormous correspondence and monumental volumes of published works. He was a popularizer, giving public lectures to huge audiences after the supernova of 1604, writing in a lucid, vulgarized Latin. No one comes as close as he does to being the Carl Sagan of his day. Not too many faculties would have granted him tenure, so vigorous was his style and so stinging his criticism, at least before his condemnation.
The God Particle: If the Universe Is the Answer, What Is the Question? Page 10
