Deep Simplicity

Chapter One

Order out of Chaos

Before the scientific revolution of the seventeenth century, the world seemed to be ruled by chaos in a quite different way from the way the term is used by scientists today, but in the same way that most people still apply the word. There was no suggestion that there might be simple, orderly laws underpinning the confusion of the world, and the nearest anyone came to offering a reason for the behavior of wind and weather, the occurrence of famines, or the orbits of the planets was that they resulted from the whim of God, or the gods. Where order was perceived in the Universe, it was attributed to the response of physical objects to a need for harmony and order to be preserved wherever possible—the orbits of the planets and the Sun around the Earth (thought to be at the center of the Universe) were supposed to be circles, because circles were perfect; things fell downwards because the center of the Earth was at the center of everything, the center of symmetry in the Universe, and therefore the most desirable place to be. Even when the philosopher Aristarchus of Samos, who lived in the third century b.c., dared to suggest that the Earth moved around the Sun, he still imagined that it must follow a circular orbit.

These examples highlight an absolutely crucial difference between the science of the Ancients and the science of post-Galilean times. The Ancient Greeks were superb mathematicians, and in particular they were superb geometers, who had a very good understanding of the relationships between stationary things. This geometry had its roots in even earlier cultures, of course, and it is easy to imagine how this first science may have arisen out of the practicalities of life in the developing agricultural societies of prehistory, through the problems associated with building houses and laying out towns, and the need, as society became more complicated, to divide up land into fields. But the Ancients had no understanding at all of how things move, or the laws of motion. You have only to look at how puzzled they were by Zeno’s famous paradoxes, such as the soldier who can never be killed by an arrow. If he runs away, then by the time the arrow reaches the position he was in he has moved; in the time it takes the arrow to cover that extra distance, he can move a little farther; and so on.

In spite of the existence of people like Aristarchus, the Earth-centered Universe remained the established image (what scientists would now call a “model”) even after Nicolaus Copernicus published his model of a Sun-centered Universe (but one still based on circles) in 1543. His book, De Revolutionibus Orbium Coelestium, had been essentially completed in 1530, and much of its contents were widely discussed before publication, leading Martin Luther to comment in 1539, “This fool wishes to reverse the entire science of astronomy; but sacred Scripture tells us that Joshua commanded the Sun to stand still, and not the Earth.” Responding to similar criticisms, Galileo later riposted: “The Bible shows the way to go to Heaven, not the way the heavens go.” It was Galileo’s contemporary Johannes Kepler, using observations painstakingly compiled by Tycho Brahe, who established, for those with open eyes, that not only did the planet Mars move around the Sun, but that it did so in an elliptical orbit, pulling the rug from under the notion that the kind of circular perfection beloved of the Ancient Greeks ruled the cosmos.

Even to people who know little about science, or the history of science, Galileo (who lived from 1564 to 1642) is famous today as the man who turned one of the first telescopes on the heavens, found evidence to support the Sun-centered Copernican model, and had a run-in with the Catholic Church, which led to his conviction for heresy and the suppression of his books in Catholic countries—which (of course) led to them selling like hotcakes everywhere else. But he did much more than this. It was Galileo, more than anyone, who laid down the principles of the scientific method of investigation, which involves comparing theories (or models) with the outcome of experiment and observation, and it was Galileo who first came to grips with motion in a scientific way.

The key to Galileo’s work on motion was a discovery he made while a medical student in Pisa in 1583. During a boring sermon in the cathedral there, he watched a chandelier swinging to and fro, and timed the swing with his pulse. Galileo realized that the time it took for the lamp to complete one swing was the same whether it swung through a wide arc or a shallow one, and later experiments showed that the time taken for a pendulum to swing depends on its length, not on how far it swings. This is the basis of the pendulum clock, but even without going so far as to build a clock (he did design one, later built by his son), Galileo was able to use a pendulum as an accurate timekeeper when he later carried out experiments to study the behavior of balls rolling down a ramp. These experiments provide another insight into both Galileo’s mind and the scientific method. He wanted to study falling objects, to investigate the effect of gravity on motion. But falling balls moved too fast for him to keep track of. So he rolled the balls down an inclined ramp, realizing that this gave him a stretched-out and slowed-down version of the way balls fall under gravity. Through these experiments, Galileo developed the idea of acceleration. The velocity (or speed) of an object tells you how far it moves in a certain amount of time—say, one second. A constant velocity of 9.8 meters per second means that in every second the moving object covers a distance of 9.8 meters. But Galileo found that falling objects (or balls rolling down a ramp) move faster and faster, with the speed increasing each second. Crucially, his experiments showed that the speed increases by the same amount every second. This is uniform acceleration, and a uniform acceleration of 9.8 meters per second per second means that, starting from rest, after one second an object has a velocity of 9.8 meters per second, after two seconds it has a velocity of 19.6 meters per second, after three seconds it has a velocity of 29.4 meters per second, and so on. I have chosen this particular example because 9.8 meters per second per second is, indeed, the acceleration caused by gravity for a falling object at the surface of the Earth; because time comes into the calculation twice, it is called a second order effect, while velocity is a first order effect. And this acceleration due to gravity explains why pendulums behave as they do.

Galileo did something else—something central to the story we tell in this book. He realized that the balls rolling down his inclined planes were being slowed down a little by friction. In fact, what he measured was not a perfectly uniform acceleration. But he took the dramatic and influential leap, astonishing for his time, of extrapolating from his actual observations to work out how his balls would move without the effect of friction, on some idealized, perfectly slippery slope. This kind of extrapolation would be at the heart of the scientific investigation of the world for the next four centuries. When scientists—physicists in particular—tried to describe the world in terms of mathematical laws, they formulated those laws to describe the behavior of mythical objects such as perfectly hard spheres, which bounce off one another without being deformed and roll along surfaces without feeling friction, and so on. But, unlike the Ancient Greek philosophers, they knew that their image of perfection did not represent the real world. Armed with those equations, they could then try to put in extra terms, correction factors, to take account of the imperfections of the real world, allowing, say, for the effect of air resistance on a falling object. Air resistance explains why on Earth a hammer and a feather fall at different rates, while on the airless Moon, as the Apollo astronauts demonstrated, they fall at the same rate.

All of this helped Galileo to cast out of science another aspect of the geometrical perfection that his predecessors had imagined in the real world. Before Galileo, it was thought that when a cannon fired its ball at some angle above the horizontal, the flight of the ball would consist of a straight line as it left the muzzle, then it would follow the arc of a perfect circle for a time, and then it would fall vertically to the ground. Only the imagined perfection of straight lines and circles was involved in the motion. Applying his discovery that gravity produces a constant downward acceleration on the cannonball, and allowing for the initial velocity of the ball out of the muzzle, Galileo showed that the flight of the ball must actually be a single smooth curve, part of a parabola, all the way to its target. The same calculations showed that the maximum range for the cannon (assuming the same charge of gunpowder and weight of shot) would always be achieved when it was fired at an angle of 45 degrees upward from the horizontal. These were practical matters of great importance in the turbulent times Galileo lived in, and this kind of military work helped establish his early reputation. Whatever philosophers and theologians might say about perfection, armies in the field had no time to quibble about the desirability of circular motion; all they wanted to know was which way to point their guns to achieve maximum effect, and Galileo told them.

It was a combination of Kepler’s discovery of elliptical orbits and Galileo’s insights into both acceleration and the scientific method that paved the way for the greatest scientific discovery of the seventeenth century, and perhaps of all time: Isaac Newton’s universal law of gravitation. Newton was born in 1642 and died in 1727. His great work Philosophiae Naturalis Principia Mathematica, or just the Principia for short, was written from 1684 to 1687, and published in 1687, but was based on ideas that he had developed twenty years earlier, when he was a young Cambridge graduate about to become a fellow of Trinity College, and had been forced to spend months at his mother’s home in Lincolnshire when the University was closed because of the plague. Like Galileo, Newton stressed the importance of comparing theories and models with experiments and observations of the real world, and always carried out relevant experiments himself, whenever possible, to test his ideas. This is so deeply ingrained as part of the scientific method today that it may seem obvious, even to nonscientists, and it is hard to appreciate the extent to which, even into the seventeenth century, many philosophers would speculate about the nature of the physical world in an abstract way, without ever getting their hands dirty in experiments. The classic example is the argument about whether two different weights dropped from the same height at the same time would hit the ground together—an argument that rumbled on for decades among those abstract thinkers even after a Flemish engineer, Simon Stevin, actually did such experiments with lead weights dropped from a height of about ten meters, found that they fell at the same rate, and published his results

in 1586.1

Newton also embraced and refined Galileo’s insight into the value of deliberately simplified models (such as Galileo’s frictionless planes) as descriptions of particular aspects of the real world. It is a key feature of Newton’s work on gravity and orbits, for example, that in his calculations of the effects of gravity he treats an object such as Mars, or the Moon, or an apple, as if all its mass were concentrated at a single point, and that provided you are outside the object of interest, its gravitational influence is measured in terms of your distance from that point, the center of mass of the object (which is also the geometrical center if the object is a sphere). The alternative would be to carry out a separate calculation for each atom in the Moon, or Mars, or whatever object you were studying. In the Principia, Newton proved that this is the case for spherical objects. He knew that the Earth is not precisely spherical (indeed, he was able to calculate how much the Earth bulges at the equator because of its rotation); but he felt it was reasonable to assume a spherical Earth (and a spherical Sun, a spherical Mars, and so on) as a first approximation, and calculate the orbits accordingly. As it happens, later calculations have established that, provided you are far enough away from them, even very irregularly shaped objects do act, gravitationally speaking, as if all their mass were concentrated at a point, but this doesn’t diminish the importance of the idea of using idealized approximations to reality where it is necessary, or helpful, to make the calculations more straightforward.

There is, though, rather more to this particular story than meets the eye. In the Principia, Newton proved the validity of treating the gravitational influence of a spherical object as if all its mass were concentrated at its center, using geometrical techniques that would have been understood by the Ancient Greeks, and were certainly familiar to Newton’s contemporaries. These calculations were difficult. But we now know that well before he wrote the Principia, Newton had developed (or discovered) the mathematical technique now called calculus, and that this proof is very easy using calculus. Some scholars suspect that Newton actually solved the problem using calculus first, and then went through the painstaking process of translating everything into classical terminology, to make sure that his contemporaries would understand it. If so, he may, in a sense, have shot himself in the foot, because by keeping quiet about his new mathematical technique, he paved the way for a bitter wrangle with the German Wilhelm Leibniz, who invented the technique independently (and gave it the name by which it is still known). Leibniz had the idea a little later than Newton did, but had the good sense to publish his work, which is partly why there were bitter rows about priority (the bitter rows were also partly because both protagonists were unwilling to make concessions about sharing credit for their discoveries, and Newton in particular was a rather unpleasant person, who had an arrogant belief in his own abilities and a spiteful reaction to anyone he perceived as an opponent). But the arguments over priority are of little concern here. What matters is that calculus is a technique that makes it possible to break up problems into tiny components that can be manipulated mathematically and the results added up to give a solution to the overall problem. In the case of the gravitational influence of a spherical object, for example, the sphere can be treated as being divided up (differentiated) into an infinite number of infinitesimally small pieces of matter, and an equation describing the gravitational influence of such a typical piece of matter in terms of its position in the sphere can be written down.