by David Orrell
Like Galileo, Newton had his first public success with an improved telescope. The invention, which increased magnification by means of a mirror he ground himself, secured Newton a place in the Royal Society, a prestigious club for eminent scientists. There, he got drawn into a discussion with Edmond Halley (for whom the comet is named) and others about whether the elliptical planetary orbits that Kepler had discovered could be the result of an attractive force from the sun. Newton claimed that he had solved the question several years earlier, during his time in Lincolnshire, but couldn’t find the proof. He wrote it up anew, starting a project that culminated in his seminal work, Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), known more simply as the Principia.
The Principia laid down three laws of motion that Newton believed governed everything from apples to planets. The first law, often known as the law of inertia, stated (in Latin, of course) that “every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.” The key phrase here is “uniform motion.” So if a person is hurtling across the ice on frictionless skates, he will continue moving in the same direction and at the same speed until he encounters an obstacle, such as an ice-hockey player.
Newton’s second law states: “The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.” As Galileo knew, the speed of a falling object changes with a constant acceleration that is proportional to the force of gravity acting on it. The third law says: “To every action there is always opposed an equal and opposite reaction.” The earth attracts the moon, but the moon also attracts the earth. The reason the moon goes around the earth, rather than vice versa, is because the earth is much larger. However, Newton showed (and Kepler had earlier argued) that the tides are the ocean’s response to the moon’s gravitational tug.
Scientists and military engineers, including Leonardo da Vinci, had long been interested in the behaviour of projectiles such as cannonballs. Newton carried out a kind of thought experiment in which he imagined a cannonball being shot horizontally from the top of a high mountain. If there were no air resistance, the ball would fall to the ground in a slow arc. If the cannon was extremely powerful, though, it could in principle send the ball over such a distance that the curve of the earth would have to be taken into account. There would come a point where the ball would continuously fall to earth but never get there, eventually performing a complete circle. In effect, this is what satellites, or the moon, do; they are always falling to earth but keep missing.
MALE VS FEMALE
To prove his results mathematically, Newton had to develop the tool of calculus, which was also independently discovered by Leibniz. Since this approach is key to how predictions are currently made in all branches of science, it is worth discussing in detail. Let’s suppose we wanted to predict how long it would take a stone to hit the ground after being dropped, Galileo fashion, from the Leaning Tower of Pisa. The motion of the stone can be described at any time (t) by its position (x) and its velocity (v); these are known as variables, since they change with time. The laws of motion then give two equations:
dx/dt = v
dv/dt = –g
In words, the first line says that the rate of change of height x (denoted dx/dt) equals the velocity. The second line says that the rate of change of velocity (the acceleration) is given by the force of gravity per unit mass, which for objects near the earth’s surface is a constant parameter g. Experiments have shown that g is about 9.8 in standard metric units (Newton showed it actually decreases with the square of the distance from the centre of the earth, but this is a small effect here). The minus sign indicates that the force is directed downwards, so height decreases with time.
The equations therefore involve two kinds of quantities: the variables x and v, which change with time, and the parameter g, which is treated as fixed. Time itself is treated as a dimension rather like distance, and is independent of the other quantities. As Newton wrote in the Principia, “Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external.” It varies in a smooth, continuous fashion, and can be infinitely divided into smaller and smaller sections.
The two equations are known as ordinary differential equations, or ODEs, and together make up a dynamical system. Note that they tell us only how the position or the velocity is changing. They are similar to the prompts given by a navigation system in a car: drive straight now; turn left now; turn right now. If you follow the instructions at each moment, you should reach your destination even if you are in unfamiliar territory. The ODEs do the same thing, except that they constantly update the information. To predict when the stone will arrive at its destination, the ground, we have to compute all the forces acting on the stone at every instant.
The ODEs therefore tell a kind of story (or ode) about what happens to the stone from the moment it is dropped. If we probe a little deeper, we discover there is also a subtext. Francis Bacon wrote in The Masculine Birth of Time that scientific laws “conquer and subdue Nature,” and they “storm and occupy her castles and strongholds.”27 In her more recent Sideways Look at Time, the author Jay Griffiths observes that “Aristotle used the terms ‘male’ and ‘female’ to describe differing understandings of time in the cosmos, calling the heavens male because he considered them eternal and immutable, while calling the earth female because it was changeable.”28 In this drama, then, the variables are the “feminine,” changeable quantities that are governed and controlled by the “masculine,” unchanging physical laws. The style is austere—the ODEs describe where the stone is and what it is doing, but they say nothing about its colour, shape, smell, texture, or weight. The senses are not engaged. As the Pythagoreans said, “’Tis mind that all things sees and hears. What else exists is deaf and blind.”29
Calculating the solution to the ODEs—the actual path followed by the stone—is a thorny problem, but fortunately for us, Newton developed his theory of calculus to help. First, we must know the initial condition for the variables: the starting position x0 (the height of the tower, which is about sixty metres) and the velocity v0 (zero, if the stone is dropped). As shown in the notes, the answer can then be expressed in terms of x0, v0, t, and the fixed parameter g.30 For any initial condition, the solution gives the height and the velocity as a function of time, so it can be viewed as a kind of machine: put in the time, turn the handle, and out comes the answer. For the solution to be accurate, the initial condition must be precisely measured and the equations must account for all the forces. Observations and theory must each be correct.
While we now have a formal answer to the problem, we still need a way to visualize and interpret the results. In his work La Géométrie, published in 1637, the French philosopher and scientist René Descartes welded together the Arab tradition of algebra, which dealt with equations, and the Greek tradition of geometry, which dealt with geometric figures, by showing that the solutions to equations could be plotted as figures on what is now known as a Cartesian grid.
THE GRID
Like many North Americans, I grew up in a Cartesian grid. The streets in Edmonton, Alberta, run north to south, while the avenues run east to west. The city is therefore divided into neat squares. This doesn’t quite hold everywhere, of course—the river valley is stubbornly non-Cartesian, as are some of the suburbs—but on the whole, the system works, making it incredibly easy for Edmontonians to find their way around. In cities designed around more organic principles, I find it hard to shake the feeling of being permanently disoriented, like an explorer without his magnetic north.
Descartes reportedly got his idea when, as a bored young engineer working for the military, he contemplated a fly buzzing around the room and realized that it would be possible to represent its location in three-dimensional space by using three diffe
rent co-ordinates.31 (The idea later turned out to have all sorts of military applications, in guidance systems, for example.) If the aim is instead to locate an office in Edmonton, the co-ordinates could be the street and building number, the avenue, and the floor number. In a dynamical system, the method can be used to plot any variable or combination of variables. Time can be treated as a separate dimension, like a spatial co-ordinate.
FIGURE 2.4 Plots of the position (x) and velocity (v) of a falling object, as a function of time (t).
Figure 2.4 shows x and v as functions of time. If you look up a certain time on the horizontal axis of the Cartesian grid, you canfind the value of the variable at that time on the vertical axis. In the left grid, the stone begins at time 0 at a height of 60 and takes about 3.5 seconds to reach height 0 (i.e., hit the ground). The right grid shows that velocity becomes increasingly negative (i.e., faster) in a linear fashion.32. In reality, the speed would taper off because of the effect of air resistance, which is not considered here.
If the equations of the dynamical system are sufficiently simple, then the methods of calculus can be applied to find the solution, as here. More complicated systems often cannot be solved using calculus, so an approximation technique must be employed. Essentially, this involves dividing the time into short steps, solving the equations for each time step, and finally stitching the answers together. This approach involves a lot of computation and can be numerically unstable, so it didn’t become feasible for general use until the invention of fast computers in the 1950s.
Newton was a Unitarian and rejected the doctrine of the Holy Trinity. (In religion as well as science, he favoured one over plurality.) If this had become known by his Cambridge employers, it would have cost him his job. But unlike Kepler and Galileo, Newton kept his views secret and avoided such fights, though the stress might have contributed to an apparent nervous breakdown he suffered in 1693. He cut back on scientific research and took a position as master of the mint at the Bank of England. (Pythagoras was not the last scientist to take an interest in coins.) He took his job seriously, especially when it came to combatting counterfeiters, several of whom he sent off to the gallows. He also made sure to protect his own image, fiercely denouncing scientists such as Robert Hooke who could claim to have contributed to his ideas. He died in 1727 a wealthy and famous man, and was buried in Westminster Abbey.Alexander Pope supplied his epitaph, which compared Newton to a kind of cosmic light bulb:
Nature, and Nature’s Laws lay hid in Night.
God said, Let Newton be! And All was Light.
Newton synthesized the work of many scientists before him and created a kind of Unitarian theory of the universe, which viewed everything from apples to planets as being united under a single set of laws. Kepler had discovered that the planets move in elliptical orbits, but he had seen this as a violation of Pythagorean symmetry. Newton showed that what counted was not the shape of the motion but the underlying dynamic. The orbit might be an ellipse, but the law is simple and square.
A consequence was that scientists suddenly seemed to have enormous predictive power at their disposal. If the motion of a planet around a star is given by a simple equation, then we can figure out its position at any time in the future just by plugging in numbers. And if phenomena here on earth obey similar principles, they too can be modelled mathematically. Kepler, it seemed, was right: we lived in a deterministic, predictable, clock-like universe.
3 DIVIDE AND CONQUER
THE GOSPEL OF DETERMINISTIC SCIENCE
The simplicity of nature is not to be measured by that of our conceptions. Infinitely varied in its effects, nature is simple only in its causes, and its economy consists in producing a great number of phenomena, often very complicated, by means of a small number of general laws.
—Pierre Simon Laplace, French mathematician and physicist
The vanity of men
they would like to retain
this passing winter moon.
—Issa, Japanese haiku poet
STRAIGHT VS CROOKED
I once worked for two years at a research institute in Saclay, near Paris. While living there, I was struck, among other things, by the enjoyment the French seemed to take in straight lines. The gardens of Paris and Versailles glory in the geometric perfection of the line that seems to go on forever. From the Louvre, you can look in an almost perfect line through the Jardin des Tuileries, down the Champs Élysées, all the way to the Arc de Triomphe and beyond. Even the bread is in the shape of a line. Perhaps it is not surprising that so much deterministic science, based on linear, cause-and-effect thinking, made its start there. The apple of deterministic science may have fallen from Newton’s tree, but it found particularly fertile soil in the country across the Channel.
In 1637, Descartes had prepared the ground with his Discours de la méthode, in which he laid out his principles for scientific discovery. These were:
Never accept anything as true unless it clearly is.
Divide every difficult problem into small parts, and solve the problem by attacking these parts.
Always proceed from the simple to the complex, looking for patterns and order.
Be as complete and thorough as possible, so that nothing is missed.1
That troublesome beast nature was to be interrogated like a witness in a court of law. To analyze a complex event, it was only necessary to break it down into separate components, work out where everything was at any given time and what it was doing, and apply the basic physical laws. Just as Baron Haussmann would soon be blasting arrow-straight boulevards down the centre of Paris, the French scientists, along with their colleagues across Europe, began applying the gospel of deterministic science to the natural world.
Newton had shown that the force of gravity varied inversely with the square of distance, and he’d used calculus to demonstrate that as a result the planets followed elliptical orbits. Charles Augustin de Coulomb then found a corresponding inverse-square law for electrical charges. If the distance between two charge-carrying spheres is doubled, he determined, the force between them decreases by a factor of four, exactly like gravity. It followed that charged bodies were as predictable as the planets.
Antoine Lavoisier pushed the Newtonian scheme into the field of chemistry. Newton had throughout his life been fascinated by alchemy, but he’d never examined the behaviour of chemicals with the same kind of rigour that he applied to the heavens. Lavoisier, whose day job as an accountant for a private tax-collection agency resulted in a date with the guillotine during the French Revolution, applied his accounting skills to chemical experiments. He showed that chemicals were of two types, elements and compounds. In a famous two-day public experiment, he demonstrated how water was made up of hydrogen and oxygen. We are now somewhat inured to the wonder that this must have caused the audience, but it is still amazing to think that water, perhaps the most important substance in our lives, is actually made up of two gases, oxygen and hydrogen, the second of which is highly explosive.
By the nineteenth century, science was on a roll—or rather, a relentless Napoleonic march down a long, straight line. The optimism was captured by Pierre Simon Laplace, who stated in 1820 that if we knew the present state of all particles of the universe, and the forces acting on them, we could, in principle, predict their future: “We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.”2 This know-it-all intellect is sometimes referred to as Laplace’s demon. Of course, Laplace didn’t think scientists could achieve such accuracy in practice—but their goal was to come as close as possible.
>
Not everyone was convinced by the power of determinism, or by the idea that what worked for planets would apply everywhere. In 1790, Immanuel Kant said: “It is absurd to hope that another Newton will arise in the future who will make comprehensible to us the production of a blade of grass according to natural laws.” Leaders of the Romantic movement, like Blake in England and Goethe in Germany, criticized the mechanical model and espoused an organic view of the world. Goethe’s epic poem Faust turned the idea of selling the soul to the devil into a metaphor for the power and the hidden, human cost of materialistic science. Blake put it even more bluntly: “May God us keep / From single vision and Newton’s sleep!”3
However, deterministic science was stunningly confirmed, at least to its practitioners, by its ability to predict previously unknown features of physical systems. James Clerk Maxwell produced four equations that showed that light was a combination of electric and magnetic oscillations, and that predicted the existence of electromagnetic waves outside the visible spectrum (a hypothesis that was later confirmed by Heinrich Hertz, and nowadays by anyone who listens to his car radio). Dmitri Mendeleyev predicted the existence of undiscovered substances to fill the gaps in his periodic table, which ordered the elements according to atomic weight. Urbain Leverrier used Newton’s laws to show that measured perturbations in planetary orbits had to be caused by the presence of another planet, Neptune, whose existence was soon confirmed by observation. (He also predicted the planet Vulcan, which too was observed for some time, despite the fact that it didn’t exist.)
Determinism wasn’t limited to purely physical systems. In December of 1831, a twenty-two-year-old med-school dropout boarded the Beagle to begin a fiveyear trip around the world. At the time, many people relied on physiognomy to divine human character, and the captain of the Beagle, Robert FitzRoy, almost didn’t select the young man because he distrusted the shape of his nose.4 However, Charles Darwin passed this hurdle, by a nose, and started a voyage that would change both his life and our understanding of life.
