It is difficult for us to see the world as Aristotle did. Our thoroughly quantitative understanding of motion has become second nature, thanks to familiar concepts like uniform speed and acceleration, to a technologically rich environment containing instruments like digital clocks and speedometers, and to our practical experience with equipment that depends on such concepts and instruments. The experience of Aristotle and his contemporaries was much different. They had neither the experimental instruments nor a mathematical framework for measuring and analyzing motion, and no urgent reason to seek them. They found it plausible to understand movement in terms of form and purpose, not of how quickly the motion takes place.
Aristotle and his contemporaries were not familiar with any of the key components of F = ma. His notion of speed or ‘quickness’ was simply that some things cover more ground in the same time than other things – what we would call average speed or overall speed, rather than instantaneous speed, or speed at a particular instant.5 His notion of acceleration was simply that some things go more quickly as they approach their natural place.6 He had no concept of mass: of a resistance to being pushed that is not identical to weight. And he had no quantitative notion of dynamis, the capacity for motion, nor any units to measure it in.
Nevertheless, it made sense to view nature as a vast ecosystem, comprised of different types of substances acting through different kinds of inner compulsions on other substances, affecting others and being affected in turn, everything with a different purpose to play, all essential to the maintenance of the ecosystem with its qualitatively different domains. Understanding nature required seeing its phenomena in their perfected state – ‘perfected’ in the sense of fully deployed or actualized (the adult tree, the mature human being, the well-functioning society), the phenomena having attained their telos, or end, for in that condition the whys and hows of phenomena are most clear.
Aristotle liked to say that the wise person seeks only as much exactitude as the subject matter allows. He described what he saw, to the most appropriate level of precision that he could. What appeared to matter in understanding the motions of nature was the role that things like form, matter, and purpose play in converting potentiality into actuality. And these ultimately referred to the unmoved mover, who communicates through love via the outer spheres to the moon and then to the sublunary world.
Steps Beyond Aristotle
Aristotle’s picture of nature had an enormous impact on Western civilization. His ideas were passed on by students at the Lyceum, the school he founded, and by commentators on his works – at first Greeks, and then, from the ninth to the twelfth centuries, Arabs, from whom later Western scholars learned about Aristotle.
But aspects of Aristotle’s picture were not completely satisfying, not even to him. He seemed puzzled, for instance, by how things such as projectiles and potters’ wheels moved after the initial push. If a mover has to be in constant contact with what it moves, why doesn’t a stone or arrow plunge to the ground after leaving the hand or bow? Aristotle considered two possibilities. One was that the mover (thrower or bow) impregnates or impresses a force on the medium (air) around the projectile (stone or arrow), which then keeps the object in motion.7 The other explanation, the doctrine of antiperistasis, was that air displaced in front of the projectile rushes around to the back to squeeze the projectile forward.8 Aristotle was not comfortable with either explanation.
Later thinkers, too, were dissatisfied by this and by other elements of Aristotle’s account of motion. Some objections were logical, some empirical, some both. The result was discussion, inquiry, modification of Aristotle’s concepts, the introduction of new concepts, and – during a journey of thousands of years – a slow shift of attention to different aspects of motion that would lead, eventually, to F = ma. We will travel a long way without seeing anything that resembles its components. But each step of the journey was essential. What follows are some of the steps.
In the third century bc, Strato (340–268 BC), a Greek from Lampsacus in Asia Minor who took over as head of the Lyceum in 287, developed and extended Aristotle’s thinking in an influential book called On Motion. Strato found he had to revise or even reject some of Aristotle’s ideas to make them consistent with logic or experience. One was the idea that there were two kinds of natural movement: up and down. Strato argued that all things naturally go down toward the earth’s centre, and that if light things like fire and smoke rise, it’s because they are displaced or ‘squeezed out’ by heavier stuff. Strato was also bothered by two observations that seemed to suggest that things pick up speed as they fall. One was that when rainwater pours off a roof, the flow is continuous at first but then breaks into droplets, which could not happen if the water weren’t moving more quickly.9 The other was that, when you drop a stone to the ground from high up, the impact is more powerful than when you drop it from just above the ground. How could this be? The stone hasn’t gotten heavier! It must have picked up speed, Strato concluded, meaning that a falling body ‘completes the last part of its trajectory in the shortest time’, a rudimentary notion of acceleration more sophisticated than Aristotle’s.
In the sixth century ad, John Philoponus (‘Lover of Hard Work’, ca. 490–570) further revised Aristotle’s ideas on motion. Philoponus argued on logical grounds that motion was possible in a vacuum (something Aristotle had rejected), and solved the problem of what happens when force equals resistance by declaring that speed is determined by an excess of force over resistance. Philoponus is also the first person known to have actually experimented with falling bodies of different weights, discovering, as Galileo would a thousand years later, that they fall at approximately the same rate. But Philoponus’s most original and far-reaching revision of Aristotle’s ideas concerned projectile motion. He rejected antiperistasis; if the mover communicates motion to the air behind the projectile, why can’t we send stones and arrows flying by stirring up the air behind them with our hands? Philoponus proposed that, when we throw a stone, our hand impresses a force not on the air but on the stone itself. This ‘impressed force’ causes the motion to continue from inside the projectile, but is slowly consumed in overcoming the resistance of the medium and the natural downward force, and eventually used up as the natural motion takes over or the stone hits the ground. This view was still faithful to Aristotle’s in that it assumed that an object did not move by itself but always required contact with some other cause, such as the weight of a falling body or the impressed force borrowed from the hand. What was new was that this cause could be internal, not external, to the moving body. This idea led Philoponus and his followers to see the world differently. They no longer needed to distinguish natural and enforced motions, nor to separate the earthly and heavenly realms. God created the heavens, and then used impressed force to keep them going, there being no medium in the heavens to exhaust them. Philoponus’s influence helped to inspire scholars trying to understand motion to shift their attention from its end point – the goal or purpose of the motion, whether on earth or in heaven – to its beginning point, or what set it in motion.
Philoponus’s modifications, especially concerning impressed force, influenced Islamic commentators on Aristotle, such as the Spanish Islamic theologian Ibn Bājja (known to the West as Avempace, ca. 1095–1138), the Spanish Islamic theologian Ibn Rushd (Averroës, who objected to Philoponus’s view, 1126–1198), and the Persian Islamic theologian Ibn Sīnā (Avicenna, 980–1037). The latter translated Philoponus’s idea of impressed force into Arabic as mail qasrī (violent inclination). Heavier bodies can retain more mail than light bodies, which is why you can throw a stone farther than a blade of grass or a feather. And the Arab commentators concocted other situations where Aristotelian explanations were dissatisfying: What would happen if a tunnel were dug through the earth and a stone dropped in it? Would a thread attached to an arrow’s tip be pushed forward? In Ibn Sīnā’s work, even more clearly than in Philoponus’s, the key to motion is to be found not in formal and
final causes but in efficient and material causes.
This shift in attention is clear in the work of John Buridan (ca. 1300–1358). Further developing the ideas of Philoponus and Ibn Sīnā, Buridan gave impressed force the technical name impetus, by which it would be known until modern times. Unlike impressed force, impetus did not use itself up but was permanent; a body could only lose it by transferring it to something else.10 Impetus may sound like our notion of inertia, but unlike inertia it was still a cause. Thus the new framework was still Aristotelian, for it retained the distinction between natural and violent motion and viewed a projectile such as a stone or arrow as continually moved by the action of a cause, though this cause (the impetus) functioned within, not as per Aristotle without, the body. But several key Aristotelian puzzles – such as the question of projectile motion and how bodies fall – had vanished, for the thrower transmits impetus to the stone rather than to the medium, while a falling body acquires impetus as it falls, explaining why it picks up speed. The idea of impetus helped produce a primitive notion of mass – of resistance in a body different from weight – because things like cannonballs can ‘hold’ more impetus than light wood. And it explained why the celestial spheres move forever without divine intervention: with no resistance in the heavens, the spheres need no intervention. God created the spheres, and then gave them impetus, which is why God could rest on the seventh day without His creation grinding to a halt. God’s role is thus reversed from the way Aristotle saw it: God is not the continually active final cause that draws the spheres into motion and toward which they strive, but the efficient cause that sets them in motion in the first place.11
For the next 300 years, scholars used the idea of impetus to understand and explain motion. It reduced, but did not eliminate, the need to make qualitative differences between natural and violent motions, different kinds of substances, and the heavens and the earth. It also allowed for the development of new conceptions of force, such as percussive force (something that could act once, such as a bat striking a ball) and force that acts continually from a distance, such as whatever pulls objects to earth. It facilitated the development of the idea of mass; some internal density of matter in a body that resists force that is related but not identical to the body’s weight. Scholars were beginning to look at motion all by itself – in what philosopher Charles Taylor calls the ‘immanent frame’ – and to study it in certain (but not all) respects without reference to the purposes, plans, and designs of the rest of the universe. They begin to see what we would call a separation between physics and metaphysics. The scientific world and the lived world were beginning to come apart.
Mathematics, meanwhile, was being applied to the world in new ways. Numbers had been used in human affairs for centuries, of course, but scholars were developing new tools to deepen and extend their use. One was Thomas Bradwardine (ca. 1300–1349), from Merton College at Oxford University, later the Archbishop of Canterbury, so renowned in his time that he is mentioned (briefly) in Chaucer’s Canterbury Tales. Bradwardine developed the foundations of a mathematical framework able to handle velocity, instantaneous velocity (velocity at any particular instant, as opposed to average velocity over a time interval), uniform velocity, uniform acceleration, and changing acceleration.12 He recast the views of Aristotle, Philoponus, and Averroës in mathematical form, displayed their limitations, and stated his own law. Bradwardine’s work was further developed by Nicholas Oresme, who showed how numbers could be applied to describe any continuously varying quantity, such as movement, heat, and so forth. You ‘pretend’, Oresme said, that you are measuring a geometrical surface.13
Bradwardine, his followers (known as the Oxford ‘calculators’), Oresme, and others of the time were not experimenters, but produced a mathematically sophisticated framework for later experimenters. Their work paved the way for the widespread application of numbers to the world by people who saw no need to pretend when using them. In the late sixteenth and early seventeenth centuries, a vast extension of numbers into the world took place in which new kinds of phenomena were quantified. William Harvey (1578–1657) quantified how the heart pumped blood; Santorio Santorio (1561–1636) quantified the intake and excretion of food by the body.14 Such quantification profoundly affected how motion was understood. Many earlier thinkers, such as St. Thomas Aquinas, had understood many kinds of change as happening through the increased or decreased participation of a body (an apple, a person) in the form of something else (redness, goodness). But the increasing mathematization encouraged the view of all change as taking place through addition or subtraction, similar to the way a line segment changes length by adding or subtracting a segment of a discrete length.
Other events introduced new dissatisfactions into what remained of Aristotle’s framework, further paving the way for F = ma. A supernova occurred in 1572, another in 1604, and astronomers were able to show that these events occurred not near the earth but in the celestial realm; evidently, things there change just as down here. In 1609, Galileo Galilei (1564–1642) used a telescope to bring the heavens closer, suggesting more similarities with this world than suspected. Such events fostered attempts to develop a physics for the entire universe. Other developments changed the way humans looked at forces. In 1600, the physician to Queen Elizabeth, William Gilbert, wrote a work on magnetism – one of the first treatises of modern science – arguing that magnets work by emitting rays. Indeed, Gilbert said, the earth itself is a magnet, emitting a force that extends in space and varies in strength with distance. This helped promote the idea of a force that could act, in a distinctly un-Aristotelian matter, without contact. Johannes Kepler (1571–1630) published two books, New Astronomy (1609) and Harmonies of the World (1619), that provided three mathematical laws governing planetary orbits, a kind of mathematical script in the world. Kepler argued that God could have caused the planets to move any way He wished, but decided to have them obey mathematical laws because He found such laws beautiful. The mathematics of the world was the script of the world, and its final cause as well.
Galileo was more radical: not only can we read the mathematical script of the world, but we should only do so and forget other kinds of causes. The ‘book of nature’, he wrote, is ‘written in mathematical figures.’ Seeking fantasies such as final causes is not worthwhile. To help read this book, Galileo introduced a brilliant thought experiment: Imagine what would happen on a plane of infinite extent and no resistances, and try to understand how things would move on it, and he proceeded to investigate by staging experiments with things like swinging pendulums and balls rolling down inclined planes. This involved treating space and time quite differently from the way Aristotle had. While Aristotle had treated space as a boundary, Galileo saw it as a container with geometrical properties. To understand motion, you look at how many units of space (Galileo measured it in cubits) an object covered in how many units of time (pulse-beats or water drops). In the process, Galileo discovered the famous law of motion of a falling body – stated by him as a ratio, though nowadays we always state it as the equation d = at2/2, rewriting Galileo in our terms the same way Bradwardine did his precursors. This was the first true mathematical law of nature, the first piece of science to be written in the same language that F = ma would be. Galileo was also able to analyse motions by such things as cannonballs, marbles, and pendulums into two components: a uniformly moving one (push sideways) and an accelerated one (downward).
Galileo Galilei (1564–1642)
But Galileo did not yet have the components of F = ma. He was still in the shadow of the Aristotelian tradition that distinguished between the natural tendencies of a body, such as free fall, and ‘violent’ pushes or pulls applied from the outside – and tended to think of force in terms of the latter. He did not, for instance, think of a falling body as accelerated by a force, which inhibited him from arriving at a general conception of force and its role in motion. Compounding this was a terminological uncertainty; Galileo was unsure about what to call force
, and often uses nearly synonymous terms such as impetus, moment, energy, and force (from the Latin fortis, for strong or powerful).15 When he spoke of force, it was generally not what we call continuous force but instantaneous force (one thing striking another, like a billiard cue a ball or a hammer a nail), or a series of them added together. And Galileo had but a dim recognition of mass – a property of bodies that resists a force, a density of matter related to weight but not identical to it, present even in the absence of gravity. Many of Galileo’s ideas, indeed, sound strange to modern ears, as his remark that circular motion is proper to ordered arrangements of parts, or that straight-line motion indicates that a thing is out of its natural state and returning to it. Science historian Richard Westfall calls Galileo’s conception of nature ‘an impossible amalgam of incompatible elements, born of the mutually contradictory world views between which he stood poised.’16
Newton
But all these elements appear clearly and systematically in the Principia (1687) of Isaac Newton (1642–1727). Newton had learned much from Galileo and other precursors, and developed a generalized and truly quantitative conception of force both continuous and instantaneous, relating it to quantitative changes in the motion of bodies. In the Principia, changes in motion are not explained by what is inside them, but only by the forces that befall them from without. This was a new way of looking at motion – not at its why, but exclusively at its how.
A Brief Guide to the Great Equations Page 5
