A Brief Guide to the Great Equations (7 page)
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.
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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.
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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.
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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
=
at
2
/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.’
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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.
This was a modest step beyond Galileo and other close contemporaries of Newton such as Leibniz and Descartes, but thanks to its significance was monumental. It changed the ontology of nature – the way we conceive the most basic units of explanation of the reality we see. Most of Newton’s contemporaries conceived of these units as the bodies themselves, which affected other bodies through various mechanisms that brought about different kinds of changes. Newton transformed that, asserting that explanations of motion had to be in terms of the forces that changed the motion of a mass. The three basic terms in the ontology of motion were now
force
,
mass
(what resisted force), and
acceleration
(change in motion). And each of these was quantitative, measurable.
Isaac Newton (1642–1727)
The
Principia
, the most revolutionary single publication in science, much like Euclid’s
Elements
, lays out its contents as if they
were deductions from self-evident axioms. It contains three Books (the first two called ‘The Motion of Bodies’, the third ‘The System of the World’) preceded by a Preface, eight Definitions, and a set of ‘Axioms, or Laws of Motion.’ In the Definitions, we see Newton, sometimes clumsily, developing the components that would be included in
F
=
ma
– especially the idea of force – from out of the ideas of his predecessors. Definition One is of mass, or quantity of matter; Definition Two of quantity of motion. Of the following Definitions, of force of various kinds, some court ambiguity and even confusion: Definition Three, for instance, is supposedly about ‘inherent force’, but is actually about what we call inertia, describing it in terms of impulse. This definition is best viewed, one commentator says, as ‘a concession to pre-Galilean mechanics.’
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Definition Four is the key one, and defines ‘impressed force’ in more modern terms as ‘an action exerted on a body to change its state either of resting or of moving uniformly straight forward.’ Newton thereby generalized the notion of force developed by his precursors, extending it from instantaneous forces to continuous forces, which was a product of Newton’s intuition, science historian I. Bernard Cohen notes, because it was ‘a step which he never justified by rigorous logic or by experiment.’
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And as Newton writes later, ‘this concept is purely mathematical, for I am not now considering the physical causes and sites of forces.’
19
The remaining four definitions are of what amount to other aspects of force. In a commentary, Newton warns that ‘although time, space, place, and motion are very familiar to everyone’, and have a popular meaning that arises from sense perception, he will give them a technical meaning, and proceeds to describe what he calls ‘absolute time’, whose flow is unchanging, and ‘absolute space’, which remains homogeneous and immovable.’ Newton then distinguishes between absolute and relative motion. ‘Absolute motion is the change of position of a body from one absolute place to another; relative motion is change of position from one relative place to another.’
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Thus the motion of a sailor on a ship is the sum of three motions: his motion relative to the ship, the ship’s
motion relative to the earth, and the earth’s motion relative to absolute space.