[Newton’s own] description: A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.
DISCOVERER
: Isaac Newton
DATE
: 1684–87
Newton’s second law of motion,
F
=
ma
, is the soul of classical mechanics.
The equation
F
=
ma
is shorthand for Newton’s second law of motion. It is the 1 + 1 = 2 of classical mechanics. It seems obvious and straightforward. The equation appears simply to translate an ordinary experience into measurable terms: push something and it either starts to move or moves differently.
Yet, like 1 + 1 = 2,
F
=
ma
erupts into mystery when looked at closely. It does not, in fact, refer to ordinary experience but to an abstract world of zero resistance: in the real world we have to continue pushing things like desks and carts to keep them moving at the same speed. The equation does not incorporate Einstein’s famous discovery
of the interchangeability of mass and energy. It gives centre stage to force – a concept absent from most formulations of contemporary theories like relativity and quantum mechanics. Finally, the equation seems, in a contradictory way, to be both a name and a description. It seems both to define force, mass, and motion, and to state an empirically discovered and testable relationship among them.
How can such an elementary equation about something as ordinary as motion conceal so many complexities? The answer can be gleaned in the remarkable historical journey that led from ancient times to the equation’s formulation in the seventeenth century. To arrive at this equation, human beings had to train themselves to look at motion in new ways – to learn to look at different aspects of it, and to change how they thought about what they saw. In the course of this vast journey, new sights slowly and progressively came into view, occupied centre stage, and then vanished off the horizon, with each familiar landscape slowly yielding to another, until the travelers found themselves in an entirely new world.
The journey began in primitive times, when human beings saw the world as ruled by deities. This was natural and inevitable, perhaps the simplest and most straightforward way to make sense of things. All humans acquire a notion of force from individual experiences of pushes and pulls of daily life, in applying our muscles to lift, squeeze, or roll things. Generalizing that experience, early humans could readily conceive everything in nature – from nearby phenomena like thunder and rain to the movements of distant bodies like the sun and stars – as the result of spirits behaving and misbehaving, exerting their particular internal forces. Thus early ideas of force were closely connected with religious ideas of the direct presence of gods in the world.
1
The earliest humans naturally tried to control nature by pleasing the spirits through ritual and prayer – the earliest forms of technology.
But that did not succeed in bringing about the desired control. A far more effective way to predict and influence nature seemed to be to pay attention to the kinds and amounts of changes in nature – the recurrence of the seasons, the various movements of the planets and stars, the behaviour of fire and floods, and so forth. But nature is so varied! Sunlight and clouds, tides and storms, plants and animals, men and women, plans and ideas, houses and cities are constantly being born and dying, rising and falling, changing colors and forms, and moving about. How could one make sense of all these motions?
The Greek philosopher Aristotle (384–322
BC
) was the earliest we know who drew up a systematic account of all kinds of motion or change – he used the same word,
kinesis
, for both. Kinesis is so important, he thought, that to understand it is tantamount to understanding nature itself, and he created a framework to include all varieties of kinesis: of animate and inanimate objects, with and without human intervention, on earth and in the heavens. He distinguished several kinds of kinesis: the substantial change of a thing being born or dying (fire consuming a log); the quantitative change of a thing growing or shrinking; the transformational change of one property changing into another (a green leaf turning brown); and local motion, or something changing its place.
Aristotle viewed these changes with biologically trained eyes. He regarded the world as a kind of cosmic ecosystem that contained many different levels of organization. Motion in this ecosystem is almost never random or chaotic, but a process of passing from one state to another in which something existing only in potential (a formal principle) is underway to being actualized. Many levels of organization are built on top of each other – human beings make up a state, organs make up a human being – so that any event is shaped by a complex network of different kinds of causes.
Aristotle understood this cosmic ecosystem in the framework of a set of key distinctions. He distinguished, for instance, between two kinds of motion: natural, or violent and forced. Natural motion was that of things moving themselves in their own habitats – acorns
growing into oaks, or eggs into chickens – where the change actualizes some innate principle in the substance itself. Forced or violent motions occur when the change is imposed from the outside, as what happens to oak trees when humans fell them to build houses, or to chickens when humans slaughter them for food.
Aristotle also thought it mattered where a change happens. In the earthly realm below the moon, substances are composed of different mixtures of earth, air, fire, and water, and objects don’t move constantly but intermittently. In the heavenly realm, objects are made of an unchanging substance called ‘ether’, and move ceaselessly and circularly. If today we find this unjustified, it is a sign of how far we have traveled since Aristotle’s time and how our sight has changed, for his ideas were based on rational argument, logical deduction, and careful observation. For hundreds of years, astronomers in Greece and elsewhere had never witnessed any changes in celestial behaviour, nor seen anything but circular motion.
2
Only circular movement can continue unceasingly, he thought, and only some special substance, not known on earth (hence the strange name ether), does not experience change.
3
In the celestial realm, motion is initiated by a so-called unmoved mover, which drew the celestial spheres into motion. This was Aristotle’s analogue to God, though it was impersonal and not something with which one could have what we twenty-first-century humans call a ‘relationship.’ The celestial spheres, by various intermediaries, transmit the motion to the terrestrial sphere. Thus all motions in the cosmic ecosystem, however tiny, are connected, in a mediated way, with the first principle of the universe, and ultimately have to be understood in that context.
When Aristotle discussed what we call motion, then, we onlookers from 2,500 years later have to be careful not to read in our own assumptions. When he speaks of local motions, it is generally in the context of events such as a horse pulling a cart on the road or shipbuilders pushing a boat. Such events arise from a complicated network of purposes, plans, and designs that are being actualized, of which the local motion is only one aspect. And when Aristotle
does discuss that aspect, he is not propounding and defending some hypothesis about local motion apart from the event itself, but rather speaking in general terms about the work required to accomplish such tasks to illustrate some other point. In these kinds of events, furthermore, the role of acceleration is almost nil, and rules of thumb such as ‘A force that moves a body against a certain resistance a certain distance in a certain time moves the same body half the distance in half the time’ work fine. Though at one point, in an assertion to become infamous two millennia later as the target of falling-body experiments, he remarked that ‘If half the weight moves the distance in a given time, its double (i.e., the whole weight) will take half the time.’
4
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.
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.