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.
Isaac Newton1
Despite their somewhat dubious status, atoms (or ‘corpuscles’) of one kind or another were fairly ubiquitous in the physical descriptions adopted by the seventeenth-century mechanical philosophers.2 Like Boyle, Isaac Newton remained relatively unperturbed by the fact that it was impossible to gain any direct evidence for their existence. The first edition of Newton’s Mathematical Principles of Natural Philosophy was published in 1687, and in a second edition published in 1713, he added four ‘rules of reasoning’. Rule III, which concerns the ‘qualities of bodies’, contains a partial summary of his atomism. In it he insists that an object’s shape, hardness, impenetrability, capability of motion, and inertia all derive from these same properties manifested in the object’s ‘least parts’.3
Like Boyle, Newton was happy to accept that the visible properties and behaviour of a macroscopic object (such as a stone) can be understood in terms of precisely the same kinds of properties and behaviour ascribed to the invisible microscopic atoms from which it is composed. The message is reasonably clear. Let’s not worry overmuch about the status of metaphysical entities that we can’t see and can’t derive any direct evidence for. Let us instead devote our intellectual energies to the task of describing the motions of material objects that we can observe and perform measurements on (a discipline called kinematics). Then let’s further seek underlying explanations for this motion (a discipline called dynamics). We do this reasonably secure in the knowledge that anything we can discover about such macroscopic objects will likely apply equally to their constituent microscopic atoms.
Where do we start? By acknowledging that any observations or measurements of the motions of objects require a framework in which these can be made. This is a framework created through agreed conventions in the measurement of distance and time intervals. And we must now bite the bullet and propose a definition of mass.* This Newton does, in the very first few words of Book I of Mathematical Principles:4
The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. … It is this that I mean hereafter everywhere under the name body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.
We interpret Newton’s use of the term ‘bulk’ to mean volume. So, the mass of an object is simply its density (the amount of mass per unit volume, measured—for example—in units of grams per cubic centimetre) multiplied by the volume of the object (in cubic centimetres). This gives us a measure of the object’s mass (in grams). This all seems perfectly logical and reasonable, although we’ll soon be coming back to pick over this definition in an attempt to understand what it really says. Let’s run with it for now.
So, an object has a certain intrinsic mass—the ‘quantity of matter’ in it—which is related to its density and volume, and which we can denote using the symbol m. Now, the property of mass that becomes its defining characteristic is that it is also the measure of the force or power of the object to resist changes in its state of motion. This is typically referred to as inertial mass. Why not simply call it ‘mass’ and acknowledge that all mass gives rise to inertia? We’ll see why it becomes necessary to be a little more careful in our choice of terminology later in this chapter.
The evidence for the property of inertia comes from countless observations and measurements of objects both at rest (not moving) and in motion. I would argue that we cannot fail to gain at least qualitative evidence for the property of inertia just by living our daily lives. Newton drew on the body of accumulated experimental evidence (perhaps most notably the work of Galileo) to frame his first law: an object will persist in a state of rest (not moving) or of uniform motion in a straight line unless and until we apply (or ‘impress’) a force on it.5
Okay, so in addition to introducing and defining the concept of mass, Newton has now introduced a second concept: that of force. But what is a ‘force’, exactly? Newton explains that this is simply an ‘action’ exerted on an object that changes its state of motion, either from rest or from uniform motion in a straight line. The force is the action, and is impressed only for as long as the action lasts. Once the action is over, the force no longer acts on the object and it continues in its new state of motion until another force is impressed.6
There is no restriction on the kind of action (and hence the kind of force) that can be applied. I can kick the object; I can shoot it from a cannon; I can whirl it around like a discus; if it’s made of conducting material I can charge it with electricity and move it with an electromagnet. All these different kinds of actions—all these different forces—act to change the state of motion of the object. This all seems clear enough, though there’s a niggling doubt that these definitions are starting to look a bit circular. Let’s keep going.
To get a real sense of what these definitions and the first law are telling us, we need to be a little clearer about what Newton means by ‘motion’, and specifically ‘uniform motion in a straight line’. Let us imagine we are somehow able to set a perfectly spherical object in motion in a vacuum, with no external force acting on it. The first law tells us that this object will persist in its straight-line motion. If the sphere has an inertial mass m, and is set moving with a certain speed or velocity v (the rate of change in time of the object’s position in space), then we determine the linear momentum of the object as its mass multiplied by velocity (m times v, or mv). This is what Newton means by ‘uniform motion’.
Obviously, an object at rest has no velocity (v = 0) and therefore no linear momentum. The first law tells us that to get it moving, we need to impress a force on it. Likewise, when we impress a force on our spherical object with linear momentum mv, we change the momentum, by an amount determined by Newton’s second law: the change in motion is proportional to the magnitude of the force and acts in the direction in which the force is impressed.7
What happens depends logically on the direction in which we apply the force. If we apply it in precisely the same direction in which the object is already moving, then we can expect to increase the linear momentum—we increase the motion, specifically the speed with which the object is moving. If we apply it in the opposite direction, we will reduce the linear momentum. Applying a force that exactly matches the linear momentum but in the opposite direction will slow the object down and bring it back to rest. Applying the force at an oblique angle may change the direction in which the object is moving.
We define the magnitude of the applied force F as the rate of change of linear momentum that results from its impression. The result is the famous statement of Newton’s second law: F = ma, force equals mass times acceleration.8 Though famous, this result actually does not make an appearance in the Mathematical Principles, despite the fact that Newton must have seen this particular formulation of the second law in German mathematician Jakob Hermann’s treatise Phoronomia, published in 1716.* It is often referred to as the ‘Euler formulation’, after the eighteenth-century Swiss mathematician Leonhard Euler.
There is something deeply intuitively appealing about the second law. Kick a stone with a certain force and it will fly through the air, accelerated to some final speed before eventually succumbing to the force of gravity and returning to the ground. A heavier stone with a greater inertial mass will require a stronger kick to get the same acceleration. This description of the subtle interplay between force, mass, and acceleration is simple, yet profound. There can surely be no doubting its essential correctness.
Indeed, Newton’s laws of motion have stood the test of time. Yes, there are circumstances where we have to abandon them because nature turns out to be even more subtle at the microscopic level of atoms and sub-atomic particles, and at speeds approaching that of light. We’ll look at these circumstances in the chapters to follow. But for most of our ‘everyday’ applications on our earthly scale, Newton’s laws work just fine.
I guess for the sake of completeness I should also state the third law: for every action we exert on an object (for every force we impress on it), there is an equal and opposite reaction.9 Kick the stone, and the stone kicks back.
Now, if these laws are intended to provide a foundation for our understanding of the properties and behaviour of material substance, we should take a sober look at what they’re telling us. The first thing we must do is acknowledge the inadequacy of Newton’s definition of mass. As Mach, the arch-empiricist, explained, Newton defines mass as the quantity of matter in an object, measured by multiplying the density of the object by its volume. But we can only define the object’s density in terms of its mass divided by its volume. This just takes us around in circles.10 Defining mass in terms of density doesn’t define mass at all.
But if we move on hurriedly and try to define mass using Newton’s laws, we find we can’t escape a vicious circularity. We could say that the inertial mass m is a measure of the resistance of an object to acceleration under an impressed force, F. But then what is the measure of the force F if not the extent to which it is resisted by the inertial mass m of the object? Try as we might, there’s no way out.
The redeeming feature of F = ma is that the impressed force F can result from many different kinds of actions. Whatever I choose to do, the measure of resistance to the different kinds of acceleration is the same. The inertial mass of the object appears to be consistently intrinsic to the object itself—it would seem to be a property of the object. But this doesn’t take us any further forward. It doesn’t tell us what mass is.
Mach sought to provide an operational definition of mass using the third law. This avoids the challenge by referring all measurements to a mass ‘standard’. By working with ratios (e.g., of accelerations induced between two objects that are in collision), it is possible to eliminate force entirely and determine the ratio of the inertial masses of the two bodies. If one of these is an agreed standard, the inertial mass of the unknown body can then be measured in reference to it.
Let’s pause to reflect on this. In the Mathematical Principles, Newton constructed the foundations for what we now call classical mechanics.* This is a structure that has proved itself time and again within its domain of applicability. Its central concepts of space and time, mass and force are intuitively appealing and appear to be consistent with our common experience. There is nothing in our familiar world of macroscopic objects moving in three-dimensional space that can’t be explained using its laws supplemented, as appropriate, with similarly derived mechanical principles. Watch a game of tennis or snooker, and you’ll quickly come to appreciate Newton’s laws of motion.
Yet this is a structure that gives us nothing more than an operational definition of inertial mass, a definition that allows us to relate the properties of this object here with that object over there. It has nothing whatsoever to say about one of the most fundamental properties of all material objects. It doesn’t tell us what mass is. And it gets worse.
Newton’s ambition was not limited to the mechanics of earthly objects or, by inference, the mechanics of the atoms from which these objects might be composed. He sought to extend his science to describe the motions of heavenly bodies and, specifically, the planets. If successful, there would be no limits to the scope of such a science. It would be applied to matter in all its forms, from microscopic atoms to the familiar objects of everyday experience on Earth to objects in the furthest reaches of the visible universe. But to complete the picture he needed another law.
Now, unlike the laws of motion, in the Mathematical Principles Newton’s famous inverse-square law of universal gravitation is not accorded the honour of a heading containing the word ‘law’. The reasons for this are partly scientific, partly political. Newton communicated the draft of Mathematical Principles through the newly appointed Clerk to the Royal Society in London, the astronomer Edmond Halley. In May 1686, Halley wrote to Newton to tell him the good news: the Royal Society had agreed its publication.* But there was also bad news. At a recent Society meeting Robert Hooke—Boyle’s erstwhile laboratory assistant, now Curator of Experiments at the Society—had claimed (loudly) that he, not Newton, had discovered the famous inverse-square law. Hooke had demanded that Newton acknowledge this.
The truth is that in 1681 Hooke had deduced the inverse-square relationship entirely empirically, from measurements he had performed with his assistant, Henry Hunt. But Hooke did not have the mathematical ability to construct a theory from which the inverse-square law could be derived.
It seems that Newton was inclined to be conciliatory, but as he received more news of Hooke’s claims from other colleagues who had attended the meeting, his temper evaporated. He argued that mathematicians who ‘do all the business’ were being pressed into service as ‘dry calculators and drudges’. Outraged by this attempt to undermine his own role in the discovery of the theory of universal gravitation, he dismissed Hooke as a man of ‘strange, unsocial temper’ and proceeded to reduce all acknowledgement to Hooke in the Mathematical Principles to the very barest minimum.11
So strong was his ire that he arranged the destruction of one (possibly two) portraits of Hooke that hung on the walls at the Royal Society’s rooms. Consequently, there is no known portrait of Hooke, although the historian Lisa Jardine believed she had tracked down such a portrait in London’s Natural History Museum, mislabelled as that of John Ray, a naturalist and Fellow of the Royal Society.
Newton was tempted to withdraw Book III (which he had called ‘The System of the World’) from the Mathematical Principles entirely. Instead, he replaced it with a section containing a very measured—and much more demanding—summary of observed astronomical phenomena followed by a sequence of propositions from which his ‘system’ can be deduced. Of the observed phenomena he described, Phaenomenon IV is the most worthy of note. At the time of writing, there were five known planets in addition to the Earth—Mercury, Venus, Mars, Jupiter, and Saturn. Newton noted that the time taken for each planet to complete its orbit around the Sun (the planet’s orbital period) is proportional to the mean distance of the planet from the Sun. Specifically, the orbital period T is proportional to the mean distance r raised to the power 3∕2.12
After some unpacking, we can recognize this is as Kepler’s third law of planetary motion. Based entirely on empirical observations of the motions of the planets that were known at the time measured relative to the fixed stars, Johannes Kepler had deduced that they move in elliptical orbits with the Sun at one focus, and that a line drawn from the Sun to each planet will sweep out equal areas in equal times as the planet moves in its orbit. The third law is a numerical relationship between the orbital period and the mean orbital distance or radius, such that T2 is proportional to r3 or, if we take the square root of both, T is proportional to r3/2.13
Newton subsequently goes on to declare Proposition VII: there is a power or force of gravity intrinsic to all objects which is proportional to the quantity of matter that they contain.14 It is then possible to use Kepler’s third law to deduce the magnitude of the force of gravity of one object (such as a planet) acting on another, leading to Proposition VIII: in a system of two uniform spherical objects mutually attracted to each other by the force of gravity, the magnitude of the force varies inversely with the square of the distance between their centres.15
Most physics textbooks interpret these propositions as follows. If we denote the masses of two uniform spherical bodies 1 and 2 as m1 and m2, and r is the distance between their centres, then the force of gravity acting between them is proportional to m1m2/r2. This is the famous inverse-square law. To complete this description, we need to introduce a constant of proportionality between this term and the measured force of gravity, which we again denote as F. This constant is a matter for measurement and convention. It is called the gravitational constant and is usually given the symbol G. It has a measured value of 6.674 × 10−11 Nm2/kg2 (Newton metres-squared per square kilogram).*
But this force of gravitation is just not like the forces generated by the various actions we considered in our discussion of the laws of motion. These latter forces are impressed; they are caused by actions involving physical contact between the object at rest or moving in a state of uniform motion and whatever it is we are doing to change the object’s motion. A stone will obey Newton’s first law—it will persist in a state of rest—until I impress on it a force generated by swinging my right foot through the air and bringing it into contact with the stone. Or, to put it more simply, until I kick it.
But precisely what is it that is impressed upon the Moon as it swoons in Earth’s gravitational embrace? How does the Moon (and the Sun) push the afternoon tide up against the shore? When a cocktail glass slips from a guest’s fingers, what impresses on it and forces it to shatter on the wooden floor just a few feet below?
Newton was at a loss. His force of gravity seems to imply some kind of curious action-at-a-distance. Objects exert influences on each other over great distances through empty space, with nothing obviously transmitted between them. Critics accused him of introducing ‘occult elements’ into his theory of mechanics.
Newton was all-too-aware of this problem. In a general discussion (called a ‘general scholium’), added in the 1713 second edition of Mathematical Principles at the end of Book III, he famously wrote: ‘I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses.’16 If that’s not bad enough, we have no real evidence to support what might be an instinctive conclusion about the nature of the masses m1 and m2 which appear in the inverse-square law. We might be tempted to accord them the same status as the m in Newton’s second law.
But physicists can be rather pedantic. In classical mechanics, motion and gravity appear to be different things, and some physicists prefer to distinguish gravitational mass (the mass responsible for the force of gravity) from inertial mass. Some go even further, distinguishing active gravitational mass (the mass responsible for exerting the force of gravity) from passive gravitational mass (the mass acted on by gravity).
For all practical purposes, empirical measures of these different kinds of mass give the same results, sometimes referred to as the ‘Galilean’ or ‘weak’ equivalence principle, and they are often used interchangeably. As we will see in Chapter 7, Einstein was willing to accept that these are all measures of an object’s inertial mass. We must nevertheless accept an element of doubt.
Let’s take stock once more. The research programmes of the mechanical philosophers had been building up to this. For sure, Newton didn’t do all this alone. He ‘stood on the shoulders of giants’ to articulate a system that would lay foundations for an extraordinarily successful science of classical mechanics, one that would survive essentially unchallenged for 200 years. This was a science breathtaking in its scope, from the (speculative) microscopic atoms of material substance, to macroscopic objects of everyday experience, to the large-scale bodies of stars and planets.
However, look closely and we see that these foundations are really rather shaky. We have no explanation for the ‘everyday’ property of inertial mass, surely the most important property—we might say the ‘primary’ property—of material substance. We have no explanation for the phenomenon of gravity, another fundamentally important property of all matter. And without explanation, we must accept that we have no real understanding of these things.
We do have a set of concepts that work together, forming a structure that allows us to calculate things and predict things, enough to change profoundly the shape and nature of our human existence. This network of concepts is much like a game of pass the parcel. It’s fun while the music plays, and we keep the parcel moving. But at some stage we know the music will stop.