Introduction

Successful scientific theorizing requires the formulation of suitable quantities. Such quantities, however, are not apparent in our sense experience, and the struggle to discover proper theoretical quantities is a pressing problem.Footnote ¹ This article sheds light on this problem by considering Isaac Newton’s discovery of the equivalence of inertial and accelerative mass.Footnote ² I will argue that insights regarding the relativity of motion led Newton to see that, in addition to the quantity measured by a body’s resistance to change in its state of motion, there is another invariant quantity that is measured by a balance. He then devised a double-pendulum experiment to provisionally establish the relationship between these quantities, further leading him to reformulate his conception of force.

Newton’s discovery of the equivalence principle is revealed by detailed study of his manuscripts leading up to the Principia (Newton Reference Whitman1999). Newton was explicitly concerned with what he saw, in the practice of natural philosophers of his time, as a grievous lack of attention paid toward establishing suitable conceptual foundations. He is thus an ideal subject, and his meticulous work to develop a conceptual framework is fortunately recorded in detailed manuscripts. Some of this concern can be found in his published works, but much more can be learned from seeing how he addressed open conceptual problems as he encountered them. Changes in definitions, for example, track changes in the fundamental concepts of Newtonian physics, for he marked significant shifts in his understanding of the foundations of his emerging science by abandoning technical terms in favor of new ones.

Copia Materiæ—the Inertial Quantity of Matter

In the summer of 1684, Edmund Halley visited Isaac Newton, seeking assistance in determining the trajectory a planet would follow when subject to an inverse square attraction toward the sun. Newton already knew that the orbit would be elliptical. After failing to locate the proof among his papers, he promised to send it later. In November, Halley received more than he was hoping for in a nine-page tract, called De motu corporum in gyrum (On motion of bodies in orbits; Newton Reference Newton1684a). Over the following 18 months, this nine-page tract would be expanded into the 500+ page Principia. In this first draft of De motu, Newton nowhere invokes quantity of matter. The third De motu draft, entitled De motu corporum in medijs regulariter cedentibus (On motion of bodies in regularly yielding media; hereafter, Reg.Ced.; Newton Reference Newton1685a), which dates to the end of January 1685, is the most interesting for my purposes here, for it marks a turn toward the clarification of fundamental concepts. The original was penned by his amanuensis, Humphrey, but the manuscript is heavily worked over, with modifications, insertions, and deletions in Newton’s hand. Like the first two De motu drafts (Newton Reference Newton1684a, Reference Newton1684b), the original version of Reg.Ced. does not contain a definition of the quantity of matter, but it does have a definition for a new quantity, the quantity of motion, that is relevant here:

This is the first occurrence of the term copia materiæ in Newton’s manuscripts on dynamics, as well as the first mention of the double pendulum experiment. The word he had been using was moles, which was Descartes’s word for bulk, or extension.Footnote ⁴ It is not entirely clear why Newton abandoned moles, which I call the mechanical quantity of matter (for more on this quantity, see Fox, Reference Foxforthcoming). What is interesting about copia—translated as “abundance” or “amount”—is that it appears to come from Lucretius’s De Rerum Natura (Carus Reference Carus and Rouse1937), a work Newton knew very well. Copia occurs in several places, but in De Rerum Natura, Book II, lines 266–83, it appears to indicate the amount of reluctance to move when pushed or pulled after colliding with another body. We know that Newton knew this passage in De Rerum well, for Casini (Reference Casini1984) has shown that Newton had prepared a series of excerpts, taken from classical writings, to include at key points in Principia, Book III. And the excerpt to be incorporated into Propositions 6 and 7 is from Lucretius and contained this passage. As I show below, both copia and the double pendulum experiment are key components of the story behind these propositions.

As for the double pendulum experiment, it is none other than the one that he describes in the Principia (Bk. III, Prop. 7) as establishing the proportionality of mass and weight. I will describe the details of the experiment below, but he clearly saw an interesting problem as arising; namely, how does this quantity of matter, this measure of heaviness, relate to the measure of heaviness we call weight?

This laconic reference to the experiment is, fortunately, not the only place he comments on it.Footnote ⁵ The next place he mentions it is in a new definition inserted on the facing folio, in which he proposes yet another type of quantity of matter. Before discussing that, however, there is another innovation in Reg.Ced. that is key to understanding this new notion. In the definition of motion, which is also new as of Reg.Ced., he distinguishes between absolute and relative motion, stating that absolute circular motion can be distinguished from relative circular motion by the endeavor to recede. What is striking for my purposes, however, is what he says at the end of this definition:

Suppose two bodies are positioned side by side. Their absolute motion is always and only changed by a net external force impressed on them. If, however, a net force is impressed on only one of the bodies, then their relative motion will change, although the absolute motion of the body not acted on will not. But if the impressed force acts on both bodies equally, then their relative motion will not change. This passage is clearly a step toward the scholium on absolute space, but it is also important for another reason. As of this De motu draft, he has not yet formulated what in the Principia is corollary 6. Corollary 5 is Newton’s formulation of Galilean relativity, that the dynamics of a system of interacting bodies is the same whether the system is at rest or moving with constant velocity in a straight line. Corollary 6 identifies yet another dynamical symmetry implicit in the laws of motion: the dynamics within a system will likewise be unchanged if it is acted on by a net external force that equally accelerates all bodies within the system. This is a striking insight, for it means that it is not just that inertial motion and rest are indistinguishable but acceleration, too, given the right kind of force. But what kind of force could do the job? On what quantity would it act?

All forces produce accelerations in proportion to the mass of the body moved. If equal forces are impressed on two bodies of different masses, the smaller mass will accelerate faster than the larger mass, in accord with the Second Law. The kind of force, however, that Newton is referring to here is different in that it accelerates all bodies the same, regardless of their mass. One such force is gravity. This was a startling discovery of Galileo’s, that in the absence of air resistance, all bodies fall at the same rate. This is counterintuitive, for we expect that more massive bodies fall faster—they are, after all, heavier. Gravity is strange insofar as its magnitude varies, depending on the mass of the body on which it is acting. This is unlike typical mechanical forces, the magnitudes of which do not vary with the size of the body acted on. Rather than the magnitude of the force varying, what varies with such forces is the action, that is, the acceleration produced. If gravity were like that, then—and this would be even more counterintuitive—smaller masses would accelerate faster than larger ones, and all bodies would have the same weight, as weight is the measure of the gravitational pull on them.Footnote ⁶ Were it not for the peculiar way in which gravity acts, our notion of weight would be radically different, if we had one at all.

Notice, however, what I have done in order to say what is so strange about gravity. I invoked the distinction between mass and weight, and one of the main arguments of this article is that Newton did not have this distinction clearly worked out yet. And what is really surprising is that, even by this time when Newton has copia, the inertial quantity of matter, what he does not yet understand sufficiently is weight. This is surprising, for weight seems like the quantity that is most immanent in our experience. This is a philosophical moral I hope comes through clearly; namely, quantities given to us rather directly in experience are sometimes the most puzzling from the perspective of empirical study. And a corollary to this is that sometimes theoretical quantities that are quite foreign to our sense experience (e.g., space-time curvature, quanta of energy, or enthalpy) can be empirically well grounded.

Pondus—the Accelerative Quantity of Matter

Shortly after Humphrey penned the original draft of Reg.Ced., Newton realized that the notion of weight needed clarification. What seems to have driven him to see this is that the conceptual framework he had at the time was inadequate for understanding the results of the pendulum experiment. He then inserted several new definitions, one of which is for a new quantity of matter, what I call the accelerative quantity. Before displaying it in English, I want to alert you that Newton is here giving a new technical definition for pondus, which is the Latin word for weight.

Given that pondus means weight, what could he have meant by “pondus even when gravitation is not being considered”? It would be wrong to assume that Newton has the concept of mass here but is simply using a different term. Notice that Newton is purposely conflating what were distinct concepts, namely, pondus, meaning weight, and copia, the inertial quantity of matter. If pondus is the abundance of matter moved apart from considerations of gravity, then it is not clear what to say about weight, which, up until this point, was signified by pondus. It is also not at all clear why he would have wanted an additional term for the inertial quantity of matter.

What he is doing, both in the original definition of copia and in the definition of pondus, is searching for quantities that are invariant, even given the relativity of motion. As we saw in Def. 8, above, he already had reason to suspect that acceleration is potentially not an invariant measure, given a suitable type of force. What led him to this definition of pondus was thinking about whether the possibility of forces that act equally on all bodies in a system has implications for the measure of matter, since such a force would also be acting on the measurement device day, the balance. And what I think Newton realized was that there may be a quantity that depends neither on one’s state of motion nor on whether forces are acting on the system of bodies, including the measurement device. He initially saw that copia was an invariant quantity, and he has now identified another.

The answer, then, to what he meant by “pondus even when gravitation is not being considered” is striking and may at first strike you as false. He saw that a balance does not measure weight. Weight is a quantity that depends on the strength of the gravitational field. In most cases, when we measure mass we are actually measuring weight, from which we then infer mass. This is true of spring scales and electronic balances. It is not true, however, of the kind of scale used in Newton’s day, the double-pan balance. Suppose I put an object on one side of a balance and then add my test weights to the other side and find that the object exactly balances six of my test weights. Now suppose I take the whole setup to the moon where, we now know, the object will weigh about one-sixth what it weighs on Earth. Although my object weighs less, so to do the test weights, so the weight, according to the balance, will not have changed. Weight, as measured by a balance, is invariant, so long as the force equally accelerates all parts of the balance, including the object being measured and the test weights. In other words, Newton saw that the notion of weight, or heaviness of bodies, is ambiguous, for the quantity we experience when lifting them is tied to the strength of Earth’s gravity, but the quantity that was actually used by natural philosophers (not to mention merchants, builders, etc.), that which is measured by a balance, is invariant. And it was this quantity that he wanted, not the quantity we have so much experience with.

I call this the accelerative quantity of matter, for it is a measure of matter that remains invariant, even when there is a force present that acts to accelerate all bodies equally. Surely this is why Newton called it a quantity of matter, for it does not vary with the strength of the acceleration field. This also clarifies what he meant by saying that it is “apart from considerations of gravitation,” for, although this measure of matter requires the presence of a special type of force, one that acts so as to accelerate all bodies equally, it is not a measure of the gravitational pull on a body, in the way that we now understand weight.Footnote ⁷

The Double Pendulum Experiment, a New Conception of Force, the Discovery of the Equivalence Principle

What remains is how Newton recovered the concept of weight as a measure of force. This partly occurs in the manuscript Liber Secundus (Newton Reference Newton, Smith and Whitman1685c), the initial draft of what came to be Book III of the Principia. Like Book III, Liber Secundus contains Newton’s argument that terrestrial and celestial gravity are one in kind—most important here—and that the solar system is heliocentric, as in the Copernican system, not geocentric, as in the Tychonic system. It also contains Newton’s argument that gravity is an inverse square attraction that holds between every pair of particles in the universe.

The argument that terrestrial and celestial gravity are one in kind occurs in Articles 18 and 19 of Liber Secundus. Newton begins by arguing that, at equal distances from the sun, all planets would have the same orbital period, regardless of their pondus.Footnote ⁸ The argument is based on Kepler’s 3/2 power rule, according to which $T\propto a^{3/2}$, where T is the orbital period and a is the planet’s mean distance from the sun. Insofar as quantity of matter plays no role in this, Newton argues that the centripetal force holding the planets in their orbits acts only in proportion to their pondus, for, otherwise, planets at equal distances from the sun would move slower or faster, depending on their constitution. Given that they do not, as evidenced by the fact that the 3/2 power rule holds for the known planets and their moons, he concludes that celestial gravity acts solely in proportion to pondus.

In Article 19, Newton takes up the case of terrestrial gravity. Although we have no 3/2 power rule for bodies moving near the earth, he uses the equality of gravitational acceleration to establish something similar, claiming “Others have long observed that all bodies descend in equal times (at least if the very small resistance of air is removed), and it is possible to discern the equality of the times to the highest degree of accuracy in pendulums” (Newton 1685b, fols. 13r–14r). Newton goes on to describe the experiment in detail. He wants to show that the circumterrestrial force, like celestial gravity, accelerates all bodies the same, regardless of their pondus and type of matter. He tells us that, by Law 2, the action of the circumterrestrial force, if it is proportional to pondus, will make bodies suspended by equal cords oscillate in equal times and that, if the force on one kind of matter is less, the period will be greater, and if the force is greater, the period will be less. The apparatus is simple. He made two equal wooden boxes, filling one with wood and the other with an equal pondus of gold. He suspended the boxes side by side with 11-foot cords, making pendulums exactly alike with respect to their length, pondus, shape, and air resistance. Setting them in motion, he observed that they kept equal oscillations for a very long time. He repeated this many more times, comparing the nine different materials, listed in Def. 7 above.Footnote ⁹ This is the double pendulum experiment.

In the context of the conceptual difficulties Newton faced in 1685, this experiment turns out to be crucial in ways obscured by his claim that it is a test of the equality of gravitational acceleration. Had that been his real purpose, he should have compared pendulums with varying pondus, placing a quantity of, say, gold in one pendulum and, say, twice as much gold in the other.Footnote ¹⁰ Newton knew that this would work in a vacuum but not in the presence of air, for air resistance is proportional to the surface area of the pendulum. This is why he fashioned the identical wooden boxes, as the surface area of the different materials varies considerably. But he also knew that air resistance is not like gravity for, holding surface area fixed, the magnitude of the force of air resistance is fixed, while the acceleration varies—by Law 2—in proportion to a body’s copia. He could not, therefore, have thought that this was a simple test of the equality of gravitational acceleration.

What, then, was the purpose of the experiment? A useful question to ask when an experiment’s purpose is unclear is what will be inferred if it fails. In this case, had the period of oscillation differed between, say, the gold and the wheat, would Newton have concluded that gravitational acceleration is not independent of weight? Fortunately, he tells us at the end of Article 19, “In these experiments, in bodies of the same pondus, a difference of matter that would be even less than a thousandth part of the whole could have been clearly noticed” (Newton 1685b, fol. 14r). In other words, had the pendulums not oscillated equally, he would have concluded that the quantities of matter differed.

Given that the pendulums had the same quantity of matter, however, in what sense could the experiment indicate a difference in quantity of matter? By closely following the changes to the manuscripts, it is clear that the problem Newton was addressing was that he had two different quantities of matter and that he saw the experiment as potentially telling whether copia and pondus are related. The experiment is elegant. It takes advantage of the presence of a force, in addition to gravity, that provides a way to compare the inertial and accelerative quantities of matter. It was not conceived for the purposes of establishing what happens in the absence of air resistance, for air resistance plays a crucial role in making the experiment what it is. The specific question the experiment answered for him, then, was whether the conceptually distinguishable accelerative and inertial quantities of matter are empirically distinguishable. And what he found was that whenever one varies the accelerative quantity of matter, one thereby varies the inertial quantity of matter by the same amount. The experiment told him this because air resistance acts in proportion to the inertial quantity. Air resists movement because it is composed of solid bodies that resist being pushed out of the way as the pendulum moves. The design of the boxes ensured that the composition of the material inside did not matter. The accelerative quantity of matter was always the same, for he used a balance to ensure that the two pendulums always had the same pondus. So the only way for the quantity of matter to differ is if the inertial quantity differed. He used the air resistance, in other words, as a measure of copia. Had the air resistance slowed some of the pendulums differently, it would have indicated that the inertial quantity differed from the accelerative quantity.

Two issues still remain. The first is that Newton needs to recover a notion of weight as a measure of force, not an invariant quantity of matter. This problem is evident earlier in Article 19, where he says, “Accordingly, the copia of gold was to the copia of wood as the action of the force [that produces motion] upon all the gold to the action of the force [that produces motion] upon all the wood—that is, as the pondus of one to the pondus of the other” (Newton 1685b, fol. 14r). The problem occurs in the final clause, for pondus is the accelerative quantity of matter, not a measure of force. The solution, however, is simple. Since the accelerative quantity just is the inertial quantity, he no longer needs to distinguish them; he can stop using pondus in the technical sense and use it to mean the measure of the gravitational pull on bodies. And dropping the technical sense of pondus turned out to be easy enough. All that he had to do was cross out the final sentence of Article 19, which was just a reminder that pondus means quantity of matter, and then go back to Article 18, changing all but one of the several instances of pondus to “quantity of matter” (see Newton 1685b, fol. 13r).

The final issue involves the notion of force. Newton had claimed above that if terrestrial gravity acted differently on different materials, then, by Law 2, the periods of oscillation would differ. This does not follow, however, from Law 2 as formulated at that time, which occurs in Reg.Ced. It states that a change in motion is proportional to the force impressed. But what this episode brought to his attention was the need to distinguish between two types of force. The problem is that, on the one hand, typical mechanical forces, like impact, are proportional to the acceleration produced. On the other hand, a force like gravity is more difficult to describe, for, like impact, it is proportional to the acceleration produced, but the magnitude of the force varies, producing the same acceleration regardless of quantity of matter. To describe such behavior, I followed Newton in saying that gravity acts in proportion to quantity of matter. But this is slightly misleading, because ordinary mechanical forces are also proportional to quantity of matter, in the sense that $F_{\mathrm{mech}}\propto ma$. The difference is that the magnitude of the action of forces like gravity varies, depending on the mass of the body it acts on. And this strange feature is what gives rise to the equality of gravitational acceleration regardless of mass. What Newton needed, then, was to distinguish these types of forces, which he came to do after Liber Secundus, by distinguishing between motive and accelerative quantities of force. The accelerative quantity of force is proportional to the acceleration it produces, while the motive quantity is proportional to mass times acceleration.Footnote ¹¹ This distinction then allows him to correct the problem with the description of the double pendulum experiment by disambiguating the notion of force in Law 2. The change occurs in the very next draft of the laws we have after Liber Secundus. The change is the addition of a single word to Law 2, which now states that a change in motion is proportional to the motive force impressed.

Conclusion

This kind of interplay between empirical and philosophical work—how conceptual questions and innovations can lead to new empirical questions, which can lead again to further conceptual innovation—is striking. Newton started this episode by questioning the quantity of matter in light of his insights regarding the relativity of motion. The search for a proper quantity led to an experiment designed to discover whether the inertial and gravitational quantities differed. This empirical work then led to a deeper philosophical understanding of the very notion of force and a change to the laws of motion, the fundamental presuppositions on which all of his physics was built. In recent times, it is common to see discussions of the distinction between inertial and gravitational mass. And, indeed, the distinction goes back at least as far as Kant, who further distinguished a third quantity of matter.Footnote ¹² What I have shown here is that Newton not only saw that the two are potentially distinct, but he also provisionally established their equivalence.

This understanding of what we now call the equivalence principle, moreover, put him in an interesting position regarding the mechanical philosophy. Since he countenanced forces abstractly, independent of the question of mechanism, his contemporaries complained that his was a theory of “occult” causes. But what has been underappreciated is that he had established that, although it is in some sense mysterious, gravity acts in accord with the same quantity of matter as mechanical forces. The requirement that proposed forces act in proportion to the inertial quantity of matter is a strong condition of intelligibility. It is more than that any proposed mechanism of gravity must be compatible with the laws of motion. What it means is that any hypotheses about the causes of motion within, say, the solar system, must be compatible with the fact that, in order to not disturb the dynamics of the system, they must equally accelerate everything in it. In other words, any force that is acting on the solar system, as in, say, an invisible cause holding the earth at rest in the center and making the sun and planets revolve around it, must be one that does not affect the dynamics so far identified by Newton. And, in order to do that, such a cause must be an accelerative force, namely, gravity.