Newton on active and passive quantities of matter

https://doi.org/10.1016/j.shpsa.2020.03.006Get rights and content

Highlights

  • Newton distinguishes active and passive quantities of matter by their measures.

  • Conditions on measures constrain evidence for the additivity of each quantity.

  • Newton analyzes Law III and analogizes magnets to infer active additivity.

  • Conditions on passive additivity frustrate this inference.

Abstract

Newton published his deduction of universal gravity in Principia (first ed., 1687). To establish the universality (the particle-to-particle nature) of gravity, Newton must establish the additivity of mass. I call ‘additivity’ the property a body's quantity of matter has just in case, if gravitational force is proportional to that quantity, the force can be taken to be the sum of forces proportional to each particle's quantity of matter. Newton's argument for additivity is obscure. I analyze and assess manuscript versions of Newton's initial argument within his initial deduction, dating from early 1685. Newton's strategy depends on distinguishing two quantities of matter, which I call ‘active’ and ‘passive’, by how they are measured. These measurement procedures frame conditions on the additivity of each quantity so measured. While Newton has direct evidence for the additivity of passive quantity of matter, he does not for that of the active quantity. Instead, he tries to infer the latter from the former via conceptual analyses of the third law of motion grounded largely on analogies to magnetic attractions. The conditions needed to establish passive additivity frustrate Newton's attempted inference to active additivity.

Introduction

Newton's heralded ‘deduction’ of the law of universal gravity occurs in the first seven propositions of Book III of Principia (1687, 1713, 1726). Roughly, the first five establish that the centrally directed force (‘gravity’) upon an orbiting body toward a central body is inversely proportional to the square of its distance from the central body; the sixth that gravity is directly proportional to the quantity of matter of the orbiting body; and the seventh that gravity is directly proportional to the quantity of matter of the central body. Newton's law of gravity thus involves three components: the two mass terms in the numerator and the distance-squared term in the denominator.

There is different empirical evidence for the claim that gravity is proportional to the quantity of matter of the orbiting body than there is for the claim that it is proportional to the quantity of matter of the central body.2 Newton establishes the former claim in part by substantiating a principle underlying the widespread observation that all bodies fall at the same rate. Bodies of different quantities of matter would fall at the same rate, Newton reasons, if gravity were to proportion itself to their quantities of matter.3 To substantiate this for terrestrial bodies, III.6 describes the two-pendulum experiment which in part rules out that gravity is proportional to bodies' qualitative properties, such as their forms and textures.4 For celestial bodies, Newton reasons in III.6 that the non-eccentric and ‘extremely regular motion’ of, e.g., Jupiter's satellites shows that the action of the sun on unequal bodies of that system produces equal ‘accelerative gravities,’ i.e. gravity proportions itself to the bodies at a given distance, and so makes unequal bodies at equal distances describe equal spaces in equal times.5 How Newton establishes the claim concerning the central body, by contrast, has been a matter of great controversy, since he relies upon an application of the third law of motion which itself seems to presuppose the law of gravity.6

Newton understands these arguments to have established more than just the proportionality of gravity to the quantities of matter of the orbiting and central bodies. In particular, he takes them to have established universal gravity: that every two particles interact directly as their masses and inversely as the square of their distance. Hence in III.6, he extends his reasoning to apply to all the parts of the attracted body: ‘But further, the weights of the individual parts of each planet toward any other planet are to one another as the matter in the individual parts.’ And after concluding III.7 with a significant ‘Q.E.D.’ (the first such declaration of Book III), Newton begins the first corollary with the claim: ‘Therefore the gravity toward the whole planet arises from and is compounded of the gravity toward the individual parts.’

This paper focuses on Newton's evidence for universality.7 In particular, I investigate the following problem. In order to secure universal, particle-to-particle interaction, the evidence establishing the proportionality of gravitational force to each mass term must do extra work. It must also establish the additivity of mass. I call ‘additivity’ the property a body's quantity of matter has just in case, if gravitational force is proportional to that quantity, the force can be taken to be the sum of forces proportional to each particle's quantity of matter. Where little m is the mass of the orbiting or attracted body, big M is the mass of the central or attracting body, S is the collection of all particles composing the relevant body, and d is the distance to the body or the particle, Newton must establish both entailments:8Fgmd2Fg=iSFiwhereeachFimidi2FgMd2Fg=iSFiwhereeachFiMidi2

What follows addresses the following two questions. First, in what sense are these entailments distinct if gravitating bodies are simultaneously attracted and attracting? Second, to what extent does Newton succeed in establishing these entailments?

In the next section, I distinguish the entailments both by how the mass terms are empirically measured and by what evidence Newton has to establish the additivity of each mass term so measured. The two-pendulum experiment, accepted principles of statics, and I.65 help establish the first entailment, but no empirical experiments are available to Newton directly to establish the second. Newton nevertheless recognizes that these entailments occasion distinct evidence problems and successfully frames experimental programs which historically come to provide support for the second entailment.

In the subsequent section, I provide an account of how Newton confronts the evidence problem for the second entailment in the manuscript version of Liber secundus, an earlier version of Principia Book III which contains his initial argument for universal gravity (composed spring 1685). In place of empirical evidence, Newton offers conceptual analyses of the third law of motion and reasoning by analogy to magnetic attractions. These efforts are not entirely satisfactory to establish the second entailment.

I focus on the initial argument for a number of reasons. The final argument in III.7 is compressed to such an extent that it partly masks its own logic; the initial argument may help unpack it. The central claims of III.6–7 use warrants from the same propositions of Book I as are cited in counterpart places of the initial argument; how the initial argument relies on these propositions may help deepen our understanding of how the final argument does. A significant portion of the initial argument consists in conceptual analyses of the third law of motion which are entirely eliminated in the final argument, even though III.7.1 recaps - in reverse order - much of the deductive sequence of the initial argument and retains an appeal to magnets which is germane to the eliminated parts. Close scrutiny of these parts of the initial argument may thus help illuminate how the final argument goes beyond it and qualifies as a ‘deduction’ of universal gravity.

Section snippets

Active and passive quantities of matter

Why should the evidence for the first entailment be different than that for the second? To answer this question, I would like to distinguish two quantities of matter. By ‘passive quantity of matter’ I mean that quantity of it in proportion to which a body is acted upon by another. By ‘active quantity of matter’ I mean that quantity of it in proportion to which a body acts upon another.9

Liber secundus and three versions of law III

Before Newton settles on a three-book form for Principia in summer 1686, he composes a two-book treatise with ‘an earlier version of book 3 in popular form, so that it might be more widely read’ (P, 793). Drafted in the early part of 1685, these are titlted De motu corporum, Liber primus and Liber secundus. After learning of Hooke's priority claim on the inverse-square proportion, Newton then ‘translate[s] the substance of the earlier version [of Book 3, i.e. Liber secundus] into propositions

Conclusion

Newton is committed to establishing the additivity of mass as a premise in his deduction of universal, particle-to-particle gravity. There are, however, two kinds of mass which enter into the expression for the force of gravity. I have distinguished these kinds of mass by the measurement procedures available to Newton. A double-pan balance can measure what I call ‘passive quantity of matter’, while acceleration field strength (indicated by Keplerian orbital periods) can measure what I call

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    1

    The author would like to thank the Du Châtelet Prize committee - Katherine Brading, Mary Domski, Andrew Janiak, and Chris Smeenk - and participants of the Prize workshop at Duke for their thoughtful suggestions. Particular thanks are due to George Smith, whose encouragement and prior scholarship shaped this paper. Thanks also to Adrian Currie, Lorraine Daston, Robert DiSalle, William Harper, Michael Friedman, Eric Schliesser, Marius Stan, and Friedrich Steinle for helpful discussions. Thanks, finally, to Jürgen Renn for an invitation to work at the Max-Planck-Institut für Wissenschaftsgeschichte (Berlin), and to the librarians there for help finding material used in this paper.

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