newton and the mechanical philosophy.pdf

NEWTON AND THE MECHANICAL PHILOSOPHY:GRAVITATION AS THE BALANCE OF THE HEAVENS

Peter Machamer, J. E. McGuire, and Hylarie Kochiras

abstract: We argue that Isaac Newton really is best understood as being in thetradition of the Mechanical Philosophy and, further, that Newton saw himself as beingin this tradition. But the tradition as Newton understands it is not that of Robert Boyleand many others, for whom the Mechanical Philosophy was defined by contact actionand a corpuscularean theory of matter. Instead, as we argue in this paper, Newtoninterpreted and extended the Mechanical Philosophy’s slogan “matter and motion” inreference to the long and distinguished tradition of mixed mathematics and the studyof simple machines.

1. INTRODUCTION

Much has been written about the Mechanical Philosophy of the seventeenthcentury and its role in the emergence of early modern science. Alan Gabbey,for instance, has offered several detailed analyses of the different uses of theconcepts of “mechanics” and of “the mechanical philosophy” among early

Peter Machamer is Professor of History and Philosophy of Science and Associate Director ofthe Center for Philosophy of Science at the University of Pittsburgh. He has written on anumber of seventeenth-century topics and is co-author with J. E. McGuire of Descartes’s ChangingMind (Princeton University Press, 2009).J. E. McGuire is Professor Emeritus in the Department of History and Philosophy of Scienceat the University of Pittsburgh. McGuire has published numerous papers on early modernscience and philosophy, many of which are collected in Tradition and Innovation: Newton’s Meta-physics of Nature (Kluwer, 1995). Recently, he co-authored Descartes’s Changing Mind with PeterMachamer (Princeton University Press, 2009) and is currently completing a book-length studyof Aristotle’s modal theories with James Bogen. In 2011, McGuire was awarded the SartonChair and medal at Ghent University, Belgium.Hylarie Kochiras was a postdoctoral fellow during 2010–11 at the University of Pittsburgh’sCenter for Philosophy of Science, and is currently a European Institutes for Advanced Study(EURIAS) Fellow at New Europe College in Bucharest. Her research focuses on Newtonand early modern philosophy of science, and her publications include “Gravity and Newton’sSubstance Counting Problem” (Studies in History and Philosophy of Science, 2009), “Locke’s Phi-losophy of Science” (Stanford Encyclopedia of Philosophy, 2009), and “Spiritual Presence andDimensional Space beyond the Cosmos” (Intellectual History Review, 2012).

The Southern Journal of PhilosophyVolume 50, Issue 3September 2012

The Southern Journal of Philosophy, Volume 50, Issue 3 (2012), 370–88.ISSN 0038-4283, online ISSN 2041-6962. DOI: 10.1111/j.2041-6962.2012.00128.x

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modern natural philosophers (see, e.g., Gabbey 1985, 1990, 1993, 2002). Inan early paper, Gabbey suggested that ‘mechanics’ has had multiple mean-ings and that there were several so-called mini-revolutions “in collisiontheory, in statics and the theory of machines, in hydrodynamics, in vibrationtheory, in theories of central forces and rigid body motions—each with itsown new principles and procedures” (Gabbey 1990, 496). More recently,Gabbey has offered a general taxonomy of these different uses, explicitlydistinguishing between ‘mechanics’, understood as “concerned in some waywith manual activity,” and ‘mechanical’, which “connoted the theory ofmachines and more generally mechanics qua the science of body in motionand rest” (2002, 336).

Now Gabbey is certainly correct that there were many uses of the concept(and word) “mechanical.” For instance, we can identify in the early modernperiod the application of the term ‘mechanical’ to a variety of differentsituations, whether mathematical (Newton, as will be indicated subsequently)or nonmathematical (Boyle), whether involving straight line collisions (Des-cartes and Huygens) or vibrating bodies (Hooke), and, more generally, asapplied to a host of situations involving the identification and explanation ofthe active principles by which change occurs. However, we hesitate to acceptGabbey’s conclusion that there are only varieties of mechanical philosophiesand no general Mechanical Philosophy in the early modern period, preciselybecause we reject his sharp distinction between ‘mechanics’ and ‘mechani-cal’. For our part, it seems beneficial to identify the various uses of ‘mechan-ics’ and ‘mechanical’ as somehow referring to versions of a general MechanicalPhilosophy, especially in those cases where the actors themselves use varioussynonymous concepts of mechanics and mechanical.1 More importantly, aswe urge below, understanding the character of this Mechanical Philosophy,as well as its development over the seventeenth century, requires understand-ing how the terms ‘mechanical’ and ‘mechanical philosophy’ are relatedamong early modern practitioners. Accordingly, we must consider how thepractice of mixed mathematics and the study of simple machines—whichGabbey would relegate to the category of “mechanics”—became integratedinto the mechanical principles used to investigate nature.

To make our case, we focus below on Isaac Newton’s relationship to theMechanical Philosophy, aiming to show how Newton’s consideration of

1 In this respect, our project is also opposed to Dan Garber’s recent claim (Garber, forth-coming) that there was no Mechanical Philosophy in the seventeenth century. The character-ization of the Mechanical Philosophy, which we forward below, also differs from that of A. R.and M. B. Hall (Newton 1962). They take the Cartesian reliance on material impact to be afirst-order explanation and take Newton’s use of mathematized forces to be a second-orderexplanation (Newton 1962, 76). We take each to be a version of the mechanical.

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mixed mathematics and simple machines allowed him to extend the basicprogram of the Mechanical Philosophy to include the mathematical treat-ment of forces. At the end, we show, pace recent commentators, that Newton’sproposal of universal gravitation is entirely consistent with his commitment tothe general tenets of the Mechanical Philosophy. First, however, we mustclarify our general account of the Mechanical Philosophy, and to do so, weconsider generally accepted uses of ‘matter’ and ‘motion’ in the MechanicalPhilosophy’s famous slogan “matter and motion.”

2. A GENERAL CHARACTERIZATIONOF THE MECHANICAL PHILOSOPHY

For Boyle, as for most other contemporary natural philosophers, there is onlyone kind of matter—an extended, divisible, and impenetrable mattercommon to all bodies. This single-matter theory for both the celestial andterrestrial realms began to be popular with the publication in 1610 of Gali-leo’s Sidereus Nuncius and spread with the growing acceptance of Copernican-ism among natural philosophers. Granting that the earth orbits the sun, it wassimply no longer clear why terrestrial matter should be considered different inkind from so-called celestial matter. In addition to Copernican pressures totreat the matter of the celestial and terrestrial realms as the same, this unity ofmatter was reinforced by the revival of ancient atomism and Epicureanism(see Machamer 2009 and Wilson 2008).

However, a unified matter theory cannot account for diversity in nature,and this is where “motion” enters the world-view of the Mechanical Philoso-phy. The parts of bodies, if left to themselves, would remain at rest relative toone another. Accordingly, motion, as the active causal principle, is necessaryto produce change. As Boyle puts it: “local motion seems to be indeed theprincipal amongst second causes, and the grand agent of all that happens innature” ([1666] 1991, 19). Thomas Hobbes is of the same opinion: “thevariety of all figures arises from the variety of motions by which they areconstructed, and motion cannot be understood to have any cause other thananother motion . . . it is unintelligible that something depart from rest or frommotion except by motion” ([1655] 1966, De Corpore, Part 1, Chapter Six).Most leading natural philosophers of the seventeenth century, such asGalileo, Descartes, Barrow, Huygens, and Newton, were in agreement on thispoint, though many also allowed that immaterial minds could cause motionin some bodies. (Moreover, unlike Hobbes many did not restrict the under-standing of efficient causation to motion alone.)

Thus, whatever else fell under its purview, the Mechanical Philosophy wasthe study of local motion. Most particularly, it was the study of local motion

372 PETER MACHAMER, J. E. MCGUIRE, AND HYLARIE KOCHIRAS

as the efficient cause of all motions that occur in the cosmos. Accordingly, toexplain variety among phenomena, it was no longer necessary to invoke thefourfold causes of the Aristotelian tradition. A single efficient cause—localmotion—was sufficient. This conception possessed great explanatoryeconomy. It tied the Mechanical Philosophy to the legacy of ancient atomism,the view that everything is composed by the local motions and shapes ofatoms moving in a void, though there would be disputes about the nature ofbodies and about whether or not there was a void.

Moreover, what became important during the seventeenth century wasnot that the traditional categories of causes were de-emphasized for explana-tory purposes (that certainly occurred) but, rather, that a new stress wasplaced on the category of law, and on laws of motion, in particular. Increas-ingly, thinkers from Galileo onward embarked on a search for the regularpatterns of motion exhibited by the behavior of phenomena. This culminatedin 1665 when the Royal Society offered a prize for the best quantitativetreatment of the phenomenon of impact. Inspired by Descartes’s attempt,based on his three natural laws, to deduce the descriptive rules that governbodies on impact, Wren, Wallis, and Huygens all submitted detailed quanti-tative descriptions of the relationship (see Wallis et al. 1669). It is hardly amatter of dispute that the concept of law became a basic, explanatory cat-egory in the natural philosophy of the seventeenth century.2 And it is thushardly surprising that Newton should preface his treatise on motion with thetitle “Axioms, or Laws of Motion.” What we propose is that an understandingof the Mechanical Philosophy involves, not only the rejection of Aristotelian-ism and the acceptance of unified matter that is sometimes conceived of ascorpuscular in nature, but also something even more important; it involvesthe view that motion is the basic efficient cause and that motion’s regularitiescan be expressed in terms of quantitative laws.

3. NEWTON AND THE MECHANICAL PHILOSOPHY

Adopting our general interpretation of how “matter and motion” was usedduring the seventeenth century, we can draw some evident connectionsbetween the Mechanical Philosophy and Newton’s philosophy of nature. Inan early notebook (Questiones quaedam Philosophica, or the Trinity Notebook),Newton declares himself in favor of a unified theory of matter, indeed ofatomism (Newton [1661–65] 2003). Drawing from Greek sources of atomism,he developed a systematic picture of the coordinated motions of atoms in

2 Why this emphasis on laws of nature occurred in the seventeenth century remains aquestion of debate. See Zilsel 1942, 245–79, and the overview of the current debate in Henry2004.


time, space, and place (336–51). Newton remained a partisan of atomismthroughout his life, opposing the categories of substance and quality that werecentral to the Aristotelian tradition. His opposition to Aristotelianism isevident in a document dating from the early 1670s. On printed sheets,probably intended for an edition of his theory of optics, Newton tells us thatby means of an “improper distinction which some make of mechanicalHypotheses, into those where light is put a body, and those where it is put theaction of a body, understanding the first of bodies trajected through amedium, the last of motion or pression propagated through it . . . whereaslight is equally a body or the action of a body in both cases” (cited in Cohen1958, 365). Newton goes on to state that “the bodies in both cases must causevision by their motion” (365). Unlike the Peripatetic view that light is aqualitative modification of a subject, mechanical hypotheses emphasize thatit arises from the “motion” of bodies. Thus, Newton’s purpose is not to put“body” in opposition to “motion,” but in opposition to “to a Peripatetickquality, stating the question between the Peripatetick and the MechanickPhilosophy by inquiring whether light be a quality or body” (365). To buttressthe point, Newton contrasts the ontology he favors with the Peripateticposition. The Peripatetics use the terms “Quality, Subject, Substance, sen-sible qualities,” whereas the Mechanical Philosophy uses the terms “Body,Modes, Actions,” leaving “undetermined the kinds of those actions (supposewhether they be pressions, strokes, or other dashings), by which light mayproduce in our minds the phantasms of colours” (365). It is important tonotice the stress that Newton places on the terms ‘action’ and ‘motion’: it ismotion—the motion of bodies—that acts on the sense organs to producesensations of color. In general, motion is taken by Newton to be a universaland physical cause, the action of which generates change in nature. Thisconception remains with Newton throughout his career and is the back-ground of the “science of motion” developed in the Principia ([1726] 1999)and the Opticks ([1730] 1952). Indeed, it is fair to say that it forms the core ofNewton’s understanding of the Mechanical Philosophy.

However, as important as these general connections between the Mechani-cal Philosophy and Newton’s philosophy are, there is more that can andshould be said. We should, in fact, resist reading Newton along purelyBoylean lines, according to which the Mechanical Philosophy is intertwinedwith the corpuscular philosophy and adherence to contact action, and con-sider Newton’s Mechanical Philosophy against the backdrop of two othertraditions that affected the way in which seventeenth-century natural philoso-phers thought of motion as the principal efficient cause acting in nature: themixed sciences and the simple machines. Earlier mechanics are in fact mostlyconcerned with the simple machines (and sometimes hydrostatics). For


instance, Guido Ubaldo says that all simple machines can be reduced to thewheel, and he stresses their practical applications, worrying mostly abouttheir utilities. He also stresses “And mechanics, since it operates againstnature or rather in rivalry with the laws of nature, surely deserves our highestadmiration” (Ubaldo [1581] 1969, 241). Optics and astronomy from theancient period to the seventeenth century used geometry to demonstrateresults established by observation and experiment. Since they combined bothphysical and geometrical principles, they were “mixed” or subalternate sci-ences (see Biener 2008). What these sciences studied was motion—themotions of light and of the planets modeled and demonstrated geometrically.Moreover, after Galileo, mechanics became the quantitative study of localmotion of bodies, and mechanics and its relation to the simple machines wasno longer conceived in opposition with nature. This became the new modelof intelligibility for understanding nature (see Machamer 1998). Indeed, in hisearly unpublished De Motu (1605), Galileo had attacked the Aristotelianprinciples of sublunar motions (light and heavy) as unintelligible by usingArchimedean hydrostatics and, more significantly, by showing that themotion of all bodies was due to “heaviness” and could be made intelligible bymodeling motion as a balance problem.

The simple machines have been studied since antiquity (see Berryman2009): the balance, the lever, the pulley, the inclined plane, and the screwbeing the principal devices. Each of these devices involves a relationship ofbalance (equilibrium) between an input and an output. They exploitmechanical advantage (leverage or input) to multiply the effect of force(output). The lever is a classic example of a device for producing a mechanicaladvantage. Depending on where the fulcrum is placed in relation to the load,mechanical advantage can be increased or decreased, such that the forceapplied to the load is equal to the work done by the force exerted on the lever.It is important to notice that the notions of action, force, and motion areinvolved. The lever is in fact a dynamical system in which the action of a forceon one end is balanced on the other by a change in position of the load in theopposite direction. So we have a balanced reconfiguration of the opposingends of the lever brought about by the action of motion. Thus motion couldbe conceived in terms of equilibria relations, as somehow resulting in somesort of stasis (its later connection to statics explicable by the fact that we canmeasure proportions only by treating them as static, even though they may becontinuously changing). And rest could be conceived as a stable equilibriumstate.

In brief, then, the study of simple machines was the study of motion andrest, and it was characteristically mathematical. The simple machines are thusgood examples of using mixed mathematics. They apply mathematical prin-


ciples to physical objects. However, it must be noted that they describe onlycertain properties of the physical bodies they treat. Many physical properties,such as the materials constituting any given simple machine, are neglectedand not taken to be relevant to the explanations.3

In what follows, we aim to show two things: first, that Newton placedhimself in this tradition of the simple machines and, second, that Newton usedthe balance as the model for treating gravitational action as a simple machine.

4. MECHANICAL PRINCIPLES IN THE AUTHOR’S PREFACE

Let us turn now to the Author’s Preface to the Principia, which appeared in thefirst edition and was retained in all subsequent editions.4 Two features of thePreface make it especially interesting for our purposes. First, Newton recon-figures the relationships among geometry, mechanics, and natural philosophyin such a way that gravity becomes part of mechanics. Second, he claims tohave discovered the gravitational force through “mechanical principles,”which he identifies with the mathematical principles of natural philosophythat provide the title for his treatise. To begin, it is worth quoting the firstparagraph of the Author’s Preface in full.

Since the ancients (according to Pappus) considered mechanics to be of the greatestimportance in the investigation of nature and science and since moderns—rejectingsubstantial forms and occult qualities—have undertaken to reduce the phenomenaof nature to mathematical laws, it has seemed best in this treatise to concentrate onmathematics as it relates to natural philosophy. The ancients divided mechanics into twoparts: the rational, which proceeds rigorously through demonstrations, and the prac-tical. Practical mechanics is the subject that comprises all the manual arts, from whichthe subject of mechanics as a whole has adopted its name. But since those who practicean art do not generally work with a high degree of exactness, the whole subject ofmechanics is distinguished from geometry by the attribution of exactness to geometry andof anything less than exactness to mechanics. Yet the errors do not come from the artbut from those who practice the art. Anyone who works with less exactness is a moreimperfect mechanic, and if anyone could work with the greatest exactness, he wouldbe the most perfect mechanic of all. For the description of straight lines and circles,which is the foundation of geometry, appertains to mechanics. Geometry does not teachhow to describe these straight lines and circles, but postulates such a description. For

3 See Bertoloni Meli 2010 for discussion of how the study of simple machines, for practi-tioners such as Galileo, was integrated with an “axiomatic tradition” in natural philosophy.

4 Additional draft prefaces, never completed, were written in the years just prior to andfollowing the publication of the second edition in 1713. See Cohen’s translation and discussionof one of these drafts—the Unpublished Preface to the Principia, as he calls the manuscript ULC Ms.3968, fol. 109—in his Guide to Newton’s Principia (Newton [1726] 1999, 49–54). See also thediscussion by Guicciardini, who refers to that same draft as the “Intended Preface” (Guicciar-dini 2009, 303–05).


geometry postulates that a beginner has learned to describe lines and circles exactlybefore he approaches the threshold of geometry, and then it teaches how problems aresolved by these operations. To describe straight lines and to describe circles areproblems, but not problems in geometry. Geometry postulates the solution of theseproblems from mechanics and teaches the use of the problems thus solved. Andgeometry can boast that with so few principles obtained from other fields, it can do somuch. Therefore geometry is founded on mechanical practice and is nothing otherthan that part of universal mechanics which reduces the art of measuring to exactproportions and demonstrations. But since the manual arts are applied especially tomaking bodies move, geometry is commonly used in reference to magnitude, andmechanics in reference to motion. In this sense rational mechanics will be the science,expressed in exact proportions and demonstrations, of the motions that result fromany forces whatever and of the forces that are required for any motions whatever.The ancients studied this part of mechanics in terms of the five powers that relate to themanual arts [i.e., the five mechanical powers] and paid hardly any attention togravity (since it is not a manual power) except in the moving of weights by thesepowers. But since we are concerned with natural philosophy rather than manualarts, and are writing about natural rather than manual powers, we concentrate onaspects of gravity, levity, elastic forces, resistance of fluids, and forces of this sort,whether attractive or impulsive. And therefore our present work sets forth math-ematical principles of natural philosophy. For the basic problem [lit. whole difficulty]of philosophy seems to be to discover the forces of nature from the phenomena ofmotions and then to demonstrate the other phenomena from these forces. It is tothese ends that the general propositions in books 1 and 2 are directed, while in book3 our explanation of the system of the world illustrates these propositions. For inbook 3, by means of propositions demonstrated mathematically in books 1 and 2, wederive from celestial phenomena the gravitational forces by which bodies tendtoward the sun and toward the individual planets. Then the motions of the planets,the comets, the moon, and the sea are deduced from these forces by propositionsthat are also mathematical. If only we could derive the other phenomena of naturefrom mechanical principles by the same kind of reasoning! For many things lead meto have a suspicion that all phenomena may depend on certain forces by which theparticles of bodies, by causes not known, either are impelled toward one another andcohere in regular figures, or are repelled from one another and recede. Since theseforces are unknown, philosophers have hitherto made trial of nature in vain. But Ihope that the principles set down here will shed some light on either this mode ofphilosophizing or some truer one. (Newton [1726] 1999, 381–83)

Although at the outset of the Preface, Newton characteristically claimssome continuity with the ancients, he then reconfigures disciplinary bound-aries by opposing a still-dominant ancient division. As he describes it, “Prac-tical mechanics is the subject . . . from which the subject of mechanics as awhole has adopted its name. But since those who practice an art do notgenerally work with a high degree of exactness, the whole subject of mechanicsis distinguished from geometry by the attribution of exactness to geometry and ofanything less than exactness to mechanics.” Part of Newton’s concern in these


and surrounding lines—most notably, his assertion that “geometry is foundedon mechanical practice and is nothing other than that part of universal mechan-ics which reduces the art of measuring to exact propositions anddemonstrations”—is to advance an epistemological view of geometry thatopposes the Cartesian one. Several commentators have recently shown thatDescartes’s La Géométrie ([1637] 1902) was indeed the Preface’s specific target5

and that identification has played centrally in debates about whether or whichsort of geometrical “constructivism” Newton accepted.6 However, we neednot engage those debates here; instead we note some more general points.

First, we take Newton to be following Isaac Barrow when he says “geometryis founded on mechanical practice.” Barrow held that local motions wereeven to be thought of as being the basis of geometry. Lecture One of hisGeometrical Lectures: Explaining the Generation, Nature and Properties of Curve Lines(1735) opens with a discussion of motion and then goes on to discuss time andthen velocity. Barrow ends the first lecture with a discussion of uniform anduniformly accelerated motion. Lecture Two is entitled “Generation of mag-nitudes by ‘local movements’: The simple motions of translation and rota-tion.” Barrow writes in the opening lines of this lecture: “Mathematicians arenot limited to the actual manner by which a magnitude has been produced;they assume any method of generation that that may be best suited to theirpurpose” (Barrow [1735] 1916, 42). In a similar vein, Newton considersmechanics to be the science of local motions, and these motions are used bythe mathematicians to construct or generate curves, but not necessarily byusing material instruments (such as rulers and compasses). Mechanics “pre-cedes” geometry because one must know about the motion of points to knowabout lines, about the motion of lines to know about planes, and about themotions of points on a plane to know about curves.7

Second, to reconfigure the relationships among mathematics, mechanics,and natural philosophy, Newton identifies erroneous presumptions in thatancient division between rational and practical mechanics that he means tooverthrow. Contrary to the presumption of rational mechanics, Newtonwants us to realize that there is exactness in the divinely created machine of theworld. Thus, the objects of mathematical methods are not confined toabstract mathematical entities, and mechanics should no longer be thought tobe a branch of mathematics. Moreover, disputing the longstanding presump-tion that practical mechanics, with its restricted domain, legitimately repre-

5 See Domski 2003 and Guicciardini 2009. See also Garrison (1987, 614) for discussion ofCartesian analysis as Newton’s target.

6 See also Garrison 1987 and Sepkowski 2005.7 See Domski (this volume) for more on the importance of geometry for Newton’s science of

bodies.


sents the science of motion, Newton wants us to realize that the science ofmotion fundamentally includes the natural powers or forces, notably “gravity,levity, elastic forces, resistance of fluids, and forces of this sort, whetherattractive or impulsive” (Newton [1726] 1999, 382).We should no longerallow our understanding of the domain of mechanics, the science of motion,to be determined by practical mechanics, which traditionally restricted itsgaze to the imperfect machines of human creation and the powers or forcesassociated with them. Natural powers and forces are also in the purview ofrational mechanics, for as Newton asserts, the world is a machine and aperfect one, with God its creator being “the most perfect mechanic of all.”8

For Newton, the principles governing this perfect machine are mathemati-cal, which leads us to our final point. These mathematical principles are oneand the same with mechanical principles, and it is by means of thosemechanical or mathematical principles that causal principles—forces—arediscovered from phenomena. Newton first denotes the process of discovery asthe goal of natural philosophy: “For the basic problem of philosophy seems tobe to discover the forces of nature from the phenomena of motions and thento demonstrate the other phenomena from these forces” (Newton [1726]1999, 382). Subsequently, in the brief remarks expressing his wish to discoverthe short range, interparticulate, attractive and repulsive forces that preoc-cupy him elsewhere, he identifies the mathematical principles facilitating suchdiscoveries as mechanical. “If only we could derive the other phenomena ofnature from mechanical principles by the same kind of reasoning!” (382). Itappears, then, that the natural philosophy that Newton introduces in thePreface qualifies as a Mechanical Philosophy in the sense set out earlier. As hecashes out his Principia project, he draws from the simple machine tradition byusing mathematical proportions that express the operations of nature.

The purpose of the Author’s Preface is to place the Principia in the traditionof the mixed sciences with the simple machines subsumed—that is, to place itin the mechanical/geometrical tradition of the ancients. But as the Prefacealso makes clear, Newton also sees himself as extending this tradition inimportant ways. The conception of mechanics that is at play in the Principiacan profitably be understood within this innovative framework.

In the Preface, rational mechanics is presented as “the science of motionsresulting from any forces whatsoever” (Newton [1726] 1999, 382). Thus

8 Newton repeats his own thoughts on the matter in his first letter to Bentley, which isincluded in Newton: Philosophical Writings (2004, 94–97). Earlier, mathematician and physicianHenri de Monantheuil had asserted the world to be “a machine, and indeed of machines, thegreatest, most efficient, most firm, most beautiful,” and its creator to be “the most accurate andincessant Geometer.” Monantheuil’s remarks are translated and discussed by Helen Hattab(2005, 113–15).


understood, mechanics demonstrates the properties of any and all motionsthat arise from the action of any and all forces. Mechanics is “rational” in thatit demonstrates the properties of motions that arise from the action of naturalpowers and not from the action of the simple machines. Accordingly, New-ton’s chief concern in the Principia is with motions caused by “natural powers”such as gravity: “therefore I offer this work as the mathematical principles ofphilosophy, for the whole burden of philosophy seems to consist in this—fromthe phenomena of motions to investigate the forces of nature, and then fromthese forces to demonstrate the other phenomena” (382).

It is clear, then, that Newton conceives of the Principia as “the science ofmotions” demonstrated by geometrical principles. It is from motions geo-metrically considered that the action of the forces producing them can beinvestigated. The laws of motion describe the relation between the action offorces and the motions they produce. But since the Principia is concerned withthe geometrical modeling of motions abstractly considered (i.e., rationalmechanics), it

use[s] interchangeably and indiscriminately words signifying attraction, impulse, orany sort of propensity towards a center, considering those forces not from a physical,but only from a mathematical point of view. Therefore, let the reader beware ofthinking that by the words of this kind I am anywhere defining a species or mode ofaction or a physical cause or reason, or that I am attributing forces in a true andphysical sense, to certain centers (which are only mathematical points) if I happen tosay that centers attract or that centers have forces. (Definition 8; Newton [1726]1999, 408)

Newton makes the same point forcibly in the opening of Section 11, Book 1,and in the Scholium to Proposition 69, Book 1.

But what does it mean to consider forces mathematically and not physi-cally? It means, in the first instance, to deal with the observable effects of aforce, a phenomenon that can be described accurately by the use of geometry.Thus, we can measure and demonstrate mathematically the change in abody’s speed or momentum that results from the action of a force. Thus, whatis captured geometrically are the observable changes of state that bodiesundergo apart from any consideration of the sorts of physical causes involved.This distinction is set out clearly in the Scholium to Proposition 69, Book 1.Newton tells us that he is not considering the

species of forces and their physical qualities, but their quantities and mathematicalproportion, as I have explained in the definitions. Mathematics requires an inves-tigation of those quantities of forces and their proportions that follow upon anyconditions that may be supposed. Then, coming down to physics, these proportionsmust be compared with the phenomena so that it may be found out which condi-


tions [or laws] of forces apply to each kind of attracting bodies. And then finally itwill be possible to argue more securely concerning the physical species, physicalcauses, and physical proportions of these forces. (Newton [1726] 1999, 588–89)

It is clear, then, that the mathematical treatment of force relates only tocertain quantities of bodies and with the geometrical proportions that pertainto them. The mathematics describes only the activities of the causes.

Consider, for instance, the second law of motion. It tells us that “A changein motion is proportional to the motive force impressed and takes place alongthe straight line in which that force is impressed” (Newton [1726] 1999, 416).Here is one of the fundamental proportions of the Principia. It tells us that thechange in a body’s motion is directly proportional to the action of animpressed force. In speaking of the body’s motion, Newton is referring to its“quantity of motion” or momentum, the measure of which is mass x velocity(Definition 2). An impressed force is the action that changes the state of abody “all at once or successively by degrees” (Law 2, Definition 4; Newton[1726] 1999, 416). Note that the notion is completely general and is notrestricted to any particular mode of action, whether attractive or repulsive.Thus, the defined proportions and quantities can be treated geometricallywithout considering the kind or species of force involved.

The generality of Newton’s “science of motion” is shown dramatically inProposition 1, Theorem 1 of the First Book of the Principia. He proves that ifa body is continually drawn toward some center, its otherwise inertial motionwill be transformed into motion along a curve, and that a line from the centerto the body will sweep out equal areas in equal times. In the proof Newtonspeaks of a “centripetal force, by which the body is continually drawn backfrom the tangent to this curve” (Newton [1726] 1999, 445). But this simplymeans that at successive instants, each of an infinite number of points on thecurve is directed to the center, regardless of the mode of action. Again, theproof says nothing about a body at a center exerting a force, and nothing ofthis sort is assumed. The center is a geometrical point around which equalareas may be found. The basic idea is intuitively simple, namely, that if a bodyis moving in purely inertial motion, then with respect to any point not on thatline of motion, the law of equal areas must apply. Newton shows this in thecase of all the curves generated by the conic sections. The generality of theproof is important. It allows Newton to speak of a centripetal force withoutassuming the physical mode of action involved, which can be determined,presumably, only in light of empirical data. Interestingly, at the beginning ofSection 11, Book 1, Newton says that he will henceforth speak of the “motionof bodies that attract one another, considering centripetal forces as attrac-tions, although perhaps—if we speak in the language of physics—they mightbe more truly called impulses” (Newton [1726] 1999, 561). If Proposition 1,


Theorem 1 were interpreted physically, the centripetal forces would consist ofdiscrete impulses successively driving the body into a curve around a center.

The element Newton draws from the simple machine tradition is his meansof discovering or identifying genuine causes in the world.9 The Principia’s thirdbook applies the principles developed in the previous books to the solarsystem; having found the gravitational force’s mathematical proportions, hedescribes it as the force “by which celestial bodies are kept in their orbits”(Newton [1726] 1999, 806). It therefore is legitimate to understand Newton’snatural philosophy as mechanical, in virtue of the mathematical approachthat he draws from the simple machine tradition and takes to identify forcesas physical causes.

5. GRAVITATION

Following our account of Newton’s relationship to the Mechanical Philoso-phy, as one filtered through his sensitivity to mixed mathematics and simplemachines, we have a novel way of understanding how the force of gravitationis itself consistent with, and also an extension of, the Mechanical Philosophy.Our approach to gravitation is opposed to now-standard readings, accordingto which material contact action is taken to be the sine qua non of the Mechani-cal Philosophy. For example, Andrew Janiak suggests that a prohibitionagainst unmediated distant action was “a crucial norm of the mechanicalphilosophy (in all its guises)” (2008, 53).10 On this account, there are severalsignificant passages that indicate Newton’s distance from the traditional

9 For more on the importance of a mechanical interpretation of mathematics for Newton’sproject, see Guicciardini, who remarks, “Hobbes and Barrow . . . maintained that a mechani-cally based geometry is a discipline endowed with scientific character insofar as it yieldsknowledge of causes. According to Newton, a mechanically based geometry achieves exactlythis end” (Guicciardini 2009, 302). “Newton wanted to inject certainty into natural philosophyvia geometry. From his viewpoint, algebra was not endowed with the certainty that character-izes geometry. He often repeated that geometrical objects, such as plane curves, are betterunderstood if the reason of their genesis is known” (319). For more on Newton’s attitude towardthe certainty of geometry, see Janet Folina’s contribution to this volume.

10 Janiak explicitly indicates that he does not claim to have identified or characterized allvariants of the mechanical philosophy but, rather, to have identified those he considers mostsalient for understanding Newton: as he calls them, strict mechanism and loose mechanism. The strictmechanist does not admit forces, except insofar as forces may be reduced to bodily properties,namely, size, shape, motion, and solidity; while the loose mechanist allows forces, such as thoseinvolving impact. Both deny unmediated distant action, and again, although Janiak does notaim to identify all variants of the Mechanical Philosophy, he appears to see a prohibition againstdistant action as common to all variants, as indicated by the quoted remark (Janiak 2008,52–53). McGuire also emphasized contact action in an earlier paper. After identifying a widerange of meanings of ‘mechanical’, McGuire writes that while there was no agreement aboutthe sufficient conditions, “all agreed that contact action was a necessary condition for amechanical explanation” (McGuire, 1972, 523n2).


Mechanical Philosophy. For instance, just prior to the most famous words ofthe General Scholium, which was added in the 1713 edition—words bywhich Newton explains that he has assigned no cause to gravity because hewill not “feign hypotheses”—he writes that the force’s cause “acts not inproportion to the surfaces of the particles on which it acts (as mechanicalcauses are wont to do)” but rather “in proportion to the quantity of solidmatter” (Newton [1726] 1999, 943). Similarly, in Query 28, first added forthe 1706 Optice (a Latin edition of his Opticks), he distinguishes himself fromthose philosophers who do feign hypotheses in their zeal to explain all things“mechanically,” that is, by “dense matter” (Newton [1730] 1952, 369).

However, we should not sever the ties between gravitation and theMechanical Philosophy on these grounds alone. As the phenomena of thenatural world include gravitational effects, a fundamental feature of theMechanical Philosophy, as understood by Newton, must be the explanationof those effects by means of something drawn from the study of the math-ematics of simple machines, which we take to be the sine qua non ofseventeenth-century mechanics. The mechanics figuring in this endeavorincreasingly expanded into problems about force and local motion. Thediscipline had traditionally been understood as mixed mathematics—ascience whose objects were physical in some sense but yet, in virtue of itsmethods, was mathematical.11 But this is not to say that mechanics whenclassified as mixed mathematics excluded all consideration of force; asMachamer and Woody (1994) have emphasized, problems within the simplemachines were not limited to statics but also included dynamics. As a notableexample, bodies on a balance will be in equilibrium if the ratio of their masses(or weights) is inversely proportional to the ratio of their distances from thefulcrum. This had been used to explain various phenomena, from pseudo-Aristotle’s Mechanica to Galileo’s De Motu, and in ways that employed theheaviness of bodies without trying to explain it. As traditionally classified,then, mechanics could employ the force of heaviness without looking into thenature of matter to determine how these forces worked.12

For those hoping that the functioning of simple machines would be anilluminating guide to the functioning of natural processes, there was animportant question of just how far that illumination could reach. For Galileo,the balance served as a model of intelligibility for a range of phenomena.Unlike the Aristotelians’ irreducible substantial forms, the balance was aphysical object, and modeling some phenomenon on it enabled one to visu-alize that phenomenon in terms of the proportions, structures, and interre-

11 See, e.g., Bertoloni Meli 2006.12 See Des Chene 2005; see also Garber 2002, esp. 189, and Berryman 2009, 244–45.


lations of clearly observable parts. In Day IV of his Dialogue Concerning the TwoChief World Systems ([1632] 1953), for instance, Galileo explained the tides byinvoking the motion of a pendulum, which he elsewhere reduces to thebalance; in other works, he also invoked the balance to treat problems of freefall.13

Understanding the balance also seems to have been in Newton’s mindwhen assessing gravitational forces. In the balance there is an equilibriumobtained between the forces on each side of the arm at different distancesfrom the fulcrum, and the physical arm of the balance is neglected. So theplanets balance each other by their attractive forces, and the connectionbetween them is neglected. In Newton’s System of the World, in fact, he imaginesthat the planets are connected as if by a rope (Newton [1728] 1969, 38–39).The taut rope might be seen as playing the role of the arm of a balance.

Newton explicitly mentions machines and their forces in Corollary II tothe Axioms or Laws of Motion. There among his talk of equipollent andequilibrium, he gives a general rule about the composition and resolutionof forces as applied in the case of a complex machine made of balances,levers, and screws. He concludes: “the whole of mechanics . . . depends onwhat has just now been said. For from this are easily derived the forces ofmachines, which are generally composed of wheels, pulleys, levers, stretchedstrings, and weights, ascending directly or obliquely, and the other mechani-cal powers . . .” (Newton [1726] 1999, 419–20). But he had used wheels andweights, and all those images in his proof. So it seems reasonable to con-clude that though he was confining his talk to real machines, he was usingthe machine language to show how forces in general work.

Later in the Scholium to the same chapter, Newton again illustrates gravi-tation of the parts of the earth by saying, “As bodies are equipollent incollisions and reflections if their velocities are inversely inherent forces [i.e.,forces of inertia], so in the motions of machines those agents [i.e., actingbodies] whose velocities (reckoned in the direction of their forces) areinversely as their inherent forces are equipollent and sustain one another bytheir contrary endeavors” (Newton [1726] 1999, 428–29). He then moves onto talk about the balance, the pulley, and other complex machines. Yet hesays, “But my purpose here is not to write a treatise on mechanics.” Realmachines are instances of the “wide range and certainty of the third law ofmotion” (430), which is the more general form for rational mechanics andindicates the reciprocal action and reaction of all phenomena on one another.

Newton’s theory of gravitational force is illuminated when seen against thisbackground of simple machines. It, too, constitutes a dynamical system, a

13 See the discussion in Machamer 1998, esp. 61.


system of mutually interacting bodies constrained by the action of omnipres-ent forces. The law states that all bodies mutually attract one another in directproportion to their masses and inversely as the square of the distances sepa-rating them. Thus the action of one body is equally balanced or put intoequilibrium by the opposing action of other bodies. Mathematically consid-ered, gravitation is a phenomenon whereby bodies act mutually on oneanother at a distance. But since their ability to attract one another is directlyproportional to their masses, gravitation is a penetrative force acting instan-taneously and reciprocally on the total mass of each body. Furthermore,gravitational action generates curves by means of motions that satisfy thegeometry of the conic sections. Thus, in Newton’s eyes, the reciprocal actionof bodies on one another generates physical curves that can be analyzedgeometrically. Newton’s gravitational theory beautifully exemplifies hisespousal of the ontology of force, action, and body. Moreover, as we havestressed throughout, it exhibits his adherence to the traditions of the mixedsciences and the simple machines, the latter understood as dynamical systems.It also reveals Newton’s extension of this tradition: the theory of gravitationclearly illustrates his claim in the Preface that he is concerned, not withmanual powers as were the ancients, but with the “natural powers,” whichinclude gravity.

6. CONCLUSION

Newton clearly fits into the tradition that, by the late seventeenth century,believed that motion may be described by regular and universal geometricallaws. This is a shift of focus from individually natured things (Aristotle) tonature itself—a turn away from things whose intrinsic inner natures oressences are the source and measure of what they can do, and toward natureas a system of laws by which individual things act in concert. So there is a shiftaway from inner and individual principles or causes of change and towardgeneral systems of change.

This is what the Mechanical Philosophy had become. This emphasis anduse of the simple machines will continue in the work of Euler and Lagrange.They will disagree with Newton in many respects but will bring back thesimple machines explicitly as the way of proving the application of theiranalytic method to the world (see Hepburn 2007, 2010).14

What we have tried to show is that the Newtonian “revolution” wassteeped in the tradition of mixed mathematics and the simple machines.

14 See Michael Friedman’s contribution to this volume for Kant’s use of the balance in histreatment of the Newtonian notion of quantity of matter.


When this is combined with a view of the unity of matter, it opens the worldto being understood by a set of universal laws that apply to all matter and tothe forces that are active in bringing about changes. Maybe we should callthis, innovation through tradition.15

REFERENCES

Barrow, Isaac. [1735] 1916. The geometrical lectures of Isaac Barrow. Trans. J. M. Child.Chicago and London: Open Court.

Berryman, Sylvia. 2009. The mechanical hypothesis in ancient Greek natural philosophy. Cam-bridge: Cambridge University Press.

Bertoloni Meli, Domenico. 2010. The axiomatic tradition in seventeenth-century mechan-ics. In Discourse on a new method: Reinvigorating the marriage of history and philosophy of science,ed. Mary Domski and Michael Dickson, 23–41. LaSalle, IL: Open Court.

———. 2006. Mechanics. In The Cambridge history of science, volume 3: Early modern science, ed.Katharine Park and Lorraine Daston, 632–72. Cambridge: Cambridge UniversityPress.

Biener, Zvi. 2008. The unity of science in early-modern philosophy: Subalternation,Demonstration, and the geometrical manner in late-Scholasticism, Galileo, and Des-cartes. PhD dissertation, University of Pittsburgh.

Boyle, Robert. [1666] 1991. The origine of formes and qualities (according to the corpuscularphilosophy). In Selected philosophical papers of Robert Boyle, ed. M. A. Stewart. Indianapolis,IN: Hackett.

Cohen, I. B. 1958. Versions of Isaac Newton’s first published paper. Archives Internationalesd’Histoire des Sciences 11: 357–75.

Descartes, René. [1637] 1902. La géométrie. In Oeuvres, ed. Ch. Adam and P. Tannery. Vol.6. Paris: L. Cerf.

Des Chene, Dennis. 2005. Mechanisms of life in the seventeenth century: Borelli, Perrault,Régis. Studies in the History and Philosophy of Biology and Biomedical Sciences 36: 245–60.

Domski, Mary. 2003. The constructible and the intelligible in Newton’s philosophy ofgeometry. Philosophy of Science 70: 1114–24.

Galileo. [1632] 1953. Dialogue concerning the two chief world systems. Trans. Stillman Drake.Berkeley: University of California Press.

Gabbey, Alan. 2002. Newton, active powers, and the mechanical philosophy. In TheCambridge companion to Newton, ed. I. Bernard Cohen and George E. Smith, 329–57.Cambridge: Cambridge University Press.

———. 1993. Descartes’s physics and Descartes’s mechanics: Chicken and egg? In Essayson the philosophy and science of René Descartes, ed. Stephen Voss, 311–23. New York: OxfordUniversity Press.

———. 1990. The case of mechanics: One revolution or many? In Reappraisals of the

15 Many special thanks to Mary Domski who helped us sort out our thoughts and gavetangible input into the restructuring of this paper. She might well be listed as a co-author.Thanks also to two anonymous referees.


scientific revolution, ed. David C. Lindberg and Robert S. Westman, 493–528. Cam-bridge: Cambridge University Press.

———. 1985. The mechanical philosophy and its problems: Mechanical explanations,impenetrability, and perpetual motion. In Change and progress in modern science, ed. JosephC. Pitt, 9–84. Dordrecht: D. Reidel.

Garber, Daniel. 2002. Descartes, mechanics, and the mechanical philosophy. MidwestStudies in Philosophy 26: 185–204.

———. Forthcoming. Remarks on the pre-history of the mechanical philosophy. In Themechanization of natural philosophy, ed. Sophie Roux and Dan Garber. Dordrecht:Springer.

Garrison, James W. 1987. Newton and the relation of mathematics to natural philosophy.Journal of the History of Ideas 48: 609–27.

Guicciardini, Niccolò. 2009. Isaac Newton on mathematical certainty and method. Cambridge,MA: MIT Press.

Hattab, Helen. 2005. From mechanics to mechanism: The Quaestiones Mechanicae andDescartes’ physics. In The science of nature in the seventeenth century: Patterns of change in earlymodern natural philosophy, ed. Peter R. Anstey and John A. Schuster, 99–130. Dordrecht:Springer.

Henry, John. 2004. Metaphysics and the origins of modern science: Descartes and theimportance of laws of nature. Early Science and Medicine 9: 73–114.

Hepburn, Brian. 2010. Euler, vis viva, and equilibrium. Studies in History and Philosophy ofScience 41: 120–27.

———. 2007. Equilibrium and explanation in 18th century mechanics. PhD dissertation,University of Pittsburgh.

Hobbes, Thomas. [1655] 1966. De Corpore, The English works of Thomas Hobbes of Malmesbury,ed. Sir William Molesworth. Aalen: Scientia.

Janiak, Andrew. 2008. Newton as philosopher. Cambridge: Cambridge University Press.Machamer, Peter. 2009. Galileo Galilei. The Stanford encyclopedia of philosophy, ed. Edward N.

Zalta. Available online at http://plato.stanford.edu/archives/spr2005/entries/galileo.

———. 1998. Galileo’s machines, his mathematics, and his experiments. In The Cambridgecompanion to Galileo, ed. Peter Machamer, 43–79. Cambridge: Cambridge UniversityPress.

Machamer, Peter, and Andrea Woody. 1994. A model of intelligibility in science: UsingGalileo’s balance as a model for understanding the motion of bodies. Science andEducation 3: 215–44.

McGuire, James E. 1972. Boyle’s conception of nature. Journal of the History of Ideas 33:523–42.

Newton, Isaac. [1730] 1952. Opticks, or A treatise of the reflections, refractions, inflections and colorsof light. Based on the fourth edition of 1730. New York: Dover.

———. 1962. Unpublished scientific papers of Isaac Newton. Ed. and trans. A. R. Hall and M.B. Hall. Cambridge: Cambridge University Press.

———. [1728] 1969. A treatise of the system of the world, anonymous translation, believed tobe by Andrew Motte. 2nd ed. with an introduction by I. Bernard Cohen. London:Dawsons Pall Mall.


———. [1726] 1999. The Principia: Mathematical principles of natural philosophy. Trans. I. B.Cohen and A. Whitman. Berkeley: University of California Press.

———. [1661–65] 2003. Certain philosophical questions: Newton’s trinity notebook. Ed. J. E.McGuire and Martin Tamny. Cambridge: Cambridge University Press.

———. 2004. Newton: Philosophical writings. Ed. Andrew Janiak. Cambridge: CambridgeUniversity Press.

Sepkowski, David. 2005. Nominalism and constructivism in seventeenth century math-ematical philosophy. Historia Mathematica 32: 33–59.

Wallis, J., C. Wren, and C. Huygens. 1669. Laws of motion and collision papers. Trans-actions of the Royal Society 4: 307–38.

Wilson, Catherine. 2008. Epicureanism at the origins of modernity. Oxford: Oxford UniversityPress.

Ubaldo, Guido. [1581] 1969. Selections from The books of mechanics. In Mechanics in sixteenth-century Italy: Selections from Tartaglia, Benedetti, Guido Ubaldo, and Galileo, trans. and ed.Stillman Drake and I. E. Drabkin. Madison: University of Wisconsin Press.

Zilsel, Edgar. 1942. The genesis of the concept of physical law. Philosophical Review 51:245–79.


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