The third law in Newton’s Waste book (or, the road less taken to the second law)

Studies in History and Philosophy of Science Stud. Hist. Phil. Sci. 36 (2005) 43–60 www.elsevier.com/locate/shpsa

The third law in NewtonÕs Waste book (or, the road less taken to the second law) Doreen L. Fraser Department of History and Philosophy of Science, 1017 Cathedral of Learning, University of Pittsburgh, Pittsburgh, PA 15260, USA Received 2 April 2004

Abstract On the basis of evidence drawn from the Waste book, Westfall and Nicholas have argued that Newton arrived at his second law of motion by reflecting on the implications of the first law. I analyze another argument in the Waste book which reveals that Newton also arrived at the second law by another very different route. On this route, it is the consideration of the third law and the principle of conservation of motion—and not the first law—that prompts Newton to formulate the second law. The existence of these two routes is significant because each employs a distinct kind of reasoning about forces. Whereas the Nicholas–Westfall route via the principle of inertia bears the mark of DescartesÕs influence, the alternative route proceeds from the action–reaction principle, which is widely regarded as an original Newtonian contribution to mechanics. In the course of exploring this alternate route to the second law, the origins and justification of the third law are examined. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Issac Newton; Waste book; Third law of motion; Second law of motion; Richard Westfall; John Nicholas.

E-mail address: [email protected] (D.L. Fraser). 0039-3681/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.shpsa.2004.12.003

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1. Introduction As a historical document, the Waste book is of special interest because it contains some of NewtonÕs earliest thoughts on dynamics. The earliest dynamical entry is dated 20 January 1665, when Newton was still an undergraduate at Cambridge. As the title of this entry—ÔOf ReflectionsÕ—indicates, NewtonÕs primary objective in the Waste book was to develop a theory of collision. NewtonÕs contemporaries also recognized the need for an adequate collision theory. In 1668 the Royal Society invited submissions on impact and Wren, Huygens, and Wallis responded (Westfall, 1971, p. 203). In the course of formulating his own collision theory, Newton presented versions of all three of the laws of motion that were later to appear in the Principia. Consequently, the Waste book affords an opportunity to probe the origins of the laws, the relationships between them, and their development at the earliest stages of NewtonÕs thinking on the matter. At the heart of NewtonÕs new collision theory is a novel concept of force. This concept of force has a fundamental characteristic that he would subsequently describe in the second law of the Principia: The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which the force is impressed. (Newton, 1962, p. 13) The novel characteristic of the second law concept of force that is already contained in the Waste book is that force is proportional to the change in quantity of motion that it causes. This characteristic of the Principia concept of force was, of course, instrumental in the formulation of the gravitational theory set out in the Principia. The pressing historical question is what prompted Newton to formulate the concept of force expressed in the second law. The Waste book contains ample evidence that Newton took as the starting point for developing his own theory the theory of collisions that Descartes presented in his Principia philosophiae (see Herivel, 1965, pp. 42–53). It would have thus been natural for Newton to have adopted DescartesÕs concept of force as a prototype. Descartes employed the concept of the force that a body has for continuing to move in a straight line or for continuing to be at rest (Descartes, 1984, p. 242; Gabbey, 1971, p. 20). This Ôforce of motionÕ is proportional to the quantity of motion of the body rather than the change in quantity of motion. Alan Gabbey has dubbed the Cartesian force of motion concept the ‘‘Ôcontestant’’ view of forceÕ because Ô[i]nteractions between bodies were seen as contests between opposing forces, the larger forces being the winners, the smaller forces being the losersÕ (Gabbey, 1971, pp. 20, 16). For example, DescartesÕs second rule for collisions dictates that if a larger body B collides with a smaller body C travelling at the same speed in the opposite direction, then after the collision body B will continue travelling in the same direction and body C will be compelled to reverse its direction (Descartes, 1963, p. 197; Gabbey, 1971, p. 23). Body B ÔwinsÕ this contest—it persists in its motion in a straight line—because its quantity of motion, and hence its force of motion, is greater. The second law con-

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cept of force, in contrast, does not lend itself to this contest model of collision because the strength of a second law force is dependent on the final motion as well as the initial motion. That is, the force of motion concept is well-suited to functioning as a determinant of the outcome of a collision (that is, the final motions) because it is dependent only on the initial quantity of motion; the second law concept of force is ill-suited to performing this function because the outcome of the collision (that is, the final motion) is contained in the concept of the force as the cause of changes of motion. This is an illustration of how closely a concept of force is bound up with a particular theory of collision. Richard Westfall and John Nicholas have charted paths from the force of motion concept to the second law force concept. Although their accounts differ in many respects, both argue that the key consideration that induced Newton to modify the force of motion concept was the principle of inertia. For example, Westfall asserts: The unique position of the Waste book in the history of dynamics derives from its recognition that a dynamics built upon the principle of inertia demands a concept of force different from the prevailing one. (Westfall, 1971, p. 344) The more he worked on impact, however, the more the incompatibility of internal force [that is, force of motion] and the principle of inertia revealed itself. (Westfall, 1980, p. 146) One statement of the principle of inertia in the Waste book is the following: ÔAx: 100. [e]very thing doth naturally persevere in that state in which it is unlesse it bee interrupted by some externall causeÕ (Herivel, 1965, p. 153). This statement of the principle of inertia closely resembles that set out by Descartes in the first and second laws in Principia philosophiae. In broad outline, the idea propounded by Westfall and Nicholas is that Newton was prompted to modify the force of motion concept by the recognition that in a collision Ôthe force of a bodyÕs motion functions in relation to the second body as the ‘‘external cause’’ mentioned in [the statement of the principle of inertia] as the sole means that can alter its state of motion or restÕ (Westfall, 1971, p. 344). The underlying abstract conception that Ôcauses applied to changes of stateÕ goes back at least as far as Aristotle (Nicholas, 1978, p. 108). The Cartesian and Newtonian principles of inertia establish the Ôdynamical parityÕ of states of rest and uniform rectilinear motion (ibid., p. 108)1; put another way, a change from rest to uniform rectilinear motion, from uniform rectilinear motion to rest, or from one uniform rectilinear motion to another constitutes a change of state. It follows from the recognition that forces are the external causes of such changes of state—Ô[i]n a context set by the introductory

1 While this is certainly true of Newton, note that for Descartes there is a significant sense in which states of rest and motion are not on par because Descartes maintains that Ôrest is the opposite of motion, and nothing can by its nature tend towards its opposite, or towards its own destructionÕ (Descartes, 1984, p. 241). See Gabbey (1971, pp. 59–61) for further discussion of this point.

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assertion of the principle of inertia and by the problem of impactÕ—that Ôthere was apparently only one quantity that could be the measure of force—change in [quantity of] motionÕ (Westfall, 1971, p. 345). Implicit in this reasoning is the principle that the strength of a cause (that is, the force of the second body) is to be measured by the strength of its effect (that is, the change in quantity of motion of the first body). Nicholas concurs with this general account and adds an additional detail: that Newton did not leap directly from the force of motion concept to the full second law concept, but rather that there was an important intermediary that acted as a bridge between these two concepts. This intermediate concept is captured in what Nicholas dubs the ÔextremalÕ second law: the Ôprinciple . . . which equates the force of motion of a body and the force which sets that body from rest into that motion or, for that matter, reduces the body from that motion to restÕ (Nicholas, 1978, p. 119). This limited ÔextremalÕ force concept bridges the gap between the force of motion and second law concepts of force by being compatible with both. Nicholas argues that Ôthe transition . . . from force as quantity of motion to force as change in quantity of motion, had motivating it not simply a recognition of the consequences of the principle of inertia as such, but very specifically, a very singular Cartesian way of representing . . . ‘‘the dynamical parity’’ of rest and motionÕ (ibid., p. 110). The relevant peculiarity was DescartesÕs insistence that Ôfor example, the action needed to move a boat which is at rest in still water is no greater than that needed to stop it suddenly when it is movingÕ (Descartes, 1984, p. 234). In addition, as we shall see, NewtonÕs attention was trained on extremal cases in which bodies are brought to rest or moved from rest by his treatment of elastic collisions. The full second law concept of force emerged as a generalization of this extremal concept (Nicholas, 1978, p. 121). As Nicholas points out, this account of the development of the second law concept of force is a widely held historical generalization (ibid., pp. 108–109). It is a natural account of how NewtonÕs concept evolved from its antecedents that seems plausible independent of any documentary evidence. As it happens, support for this account can be found in a section of the Waste book. However, this section contains only one of the strands of argument in the Waste book that bear on the development of NewtonÕs concept of force. One of the other strands of argument reveals that Newton also forged another very different path to the second law. In this case, the catalyst was not the principle of inertia—the first law in the Principia—but the action–reaction principle—the third law. In the Waste book, the action–reaction principle is that at the moment that a body (a) collides with a body (b), Ô(a), presses [(b) as much] as (b) presses (a)Õ (Herivel, 1965, p. 142) and these pressures are exerted in opposite directions. Newton arrives at the second law concept of force by juxtaposing this action–reaction principle with his principle of conservation of motion. The principle of conservation of motion implies that the quantity of motion gained (or lost) by (a) in the collision between (a) and (b) is equal to the quantity of motion lost (or gained) by (b). That is, neglecting the directions of these quantities for the moment, these two principles imply that the change in quantity of motion of (a)

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equals the change in quantity of motion of (b) and that the magnitude of the pressure or force on (a) equals the magnitude of the pressure or force on (b). It then seems natural to identify the forces as the equal causes and the equal changes in quantity of motion as the equal effects generated. This step yields the second law concept of force: force is proportional to the change of motion that it causes. The Waste book provides evidence that Newton actually followed this chain of reasoning to arrive at the second law concept of force. This route to the second law via the third law and conservation of motion is of particular interest because it seems to reflect a distinctively Newtonian way of thinking about dynamics. It is widely agreed that the third law is one of NewtonÕs original contributions to dynamics.2 On the inertial route to the second law, DescartesÕs influence is instrumental. The principle of inertia and the peculiar presentation of the dynamical parity of rest of motion in terms of the activity required to generate these states are not DescartesÕs only contribution. In the Cartesian contest model of collisions, forces are relevant to the analysis of collisions only insofar as they function as the agents that cause the post-collision motions. This is because the ultimate aim of Cartesian collision theory is not to investigate the underlying forces, but only to determine the resultant states of motion of the bodies. Forces perform a similar function on the inertial route to the second law: forces are linked to changes of motion by the abstract principle that external causes occasion changes of state; the focus of the investigation of collision is not the forces—the changers of motion—but the changes of motion. In contrast, as we shall see, on the action–reaction route forces are not conceived as abstract causal agents, but as concrete physical entities and the primary focus is on understanding the forces rather than deriving the changes of motion. To this end, Newton introduced several physical models of the forces, including one based on springs. The evidence that Newton travelled the action–reaction path to the second law is found in one argument in one section of the Waste book. However, to fully understand NewtonÕs reasoning it is necessary to set this argument in context. The relevant broader context is the development of a theory of collision. More narrowly, an investigation of the origins of the third law and the status which Newton accorded it in the Waste book is also relevant. Finally, the other arguments that pertain to the evolution of the second law concept of force are relevant. These arguments will give us a sense of the kind of collision theory that Newton sought. They will also shed light on NewtonÕs views regarding the relationship between the inertial and action–reaction routes to the second law.

2

While the notions of action and reaction were employed and analyzed by NewtonÕs predecessors and contemporaries, it is widely agreed that Newton was the first to formulate a complete version of the third law (Dugas, 1958, p. 350; Home, 1968, pp. 42, 50–51; Westfall, 1971, pp. 348, 106, 113).

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2. The development of a collision theory Throughout the Waste book, Newton demonstrates his desire to obtain a completely general theory of collisions, one that could treat elastic as well as inelastic bodies and oblique as well as collinear paths. He takes the first3 steps to this general theory in the section that Herivel labels IIb. In this section, he considers the special case of perfectly inelastic collisions between bodies that Ôhave noe vis elastica to reflect the one from the otherÕ4 and thus Ôdoe not reflect one another but conjoyne at their meetingÕ (Herivel, 1965, p. 133).5 Following Descartes, Newton solves this collision problem by applying a principle of conservation of motion. However, his treatment diverges from DescartesÕs in a significant respect: Newton treats the quantity of motion as a directional quantity that is governed by the principle of conservation of motion. This is indicated by the employment of a minus sign in the statement of conservation of motion in Section 5. While Descartes does deal with directions of motions, his remarks suggest that the quantity of motion is not a directed quantity, and that, consequently, the directions of motions are not governed by his principle of conservation of motion, but treated separately (Descartes, 1984, pp. 242–243; Garber, 1992, pp. 208, 188–193). NewtonÕs modification served to rectify well known problems in DescartesÕs collision theory. It appears that NewtonÕs strategy in Section IIb is to tackle the simplest case of collision first. Perfectly inelastic collisions are the simplest case in which to apply the principle of conservation of motion because the two bodies travel with the same velocity after the collision, decreasing the number of variables. This would have been helpful if, as Herivel suggests (Herivel, 1965, p. 135 n. 7), Newton intended to use these calculations in an experimental test of the principle of conservation of motion. More significantly, perfectly inelastic collisions are particularly simple on NewtonÕs approach to collision problems because elastic forces are not involved and therefore a treatment of these is not required; put another way, since the bodies do not separate after the collision, the process of separation need not be studied. Westfall characterizes the cancelled and retained passages in IIb as Ôfalse startsÕ in which Newton Ôhad tried to come directly to grips with [the problem of impact]Õ before setting himself on the right

3 Here and elsewhere I follow HerivelÕs dating of passages from the Waste book. His temporal ordering of the passages discussed in this paper coincides with the order in which they appear in the Waste book (Herivel, 1965, p. 128). A date entered in the margin indicates that the first of these entries was made in January 1665 (in the modern calendar) (ibid., p. 129). These entries appear to be final drafts; less refined versions may have been composed slightly earlier (ibid.). 4 This quotation was taken from a passage that Newton cancelled. 5 Unless otherwise noted, all quotations are taken from the reproduction of the Waste book printed in Herivel (1965).

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course and laying down his definitions (Westfall, 1971, p. 343). But Section IIb should not be regarded as merely a failed attempt. It does not contain the more sophisticated dynamical analysis found in subsequent passages because it treats the special case which is so simple that this is not required. Westfall also overlooks the achievement of Section IIb: the correction of DescartesÕs principle of conservation of motion by treating motion as a directed quantity. This is an important prerequisite for the development of the collision theory presented in later sections. In the next section—Section IIc, entitled ÔDefinitionsÕ—Newton commences a more general treatment of collisions that covers all cases, including elastic collisions in which bodies separate after impact. In Definition 5 he defines reflection as follows: Ô[a] quantity is reflected when meeting with another quantity it looseth the determination of its motion by rebounding from itÕ (p. 138). He goes on to analyze the dynamical process underlying rebounding: As if the bodys a, b meete one another in the point c they are parted either by some springing motion in themselves or in the matter crouded betwixt them. and as the spring is more dull or vigorous/quick soe the bodys will bee reflected with with [sic] more or lesse force: as if it endeavour to get liberty to inlarge itself with as greate strength and vigor as the bodys a, b, pressed it together, then the motion of the body a from b will bee as greate after as before that reflection, but if the spring have but halfe that vigor, then the distance twixt a and b, at the minute after the reflection shall bee halfe as much as it was at the minute before the reflection. (Pp. 137– 138) The hypothetical Ôas ifÕ indicates that, for dynamical purposes, rebounding can be modelled by a spring, but that a springing motion may not be the true cause of rebounding. At the time Newton composed the Waste book, the modelling of phenomena using springs was in vogue. A prominent example is BoyleÕs theory of gases, which uses springs to model the behaviour of gases. Because Newton recorded notes on BoyleÕs New experiments physico-mechanicall, touching the spring of the air in the Trinity Notebook and refers to this work in the Questiones quaedam philosophicae (McGuire & Tamny, 1983, p. 24), we know that Newton was familiar with BoyleÕs spring model when he wrote this passage. In Definition 9 Newton formalizes this informal discussion of force by explicitly defining ÔforceÕ as Ôthe pressure or crouding of one body upon anotherÕ (p. 138). This definition is not abstract, but concrete and physical. In fact, Newton is so far from abstractly conceiving of forces as merely causes of changes of state that he does not even mention the effects of forces. However, the context of elastic collisions in which this definition is employed does furnish some insight into NewtonÕs conception of the causal function of forces. On NewtonÕs conception, the change of state caused by a force is not a change in motion directly; rather, ÔcroudingÕ or ÔpressureÕ causes the compression of elastic bodies. Again, this reflects the employment of a concrete, physical conception rather than an abstract, metaphysical one.

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3. The origins and justification of the third law Section IId, entitled ÔAxiomes and PropositionsÕ,6 contains more comprehensive dynamical analyses of collisions. In Axiom 7 Newton again begins with the simplest case, that in which the bodies do not rebound after impact: If two bodies [a and b moving?] against one another the same way towards O, (a) overtaking (b) none of their motion shall be lost, for (a), presses [(b) as much] as (b) presses (a) and therefore the motion of (b) shall increase [as much] as that of (a) decreaseth. (P. 142) Since force has been defined as ÔpressureÕ, the statement that Ô(a) presses [(b) as much] as (b) presses (a)Õ is an instance of the action–reaction principle for forces. This is the first appearance of the action–reaction principle in the Waste book. This passage is also noteworthy because this argument employs the second law concept of force, that forces cause changes in motion. This is also the first appearance of the second law concept of force in the Waste book. To understand properly the relationship between the second and third laws in this argument it is therefore necessary to investigate the origins and the status of the third law in the Waste book. While Newton does justify his inference that Ôthe motion of (b) shall increase [as much] as that of (a) decreasethÕ in the above passage by appealing to Axiom 4,7 he does not justify his assertion that Ô(a), presses [(b) as much] as (b) presses (a)Õ. The instances and versions of the action–reaction principle that appear in subsequent passages are not accompanied by any explicit statements about its origin or justification either. In Axiom 8 Newton again states without explanation or justification that Ôat their occursionÕ two bodies moving towards one another Ôpresse equally uppon one anotherÕ (p. 142). In the context of another analysis of the reflection of soft bodies in Axiom 9, he refers to Ôtheire pression one upon the otherÕ (ibid.). In the same axiom he justifies his conclusion about the motions of hard bodies with the assertion that Ôthere cannot bee/succeede diverse degrees of pressure twixt two bodies in one momentÕ (p. 143). A version of the third law is again baldly stated in Axiom 121 of Section IIe: Ô[i]f 2 bodys p and r meet the one the other, the resistance in both is the same for soe much as p presseth upon r so much r presseth on pÕ (p. 159). In the order in which they were entered in the Waste book, this axiom is followed by Axiom 119: If r [Fig. 5] presse p towards w then p presseth r towards v. Tis evident without explication. (Ibid.) In the diagram, points w and v and the centres of bodies p and r are collinear and w and v are on opposite sides of the bodies. Home has interpreted these axioms as indicating that when Newton composed the Waste book he believed that the third law is true a priori (p. 49). However, there are good reasons to doubt this interpretation. 6 7

For the sake of brevity, I will refer to the ÔAxiomes and PropositionsÕ as simply ÔAxiomsÕ. This appeal is contained in a footnote in HerivelÕs reproduction.

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First, considering these two axioms in isolation, the only aspect of the third law that Newton characterizes as Ôevident without explicationÕ is that the pressures are in opposite directions; when two bodies are pressing against one another this does seem intuitively obvious. What is less obvious, and is essential content of the third law, is that these pressures are of equal magnitude. Second, in the context of the Waste book as a whole, the version of the third law stated in axioms 121 and 119 is part of the analysis of collision that is gradually developed in Sections IIb through IIe. Therefore, it seems more likely that the justification for the third law is in some way related to this analysis. In all these cases, Newton appeals to the action–reaction principle as an unjustified premise to support his dynamical analysis of collisions. This context contains some clues that point to the origins and justification of the action– reaction principle. Definition 5, which is prior to all these appeals to the action–reaction principle, stipulates that reflections—collisions in which the bodies rebound—are to be modelled by springs. Taking this suggestion literally, we can imagine that there is a spring between the colliding bodies that gets compressed as the bodies approach one another. At the moment at which the bodies collide the spring is fully compressed. When the spring Ôendeavours to get liberty to inlarge it selfeÕ (p. 138), it presses against the bodies equally and in opposite directions. Thus, on the spring model of collisions the action–reaction principle is true. Given that the action–reaction principle first appears in the context of a dynamical analysis of collisions, it seems reasonable to speculate that the principle originated in the spring model and that Newton took the spring model to be the justification for the principle. Questiones quaedam philosophicae, a series of entries contained in another notebook that Newton kept before and during the period in which he entered ÔOf ReflectionsÕ into the Waste book, lends some support to this hypothesis. In a section entitled ÔOf Rarity and Density. Rarefaction and CondensationÕ Newton describes an experiment designed to determine which of two bodies is more dense. McGuire and Tamny date this section to early 1664, prior to the composition of ÔOn ReflectionsÕ in the Waste book (McGuire & Tamny, 1983, p. 12).8 The experiment involves suspending two bodies (d and e) from strings and separating them by a compressed spring. When the spring is allowed to expand it Ôshall cast both ye body d and e from it and they receve alike swiftnes from ye spring if there be ye same quantity of body in bothÕ (ibid., p. 358). That is, applying Axioms 3 and 4 of the Waste book, in this case the spring presses on the bodies equally because the pressures generate equal motions. When the quantities of the bodies differ, Ôye body bo (being fastened to ye spring) will move towards ye body wch has less body in itÕ (ibid.). Newton gives no further explanation, but this result is consistent with the spring exerting equal forces on the two bodies and thus, by Axioms 3 and 4 of the Waste book, imparting a greater speed to the lesser body and causing the spring to move in that direction as it expands. This section of the

8

See n. 3 above.

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Questiones makes it seem plausible that at the time Newton wrote the Waste book he knew that a decompressing spring exerts equal pressures on adjacent bodies. In addition, Newton was familiar with BoyleÕs New experiments physico-mechanicall, touching the spring of the air (McGuire & Tamny, 1983, p. 24). Boyle does not state that both sides of a spring exert equal pressures here, but this work would have provided Newton with some background on the pressures associated with springs (see, for example, Boyle, 1965, pp. 11–12). Some support for the hypothesis that the consideration of elasticity led Newton to his third law can also be found in a preliminary draft of the definitions and laws of the lectures De motu (Herivel, 1965, MS Xa). According to Herivel, this is the first appearance of the third law after the Waste book (ibid., p. 31): Law 3. As much as any body acts on another so much does it experience in reaction. Whatever presses or pulls another thing by this equally is pressed or pulled. If a bladder full of air presses another equal to itself both yield equally inwards. (Ibid., pp. 312–313) Like springs, air bladders are elastic. A collision between air bladders would be an elastic collision between soft bodies. This case of collisions is special because springiness—the springiness of the bodies—is genuinely the cause of the rebounding of the bodies after impact. Newton describes this case in the first part of Axiom 9 in the Waste book: Ôsuppose the bodies to have a springing or elastic force soe that meeting one another they will relent and be pressed into a sphaeroidicall figure, and in that moment in which there is a period put to theire motion towards one another theire figure will be most sphaeroidall and their pression towards one another is at the greatestÕ (p. 142). That the air bladders are deformed by equal amounts indicates that they exert equal pressures upon one another. Strictly, deformations can only be observed in deformable bodies, so this conclusion only applies to deformable bodies. An assumption like that of the spring model—that the dynamics of all collisions are Ôas ifÕ they had been caused by elastic forces—is needed to extend this conclusion to perfectly hard bodies. The fact that the De motu version of the third law fits so seamlessly with the Waste book analysis of elasticity and forces as ÔpressuresÕ or ÔcroudingsÕ suggests that this later evidence is indeed relevant to the origin and status of the action– reaction principle in the Waste book. The passage from the lectures De motu is also interesting because it suggests that empirical factors may have played a direct role in the discovery of the third law: depressions in bladders of air are observable. Because the degree of elasticity9 of the bladder is a factor in how much it depresses—as Newton notes, the bladders must be equal to yield equally inwards—it would be difficult to test the third law by measuring depressions in air bladders. However, the observation that approximately equal bladders yield approximately equally inwards could

9 In modern terms, the Ôdegree of elasticityÕ corresponds to the spring constant, k, that appears in HookeÕs Law, F =
kx.

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have set Newton on the track to the third law. In any case, the evidence suggests that the third law originated in the empirical or theoretical investigation of elasticity and that, for Newton, its justification lay in his application of an elastic model of collisions. Westfall advances a different hypothesis about the origin of the third law: Except for the special case in which p and r have equal motions, the conclusion that they press each other equally had seemed obviously false to the great majority of those who before 1665 had attempted to analyze the force of percussion. The Waste book suggests that Newton came to deny the intuitively obvious by recognising that for every impact there is a frame of reference, that of the common centre of gravity, in which the two bodies do have equal forces. That is, his treatment of impact depended on his accepting a relativity of motion in terms of which the idea of an absolute force of a bodyÕs motion is meaningless. (Westfall, 1971, p. 348) On my reading of the Waste book, the genesis of the third law is entirely independent of the centre of motion analysis. That this analysis occurs after the introduction of the third law suggests that this analysis was not instrumental in the formulation of the third law. However, since there are indications that the Waste book was not composed extemporaneously (Herivel, 1965, p. 141), this is merely suggested. Another shortcoming of WestfallÕs interpretation is that there is no evidence that Newton is employing the force of motion concept in the passages in which he presents the third law. In fact, he seems to be using either the concrete conception of force set out in Definition 9 or the second law concept of force. WestfallÕs account is, of course, inapplicable to the second law concept of force because force in this sense is not relative to frames of reference. Similarly, WestfallÕs account is not applicable to the concrete conception of force defined in Definition 9 because, presumably, Newton does not take the magnitudes of the physical ÔpressuresÕ or ÔcroudingsÕ in collisions to be relative to any frame of reference. For these reasons, WestfallÕs explanation of the genesis of the third law seems inadequate. In fact, as we shall see, the third law influences NewtonÕs views on the concept of force, and not the other way round.

4. The action–reaction route to the second law As we have seen, in the Waste book the third law of motion is an unjustified premise in arguments that pertain to the relationship between the forces that act in collisions and the resultant motions of the bodies. These arguments shed light on the evolution of the second law concept of force. Again, a preliminary analysis of the relationship between forces and motions can be found in Definition 5, which was quoted above. The upshot of the reasoning in this passage is that force is proportional to the motion generated in a body set into motion from rest or the motion destroyed in a body brought to rest. This is part way to what Nicholas calls the ÔextremalÕ second law: the Ôprinciple . . .

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which equates the force of motion of a body and the force which sets that body from rest into that motion or, for that matter, reduces the body from that motion to restÕ (Nicholas, 1978, p. 119). In Definition 5 the extremal force concept is only partially realized because Newton has not introduced the force of motion concept. At this point, he is working with the physical definition of force as ÔpressureÕ or ÔcroudingÕ. It is force in this sense that is held to be proportional to the motion destroyed or generated. The first application of the full second law concept of force occurs in Axioms 7 and 8, axioms that were examined in the previous section because they also contain the first instances of the third law. In Axiom 8 Newton argues that Ô[i]f two quantities (a and b) move towards one another and meete in O, [t]hen the difference of theire motion shall not bee lost nor loose its determinationÕ (p. 142). Since the bodies move in opposite directions, the addition of the vector motions is equivalent to the difference of the scalar motions; hence, Newton is arguing that the total motion is conserved in this special case. His argument for this conclusion is that Ôat their occursion they presse equally uppon one another and therefore one must loose noe more motion than the other dothÕ (ibid.). That is, each body loses the same quantity of motion because the forces that cause this loss of motion are of equal magnitude. This reasoning can be reconstructed algebraically as follows: paf
pbf ¼ paf þ pbf ¼ ðpai þ DpÞ þ ðpbi þ ð
DpÞÞ ¼ pai þ pbi ¼ pbi
pai ; where there is a minus sign in front of the second Dp term because the forces, and therefore the changes in motion, are in opposite directions, that is, Fba =
Fab implies Dpa =
Dpb. This argument clearly relies on the full second law concept of force, according to which force is proportional to change in quantity of motion. The force exerted on each of the bodies is proportional to the difference in each bodyÕs initial and final quantities of motion. Since both of these quantities of motion are non-zero (that is, the body is not at rest), this is not merely the extremal second law force concept, but the full second law concept. Put another way, it is obvious that Newton is not employing the extremal force concept here because only one force affects the motion of body a; on an extremal analysis, there would be two forces—one that destroys the initial motion of a and another that generates the resultant motion of a. The argument presented in Axioms 7 and 8 also reveals a path that lead Newton to the second law concept of force. The argument has the following simple structure: the third law in combination with the second law concept of force yields the conservation of motion. This is a case in which Newton certainly would have known the correct conclusion of the argument—that motion is conserved—before he formulated it. Newton was familiar with DescartesÕs account of collisions in Principia philosophiae, in which a principle of conservation of motion played a prominent role. As

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we noted, Newton had corrected DescartesÕs conservation principle in an earlier section of the Waste book. We have also seen that he regarded the third law as an established fact. Consequently, it seems likely that he tinkered with the force concept to obtain the desired conclusion that motion is conserved from the premise of the third law.10 It is not a large leap to the second law from this starting point. Consideration of the conservation of motion would have trained his attention on the differences between the total initial and final quantities of motion. When the fact that the forces are of equal magnitude and in opposite directions is added, it is a small step to force being proportional to change in motion via the principle that equal changes of motion in opposite directions cancel, conserving the total quantity of motion. The result must be the full second law concept of motion; the extremal second law will not do because the quantities of significance for the conservation of motion are the (in general) non-zero initial and final motions. There is some evidence for this interpretation in a marginal note inscribed next to Definition 5: noe motion is lost in reflection. For then circular motion being made by continuall reflection would decay. (P. 139 n. f) This note accompanies the discussion of the forces that act in collisions and the resultant motions of bodies. The implication is not only that Newton knew that motion had to be conserved, but that he sought to reconcile this principle with this dynamical analysis of reflections. On this path to the second law, there is no transition from the force of motion concept to the second law concept; rather, Newton starts out with the concrete, physical conception of force set out in Definition 9 and then imbues it with an additional abstract significance to arrive at the second law concept of force. Indeed, it is critical that Newton start out with a concrete physical concept of force rather than the force of motion concept. The third law governs force or pressure, so some concept of force or pressure is required. Of course, the physical conception of force as pressure is sufficient for this purpose. However, the force of motion concept could not serve this purpose because the only case in which it would be true that the forces of motion of bodies are of equal magnitude upon impact is when the bodies have equal motions; in all other cases the third law would be false. Put another way, had Newton started with the force of motion concept, he would have landed himself in WestfallÕs dilemma about how the forces of motion in a collision could be considered equal; WestfallÕs dilemma never arises for Newton because he starts out with a concrete, physical conception of force.

10

The use of the action–reaction principle embodied in the third law originates with Newton, but others had appealed to conservation of motion to define force. Gabbey maintains that ÔDescartes quantifies the resisting force [of a body at rest] by setting it equal to the total change in motion which it would receive if the collision were seen simply as an occasion for a redistribution of motion according to the conservation law, without recognizing in addition that a contest of forces is involvedÕ (Gabbey, 1971, p. 25).

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5. Another argument for the conservation of motion Axioms 7 and 8 are followed by a much different argument for the conservation of motion in Axioms 9 and 10. This argument is only applicable to perfectly hard and perfectly elastic bodies. Prima facie, it seems a bit surprising that, after presenting a completely general demonstration of the conservation of motion in Axioms 7 and 8, Newton would proceed to introduce a different argument that only applies to certain types of bodies. In Axiom 9, Newton considers the special case of Ôtwo equall and equally swift bodysÕ (p. 142). In this special case, the conservation of motion implies that the bodies Ômove as swiftly frome one another after the reflection as they did to one another before itÕ (ibid.). He considers two types of bodies: Ôsphaericall bodysÕ that Ôhave a springing or elastic forceÕ and Ôsphaericall and absolutely sollidÕ bodies (pp. 142–143). In the former case, the argument begins with the assertion that Ôin the moment in which there is a period put to theire motion towards one another theire figure will be most sphaeroidical and theire pression one upon the other is at the greatestÕ (p. 142). Newton then further restricts his attention to the perfectly elastic case in which the Ôendeavour to restore theire sphaericall figure bee as much vigorous and forcible as theire pressure upon one another was to destroy itÕ (ibid.). The conclusion drawn is that the bodies will Ôgaine as much motion from one another after their parting as they had towards one another before theire reflectionÕ (ibid.). Newton is implicitly employing the (partial) extremal concept of force here: the conclusion follows from the premises in virtue of the fact that each of the extremal forces that reduces the bodies to rest is equal in magnitude to the extremal force that generated the motion from rest. The argument for the second case—that of Ôsphaericall and absolutely sollidÕ bodies—parallels the argument for the first case, with a small twist. In this case he appeals to the action–reaction principle—Ôthere cannot bee/succeede divers degrees of pressure twixt two bodies in one momentÕ (p. 143). However, unlike in Axioms 7 and 8, Newton does not move from this premise directly to the conclusion that motion is conserved. Instead, he uses it to justify a minor assumption and arrives at his ultimate conclusion via the (partial) extremal concept of force: Ôsoe much force as deprived the bodys of theire motion towards one another soe much doth now urge them from one another and therefore they shall move from one another as much as they did towards one another before theire reflectionÕ (ibid.). Although, prima facie, it does seem a bit surprising that Newton would introduce this restricted argument for conservation of motion at this point in the Waste book, the passage suggests his motivation: he was seeking a deeper understanding of the forces acting in collisions. Axioms 7 and 8 do not provide a detailed description of the particular forces that oppose one another at each stage of the collision process. Axiom 9 appears to be an attempt to fill in these details. There is further evidence that Newton desires a deeper account of the forces acting in collisions in Section IIe, discussed below. At the same time, it would certainly not have escaped Newton that the argument in Axioms 9 and 10 is not entirely successful because it is restricted to the

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case of perfectly elastic collisions. For one thing, he explicitly limits his discussion to this case in Axiom 9 and he is certainly aware of the existence of inelastic collisions because he considers them in Definition 5. Also, his marginal note to Definition 5, quoted above, reads like a worry that motion is not conserved in the inelastic collisions that he describes. It is also telling that the argument for the conservation of motion given in Axioms 7 and 8 resurfaces in Section IIe, but the argument in Axiom 9 does not. However, Newton certainly does not draw the moral that the (partial) extremal concept of force employed in Axiom 9 is flawed: the extremal concept of force, as well as the force of motion concept, continues to be employed in Section IIe. Axiom 10 extends the argument given in Axiom 9 to the more general case of Ôunequall and unequally moved bodiesÕ that are perfectly hard and elastic. The claim defended in Axiom 10 is that Ô[t]here is the same reason when unequall and unequally moved bodys reflect, they should separate from one another with as much motion as they came togetherÕ (p. 143). Herivel finds this axiom problematic: The result given in this number is true for a perfectly elastic collision, as follows, for example, either from NewtonÕs law of relative velocities with e = 1, or by simultaneous application of conservation of momentum and conservation of energy. But it is not clear how Newton supposed that the general result followed from the very special case considered in the previous number . . . . Notice that the term ÔmotionÕ is here employed in the sense of movement rather than in the usual sense of quantity of motion (=momentum). (Ibid., p. 151 n. 5) There is a problem with this axiom, but it lies elsewhere. NewtonÕs phrase Ôthere is the same reasonÕ has the effect of restricting this axiom to perfectly elastic collisions because these are the only cases treated in the argument in Axiom 9. Also, ÔmotionÕ should be interpreted in the technical sense of quantity of motion here. This is a charitable reading because it makes better sense of the application of the argument in Axiom 9 to the more general case treated in Axiom 10. It is also a reasonable interpretation because in the axioms preceding Axiom 9 Newton argues for the conservation of motion; because in the special case in which the bodies are equal and have equal speeds the principle of conservation of motion dictates that their resultant speeds will be equal, this seems to be a continuation of that discussion. The problem arises when the argument for hard bodies is applied to the case of equal and unequally moved bodies. In the course of this argument Newton appeals to the principle that Ôthere cannot bee/succeede divers degrees of pressure twixt two bodies in one momentÕ (p. 143). If this is interpreted as an expression of the third law principle that the pressures that the bodies exert upon one another are equal, then there is a problem reconciling this principle with the extremal forces of the bodies. In the case of equal and unequally moved bodies, the magnitudes of the initial motions of the bodies are not necessarily equal; consequently, in general unequal degrees of pressure are required to destroy the motions of the bodies. Since the total motion is non-zero, and equal forces acting in opposite directions would generate

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zero total motion, the conservation of motion is strictly inconsistent with the action– reaction principle on the extremal analysis of force. Several conclusions could be drawn from this contradiction. A possible explanation is that Newton did not regard the statement in Axiom 9 as a complete version of the third law. It is possible that in Axiom 9 he is using ÔpressureÕ in the sense that there exists a single pressure between two bodies rather than the distinct pressures that each exerts on the other (for a total of two pressures), which would dissolve the problem. However, this would be inconsistent with the principle to which he appeals in Axioms 7 and 8 and with his use of the term ÔpressureÕ elsewhere in the Waste book. It is more likely, however, that since he did not apply the argument from Axiom 9 to the case in Axiom 10 in detail, he was unaware of this problem. As a consequence, he missed an opportunity to discover that the extremal concept of force is incompatible with the third law and the principle of conservation of motion.

6. Conclusion In the passages that have been examined to this point, no direct appeal has been made to the principle of inertia. The principle is stated in Axioms 1 and 2 of Section IId, but it is not cited in either Definition 5 or Axioms 7–10. In Section IIe Newton returns to the principle of inertia: the section begins with a recapitulation of Axioms 1 and 2 and continues with an investigation of their implications for collision theory. This is the section that Westfall and Nicholas focus on when arguing that the principle of inertia played a pivotal role in the genesis of the second law concept of force. In the very first axiom of Section IIe Newton introduces a contest model of collisions in which the ÔadvantageÕ that one body has over another is a function of their respective powers to persevere in their states and in which the bodies Ômutually hinder their perseverance in their statesÕ (pp. 153–154). The force of motion enters as the ÔpowerÕ of a body to Ôpersevere in its stateÕ. However, there is a crucial difference between NewtonÕs use of the contest model and DescartesÕs. While Descartes employs the contest model to determine the outcomes of collisions, Newton only considers whether or not the collision of two bodies will reduce both to rest. This is because Newton has already developed a theory of collisions that governs the resultant motions of bodies: the theory adumbrated in Axioms 7 and 8 of IId. Newton demonstrates his confidence in this account by proceeding to analyze more complex collision scenarios in the latter part of Section IId. But if Newton does not employ the contest model to derive the resultant motions of colliding bodies, then why does he introduce the contest model? NewtonÕs aim in Section IIe seems to be the attainment of a deeper understanding of the forces exerted in collisions. As we have seen, the presence of Axioms 9 and 10 signals a similar desire, but the argument contained in these axioms is not entirely satisfactory because it is restricted to the special case of perfectly elastic collisions. Because the spring model is only hypothetical—the forces behave as if they were due to springs—the true source of the forces remains a mystery. By employing the contest

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model and applying the associated principle of inertia, Newton can fill this lacuna: in Axiom 106, he identifies the source of the ÔpressureÕ on a body a in a collision between bodies a and b as Ôthe power which b hath to persever in its velocity or stateÕ, which Ôis usually called the force of the body bÕ (p. 156). The implication is that, on the inertial route, the Cartesian principle of inertia, the Cartesian representation of the dynamical parity of rest and motion, and an abstract conception of forces as causal agents were not the only factors that pushed Newton to the second law. NewtonÕs interest in the forces themselves was also an important factor. Perhaps Nicholas is recognizing the importance of this same factor but describing it in a slightly different way when he says that ÔNewtonÕs defection from Mechanical Philosophy strictly conceivedÕ in his Ônon-Mechanical commitment to elastic forces in matterÕ is one of the ÔpressuresÕ that Ôforced him towards an enlarged force theory from the Inertial Contest ViewÕ (Nicholas, 1978, pp. 127–128). There is an inconsistency in Section IIe. In Axiom 105 Newton considers the case of equal and equivelox bodies and remarks that Ôsince these bodys have noe advantage the one over the other the hindrance on both parts will be equallÕ (p. 156). The implication is that if the bodies had unequal motions—and therefore unequal forces of motion—then the hindrances on both parts would not be equal. This contradicts Axiom 121, which states that Ô[i]f 2 bodies p and r meet the one the other, the resistance in both is the sameÕ (p. 159). This is the same contradiction entailed by the force of motion concept and the action–reaction principle that Westfall raised (see my Section 3 above). Section IIe does not contain any indication that Newton convinced himself that this contradiction is merely apparent by appealing to the relativity of motion. The more probable explanation of the presence of this inconsistency in IIe is that Newton was simply not aware of it. Throughout the Waste book, Newton keeps the analysis of collisions based on the principle of inertia separate from that based on the action–reaction principle. In the sections prior to Section IIe, the principle of inertia is stated, but not invoked in the analysis of collision. In Section IIe, Newton employs an inertial analysis up to and including Axiom 118. The action–reaction principle does not appear in any of these axioms. In the final four axioms of Section IIe he recapitulates the essence of the argument for the conservation of motion given in Axioms 7 and 8 and then continues in this vein. The one exception to this pattern of dissociation is Axioms 9 and 10, where Newton combines the extremal concept of force and the action–reaction principle. However, he does not discover the inconsistency lurking in these axioms because he does not delve into the details of the problematic case in which the colliding bodies have unequal motions. The fact that the principle of inertia and the action–reaction principle are for the most part treated separately makes it seem plausible that Newton was unaware of the inconsistency in IIe. NewtonÕs compartmentalization of the two aspects of his collision theory has several further implications. In WestfallÕs view, Ô[t]he Waste book suggests that rather than seeing the one concept of force [that is, the force of motion concept] as the denial of the other [that is, the second law force concept], Newton sought to reconcile them in a unitary dynamicsÕ (Westfall, 1971, pp. 346–347). The inconsistency in IIe suggests that, on the contrary, Newton was far from integrating the components of

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his theory connected with the principle of inertia and those connected with the third law. As a result, Newton does not attain a unitary dynamics in the Waste book. Since Newton considered the implications of the principle of inertia in isolation from the analysis of collision involving the action–reaction principle and the conservation of motion, it is plausible that he did in fact arrive at the second law by two different routes. In Sections IIc and IId, the second law concept of force evolves out of the physical conception of force as ÔpressureÕ or ÔcroudingÕ. The second law force concept is the bridge in the argument from the premise of the action–reaction principle to the conclusion that motion is conserved. In Section IIe, the second law concept of force emerges from an inertial analysis of collisions. Although Newton treated these two routes to the second law separately, they are in fact united by a common thread in NewtonÕs thought: both are products of his desire for a more fundamental dynamical understanding of the process of collision.

Acknowledgements For helpful discussions and comments, I thank John Nicholas, Peter Machamer, participants in the Newton seminar lead by Ted McGuire and Bernard Goldstein at the University of Pittsburgh in Spring 2002, and especially Ted McGuire.

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