Abstract
As previously discussed, Newton singled out Descartes’ espousal of an Aristotelian/Scholastic theory of place and motion as the primary target of his anti-relationalist assault. As mentioned in the Introduction, there would seem to exist two general strategies of countering Newton’s allegations: (1) accept the contention that a fixed reference frame is incompatible with the tenets of spacetime relationalism (and attempt to provide an alternative foundation for Cartesian dynamics based on one of the laws of nature, such as the third law of motion); or (2) on the contrary, insist that such frames can be successfully and coherently established in a continuously changing matter-filled universe. This chapter, like the dynamics-based “rest” force procedure reviewed in chapter 6, will take the former option. That is, we shall investigate, once again, the feasibility of providing a framework for a consistent set of material-interaction laws without recourse to permanent reference frames. In this regard, Huygens’ concept of a center-of-mass reference frame will prove invaluable, for it constitutes the a further means of preserving the basic content of Descartes’ collision rules without jeopardizing the Cartesian conservation law (i.e., quantity of motion). In short, on this version of Cartesian science, we can one again retain quantity of motion and Descartes’ theory of impact (from his third natural law). Unlike the method developed in chapter 6, however, Huygens’ approach to the problem attempts to utilize the predictions of just one of the seven collision rules, namely, the first rule, as a means of reconstructing a relationally compatible version of all bodily impact.
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Endnotes
Huygens’ relationalism seems to have remained largely (R1), as is evident in the following passage: “there is nothing to distinguish straight motion from rest, and both one and the other are relative….” Huygens (1950, vol. 16, 183), trans. by A. Elzinga (1972, 96). But, the center-of-mass reference frame seems to usher him into the realm of (R1*) relationalism, as will be discussed below. See, also, endnote 2.
For a discussion of Huygens’ role in the history of relational spacetime theories, see Stein (1977). Stein suggests that Huygens’ treatment of rotation favors a much weaker commitment to relationalism than has normally been acknowledged by commentators: that is, using our classification scheme, Stein holds that Huygens’ theory of rotation would fall under (R2), or (R1*), and not (R 1). However, I concur with Earman (1989, 67–71), who regards Huygens’ brief statements on rotation as too vague and underdeveloped to constitute a clear instance of (R2)/(R1*) relationalism. For various aspects of Huygens’ natural philosophy, see Elzinga (1958) and Dugas (1958, chap. 10 ).
Huygens (1950, vol. 16, 186), trans. by Westfall (1971, 149). Unless otherwise noted, all following translations will be based on Westfall, but checked against the Oeuvres Complètes.
C. Huygens, “De Motu Corporum ex Percussione,” in Oeuvres Completes, vol. 16 (1929, written 1656), 31; trans. R. J. Blackwell in Isis, 68, 574–597 (1977).
I owe many of the details of the following discussion to Barbour (1989, 473–478).
In this context, I am using “invariant” to signify a quantity of a physical law that retains the same value over time. Recalling our discussion of differential geometry from chapter 2, ‘invariant’ can also be used to describe a quantity of a physical theory that retains the same value in all the reference frames formed by an admissible coordinate transformation (i.e., the covariance group of the theory).
In addition, it is important to note the possible influence of the “virtual velocity” tradition in Huygens’ decision to use of the center-of-mass frame to conserve quantity of motion. On this theory, the equilibrium of suspended weights is caused by the balancing-out of their products of downward-directed speed, possibly as some sort of potential speed equivalent to weight, and distance from the center point. This view is nicely presented in one of Galileo’s lesser known works: “Two weights equall in absolute Gravity, being put into a Ballance of equall Arms, they stand in Equilibrium, neither one going down, nor the other up: because the equality of the Distances of both, from the Centre on which the Ballance is supported, and about which it moves, causeth that those weights, the said Ballance moving, shall in the same Time move equall Spaces, that is, shall move with equall Velocity, so that there is no reason for which this Weight should descend more than that, or that more than this; and therefore they make an Equilibrium, and their Moments continue of semblable and equall Vertue.” Discourse on Bodies in Water, trans. T. Salusbury, ed. S. Drake (Urbana: University of Illinois Press, 1960) 6–7. Rohault’s important Cartesian text also presents this notion. See, A System of Natural Philosophy, vol. 1 (written 1671), trans. J. Clarke and S. Clarke (1723), (New York: Johnson Reprint Corp., 1969), 43–44. However, it is not known if the virtual velocity concept actually played a role in Huygens’ approach to quantities conserved in collisions, a set of circumstances somewhat different than wieghts held in equilibrium. For more on virtual velocities, see chapter 5.
This proof appeared in 1656, four years after his initial discovery of the center-of-mass method for preserving quantity of motion. See Westfall 1971, 148–159.
C. Huygens, Oeuvres Completes, vol. 16, 233; trans. J. B. Barbour, 1989, ibid., 474.
This conceptualization of impact is brought out in precise detail by Rohault, an influential Cartesian of the later seventeenth century. With respect to a body whose direction is reversed after impact, he states: “Because the Notion we have of reflected Motion is not different from the Notion we have of direct Motion, we ought not to think that these Motions are contrary to each other, but that the one is only a Continuation of the other, and consequently, that there is not any Moment of Rest in the point of Reflection,…. Besides, if a body which was in Motion, comes to be but one Moment at Rest, it will have wholly changed its manner of existing into the contrary, in which there will be as much Reason for its continuing, as if it had been at Rest a whole Age;….” To these views, Smith’s footnotes offer a Newtonian reply: “There may be a Moment of Rest, in the point of Reflection; because the reflected Motion, is not a Continuation of the Direct; but a new Motion impressed by a new Force, viz. the Force of Elasticity.” J. Rohault 1969, 81.
Despite Newton’s third law and his acknowledgement of elastic force, it must be admitted that his analysis of imperfectly elastic collisions (where the bodies do not recover their initial speeds) displays a strong kinematical bias, since he merely compares the pre-impact and post-impact velocities to determine their relative ratio. If the bodies are perfectly elastic or perfectly inelastic, this ratio will equal 1 and 0 respectively (i.e., in the latter case, the bodies become attached during impact). For all bodies with an internal constitution in-between these values, a factor k (0 s k s 1) is needed, the “coefficient of restitution,” to convert the post-impact velocity to the value that obtained before the collision. Hence, Newton’s measurement of the elastic force of a material body, an event allegedly involving forces, is conceived from the kinematic standpoint of a factor required to restore perfectly elastic collisions—and hence satisfy his momentum conservation law. See Newton, 1962b, ibid., 25.
Snelders (1980, 120). Although it is beyond the scope of this paper to cover this issue in depth, see also, Westman (1980) and Shapiro (1980) for more details of Huygens’ physics and its relation to Cartesianism.
See, e.g., Friedman (1983, chapters 2 and 3).
Sklar (1974, 200). Mach’s theory is presented in (1942, 280–286).
See Earman (1989, chap. 4), for an analysis of the various attempts by relationalists, including Huygens, to explicate circular motion and its dynamic effects; and chapter 6, for an examination of the viability of (R2), to be discussed below. Earman gives a nice exposition of how Huygens, at least in his later years, tried to preserve (R1) and still account for rotation (see endnote 2). Furthermore, there is the related problem of whether (R1) is implicitly contained in (R2), but there does not appear to be any overt reason for accepting this entailment relationship.
Mach seems to suggest that the Newtonian predictions are simply wrong; i.e., that the rotation of the stars relative to the fixed bucket would likewise result in the centrifugal effects being experienced only by the latter. (see, Mach 1942, 283–284) However, if Mach is correct, so that local inertial effects depend on distant massive bodies, then this fact should be in principle verifiable through available experimental or observational evidence (possibly by studying large stellar bodies; see, Sklar 1974, 201)—but, Mach left no hints as to how his theory could be verified by anything less than rotating the entire set of fixed stars.
Westfall (1971, 156–158); Gabbey (1980, 178–181). Huygens’ treatment of centrifugal force can be expressed in the modem formulation, mv2/r, where r is the radius of the circle (Westfall 1971, 170), although Huygens did not carry his discovery that crucial step further, as Newton did, and postulate the force of gravity required to offset this force—the influence of, or devotion to, the Cartesian vortex was crucial, here—see, e.g., Hall (1976).
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Slowik, E. (2002). Constructing a Cartesian Dynamics without “Fixed” Reference Frames: Collisions in the Center-of-Mass Frame. In: Cartesian Spacetime. International Archives of the History of Ideas / Archives Internationales d’Histoire des Idées, vol 181. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0975-0_9
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