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ABSTRACT
This article situates Descartes’ physical thinking within the nexus of machine science, which rests upon different foundational piers than regular classical mechanics of a Newtonian stripe. In particular, connected cyclic processes of the sort encountered in clockwork mechanisms (and Descartes’ own vortices) become central rather than impactive collisions of any kind. Such a placement supplies a more sympathetic understanding of many of his most notorious claims: conservation of ‘quantity of motion’, relationalism with respect to space, relative rest as an explanation of particle cohesion, etc. Of course, the resulting system of ideas is not perfect, but the mistakes are more subtle than usually presumed.
Disclosure Statement
No potential conflict of interest was reported by the author.
Notes
1 Ferguson [Citation1962] provides a brisk overview of the developmental history of this topic.
2 All of the figures have been modified from either Descartes [Citation1644] or Reuleaux [Citation1876], with some influence from Aiton [Citation1972] in .
3 As Reuleaux showed, the functional relationships within a very large field of mechanical devices can be duplicated locally by gizmos that consist entirely of links together with frictionless bolts (which is why theoretical discussions of mechanical capacity frequently concentrate specifically upon linkages). Mathematically, such a linked assembly corresponds to an obvious set of linear equations that stipulate that the link R maintains a constant length between its two hinged points L and R. The resulting equations for each distinct link become coupled together through their overlapping citations of the locations L and R, and give rise to transformation matrices between input and output motions that will generally prove non-singular (except for occasional ‘dead spots’) in the special situation of a ‘closed kinematic chain’. This non-singular behaviour in transformation matrices represents another expression of the special internal characteristics that render the ‘realm of mechanism’ descriptively special. In our opening epigraph, Sylvester uses the term ‘complete linkage’ to capture the same mathematical idea.
4 The exact requirements are captured within a relationship called ‘Grübler’s formula’ which also embraces broader mechanisms (such as hydraulic cranes) that possess multiple degrees of internal freedom. In the exact manner that Sylvester illustrates within our opening epigraph, Descartes presumes that the freedoms of naturally occurring mechanisms appear indeterminate only because their further attachments to the surrounding ‘system of the world’ have not yet been fully diagnosed (in Sylvester’s jargon, these unseen connections make the linkage ‘complete’, possibly on an infinite basis). Such considerations cast Descartes’ celebrated reservations with respect to ‘mechanical curves’ in a revealing light [Wilson Citation2019]. With respect to the mathematical issues active in this setting, each link in a mechanism corresponds to a linear equation running through the positions of its two hinges. Collecting these equations together as a system, we can frame the transformation matrices M that maps a given position (a,b,c)T to other n-tuples. If Grübler’s requirements are satisfied, the transformation M will prove (generically) unique, mathematically capturing the exceptional manner in which true mechanisms are internally deterministic. However, we should also observe that in certain positions such an M will occasionally turn singular, leading to the failures of determinism that engineers label as ‘dead points’ (this is why I needed to add the qualifier ‘generically’ to the characterization in the previous paragraph). I am not aware of any passage where Descartes worries about these exceptional breakdowns.
5 To articulate these arrangements in Cartesian terms, if vortical flows A and B slip past one another without friction, A’s presence will alter the determinations of B’s particles (and vice versa), but A and B will not exchange any of their stored quantity of motion with each other. Conserving quantity of motion always remains a higher priority for Descartes than preserving determination does. His pineal gland is constructed so that the soul adjusts only the determinations of the little ball therein, without any alteration in work capacity.
6 In Pierre Duhem’s assessment, Descartes is one of the first writers who clearly separated the statics of such devices from the faulty dynamical assumptions with which they had been heretofore muddled. He further observes [Citation1991: 241]:
Descartes clearly understood and underscored the infinitesimal property of the Principle of Virtual [Work] and he states what nobody had explicitly formulated before him: the necessity to apply this principle to an infinitely small displacement originating from a state of equilibrium.
7 Descartes’ follower Jacques Rohault amplifies this reasoning as follows [Citation1671: 81]:
When a Body falls perpendicularly upon another, which is hard and immoveable, it is evident, that the Reflexion ought to be made in the same line in which the body moved before, there being no Reason why it should incline one way rather than another. Wherefore there is no difficulty in this manner, except when the Line in which the Body begins to move makes oblique Angles with the Superficies of the Body against which it strikes. But the Judgement we are to make of this depends upon what we are going to say concerning the Composition of Motion, and of its Determination.
8 I know of no Cartesian passages that explicitly treat device engagement in quite these terms, but I believe that Cartesian-like principles of energetic apportionment across coupled systems are better rendered in these mechanism-focused terms, rather than by attempting to treat them in two-body impactive terms, as Descartes in fact does, with rather unfortunate results. Because ‘forces’ transmit across rigid bodies instantaneously, Descartes often needs to appeal to escapement-like mechanisms to slow down transmission speeds.
9 Engineers label these striking symmetries as ‘inversions of the mechanism’.
10 Here’s another way to understand the delicate mathematical issues in play. I have placed ‘initial conditions’ in scare quotes, because they normally require the positions and velocities of the system at t0 be specified. By replacing the latter by the internal energy E, we lose the capacity to calculate the elapsed time required for the device to shift to a new configuration (in other words, we can’t calculate the rates at which the device will pass through its anointed positions). Any method that I know for computing these speeds requires knowing the masses of the component links and their movements with respect to an Absolutist inertial frame. But these matters are subtle, and it is not surprising that they were overlooked by philosophical relationalists such as Descartes. There is a further behavioural symmetry relevant here, called ‘material frame indifference’, but I’ll not attempt to explicate it in this essay. I suspect that a significant portion of the widely held ‘philosophical’ convictions in favour of ‘relationalism’ depend covertly upon some admixture of these two behavioural ingredients.
11 For a survey of the difficulties attending ‘impact’, see Wilson [Citation2020]. Newton’s own ‘coefficient of restitution’ approach qualifies as such a ‘crude empirical rule of thumb’. Leibniz’s contention that ‘nature doesn’t make jumps’ represents an early attempt to ban instantaneous recoil from the vocabularies of science.
12 Although it has been commonly recognized that Descartes advances an early form of conservation of energy that relies upon hidden internal motions, this account is mechanism-based, instead of relying upon the kinetic energies of disconnected swarms of corpuscles, as occurs in modern kinetic theory. Heinrich Hertz’s ‘hidden masses’ approach [Citation1956] is much closer to Descartes, as perceptive commentators such as Ernst Mach have noted. Our identification of ‘quantity of motion’ with internally stored potential energy casts the venerable vis viva controversy in a revealing light. With respect to a pendulum, Leibniz argues that its inherent ‘force’ is best measured when it becomes maximally registered as kinetic motion (i.e. when the bob passes through the bottom point of its swing). Descartes, conversely, quantifies this internal ‘force’ in potential energy terms, (viz. as the maximal height to which the system can raise the bob, now regarded as an external load).
13 Descartes here distinguishes between ‘force’, as it pertains to the stored quantity of motion that causes a mechanism to move through its internal cycles, and a rigid body’s non-stored capacity for transmitting a locally applied push F instantaneously across its interior as a kind of potential pressure. Descartes appears to wobble somewhat inconsistently between insisting that this induced ‘pressure’ F can be any extracted from any site along B’s exterior and demanding that F’s determination ‘pressure’ will prove optimally salient at the exit locus marked out by the straight-line ray that departs from F’s insertion point with the same angle of determination (this discrimination is required by his theory of light). I might further observe that virtually all of Early Modern thinking about physics struggles with the difficulty of addressing ‘internal pressure’ in a coherent manner.
14 A qualification: from any part that is not at rest with respect to our own reference frame, for nothing can be drained from an extraction pin located upon a link currently at rest with respect to ourselves.
15 Alan Nelson [Citation1995] labels these calls incoherent motions as ‘micro-chaos’.
16 The authors extend these ‘unities’ to a much wider spectrum of Cartesian assembly—not only to his sundry vortices, but to the human mind/body combination and the interpersonal dependencies that become forged through the bonds of love and society. These extended claims offer deep insight into Descartes’ wider expanses of thought, but here I am content to emphasize the specialized ‘unity’ inherent within Reuleaux’s diagnosis of the structural relationships required within a mechanism proper. Through Grübler’s formula, the Brown/Normore ‘unity’ required acquires a precise mathematical characterization. Des Chene [Citation2001] includes an interesting discussion of these issues, although it overlooks the connections between Descartes’ ‘unities’ and Reuleaux’ closed kinematic chains. Descartes characterizes the cyclings of machines and vortices as ‘equilibriums’ and highlights their importance within his thought [Citation1896: 44]:
In this way the entire system is in a state of equilibrium. But this is a very difficult thing to conceive of, because it is a mathematical and mechanical truth. We are not sufficiently accustomed to thinking of machines, and this has been the source of nearly all error in philosophy.
17 Descartes [Citation1645] expands upon these observations in a letter to Clerselier.
18 I made such an attempt long ago [Citation1997]. This discussion, while imperfect, supplies a general portrait of how such a reconstruction might work.
19 I’d like to thank Alan Nelson, Calvin Normore, and the referees for helpful suggestions.