Cohesion, Division and Harmony: Physical Aspects of Leibniz’s Continuum Problem (1671–1686)
Leibniz often claimed that his struggles with the problem of the composition of the continuum and its solution were formative for his theory of substance. As has long been recognized, mathematical considerations— especially his creation of the differential calculus and the work on the summation of infinite series—were highly relevant. But the role of physical considerations has been comparatively neglected, and it is this I want to address in this paper by discussing three topics from physics which appear to have been particularly important for Leibniz in formulating his solution: the problem of cohesion, the problem of “the solid and the liquid,” and the implications of the relational nature of motion.
Of course, if the composition of the continuum is understood as a purely mathematical problem, one may well wonder what bearing physical considerations could have on it. But for Leibniz and his contemporaries, the problem was not restricted to the composition of purely mathematical entities—such as whether a line is composed out of points or infinitesimals or neither—but was understood as applying to all existing quantities and their composition. In this wider sense, the continuum problem is: what (if any) are the first elements of things and their motions? Are there atoms or indivisible elements of substance? Is space composed of points, or time of moments? These metaphysical questions are in turn linked to some pressing problems of physics: for Descartes, the actual division of (at least some parts of) matter into indefinitely small particles is a necessary condition for motion through unequal spaces in the plenum; for Galileo, the supposition of indivisible voids between the indivisible parts of matter explains the cohesion of bodies; for Hobbes, the Galilean analysis of the continuous motion of bodies into infinite degrees requires a foundation in terms of [End Page 110] endeavors, infinitely small (but unequal) beginnings of motion, and these same endeavors are the cornerstones of his materialist psychology.
Leibniz inherits this wider conception of the continuum problem, and it is the whole cluster of problems concerning infinite divisibility, the actual infinite, the existence of atoms of matter or substance, and the analysis of continuous space, time, and motion to which his characteristic allusions to the “labyrinth of the continuum” refer. 1 Given this complexity it is not at all easy to summarize what Leibniz took his solution to the problem to be. But in broad brush strokes, it involves at least the following: insofar as anything is continuous, its parts are indiscernible from one another, and thus indefinite. The continuum is therefore not an actually existing thing, a whole composed of determinate parts, but an abstract entity. In existing things, by contrast, the parts are determinate, and are prior to any whole that they compose. Matter, for example, considered abstractly (i.e., as primary matter), is a homogeneous, continuous whole, consisting in a pure potentiality for division; but taken concretely (i.e., as secondary matter), it is at any instant not only infinitely divisible, but actually infinitely divided by the differing motions of its parts. Thus no part of matter, however small, remains the same for longer than a moment; even shape or figure is evanescent, and a body with an enduring figure is something imaginary. Similarly, there is no stretch of time, however small, in which some change does not occur. Change, on the other hand, can only be understood in bodies as an aggregate of two opposed states at two contiguous or “indistant” moments; but again nothing can remain in precisely the same state for longer than a moment, so the supposed enduring states of bodies must themselves be to some extent imaginary. Thus the perduring element in matter is not something material, i.e., explicable in terms of the extended, motion and figure. There must, however, be such a perduring element in any part of matter however small, which is the principle of all the changes occurring in it. This immaterial principle, Leibniz concludes, consists in a primitive force of acting. It bestows unity on a substance by taking that thing through all its states in a lawful way: it thus encompasses the laws that govern the series of states of the thing, as well as an endeavor or appetition, taking it into the next state of the series. Finally, this (immaterial) principle of unity of any given substance is the complement of its primitive passive power, or (material) principle of diversity, [End Page 111] and substance is constituted by these primitive active and passive powers together.
I would not presume such a highly condensed account to be readily intelligible, and indeed there is much that is missing (such as the whole question of the status of infinitesimals and the calculus, Leibniz’s philosophy of the infinite, and the relation of mathematics to reality), but I hope it is enough at least to set the scene, and to give some of the flavor and richness of Leibniz’s assault on the continuum problem. Certainly I have no intention of giving a complete account of the latter here. But I hope to say enough to illuminate some of the main turns Leibniz took en route to his own distinctive solution.
Leibniz did once give his own account of his exit from the labyrinth, as a kind of proem to the discussion of dynamics in the second dialogue of the Phoranomus: Or, on Power and the Laws of Nature (July 1689), written just prior to his Dynamica. 2 This is interesting not least for the insight it gives into his changing views on the status of atoms, absolute space, and motion (on which I shall quote him later). Notably too, though, Leibniz there identifies a single clew that led him out of his labyrinth, in the words: “Accordingly I discovered no other Ariadnean thread that would finally extricate me from that labyrinth than the calculation of powers, assuming this metaphysical principle, that the total effect must always be equal to its full cause” (Leibniz  1991, p. 811). 3
Of course, it may be that the labyrinth Leibniz is referring to here is not that of the continuum pure and simple, despite his references to many features that are constitutive of his solution to the latter. 4 But if it is the latter he means, I think we would do well not to take this description of the “Ariadne’s thread” too literally; at any rate, I have found no convincing evidence that Leibniz’s exit from the labyrinth of the continuum was immediately illuminated by this metaphysical principle, which he first announced in the summer of 1676, whereas he persisted in his labors in that labyrinth (even on a conservative estimate) until around 1683–86, when the mature solution outlined above emerged in a series of papers written at that time. On studying his papers from 1676 to 1686 that are of immediate relevance to his solution, one finds not so much one thread that delivers Leibniz from his labyrinth, as several converging strands of [End Page 112] thought, some predominantly mathematical, others deriving from metaphysical, epistemological or physical concerns, all tugging on each other in an extremely complex web of mutual influence. Nevertheless, whatever Leibniz’s rhetorical purposes in so promoting the role of the principle of equality of cause and effect, it is unlikely that the account he presents in the Phoranomus of the emergence of his views is wholly misleading. It gives a point of view, at any rate, and to the extent that it highlights the role of physical considerations in Leibniz’s solution, it is very congenial to my purposes here. Accordingly, I shall structure my essay around the narrative link provided by this account, although I shall not center it on the principle of the equality of cause and effect, which I shall only treat in passing. Nor will I have the space here to examine the emergence of Leibniz’s doctrine of the conservation of vis viva, or its connection with the Principle of Continuity, despite the obvious relevance of these matters.
The Problem of Cohesion
According to his testimony in the Phoranomus, Leibniz first began to wrestle with the problems of the continuum as an adolescent, when, emerging from “the prickly thornbrakes of the scholars into the pleasanter pastures of more recent philosophy,” he found himself “wonderfully taken in by that flattering easiness of understanding, by which it seemed that everything which had previously been shrouded in murky notions could be comprehended by a lucid imagination. Thus after deliberating on this long and hard, I finally came to condemn forms and qualities in material things, and reduced everything to purely mathematical principles; but since I was not yet versed in geometry, I persuaded myself that the continuum consists of points, and that a slower motion is one interrupted by small intervals of rest” (Leibniz  1991, p. 803).
But Leibniz’s attachment to these doctrines did not long survive his university years. In his first major attack on the continuum problem, 5 the Theoria Motus Abstracti of 1671, Leibniz disowns the thesis that motion is interrupted by rests, although the composition of the continuum from points of a certain kind is upheld. The linchpin of this account is the notion of conatus or endeavor, a notion Leibniz informs us he inherits under the name tendentia (tendency) from Weigel, his teacher at Jena, as [End Page 113] well as (more obviously) from Hobbes. 6 The latter had attempted to assimilate Cavalieri’s “indivisibles” to Euclid’s points by redefining a point as “that whose quantity is not considered” (Hobbes 1668, pp. 98–99). Building on these insights, but at the same time heavily amending them, Leibniz defines a point as “that which has no extension, that is, that whose parts are indistant, whose magnitude is inconsiderable, unassignable, is smaller than can be expressed by a ratio to another sensible magnitude unless the ratio is infinite, smaller than any ratio that can be given.” 7 Unlike a minimum, a “thing which has no magnitude or part,” 8 whose existence Leibniz denied, a point would have a magnitude: only it would not be assignable. Endeavor is now defined by analogy: “Endeavor is to motion as a point is to space, or as one to infinity, for it is the beginning and end of motion.” 9 Thus whereas Hobbes’s endeavors are true motions, being motions through a line whose quantity is not considered, Leibniz’s are not true motions, but what might be called non-Archimedean infinitesimals: they have a quantity smaller than any that can be assigned (i.e., than can be expressed by a finite ratio to a finite motion).
So far this seems exclusively metaphysical. Indeed, one of the main motivations for his work in this period is to prove that “bodies left to themselves”—i.e. bodies with no minds—would dissolve to nothing. 10 But for Leibniz metaphysics was never exclusive of physics, since it provided the latter’s foundation, and he had long had his eye on a problem in physics that was grist for his mill about the inadequacy of bodies by themselves. This was the problem of cohesion, how it is that bodies hold together, rather than dispersing into the void or plenum.
The problem of cohesion was very much an open problem in the latter half of the seventeenth century. It had in a sense been bequeathed by Galileo, who had shown how to calculate empirically how much of a solid’s [End Page 114] cohesion was due to nature’s horror of the vacuum alone, and how much was due to another cause (by ascribing the cohesion of water solely to horror of the vacuum, and equating the force of cohesion of any other material with its resistance to breaking under its own weight); he himself speculated that this excess cohesion was due to the vacuum within, the infinitely many interstitial voids (non quante) that could be conceived as holding together the infinite indivisibles of a given substance (Galilei 1638, pp. 65–80). Translated into the terms of some twenty years later, he had shown how much of a body’s cohesion was due to the pressure of the air, and how much due to another cause. Explanations of this other cause varied. For believers in atoms and vacuum like Gassendi, the cohesion of macroscopic bodies was to be explained by the hooks and angles by which the constituent atoms held onto each other; whereas atom-and-plenum physicists like Huygens, having substituted atmospheric pressure for nature’s abhorrence of an external vacuum, followed Galileo in assuming like causes for like effects, and ascribed this excess cohesion to the pressure of a fluid more subtle than air.
Here Leibniz saw the possibility of a novel solution. For in all such atomist explanations, the cohesion of the atoms themselves remains unexplained, so that one has only really succeeded in pushing the problem further down, exposing oneself to the threat of an infinite regress. This provides an opening for introducing something from outside the regression, just as in Aristotle’s argument for the prime mover, where bodies’ motions cannot be caused by other bodies in infinitum. (Indeed Leibniz often mentions the example of cohesion in the context of infinite regress arguments.) But with the notions of endeavor and physical point defined as above, Leibniz believes he can offer a satisfactory technical solution to the problem of cohesion. This he gives as a demonstration from his definitions: If one body impels another, or endeavors to move it, it has already begun to penetrate it. Therefore the two bodies overlap at the point of impact, and consequently share an extremum, it being a place smaller than any determinable by us. But “things whose extremities are one are continuous, that is, cohering, by Aristotle’s definition too, since if two things are in one place, one cannot be impelled without the other” (Leibniz [1663–72] 1966, p. 266). 11
The resulting cohesion, however, will only be momentary. For things which move in such a way that one part endeavors into another’s place, cohere only while the endeavor lasts, 12 and “no endeavor lasts longer than a [End Page 115] moment without motion, except in minds” (Leibniz [1663–72] 1966, p. 266). 13 Therefore a lasting coherence requires a continuous motion across the boundary in question, and thus across the boundaries of all the parts of any body that is to constitute a cohering whole. An obvious candidate for a continuous motion of this kind, underwritten by a tradition going back to Anaximander, Democritus, and Anaxagoras, is vortical motion; and Leibniz follows Hobbes (while believing he is correcting him) in giving circular motions as the cause of cohesiveness. 14 His “New Physical Hypothesis” is that the solar and terrestrial globes each possess “a motion around their own center,” which, when combined with the perpendicular action of light rays from the sun falling upon the earth produces bullæ or “bubbles” like those of a glassmaker (Leibniz [1663–72] 1966, pp. 223–27) (or, in a later version, terrellæ, that is, little magnetic “earthlets,” which, if solid, are “globules,” and if hollow, are bullæ), whose spinning motion along their meridians renders them cohesive (Leibniz [1663–72] 1980, pp. 29ff.). Accordingly, everything that is firm or cohesive is composed of these spinning corpuscles, and the infinite regress afflicting atomist accounts is forestalled.
Now it may be thought that this theory is of only limited interest, and Leibniz would have abandoned it as soon as he “became a geometer,” the notions of non-Euclidean points and endeavors not being such as a serious geometer could entertain. 15 But in my view critics (including Leibniz himself) have been unduly harsh on this early theory. For, putting aside the infelicity of naming them “indivisibles,” 16 these neo-Hobbesian points that are less than any assignable quantity are clear forerunners of Leibniz’s infinitesimals, and of their justification—although this is not the place to enter into a digression on that subject. Moreover, the second thesis that Leibniz dismisses in the Phoranomus as being from before he “was versed in Geometry,” the Gassendian analysis of motion as interrupted by little intervals of rest, is explicitly repudiated in the Theoria Motus Abstracti (TMA), of which the seventh fundamentum praedemonstrabile (demonstrable first principle) is: “(7) Motion is continuous, or not interrupted by any [End Page 116] little intervals of rest.” The demonstration depends only on the “inertial” principle for rest (found also in Hobbes and even Aristotle): “(8) For once a thing comes to rest, it will always be at rest, unless a new cause of motion occurs.” This is itself the converse of what we now call the principle of inertia: “(9) that which is once moved will, insofar as it is able (quantum in ipso est), always move with the same velocity and in the same direction” (Leibniz [1663–72] 1966, p. 265). Thus the continuity of motion, on this reading, is a consequence of Gassendi’s inertial principle of motion itself. But what enables Leibniz to claim this consequence—over and against the discontinuist analysis he had previously upheld along with Gassendi and Arriaga—is the foundation provided by the theory of endeavors. For since there is nothing in a body (conceived purely geometrically) to offer any resistance to an impinging endeavor, every endeavor is propagated indefinitely, becoming weakened only in the sense that it diffuses into components along different directions (which, if added together, would recompose into the original endeavor). Thus, as Aristotle had taught, whatever moves will keep on moving.
This foundation seems to have remained intact for some time. At any rate, we can find Leibniz upholding the continuity of motion in similar terms as late as December 1675. 17 But in a piece written in April 1676 ([1672–76] 1980, p. 492) Leibniz announces that he has “just demonstrated that endeavors are true motions, not infinitely small ones,” 18 a result which undermines the foundation of the theory of the TMA. For the latter depends on the conception of endeavor as an infinitesimal motion, as the “beginning of entering a place,” so that at the instant of impact the moving body is in more than one place, both against the other body and beginning to penetrate it. But if there is no such infinitesimal motion, no body will ever be in more than one place at a time. Thus “when one body is pushed by another” the only explanation is “that they are incompatible” ([1672–76] 1980, p. 493), and the whole account falls through. Consequently, after some further reflection, Leibniz finds himself “forced to conclude that motion is not continuous, but happens by a leap” ([1672–76] 1980, p. 494)—a conclusion we shall examine further below.
But if we turn from the metaphysical underpinning and ask what becomes of this early theory of cohesion itself, things are very curious. It [End Page 117] survives a further detailed reading of Galileo’s Discorsi, Leibniz’s notes for which date from shortly after his arrival in Paris (see [1672–76] 1980, pp. 167–68, 13–16), and also his readings of Huygens’s and Boyle’s latest pronouncements on the subject in the fall of 1672 (see [1672–76] 1980, pp. 94–96, 16, 44–47). This much is understandable, given the explanation of “primary cohesion” provided by the theory, Leibniz was able to avail himself of the usual accounts in terms of the “pression” of the air or of a subtle fluid to explain what he called “secondary cohesion,” that is, such phenomena as the difficulty of parting well polished marble slabs once they have been pressed together. Moreover there is a consequence of the endeavor theory that is independent of its foundation, namely that cohering bodies are those that move together with the same motion. Later, after abandoning his early theory of the continuum, Leibniz will take this as definitional. But the notions of harmony and sympathy in terms of which his mature theory would be framed may already be found in Leibniz’s earlier writings. Thus, as he writes in a cancelled draft of A Demonstration of Incorporeal Substances (fall 1672?):
cohering bodies are co-moving i.e. one cannot be impelled without the other, by [a previous] definition. If one cannot be impelled without the other, also one cannot be acted on without the other. For every passion of bodies is a being moved, i.e. impelled, by another. Therefore it is necessary that cohering bodies sympathize([1672–76] 1980, p. 80)
This anticipates his later view of the Specimen Inventorum (c. 1686) that “cohesion arises from motion insofar as it is harmonious,” for which he claims experimental corroboration. 19 In the same vein, in a piece dating from between 1678 and 1682 he writes that “The cohesiveness of a body, or the cohesion of its parts, arises from the fact that they are set in motion in such a way that they are hardly separated at all, and since they are borne along with a [common] motion by the whole surrounding system, they cannot be separated without force, that is without some disturbance of the system” (Leibniz 1982–89, p. 2041). Thus Leibniz’s theory of cohesion as motus conspirans seems to have gone on to enjoy a life of its own independent of the vagaries of the foundations from which Leibniz had originally derived it. [End Page 118]
Yet perhaps this continuity is illusory. For in a series of papers from 1676, some written in February and March before the demise of the endeavor theory, we find Leibniz toying with perfectly solid or indestructible atoms of the kind his earlier theory had been designed to displace. A partial explanation of this reversal is given in the Phoranomus, where Leibniz testifies that in his attempt to reconcile his laws of collision from the TMA with the system of things, “I imagined elementary or primitive bodies to be equal to each other. Thence there now appeared to be a way of making a greater body offer greater resistance, if only the elements in it were assumed to be not continuous but interrupted” ( 1991, p. 808). But in order to throw more light on this abrupt change of heart, it will help to proceed to another physical problem that was preying on Leibniz’s thoughts at that time.
Actually Infinite Division
In the rest of the passage quoted from the Phoranomus at the head of the last section, Leibniz continues his reminiscences about his early views as follows:
And I indulged other dogmas of this kind, to which people are prone when they wish to entertain every imagination, and do not notice the infinity lurking everywhere in things. But although when I became a geometer I relinquished these opinions, atoms and the vacuum held out for a long time, like certain relics in my mind rebelling against the idea of infinity; for although I conceded that every continuum could be divided to infinity in thought, I still did not grasp that in reality there were parts in things exceeding every number, which follow from motion in a plenum.(Leibniz  1991, p. 803)
This colorful image of atoms and the void as Gassendian relics, stubbornly holding out against the idea of the actual infinite, is in fact quite apt, and is borne out by a scrutiny of his surviving papers. For the doctrine of the actually infinite division of matter appears early in the Leibnizian corpus as an objection to atoms. For instance, in “On Primary Matter,” dated 1671, Leibniz states unequivocally: “Matter is actually divided into an infinity of parts. There is in any body whatever an infinity of creatures. . . . There are no atoms, or bodies whose parts are never separated” ([1663–72] 1966, p. 280). Yet in the same year, in the Hypothesis de Systemate Mundi, he describes the world as “a space full of globes,” or spherical atoms, “touching each other only at points,” and with voids in their interstices, so that “[t]he only whole bodies are Atoms” ([1663–72] 1966, p. 294). It may be that these texts do not represent so much incompatible views, as incompatible [End Page 119] terminology. In upholding both actually infinite division and the existence of “atoms,” Leibniz did after all have a precedent in Hobbes, and one could make a case that atoms in this sense are merely very small corpuscles that are contingently unseparated, yet still “divided,” in the sense of having parts that cohere. This would make sense of his claim in the Hypothesis Physica Nova, impossible on an Epicurean or Gassendian reading, that “since the continuum is divisible to infinity, any atom will be of infinite kinds, like a sort of world, and there will be worlds within worlds to infinity” (Leibniz [1663–72] 1966, p. 241).
But this will not explain away Leibniz’s appeals to atoms in 1676, where he is as committed to their indestructibility as Gassendi had been, and there is also a return to the interruptedness of motion. Here the argument “from motion in the plenum” is an argument for atoms, rather than against them. In its general form the argument is the old one found in Lucretius that, unless there is some ultimately unbreakable solid, then the action of the surrounding matter will eventually dissolve any given body into nothing: thus “If an atom should once subsist, it will always subsist. For the liquid matter of the surrounding plenum will endeavor to break it up immediately, since it disturbs its motion, as can easily be shown” (Leibniz [1672–76] 1980, p. 473) and “If there were no atoms, everything would be dissolved, given the plenum” (Leibniz [1672–76] 1980, p. 525). (This was also a major motivation for Cordemoy’s deviation from a strict Cartesianism in favor of atoms, as Leibniz was later to point out.)
But Leibniz has in mind a more specific argument “from motion in the plenum,” one deriving from Descartes’s argument in his Principles of Philosophy of 1644 for the genuine division of certain parts of matter into indefinite particles. And from the moment in December 1675 when Leibniz again sits down with this work in order to assess Descartes on his own terms, it gradually assumes greater and greater importance in his work. Descartes gives the argument while rebutting the objection raised by the ancient atomists to motion in a plenum: that, unless there were an empty space for a body to move into, it would be unable to move. He countered that this would not rule out circular motion, since a whole ring of matter could start to move simultaneously. Of course, not all circulating matter moves at the same speed, so Descartes felt obliged to explain that this too would be possible, provided the matter moving through a narrower space were to move with a proportionately faster motion. But this has the consequence that at least some part of matter must “accommodate its own figure to the innumerable measures of the [progressively narrower] spaces,” which involves “all its innumerable particles” being to some degree [End Page 120] mutually displaced—and thereby actually divided—from each other. 20
But if “cohering bodies are co-moving,” then Descartes’s argument—which Leibniz later describes as “most beautiful and worthy of his genius” (Loemker 1969, p. 393)—confronts him with two consequences fatal to his own earlier theory: the fluid matter will have to be perfectly fluid, i.e., such that no part, however small, will be left cohering with any other; 21 and whatever solid corpuscles are floating in it will have to be perfectly solid, since, as we already saw him arguing, the action of the surrounding liquid matter will endeavor to dissipate them from the beginning. Thus in accounting for solids, Leibniz reverts to irrefrangible atoms, which, like his earlier bullae and terrellae, are constituted by mini-vortices (a view echoed by Malebranche in his then recently published Recherche de la Verité). Only now their cohesiveness is not attributed to their circular motion, but to their possessing minds: “the solidity or unity of a body comes from a mind, . . . there are as many minds as vortices, and as many vortices as solid bodies” and “there is some portion of matter that is solid and unbreakable, . . . thought enters into the formation of this portion, and, whenever it has a single mind, it becomes a single and indissectible body, i.e. an atom, of whatever magnitude.” 22
But, even while speculating that the “perfect points” into which a fluid may be divided might be identifiable with the infinitesimals of his calculus, Leibniz does not appear comfortable with this interpretation of infinite division, contradicting as it does all his proofs that there are no minima in [End Page 121] bodies. Thus as he explores the analogy between infinite series and unbounded lines, we see him gradually becoming aware that another understanding of actual infinite division is possible: “if we supposed that any body whatever is actually resolved into smaller bodies—that is, if it is supposed that there are always some worlds within others—would the body thereby be divided into minimum parts? Accordingly, being divided without end is different from being divided into minima, in that [in such an unending division] there will be no last part, just as in an unbounded line there is no last point” ([1672–76] 1980, p. 513).
This trend of thought culminates in the view put forward in the Pacidius Philalethi, or “First Philosophy of Motion,” of November 1676, when Pacidius (speaking for Leibniz) proposes it with an inspired analogy:
If a perfectly fluid body is assumed, a finest division, i.e. a division into minima, cannot be denied; however, a body that is indeed everywhere pliant, but not without a certain and everywhere unequal resistance, still has cohering parts, although these are opened up and folded together in various ways. Accordingly the division of the continuum must not be considered to be like the division of sand into grains, but like that of a sheet of paper or tunic into folds. And so although there occur some folds smaller than others infinite in number, a body is never thereby dissolved into points, or minima. On the contrary, every liquid has some tenacity, so that although it is torn into parts, not all the parts of the parts are so torn in their turn; instead at any time they merely take shape, and are transformed; and yet in this way there is no dissolution all the way down into points, even though any point is distinguished from any other by its motion.(Leibniz [1672–76] 1980, p. 555)
Thus we are to imagine matter as infinitely divided, but with parts that are only ever finite, on the model of a converging infinite series, such as 2/3 + 2/9 + 2/27 + . . . = 1. This infinite division therefore consists in a sort of partition, but with a different partition from one instant to the next. Bodies may remain cohering while not keeping exactly the same figure and magnitude from one moment to the next. Thus the atomist argument is not cogent: it is not necessary for there to be bodies with absolute coherence, and which always have the same shape and size. One can see the beginnings of this view in the Paris writings, 23 but by 1683 [End Page 122] Leibniz has realized its significance: “there are no precise figures in the nature of things. . . . Corporeal substance has no definite extension.” 24 Thus, he will claim in the same piece, it is not an atom of constant shape and size that is held together by a mind, but rather a “substantial atom or corporeal substance,” whose self-identity is constituted by a substantial form or soul.
But in the contest of arguments it takes some time for this interpretation of actually infinite division to become entrenched as the victor. Indeed, we find Leibniz promoting atoms (his Gassendian “relics”?) as a consequence of motion in a plenum, in a paper written barely a month later, the Catena mirabilium demonstrationum de Summa rerum of 12 December 1676 ([1672–76] 1980, p. 585). Of course, it is hardly the case that all problems concerning continuity are resolved in the Pacidius to his final satisfaction. In particular, the main conclusion of the dialogue is a reaffirmation of the discontinuity of motion he had found himself compelled to revert to in April as a result of the collapse of the foundation of motion in terms of endeavors. But this eclipse of the continuity of motion seems only to have been temporary, its reinstatement depending on his coming to terms with two problems left out of the discussion there: the relativity of motion and the correct identification of the cause of motion or motive force. 25
The Relativity of Motion
Returning to Leibniz’s autobiographical account in the Phoranomus, he there relates that as he struggled to formulate the correct laws of motion, he
began to have great misgivings about the nature of motion. For when formerly I had conceived space as a real immobile place, endowed only with extension, I was able to define absolute motion as change of this real space. But gradually I began to doubt whether there existed in nature such an entity as we call space, and from this it followed that there could also be doubts about absolute motion. Certainly Aristotle had said that place is nothing but the surface of the surrounding bodies, and Descartes had followed him, defining motion (i.e. change of place) as change of vicinity. Hence [End Page 123] it seemed to follow that what is real26 and absolute in motion does not consist in what is purely mathematical, such as change of neighborhood or situation, but in motive power itself; and if there were no such thing, this would mean that absolute and real motion would be abolished.(Leibniz  1991, pp. 810–11)
Again let us see whether the extant manuscripts bear out these recollections of Leibniz’s in 1689. Near the end of 1675, commenting on a diagram in Descartes’s Principles showing two bodies translated in opposite directions along the Earth’s surface, he at first concludes that either the Earth may be regarded as being at rest, or even “one of the smaller bodies” if he imagines himself as being on it. He continues: “Therefore it is clear that any body may be imagined to be at rest, if only a sentient being is understood to be on it” ([1672–76] 1980, p. 215). Here he adds what sounds a lot like Mach’s response to Newton’s “rotating vessel” argument for absolute motion, though the implicit allusion is to Seneca: “being thrown off along the tangent does not argue the real motion of the rotating thing, since it would be the same if everything moved around it” ([1672–76] 1980, p. 215). 27 Our usual attributions of motion and rest may be saved, however, by appeal to the principle that “we humans are accustomed to attribute rest to the larger bodies, and this for the sake of abbreviation or the ordering of thought” ([1672–76] 1980, p. 215). This, of course, would make the Earth at rest. On further reflection, he provides a principle which encapsulates the greater simplicity of the Copernican hypothesis: “when things are mutually changing their situation and it is asked which of them should be said to be moving, motion should always be ascribed to a certain finite thing, rather than to the rest of the entire world outside it” ([1672–76] 1980, p. 217). Notably, however, at this time he does not draw any further conclusion concerning the implications of this for the reality of motion itself.
A little over a year later, though, Leibniz is ready to invest the relativity of motion with greater significance. In a note dated February 1677, he writes:
A wonderful fact: motion is something relative [quiddam respectivum], and we cannot distinguish just which of the bodies is moving. Therefore if it is an affection, its subject will not be any one [End Page 124] particular body, but the whole world. Hence it is necessary that all its effects also be relative [respectivos]. The absolute motion we imagine [fingimus] to ourselves, on the other hand, is nothing but an affection of our soul while we regard ourselves or other things as immobile, since we are able to understand the whole thing more easily when these things are regarded as immobile.(1982–89, (no. 145) p. 654)
Here (a decade before the publication of Newton’s Principia) we see absolute motion firmly relegated to the status of an imaginary entity: we feign or imagine a given body or bodies to be at rest as suits our epistemological purposes. There is no fact of the matter, considering motion as it is in itself, whether the Earth or the Sun is in motion, and we are at liberty to choose the hypothesis that reduces things to a more intelligible order. Nevertheless, Leibniz adds, when we instead “consider motion not formally as it is in itself, but with respect to cause, it can be attributed to the body by whose contact change occurs” (1982–89, p. 654). 28 This causal criterion is variously elaborated later. As I have argued elsewhere (Arthur 1994, pp. 230–35), it enables Leibniz to save the phenomena of our usual attributions of motion; but nothing breaks the equivalence of all hypotheses concerning motion considered formally. 29
In the passage from the Phoranomus quoted above, Leibniz attributes this hardening of his position on the relativity of motion to his growing doubts over the reality of space. This seems consistent with the manuscript evidence. For although one can find him making remarks critical of a container conception of space as early as 1672, 30 it is not until 1676— between the two moments of his thought just considered—that something like his mature conception of space can be seen emerging. “Supposing space has parts,” he reasons in a journal entry dated 18 March 1676, “—that is to say, so long as it is divided by bodies into empty and full parts of various shapes—then it follows that space itself is a whole or [End Page 125] entity per accidens; it continuously changes, and becomes first one thing then another: to wit, when its parts change and are extinguished and supplanted by others” ([1672–76] 1980, p. 391). Consolidating these reflections a month later, Leibniz calls the space aggregated from these parts “universal space”; it is “like a net,” and is as ephemeral as are the figures and shapes into which it is divided, which change from one moment to the next. 31 What persists through these continuous changes, on the other hand—“the basis of space”—is not matter (which is just as ephemeral as space in this regard [(1672–76) 1980, p. 392]), but “the extended per se,” pure extendedness, “the immensum itself,” i.e., “God insofar as he is thought to be everywhere.” 32 But “this immensum neither has nor can have bounds, and is one and indivisible” ([1672–76] 1980, p. 519).
These considerations would naturally incline Leibniz to take a stronger stand on the relativity of motion. For there is clearly no way of defining absolute place on this account of space: universal space is an aggregate of places, but it does not remain in existence from one instant to the next; whereas the extended per se, which does persist, is an indivisible whole in which no places may be defined. And without absolute place, as he reasoned in the Phoranomus, there can be no absolute motion.
Nevertheless, when we examine the manuscripts of 1676 we do not find Leibniz arguing directly from the absence of absolute place to the relativity of motion in this way. Instead we find him exploring a more subtle connection between the two, prompted by his concerns about the continuity of motion, and his persistent efforts to derive the laws of motion—that is, the laws of impact, especially the law of conservation of quantity of motion—from the nature of matter itself. For with respect to the first concern, the view of space as evanescent is intimately tied up with his analysis of motion as discontinuous. In “On Motion and Matter,” written at the beginning of April between his reflections on space quoted above, Leibniz had argued that apparently continuous motion was to be analysed in terms of the moving body being annihilated in one place, and recreated immediately afterwards in another “indistant” from it. Yet in order for the body to be said to be the same individual, it must in its later state retain [End Page 126] the effect of its prior state; but he had just concluded in his “Meditation on the principle of the individual” (dated 1 April) that this can only happen by the persistence in it of a mind. Accordingly he concludes here that in the annihilation and recreation of a body, “a mind always remains intact that assists it.” 33 That is, just as the continued self-identity of a body is guaranteed by its containing a mind which “remembers” its prior states, so the appearance of continuity of motion is guaranteed by an enduring mind that “assists” the body by recreating its states so that they follow one another lawfully.
This appears to bring us close to the monads or simple substances of Leibniz’s mature philosophy. But he has not arrived there yet. For the minds in individual bodies are inadequate to ground the law of conservation of quantity of motion, according to which “magnitude compensates for speed, as if they were homogeneous things”; and this, Leibniz argues in the same piece, “is an indication that matter itself is resolved into something into which motion is also resolved, namely a certain universal intellect. For when two bodies collide, it is clearly not the mind of each one that makes it follow the law of compensation, but rather the universal mind assisting both, or rather all, equally” ([1672–76] 1980, p. 493). That is, the minds of the bodies taken singly would not “know” to adjust their speeds so that the sum of the quantities mv (“bulk”—or later, mass—times speed) is the same before and after collision. Thus the “assisting mind” that guarantees this is not the individual mind, but God, as Leibniz duly (and triumphantly) concludes. 34 This, then, is the mind that “remains intact” so as to provide for the continuity of motion, and which, insofar as it is everywhere, constitutes the immensum, the “basis of space” that remains during its changes.
But this is not all. For the quantity of motion is not in any case conserved in individual bodies. As Leibniz knew, what is conserved in a collision is not the sum of all the individual products, mass times speed (their sum, being what we would now call a scalar quantity), but that of the products of mi with the directed speeds (velocities) vi (what we now call the vectorial sum, Thus a given velocity may be added or subtracted from all the velocities before and after the collision, and the sum will still be the same (given the conservation of mass). Another way of [End Page 127] putting this is that it is only the relative velocity (or velocity difference) of the two bodies that matters. In Leibniz’s own words:
On the other hand, it is not necessary that the same quantity of motion is always conserved in the world, since if one body is carried by another in a certain direction, but is moving of its own accord equally in the contrary direction, it will certainly come to rest, i.e. it will not leave its place. From this it follows that the conservation of the quantity of motion must be asserted of the action, or relative motion [de actione, sive motu respectivo], by which one body is related to or acts on another.([1672–76] 1980, p. 493)
Here we can see the grounds for his assertion from the following February quoted above, that “motion is something relative,” whose “subject will not be any one particular body, but the whole world.” For if it is only the relative motion or action that is conserved (by the assistance of the divine mind), then there is no reality to motion in bodies taken individually. The link between the two pieces is highlighted by the fact that in both of them Leibniz uses the Latin word respectivus to qualify motion. This is significant in light of the connection (lost in my translation of the word as “relative”) with respectus, or point of view. Motion is relative to, or with respect to, the point of view we adopt. The world will look different from the various perspectives that result from holding different bodies to be at rest (preeminently, one’s own); yet the laws are such that there is a harmony between these points of view. Motion therefore, is phenomenal. Its reality resides not in the body itself, but in its relations with all other bodies, and in God’s underwriting the conservation of relative motion by his immediate operation.
This conclusion of the spring of 1676 that bodies depend directly on God for the continuation of their motion is further strengthened in the fall of that year by Leibniz’s close analysis of the continuity of motion in the dialogue Pacidius Philalethi. There he concludes that bodies are incapable of action while they are in motion, “because there is no moment of change common to each of two states, and thus no state of change either, but only an aggregate of two states, old and new” ([1672–76] 1980, p. 566). Consequently:
there is no state of action in a body, that is to say, no moment can be assigned at which it acts. For by moving the body would act, and by acting it would be changed or acted upon; but there is no moment of being acted upon, that is, of change or motion, in the [End Page 128] body. Thus action in a body cannot be conceived except through a sort of aversion (non nisi per aversionem quandam). If you really cut to the quick and inspect every single moment, there is no action. Hence it follows that proper and momentaneous actions belong to those things which by acting do not change.([1672–76] 1980, p. 566)
The plural here makes it look as if Leibniz is ready to countenance a multiplicity of things capable of acting. But the “minds” that were so prevalent earlier in the year are nowhere mentioned in the Pacidius. Instead, after reaffirming that the action by which a moving body is transferred from one place to another contiguous to it “does not belong to the very body e which is to be transferred” ([1672–76] 1980, p. 566), Leibniz concludes that “what moves and transfers the body is not the body itself, but a superior cause which by acting does not change, which we call God. Whence it is clear that a body cannot even continue its motion of its own accord, but needs to be continuously propelled by God, who, however, acts constantly and in accordance with certain laws by virtue of his supreme wisdom” ([1672–76] 1980, p. 567).
But Leibniz is troubled by a further problem concerning the action of bodies which the appeal to “certain laws” and the constancy of divine action does not resolve. For even if God conserves relative motions, the total quantity of motion in the universe is not thereby conserved: some action is always lost in a head-on collision, unless it is perfectly elastic. Consequently: “This meant that velocity would always be diminished, and would never be restituted” ( 1991, p. 808). And try as he might, Leibniz was unable to derive a “law of compensation” for such velocity losses from the nature of matter alone. As he recollects in the Phoranomus:
But even when everything is taken into account, I noticed that in the end no exit could be found through such rules. For, granted it may happen that an action that was destroyed or diminished in matter by a head-on collision could in turn be increased or restored by a body running into the former body or one at rest; still, that there should always occur in nature a compensation, either somewhere else, or rather, such as we experience in those very bodies which are colliding or very close together, this, I began to see, really could not be obtained from these rules of motion alone, however they were combined. . . . From these and many other arguments that I finally put together, it is established that the nature of matter is not yet sufficiently known to us, and that no reason can be given for either the inertia or the power of bodies unless there is [End Page 129] something other than extension and impenetrability in bodies.( 1991, pp. 808–9)
We know how this story ends. In January of 1678, Leibniz finally discovers that the true measure of activity or power in bodies is given not by mv but by mv2, and that this quantity is always conserved. It is this that allows him to relocate activity in bodies in the simple substances (replacing the former “minds”) that are in them, and that allows him to abandon the occasionalist position that there is only one thing “which by acting does not change” for the view he seems to have preferred all along, that such substances are to be found everywhere. Thus where in February 1677 motion and its effects are said to be “affections of the whole world,” by the following February Leibniz is able to talk of effects of motion as powers in bodies: “there seem to be two effects of motion, one in the mind, namely appearances, and the other in another body, namely powers” (Leibniz 1906, p. 115). Where previously God had been required to guarantee agreement among the phenomena of motion seen from different respects, according to laws of motion governing his actions, now Leibniz can describe the laws of motion as emanating from the constitutions of individual things, that is, from bodies insofar as they are united by a principle of activity, a force or power operating within them. As Leibniz wites in the Mira of 1683:
And just as color and sound are phenomena, rather than true attributes of things containing a certain absolute nature without relation to us [sine respectu nobis], so too are extension and motion. For it cannot really be said just which subject the motion is in. Consequently nothing in motion is real besides the force and power vested in things, that is to say, beyond their having such a constitution that from it there follows a change of phenomena constrained by certain rules.(1982–89, p. 294)35
These rules, of course, would be the laws of motion: conservation of quantity of relative motion, and of vis viva. But as he explains in another [End Page 130] manuscript of this period, although these laws are “in body,” they cannot be derived from material considerations: “There are certain things in body which cannot be explained by the necessity of matter alone; such are the laws of motion; which depend on the metaphysical principle of the equality of cause and effect” (1982–89, p. 51).
The latter, of course, is what Leibniz had referred to as his “Ariadne’s Thread.” Others have shown how it facilitated his escape from the maze involving the laws of motion through his discovery of vis viva; 36 and it is perhaps to this maze that the figure of Ariadne’s thread alludes. At any rate, its connection with the continuum problem appears to consist simply in the fact that the discovery of conserved powers in bodies allowed Leibniz to reposit activity in them, and thus to abandon the discontinuist position of 1676–77 in favor of a real continuity of motion. But this itself deserves some explanation. For the discovery of the true measure of force does not block Leibniz’s inference from the relativity of motion that only relative action can be discerned, nor is it easy to see how it does anything to offset Leibniz’s penetrating conclusion concerning momentaneous action, that it cannot be found in a body. In what follows I shall try to sketch an explanation, and at the same time tie together the main themes of this essay.
We saw earlier that according to Leibniz bodies cohere by virtue of their parts having a motion in common, motus conspirans; and also that, because of the actions of the surrounding plenum, every body is actually infinitely divided by the differing motions created within it. Again we saw how Leibniz interpreted this actually infinite division in the Pacidius as a partition of matter “like a tunic into folds,” so that although some parts might retain their form for a while, there would be no time in which some change did not occur. In a manuscript of around 1686, Leibniz subjects this view to an interesting reexamination, arguing that one consequence of the actual infinite division of matter is that “there is no body that has any figure for a definite time, however small that might be.” For “although it is true that one could always draw an imaginary line each instant, this line will endure in the same parts only for this instant, because each part has a movement different from every other, since it expresses the whole universe in a different way” (1982–89, p. 1478). 37 Yet “what exists only at a moment has no existence, since it starts and finishes at the same time” [End Page 131] (1982–89, p. 1478). Alluding now to his proof in the Pacidius that “there is no middle moment, or moment of change, but only the last moment of the preceding state and the first moment of the following state,” he objects that “this supposes an enduring state” and “all enduring states are vague; there is nothing precise about them” (1982–89, p. 1478). Consequently, he reasons:
In an instant, without motion being considered, it is as if the mass were all one; and thus one can give it any such figures one wants. But also all variety in bodies ceases; and, consequently, all bodies are destroyed. For motion or endeavor makes their essence or difference. And in this moment everything reverts to chaos. Endeavor cannot be conceived in mass by itself.(1982–89, p. 1479)38
That is, to put it positively, the endeavor or momentaneous action that cannot be found in body except by a sort of aversion must nevertheless exist, since it is the very essence of body. Instantaneous actions or tendencies to change state must exist, or “all variety in bodies would cease”; indeed, all phenomena would cease, since whatever has no variety cannot be sensed, and whatever is in principle imperceptible does not exist. But it is substance, and by the above arguments not anything phenomenal, that consists in an endeavor or instantaneous tendency to change its relations to all other things according to certain laws contained in it. Thus even if continuity cannot be found in the phenomena of motions, all such motions presuppose a continuity of activity that is resolvable into instantaneous changes of state or endeavors (what Leibniz later calls “appetitions”).
This, I believe, is the key to understanding Leibniz’s emergence from his labyrinth. When he relocates activity in bodies, it is an inferred activity, not the observable one, that he has in mind. The endeavors are no longer those of bodies; they cannot be conceived in mass by itself. Rather they belong to the substances presupposed in body. For there must be subjects of change and activity, even if they cannot be discerned from the phenomena of motion. Thus, continued motion presupposes beings which by acting do not change, and whose action consists in their changing relations to all other such beings, and the continuity of whose actions requires an endeavor or appetition at each instant.
In conclusion, I hope I have shown that considerations concerning the physical problems of the cohesion of bodies, the actually infinite division [End Page 132] of matter, and the continuity and relativity of motion, were of vital importance to Leibniz’s escape from the labyrinth of the continuum. The cohesion of bodies, originally explained in terms of an endeavor to penetrate one another, and a resultant continuity, reduces to a harmony of motions; the problem of fluidity and solidity is resolved by a new interpretation of actually infinite division, according to which each ‘part within part’ is moving with a different motion, and endowed with a different degree of tenacity. Thus the material world at an instant is a particular partition, and space the aggregate of the resulting parts, and it is necessary to have recourse to a universal mind to restore temporal continuity. Motion, however, is only relative, and therefore an affection of the whole system of relations among bodies, rather than residing in them individually. Moreover, instantaneous actions can only be conceived in bodies through a sort of aversion. But Leibniz’s discovery of his measure of force or activity in bodies enables him to reposit continuity in individual things, each substance being a certain force of acting or endeavor to change its relations to all others, in accordance with internal laws. Now the harmony of motions resides in the system of these mutual relations; as these change, the body associated with the substantial form folds and unfolds. The phenomenal world at any instant is constituted by the harmony among the points of view of all the constituent substances, especially the points of view corresponding to different attributions of rest and motion. Leibniz sums all this up very nicely in a passage from another manuscript of around 1686, “Motion is not Absolute,” so let me give him the final word:
And indeed every single substance is a certain force of acting, or an endeavor to change itself with respect to all the others according to certain laws of its own nature. Whence any substance whatever expresses the whole universe, according to its own point of view. And in the phenomena of motions this fact is especially apparent, since every single body that one posits there must have a motion in common with some other, as if they were in the same ship, as well as its own motion, reciprocal to its bulk; how this could be so could not be imagined if motions were absolute and every single body did not express all others.(1982–89, (Ve447) p. 2047)39
Richard Arthur is professor of philosophy at Middlebury College in Middlebury, Vermont. His current research interests include the history of philosophy and science, the philosophy of time and space, and cosmology. He is the author of the forthcoming Leibniz: The Labyrinth of the Continuum, Writings of 1672 to 1686. He is also currently at work on Matters of Moment: Studies on Time in Early Modern Natural Philosophy, a treatment of time in the mathematics, physics, and philosophy of Galileo, Torricelli, Descartes, Gassendi, Barrow, Newton, Huygens, and Leibniz.
1. Cf. the seventh item in his prospectus for an encyclopædic work, Guilelmi Pacidii de Rerum Arcanis of 1676: “Labyrinthus Posterior, seu de Compositione continui, tempore, loco, motu, atomis, indivisibili et infinito”—i.e. “the second labyrinth, or on the composition of the continuum, on time, place, motion, atoms, the indivisible and the infinite” (Leibniz [1672–76] 1980, p. 527). All translations given in this paper are my own.
2. These Latin dialogues have recently been transcribed and annotated in a critical edition by André Robinet in Leibniz ( 1991).
3. “Ut igitur ex illo Labyrintho me tandem expedirem, non aliud filum Ariadnæum reperi, quam æstimationem potentiarum assumendo Principium, Quid Effectus integer sit semper æqualis causæ suæ plenæ.”
5. I say first major attack, for, as some excellent recent studies have shown, by 1670 Leibniz’s engagement with the continuum problem was already well developed. Philip Beeley (1996) has shown its origins in Leibniz’s student dissertations of 1663 and 1666, in connection with the problem of distinguishing discrete from continuous wholes, and in laying a foundation for quantity. Christia Mercer, in her essay with Bob Sleigh (1995), has shown how it is intimately connected with his early theological projects, particularly the Catholic Demonstrations of 1668.
6. “videbam corpus motum ab eodem quiescente singulis momentis eo saltem differe quod corpus in motu positum semper habet conatum quendam, seu (ut verbo Erhardi Weigelii insignis in Saxonia Mathematici utar) tendentiam, hoc est initium pergendi” (Leibniz  1991, p. 805).
7. “Punctum . . . [est] cujus extensio nulla est, seu cujus partes sunt indistantes, cujus magnitudo est inconsiderabilis, inassignabilis, minor quam quae ratione, nisi infinita ad aliam sensibilem exponi possit, minor quam quae dari potest” (Leibniz [1663–72] 1966, p. 265).
9. “(10.) Conatus est ad motum, ut punctum ad spatium, seu ut unum ad infinitum, est enim initium finisque motus” (Leibniz [1663–72] 1966, p. 265).
10. See, for example, Proposition 36 of Propositiones Quædam Physicæ: “Si materia sibi relicta esset, omnia magis magisque accederent ad æquilibrium universale”, and its corollary: “Si omnes substantiæ corporeæ sunt, motus sensibilis cessavit ante datum quodlibet, id est ab æterno, ac proinde nunquam fuit” (Leibniz [1672–76] 1980, pp. 65–66).
11. Cf. Aristotle’s definitions in his Physics VI 1, 231a 19, and Metaphysics XI, 1069a 1–15.
12. Cf. Leibniz to Oldenburg, 28 September 1670: “Statuo igitur: quæcunque ita moventur ut unum in alterius locum subire conetur, ea durante conatu inter se cohærent” (Leibniz [1663–85] 1926, p. 64).
13. “(17.) Nullus conatus sine motu durat ultra momentum præterquam in mentibus.”
14. Here I am indebted to an unpublished paper by Catherine Wilson, delivered at the same Dibner Symposium as this was: “The significance of inertial circular motions in Leibniz’s Paris Notes, with reference to Aristotle, Hobbes and Descartes.”
16. —an infelicity which Leibniz in any case corrected in “De Minimo et Maximo” of fall 1672, where indivisibles are no longer identified with the inassignable parts of the continuum, but are instead identified with minima, and rejected accordingly as nonexistent ([1672–76] 1980, [no. 5] pp. 97–98).
17. Thus in De Materia, de Motu, de Minimis, de Continuo of December 1675: “(2) Change cannot cease, or whatever is moved will go on moving. And by the same token, it cannot begin. (3) Every body is in motion. For every body is movable, and whatever is movable has been moved. (4) To be in a place is to traverse a place, because a moment is nothing; and every body is in motion” ([1672–76] 1980, [no. 58] p. 470).
18. “[A]libi enim demonstravi, nuperrime, conatus esse veros motus, non infinite parvos.”
19. “Ex motu quatenus conspirat oriri cohæsionem, duobus habemus experimentis gypsi fusi, quod bullas agit, et limaturæ chalbydis cui admovetur magnes quæ abit in fila. Ut nil dicam de vitrifactione . . . “ (That cohesion arises from motion insofar as it is harmonious, we have from two experiments: that of plaster when poured, which forms bubbles; and that of iron filings, which, when a magnet is moved towards them, turn into threads—to say nothing of vitrification . . . ) (1982–89, p.492).
20. See Descartes, Principia Philosophiae, part 2, § 33–35. Descartes had tried to mitigate this actual infinite by claiming that it was the parts of matter that were indefinitely many (see part 1, § 26–27, as well as part 2, § 34–35). But Leibniz had already argued in 1671 for the actual infinitude of the parts of the continuum on the grounds that “Descartes’s ‘indefinite’ is not in the thing, but the thinker” ([1663–72] 1966, p. 264).
21. “It seems to follow from a solid in a liquid that a perfectly fluid matter is nothing but a multiplicity of infinitely many points, i.e. bodies smaller than any that can be assigned. . . . Perhaps it follows from this that matter is divided into perfect points, i.e. into all parts into which it can be divided.”—“On the Secrets of the Sublime,” 11 Feb. 1676 (Leibniz [1672–76] 1980, [no. 60] p. 474).
22. From 15 April 1676 ([1672–76] 1980, [no. 71] p. 509); and from “Notes on Science and Metaphysics,” 18 March 1676 ([1672–76] 1980, [no. 36] p. 393), which reads in full: “Since, therefore, I have established on other grounds that there is some portion of matter that is solid and unbreakable—for no adhesive can be allowed in the primary origins of things, as I judge to be easily demonstrable—and since, moreover, connection cannot be explained in terms of matter and motion alone, as I believe I have shown satisfactorily elsewhere; [—Above this Leibniz has written ‘error’.—] it follows that thought enters into the formation of this portion, and, whenever it has a single mind, it becomes a single and indissectible body, i.e. atom, of whatever magnitude.”
23. On 15 April, for example, Leibniz writes “Corpus incorruptibile pariter ac mens, varia circa ipsum organa varie mutantur” (The body is just as incorruptible as the mind, but the various organs around it are variously changed) ([1672–76] 1980, [no. 71] p. 509); and he has already relegated pure figures like the circle and the parabola to the realm of the fictitious ([1672–76] 1980, [no. 69] pp. 497ff.).
24. Mira de natura substantiae corporeæ (“Wonders concerning the Nature of Corporeal Substance”), c. March 1683; Leibniz (1982–89 [no. 82], p. 294).
25. In a note added at the top of the Pacidius, Leibniz remarked: “Here are considered the nature of change and the continuum, insofar as they are involved in motion. Still to be treated are, first, the subject of motion, so that it may be clear which of two things changing their mutual situation we should ascribe the motion to; and second, the cause of motion, or motive force” ([1672–76] 1980, p. 529).
26. Reading reale with Gerhardt for Robinet’s reali.
27. The phrase omnia circa ipsum agantur evokes Seneca’s circa nos deus omnia an nos agat (“whether god moves everything around us, or moves us around”), which Leibniz quotes in a letter to Perrault of this year, as well as in a later piece on the relativity of motion. The cited text occurs at the end of a passage in Seneca’s Natural Questions (Seneca 1972, pp. 230–31) in connection with the heliocentric hypothesis of Aristarchus of Samos.
28. “Notandum tamen motum non in se formaliter, sed ratione causae considerando, posse attribui eius corpori a cuius contactu provenit mutatio.”
29. In his contribution to the Dibner Symposium, Daniel Garber argued that Leibniz moved from a position of “weak equivalence of hypotheses,” where one can actually assign motion and rest on the basis of forces, to a “strong equivalence of hypotheses,” where this is no longer considered possible. Reluctantly I cannot agree with this analysis. On my view, although Leibniz offered several not entirely satisfactory accounts intended to “save” our normal attributions of motion, he never thought that one could discern from the phenomena which body is in motion “in full metaphysical rigor.”
30. In his Propositiones Quædam Physicæ of 1672 Leibniz says: “Notio Vacui a falsa Loci seu Spatii, . . . notione oritur, quæ me quoque diu decepit. Ita nimirum animis primo obtutu occurrit, spatium esse substantiam quandam immobilem, instar vasis universalis” ([1672–76] 1980, pp. 55–56).
31. “[E]x additione molis et massæ resultant spatia, loci, intervalla, quorum aggregata dant Spatium Universum, sed hoc spatium universum est Ens per aggregationem, continue variabile; compositum scilicet ex spatiis vacuis plenis[que], ut rete, quod rete continuo aliam accipit formam, adeoque mutatur . . . “ ([1672–76] 1980, p. 519).
32. “[A]t basis spatii, ipsum per se extensum, indivisibile est, manetque durantibus mutationibus. . . . Immensum itaque est, quod in continua spatij mutatione perstat. . . . Ipsum autem immensum est Deua quatenus cogitatur esse ubique . . . “ ([1672–76] 1980, p. 519).
33. “Concludere cogar . . . corpus aliquandiu durans in uno loco, immediate post fore in alio licet dissito, id est materiam hic extinctam, alibi reproduci. Mens vero salva semper manet, quæ ipsi assistit” ([1672–76] 1980, p. 494).
34. “Posito Motum esse reproductione de distantia in distantiam, jam illud præterclare patet, multo magis, quomodo Deus sit immediata omnium causa. . . . Hinc demum patet admirabiliter causa rerum, et productio ex nihilo. Mens tamen semper persistit” ([1672–76] 1980, p. 494).
35. “Et quemadmodum color et sonus, ita etiam extensio et motus sunt phænomena potius quam vera rerum attributa quæ sine respectu ad nos absolutam quandam naturam contineant. Revera enim dici non potest cuinam subjecto insit motus, et proinde nihil in motu reale est, præter vim et potentiam in rebus inditam, seu talem earum constitutionem, ut inde sequatur phænomenorum mutatio certis regulis alligata.” In a similar vein, Leibniz writes in De modo distinguendi phænomena realia ab imaginariis (c. 1683–1686), “Concerning bodies I can demonstrate that not only light, heat, color and similar quantities are apparent, but also motion, figure, and extension. And if anything is real, it is only the force of acting and being acted upon, and so the substance of body consists in this (as if in matter and form). But those bodies which do not have a substantial form are merely phenomena, or at any rate aggregates of true bodies” (1982–89, [no. 124] p. 481).
37. “Il est vray qu’on pourra tousjours mener une ligne imaginaire chaque instant, mais cette ligne dans les mêmes parties ne durera que cet instant, parce que chaque partie a un mouvement different de toute autre puisque elle exprime autrement tout l’univers. Ainsi il n’y a point de corps qui ait aucune figure durant un certain temps quelque petit qu’il puisse estre.”
38. “In instanti le mouvement n’estant pas consideré, c’est comme si la masse estoit toute unie; et alors on peut luy donner telles figures qu’on veut. Mais aussi toute la varieté dans le corps cesse; et par consequent tous les corps sont detruits. Car le mouvement ou l’effort fait leur essence ou difference. Et dans ce moment tout revient en caos. L’effort ne sçauroit estre concû dans la masse seule.”
39. “Et vero unaquæque substantia est vis quædam agendi, seu conatus mutandi sese respectu cæterorum omnium secundum certas suæ naturæ leges. Unde quælibet substantia totum exprimit universum, secundum aspectum suum. Et in phænomenis motuum res egregie apparet, unumquodque enim corpus ibi supponi debet habere motum communem cum altero quovis, quasi in eadem navi et proprium moli reciprocum; quale quid fingi non posset, si motus essent absoluti nec corpus unumquodque omnia alia exprimeret.”