Leibniz's Theoretical Shift in the Phoranomus and Dynamica de Potentia - Francois Duchesneau

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rspectives on Science 6.1&2 (1998) 77-109

Leibniz's Theoretical Shift in the Phoranomus and Dynamicade Potentia

François Duchesneau

Figures

This paper is concerned with the nal stage in the invention of the dynamics or theorwhich forms the core of Leibniz's contribution to physics. It tries to determine the rethis nal stage took place and to unravel the arguments Leibniz developed to establish version of his theory. In his article De primæ philosophiæ emendatione et notione substantipublished in 1694 in the Acta Eruditorum, Leibniz mentions for the rst time in print that set the ground for a new science calleddynamica and centered on the notion of force (GP I468-70). He adds that he expects the notion of force will bring much light to our understhe notion of substance (GP IV, p. 469).1 Indeed, his use of the term "dynamics"(dynamidynamice) can be traced back in various correspondences to an apparently initial occurreexchange of letters, begun in 1689, with Rudolf Christian von Bodenhausen. It is evident that the principles of the new science form the subject matter of the Dynamica de poten

et legibus naturæ corporeæ, a major scientic work Leibniz started working on in late 1689. Just previous to setting on that ambitious project, Leibniz, in July 1689, had undwrite thePhoranomus, a dialogue which was left incomplete, but wherein he sketched somarguments that, appropriately modied, found their way into the Dynamica. My larger project iunderstand the genesis, nality, and argumentative structure of the theoretical construform the dynamics. But this paper will focus more specically on describing the methshift in the analysis of force that took place in thePhoranomus. The shortcomings of this inattempt seem to have occasioned[End Page 77] the arguments Leibniz recast and developthe Dynamica. 2 ThePhoranomus may thus serve as a key opening to one of Leibniz's majorless known intellectual achievements in the area of physical science.

First, I shall briey recall the initial steps, beginning in 1676, that led to the 1689 prpresent the circumstances, structure and objectives of thePhoranomus. Second, I intend to shthat the main theoretical challenge Leibniz was faced with at the time he wrote thePhoranomwas that of counteracting the Cartesians' objections against his demonstration of the pconservation ofvis viva. Third, I shall try to characterize the new way of proof Leibnizplace which abstracts from our particular system of things in an effort to determine a the moving force is expressed in uniform unconstrained motion. Certain methodologicaguide thisattempt, especially the principle of the equality between full cause and entiFourth, Ishall investigate the concepts and models framing themathesis mechanica which

supposed to uncover the "formal effect" of moving force as it expresses the form of the

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self-restoring agent. The main problem in this case is to nd means of equating maexpressions and metaphysical reasons and to provide models more general than those pethe experience of falling bodies and gravitational effects. The strategy to cope with thiconsists in forming a system of abstract denitions according to which models can be cso to speak, a priori. Finally, it seems appropriate to turn to the Dynamica de potentia to discovhow the same methodological pattern overcomes the shortcomings of thePhoranomus. Thcentral ingredient of the new theory is a concept of "formal action" which grounds tmodels of the dynamics dened as the science of "power and action." This concept Leibnizian invention which determines the structure and signicance of the mature dyna

1. Leibniz's Reformed Mechanics and the Project of the Phoranomus

A complete genealogy of Leibniz's views on mechanics should take into account the initwhich he formulated in theTheoria motus abstracti and Hypothesis physica nova (1671). Bthose views were to be drastically revised at a later time. The real story begins when Leihis return to Germany from France in 1676, reconsiders the laws of motion betweenbodies. His ambition then was to go beyond the relativist statements of laws proChristiaan Huygens, John Wallis, and Edme Mariotte, and to set the system of mechaunder the aegis of a geometrical demonstrative framework (Duchesneau 1994). He b

had found the key to a possible conciliation between the[End Page 78] empirical laws of moand an a priori principle of conservation: this key was the regulative principle of eqbetween full cause and entire effect. In the De corporum concursu, dated January-February 1a text that Michel Fichant has edited recently for the rst time (Leibniz 1994), Leibnizto systematically deduce the various cases of motion between colliding bodies in accora principle akin to Descartes's principle of the conservation of quantity of motion measuproduct of mass and speed(mv). He compared the calculations derived from his theoreticawith the results of an experiment based on the properties of the pendulum and destined respective displacements following the collision of unequal moving bodies--equaconsidered only as the limit of a progressive series of unequalities. One body being a

motion of the other would be produced by a fall from various heights. Leibniz couwitness that the inferences drawn from that model did not match the data collectedexperiment. So he decided to modify and recast the deduction by basing it on the conservation of the product of mass and height of fall before collision--a methodologicwould call areformatio. This time, the agreement between the various cases tted the requof the equality between full cause and entire effect in a continuous progression. In ouuniverse, the system would be organized in such a way that the equivalence is maintainphenomenal changes. However, even if, from then on, Leibniz was in possession of his conservation ofvis viva according to which the moving force conserved in mechanical inteis measured by the productmv 2 , he still could not account adequately for the subjacent for

Signicantly, Leibniz only makes his reformed mechanics public when he attains a formulation of his philosophical system. Thus, his rst ofcial challenge to Cartesian took place in the Brevis demonstratio erroris memorabilis Cartesii published in 1686 in the Ac Eruditorum (GM VI, pp. 117-23), the arguments of which were integrated into the Discours métaphysique in the same year (GP IV, §§17-18, pp. 442-44). Opposing those who supmechanics strictly consisting of empirical laws, Leibniz joined with Descartes in acknthat a principle of conservation of potentia motrix needs to belong among the principal ingreof physical explanation. But, relying upon Galileo's law of free fall and on axioms adCartesians, he established the disparity between the Cartesian measure of quantity of m

the true requisites of the concept of moving force, and he showed that the power cons



be measured by the productmv 2 .

Leibniz's demonstration proceeds this way. It starts from two premises Cartesiaadmit: (1) A body falling from a certain height acquires[End Page 79] by its fall force to lift itself back to its initial position; and (2) As much force is require

body A of one pound to a height CD of four feet as is required to lift a body B of four pheight of one foot (seeg. 1). Combining the two propositions makes it possible to consthe forces gained by lifting A to C and B to E are equal to each other and proportioheights of fall CD and EF. Then, Leibniz resorts to Galileo's (empirical) law of free f

fall, the spaces run through are proportional to the squares of the times spent to that efthe speeds are proportional to the square roots of the spaces, i.e., the heights of fall. Rdistance CD, body A has acquired a speed proportional to 2, while running the distanceB has acquired a speed proportional to 1. Considering the productsmv on both sides, we wouldquantities of motion proportional to 2 on the one hand, to 4 on the other. Hence conservation of quantity of motion. If we take the heights of fall to be respectively propthe masses of the moving bodies, and if we take the product of mass and height to eforce as it is entirely spent in the effect of lifting, we get a measure of force based on tmv 2 . This estimation according to the effect produced presumes that the body lifting itsits initial position exhausts all the force it has acquired previously by its fall in conform

principle of the equality between full cause and entire effect. Thus, this principle ruleanalytic demonstration of the principle of conservation ofvis viva.

At this stage, Leibniz has not set forth the structure of the dynamics[End Page 80] yet. But explicitly suggests that the theory should conciliate empirically grounded modesubstratum of entities expressing the immanent order of nature. Force is presented as a concept exceeding the intelligibility of geometrical concepts. And this new concept is pown considerable regulative power for unifying the various empirical laws.

The next stage concerns the invention of the dynamics proper. In the period 1686-89, L

been busy counteracting the Cartesians' objections voiced by François Catelan anMalebranche.3 Through the polemics, Leibniz was confronted with the problem of justiprinciple of conservation as a true foundation for the system of laws of nature, since hiwould depend on an empirical generalization, Galileo's law of free fall, as well as on features of our physical world such as elasticity and gravity. For Leibniz, a true sciencestructure of demonstrative arguments that can be exposed synthetically. In the mechanics, Leibniz used analytic devices which provided a sort of a posteriori unfoldnew conservation principle. It seems appropriate now, both for polemical and methreasons, to verify the capacity of this principle for grounding a deductive synthesis of tnature (by combination of equations). We should remind ourselves at this stage that o

main tenets of Leibniz's methodology as formulated in his numerous projects for a demencyclopedia is thede jure correspondence between analysis and synthesis, orcombinatoria, evidenced both in thears demonstrandi and ars inveniendi (Duchesneau 1993). The anaprocesses elaborated to prove thevis viva principle should normally be replaced by a syordering of reasons that would illustrate the role of this principle as a keystone in physiThis goal seems to have provided the chief motivation for Leibniz's enquiries leaddynamics.

In March 1689, Leibniz started on a year-long journey to Italy (Robinet 1988). His Italiafforded a rather propitious context for attempting a synthetic recasting of the

mechanics. Throughout his encounters with Italian scientists, the challenge was indeed


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to show that he possessed the means to supersede both Cartesian and Galilean models fphysical theory, and that he was on his way to a new system of natural philosophy baformal nature of force. It is most probably with this objective in mind that Leibniz unwrite thePhoranomus (1689) and Dynamica de potentia (1689-90).

Until very recently, thePhoranomus seu De potentia et legibus naturæ was[End Page 81] knowonly from the mention of its title by Louis Couturat in connection with another fragmtranscribed--which has nothing to do with it except being the next piece according to Bcatalogue of Leibniz's Handschriften (C, p. 590)--and from a fragment of Dialogue II thGerhardt had transcribed and edited (Gerhardt 1888). André Robinet has recently editedPhoranomus (Leibniz 1991a; also 1991b). We know for sure that this important work wain a very short timespan during the summer of 1689, while Leibniz was staying in having regular meetings with representatives of the Roman intelligentsia. It is also clecomplete, but not fully recopied,Phoranomus was soon to be left aside by Leibniz whose ihad shifted to writing the Dynamica de potentia. The various chapters of the latter workthrough a series of drafts during Leibniz's stays in Rome, Florence, and Venice. And Bodenhausen in Florence was sent those drafts, a neat version of which he was suprepare for publication. After his return to Germany in spring 1690, Leibniz felt lesinclined to complete and publish the work, and after Bodenhausen's death on 9 Aprirequested his papers back. The Bodenhausen copy was published by C. I. Gerhardt in VI, pp. 281-514).

Like thePacidius Philalethi (1676), also a transition piece, thePhoranomus is cast in the forma dialogue. It possesses strong realist overtones, and it seems to reect discussions Leihave had in Rome with members of the Accademia sicomatematica, whose sessions he attended. In fact, in two successive sessions, narrated respectively in Dialogues I aPhoranomus presents Leibniz and some members of that academy pursuing arguments ocentral to the reformed mechanics and to the dynamics to be. The introduction is affoletter supposedly addressed by Leibniz to Melchisédech Thévenot, a member of the AcaSciences in Paris, the emphasis of which concerns a critical evaluation of Descartes's pand science. Leibniz had received news from Simon Foucher that Daniel Huet had just pCensura philosophiæ cartesianæ (1689). He recalls that he had himself contributed to theappraisal of Descartes's physical hypotheses. Such a critique may be found for instaletters to Hermann Conring of 3/13 January, 19/29 March, and June 1678 (A II, 1, p397-404, and 418-20). Concerning physical hypotheses, he had felt the need to cocomplete Descartes's work. In particular, the errors in Descartes's laws of nature had oLeibniz to undertake such revisions as thePhoranomus tries to account for: "Indeed, as Descfailed in setting out the laws of nature, he provided me with the occasion for establishiones" (I, §1, Leibniz 1991a, pp. 446-47).

Left incomplete and faulty, thePhoranomus affords a preliminary integration[End Page 82] the mechanics of force which the Dynamica de potentia develops and articulates into a systewhole following on the work done during the summer of 1689. Previously, e.g., i Dcorporum concursu and Brevis demonstratio, Leibniz had grounded his new principconservation of force measured bymv 2 on an at least partly a posteriori demonstrationdemonstration relied on Galileo's law of free fall and it pertained to a system of the univelasticity and gravity would be found--properties which Leibniz and his contemporariexplain by resorting to hypothetical models. Indeed, Leibniz relied also on axioms infArchimedean statics and reformulated by seventeenth century mechanists. Above all, headvantage of an architectonic principle, the principle according to which the entire effecause must be considered equivalent through mechanical interactions. Such a principle



the requirement of sufcient reason as it would apply to the phenomenal realm of motiothe moving force generated by a body falling from a given height is fully exhaussubsequent lifting of that body to its initial height, the product of mass and space of traproduct which is proportional with the square of speed, serves to measure the force tthrough a sequence of equivalent generations and exhaustions. But is not such a conserrestricted to applications within the bounds of a specic system of phenomena? Dfeatures of this system pertain to an empirically grounded model? If this were the case--to be--the Leibnizian reformed mechanics would be justied by a set of proofs built anda posteriori, at least for a signicant portion of the premises involved.

The challenge Leibniz sets himself in thePhoranomus consists in trying to generalize his meof moving force so that it can apply to all phenomena involving motion. The program reequivalent causal circumstances be identied, both in violent motion manifestingcollisions with generation and exhaustion of the moving force, and in unconstrain(innocuus motus) manifesting itself in uniform translation without interference of any faor impeding extrinsic determination. The latter is what Leibniz calls"motus æquabilis qualis pse est" (II, §§F and G, Leibniz 1991b, pp. 820-21).4 As he attempts to account for the underlying such unconstrained motions, Leibniz develops a system of a priori argumentpremises of the empirical[End Page 83] sort would serve only as corroborative instances.

this project is complete, he is in a position to claim that he has founded a science of whose specicity and theoretical potential he can spell out. While thePhoranomus still speaks itentative fashion of a new science concerning force and effect,"nova de potentia et effscientia" (II, §G, Leibniz 1991b, p. 826), the subsequent major text, the Dynamica de potendeals demonstratively with a new science concerning force and action, and this new henceforth properly identied by the name and concept of dynamics. Thus in theSpecim præliminare of the Dynamica Leibniz states:

I judged that it was worth the trouble to muster the force of my reasonings throughdemonstrations of the greatest evidence, so that, little by little, I might lay the

foundations for thetrue elements of the new science of power and action, which onemight calldynamics. I have gathered certain preliminaries of this science for specialtreatment, and I wanted to select a ready specimen from these in order to excite cleminds to seek truth and to receive the genuine laws of nature, in place of imaginaryones. (GM VI, p. 187; Leibniz 1989, p. 107)

Similarly, at the end of the rst section of the second part, Leibniz mentions: "It seems thave thus disclosed the sources yet quite unexplored of thedynamical science of power action" (GM VI, p. 464).5 The parallel expressionsscientia de potentia et effectu and scientia potentia et actione clearly indicate the change of focus between thePhoranomus and t

Dynamica de potentia: the main shift takes place with the concept of action substitutingconcept of effect and providing a basic theoretical equivalent for the formal identicatioIn the former work, effect means the distance run through conjoined with the speed of tas if this effect would exhaust the force involved while conserving it virtually all alolatter work, formal effect as measured by the distance run through does not entail exhaonly extensive application of the immanent force, while formal action combines formal a consideration of the time spent, which manifests intensive application of the same force. We shall discuss these distinctions further later on.

Prior to my own analysis (Duchesneau 1994), thePhoranomus had only been accounted fo

André Robinet in the notes of his edition (Leibniz 1991a; also 1991b). He had based







bodies in achieving that result. If the effects involving colliding bodies were comprisedsame measure of time, would not the conservation of quantity of motion prevailconservation ofvis viva? This is indeed the case in the rst instant of summation of without subsequent integration of conatus: in this rst instant, the moving force haaccumulated to be spent in an equivalent exhaustion effect. Such a case is illustratedwhen bodies are xed at the extremities of a balance according to a ratio of distances that of the respective masses; then, the vertical constrained motions effected during theelicit the conservation ofmv, rather thanmv 2 . The Cartesians' argument was that if the effforce is considered in relation to time, it should be measured according to its value in

time, which is analogous to its value in the instant, itself measured by the productmv.Through Baldigiani, another participant in the dialogue, Leibniz has already seized the oof denouncing the paradox which Cartesians fall into as a consequence of Galileo's accofall. In the case of falling bodies, the equivalence between full cause and entire effeconstancy ofmv 2 . But why, then, do people tend to relate the lifting of bodies and their rspeeds? This mistake derives from a prime theorem of statics: two bodies are in equilibtheir respectivestatus are such that if one begins to descend, the motions thus provokespeeds that are proportional to the respective masses of bodies. Hence the unwarranteto infer, as if from purely rational evidence, that "forces of bodies are equal when s

inversely proportional[End Page 86] with bodies" (II, §I, Leibniz 1991b, p. 829). Here, Leibargues, statics provides only a particular case in which heights are proportional to speedthe general case in which the effect of force is truly actualized implies a proportionasquare of speeds. In any case, the powers are equal when the masses are inversely propthe heights at which they may rise. The Cartesian error was to conate the special casebound together and therefore mutually impeding, with the more general case of bodiesbondage in their generating and exhausting potentia motrix:

Hence we get a universal principle which will succeed in statics as well as in the reof general phoronomy, namely that the powers are equal when the weights that can

drawn by the force of those powers are reciprocally as the heights at which they cabe lifted. . . . This was indeed known to the learned, but, as they had envisagedcomparative lifting only in bodies bound together, thus not ascending freely, and athey had not therefore examined bodies in motion with that touchstone, they weredeprived of the excellent fruit of a very general truth, which throws most light upothe communication of motion and interaction of bodies. (II, §I, Leibniz 1991b, p. 8

However, in thePhoranomus, the principal counterattack against the Cartesians' spobjection that the entire effect should be assessed in the unit of time consists in demonsthe temporal restriction imposed on the integral expression of moving forces proves in

with the architectonic rule of continuity. One should acknowledge the apagogic conaccruing from the incompatibility between the temporal clause and the rule accordingneither by a force of gravity nor by any other force the common center of gravity of abodies can rise more in the effect than in the cause.6 This rule derives directly from Huygboat model as the means to establish empirically warranted laws of motion. Huygenentailed two principles of conservation, that of the directional translation of the commogravity for any system of colliding bodies, and that of the relative speeds of those bodand after collision. Leibniz, as well as his Cartesian opponents, admitted these relativeexcept that they differed on the absolute conservation principle they ought to match tThe [End Page 87] rst of Huygens's rules is equivalent to a principle of conservationprovided the productmv is considered vectorially, not scalarly, as with Descartes's rules of




The beauty of Leibniz's argument consists in turning the consideration of time back aobjectors, by stating that their account of time for a scalar measure of force leads themHuygens's model, which is also based on the unit of time but entails no conictconservation of . On the one hand, Leibniz's demonstration relates directly to the ecorroborated hypothesis about conservation of the common center of gravity. But then, raise this auxiliary principle to an epistemological standing of the a priori sort, Leibnizthe counterfactual supposition that, if there were a divergence from conservation of thcenter of gravity, the implicit necessary consequence would be an admission of mperpetual motion. The same type of apagogical demonstration makes it possible to ddisparities that would presumably result from more or less oblique translations of movbecause of the different times spent in producing the same total effect. Finally, Lehimself to the task of systematically overthrowing Catelan's and the Cartesians' main obpressing that in violent motion, the space run through, namely the height attained by mfull expense of force, already implies an account of time in the measure of the squarcorrelative with that height: hence the presumed redundancy of an account of time baccount of the space run through. Indeed, this is the inverse case from that of a motion formal effect without external constraint (II, §R, Leibniz 1991b, p. 848). In short, it issystematic working out of arguments against the Cartesians in the aftermath of the poleCatelan and Malebranche that determines the theoretical construction Leibniz is attempPhoranomus.

3. The A Priori Method of Demonstration

The second point highlighted by Robinet deals with the paradoxes of an a priori modethe formal effect of unconstrained motion, as this effect would jointly be measureexhaustion and entire self-preservation. This model developed in thePhoranomus can be statedfollows. The architectonic premise consists in the principle of equivalence between fullentire effect, but applied outside of the specic case of our world system. In that systeonly interpret the phenomenal intercourse of moving bodies by means of empirically co

auxiliary hypotheses which apply to elastic bodies in the context of impetus generated According to the system of constraints on colliding bodies, moving force is measured tviolent effect it produces and wherein it entirely consumes itself. In the alternative dem[End Page 88] approach Leibniz sought, we should consider force abstractly as it runconstrained motion; that motion would express its causation in and through itunfolding alone. Indeed, this new approach stands as an abstraction from the conditionsto our system of thingshic et nunc. But the same fundamental presupposition, the same Athread previously identied, namely the architectonic principle of equivalence betweenand entire effect, would apply. Though originally introduced in a context of Archimed(I, §18, Leibniz 1991a, p. 477) and of empirically corroborated phoronomy of violent mo7 is now granted a more general meaning and role.In fact, the argumentation Leibniz tries to develop in the a priori mode in thePhoranomus stemfrom Archimedean statics. His interlocutor Charinus resorts to the equivalence beinverse ratio of mass and speed for unequal bodies in equilibrium and transposes it to afree translation on a horizontal surface. That analogy from statics to kinetics in assmoving force generates an aporia. This aporia relates to the "modal" status of speed, with the "substantive," therefore real, status of bodies. This is seemingly what can bfrom the very tortuous passage inPhoranomus, II, §E (Leibniz 1991b, pp. 815-16). If a body two mass units can be granted the double power of a body with one mass unit movin

same speed, this cannot be the case if a body moves with double the speed, compared




body of the same mass moving with one speed unit. The power of the rst body measured as equivalent to twice the power of the second according to themv parameter, when speeds actualized in a given time are measured according to the spaces run through.

As underlined by Lubinianus (Leibniz), it is only by acknowledging a proportionality bainverse ratio of the times spent in motion and the direct ratio of the products of masses run through that a system of equivalences may be restored at the level of the movinvolved. Given the same bodies and the same spaces, the powers will be inversely asspent in translation. It follows that in a given unit of time, the forces will be measured acthe productmsv, or, becauses is equivalent tov in the unit of time, they will be meaaccording to the productmv 2 . But this argument may easily be considered a paralogishence, it will not have much more value than the paralogism according to which a bomass unit with two speed units would be equivalent to a body of two mass units with unit. The defect[End Page 89] in such an equation consists in the absence of congruence bcases, since those cases(casus) would combine states(status) and things(res) disparately, thpreventing the eliciting of homologous relations: "What is in question is not what a mocan do, whatever be the time granted, but how a proposed case may be estimated from and space, and how a given case may be resolved into two equivalent ones, of which icomposed" (II, §F, Leibniz 1991b, p. 817). The thesis Leibniz asserts a priori, rather than

consists in postulating that the combined consideration of the spaces run through ainverse of times renders cases congruent. The twofold spatio-temporalstatus (measured accordto v 2 ) combine homogeneously with theres and would thus determine a measure of the mforce as it is expressed in uniform unconstrained motion:

Thus, through the resolution of bodies into parts, the speed, or space and time, beiconserved, we had inferred, demonstrated, that given the same speeds the powerswere proportional to the bodies. Similarly, we have demonstrated, which isparadoxical, but absolutely true, that, the body being conserved, time and space beresolved jointly (for otherwise the case given could not be divided in several cases

congruent with each other while different),given the same bodies, the powers are proportional to the square of speeds. (II, §E, Leibniz 1991b, p. 816)

So, the justication for this system of equivalences consists in a combination of ratios rmodal parameters: by homologous combinations, these ratios represent the effect of exerts itself freely. On the one hand, no other element of motion could be taken accounsynthetic representation of cases. On the other hand, this combination of modal elemesufce in providing an analytic equivalent for those forces whose effects are expressunconstrained uniform motions. In contrast, notwithstanding Descartes's appraisal wmodelled after Archimedean statics, speed alone could not sufce in expressing this

moving force, nor would it be the case for space run through if considered as the onlparameter.8

Ultimately, as summarized by Baldigiani, one of the participants, Leibniz's thesis boilasserting that the effect equivalent to force in the case of uniform unconstrained moti(motæquabilis qualis per se est) is [End Page 90] determined by the combining of two "requithe space moved through and the speed of motion (II, §F, Leibniz 1991b, p. 820). Robiinterprets this thesis as paralogistic since the effect is not decomposed, as it is in the Dynamica potentia, into formal effect and formal action: in fact, not yet being provided with this dLeibniz cannot justify his measure of the effect otherwise than by combining the exhau

considered now in the unit of time with the same effect transposed in the shape of




equivalent virtual effect that accompanies the actual effect as it exhausts itself. Personmore struck by the fact that the defect in Leibniz's analysis consists in an inadequate between the "extensive" and "intensive" components which would combine in the reactualized effect of a force that is kept intact through the motion it accomplishes. Manything else, Leibniz lacks a causal model to account "metaphysically," that is abstractintegrative combination of intensive and extensive dimensions. Due to that deciency, of congruent cases remains aporetic, and cannot be ultimately warranted but on anderived from the a posteriori demonstration of the principle ofvis viva conservation. Timpression of paradox and the tortuous writing which affect the passages concerning tdemonstration in thePhoranomus are probably due to an insufciently rened mod justifying the combination of parameters pertaining to the action considered in itsexpression.

Despite the shortcomings of the notion of effect on that account, the model deployPhoranomus seems to suggest a demonstrative pattern for generalizing the principlconservation ofvis viva to such a force as would underlie uniform unconstrained motionpower exhaustion. This model appeals jointly to the principle of the equality betweenand entire effect and to a sought for combination ofstatus that, correlated with theres involvemight warrant the congruence of homologous cases. In my view, the very requirememethodological model determine the conceptual renements underpinning the tconstruction the Dynamica de potentia illustrates shortly thereafter. By emphasizing the aelements of a notion of effect that combines the modes of expression of force in both cand unconstrained motion, Robinet fails to see that Leibniz is seeking a way of concepriori construction how the modal elements of force might be combined to acconservation of the substantive elements underpinning both constrained and uncmotions.

4. The New Mathesis Mechanica

In this section, I intend to challenge Robinet's thesis that, in the course of writing thePhoranomuLeibniz would have felt the need to drastically[End Page 91] "reform" his theoretical approIn contrast, I shall develop the view that Leibniz was already clear then about threquirements of what he conceived as an essentially a priori mode of demonstration. my intent is to assess the demonstrative scheme thePhoranomus establishes for the forthcomdynamics. The principle of equivalence is still at the center of the Leibnizian strategy, ban abstract understanding of what force means as the causal element for both construnconstrained motions, the newmathesis mechanica must nd a means of concilimathematical and metaphysical reasons.

I shall start by considering Robinet's point about the causal connection of thePhoranomus withe Dynamica de potentia. Robinet sees in the former text the same type of inventive pronds in the De corporum concursu. 9 In the 1678 manuscript wherein Leibniz rst adopconservation ofmv 2 , Leibniz began by performing an analysis of the cases ofconcursaccording to the norm specied by the principle of conservation of quantity of motio1994; Fichant 1990; Duchesneau 1994, pp. 95-132). On the way, he had been struinconsistency between the theoretical inferences thus conceived and the data providexperiment on the collision of bodies endowed with pendular motions. He had reconsidered the main presupposition--Descartes's absolute conservation rule--in ligcausal equivalence principle and he had proceeded to revise(reformari) the analysis of caaccording to a new conception of the measure of force; this conception thereafter




canonical expression in thevis viva conservation principle. Hence a redrawing of the a"post reformationem." Robinet attempts to uncover an analogous revolution in theapproach in the rather abrupt transition from thePhoranomus, probably written in the secondof July 1689, to the Dynamica de potentia whose initial drafts can be traced back to ASeptember of the same year. The dialectical rupture he sees resides in a two-term csystem(power and effect) in thePhoranomus being supplanted by a three-term system(poweffect, andaction) in the Dynamica de potentia. Only the theoretical framework provided blatter would bring about a complete theory of force, and correlatively a new science, desthe termdynamica. This view of the transition between the two texts hinges on a "revolucausal scheme: in light of his latest theoretical invention, that of the concept of actiowould have felt compelled to revise radically his previous mechanical conceptions.

Considerable nuances should be brought to this historiographical reconstruction[End Page 92]which imagines too drastic a shift in the two-step process resulting in the inventidynamics. It seems that Leibniz's methodological preoccupations following the discovvis viva theorem have justied more directly his search for an a priori demonstration. reasons, at once epistemological and metaphysical, may help explain the contents of thof unconstrained effect. Henceforth, the analysis leads to a new principle, that of the coof quantity of action, synthetically grounded in an abstractly conceived system of repreAs a consequence, that principle and this system make it possible to enlarge the tframework of the reformed mechanics. Thus, the principle of conservation of quantityand the abstract system of force, on the one hand, and the principle ofvis viva conservation the more concrete and empirical system concerning phenomena this principle refers other hand, are integrated under the same theory. Inasmuch as the Leibnizian methoproject should be taken direct account of, thePhoranomus, in its very state of incompleteprovides signicant evidence on the combinative models underpinning the enlargemreformed mechanics in the Dynamica. On that account, my analysis of Leibniz's demonsscheme will mark its distance from the notion of a methodological rupture.

In thePhoranomus, Leibniz situates his program by reference to Galileo's attempt at apmore geometrico demonstrative structure to phenomenal reality. The actors in the dialforth the requirement that experiments should be linked with reasons, and that, to thishould resort to mathematical analysis according to the several techniques available. Buof demonstrative structure means resorting to principles. Beyond statics in the Arctradition, the principles did not seem to have been identied yet that might ground ansystem of demonstrative inferences about phoronomy, that part of mechanics deacommunication of motion and interaction of colliding bodies (I, §5, Leibniz 1991a, pp. 452-5This is the challenge Leibniz wishes to take up in thePhoranomus.

To attempt this demonstrative extension of mechanics, Leibniz uses a key belonging to mathemechanica yet to be completed. He states: "I shall therefore bring forth clearer demothan anymathesis mechanica may have ever seen" (II, §M, Leibniz 1991b, p. 837). This happin a passage where Leibniz tries to answer Catelan's and the Cartesians' objections on tof an extended "geometry." This Leibnizianmathesis mechanica combines several conceptools which Leibniz had used somehow in his previous work on the reformed mechanarchitectonic principle, the principle of equivalence between full cause and entiunderpinning the arguments previously used relative[End Page 93] to the reformed mechan(2) models directly borrowed from innitesimal geometry and used to transcribe stphoronomical phenomena involving transition to a limit; (3) combinatorial models emunravel the contents of theoretical concepts representing the order of causes and effecabstract denitions relating to the order of causes and effects: these denitions are "met



insofar as they exceed the level of what is geometrically representable, and cannot thbased on imaginative analogies. In less analytic fashion, this methodological progannounced in Baldigiani's introductory remarks to Lubinianus (Leibniz):

As . . . geometry is subjected to an analytic calculus by means of the equality betwthe whole and all its parts, so in mechanics by means of the equality between theeffect and the whole of its causes, or between the cause and the whole of its effectscertain so-called equations and a sort of mechanical algebra are reached through thuse of this axiom. Hence you will conjoin a science most useful to life with greatpersonal benet, if you bring us such bright light in the great darkness we are in, aimpose laws not only on statics, which Archimedes had formerly put under bondagbut also on universal phoronomy and the explanation of moving forces. (I, §6, Lei1991a, p. 454)

Charinus's arguments for launching the discussion point to the same methodologicdeveloping amathesis mechanica that would reach the level of abstract intelligibility requiradically new theory of force as potentia motrix:

In geometry and numbers I observe evident principles of unavoidable necessity.Everything gets explained by parts of the same magnitude variously transposed. Bthe moving forces seem to me to possess something incorporeal I do not know of, very little subject to the imagination. Therefore, everytime I would conceive thepowers of machines, I was confronted with something unexplored and not admittinof any image. (I, §8, Leibniz 1991a, p. 457)

The various elements ofmathesis mechanica thus brought onto the stage need investigatiothe question is above all to what degree they combine into an adequate demonstrative st

The principle of equivalence between full cause and entire effect affords a leitmotiv aPhoranomus arguments. Since this very principle ruled over the demonstrations in Dcorporum concursu and Brevis demonstratio, it does not assume now a wholly original, buta more extensive, role. But here Leibniz undertakes a reinterpretation of the whole[End Page 94]of Archimedean statics under the ægis of that principle, and in so doing, he subjects thethe geometrizable to a causal approach in terms of moving forces. The generalizing funprinciple should be stressed in contrast with its previously more restricted usages. An arprinciple serves theoretical explanations in an essential way by providing heuristicalThis heuristical function is twofold: the principle is used, on the one hand, to discmodels, and, on the other, to determine and build models that may optimally satisfy thereason requirement. According to that pattern, the principle of causal equivalencdetermining role in extending demonstrative models from statics to phoronomy, as witsought-for transition from equilibrium and conservation of the common gravity centedisplayed in cases of impact, and further on, to force exerting itself in unconstrained §18, Leibniz 1994, p. 477). While phenomena in statics can generally be regeometrically restricting the expression of causes to mere extensive parameters, the trthe theory of force implies that we deal with notions of cause that need to t the process taking place either in "violent" or "innocuous" motion.10

Framing those determining causes abstractly through adequate models that pertain at omathematics of the innite and to certain theoretical denitions warrants applying the eprinciple to this new domain. In the passage immediately preceding the a priori argumuniform unconstrained motion, Leibniz notes that this "metaphysical" principle has pro




with an Ariadne's thread for the estimation of forces. Insofar as this principle alinferences conformable to the data of experience and free from intrinsic aporias, he hassystem of causes congenial to his underlying commitments, causes which are neither "purely mathematical," as would be the intercourse of atoms deprived of any intelligdiversity and thus entailing some "blind property of nature," but which reect an "order," analyzable in terms of "metaphysical reasons" (II, §10, Leibniz 1991b, p. 811). Workout models, in particular theoretical models to represent force as cause, should make it apply the principle in the area of phoronomy. Hence the critical importance of those mconceiving[End Page 95] how themathesis mechanica can be achieved. Instances of this tymodel are provided in later Leibnizian texts, for instance in theSpecimen dynamicum (169where Leibniz species his typology of primitive and derivative, active and passive forcforth a system of representation for the integration of conatus and impetus that draanalogy of a stepwise summation of innitesimals (GM VI, pp. 234-54).

As noted, such models relate, on the one hand, to a geometrical analysis inspired by incalculus, on the other, to a system of concepts capable of expressing the causal aelements beneath constrained and unconstrained motion. Let us consider the aspect cgeometrical innitist models. Even when he resumes Archimedean statics, and in partheory of barycenters, Leibniz takes advantage of the technique of transition to threpresenting innitesimal ratios. But building this type of model is particularly meanindealing with the representation of accelerated and decelerated motions as effects resucentrifugal or gravitational force. Reciprocally, the models resorted to in accouphenomena of resistance rely on the same logic of analysis in terms of series of inratios.

Leibniz starts with his distinction betweenvis viva and vis mortua. He presumes that the relabetween "dead" and "living force" is analogous with that between the nite and the innbetween the point as the beginning of a line and the line itself. Also, conatus stands to das impetus does to living force (I, §18, Leibniz 1991a, p. 478). Such is the starting poireformed mechanics for the distinctions Leibniz renes throughout subsequent presentatdynamics. The impetus is conceived as generated by a continuous summation of conmoving body. So one passes from an embryonic to a developed dimension of forcconnection, Leibniz recalls that Galileo and Giovanni Alfonso Borelli after him had conforce of percussion to that of gravity, as the innite to the nite.11 Every body in motion possesses an impetus, and that impetus helps express the existence of avis viva generated throua summation of conatus. It is noteworthy that the proposed model, if it draws adeqrelationship of conatus to impetus, fails to indicate clearly how the mathematical exprimpetus differs from that forvis viva. However, in line with his more denitive theses, Ldevelops the relationship of inertia with the[End Page 96] action of conatus, since the resis

of bodies to motion implies inverse summative processes. If any motion, be it so whatever body can act on any other body, be it so large, the speed communicated to tbody and relative to the impetus of the rst will be the smaller, the larger the affecfollowing the ratio between their respective impetus: "If the surrounding bodies do impediment, it is certain that a body of whatever magnitude at rest can be moved by whatever smallness. And the inertia of matter does not consist in that it is absolutely remotion, but in that it will receive less speed, the larger the matter which receives itLeibniz 1991a, p. 480). But, with explanatory reasons derived from the innite summatioelementary ingredients as conatus, how can we get an adequate representation of thegravity so as to account for accelerated motion in fall, and indirectly for the force conse

interaction of bodies? In response, the explanation of gravity is based on a controlled an




the summation of impressions in centrifugal force as conceptualized by Huygens. Aninterprets that analogy according to the specications of his algorithm about the inteinnitesimal quantities; so he distinguishes in centrifugal force thevis impressa and its cumulateffect in theimpetus centrifugus (I, §21, Leibniz 1991a, p. 482). But how can one get from to a model for gravitational acceleration?

Leibniz insists on the fact that gravity is a physical phenomenon whosecauses remain obscure and controversial. Indeed, he refuses to subsunattainable explanation of gravity under a force of attraction God wo

endowed matter with. Rejecting Newtonian attraction--though Newton is not namedsuch an occult quality would contradict the formal requirement of sufcient reason, Leibhimself with Kepler, Descartes, and Huygens in hypothesizing some mechanical cpresumes that by a cause similar to centrifugal impressions and their summation in thimpetus, some very dense imperceptible bodies tend to swerve from the center antowards it less dense bodies endowed therefore with lesser centrifugal propensity (I, §21991a, p. 481). He even proposes an experiment to compare centrifugal force with thgravity directly (I, §22, Leibniz 1991a, p. 482).12 Given a tube in oblique rotation whose end is immersed in a water-lled container, water can rise in the tube to a given heicentrifugal force issuing from rotation (seeg. 2). This makes for an effect correspondi

gravity and can therefore be used to measure gravity. On that ground, Leibniz recastsaccount for the empirical law of fall according to his own model of innitesimal suFurther on, the[End Page 97] same type of model could be applied in trying to establish progression concerning the elasticity and resistance of bodies.13

The third component ofmathesis mechanica consists in using conceptual models ocombinative type. Thus, one nds at the background of the a priori argument a moexplicit combinatorial account of the notion of effect in uniform unconstrained maccordance with the analytic requirements of a similar model, the Dynamica de potentia replacthe incompletely analyzed notion of effect by a notion of formal action combining an

with an extensive parameter--thus will the formal effect be combined with the velocity accomplishment. In his yet imperfect analysis of formal effect in thePhoranomus, Leibniz triescombine several relations that could add up to a determining reason for the exhaustivforce as maintained in a continuous process. One thus gets a series of laws ofmotus æquabilis (I§G, Leibniz 1991b, pp. 821-26), which equate the dynamic factors of mass and forceintegration of the kinetic factors of time, speed and distance run through. That verytransposed and revised in the Dynamica de potentia, so as to make up for the analytic ingreof formal action and thus express a power that restores itself constantly.

The demonstrative value of conceptual combinative models is underlined by the type of

they allow. In the case at hand, Leibniz appeals to a scholastic precept which expresses tmodality of apagogic demonstrations as applied in mathematical physics, starting with[End Page98] Archimedean statics: "The conclusion is true, and no other possible reason is foundthe reason premised is true" (II, §G, Leibniz 1991b, p. 821). In this instance, the truth oconclusion is established a posteriori. But the combinative hypothesis making for an ereason is conceived a priori and helps eliminate any other set of projected sufcient rshort, any other hypothesis but Leibniz's a priori construction would prove defective bsome implicit contradiction in the combination of conceptual ingredients involdemonstrative link depends on the indirect condition that the contrary is deemed impossthe a posteriori argument seems to present the advantage of an almost direct empirical

But, in this latter case, if the validity is immediately assessed, it is restricted to a particu


http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1duchesneau_fig02.html




the system of nature experience reveals. Extending the conclusion so as to reach a gewould require, in addition, resorting to a kind of apagogical argument that could excludcombinations of possible reasons on the ground that they would entail some contradwhen reviewing in thePhoranomus the arguments that formed the essentials of his remechanics, Leibniz tends to establish them through apagogical demonstrations that alternative in terms of explanatory reasons would result in mechanical perpetual motithe combinatorial pattern can be said to rule over both styles of argumentation. But wpriori approach, it would provide a direct abstract representation for the inner naturewith the a posteriori strategy, it would offer an indirect representation for force ingredient through a hypothesis concerning phenomenal equivalents. The conjunction ways witnesses to this dual aspect of the combinatorial model.14

Applying this model in compliance with the principle of causal equivalence and the aninnitesimal geometry requires a set of abstract denitions, supporting a coherent construction. These denitions undergird the reasons for the order of phenomena posteriori way. But, at the same time, they are meant to dene concepts on the a priori wwe nd in Dialogue I of thePhoranomus a strategic distinction between the two types of fovmortua and vis viva. This distinction gives rise to a parallel one about conatus and imprespective expressions for dead and living force. Between conatus and impetus, the s

relation builds up from the nite to the innite,[End Page 99] as in the transition from a poithe virtual beginning of a line, to the line duly actualized (I, §18, Leibniz 1991a, p. 47However, the impetus represents a measure of living force in the instant and not in theeffect by which it gets exhausted. Correlatively, the impetus is negatively measured by of the body whose mass absorbs the summation of instantaneous conatus. These sdistinctions are developed and rened in later texts, especially theSpecimen dynamicum (169(GM VI, pp. 234-54). In thePhoranomus, Leibniz is content to draw his distinctions compass of a particular context: the problem at hand is to get beyond merely geometricain xing the causes or reasons for the communication of impulse and conservation force. The reality of force has to be accounted for by "ideal principles of metaphysi

Leibniz 1991a, p. 458). The representation of underpinning causes exceeds our capgeometrical schematization: it requires a reference to theoretical entities to be conceivthe geometrical conception of bodies.

Already, if one follows the initial considerations in Dialogue II of thePhoranomus, accounting inertia as it is involved in the transition from conatus to impetus means going bCartesian and Democritean (atomistic) notions about the essence of body, conceiveextension or as extension conjoined with impenetrability. In one of the more meaningpassages about his own early mechanics of theTheoria motus abstracti (1671), Leibniz explthat he had progressively withdrawn from the ctitious laws of pure phoronomy. At th

had proposed a purely rational theory of motion, based on abstract notions of cindivisibles of motion: these were supposed to combine in algebraic fashion, and, when with regard to direction, their combination would produce conatus equal to their diffecombination laws for such conatus would not take into account such physical characmass, resistance and elasticity (A VI, II, pp. 258-76; Duchesneau 1994, pp. 35-67). Aftethat early theory, he was looking for an alternative causal foundation for inertia and mo jointly considered. Further, such a foundation was needed to warrant the conservationsystem of quantity of motion, or post reformationem, of living force, in the universe (II, §§Leibniz 1991b, p. 809). Such a theoretical foundation cannot be provided by any entity mby motion as such, since motion is essentially relative. The Leibnizian solution which i

to both the a priori and a posteriori ways consists in setting out sufcient reason




production of effects that may satisfy the architectonic principle of causal equivalencwords, the strategy is to hypothesize theoretical entities in accordance with that princiwhat [End Page 100] Leibniz means when he states that the "decrees of the new sciencpower and effect . . . prescribe laws to the universe itself" (II, §G, Leibniz 1991b, p. 826). Butframe and express those theoretical hypotheses, one must frame models that will satisfyof extended geometrical intelligibility. This intelligibility implies the resources of inanalysis as well as a combination of abstract formal denitions beyond mere quanalogies. In counterpart, Leibniz condemns any attempt at resorting to scholastic occuNotwithstanding its imperfections, thePhoranomus exemplies the requirement that exte"geometrical" models and abstract denitions must t together for the framing oftheoretical hypotheses. The means to get such a tting arrangement resides in tharchitectonic principles. Such principles are instrumental in blending together the severaof a mathesis mechanica that may actualize the theory of force into a set of demonpropositions.

Contrary to what Robinet underlined as a drastic theoretical shift subsequent to thePhoranomusclose analysis of the text reveals that Leibniz sets up therein a sophisticated methodolwill directly inuence the nal stage of the dynamics. This methodology combines thearchitectonic principles such as the principle of the equality between full cause and enwith signicant attempts at extending innitist mathematical models beyond ArchimCartesian statics to physical processes involving forces. Above all, the newmathesis mechantries to frame up new conceptual combinative models to account for the formal structuras causes of unconstrained as well as constrained motions. This explains why Leibniz form combinative denitions that may determine the complex structure of dynamical ground a priori, so to speak, the various mathematical models of the new physics.

5. Transition to the Dynamica de potentia and Conservation of Formal Action

The methodological pattern set forth in thePhoranomus underlies the analyses that form

dynamics proper in the period just subsequent. In line with that pattern, the Dynamica de potenintegrates the theorem of the conservation ofvis viva within a theoretical framework that athe deciencies of the former presentation. As "a science of power and action" (GM V464), dynamics arises when Leibniz proposes an adequate architectonic of laws for conswell as unconstrained, motions. At the basis of the argument about force exerting itseconstraint, the revised theoretical denitions now concern formal effect and formal ac[EnPage 101]

The quantity of formal effect in motion is that whose measure consists in a certainquantity of matter (motion being equidistributed) being moved through a certain

length.The quantity of formal action in motion is that whose measure consists in a certainquantity of matter being moved through a certain length (motion being uniformlyequidistributed) within a certain time (GM VI, pp. 345-46).

The designation "formal" applies to properties that are judged "essential" by contrast wfeatures which depend on the contingentsitus of bodies in a specic physical systemdistinction between the two types of effects derives from the fact that the former reveal directly in unconstrained motion, and hence partake of a rational and "metaphysical" apof corporeal reality, while the latter manifest themselves through the resistance of motion--Leibnizian inertia--according to the experience of mechanical changes wh



phenomenal bodies. The concepts of formal effect and action are presented as distinwhich combine conceptual elements so as to signify the essential ingredients of force, the essential "form" of bodies, but in such a way that the relations involved may be tracombinations of quantitative parameters. This conception needs to t architectonic reqwhich imply that the "requisites" of those concepts should ideally combine to form real of the active and substantive elements beneath phenomenal realities and interactions. Scombinations of requisites would form the proper condition for a priori intelligibility.we may postulate by extension some more hypothetical combinations of requisites beywhich would be reached through a direct and fully adequate analysis of notions; these oadmitted, provided they t the relevant architectonic requirements and afford truequivalence for their objects. Under such conditions, hypothetico-deductive abstract care, and should be, called upon to guide our mathematical explanation of phenomenprecisely the case with the main concepts of Leibnizian dynamics.

Thus, the methodological pattern for the dynamics entails building from "metadenitions, representing abstractly the inner order of physical realities, such a system ofas may converge and evince the implications ofactio motrix. The axiom which binds togethesystem of equations states: "The same quantity of matter moving through the same lentime forms a greater action" (GM VI, p. 349). This species the power to act which is direcproportional to the quantity of matter times the spatial displacement, and inversely propthe time wherein action unfolds.[End Page 102]

The demonstrative argument for this theoretical proposition is at times phrased in syllogat times presented in the form of a "mathematical" calculus based on the substdenitional equivalents. For instance, the syllogistic form is to be found in the letter to BVolder of 23 March/3 April 1699:

In the uniform motions of the same body:(1) The action accomplishing the double in double time is twice the actionaccomplishing the simple in simple time. . . .(2) The action accomplishing the simple in simple time is twice the actionaccomplishing the simple in double time. . . .Hence the conclusion:3) The action accomplishing the double in double time is four times the actionaccomplishing the simple in double time. (GP II, p. 173)

The letter to Denis Papin of 14 April 1698 provides an equivalent formulation which cthe calculus pattern:

In the uniform motions of the same body, given the times,t ; the speeds,v; the spaces,s; the actions,a. . . . We shall get:(1)s asvt ; or else the spaces run through are in combined ratio of the times spent anthe speeds.(2)a assv: or else the actions are in combined ratio of the spaces run through and thspeeds with which they have been run through.(3) Therefore (in art. 2, substitutingtv fors according to art. 1)a as tvv. Or else: theactions are in combined ratio of the times and the square of speeds. (LBr 714, fol.136v, Ranea 1989, p. 53)

To establish his system of equivalences and combine the different requisites repres

notion of formal action, Leibniz must make use of a distinction that implies a twofold



signicance for the factorv. On the one hand, action is considered from the viewpoint of iteffect in spatial displacement; on the other, its intensive aspect is considered from the viinstantaneous production of that formal effect. In the Dynamica de potentia, a strategic denittranslates this twofold relation ofintensio and extensio (or diffusio) of action: "The diffusioaction in motion or theextensio of action is the quantity of formal effect in motion. Theintensio the same action is the quantity of speed by which the effect is produced or the matterthrough length" (GM VI, p. 355). Leibniz combines an embryonic effect in the instant, rin factorv, with an effect deployed in the translation space for the body in unconstrainerepresented in factors, therefore implicitly in factorv as implied bys. It is thus presumed thaembryonic factor reiterates[End Page 103] its intervention constantly as the formal effaccomplished. We are presented with a translation of motive action as a sort of activecausal agent involving at once the propensity to act and the motive effect translpropensity to actuality in duration. Since all resistance to motion is suspended, this prpresumed to conserve itself integrally through formal effect, "in such a way that it can bthat effect as a permanent gain in terms of virtual translation" (Duchesneau 1994, p. 186combining the extensive and intensive ingredients expressed in the motive effectconserved virtual effect, one gets a theoretical representation of the power to act. On tha Dynamica de potentia tends to focus on such a concept as the keystone for an architereasons that would integrate the theorems of conserved action as well as those of convviva.

The nodal role of theintensio-extensio couple in the analysis of "essential action" refermetaphysical notion signifying the immanent nality of centers of force. At the sameconstruction gets expressed in mathematical models, and reason monitors the whole pursuance of the architectonic requirements for theory building. A corroborative instastyle of argument at the basis of theintensio-extensio combination can be found incorrespondence with De Volder, when Leibniz tries to specify what causal perfection(præstantof action may consist in. De Volder's problem stems from the fact that he had red præstantia of action to the sole intensity of action as measured by instantaneous spcannot considerintensio as representing force completely nor measure it according to speinstant alone. Leibniz suggests distinguishing the extensive and intensive parametersreducing them to their intelligible terms, and setting forth the combination of requisiteunite them in architectonic mode. So we get two possible relations: (1) actions arproportional to the product of powers and times; and (2) actions are directly proportioproduct of speeds and spaces run through. The second relation can be further reconsidering that spaces are measured by the product of speeds and times. Provided thare reduced to the time unit, both assessments of the value of action agree in equating pthe product of mass and the square of speed. This is, it seems, "the mark of anconstruction that meets the norms of architectonic combination" (Duchesneau 1994, p. 2

This interpretation gets clear support from Leibniz's epistemological commecombinatorial resolution of the formal action concept in his letter to De Volder of 9/21700: "Thus you may see how everything conspires once more with beauty and gaccording to indubitable reason" (GP II, p. 203). Concerning the same notion, he writes May 1699: "Perfection, or the degree of reality in things,[End Page 104] particularly in motcan be estimated according to two reasons, namely extension which is here the magnitchanged place or space, and intension which is here the promptness or speed in change (LBr 714, fol. 310r, Ranea 1989, p. 56).

A. G. Ranea's interpretation concerning the conjoined notions ofintensio and extensio suggethat Leibniz resumes categories borrowed from the fourteenth centurycalculationes and relative



latitudo formarum. 15 According to him, Leibniz, out of speculative audacity, confers a sessential qualitative properties on quantitative factors whose objective meaning could omodes of extension. To be sure, we know from Michel Fichant that Leibniz was intereCalculationes de motu et intensionibus et remissionibus formarum seu qualitatum of RichaSwineshead, one of the theoreticians at Merton College, Oxford in the early fourteen(Swineshead 1520; Sylla 1991). Leibniz had access to that book while in Florence at thproject of the Dynamica de potentia was taking shape. Later, he had a copy made from oneeditions held by the Bibliothèque du Roi in Paris. But the exact connection between Swand Leibniz's theses still remains to be determined. My initial impression is that the

developped by theCalculatores in Oxford, then in Paris, and culminating in Nicole OTractatus de congurationibus qualitatum et motuum (Clagett 1968), aimed essentially geometrical representation of accelerations and other such intensive properties, so that cases at hand could be accommodated for by calculation. In contrast, the Leibnizian circare very different. Leibniz aimed, as it seems, at setting an expressive correspondence balgebraic representation of the parameters of force as it conserves itself through actioprojection of formal sufcient reasons that would represent the underpinning causes. the historical connection between Leibniz and theCalculatores, especially Swineshead, mighless signicant than suggested by Ranea. Personally, I would stress that by joining theand extensive dimensions of action, Leibniz was attempting a theoretical constr

combinatorial style: his goal was tosymbolize the order of efcient causes at the backgrouthose effects[End Page 105] associated with unconstrained action. The force thus dentheoretical entity was characterized byactio in se ipsum, an immanent activity of the movingreproducing itself in motion, so to speak. This is what Leibniz explains to De Volder after August 1699:

In the free or formal action of the mover, when conceived as acting on itself (in seipsum), we can conceive analogically a real effect that will not be the change of pla(which we consider only a modal effect), but the mover itself preparing with a givspeed for producing itself the next moment, self-generating by itself with the same

speed exerted the moment before. (GP II, p. 191)Such a project for theoretical construction could not be achieved, if it were not suparchitectonic principles, including the principle of nality. For these principles had toupon for guiding the combination of "metaphysical" concepts that would represent cause of action and force, and for framing the mathematical models that would best expcombination of concepts in the geometrical order. They were thus needed to form texplanations concerning a specic system of contingent truths that would comfundamental laws of physical nature. Such an epistemological device, tentatively launcPhoranomus and more adequately articulated in the Dynamica de potentia, helps explain h

Leibniz worked out the principles of his dynamics and proposed them as the originalconsistent theoretical basis for his physics. Correlatively, this theoretical framework dsignicant shifts in Leibniz's natural philosophy of the later period.

* * *

Before thePhoranomus, written in July 1689, Leibniz's reformed mechanics, which hafrom the De corporum concursu (1678), comprised a demonstrative system that was at leaa posteriori: among its premises, one would nd empirical laws--in particular, Galileo's fall--and contingent systemic conditions, such as elasticity and gravity. Under those cthe axioms of statics and the architectonic principles--for instance, the principle equivalence and the law of continuity--seemed to apply only to a given state of the




universe. In an attempt to surpass Galilean science, and indeed Cartesian physics, thePhoranomenvisages an immediate application of the more a priori premises, such as the prequivalence between full cause and entire effect, in analyzing the causal element inunconstrained motion. The formal ingredient of cause beneath such a motion does accumulation and exhaustion of forces, but it involves a constant self-reproduction[End Page106] of forces. Leibniz presumes that there must be a strict congruence between tcharacterizing such motions--measured by the product of masses and distances rudivided by times--and the measure of force as fully consumed in cases of impact. Tbehind this type of presupposition is due to the fact that Leibniz has not yet succeeded ina mode of synthetic combination of the exhausting effect with its preservation in uncmotion, a mode of combination that may warrant the equivalence. While he "substantive" with "modal" elements, namely theres with a twofold spatio-temporalstatus, account for the variouscasus, he still lacks an adequate combinative model such as tha Dynamica de potentia (1689-90); this later model represents action through its combined and extensive formal features (Ranea 1989; Duchesneau 1994). On the other hand, the jhe brings forward in the former text is mainly apagogical: it is based on the presumed of the proposed combination correlated with the impossibility of conceiving any oadequate alternative. In the later text, Leibniz tries to ground the sufciency of thcombination on a more systematic and direct analysis of its various implications. Are wfaced with a profound dialectical rupture when passing from the less perfect theoPhoranomus to that of the Dynamica de potentia? The fact is that thePhoranomus spells out methodological requirements of amathesis mechanica, combining mathematical models wappealed to the resources of innitesimal geometry, with abstract denitional construcmore metaphysical kind, under the ægis of architectonic principles. With the tools of thimathesthe new objective is to offer a causal theory of force in cases of unconstrained, aconstrained, motions. There is no doubt that this objective opens up the main perspectnew science of dynamics. Thus, a relatively continuous methodological transition, whconsiderably from the dramatic shift that took place in the De corporum concursu (1678), seeto link thePhoranomus to the Dynamica de potentia wherein the new science achieves it

formal expression.Université de Mont

François Duchesneau is professor of philosophy and vice-rector of planning at the UnivMontreal. He is presently serving as president of the Canadian Philosophical Associresearch interests relate to the history and philosophy of science and to early modern pHis recent published work includes La dynamique de Leibniz (1994),Philosophie de la biolo(1997), and Les modèles du vivant de Descartes à Leibniz (1998).

Notes1. "Cujus rei ut aliquem gustum dem, dicam interim, notionemvirium seu virtutis (quam Germvocant Krafft Galli la force) cui ego explicandæ peculiarem Dynamices scientiam destinplurimum lucis afferre ad veramnotionem substantiæ intelligendam."

2. On the relation between the two texts, see Robinet (1988, pp. 81-95); also (1989).

3. On thevis viva controversy, see Laudan (1968), Iltis (1971), Gale (1973), Papineau (19

4. In this context, A. Robinet uses the phrase "mouvement essentiel". Such an exp

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somewhat misleading for Leibniz would not take motion, which is always relative, for tof any reality whatsoever. According to Leibniz, extension itself, which for the Cartesiunderpin the modes of motion as the essence ofres corporea, belongs to the phenomena nonessential property. It may however be well-founded as it expresses the inner activimmaterial nite substances in orderly fashion.

5. In a preliminary text entitledConspectus operis (GM VI, pp. 284-85), Leibniz was alusing the termdynamica in a context wherein it related both to abstract and concrete abearing respectively on force and action, and on active causes and effects as they ope"system of things."

6. Phoranomus, II, §M (Leibniz 1991b, pp. 837-88): "Demonstrationes igitur afferam, fortasse nihil unquam clarius Mathesis mechanica vidit. Et quidem diversis modis consePrincipio autem facile mihi opinor concedetis hanc Hypothesin, cui similes usurpant erunimirumnon posse eri ut vi gravitatis, et earum actionum inter corpora quæ ex ipsequuntur, commune corporum centrum gravitatis altius in ne reperiatur quam initio fui"

7. Phoranomus, II, §D (Leibniz 1991b, p. 811): "Ut igitur ex illo Labyrintho me taexpedirem, non aliud lum Ariadnæum reperi, quam æstimationem potentiarum aPrincipium,Quod Effectus integer sit semper æqualis causæ suæ plenæ. Id vero cum expe perfecte consentire et omnibus dubitationibus satisfacere deprehenderem. . . . "8. Phoranomus, II, §F (Leibniz 1991b, p. 818): "Quare et potentiarum eadem erit æstimatnihil aliud sint casus, quos attulimus, quam potentiarum effectus. Et quam hæc rationi csint vel hinc apparet, quod in motu æquabili ex dato spatio percurso non ideo detvelocitas, aut contra. Itaque rationes ambæ sunt conjungendæ, quæ in casu temporumdant duplicatam [velocitatum rationem]."

9. See A. Robinet's presentation (Leibniz 1991a, p. 437): "LePhoranomus fut à la Dynamica cque la première version du De corporum concursu fut à la seconde: on y saisit la créleibnizienne à l'œuvre."10. Even if Leibniz refers the use of the term"phoronomica" to Joachim Jungius'sPhoranomicid est de motu locali (1679, 1689), he insists, as against Jungius's position, that phoronomreach beyond mere geometrical considerations: "Sed vires vivæ seu impetus perPhoranomicen strictius sic dictam,Phoranomices tamen nomine non intelligo quod Junglibello qui hoc titulo prodiit, ubi tantum investigat lineas tanquam motuum vestigia,doctrinæ motus pure Geometrica est, sed ipsas naturæ leges, quæ circa communication[et?] vires motrices observantur" (I, §23, Leibniz 1991a, p. 483).

11. See Leibniz's notes on his reading G.A. Borelli's De vi percussionis (1686), LH XXXV, XIVf.2, cited by A. Robinet in his annotations on thePhoranomus: "Et in ne Dialoghi 4to de mGalilæus dicit theoriam energiæ percussionis esse perobscuram . . . et vim percussinterminatam. Promittebat alibi hoc demonstrare, sed nihil tale repertum Torricellinprofectum non demonstrare, sed specturus [?] conrmat vim percussionis esse innitam1991a, pp. 527-28).

12. This experiment is also presented in the Dynamica de potentia (GM VI, p. 452).

13. As stated by A. Robinet (Leibniz 1991a, p. 538), Leibniz reiterates on that account argu

he had just published in hisSchediasma de resistentia medii et motu projectorum in












resistante, published in the Acta Eruditorum in January 1689 (GM VI, pp. 135-44).

14. See II, §G (Leibniz 1991b, p. 826): "et ideo ad amœniora et imaginationi magis satifestino, quæ licet a posteriori sumantur, hoc tamen præstans habent, quod mentepersuasione convincunt, cum illæ demonstrationes natura priores, cogant reluctantem. ipsemet interiora ne suspicione quidem attigissem, contentus vulgari motus æquabilis aper ista quæ nunc afferam exotericotera veritatem prædetexissem."

15. See Ranea (1989, p. 57): "[Leibniz's inconsequent stratagem] suggests that the'space/time' does not exhaust the meaning of velocity in thea priori argument. I think we cget a clue to this question. . . . Within the framework of the scholastic Physics these[extensio anintensio] allude to, velocity also has two different meanings: either it means the quotienand time, or the intensity of theaccidens intrinsecum of the moving body, i.e. its local notiothis way, velocity becomes a metaphysical or 'quasi-physical' sign of the inner permotion, a magnitude quite independent of any quantitative viz. extensive treatmenechoes this basic assumption of fourteenth century physics when he states that a fasteressentially more perfect than a slower one."

References

Bertoloni Meli, Domenico. 1993. Equivalence and Priority: Newton versus Leibniz. Inc Leibniz's Unpublished Manuscripts on the Principia. Oxford: Clarendon Press.

Bodemann, Eduard. (1895) 1966. Die Leibniz-Handschriften der königlichen öffen Bibliothek zu Hannover. Reprint, Hildesheim: Olms.

Clagett, Marshall. 1968. Nicole Oresme and the Medieval Geometry of Qualities and MMadison: University of Wisconsin Press.

Duchesneau, François. 1993. Leibniz et la méthode de la science. Paris: Presses UniversitaireFrance.

------. 1994. La dynamique de Leibniz. Paris: Vrin.

Fichant, Michel. 1990. "Neue Einblicke in Leibniz' Reform seiner Dynamik (1678Stud Leibnitiana 22:38-68.

Gale, George. 1973. "Leibniz's Dynamical Metaphysics and the Origins of the Controversy."Systematics 11:184-207.

Gerhardt, C. I. 1888. "Zu Leibniz' Dynamik." Archiv für Geschichte der Philosophie 1:566-81.

Gueroult, Martial. 1967. Leibniz: Dynamique et métaphysique. Paris: Aubier-Montaigne.

Iltis, Carolyn. 1971. "Leibniz and the Vis Viva Controversy." Isis 62:21-35.

Laudan, Larry. 1968. "The Vis Viva Controversy: A Post-Mortem." Isis 59:151-63.

Leibniz, Gottfried Wilhelm. 1923-.Sämtliche Schriften und Briefe. Edited by the Akademie Wissenschaften. Darmstadt and Berlin: Akademie-Verlag (Cited as A).





------. (1875-1890) 1965. Die philosophischen Schriften. Edited by C. I. Gerhardt. RepHildesheim: Olms (Cited as GP).

------. 1971. Mathematische Schriften. Edited by C. I. Gerhardt. Hildesheim: Olms (Cited as

------. 1988.Opuscules et fragments inédits. Edited by Louis Couturat. Hildesheim: Olms as C).

------. 1989.Philosophical Essays. Edited and Translated by Roger Ariew & Daniel GIndianapolis and Cambridge: Hackett.

------. 1991a. "Phoranomus seu De potentia et legibus naturæ. Dialogus I."Physis 28:429-541.

------. 1991b. "Phoranomus seu De potentia et legibus naturæ. Dialogus II."Physis 28:797-885.

------. 1994. La réforme de la dynamique: De corporum concursu (1678) et autres textes. Éditioprésentation, traduction et commentaire par Michel Fichant. Paris: Vrin.

Papineau, David. 1981. "The Vis Viva Controversy." Pp.139-56 in Leibniz: Metaphysics Philosophy of Science. Edited by R. S. Woolhouse. Oxford: Oxford University Press.

Ranea, Alberto Guillermo. 1989. "Thea priori Method and theactio Concept Revised: Dynamand Metaphysics in an Unpublished Controversy between Leibniz and Denis PapiStud Leibnitiana 21:42-68.

Robinet, André. 1988. Iter italicum. Firenze: Olschki.

------. 1989. "Les surprises duPhoranomus." Les Études Philosophiques (April-June), pp. 171-

Swineshead, Richard. 1520.Calculationes, noviter emendate atque revise. Venetiis: Cura heredScoti ac sociorum.

Sylla, Edith. 1991.The Oxford Calculators and the Mathematics of Motion 1320-1350. and the Measurement by Latitudes. New York: Garland Publishing.



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Leibniz's Theoretical Shift in the Phoranomus and Dynamica de Potentia - Francois Duchesneau

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