This paper is concerned with the final stage in the invention of the dynamics or theory of force, which forms the core of Leibniz’s contribution to physics. It tries to determine the reasons why this final stage took place and to unravel the arguments Leibniz developed to establish the mature version of his theory. In his article *De primæ philosophiæ emendatione et notione substantiæ,* published in 1694 in the *Acta Eruditorum,* Leibniz mentions for the first time in print that he has set the ground for a new science called *dynamica* and centered on the notion of force (GP IV, pp. 468–70). He adds that he expects the notion of force will bring much light to our understanding of the notion of substance (GP IV, p. 469). ^{1} Indeed, his use of the term “dynamics” *(dynamica, dynamice)* can be traced back in various correspondences to an apparently initial occurrence in the exchange of letters, begun in 1689, with Rudolf Christian von Bodenhausen. It is also quite evident that the principles of the new science form the subject matter of the *Dynamica de potentia et legibus naturæ corporeæ,* a major scientific work Leibniz started working on in late summer 1689. Just previous to setting on that ambitious project, Leibniz, in July 1689, had undertaken to write the *Phoranomus,* a dialogue which was left incomplete, but wherein he sketched some of the arguments that, appropriately modified, found their way into the *Dynamica.* My larger project is to understand the genesis, finality, and argumentative structure of the theoretical constructs which form the dynamics. But this paper will focus more specifically on describing the methodological shift in the analysis of force that took place in the *Phoranomus.* The shortcomings of this initial attempt seem to have occasioned ** [End Page 77] **the arguments Leibniz recast and developed in the *Dynamica.* ^{2} The *Phoranomus* may thus serve as a key opening to one of Leibniz’s major and less known intellectual achievements in the area of physical science.

First, I shall briefly recall the initial steps, beginning in 1676, that led to the 1689 project, and present the circumstances, structure and objectives of the *Phoranomus.* Second, I intend to show that the main theoretical challenge Leibniz was faced with at the time he wrote the *Phoranomus* was that of counteracting the Cartesians’ objections against his demonstration of the principle of conservation of *vis viva.* Third, I shall try to characterize the new way of proof Leibniz puts in place which abstracts from our particular system of things in an effort to determine a priori how the moving force is expressed in uniform unconstrained motion. Certain methodological principles guide this attempt, especially the principle of the equality between full cause and entire effect. Fourth, I shall investigate the concepts and models framing the *mathesis mechanica* which is supposed to uncover the “formal effect” of moving force as it expresses the form of the subjacent, self-restoring agent. The main problem in this case is to find means of equating mathematical expressions and metaphysical reasons and to provide models more general than those pertaining to the experience of falling bodies and gravitational effects. The strategy to cope with this problem consists in forming a system of abstract definitions according to which models can be constructed, so to speak, a priori. Finally, it seems appropriate to turn to the *Dynamica de potentia* to discover how the same methodological pattern overcomes the shortcomings of the *Phoranomus.* The central ingredient of the new theory is a concept of “formal action” which grounds the various models of the dynamics defined as the science of “power and action.” This concept is a truly Leibnizian invention which determines the structure and significance of the mature dynamics.

# 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 initial physics which he formulated in the *Theoria motus abstracti* and *Hypothesis physica nova* (1671). But those views were to be drastically revised at a later time. The real story begins when Leibniz, upon his return to Germany from France in 1676, reconsiders the laws of motion between colliding bodies. His ambition then was to go beyond the relativist statements of laws provided by Christiaan Huygens, John Wallis, and Edme Mariotte, and to set the system of mechanics back under the aegis of a geometrical demonstrative framework (Duchesneau 1994). He believed he had found the key to a possible conciliation between the ** [End Page 78] **empirical laws of motion and an a priori principle of conservation: this key was the regulative principle of equivalence between full cause and entire effect. In the *De corporum concursu,* dated January-February 1678, a text that Michel Fichant has edited recently for the first time (Leibniz 1994), Leibniz undertook to systematically deduce the various cases of motion between colliding bodies in accordance with a principle akin to Descartes’s principle of the conservation of quantity of motion measured by the product of mass and speed *(mv).* He compared the calculations derived from his theoretical model with the results of an experiment based on the properties of the pendulum and destined to measure respective displacements following the collision of unequal moving bodies—equality being considered only as the limit of a progressive series of unequalities. One body being at rest, the motion of the other would be produced by a fall from various heights. Leibniz could not but witness that the inferences drawn from that model did not match the data collected from the experiment. So he decided to modify and recast the deduction by basing it on the presumed conservation of the product of mass and height of fall before collision—a methodological move he would call a *reformatio.* This time, the agreement between the various cases fitted the requirement of the equality between full cause and entire effect in a continuous progression. In our physical universe, the system would be organized in such a way that the equivalence is maintained through phenomenal changes. However, even if, from then on, Leibniz was in possession of his theorem of conservation of *vis viva* according to which the moving force conserved in mechanical interactions is measured by the product *mv*^{2}, he still could not account adequately for the subjacent forces.

Significantly, Leibniz only makes his reformed mechanics public when he attains a first global formulation of his philosophical system. Thus, his first official challenge to Cartesian mechanics took place in the *Brevis demonstratio erroris memorabilis Cartesii* published in 1686 in the *Acta Eruditorum* (GM VI, pp. 117–23), the arguments of which were integrated into the *Discours de métaphysique* in the same year (GP IV, §§17–18, pp. 442–44). Opposing those who supported a mechanics strictly consisting of empirical laws, Leibniz joined with Descartes in acknowledging that a principle of conservation of *potentia motrix* needs to belong among the principal ingredients of physical explanation. But, relying upon Galileo’s law of free fall and on axioms admitted by Cartesians, he established the disparity between the Cartesian measure of quantity of motion and the true requisites of the concept of moving force, and he showed that the power conserved is to be measured by the product *mv*^{2}.

Leibniz’s demonstration proceeds this way. It starts from two premises Cartesians would admit: (1) A body falling from a certain height acquires ** [End Page 79] **by its fall the force to lift itself back to its initial position; and (2) As much force is required to lift a body A of one pound to a height CD of four feet as is required to lift a body B of four pounds to a height of one foot (see fig. 1). Combining the two propositions makes it possible to consider that the forces gained by lifting A to C and B to E are equal to each other and proportional to the heights of fall CD and EF. Then, Leibniz resorts to Galileo’s (empirical) law of free fall. In free fall, the spaces run through are proportional to the squares of the times spent to that effect, while the speeds are proportional to the square roots of the spaces, i.e., the heights of fall. Running the distance CD, body A has acquired a speed proportional to 2, while running the distance EF, body B has acquired a speed proportional to 1. Considering the products *mv* on both sides, we would get quantities of motion proportional to 2 on the one hand, to 4 on the other. Hence the non-conservation of quantity of motion. If we take the heights of fall to be respectively proportional to the masses of the moving bodies, and if we take the product of mass and height to express the force as it is entirely spent in the effect of lifting, we get a measure of force based on the product *mv*^{2}. This estimation according to the effect produced presumes that the body lifting itself back to its initial position exhausts all the force it has acquired previously by its fall in conformity to the principle of the equality between full cause and entire effect. Thus, this principle rules over the analytic demonstration of the principle of conservation of *vis viva.*

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

The next stage concerns the invention of the dynamics proper. In the period 1686–89, Leibniz had been busy counteracting the Cartesians’ objections voiced by François Catelan and Nicolas Malebranche. ^{3} Through the polemics, Leibniz was confronted with the problem of justifying his principle of conservation as a true foundation for the system of laws of nature, since his argument would depend on an empirical generalization, Galileo’s law of free fall, as well as on contingent features of our physical world such as elasticity and gravity. For Leibniz, a true science implies a structure of demonstrative arguments that can be exposed synthetically. In the reformed mechanics, Leibniz used analytic devices which provided a sort of a posteriori unfolding of the new conservation principle. It seems appropriate now, both for polemical and methodological reasons, to verify the capacity of this principle for grounding a deductive synthesis of the laws of nature (by combination of equations). We should remind ourselves at this stage that one of the main tenets of Leibniz’s methodology as formulated in his numerous projects for a demonstrative encyclopedia is the *de jure* correspondence between analysis and synthesis, or *combinatoria,* as evidenced both in the *ars demonstrandi* and *ars inveniendi* (Duchesneau 1993). The analytic processes elaborated to prove the *vis viva* principle should normally be replaced by a synthetic ordering of reasons that would illustrate the role of this principle as a keystone in physical theory. This goal seems to have provided the chief motivation for Leibniz’s enquiries leading to the dynamics.

In March 1689, Leibniz started on a year-long journey to Italy (Robinet 1988). His Italian sojourn afforded a rather propitious context for attempting a synthetic recasting of the reformed mechanics. Throughout his encounters with Italian scientists, the challenge was indeed for Leibniz to show that he possessed the means to supersede both Cartesian and Galilean models for building physical theory, and that he was on his way to a new system of natural philosophy based on the formal nature of force. It is most probably with this objective in mind that Leibniz undertook to write the *Phoranomus* (1689) and *Dynamica de potentia* (1689–90).

Until very recently, the *Phoranomus seu De potentia et legibus naturæ* was ** [End Page 81] **known only from the mention of its title by Louis Couturat in connection with another fragment he had transcribed—which has nothing to do with it except being the next piece according to Bodemann’s catalogue of Leibniz’s *Handschriften* (C, p. 590)—and from a fragment of Dialogue II that C. I. Gerhardt had transcribed and edited (Gerhardt 1888). André Robinet has recently edited the entire *Phoranomus* (Leibniz 1991a; also 1991b). We know for sure that this important work was written in a very short timespan during the summer of 1689, while Leibniz was staying in Rome and having regular meetings with representatives of the Roman intelligentsia. It is also clear that the complete, but not fully recopied, *Phoranomus* was soon to be left aside by Leibniz whose interest had shifted to writing the *Dynamica de potentia.* The various chapters of the latter work went through a series of drafts during Leibniz’s stays in Rome, Florence, and Venice. And Baron von Bodenhausen in Florence was sent those drafts, a neat version of which he was supposed to prepare for publication. After his return to Germany in spring 1690, Leibniz felt less and less inclined to complete and publish the work, and after Bodenhausen’s death on 9 April 1698, he requested his papers back. The Bodenhausen copy was published by C. I. Gerhardt in 1860 (GM VI, pp. 281–514).

Like the *Pacidius Philalethi* (1676), also a transition piece, the *Phoranomus* is cast in the form of a dialogue. It possesses strong realist overtones, and it seems to reflect discussions Leibniz could have had in Rome with members of the *Accademia fisicomatematica,* whose sessions he had attended. In fact, in two successive sessions, narrated respectively in Dialogues I and II, the *Phoranomus* presents Leibniz and some members of that academy pursuing arguments on themes central to the reformed mechanics and to the dynamics to be. The introduction is afforded by a letter supposedly addressed by Leibniz to Melchisédech Thévenot, a member of the Académie des Sciences in Paris, the emphasis of which concerns a critical evaluation of Descartes’s philosophy and science. Leibniz had received news from Simon Foucher that Daniel Huet had just published a *Censura philosophiæ cartesianæ* (1689). He recalls that he had himself contributed to the critical appraisal of Descartes’s physical hypotheses. Such a critique may be found for instance in his letters to Hermann Conring of 3/13 January, 19/29 March, and June 1678 (A II, 1, pp. 385–89, 397–404, and 418–20). Concerning physical hypotheses, he had felt the need to correct and complete Descartes’s work. In particular, the errors in Descartes’s laws of nature had occasioned Leibniz to undertake such revisions as the *Phoranomus* tries to account for: “Indeed, as Descartes failed in setting out the laws of nature, he provided me with the occasion for establishing the true ones” (I, §1, Leibniz 1991*a*, pp. 446–47).

Left incomplete and faulty, the *Phoranomus* affords a preliminary integration ** [End Page 82] **of the mechanics of force which the *Dynamica de potentia* develops and articulates into a systematic whole following on the work done during the summer of 1689. Previously, e.g., in the *De corporum concursu* and *Brevis demonstratio,* Leibniz had grounded his new principle of conservation of force measured by *mv*^{2} on an at least partly a posteriori demonstration. That demonstration relied on Galileo’s law of free fall and it pertained to a system of the universe where elasticity and gravity would be found—properties which Leibniz and his contemporaries tried to explain by resorting to hypothetical models. Indeed, Leibniz relied also on axioms inferred from Archimedean statics and reformulated by seventeenth century mechanists. Above all, he had taken advantage of an architectonic principle, the principle according to which the entire effect and full cause must be considered equivalent through mechanical interactions. Such a principle expressed the requirement of sufficient reason as it would apply to the phenomenal realm of motion. Because the moving force generated by a body falling from a given height is fully exhausted in the subsequent lifting of that body to its initial height, the product of mass and space of translation, a product which is proportional with the square of speed, serves to measure the force that is kept through a sequence of equivalent generations and exhaustions. But is not such a conservation law restricted to applications within the bounds of a specific system of phenomena? Do not the features of this system pertain to an empirically grounded model? If this were the case—as it seems to be—the Leibnizian reformed mechanics would be justified by a set of proofs built and grounded a posteriori, at least for a significant portion of the premises involved.

The challenge Leibniz sets himself in the *Phoranomus* consists in trying to generalize his measure of moving force so that it can apply to all phenomena involving motion. The program requires that equivalent causal circumstances be identified, both in violent motion manifesting itself in collisions with generation and exhaustion of the moving force, and in unconstrained motion *(innocuus motus)* manifesting itself in uniform translation without interference of any facilitating or impeding extrinsic determination. The latter is what Leibniz calls *“motus æquabilis qualis per se est”* (II, §§F and G, Leibniz 1991*b*, pp. 820–21). ^{4} As he attempts to account for the force underlying such unconstrained motions, Leibniz develops a system of a priori arguments, in which premises of the empirical ** [End Page 83] **sort would serve only as corroborative instances. When this project is complete, he is in a position to claim that he has founded a science of dynamics, whose specificity and theoretical potential he can spell out. While the *Phoranomus* still speaks in a tentative fashion of a new science concerning force and effect, *“nova de potentia et effectu scientia”* (II, §G, Leibniz 1991*b*, p. 826), the subsequent major text, the *Dynamica de potentia,* deals demonstratively with a new science concerning force and action, and this new science is henceforth properly identified by the name and concept of dynamics. Thus in the *Specimen præliminare* of the *Dynamica* Leibniz states:

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

true elements of the new science of power and action, which one might calldynamics. I have gathered certain preliminaries of this science for special treatment, and I wanted to select a ready specimen from these in order to excite clever minds to seek truth and to receive the genuine laws of nature, in place of imaginary ones.(GM VI, p. 187; Leibniz 1989, p. 107)

Similarly, at the end of the first section of the second part, Leibniz mentions: “It seems to me that I have thus disclosed the sources yet quite unexplored of the *dynamical science of power and action*” (GM VI, p. 464). ^{5} The parallel expressions *scientia de potentia et effectu* and *scientia de potentia et actione* clearly indicate the change of focus between the *Phoranomus* and the *Dynamica de potentia:* the main shift takes place with the concept of action substituting for the concept of effect and providing a basic theoretical equivalent for the formal identification of force. In the former work, effect means the distance run through conjoined with the speed of translation, as if this effect would exhaust the force involved while conserving it virtually all along. In the latter work, formal effect as measured by the distance run through does not entail exhaustion, but only extensive application of the immanent force, while formal action combines formal effect with a consideration of the time spent, which manifests intensive application of the same immanent force. We shall discuss these distinctions further later on.

Prior to my own analysis (Duchesneau 1994), the *Phoranomus* had only been accounted for by André Robinet in the notes of his edition (Leibniz 1991*a*; also 1991*b*). He had based his interpretation on three main tenets: ** [End Page 84] **(1) a certain conception of the authors and theses criticized by Leibniz in his attempt at generalizing the reformed mechanics; (2) a dramatic vision of the paradoxes affecting Leibniz’s first attempt at a theory of force built a priori from the effects of unconstrained motion; and (3) a “causal” representation connecting the failure of the *Phoranomus* with the genesis of the *Dynamica de potentia* and the identification of dynamics as a science. In the latter text, Leibniz resumes the attempt at setting up an a priori demonstration, but he grounds his arguments on a concept of action that goes beyond his previous notion of formal effect; thus he avoids the paradoxes he was confronted with during the summer of 1689. These three elements in Robinet’s interpretation deserve different appraisals and adjustments. The issues involved will be taken up in the next three sections

# 2. The Theoretical Context

Let us examine first the issue of theoretical context. Most interpreters of the dynamics, like Martial Gueroult (1967), ignoring or neglecting the *Phoranomus,* would tie the writing of the *Dynamica* with the objective of proposing a deductive system to counterbalance the Newtonian mechanics which was explicitly built on a “deduction from phenomena.” Indeed, at the time he undertook to establish the dynamics, Leibniz knew of Newton’s *Philosophiæ naturalis principia mathematica* (1687) and he was already organizing the defense of his harmonic circulation hypothesis against Newtonian gravitation theory (Bertoloni Meli 1993). But, as revealed by Robinet, the *Phoranomus* shows a rather different ambition (Robinet 1989). Robinet is right in stressing the various elements of Galilean tradition in the two dialogues. Those elements provided Lubinianus’s (Leibniz’s) interlocutors in the dialogue with their main references in matters of mechanics and mathematical physics. Leibniz tries therefore to show the superiority of his theoretical models relative to those of Galileo and his disciples, in view of the fact that those scientists tended to fall back on empirical evidence whenever geometrical proofs seemed out of reach. Indeed, the text instances a critical resuming of Galilean mechanics and Archimedean statics. But, at the same time, Leibniz wishes to pursue the refutation of Descartes’s mechanical principles in view of the latter’s errors concerning the conservation of quantity of motion.

Even if this twofold project catches Robinet’s attention, two complementary remarks seem in order. First, it is worth underlining that the Leibnizian methodological frame introduced in the *Phoranomus* differs significantly from the hypothetico-deductive pattern he had appealed to in his previous demonstration of the principle of conservation of *vis viva.* More clearly than in any other texts perhaps, Leibniz is planning here for the development of a demonstrative system based on principles and proofs ** [End Page 85] **in his own a priori style. We shall further consider this point in the next section.

Second, Robinet misses the point that the immediate context for the critical assessment of Cartesian mechanics is provided by the recent polemics with Catelan and Malebranche following the publication of the *Brevis demonstratio,* polemics in which Leibniz defended his position by appealing to the principle of continuity and its role in theory building, as instanced in the *Lettre de M. L. sur un principe général utile à l’explication des lois de la nature par la considération de la sagesse divine, pour servir de réplique à la réponse du R. P. D. Malebranche* (GP III, pp. 51–55). Through the characters of Ciampini and Auzout in the *Phoranomus,* Leibniz introduces precisely the Cartesians’ main objection, that time should be taken account of in comparing the moving forces involved in lifting bodies, even if those forces result from free fall (II, §M, Leibniz 1991*b*, p. 837). Indeed, in the *Brevis demonstratio,* Leibniz had established his principle of the conservation of *vis viva* on the consideration of such effects as would exhaust entirely the moving force that had previously accumulated, but with no regard for the relative time spent by different bodies in achieving that result. If the effects involving colliding bodies were comprised under the same measure of time, would not the conservation of quantity of motion prevail over the conservation of *vis viva?* This is indeed the case in the first instant of summation of impetus without subsequent integration of conatus: in this first instant, the moving force has not yet accumulated to be spent in an equivalent exhaustion effect. Such a case is illustrated in statics when bodies are fixed at the extremities of a balance according to a ratio of distances inverse to that of the respective masses; then, the vertical constrained motions effected during the same time elicit the conservation of *mv,* rather than *mv*^{2} . The Cartesians’ argument was that if the effect of force is considered in relation to time, it should be measured according to its value in the unit of time, which is analogous to its value in the instant, itself measured by the product *mv.*

Through Baldigiani, another participant in the dialogue, Leibniz has already seized the opportunity of denouncing the paradox which Cartesians fall into as a consequence of Galileo’s account of free fall. In the case of falling bodies, the equivalence between full cause and entire effect implies constancy of *mv*^{2}. But why, then, do people tend to relate the lifting of bodies and their respective speeds? This mistake derives from a prime theorem of statics: two bodies are in equilibrium when their respective *status* are such that if one begins to descend, the motions thus provoked imply speeds that are proportional to the respective masses of bodies. Hence the unwarranted tendency to infer, as if from purely rational evidence, that “forces of bodies are equal when speeds are inversely proportional ** [End Page 86] **with bodies” (II, §I, Leibniz 1991*b*, p. 829). Here, Leibniz argues, statics provides only a particular case in which heights are proportional to speeds, whereas the general case in which the effect of force is truly actualized implies a proportionality to the square of speeds. In any case, the powers are equal when the masses are inversely proportional to the heights at which they may rise. The Cartesian error was to conflate the special case of bodies bound together and therefore mutually impeding, with the more general case of bodies free from bondage in their generating and exhausting *potentia motrix:*

Hence we get a universal principle which will succeed in statics as well as in the rest of general phoronomy, namely that the powers are equal when the weights that can be drawn by the force of those powers are reciprocally as the heights at which they can be lifted. . . . This was indeed known to the learned, but, as they had envisaged comparative lifting only in bodies bound together, thus not ascending freely, and as they had not therefore examined bodies in motion with that touchstone, they were deprived of the excellent fruit of a very general truth, which throws most light upon the communication of motion and interaction of bodies.

(II, §I, Leibniz 1991b, p. 830)

However, in the *Phoranomus,* the principal counterattack against the Cartesians’ specious objection that the entire effect should be assessed in the unit of time consists in demonstrating that the temporal restriction imposed on the integral expression of moving forces proves incompatible with the architectonic rule of continuity. One should acknowledge the apagogic consequences accruing from the incompatibility between the temporal clause and the rule according to which neither by a force of gravity nor by any other force the common center of gravity of a system of bodies can rise more in the effect than in the cause. ^{6} This rule derives directly from Huygens’s boat model as the means to establish empirically warranted laws of motion. Huygens’s model entailed two principles of conservation, that of the directional translation of the common center of gravity for any system of colliding bodies, and that of the relative speeds of those bodies before and after collision. Leibniz, as well as his Cartesian opponents, admitted these relative principles except that they differed on the absolute conservation principle they ought to match them with. The ** [End Page 87] **first of Huygens’s rules is equivalent to a principle of conservation of
provided the product *mv* is considered vectorially, not scalarly, as with Descartes’s rules of motion. The beauty of Leibniz’s argument consists in turning the consideration of time back against his objectors, by stating that their account of time for a scalar measure of force leads them astray of Huygens’s model, which is also based on the unit of time but entails no conflict with the conservation of
. On the one hand, Leibniz’s demonstration relates directly to the empirically corroborated hypothesis about conservation of the common center of gravity. But then, in order to raise this auxiliary principle to an epistemological standing of the a priori sort, Leibniz resorts to the counterfactual supposition that, if there were a divergence from conservation of the common center of gravity, the implicit necessary consequence would be an admission of mechanical perpetual motion. The same type of apagogical demonstration makes it possible to discard the disparities that would presumably result from more or less oblique translations of moving bodies, because of the different times spent in producing the same total effect. Finally, Leibniz sets himself to the task of systematically overthrowing Catelan’s and the Cartesians’ main objection by pressing that in violent motion, the space run through, namely the height attained by means of the full expense of force, already implies an account of time in the measure of the square of speed correlative with that height: hence the presumed redundancy of an account of time beyond the account of the space run through. Indeed, this is the inverse case from that of a motion reduced to formal effect without external constraint (II, §R, Leibniz 1991*b*, p. 848). In short, it is the systematic working out of arguments against the Cartesians in the aftermath of the polemics with Catelan and Malebranche that determines the theoretical construction Leibniz is attempting in the *Phoranomus.*

# 3. The A Priori Method of Demonstration

The second point highlighted by Robinet deals with the paradoxes of an a priori model based on the formal effect of unconstrained motion, as this effect would jointly be measured by full exhaustion and entire self-preservation. This model developed in the *Phoranomus* can be stated as follows. The architectonic premise consists in the principle of equivalence between full cause and entire effect, but applied outside of the specific case of our world system. In that system, we can only interpret the phenomenal intercourse of moving bodies by means of empirically corroborated auxiliary hypotheses which apply to elastic bodies in the context of impetus generated by gravity. According to the system of constraints on colliding bodies, moving force is measured through the violent effect it produces and wherein it entirely consumes itself. In the alternative demonstrative ** [End Page 88] **approach Leibniz sought, we should consider force abstractly as it results in unconstrained motion; that motion would express its causation in and through its uniform unfolding alone. Indeed, this new approach stands as an abstraction from the conditions pertaining to our system of things *hic et nunc.* But the same fundamental presupposition, the same Ariadne’s thread previously identified, namely the architectonic principle of equivalence between full cause and entire effect, would apply. Though originally introduced in a context of Archimedean statics (I, §18, Leibniz 1991*a*, p. 477) and of empirically corroborated phoronomy of violent motion, ^{7} it is now granted a more general meaning and role.

In fact, the argumentation Leibniz tries to develop in the a priori mode in the *Phoranomus* stems from Archimedean statics. His interlocutor Charinus resorts to the equivalence between the inverse ratio of mass and speed for unequal bodies in equilibrium and transposes it to account for free translation on a horizontal surface. That analogy from statics to kinetics in assessing the moving force generates an aporia. This aporia relates to the “modal” status of speed, in contrast with the “substantive,” therefore real, status of bodies. This is seemingly what can be inferred from the very tortuous passage in *Phoranomus,* II, §E (Leibniz 1991*b*, pp. 815–16). If a body with two mass units can be granted the double power of a body with one mass unit moving with the same speed, this cannot be the case if a body moves with double the speed, compared to another body of the same mass moving with one speed unit. The power of the first body cannot be measured as equivalent to twice the power of the second according to the *mv* parameter, when the 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 based on the inverse ratio of the times spent in motion and the direct ratio of the products of masses and spaces run through that a system of equivalences may be restored at the level of the moving forces involved. Given the same bodies and the same spaces, the powers will be inversely as the times spent in translation. It follows that in a given unit of time, the forces will be measured according to the product *msv,* or, because *s* is equivalent to *v* in the unit of time, they will be measured according to the product *mv*^{2}. But this argument may easily be considered a paralogism, and hence, it will not have much more value than the paralogism according to which a body of one mass unit with two speed units would be equivalent to a body of two mass units with one speed unit. The defect ** [End Page 89] **in such an equation consists in the absence of congruence between cases, since those cases *(casus)* would combine states *(status)* and things *(res)* disparately, thus preventing the eliciting of homologous relations: “What is in question is not what a moving body can do, whatever be the time granted, but how a proposed case may be estimated from given time and space, and how a given case may be resolved into two equivalent ones, of which it would be composed” (II, §F, Leibniz 1991*b*, p. 817). The thesis Leibniz asserts a priori, rather than proves, consists in postulating that the combined consideration of the spaces run through and of the inverse of times renders cases congruent. The twofold spatio-temporal *status* (measured according to *v*^{2}) combine homogeneously with the *res* and would thus determine a measure of the moving force as it is expressed in uniform unconstrained motion:

Thus, through the resolution of bodies into parts, the speed, or space and time, being conserved, we had inferred, demonstrated, that given the same speeds the powers were proportional to the bodies. Similarly, we have demonstrated, which is paradoxical, but absolutely true, that, the body being conserved, time and space being resolved 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 justification for this system of equivalences consists in a combination of ratios referring to modal parameters: by homologous combinations, these ratios represent the effect of force as it exerts itself freely. On the one hand, no other element of motion could be taken account of in the synthetic representation of cases. On the other hand, this combination of modal elements would suffice in providing an analytic equivalent for those forces whose effects are expressed through unconstrained uniform motions. In contrast, notwithstanding Descartes’s appraisal which was modelled after Archimedean statics, speed alone could not suffice in expressing this effect of moving force, nor would it be the case for space run through if considered as the only essential parameter. ^{8}

Ultimately, as summarized by Baldigiani, one of the participants, Leibniz’s thesis boils down to asserting that the effect equivalent to force in the case of uniform unconstrained motion *(motus æquabilis qualis per se est)* is ** [End Page 90] **determined by the combining of two “requisites”: the space moved through and the speed of motion (II, §F, Leibniz 1991*b*, p. 820). Robinet interprets this thesis as paralogistic since the effect is not decomposed, as it is in the *Dynamica de potentia,* into formal effect and formal action: in fact, not yet being provided with this distinction, Leibniz cannot justify his measure of the effect otherwise than by combining the exhaustion effect considered now in the unit of time with the same effect transposed in the shape of a sort of equivalent virtual effect that accompanies the actual effect as it exhausts itself. Personally, I am more struck by the fact that the defect in Leibniz’s analysis consists in an inadequate distinction between the “extensive” and “intensive” components which would combine in the constantly reactualized effect of a force that is kept intact through the motion it accomplishes. More than anything else, Leibniz lacks a causal model to account “metaphysically,” that is abstractly, for this integrative combination of intensive and extensive dimensions. Due to that deficiency, the model of congruent cases remains aporetic, and cannot be ultimately warranted but on an analogy derived from the a posteriori demonstration of the principle of *vis viva* conservation. The impression of paradox and the tortuous writing which affect the passages concerning the a priori demonstration in the *Phoranomus* are probably due to an insufficiently refined model for justifying the combination of parameters pertaining to the action considered in its essential expression.

Despite the shortcomings of the notion of effect on that account, the model deployed in the *Phoranomus* seems to suggest a demonstrative pattern for generalizing the principle of the conservation of *vis viva* to such a force as would underlie uniform unconstrained motion without power exhaustion. This model appeals jointly to the principle of the equality between full cause and entire effect and to a sought for combination of *status* that, correlated with the *res* involved, might warrant the congruence of homologous cases. In my view, the very requirements of this methodological model determine the conceptual refinements underpinning the theoretical construction the *Dynamica de potentia* illustrates shortly thereafter. By emphasizing the aporetic elements of a notion of effect that combines the modes of expression of force in both constrained and unconstrained motion, Robinet fails to see that Leibniz is seeking a way of conceiving by a priori construction how the modal elements of force might be combined to account for conservation of the substantive elements underpinning both constrained and unconstrained motions.

# 4. The New *Mathesis Mechanica*

In this section, I intend to challenge Robinet’s thesis that, in the course of writing the *Phoranomus,* Leibniz would have felt the need to drastically ** [End Page 91] **“reform” his theoretical approach. In contrast, I shall develop the view that Leibniz was already clear then about the specific requirements of what he conceived as an essentially a priori mode of demonstration. Above all, my intent is to assess the demonstrative scheme the *Phoranomus* establishes for the forthcoming dynamics. The principle of equivalence is still at the center of the Leibnizian strategy, but to yield an abstract understanding of what force means as the causal element for both constrained and unconstrained motions, the new *mathesis mechanica* must find a means of conciliating mathematical and metaphysical reasons.

I shall start by considering Robinet’s point about the causal connection of the *Phoranomus* with the *Dynamica de potentia.* Robinet sees in the former text the same type of inventive process one finds in the *De corporum concursu.* ^{9} In the 1678 manuscript wherein Leibniz first adopted the conservation of *mv*^{2}, Leibniz began by performing an analysis of the cases of *concursus* according to the norm specified by the principle of conservation of quantity of motion (Leibniz 1994; Fichant 1990; Duchesneau 1994, pp. 95–132). On the way, he had been struck by the inconsistency between the theoretical inferences thus conceived and the data provided by his experiment on the collision of bodies endowed with pendular motions. He had therefore reconsidered the main presupposition—Descartes’s absolute conservation rule—in light of the causal equivalence principle and he had proceeded to revise *(reformari)* the analysis of cases according to a new conception of the measure of force; this conception thereafter found its canonical expression in the *vis viva* conservation principle. Hence a redrawing of the analysis *“post reformationem.”* Robinet attempts to uncover an analogous revolution in theoretical approach in the rather abrupt transition from the *Phoranomus,* probably written in the second half of July 1689, to the *Dynamica de potentia* whose initial drafts can be traced back to August-September of the same year. The dialectical rupture he sees resides in a two-term conceptual system *(power* and *effect)* in the *Phoranomus* being supplanted by a three-term system *(power, effect,* and *action)* in the *Dynamica de potentia.* Only the theoretical framework provided by the latter would bring about a complete theory of force, and correlatively a new science, designated by the term *dynamica.* This view of the transition between the two texts hinges on a “revolutionary” causal scheme: in light of his latest theoretical invention, that of the concept of action, Leibniz would 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 invention of the dynamics. It seems that Leibniz’s methodological preoccupations following the discovery of the *vis viva* theorem have justified more directly his search for an a priori demonstration. Complex reasons, at once epistemological and metaphysical, may help explain the contents of the concept of unconstrained effect. Henceforth, the analysis leads to a new principle, that of the conservation of quantity of action, synthetically grounded in an abstractly conceived system of representations. As a consequence, that principle and this system make it possible to enlarge the theoretical framework of the reformed mechanics. Thus, the principle of conservation of quantity of action and the abstract system of force, on the one hand, and the principle of *vis viva* conservation and the more concrete and empirical system concerning phenomena this principle refers to, on the other hand, are integrated under the same theory. Inasmuch as the Leibnizian methodological project should be taken direct account of, the *Phoranomus,* in its very state of incompleteness, provides significant evidence on the combinative models underpinning the enlargement of the reformed mechanics in the *Dynamica.* On that account, my analysis of Leibniz’s demonstrative scheme will mark its distance from the notion of a methodological rupture.

In the *Phoranomus,* Leibniz situates his program by reference to Galileo’s attempt at applying a *more geometrico* demonstrative structure to phenomenal reality. The actors in the dialogue set forth the requirement that experiments should be linked with reasons, and that, to this end, one should resort to mathematical analysis according to the several techniques available. But speaking of demonstrative structure means resorting to principles. Beyond statics in the Archimedean tradition, the principles did not seem to have been identified yet that might ground and justify a system of demonstrative inferences about phoronomy, that part of mechanics dealing with communication of motion and interaction of colliding bodies (I, §5, Leibniz 1991*a*, pp. 452–53). This is the challenge Leibniz wishes to take up in the *Phoranomus.*

To attempt this demonstrative extension of mechanics, Leibniz uses a key belonging to a *mathesis mechanica* yet to be completed. He states: “I shall therefore bring forth clearer demonstrations than any *mathesis mechanica* may have ever seen” (II, §M, Leibniz 1991*b*, p. 837). This happens in a passage where Leibniz tries to answer Catelan’s and the Cartesians’ objections on the ground of an extended “geometry.” This Leibnizian *mathesis mechanica* combines several conceptual tools which Leibniz had used somehow in his previous work on the reformed mechanics: (1) an architectonic principle, the principle of equivalence between full cause and entire effect, underpinning the arguments previously used relative ** [End Page 93] **to the reformed mechanics; (2) models directly borrowed from infinitesimal geometry and used to transcribe statical and phoronomical phenomena involving transition to a limit; (3) combinatorial models employed to unravel the contents of theoretical concepts representing the order of causes and effects; and (4) abstract definitions relating to the order of causes and effects: these definitions are “metaphysical” insofar as they exceed the level of what is geometrically representable, and cannot therefore be based on imaginative analogies. In less analytic fashion, this methodological program was announced in Baldigiani’s introductory remarks to Lubinianus (Leibniz):

As . . . geometry is subjected to an analytic calculus by means of the equality between the whole and all its parts, so in mechanics by means of the equality between the effect and the whole of its causes, or between the cause and the whole of its effects, certain so-called equations and a sort of mechanical algebra are reached through the use of this axiom. Hence you will conjoin a science most useful to life with great personal benefit, if you bring us such bright light in the great darkness we are in, and impose laws not only on statics, which Archimedes had formerly put under bondage, but also on universal phoronomy and the explanation of moving forces.

(I, §6, Leibniz 1991a, p. 454)

Charinus’s arguments for launching the discussion point to the same methodological goal of developing a *mathesis mechanica* that would reach the level of abstract intelligibility required by a radically 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. But the moving forces seem to me to possess something incorporeal I do not know of, and very little subject to the imagination. Therefore, everytime I would conceive the powers of machines, I was confronted with something unexplored and not admitting of any image.

(I, §8, Leibniz 1991a, p. 457)

The various elements of *mathesis mechanica* thus brought onto the stage need investigation. But the question is above all to what degree they combine into an adequate demonstrative strategy.

The principle of equivalence between full cause and entire effect affords a leitmotiv among the *Phoranomus* arguments. Since this very principle ruled over the demonstrations in the *De corporum concursu* and *Brevis demonstratio,* it does not assume now a wholly original, but rather a 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 the domain of the geometrizable to a causal approach in terms of moving forces. The generalizing function of the principle should be stressed in contrast with its previously more restricted usages. An architectonic principle serves theoretical explanations in an essential way by providing heuristical schemes. This heuristical function is twofold: the principle is used, on the one hand, to discard faulty models, and, on the other, to determine and build models that may optimally satisfy the sufficient reason requirement. According to that pattern, the principle of causal equivalence plays a determining role in extending demonstrative models from statics to phoronomy, as witnesses the sought-for transition from equilibrium and conservation of the common gravity center to force displayed in cases of impact, and further on, to force exerting itself in unconstrained motion (I, §18, Leibniz 1994, p. 477). While phenomena in statics can generally be represented geometrically restricting the expression of causes to mere extensive parameters, the transition to the theory of force implies that we deal with notions of cause that need to fit the dynamical process taking place either in “violent” or “innocuous” motion. ^{10}

Framing those determining causes abstractly through adequate models that pertain at once to the mathematics of the infinite and to certain theoretical definitions warrants applying the equivalence principle to this new domain. In the passage immediately preceding the a priori argument about uniform unconstrained motion, Leibniz notes that this “metaphysical” principle has provided him with an Ariadne’s thread for the estimation of forces. Insofar as this principle also allows inferences conformable to the data of experience and free from intrinsic aporias, he has set forth a system of causes congenial to his underlying commitments, causes which are neither “surd” nor “purely mathematical,” as would be the intercourse of atoms deprived of any intelligible inner diversity and thus entailing some “blind property of nature,” but which reflect an “intelligent order,” analyzable in terms of “metaphysical reasons” (II, §10, Leibniz 1991*b*, p. 811). Working out models, in particular theoretical models to represent force as cause, should make it possible to apply the principle in the area of phoronomy. Hence the critical importance of those models for conceiving ** [End Page 95] **how the *mathesis mechanica* can be achieved. Instances of this type of model are provided in later Leibnizian texts, for instance in the *Specimen dynamicum* (1695), where Leibniz specifies his typology of primitive and derivative, active and passive forces and sets forth a system of representation for the integration of conatus and impetus that draws on the analogy of a stepwise summation of infinitesimals (GM VI, pp. 234–54).

As noted, such models relate, on the one hand, to a geometrical analysis inspired by infinitesimal calculus, on the other, to a system of concepts capable of expressing the causal and formal elements beneath constrained and unconstrained motion. Let us consider the aspect concerning geometrical infinitist models. Even when he resumes Archimedean statics, and in particular the theory of barycenters, Leibniz takes advantage of the technique of transition to the limit in representing infinitesimal ratios. But building this type of model is particularly meaningful when dealing with the representation of accelerated and decelerated motions as effects resulting from centrifugal or gravitational force. Reciprocally, the models resorted to in accounting for phenomena of resistance rely on the same logic of analysis in terms of series of infinitesimal ratios.

Leibniz starts with his distinction between *vis viva* and *vis mortua.* He presumes that the relation between “dead” and “living force” is analogous with that between the finite and the infinite, or that between the point as the beginning of a line and the line itself. Also, conatus stands to dead force, as impetus does to living force (I, §18, Leibniz 1991*a*, p. 478). Such is the starting point in reformed mechanics for the distinctions Leibniz refines throughout subsequent presentations of the dynamics. The impetus is conceived as generated by a continuous summation of conatus in the moving body. So one passes from an embryonic to a developed dimension of force. In this connection, Leibniz recalls that Galileo and Giovanni Alfonso Borelli after him had contrasted the force of percussion to that of gravity, as the infinite to the finite. ^{11} Every body in motion thus possesses an impetus, and that impetus helps express the existence of a *vis viva* generated through a summation of conatus. It is noteworthy that the proposed model, if it draws adequately the relationship of conatus to impetus, fails to indicate clearly how the mathematical expression for impetus differs from that for *vis viva.* However, in line with his more definitive theses, Leibniz develops the relationship of inertia with the ** [End Page 96] **action of conatus, since the resistance of bodies to motion implies inverse summative processes. If any motion, be it so small, of whatever body can act on any other body, be it so large, the speed communicated to the second body and relative to the impetus of the first will be the smaller, the larger the affected body, following the ratio between their respective impetus: “If the surrounding bodies do not create impediment, it is certain that a body of whatever magnitude at rest can be moved by another of whatever smallness. And the inertia of matter does not consist in that it is absolutely repugnant to motion, but in that it will receive less speed, the larger the matter which receives it” (I, §20, Leibniz 1991*a*, p. 480). But, with explanatory reasons derived from the infinite summation of such elementary ingredients as conatus, how can we get an adequate representation of the effect of gravity so as to account for accelerated motion in fall, and indirectly for the force conserved in the interaction of bodies? In response, the explanation of gravity is based on a controlled analogy with the summation of impressions in centrifugal force as conceptualized by Huygens. And Leibniz interprets that analogy according to the specifications of his algorithm about the integration of infinitesimal quantities; so he distinguishes in centrifugal force the *vis impressa* and its cumulative effect in the *impetus centrifugus* (I, §21, Leibniz 1991*a*, p. 482). But how can one get from there to a model for gravitational acceleration?

Leibniz insists on the fact that gravity is a physical phenomenon whose cause or causes remain obscure and controversial. Indeed, he refuses to subsume the unattainable explanation of gravity under a force of attraction God would have endowed matter with. Rejecting Newtonian attraction—though Newton is not named—because such an occult quality would contradict the formal requirement of sufficient reason, Leibniz aligns himself with Kepler, Descartes, and Huygens in hypothesizing some mechanical cause. He presumes that by a cause similar to centrifugal impressions and their summation in the form of impetus, some very dense imperceptible bodies tend to swerve from the center and to push towards it less dense bodies endowed therefore with lesser centrifugal propensity (I, §21, Leibniz 1991*a*, p. 481). He even proposes an experiment to compare centrifugal force with the force of gravity directly (I, §22, Leibniz 1991*a*, p. 482). ^{12} Given a tube in oblique rotation whose lower end is immersed in a water-filled container, water can rise in the tube to a given height by the centrifugal force issuing from rotation (see fig. 2). This makes for an effect corresponding to gravity and can therefore be used to measure gravity. On that ground, Leibniz recasts Galileo’s account for the empirical law of fall according to his own model of infinitesimal summations. Further on, the ** [End Page 97] **same type of model could be applied in trying to establish laws of progression concerning the elasticity and resistance of bodies. ^{13}

The third component of *mathesis mechanica* consists in using conceptual models of the combinative type. Thus, one finds at the background of the a priori argument a more or less explicit combinatorial account of the notion of effect in uniform unconstrained motions. In accordance with the analytic requirements of a similar model, the *Dynamica de potentia* replaces the incompletely analyzed notion of effect by a notion of formal action combining an intensive with an extensive parameter—thus will the formal effect be combined with the velocity of its own accomplishment. In his yet imperfect analysis of formal effect in the *Phoranomus,* Leibniz tries to combine several relations that could add up to a determining reason for the exhaustive effect of force as maintained in a continuous process. One thus gets a series of laws of *motus æquabilis* (II, §G, Leibniz 1991*b*, pp. 821–26), which equate the dynamic factors of mass and force with an integration of the kinetic factors of time, speed and distance run through. That very series is transposed and revised in the *Dynamica de potentia,* so as to make up for the analytic ingredients of formal action and thus express a power that restores itself constantly.

The demonstrative value of conceptual combinative models is underlined by the type of validation they allow. In the case at hand, Leibniz appeals to a scholastic precept which expresses the specific modality of apagogic demonstrations as applied in mathematical physics, starting with ** [End Page 98] **Archimedean statics: “The conclusion is true, and no other possible reason is found, therefore the reason premised is true” (II, §G, Leibniz 1991*b*, p. 821). In this instance, the truth of the conclusion is established a posteriori. But the combinative hypothesis making for an explanatory reason is conceived a priori and helps eliminate any other set of projected sufficient reasons. In short, any other hypothesis but Leibniz’s a priori construction would prove defective because of some implicit contradiction in the combination of conceptual ingredients involved. The demonstrative link depends on the indirect condition that the contrary is deemed impossible, while the a posteriori argument seems to present the advantage of an almost direct empirical inference. But, in this latter case, if the validity is immediately assessed, it is restricted to a particular state of the system of nature experience reveals. Extending the conclusion so as to reach a general law would require, in addition, resorting to a kind of apagogical argument that could exclude all other combinations of possible reasons on the ground that they would entail some contradiction. So, when reviewing in the *Phoranomus* the arguments that formed the essentials of his reformed mechanics, Leibniz tends to establish them through apagogical demonstrations that any other alternative in terms of explanatory reasons would result in mechanical perpetual motion. Hence the combinatorial pattern can be said to rule over both styles of argumentation. But with the a priori approach, it would provide a direct abstract representation for the inner nature of things; with the a posteriori strategy, it would offer an indirect representation for force as causal ingredient through a hypothesis concerning phenomenal equivalents. The conjunction of the two ways witnesses to this dual aspect of the combinatorial model. ^{14}

Applying this model in compliance with the principle of causal equivalence and the analogies of infinitesimal geometry requires a set of abstract definitions, supporting a coherent theoretical construction. These definitions undergird the reasons for the order of phenomena on the a posteriori way. But, at the same time, they are meant to define concepts on the a priori way. Thus, we find in Dialogue I of the *Phoranomus* a strategic distinction between the two types of force, *vis mortua* and *vis viva.* This distinction gives rise to a parallel one about conatus and impetus, as respective expressions for dead and living force. Between conatus and impetus, the summation relation builds up from the finite to the infinite, ** [End Page 99] **as in the transition from a point as the virtual beginning of a line, to the line duly actualized (I, §18, Leibniz 1991*a*, p. 478). However, the impetus represents a measure of living force in the instant and not in the unfolding effect by which it gets exhausted. Correlatively, the impetus is negatively measured by the inertia of the body whose mass absorbs the summation of instantaneous conatus. These systematic distinctions are developed and refined in later texts, especially the *Specimen dynamicum* (1695) (GM VI, pp. 234–54). In the *Phoranomus,* Leibniz is content to draw his distinctions within compass of a particular context: the problem at hand is to get beyond merely geometrical concepts in fixing the causes or reasons for the communication of impulse and conservation of moving force. The reality of force has to be accounted for by “ideal principles of metaphysics” (I, §9, Leibniz 1991*a*, p. 458). The representation of underpinning causes exceeds our capacity for geometrical schematization: it requires a reference to theoretical entities to be conceived beyond the geometrical conception of bodies.

Already, if one follows the initial considerations in Dialogue II of the *Phoranomus,* accounting for inertia as it is involved in the transition from conatus to impetus means going beyond the Cartesian and Democritean (atomistic) notions about the essence of body, conceived as pure extension or as extension conjoined with impenetrability. In one of the more meaningful critical passages about his own early mechanics of the *Theoria motus abstracti* (1671), Leibniz explains that he had progressively withdrawn from the fictitious laws of pure phoronomy. At the time, he had proposed a purely rational theory of motion, based on abstract notions of conatus as indivisibles of motion: these were supposed to combine in algebraic fashion, and, when conflicting with regard to direction, their combination would produce conatus equal to their difference. The combination laws for such conatus would not take into account such physical characteristics as mass, resistance and elasticity (A VI, II, pp. 258–76; Duchesneau 1994, pp. 35–67). After rejecting that early theory, he was looking for an alternative causal foundation for inertia and moving force jointly considered. Further, such a foundation was needed to warrant the conservation of a total system of quantity of motion, or *post reformationem,* of living force, in the universe (II, §§C-D, Leibniz 1991*b*, p. 809). Such a theoretical foundation cannot be provided by any entity measured by motion as such, since motion is essentially relative. The Leibnizian solution which is common to both the a priori and a posteriori ways consists in setting out sufficient reasons for the production of effects that may satisfy the architectonic principle of causal equivalence. In other words, the strategy is to hypothesize theoretical entities in accordance with that principle: this is what ** [End Page 100] **Leibniz means when he states that the “decrees of the new science about power and effect . . . prescribe laws to the universe itself” (II, §G, Leibniz 1991*b*, p. 826). But to frame and express those theoretical hypotheses, one must frame models that will satisfy the norms of extended geometrical intelligibility. This intelligibility implies the resources of infinitesimal analysis as well as a combination of abstract formal definitions beyond mere quantitative analogies. In counterpart, Leibniz condemns any attempt at resorting to scholastic occult qualities. Notwithstanding its imperfections, the *Phoranomus* exemplifies the requirement that extended “geometrical” models and abstract definitions must fit together for the framing of adequate theoretical hypotheses. The means to get such a fitting arrangement resides in the use of architectonic principles. Such principles are instrumental in blending together the several elements of a *mathesis mechanica* that may actualize the theory of force into a set of demonstrative propositions.

Contrary to what Robinet underlined as a drastic theoretical shift subsequent to the *Phoranomus,* a close analysis of the text reveals that Leibniz sets up therein a sophisticated methodology which will directly influence the final stage of the dynamics. This methodology combines the appeal to architectonic principles such as the principle of the equality between full cause and entire effect with significant attempts at extending infinitist mathematical models beyond Archimedean and Cartesian statics to physical processes involving forces. Above all, the new *mathesis mechanica* tries to frame up new conceptual combinative models to account for the formal structure of forces as causes of unconstrained as well as constrained motions. This explains why Leibniz is trying to form combinative definitions that may determine the complex structure of dynamical effects and 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 the *Phoranomus* underlies the analyses that form the dynamics proper in the period just subsequent. In line with that pattern, the *Dynamica de potentia* integrates the theorem of the conservation of *vis viva* within a theoretical framework that avoids the deficiencies of the former presentation. As “a science of power and action” (GM VI, pp. 287, 464), dynamics arises when Leibniz proposes an adequate architectonic of laws for constrained, as well as unconstrained, motions. At the basis of the argument about force exerting itself without constraint, the revised theoretical definitions now concern formal effect and formal action: ** [End Page 101] **

The quantity of formal effect in motion is that whose measure consists in a certain quantity 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 certain quantity of matter being moved through a certain length (motion being uniformly equidistributed) within a certain time (GM VI, pp. 345–46).

The designation “formal” applies to properties that are judged “essential” by contrast with modal features which depend on the contingent *situs* of bodies in a specific physical system. The distinction between the two types of effects derives from the fact that the former reveal themselves directly in unconstrained motion, and hence partake of a rational and “metaphysical” apprehension of corporeal reality, while the latter manifest themselves through the resistance of bodies to motion—Leibnizian inertia—according to the experience of mechanical changes which affect phenomenal bodies. The concepts of formal effect and action are presented as distinct notions, which combine conceptual elements so as to signify the essential ingredients of force, and hence, the essential “form” of bodies, but in such a way that the relations involved may be translated as combinations of quantitative parameters. This conception needs to fit architectonic requirements, which imply that the “requisites” of those concepts should ideally combine to form real definitions of the active and substantive elements beneath phenomenal realities and interactions. Such “pure” combinations of requisites would form the proper condition for a priori intelligibility. However, we may postulate by extension some more hypothetical combinations of requisites beyond those which would be reached through a direct and fully adequate analysis of notions; these ought to be admitted, provided they fit the relevant architectonic requirements and afford true analytic equivalence for their objects. Under such conditions, hypothetico-deductive abstract correlations are, and should be, called upon to guide our mathematical explanation of phenomena. This is precisely the case with the main concepts of Leibnizian dynamics.

Thus, the methodological pattern for the dynamics entails building from “metaphysical” definitions, representing abstractly the inner order of physical realities, such a system of equations as may converge and evince the implications of *actio motrix.* The axiom which binds together that system of equations states: “The same quantity of matter moving through the same length in less time forms a greater action” (GM VI, p. 349). This specifies the *power to act* which is directly proportional to the quantity of matter times the spatial displacement, and inversely proportional to the time wherein action unfolds. ** [End Page 102] **

The demonstrative argument for this theoretical proposition is at times phrased in syllogistic form, at times presented in the form of a “mathematical” calculus based on the substitution of definitional equivalents. For instance, the syllogistic form is to be found in the letter to Burcher De Volder 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 action accomplishing the simple in simple time. . . .

(2) The action accomplishing the simple in simple time is twice the action accomplishing the simple in double time. . . .

Hence the conclusion:

3) The action accomplishing the double in double time is four times the action accomplishing the simple in double time.

(GP II, p. 173)

The letter to Denis Papin of 14 April 1698 provides an equivalent formulation which conforms to the 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)

sasvt; or else the spaces run through are in combined ratio of the times spent and the speeds.(2)

aassv: or else the actions are in combined ratio of the spaces run through and the speeds with which they have been run through.(3) Therefore (in art. 2, substituting

tvforsaccording to art. 1)aastvv. Or else: the actions 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 representing the notion of formal action, Leibniz must make use of a distinction that implies a twofold conceptual significance for the factor *v.* On the one hand, action is considered from the viewpoint of its formal effect in spatial displacement; on the other, its intensive aspect is considered from the viewpoint of instantaneous production of that formal effect. In the *Dynamica de potentia,* a strategic definition translates this twofold relation of *intensio* and *extensio* (or *diffusio*) of action: “The diffusion of action in motion or the *extensio* of action is the quantity of formal effect in motion. The *intensio* of the same action is the quantity of speed by which the effect is produced or the matter is carried through length” (GM VI, p. 355). Leibniz combines an embryonic effect in the instant, represented in factor *v,* with an effect deployed in the translation space for the body in unconstrained motion, represented in factor *s,* therefore implicitly in factor *v* as implied by *s.* It is thus presumed that the embryonic factor reiterates ** [End Page 103] **its intervention constantly as the formal effect is accomplished. We are presented with a translation of motive action as a sort of active form, of causal agent involving at once the propensity to act and the motive effect translating that propensity to actuality in duration. Since all resistance to motion is suspended, this propensity is presumed to conserve itself integrally through formal effect, “in such a way that it can be added to that effect as a permanent gain in terms of virtual translation” (Duchesneau 1994, p. 186). Through combining the extensive and intensive ingredients expressed in the motive effect and the conserved virtual effect, one gets a theoretical representation of the power to act. On that basis, the *Dynamica de potentia* tends to focus on such a concept as the keystone for an architecture of reasons that would integrate the theorems of conserved action as well as those of conserved *vis viva.*

The nodal role of the *intensio-extensio* couple in the analysis of “essential action” refers to a metaphysical notion signifying the immanent finality of centers of force. At the same time, this construction gets expressed in mathematical models, and reason monitors the whole process in pursuance of the architectonic requirements for theory building. A corroborative instance of this style of argument at the basis of the *intensio-extensio* combination can be found in the correspondence with De Volder, when Leibniz tries to specify what causal perfection *(præstantia)* of action may consist in. De Volder’s problem stems from the fact that he had reduced the *præstantia* of action to the sole intensity of action as measured by instantaneous speed. One cannot consider *intensio* as representing force completely nor measure it according to speed in the instant alone. Leibniz suggests distinguishing the extensive and intensive parameters of force, reducing them to their intelligible terms, and setting forth the combination of requisites that can unite them in architectonic mode. So we get two possible relations: (1) actions are directly proportional to the product of powers and times; and (2) actions are directly proportional to the product of speeds and spaces run through. The second relation can be further resolved by considering that spaces are measured by the product of speeds and times. Provided the relations are reduced to the time unit, both assessments of the value of action agree in equating power with the product of mass and the square of speed. This is, it seems, “the mark of an analytic construction that meets the norms of architectonic combination” (Duchesneau 1994, p. 298).

This interpretation gets clear support from Leibniz’s epistemological comment about combinatorial resolution of the formal action concept in his letter to De Volder of 9/20 January 1700: “Thus you may see how everything conspires once more with beauty and gets united according to indubitable reason” (GP II, p. 203). Concerning the same notion, he writes Papin on 7 May 1699: “Perfection, or the degree of reality in things, ** [End Page 104] **particularly in motion, can be estimated according to two reasons, namely extension which is here the magnitude of the changed place or space, and intension which is here the promptness or speed in change or motion” (LBr 714, fol. 310r, Ranea 1989, p. 56).

A. G. Ranea’s interpretation concerning the conjoined notions of *intensio* and *extensio* suggests that Leibniz resumes categories borrowed from the fourteenth century *calculationes* and relative to *latitudo formarum.* ^{15} According to him, Leibniz, out of speculative audacity, confers a status of essential qualitative properties on quantitative factors whose objective meaning could only refer to modes of extension. To be sure, we know from Michel Fichant that Leibniz was interested in the *Calculationes de motu et intensionibus et remissionibus formarum seu qualitatum* of Richard Swineshead, one of the theoreticians at Merton College, Oxford in the early fourteenth century (Swineshead 1520; Sylla 1991). Leibniz had access to that book while in Florence at the time the project of the *Dynamica de potentia* was taking shape. Later, he had a copy made from one of the editions held by the Bibliothèque du Roi in Paris. But the exact connection between Swineshead’s and Leibniz’s theses still remains to be determined. My initial impression is that the techniques developped by the *Calculatores* in Oxford, then in Paris, and culminating in Nicole Oresme’s *Tractatus de configurationibus qualitatum et motuum* (Clagett 1968), aimed essentially at a geometrical representation of accelerations and other such intensive properties, so that the several cases at hand could be accommodated for by calculation. In contrast, the Leibnizian circumstances are very different. Leibniz aimed, as it seems, at setting an expressive correspondence between the algebraic representation of the parameters of force as it conserves itself through action, and the projection of formal sufficient reasons that would represent the underpinning causes. Therefore, the historical connection between Leibniz and the *Calculatores,* especially Swineshead, might be less significant than suggested by Ranea. Personally, I would stress that by joining the intensive and extensive dimensions of action, Leibniz was attempting a theoretical construction in combinatorial style: his goal was to *symbolize* the order of efficient causes at the background of those effects ** [End Page 105] **associated with unconstrained action. The force thus defined as a theoretical entity was characterized by *actio in se ipsum,* an immanent activity of the moving body reproducing itself in motion, so to speak. This is what Leibniz explains to De Volder in a letter after August 1699:

In the free or formal action of the mover, when conceived as acting on itself (

in se ipsum), we can conceive analogically a real effect that will not be the change of place (which we consider only a modal effect), but the mover itself preparing with a given speed 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 supported by architectonic principles, including the principle of finality. For these principles had to be called upon for guiding the combination of “metaphysical” concepts that would represent the formal cause of action and force, and for framing the mathematical models that would best express such a combination of concepts in the geometrical order. They were thus needed to form theoretical explanations concerning a specific system of contingent truths that would comprise the fundamental laws of physical nature. Such an epistemological device, tentatively launched in the *Phoranomus* and more adequately articulated in the *Dynamica de potentia,* helps explain how Leibniz worked out the principles of his dynamics and proposed them as the original and quite consistent theoretical basis for his physics. Correlatively, this theoretical framework determines significant shifts in Leibniz’s natural philosophy of the later period.

Before the *Phoranomus,* written in July 1689, Leibniz’s reformed mechanics, which had arisen from the *De corporum concursu* (1678), comprised a demonstrative system that was at least partly a posteriori: among its premises, one would find empirical laws—in particular, Galileo’s law of free fall—and contingent systemic conditions, such as elasticity and gravity. Under those conditions, the axioms of statics and the architectonic principles—for instance, the principle of causal equivalence and the law of continuity—seemed to apply only to a given state of the physical universe. In an attempt to surpass Galilean science, and indeed Cartesian physics, the *Phoranomus* envisages an immediate application of the more a priori premises, such as the principle of equivalence between full cause and entire effect, in analyzing the causal element in uniform unconstrained motion. The formal ingredient of cause beneath such a motion does not entail accumulation and exhaustion of forces, but it involves a constant self-reproduction ** [End Page 106] **of forces. Leibniz presumes that there must be a strict congruence between the effect characterizing such motions—measured by the product of masses and distances run through divided by times—and the measure of force as fully consumed in cases of impact. The aporia behind this type of presupposition is due to the fact that Leibniz has not yet succeeded in assigning a mode of synthetic combination of the exhausting effect with its preservation in unconstrained motion, a mode of combination that may warrant the equivalence. While he combines “substantive” with “modal” elements, namely the *res* with a twofold spatio-temporal *status,* to account for the various *casus,* he still lacks an adequate combinative model such as that of the *Dynamica de potentia* (1689–90); this later model represents action through its combined intensive and extensive formal features (Ranea 1989; Duchesneau 1994). On the other hand, the justification he brings forward in the former text is mainly apagogical: it is based on the presumed sufficiency of the proposed combination correlated with the impossibility of conceiving any other more adequate alternative. In the later text, Leibniz tries to ground the sufficiency of the chosen combination on a more systematic and direct analysis of its various implications. Are we therefore faced with a profound dialectical rupture when passing from the less perfect theory of the *Phoranomus* to that of the *Dynamica de potentia*? The fact is that the *Phoranomus* spells out the methodological requirements of a *mathesis mechanica,* combining mathematical models which appealed to the resources of infinitesimal geometry, with abstract definitional constructions of a more metaphysical kind, under the ægis of architectonic principles. With the tools of this *mathesis,* the new objective is to offer a causal theory of force in cases of unconstrained, as well as constrained, motions. There is no doubt that this objective opens up the main perspective for the new science of dynamics. Thus, a relatively continuous methodological transition, which differs considerably from the dramatic shift that took place in the *De corporum concursu* (1678), seems to link the *Phoranomus* to the *Dynamica de potentia* wherein the new science achieves its true formal expression.

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

# Footnotes

1.
“Cujus rei ut aliquem gustum dem, dicam interim, notionem *virium* seu virtutis (quam Germani vocant *Krafft* Galli *la force*) cui ego explicandæ peculiarem *Dynamices* scientiam destinavi, plurimum lucis afferre ad veram *notionem substantiæ* intelligendam.”

3.
On the *vis viva* controversy, see Laudan (1968), Iltis (1971), Gale (1973), Papineau (1981).

4.
In this context, A. Robinet uses the phrase “mouvement essentiel”. Such an expression is somewhat misleading for Leibniz would not take motion, which is always relative, for the essence of any reality whatsoever. According to Leibniz, extension itself, which for the Cartesians would underpin the modes of motion as the essence of *res corporea,* belongs to the phenomena as a nonessential property. It may however be well-founded as it expresses the inner activity of the immaterial finite substances in orderly fashion.

5.
In a preliminary text entitled *Conspectus operis* (GM VI, pp. 284–85), Leibniz was already using the term *dynamica* in a context wherein it related both to abstract and concrete analysis, bearing respectively on force and action, and on active causes and effects as they operate in the “system of things.”

6.
*Phoranomus,* II, §M (Leibniz 1991*b*, pp. 837–88): “Demonstrationes igitur afferam, quibus fortasse nihil unquam clarius Mathesis mechanica vidit. Et quidem diversis modis consequar idem. Principio autem facile mihi opinor concedetis hanc *Hypothesin,* cui similes usurpant eruditi, nimirum *non posse fieri ut vi gravitatis, et earum actionum inter corpora quæ ex ipsa sola sequuntur, commune corporum centrum gravitatis altius in fine reperiatur quam initio fuit.*”

7.
*Phoranomus,* II, §D (Leibniz 1991*b*, p. 811): “Ut igitur ex illo Labyrintho me tandem expedirem, non aliud filum Ariadnæum reperi, quam æstimationem potentiarum assumendo Principium, *Quod Effectus integer sit semper æqualis causæ suæ plenæ. Id vero cum experimentis perfecte consentire et omnibus dubitationibus satisfacere deprehenderem. .* . . “

8.
*Phoranomus,* II, §F (Leibniz 1991*b*, p. 818): “Quare et potentiarum eadem erit æstimatio cum nihil aliud sint casus, quos attulimus, quam potentiarum effectus. Et quam hæc rationi consentanea sint vel hinc apparet, quod in motu æquabili ex dato spatio percurso non ideo determinatur velocitas, aut contra. Itaque rationes ambæ sunt conjungendæ, quæ in casu temporum æqualium dant duplicatam [velocitatum rationem].”

9.
See A. Robinet’s presentation (Leibniz 1991*a*, p. 437): “Le *Phoranomus* fut à la *Dynamica* ce que la première version du *De corporum concursu* fut à la seconde: on y saisit la création leibnizienne à l’œuvre.”

10.
Even if Leibniz refers the use of the term *“phoronomica”* to Joachim Jungius’s *Phoranomica, id est de motu locali* (1679, 1689), he insists, as against Jungius’s position, that phoronomy should reach beyond mere geometrical considerations: “Sed vires vivæ seu impetus pertinent ad Phoranomicen strictius sic dictam, *Phoranomices* tamen nomine non intelligo quod Jungius in libello qui hoc titulo prodiit, ubi tantum investigat lineas tanquam motuum vestigia, quæ pars doctrinæ motus pure Geometrica est, sed ipsas naturæ leges, quæ circa communicationes motuum [et?] vires motrices observantur” (I, §23, Leibniz 1991*a*, p. 483).

11.
See Leibniz’s notes on his reading G.A. Borelli’s *De vi percussionis* (1686), LH XXXV, XIV, 2 f.2, cited by A. Robinet in his annotations on the *Phoranomus:* “Et in fine Dialoghi 4to de motu Galilæus dicit theoriam energiæ percussionis esse perobscuram . . . et vim percussionis esse interminatam. Promittebat alibi hoc demonstrare, sed nihil tale repertum Torricellinis eadem profectum non demonstrare, sed specturus [?] confirmat vim percussionis esse infinitam” (Leibniz 1991*a*, 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 1991*a*, p. 538), Leibniz reiterates on that account arguments he had just published in his *Schediasma de resistentia medii et motu projectorum in medio resistante,* published in the *Acta Eruditorum* in January 1689 (GM VI, pp. 135–44).

14.
See II, §G (Leibniz 1991*b*, p. 826): “et ideo ad amœniora et imaginationi magis satisfactura festino, quæ licet a posteriori sumantur, hoc tamen præstans habent, quod mentem blanda persuasione convincunt, cum illæ demonstrationes natura priores, cogant reluctantem. Et fortasse ipsemet interiora ne suspicione quidem attigissem, contentus vulgari motus æquabilis analysi; nisi per ista quæ nunc afferam exotericotera veritatem prædetexissem.”

15.
See Ranea (1989, p. 57): “[Leibniz’s inconsequent stratagem] suggests that the quotient ‘space/time’ does not exhaust the meaning of velocity in the *a priori* argument. I think we could get a clue to this question. . . . Within the framework of the scholastic Physics these *[extensio* and *intensio]* allude to, velocity also has two different meanings: either it means the quotient of space and time, or the intensity of the *accidens intrinsecum* of the moving body, i.e. its local notion. In this way, velocity becomes a metaphysical or ‘quasi-physical’ sign of the inner perfection of motion, a magnitude quite independent of any quantitative viz. extensive treatment. Leibniz echoes this basic assumption of fourteenth century physics when he states that a faster motion is essentially more perfect than a slower one.”

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