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 notionesubstantiæ, 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 mathesismechanica 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
1991a, 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
Decorporum 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 æquabilisqualis per se est"
(II, §§F and G, Leibniz 1991b, 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 effectuscientia" (II,
§G, Leibniz 1991b, 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 call dynamics. 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 andaction"
(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 1991a; also 1991b). 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 deM. L. sur
un principe général utile à l'explication des
lois de la nature par la considération de lasagesse
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
1991b, 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 1991b, 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
1991b, 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 1991a,
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 1991b, 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 1991b,
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 1991b, 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 1991a,
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
1991b, 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 1991b, 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 1991a, 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
1991a, 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 1991a,
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 1991a, p. 481). He even proposes
an experiment to compare centrifugal force with the force of gravity
directly (I, §22, Leibniz 1991a, 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
1991b, 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 1991b, 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 1991a, 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 1991a,
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 1991b, 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 1991b, 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) s as vt; or else the spaces run through are in combined ratio of
the times spent and the speeds.
(2) a as sv: 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 tv for s according to art. 1) a
as tvv. 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 depotentia (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.

Université de Montréal

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).

Notes

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."

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

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 1991b, 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 1991b, 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 1991b, 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 1991a, 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 1991a, 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 1991a, 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 arguments he had just published in his Schediasma de
resistentia medii et motu projectorum in medioresistante,
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 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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