Leibniz’s Dynamics: Origin and Structure of a New Science 3515135200, 9783515135207

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François Duchesneau

Leibniz’s Dynamics Origin and Structure of a New Science

Philosophie Franz Steiner Verlag

Studia Leibnitiana – Sonderhefte | 58

studia leibnitiana sonderhefte Im Auftrage der Gottfried-Wilhelm-Leibniz-Gesellschaft e.V. herausgegeben von Herbert Breger, Michael Kempe, Wenchao Li (verantwortlich) und Pauline Phemister In Verbindung mit Maria Rosa Antognazza †, Stefano di Bella, François Duchesneau, Michel Fichant, Emily R. Grosholz, Martina Hartmann, Nicholas Jolley, Eberhard Knobloch, Massimo Mugnai, Arnaud Pelletier, Nicholas Rescher und Catherine Wilson Band 58 https://www.steiner-verlag.de/brand/Studia-Leibnitiana

Leibniz’s Dynamics Origin and Structure of a New Science François Duchesneau

Franz Steiner Verlag

La dynamique de Leibniz © Librairie Philosophique J. Vrin, Paris, 1994. www.vrin.fr

Bibliografische Information der Deutschen Nationalbibliothek: Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über dnb.d-nb.de abrufbar. Dieses Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsgesetzes ist unzulässig und strafbar. © Franz Steiner Verlag, Stuttgart 2023 www.steiner-verlag.de Layout und Herstellung durch den Verlag Satz: DTP + TEXT Eva Burri, Stuttgart Druck: Beltz Grafische Betriebe, Bad Langensalza Gedruckt auf säurefreiem, alterungsbeständigem Papier. Printed in Germany. ISBN 978-3-515-13520-7 (Print) ISBN 978-3-515-13526-9 (E-Book) https://doi.org/10.25162/9783515135269

Acknowledgments This book is a second edition of La Dynamique de Leibniz (Paris: Vrin, 1994). It includes several modifications brought to the original version and takes account of various advances in scholarship concerning Leibniz’s philosophy of nature in the recent years. It is offerred to the readers in English, thanks to the talent and dedication of Paul Jackanich who completed most of the translations. Over the years, my reconstitution and interpretation of Leibniz’s dynamics has benefited from the exchanges I had with Leibnizian scholars and friends on this precise topic. I express my gratitude to all of them, but especially to Michel Fichant, Richard Arthur, Daniel Garber, Anne-Lise Rey, Laurence Bouquiaux and Christian Leduc. What is most challenging for whoever explores the essence of Leibniz’s science is that, notwithstanding the best efforts deployed, there is no chance of reaching a final and definite view on the contents of it, because Leibniz himself never put a stop to his enquiries, nor ceased to work out improved versions of his developing theses from various perspectives. This is especially true about the dynamics, whose major piece, the 1689 Dynamica went through many rewritings and remained uncompleted in light of Leibniz’s own objectives. This shall be my excuse for suggesting that this book of mine be considered as a mere attempt at providing a provisional synthesis on what formed the principal contribution of Leibniz to natural philosophy, in his own time and for his posterity.

Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter I The Initial Scientific Project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.. Analyzing Reality in Terms of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2..The Theoria motus abstracti: “Conatus” as Motion Indivisible. . . . . . . . . . . . . . . . 29 3..The Hypothesis physica nova: Unifying the Models. . . . . . . . . . . . . . . . . . . . . . . . . . 50 4..Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Chapter II Reforming Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 1.. The First Milestones of the Reformatio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2..The Reformatio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3..The Brevis demonstratio: Living Force as a Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4..Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Chapter III The Structure of Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 1..The Phoranomus: A Turning Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 2..The Dynamica de potentia: Implementing a Theory. . . . . . . . . . . . . . . . . . . . . . . . . 126 3..The Specimen dynamicum: Presuppositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4..The Essay de dynamique: Combining Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5..Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Chapter IV The a priori Analytic Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 1..Unveiling a priori Arguments to Johann Bernoulli. . . . . . . . . . . . . . . . . . . . . . . . . . 194 2.. The Parameters of Action: The De Volder Correspondence . . . . . . . . . . . . . . . . . 205 3.. Justifying the a priori Way for Papin, Bayle, Jacob Bernoulli, Wolff, and Hermann. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 4..Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8

Table of Contents

General Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 1.. Primary Sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 2.. Secondary Sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Introduction One cannot deny the significance of the contributions that Gottfried Wilhelm Leibniz (1646–1716) made to science in his day. Nor can one deny their impact on posterity. Even in an era dominated by Newtonianism, the discovery of infinitesimal calculus and the invention of dynamics, to give two examples, certainly could not have been overlooked. The influence of Leibnizian science was profoundly felt throughout the 18th and 19th centuries, even when Newton’s model reigned supreme. Following a complete overhaul of physics in the beginning of the 20th century and the gradual displacement of the Newtonian model, Leibniz appeared more and more as someone offering an original and, in many ways, relevant conception of the scientific method, as someone whose scientific theories had played a major role in the history of science. Contemporary scholars have indeed worked on reconciling certain, more recent theses with earlier formulations that can be traced back to Leibniz, a precursor of the Enlightenment. As such, they have drawn diachronic comparisons on themes like relativist analyses of motion, the rational ideality found within the foundations of the cosmological categories of time and space, the need for a concept of force for theorizing energy and field effects, and the role of principles of conservation in deciding upon explanatory models. All of these themes might serve as subjects of analysis and comparison, so long as we avoid the risks of altering their contents or introducing anachronisms. All of these also possess references and convergent meanings that today’s science shares with Leibniz’s, which clearly distinguished itself in its time from the argumentative style and the theses developed by Newton and the Newtonians. Herbert Breger rightly points out that the reasons why scientific Leibnizianism ceded its place to Newtonian physics are the very same ones that have, in our time, renewed interest in Leibniz’s approach to science.1 The theoretical Newtonian model would present itself as a strictly mathematical system, “deduced from phenomena” and designed to account for a vast host of empirical problems. By contrast, Leibniz endeavoured to study the “metaphysical” foundations of physics, and thereby the development of principles of explanation and analysis that would allow for approaching empirical problems in

1

“Symmetry in Leibnizian Physics,” in Breger (2016), 13–27.

10

Introduction

a way consistent with the demands of theoretical sufficient reasons. This explanatory approach relies on architectonic principles and hypothetical-deductive inferences for analyzing phenomena. Such is the specific content of the Leibnizian style, which not only demonstrates its relationship with contemporary methodology, but also underscores the historical uniqueness of its concepts, theories and models for representing phenomena. Ultimately, the recent interest of scientists and philosophers in Leibniz’s work seems to be epistemological in nature more than anything else. The way in which Leib­ nizian theories are modeled is intriguing and can offer insight into fundamentals of the philosophy of science such as the formulation of models for analyzing phenomena, the demands of causal explanation, the relativity of concepts of sufficient reason that represent the empirical world, and the invention and justification of theories in conformity with architectonic principles. We have elected to focus here on the origins and structure of the most elaborate theoretical body in Leibniz’s science, the dynamics. Leibniz himself planed for the dynamics to become the heart of his physics. The dynamics unfolds against a historical backdrop. Its starting point is a particular theory of the combination of motions placed within the framework of a mechanistically conceived natural philosophy. Even before his stay in Paris (1672–1676), Leibniz had elaborated an abstract mechanics in the Theo­ria motus abstracti (1671), whose counterpart in the Hypothesis physica nova (1671) consisted of a hypothetical physical theory involving material structures, which were themselves the products of combinations of motions. Faced with the inconsistency of this first synthesis, Leibniz must then take into account the empirical laws of impact as established, from 1668 on, by Huygens, Wallis, Wren and Mariotte. At the end of his stay in Paris, he sets out to reconcile these laws as well as the theorems concerning percussive forces with the help of the principle of conservation of quantity of motion for which Descartes had provided the formula in 1644. Following a remarkable demonstration of combinatorial analysis and with the help of sui generis methodological rules, Leibniz thus succeeds in formulating a new principle of conservation. As the work of Michel Fichant has taught us,2 this systematic reform occurs at the beginning of 1678. But public announcements of the discovery do not emerge until 1686 with the publication of the Brevis demonstratio erroris memorabilis Cartesii. A host of arguments and texts, some published and others not, will provide the foundations of a complicated theoretical structure for this new science that will soon blossom. The Dynamica de po­ tentia (1689–1690), the Specimen dynamicum (1695), and the later Essay de dynamique (c. 1700) represent significant steps of this process. Some famous controversies and correspondences on the demonstrative arguments of the theory, particularly on the so-called a priori approach, yield a complex and disparate body of work, across which

2

Fichant (1990); Fichant, Introduction, in Leibniz (1994), 9–65.

Introduction

11

one of the major discoveries of modern science takes shape. Focusing on these significant texts, and with a view to the series of arguments that they contain, our objective will be to analyze the methodological procedures and methods of theorization at work in what constitutes Leibniz’s dynamics. Certain difficulties beset this type of research. As a whole, the scientific work of Leibniz has suffered from several partial interpretations or distortions. The works that the librarian of Hanover published during his lifetime only represented a small part of his corpus; these often included allusive texts that only indirectly reflected the magnitude of fully completed analyses. The publication of manuscripts has been spaced out up until our own time and still continues; hence, all accounts, including those of the recent past, were doomed to produce some necessarily truncated picture of a multifaceted subject. Responding to this problem, Michel Fichant worked for several years to reconstruct the text of the De corporum concursu (1678) and provide us with an edition (Leibniz 1994). This very important draft contains the birth of the reformed mechanics; it reveals how Leibniz perceived the necessity of reforming Cartesian mechanics and substituting the principle of conservation of quantity of motion (measured by the product mv) with a new principle of conservation that would account for living force (measured by the product mv2). Leibniz implements this reform with the help of methodological and epistemological tools that none of the great interpreters of his work before Fichant had identified as reaching this stage of development. Such a lacuna proves significant. However, in the majority of cases, the difficulty of constructing a representation of Leibniz’s science reveals itself in a more subtle way. The texts are numerous, often fragmented, and sometimes divergent, addressing diverse questions. The sui generis coherence of the whole tends to escape us. It is easy, if not inevitable, to stray from a sufficiently faithful analysis for an arbitrary reconstruction of a larger than life Leibnizian model. Several important works have directly contributed to establishing a more adequate vision of Leibniz’s science and the philosophy of science that accompanies and underpins it. But this is above all true for studies focusing on the contributions of Leibniz to the formal sciences, such as the analyses of Couturat, Kauppi, Ishiguro, Burkhardt, several others on the logic, and the analyses of Hofmann, Belaval, Knobloch, Serfati, and their present successors on Leibniz’s mathematics.3 Leibniz’s epistemology of the natural sciences is a different story, notably with regard to the mechanics. Indeed, numerous commentators have taken an interest in this sort of research on Leibniz, but in our view, only one remarkable work on the dynamics proper had been published, Martial Gueroult’s Leibniz. Dynamique et métaphysique (Gueroult 1967; first edition: 1934). Gueroult placed the neo-Kantian and positivist interpreters of Leibniz’s science back to back. He established the particular coherence

3

All these works are cited in the bibliography.

12

Introduction

of a scientific approach that aimed to construct a sufficient and autonomous repre­ sentation of the phenomenal order by combining inductive and deductive methods; he showed at the same time how this explanatory construction justified itself within the metaphysical context that defines the monadological system, and how it required theoretical foundations more profound than the laws governing the interaction of phenomena. The picture that Gueroult paints is remarkable for a host of reasons, and far surpasses in depth and scope all of the partial reconstructions of the dynamics with which interpreters had busied themselves. We must however push back against the master on certain points. Gueroult approaches all of the Leibnizian texts known to him as if they ought to form so to speak a coherent, atemporal whole. But the changes that Leibniz’s work underwent are significant and must necessarily be taken into account. The lack of appreciation for the evolutive character of Leibniz’s scientific thinking is apparent, first, in the absence of any reference to the methodological styles that were successively developed in the texts that prepared the way for, and then dedicated themselves to the dynamics. Indeed, Gueroult never identified the seminal text of the reform, De corporum concursu; but more generally, all of the Leibnizian methodology lay hidden in the shadow of this brilliant analysis of the normative structure of the dynamics. From this approach there followed a categorial denunciation of the so-called a priori method for demonstrating the principles of conservation of living force and formal action, and by contrast, an equally categorical prioritization of the so-called a posteriori method. In this regard, it seemed that Gueroult understated the complexity of the theoretical constructions and relied on the surreptitious resurgence of a model dominated by the Newtonian paradigm. Comprising knowledge procedures capable of founding and justifying dynamics as a science, it was still a Kantian epistemological model that served as a point of reference and allowed the originality of the Leibnizian position to be inferred. Finally, it was difficult to imagine, beyond the limits of the dynamics, how Leibniz could have conceived of the structure of a science of complex phenomena. The reconstituted coherence of Leibniz’s mechanical system obscured in a sense the broad spectrum of possible theoretical models, which owed to the degree of complexity of the phenomena being considered. If Gueroult gave expression to the “metaphysical form” of the dynamics, he sidestepped every analysis of the methodological profile of Leibniz’s science. In conformity with a post-Kantian tendency, the conception of the system took precedence over the conception of the method when it came to framing the argumentative structure of Leibniz’s physics. By contrast, a number of more recent studies on Leibniz’s science outline the first steps of an approach similar to the one that we intend to adopt.4 These studies will

4

Cf. Bouquiaux (1994); Fichant (1998); Garber (2009); Tho (2017); Arthur (2018); Garber and Tho (2018).

Introduction

13

buttress the establishment of a framework of analysis that we regard as more suitable for representing the epistemological interpretation of Leibniz’s science. We shall therefore not inscribe Prolem sine matre creatam upon the frontispiece of this work, as Montesquieu did for his L’Esprit des Lois. In the contemporary context of the history and philosophy of science, we may content ourselves with being well-informed, critical successors, apt to explore some new avenue of research that might prove original and promising. Our means are no doubt furnished by the riches accumulated by our predecessors, distant and close. Such a debt merits recognition. In another work on Leibniz et la méthode de la science,5 I drew from the analysis of the dynamics to bring to light some more general perspectives relating to Leibnizian science as a whole. I focused on the programmatic aspects of Leibniz’s conception of science, and examined the methodological considerations that it furnished philosophical analysis. I therefore addressed epistemological topics such as the creation of an innovative methodology that proves to be not only combinatorial and analytic in nature but also rational and empirical, the relationship between the various categories of truth, the fundamental and complex role of conditionally necessary truths, the structure and function of scientific hypotheses the model for which is both analytic and heuristic, and finally, the specific role assigned to architectonic principles, namely finality, the identity of indiscernibles, and continuity. One cannot however deny that the invention of the dynamics dictates all reflection on the scientific methodology of Leibniz. The true method develops over the course of his scientific labors, and reveals itself through the demands of the science’s development. This is true of the Leibnizian method both in terms of its evolution and its content. Our plan here is therefore to locate the genesis of the dynamics as science, and to retrace the main steps of the argumentative structure that emerges and unfolds across Leibniz’s works. The chapters that we shall devote to this scientific project include the genesis of the reformed mechanics, the structure of the theoretical corpus of the dynamics, and the meaning that we must attribute to one of the most problematic methodological aspects of such a construction, the a priori analytic model.

5

Duchesneau (1993); Duchesneau (2022).

Chapter I The Initial Scientific Project Commentators agree in general that Leibniz’s philosophical thought process undergoes a significant transformation during his stay in Paris (1672–1676). Examining the philosopher’s career, is the Parisian phase not characterized by the influence of the novel mathematics and experimental physics to which his new mentor, Christiaan Huygens, exposed him? The fruits of this crucial period are well known: the infinitesimal calculus on the one hand, and on the other, the first critical reflections on natural philosophy that will lead to the reformed mechanics (1678). Indeed, these discoveries will not see the light of day until later on, but they had already begun to transform the orientation of Leibniz’s research when he undertook his new librarian duties in Hano­ ver. There is every reason to believe that this revised scientific project guides the establishment of his metaphysical system, just as it directs Leibniz’s interests in empirical investigations up until the publication itself of the principles discovered in the period of 1685–1700. But what then of the first philosophy of Leibniz? Is it relegated to simply being an object of historical curiosity? Does it have no relation to the philosophy of science whose components we wish to identify?1 The period of the initial scientific project begins with the Dissertatio de arte com­ binatoria (1666) and ends in 1671 with the publication of two treatises, Theoria motus abstracti and Theoria motus concreti (or Hypothesis physica nova), respectively dedicated to the Académie des Sciences of Paris and to the Royal Society of London. But Leibniz will never entirely repudiate the contributions of this period; instead, he will emphasize the inherent limits of the algorithms used in the Theoria motus abstracti and the errors that it might have produced in the establishment of an abstract mechanics. Consequently, the arguments of the Hypothesis physica nova seem incomplete, but might suffice as a theoretical framework that prepares the way for research that will respond to the provisional and revisable systematization of that science. After the Parisian period, Leibniz eloquently attests to this in the letter that he addresses to Honoré Fabri in 1676:

1

On these issues, we can rely on Beeley (1996).

The Initial Scientific Project

15

You see what I had undertaken when I founded my Physical Hypothesis, in which I only scratched the surface of the subject, but it nevertheless seems that my work was useful. For I called the attention of men to a new, and if I dare say, truer way of reasoning about nature. If I, young man that I was, practically a novice in these matters when I wrote about them, was too daring, I would ask that you hold onto, not my example, but my design, which others endowed with greater intelligence and experience might pursue with superior success. There is however no reason to be dissuaded by the outcome of a first attempt, and I do not doubt that one day I shall be able to say things more refined than even now, but I save that for another time, since deeper investigations are in order. (A II 12 443–44)

One must surely disabuse oneself of the impression of deficiency that commentators often highlight when examining these texts, and instead keep an eye out for the determinative foundations and beginnings of a definitive systematization. In the Specimen dynamicum (1695), Leibniz will indeed emphasize the deficiencies of the first mechanics that are attributable to the Theoria motus abstracti, but also the vision of a more “systematic” explanation of things that further research would address.2 If Leibniz goes beyond his initial perspectives and theses, then it is always with relative methodological continuity. Regarding the theses of De arte combinatoria, this is no doubt what the interpreters of Leibniz’s logic tend to underscore.3 With respect to the texts on physics, one has often fallen victim to the critical rigor that comes with retrospectively judging from the more advanced theses of the dynamics. Thus, Gue­ roult easily exposes the lacunae of the first mechanics in light of the remarkably well articulated relations between the metaphysics and mechanics of vis viva that one finds in the later phase of scientific completion.4 This recurring rubric is far too restrictive however, for it undoubtedly overlooks the characteristics of the research program as they are formulated in 1670–1671, when the abstract and concrete theories of motion come together in a remarkable way. Furthermore, Gueroult does not sense the profound affinity that the scientific project has with empiricist methodology, not only as it was conceived of by thinkers like Hobbes, but also as what Leibniz describes in a somewhat general manner so as to include the experimental research of the Royal So-

2 Cf. Specimen dynamicum (1695), GM VI 240–41, L 440 (modified): “When I was still a youth and followed Democritus, along with his disciples in this matter, Gassendi and Descartes, who held that the nature of a body consists in inert mass alone, I brought out a small book entitled A Physical Hypothesis, in which I expounded a theory of motion that both abstracted from the system and blended with the system (systemati concretam). This writing seems to have pleased many distinguished men far more than its mediocrity deserved. […] Later, however, after I had examined everything more thoroughly, I saw what the systematic explanation of things consists in and discovered that my earlier hypothesis about the definition of body was incomplete.” 3 Cf. Couturat (1969), 33–50; Kauppi (1960), 129–144; Ishiguro (1990), 44–60. 4 Gueroult (1967), 8–20.

16

The Initial Scientific Project

ciety’s virtuosi.5 One of the best general representations of Leibniz’s first philosophy remains undoubtedly Hannequin’s.6 However, taking specific interest in the rational analysis that Leibniz develops for a metaphysics of motion (motus) and thought (cogi­ tatio), Hannequin tends not to grasp the overlap between the theory of knowledge and the scientific methodology that one finds in the theses of this proto-Leibnizian metaphysics. Likewise, Kabitz’s7 work underestimates the specific interest that the Hypothesis physica nova presents, as well as Leibniz’s overall project for advancing the methodology of empirical science. For their part, the more recent studies of Konrad Moll attempt to systematize the complex influences that Weigel and Gassendi had on the establishment of the first philosophy,8 but Moll’s reconstructions are too unilateral and not sufficiently focused on the epistemological profile of the works that crown the youthful period. Given the current state of affairs, it seems that one may benefit from further investigating the elements of the philosophy of science that can be found in the treaties of 1671, as well as in the texts that prepare, prefigure and complete them. 1. Analyzing Reality in Terms of Motion In his study on “La première philosophie de Leibniz”,9 Hannequin traced the development of Leibniz’s views on the theory of matter back to his time at the University of Leipzig, when he rejected the substantial forms of the Scholastics in favor of a corpuscular theory of the Gassendi variety.10 He notes in particular how Leibniz increasingly criticized the architectonic notions of theories of this kind, and therefore sensed the need for a causal reason for motion that differs from reality as extension and shape, even though motion furnishes the key for generating physical properties of size, extension and shape. Thus, in 1669 in the Confessio naturæ contra atheistas, Leibniz succeeds

5 6 7 8 9 10

Loemker (1973), 248–275. Hannequin (1908), II, 17–226. Kabitz (1909). Moll (1978–1982). Hannequin (1908), II, 17–226. Cf. for instance, the letter to Remond on 10 January 1714, GP III 606; L 654–55: “I discovered Aristotle as a boy, and was not even discouraged by the Scholastics; even now I do not regret this. But then Plato too, and Plotinus, gave me some satisfaction, not to mention other ancient thinkers whom I later consulted. After freeing myself from the trivial schools, I fell upon the Moderns, and I recall walking in a grove on the outskirts of Leipzig called Rosental, at the age of fifteen, and deliberating whether to hold onto substantial forms or not. Mechanism finally prevailed and led me to apply myself to mathematics. It is true that I did not penetrate its depths until after some conversations with Mr. Huygens in Paris. But when I looked for the ultimate reasons for mechanism, and even for the laws of motions, I was greatly surprised to see that they could not be found in mathematics, but that I should have to return to metaphysics.”

Analyzing Reality in Terms of Motion

17

in justifying the foundations of physical properties via the activity of a mens, which itself is conceived of in terms of the analogical division between finite minds and the infinite, guiding Intelligence of the universe. From then on, and following the letters to Jacob Thomasius in 1669–71, in which he attempts to forcibly reconcile his mechanical physical theory with Aristotelianism, Leibniz develops a theory of conatus inspired by Hobbes, which will serve as the foundation of the Theoria motus abstracti (1671). The Hypothesis physica nova intercedes to secure the connection between the theory of conatus and auxiliary hypotheses subordinate to the metaphysics of the mens, which maintain the fragile harmony of the theoretical model. Daniel Garber, for his part, analyzes the same sequences in a somewhat different fashion.11 According to his interpretation, in the period between 1668 and 1670, Leibniz invokes notions of God and continuous creation to provide a substantial foundation for the motion of bodies, beyond their association with finite spirits. And during the period of 1670–1672, Leibniz advances, as the Theoria motus abstracti attests, a veritable “mentalization of bodies”, a thesis according to which every corporeal entity is likened with a mens momentanea.12 To be sure, these interpretations are coherent and instructive, but one must take stock of the epistemological elements that they ignore or underestimate. One of the most significant texts in this regard is the Nova methodus discendæ do­ cendæque jurisprudentiæ (1667). There one finds methodological assertions that are essential for elaborating upon natural science. Leibniz rejects here the Cartesian precept of evidence in favor of two criteria that seem to him capable of avoiding every possible deception that might arise when framing concepts and judgements: “Analysis, or the art of judgement, seems to me to be almost entirely accomplished by the following rules: 1) never accept a single word that is not explained; 2) never accept a single proposition if it is not proven.” (A VI 1 279) These two criteria govern analysis. Four disciplines provide the principles that permit the faculties of knowledge to be orientated toward the appropriate habitus mentis, once the propositional character of the mind’s operations has been accepted. Mnemonics depends on the signs that one must use to determine the relations of comparison or connection without which the working of the mind is impossible: Leibniz seems to borrow his notion of signs from Hobbes’ De corpore (1655).13 The goal of topos theory, or the art of inventing by associating topoi or transcendental relations such as totality, causality, matter and similarity, is to govern the combination of terms, represented by signs, from which propositions are formed. 11 Garber (1982), 160–184. 12 Cf. Theoria motus abstracti, § 17, A VI 2 266; letter to Oldenburg on 11 March 1671, A II 12 147: “[…] every body is an instantaneous mind (mentem momentaneam), and thus without consciousness, sense and memory.” Regarding this mentalization of the body, cf. Garber (1982), 171–72: “Leibniz […] has put motion, now conatus, into the bodies themselves. But since mere extended things cannot properly speaking have conatus or motion, and since the essential property of body is motion, the bodies must not be mere extended things: they must be minds of a sort.” 13 Cf. Hobbes, De Corpore, I, ii, §§ 4–5, OL I 14–15; EW I 16–17.

18

The Initial Scientific Project

Analysis is itself the art of judgement, and its goal is to arrange terms in such a way as to produce scientific propositions. Methodology, for its part, is the art of assembling propositions either in a natural way, that is, by order of demonstrative dependency, or in a sequential way, when one is unable to conceive of absolute demonstrative dependency and must content oneself with an order that seeks to grant verisimilitude. The logical classification of propositions should be conducted by regarding the terms in question as the material of relational propositions. This idea was determinative in De arte combinatoria (1666),14 but Leibniz makes use of it here to account for the architectonics of scientific understanding. Referring to Francis Bacon’s De augmentis scientiarum and Novum organum, Leibniz develops a three-level conception of science that would focus on “histories, observations and theorems.” (A VI 1 285)15 History is situated on the level of describing facts and is expressed through singular propositions such as: Francis Hall (Franciscus Linus) fixed, without any visible connection points, an iron marble at the center of a glass sphere supposedly filled with water – which he could have accomplished with a magnet and by pouring two different liquids, immiscible but indistinguishable to the eyes, into the sphere. History can include describing fictive facts such as when one is confronted with the assertion: A magnet holds Mohamed’s casket suspended in the sky above Mecca. Observations are formulated via contingent and universal propositions, and follow from an induction based on singular propositions, for example: Every magnet attracts iron. There remains the case of theorems, or maxims, which correspond to science per se. These are composed of necessary and universal propositions, for example: Everything that is moved, is moved by another thing, or If the magnet attracts iron, there must be bodily effluxes transferred from the magnet to the body. Following a Baconian sort of conception, Leibniz conceives of science in a general sense as a controlled combination of these three propositional tiers, which implies that the constructive materials, i. e., the terms, are located at the source of various nested propositional connections. More precisely, Leibniz outlines a classification of simple terms that make up complex terms, a classification that is ultimately dependent on the empirical order. Simple terms present themselves immediately to sensory perception, and comprise sensory qualities as such, which are mediately perceived via bodily organs, as well as powers that the mind perceives within itself. In 1667, Leibniz identifies the latter with thinking and causality – during his revisions in 1697–1700, he will speak of perceptivity and activity.16 Mediated by sensation, the being represented is understood as referring either to the object that is directly perceived or inferred from another perception. Its essence

14 15 16

Cf. for instance Couturat (1967), 35. Bacon’s three-level conception of scientific propositions is in particular analyzed in Pérez-Ramos (1988), 239–69. A VI 1 286, z.2–8 D: “By the mind, only two things are perceived: perceptivity (that is the power to perceive) and activity, that is the power to act.”

Analyzing Reality in Terms of Motion

19

is comprised of qualities that are instantiated together for sensory apprehension. Its existence corresponds to the possibility of being an object of sensation. Recognizing several beings together gives rise to relations of co-imagination or co-essence (sameness, difference, similarity, dissimilarity, opposite, genus, species, universal, singular) and those of co-sensibility or co-existence (whole, part, order, one, several, necessary, contingent, together, cause, etc.). Regarding the qualities arrived at through the mediation of bodily organs, Leibniz conceives of them as either specific or shared. But he distances himself considerably from the Aristotelian view by identifying shared qualities only with number and extension; the abstract representation of these qualities in accordance with their various modes yields the object of arithmetic on the one hand, and that of geometry on the other. And he adds: “Everything that possesses another sensory quality beyond extension and number is called a body. Everything that does not possess it is called vacuum. It is here that physics comes into being.” (A VI 1 287) Among these qualities, motion perceived through the medium of touch plays a fundamental role insofar as all other qualities can be analyzed into subtle motions once their extension is determined. In order to implement such an analysis, one requires facts of experience that are capable of furnishing the most adequate model within the framework of general extensive determinations: “Touch also has special qualities: solidity, fluidity, tenacity (resistance), softness, etc. whose history must be very diligently compiled like that of light, colors, sounds, smells, and flavors, such that one can more easily establish their cause by matter and motion.” (A VI 1 287–88) The revision notes of the 1697–1700 period suggest certain modifications regarding the typology of common qualities, and regarding the concepts on which theorization (the third level of the science) depends. Number belongs to internal sensation just as much as it belongs to various external sensations. The modes that characterize it give rise to an algorithmic theory that directly bears on how the contents of perception are combined when they are transposed into logical combinations; what is more, this logical transposition furnishes “metaphysical theorems” capable of shedding light on other fields of truths. Likewise, the category of causality or mental activity gives rise to a positive concept of conatus that, transposed within the study of compound bodies, yields a concept of motive force and thereby paves the way for establishing the mechanics. Arithmetic engenders geometry by adding situs to number: situs is the co-existent order that introduces quality within quantity. Thus, in these belated notes one finds the idea that the reflexive perception of the mind yields concepts essential to theorizing nature. The categories generated by external sensory experience would therefore need to be formulated in consequence of the powers of the mind and the concepts that reveal themselves to the apperception of the knowing subject. To be sure, the doctrine of 1667 does not surpass the limits of what is essentially derived from sensory qualities, even if the category of causality presupposes an analogous shift from reflexive to perceptive experience. Leibniz nonetheless distinguishes between abstract and concrete philosophy when it comes to these qualities. It seems that ab-

20

The Initial Scientific Project

stract philosophy proceeds via a priori constructions based on simple terms; concrete philosophy, by contrast, proceeds via the analysis of complex subjects, or phenomenal realities, in terms of primordial qualities and therefore via the combination of simple terms. “Here one is doing nothing more than historically [i. e., empirically] identifying the qualities of things; and one demonstrates nothing new, but simply subsumes what was previously demonstrated in abstract philosophy”. (A VI 1 288) In a way similar to Hobbes’, in 1667 Leibniz conceives of physics as the study of complex motions within an extended space that can be divided by number relations. At the same time, he senses that the empirical analysis of sensory qualities, and of the subtle motions that they represent, can, by means of abstract signs referring to simple mathematically expressed terms, provide an opportunity for subsuming everything under universal and necessary propositions. Already, this thesis deviates away from the Gassendian corpuscular hypothesis, and toward the geometric representation of observable motions and subtle motions – this being the only conceptually recommendable means of analyzing phenomena. It is particularly apparent that the solidity of bodies, i. e., their antitypy or impenetrability, temporarily ceases to appear as a positive quality since it ought to be reduced to subtle motions that suffice for producing such an effect. The notes of 1697–1700 will reestablish resistance, like extension, as being at the very least an essential category of the phenomenal universe. But only then will the concept of vis viva come to justify a typology of derivative active and passive forces. The model that Leibniz initially relied on to account for material essences was one that abstractly represented these essences in terms of subtle motions. This strategy goes hand in hand with the critique of atomism, as the fragment Con­ fessio naturæ contra atheistas (1668–1669) attests. (A VI 1 489–93) In keeping with the criteria of the Moderns, one relies on the primary qualities of size, shape and motion in order to explain phenomena. This involves determining if one can thereby account for phenomena without presupposing incorporeal causes. But the primary qualities of phenomenal realities cannot be derived from the definition of body as that which “exists in space”. Most of all, the reason for one shape or size cannot derive from this postulation of the nature of bodies; likewise, bodies are movable in virtue of their existence in space, but the reason for motion cannot, it seems, derive from the bodies themselves. From this critical approach, Hannequin concluded that Leibniz relies on what is undoubtedly a conceptual analysis that could be situated within the trajectory of a pronounced rationalism.17 Indeed, this may not be the case. The concepts in question are simple terms that represent shared sensory qualities in their immediacy. If the analysis works this way, it is because it does not seem to account for the sensible world beyond its phenomenal shapes and motions. The solution that Leibniz favors differs from a phenomenalization of the sensory universe analogous to what one finds in

17

Hannequin (1908), II, 34–39.

Analyzing Reality in Terms of Motion

21

Hobbes for example. Leibniz’s solution deviates theoretically from the latter owing to its reliance on the mens infinita as the appropriate sufficient reason for the foundation of the phenomenal universe. Integrating this theoretical sufficient reason into a geometry of concrete motions to justify its specific order and qualitative determinations constitutes the anti-Hobbesian project of Leibniz according to the Confessio. Nevertheless, we certainly do not wish to deny that the orientation itself of the analysis of phenomena conforms to an empiricist approach similar to Hobbes’ own in De corpore. The critique of the reductionist viewpoint falls within this approach. It relates to the problem of solidity and its three effects, resistance, cohesion and reflection, which bodily displacements and impacts reveal. Can one adequately account for this basic physical quality in terms of the shapes, magnitudes and motions of the implicated bodies? The strategy of Leibniz consists in pointing to the limits of the best hypothesis, that of Hobbes, for whom the resistance of bodies and the maintenance of their cohesion via an endogenous reaction to exogenous percussions owe to the compensatory effects of the conatus.18 This might apply to perpendicular impacts that cause imperceptible compensatory motions on the affected surface. But what of oblique impacts that cause reflections? In the span of an instant, the reacting motion dissipates without contributing either to the cohesion or the reverberation. One might then rely on more specific models, devised as simple modalities of the corpuscular hypothesis of the atomists. The elementary parts would produce solidity by various kinds of specific shapes, hooks, barbs, rings, etc. by which they would attach to one another. Leibniz asserts from the outset that the question thus concerns the cohesion of the parts inherent to these features themselves. The inquiry into the sufficient reason of solidity would thereby be inscribed within an infinite process of analysis. Modern philosophers such as Gassendi can only find refuge in atoms that are by definition indivisible. But then the explanation is founded on no other sufficient reason for the solidity of atoms than the free will of God. And, since the concept itself of matter offers no other sufficient reason for the shape, magnitude and motion of any part whatsoever, one must postulate that this infinite incorporeal cause, the origin of the firmitas, also imposes a design that is necessary for determining the modes or properties of natural realities; and, since these determinations form a combined whole, this involves a comprehensive and complete structural design. Such is the inspiration behind this metaphysical argument: It is impossible to understand the reason why this incorporeal being chooses this size, shape, or motion rather than another, unless it is intelligent and wise in virtue of the beauty of things, and powerful in virtue of their obedience to its will. Such a being would therefore be the Mind governing the entire universe, that is, God. (A VI 1 492)

18 Hobbes, De corpore, III, xv, § 2, and xxii, § 18–19, OL I 178–79. 283–84; EW I 211, 347–48.

22

The Initial Scientific Project

This is clearly a departure from the path embarked upon by Hobbes, who sought to explain solidity in physical terms. Even if this way of theorizing in terms of motion obliges us to suppose that such and such motion extrinsically acts upon the material parts, and thus prevents us from accessing the ultimate physical causes of things, Leibniz seems to venture in this direction in his quest for a sufficient reason for basic phenomena. It is in this context that one finds the suggestions for amending Hobbes’ system and for harmonizing the method of the new physics with a metaphysics of nature that responds to a so-called Aristotelian framework. Such are the goals that one can infer from the letter to Hobbes on 13/22 July 1670, (A II 12 90–94) and from the letters to Jacob Thomasius on 26 September/6 October 1668 and 20/30 April 1669. (A II 12 17–19, 23–38) In the first place, Leibniz is interested in the “general principles” or “abstract reasons” of motion that Hobbes had developed in De corpore. Thus, the Hobbesian version of the principle of inertia receives the stamp of approval: no body begins to move itself if it is not moved by another adjacent body that is itself in motion; and, once moving, the body continues to move so long as nothing hinders it. Leibniz also expresses agreement with a principle that apparently contradicts the facts of sensory experience, one to which we will have to return when examining the paradoxes of the Theoria motus abstracti: a body at rest, no matter what size it is, can be displaced by the motion of another body, no matter how small this body is. Against the protestations of experience, Leibniz interprets, following Hobbes, the visible rest of a body as imperceptible motion. Already the principle of continuity is being appealed to in order to understand the correlation between the contrasting phenomenal appearances of motion and rest. However, more directly, Leibniz intends to develop a kinetic model to account for solidity. He thus questions Hobbes’ interpretation, which reduced the cause of solidity to the body’s reaction to impacts alone. Does visible solidity not endure in the absence of an impact presently affecting the body? And does it not endure despite the fact that, if one were dealing with a sui generis centrifugal motion, the presently unconstrained parts of the body should dissipate? Conversely, when constrained, the interference of an obstacle could only dissipate the reacting motive action of the parts in an instant. And there are other insufficiencies in Hobbes’ position. According to this doctrine, the motive disposition from the center to the periphery that generates resistance to the impact would result from the concentration of a cause of motion in some given point; this remains to be explained. Moreover, the proportionality of the reaction to the action must render the solidity of bodies relative to eventual impacts; this does not seem to agree with experience, and the rational argument would instead force one to postulate that the more powerful the initial action is, the smaller the reaction must be. In place of the paradoxical theory of Hobbes, which proves insufficient, Leibniz furnishes the elements of a physical theory of motion and its cohesive effects:

Analyzing Reality in Terms of Motion

23

I thought that, to produce the cohesion of bodies, the conatus of the parts orientated toward one another, that is, the motion by which they press upon one another, would have sufficed. For bodies exerting pressure on one another are in a conatus penetrating one another. The conatus is the starting point; the inter-penetration is the union. They therefore are at the beginning of a union. But when bodies begin to unite, their beginnings or limits become one with one another. Bodies whose limits reduce to a single one, or τὰ ἔσχατα ἔν […], are not according to Aristotle’s definition merely contiguous, but also continuous, and in truth one body whose movement is reducible to a single motion. If there is any truth to these reflections, you will appreciate that they cast the theory of motion in an original light. That bodies pressing upon one another are in a conatus penetrating one another remains to be proven. To exert pressure is to strive to invade a place still occupied by another body. Conatus is the first phase of motion, and therefore the first phase of existence in a place toward which the body is striving. To exist in the place where another body exists is to have penetrated it. Pressure therefore is the conatus of penetration. (A II 12 93–93; L 107)

Clearly, the notion of conatus introduced here refers to the motion that affects a part of a body in the initial state: it is an imperceptible incepting motion, whose effect is an imperceptible or so to speak virtual displacement of the material part subjected to the impact. In virtue of the Hobbesian principle of inertia, this motion must be produced via an effective displacement, and thus via an observable principle; but an instantaneous obstacle is produced by the conatus opposed to the part subjected to the impact, which creates an antagonistic effect. The conatus is endowed with a causal capacity for “penetrating” the extension of the part opposing it. To admit this possibility, one must suppose that there is no specific resistance to penetration that is extrinsic to the conatus, that is, to the incepting motion. Hence, the phenomenon of cohesion is represented in a purely kinetic way: the effect of extensive continuity is the result of conatus being united via their extensive elements, the physical points that generate the solid continuum. It obviously follows from this “phenomenist” framework that the productive cause itself, lurking in the background of the conatus as incepting motion, escapes sensory apprehension, if not also imaginative representation. Leibniz therefore ultimately questions the postulate of Hobbes, Omnis motor est corpus,19 which does not seem justified to him in this context, and in light of the kinetic and purely phenomenal theorization of experience. If he praises Hobbes for having interpreted sensio as a permanent reaction, then the permanence of such an activity could never be founded as such in the physical world. This would involve developing a theory of mind (mens) that accounts for what is substantial in the background of corresponding physical activity.

19

Cf. in particular Hobbes, De corpore, II, ix, § 9 and x, § 0, OL I 111–12, 116–17, EW I 126, 131.

24

The Initial Scientific Project

At the same time, in 1670, Leibniz distances himself somewhat from the Hobbesian philosophy of knowledge, a shift that will have a major impact on his theory of science. The Preface to Nizolius (1670) (A VI 2 398–444) suggests a very empiricist conception of the truth of propositions, since the analysis of statements via the understanding can only determine their clarity, and thus the combination of more or less simple terms that comprise them. Truth is the business of sensory apprehension, or the meaning of terms referring to sensory perception: “An utterance whose meaning is perceived by a percipient with the right disposition in the right medium is true, because clarity is measured by understanding, and truth by sense.” (A VI 2 408–409; L 121) In addition, Leibniz adheres to a nominalist conception of the explanatory principles. He interprets Occam’s maxim, Entia non esse multiplicanda præter necessitatem, as a parameter of the economy of presuppositions when it comes to determining the causes of phenomena. The best hypothesis is the simplest one, which is demonstrated by the prevalence in astronomy of hypotheses that only invoke simple motions. For this reason, Leibniz maintains, nominalists would have refused to rely on universals and realized abstract forms for their explanations. This amounts therefore to a rejection of the Aristotelian metaphysics of nature, which Leibniz endorses. He discerns in this interpretation of nominalism the foundation of the philosophy of the Moderns. Nonetheless, he criticizes the “ultra-nominalism” of Hobbes. Unsatisfied with reducing the universals to simple terms, the latter assumes that the definition itself of these terms refers to purely arbitrary groupings of names.20 Hobbes thus disputes that the truth of propositions can depend on an objective order of phenomena represented by abstract terms. In opposition to this position, Leibniz asserts that changes to algorithmic notation in no way influence mathematical truths, and that one can in fact express them otherwise, so long as there is no disparity in the meaning of the truths that have been transcribed. When one is moving from the world of logico-mathematical propositions to that of propositions referring to sensory phenomena, the problem of identifying the foundations of induction presents itself. How can moral certainty be ascribed to the truth of a conclusion whose scope is presumed to be universal, when the implications of the propositions are limited to an aggregation of a finite number of particular cases? Leibniz’s solution consists in postulating principles or rules of inference. These correspond to universal ideas or definitions of terms that describe how empirical facts are organized in order to formulate hypothetically necessary propositions. These definitions relate to causality and the sensory references of theoretical concepts. Thus, Leibniz formulates three rules of inference: (1) If the cause is identical or similar in every way, then the effect is identical or similar in every way. (2) The existence of a thing that is not perceived cannot be presumed. Finally, (3) everything that is not assumed must, in practice, be regarded as invalid before it is proven. (A VI 2 431) 20

Ibid., I, vi, § 14, OL I, 73; EW I, 83–84.

Analyzing Reality in Terms of Motion

25

At this stage of development in his philosophy, Leibniz had clearly thought that scientific propositions can never be perfectly certain. Indeed, this would require that one operate strictly within the definitions of the terms by abstracting from every conceptual element referring to the experience of phenomena. But the norm for accepting explanatory concepts is precisely sensory experience. This goes hand in hand however with recognizing the rules of inference that guarantee demonstrations via prolepses, that is, anticipatory explanatory reasons. Even if in principle they ought to have been derived a priori by logically articulating propositions in simple terms, the justification for these prolepses depends a posteriori on the success of the explanatory schema. For these to translate the apprehension of phenomena via sensio in an adequate way, one must in effect rely on what Leibniz designates as auxiliary propositions founded on universal reason, which themselves could never be constructed inductively. With thinking and activity, or causality, having initially proven to be qualities or powers of the mind, everything suggests that in 1670 Leibniz is ready to connect the proleptic rules of induction to the reflexive experience of the mens, even if elsewhere he remains faithful to his empiricist conception of the truth. The letters to Jacob Thomasius on 26 September/6 October 1668, and 20/30 April 1669, indirectly confirm the elements that will be relied upon to establish the new physics, which will take a nominalist approach to principles, like the one we just described. The aim of these letters is to reconstruct the notions of Aristotelian physics in such a way as to make them compatible with explaining bodies in terms of size, shape and motion, and in conformity with the approach of the Moderns. Herein lies the goal of the philosophia reformata: reconciling Aristotle with the mechanist natural philosophy of the Moderns. An important doctrinal movement in the 17th century, the philosophia reformata, included such major figures as Kenelm Digby, Thomas White, Jean-Baptiste Du Hamel and Jan De Raey in particular, whose Clavis philosophiæ naturalis seu introductio ad naturæ contemplationem aristotelico-cartesianam (1664) was particularly influential, as well as Erhard Weigel, whose teachings Leibniz had followed at the University of Jena.21 Leibniz counts himself among the proponents of this reformed philosophy, but at the forefront so to speak, insofar as he truly imagines reinterpreting Aristotle’s physics by applying the mechanical categories of the Moderns. In the Pre­ face to Nizolius, he summarizes this reform program as follows: It seems that I have sufficiently reconciled Aristotelian philosophy with the reformed one. It is true that I must briefly consider the evident truth of the reformed philosophy. The following must be proven: that nothing exists in the world, but mind, space, matter and motion. […] Now, it must be shown that nothing else is needed to explain the phenomena of the world and account for their possible causes, and above all, that there cannot be anything else. Besides, if we show that we need nothing else but mind, space, matter and 21

Cf. Mercer (1990); Bodéüs (1991).

26

The Initial Scientific Project

motion, it will follow ipso facto that the hypotheses of the Moderns, which rely only on these realities to account for phenomena, are better. (A VI 2 441).

By representing primary matter as mass (i. e., as that which is extended and resistant), form as shape, and efficient cause as motion, there seems to be a forced compromise that works in favor of the Moderns, to the extent that reducing the problem to the clearest and simplest hypothesis eliminates substantial forms as incorporeal entities. In a continuous mass (massa) of indeterminate quantity, motion introduces discontinuity and delimitates parts, from whence come shapes and thus, by analogy, the distinct forms in which all the various phenomena are expressed. As such, all sensory qualities would be reducible to subtle motions whose degrees of variation depended on the sites of application. All change is in effect reducible to the motion affecting the material parts. This holds true for generation, corruption and every alteration, and is demonstrated by the observations of micrography specialists such as Hooke, who studied for instance decomposition and the formation of rust. But, more fundamentally, the elementary properties of physics find themselves linked to the modalities of motion: “And it follows from this that the colors are generated solely by altering shapes and the layout of their surface; and one can easily furnish the same explanation for light, heat and all qualities given the opportunity.” (Letter to J. Thomasius, 20/30 April 1669, A II 12 28) If the qualities are generated and modified by motion, such might also be the case with substance itself, since the essence of the latter differs from that of the former only by virtue of the different relationship that the two have with sensory perception. To this end, Leibniz mobilizes the metaphor of points of perspective for the first time: Just as the city looks different if one observes it from a tower at its center (in Grund gelegt), so does an object if you intuit its essence. Just as the city appears different if one approaches it from the outside, so does a body perceived by its qualities. And just as the external appearance of the city depends on the variety of angles of approach, west and east, so do the qualities of a body depend on the variety of our sensory organs. From this it is clearly evident that all change can be explained by motion. (A II 12 28–29; L 97)

The thesis presented here implies a fundamental epistemological assertion: the variations in the expression of the essence of a phenomenal substance owe to the physical motions relating to the points of perspective from which these sensible qualities are grasped. At the same time, this essence is determined by the sum of specific motions gleaned from different points of perspective, at least with regard to the analysis of phenomena. The perspective that we already identified in the letter to Hobbes is therefore anticipated: phenomenal realities consist precisely of coordinated motions, interacting in unison in relation to the impacts of the surrounding physical points. In the letter to Thomasius in 1669, Leibniz’s position nonetheless remained more relativist, insofar as the combination of extension and resistance to penetration seemed to be an essential and irreducible given of sensory apprehension. Already, Leib-

Analyzing Reality in Terms of Motion

27

niz conceives of extension as belonging to the world of appearances (phantasma); but antitypy verified by touch confirms the phenomenal reality of extended, visually identifiable bodies. Moreover, it is interesting to note that Leibniz introduces, with respect to this basic notion of body, an epistemological thesis concerning the derivation of properties, via analysis or definition, from the propositional contents of the terms “extension” and “antitypy”: Since nothing exists without a cause, nothing should be presumed to exist in bodies if its cause cannot be explained by its basic components. And the cause cannot be explained unless these principles are well defined. Nothing therefore should be placed in bodies that does not follow from the definition of extension and antitypy. Only from these does one derive magnitude, shape, situation, number, mobility. (A II 12 36; L 101–102)

With these properties, implied in the basic notion of body, one can go about constructing models of specific phenomena. Furthermore, this rules out comprising the materials of these models from something other than those properties. Nonetheless, the causality of motion itself goes beyond the limits of such an analysis. To this end, Leibniz implements a concise thesis according to which the origin of motion derives from incorporeal realities. Following the passage cited above, he specifies that: “motion is not derived from them [the properties of bodies]. Hence there is no motion as a real entity in bodies.” (A II 12 36; L 102) This is why the strategic terms of the system of nature are mens, spatium, materia, and motus. Shape, with which a connection to Aris­ totelean form is drawn in a rather arbitrary fashion, can only be a principle of motion in a secondary sense. A sphere placed on a plane does not owe to its sphereness the motion that moves it; but its displacement is determined by its specific shape. Despite the analogy of form to shape, the first principle of motion can be no other than a form that is really distinct from matter and strictly efficient: this can only be a mens, the sole substrate of spontaneity. For Leibniz, the relation between the basic terms defines the explanatory framework of natural phenomena. But one must still conceive of this theoretical framework without relying on any redundant hypothesis whatsoever – as per Occam’s principle. This is precisely the reason for which beings of reason from scholastic physics find themselves excluded: The human mind can in fact imagine nothing other than mind (when it thinks of itself), space, matter, motion, and the things which result from the relations of these terms to one other. Whatever else one adds to these can only be names, uttered and variously combined, that one cannot explain or understand. Who can imagine a being that partakes neither of extension nor of thought? (A II 12 35; L 100)

Thus, Leibniz specifically excludes substantial forms, which extend the analogy of mental acts to the world of material phenomena. If phenomenal nature represents a teleological order, then it does so by referring to a transcendent design that is irreducible to the blind appetites and instincts of the body; these psychological properties

28

The Initial Scientific Project

attributed to physical realities would lack justification in the economy of basic terms generated from sensory experience, both internal and external. Whatever appears teleo­logical owes to the hypothesis of there being overlap between sequential motions and rational causes, which depend on the infinite mind and are reflected by the activity of finite minds. As Leibniz will write to Jacob Thomasius on 19/29 December 1670: “One is in need of natural philosophers who do not content themselves with deriving a geometry from physics (geometry in effect lacks a final cause), but who uncover a certain civil science within natural science.” (A II 12 119) But these statements are above all programmatic and aim to introduce the type of model that the Hypothesis physica nova will represent in its dependence on the Theoria motus abstracti. In the methodological directives of the texts preceding the Hypothesis, we seem unable to detect any significant attempt to distance oneself from the geometrization of phenomena in the name of some teleology. The dominant idea is that of a “horological” framework that allows one to dispense with any reliance on psychological pseudo-entities that function like substantial forms. Because in truth there is neither wisdom nor appetite in nature, and yet a beautiful order is created through God’s timepiece, it follows that the hypotheses of the reformed philosophy are superior to those of the Scholastics, in that they are not superfluous but rather, to the contrary, clear. (A II 12 35; L 101)

Under these conditions, applying Occam’s principle to hypotheses requires us to reduce theoretical concepts to simple terms that serve as the basis of geometricizing phenomena: everything in nature must be represented adequately by combining sizes, shapes and motions. This mechanism, as we have seen, presupposes more and more that motion determines the entirety of geometrizable effects. What this conception of the mechanist framework fails to provide is an adequate representation of the natural causality responsible for the arrangements of subtle motions in which the essence of phenomenal realities could be analyzed. This causality is explained with reference to the experience of internal sense, and thus by postulating mens as its substantive ground. The teleological order concerns the layout of these subjects of causality, whereas phenomena transcribe this design into mechanical terms. As a result, there is noticeable heterogeneity between the causal order and that of the physical effects that tend to be represented mechanically, that is, in an essentially geometrical way.

The Theoria motus abstracti: “Conatus” as Motion Indivisible

29

2. The Theoria motus abstracti: “Conatus” as Motion Indivisible In Principia philosophiæ,22 Descartes formulated laws of rectilinear impact. These “abstract” rules were presented as having been deduced from the principles of inertia and the conservation of the quantity of motion. Apart from the first rule, which expressed the law of reflection for bodies of the same size colliding at the same speed, these rules proved contrary to experience. Empirical objections would not however suffice for dethroning them, since they were put forth as the realization of a theoretical model that conformed with the rational concepts of moving matter. Since 1656, Christiaan Huygens had been drafting a treatise, De motu corporum ex percussione (which would remain unpublished until 170323) wherein he established new rules of motion conforming to experience by mapping the vectors of relative speeds in the chosen system of reference.24 In 1668, the Royal Society of London launched a competition for discovering the laws of motion in accordance with the empirical conditions of bodily impact. All at once, Wallis, Wren and Huygens offered their respective formulations of these laws. Leibniz discovered the solutions of Wren and Wallis in the Philosophical Transac­ tions of January 1669. With regard to that of Huygens, which had already appeared in the Journal des sçavans on 18 March 1669, Leibniz took note of them in the Philosophical Transactions of that April. In De rationibus motus, the successive drafts (1669–70) of which prepared the way for the publication of Theoria motus abstracti (1671), Leibniz starts off by transcribing Huygens’ rules and reproducing the table of the various cases of impact. (De rationibus motus, A VI 2 157–59) The formulation of the rules is dictated by the principle of inertia, by the postulate that two hard (i. e., elastic) bodies of equal size and speed directly colliding with one another rebound with the same speed they initially had, and by the principle of relativity illustrated in the famous model of the ship. Leibniz notes the principal propositions, including those that Huygens provided without demonstration, which prefigure the essential propositions of the Leibnizian dynamics to come: the conservation of the total quantity of progress, measured in terms of mv where v is treated as a vector, and the conservation of forces measured in terms of mv2. The general rule of calculus for displacement is described as follows: The general rule for determining motion is the following. Suppose that there are two bodies A and B. A moves with speed AD; B collides with it, and either it moves in the same direction with speed BD, or comes to rest, that is to say that point [D] falls at B. The line AB having been divided at C (center of gravity for the bodies AB), it is assumed that CE

22 Descartes, Principia philosophiæ, II, § 45–52, AT VIII-1, 67–70. 23 Huygens, Œuvres complètes, XVI, 30–91. 24 On the development of Huygens’ thoughts on this matter, cf. Chareix (2006).

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The Initial Scientific Project

equals CD. I say that EA represents the speed of body A after the encounter, and EB that of body B, each of them going in the direction that the order of the points demonstrates EA, EB. If E falls at points A or B, body A or B is brought to rest. (A VI 2 158)

The table of cases simply serves to illustrate this general rule.

Fig. 1 Table of cases of impact between two bodies, represented following Huygens’ general rule (Christiaan Huygens, Œuvres complètes, XVI, 180).

However, from the outset, Leibniz will disagree with this algorithm, which is limited to transcribing kinetic data. The first proposition of De rationibus motus makes this attitude apparent: Motion can be grasped in two ways: by reason and by sensory perception (sensu), and sensory perception cannot prejudice reason, but reason can prejudice sense perception. Thus, when sense perception appears to contradict reason, one must conclude that something intervened that is not perceptible, except by its sensory effect (ἐναντιοφαινείᾳ). (A VI 2 159)

In fact, Leibniz will adopt a system of a priori concepts capable of furnishing a purely geometric representation of impact. The structure of the theory must be deductive and based on definitions of the sort that Hobbes elaborated in De corpore, which would have the advantage of giving rise to a sort of calculus. The Leibnizian project can only appear paradoxical if we consider how thinkers of the time negatively judged Cartesian models, which in effect fail to prove to be entirely in conformity with the facts of experience. However, at the same time, one must recognize that the Euclidian model of more geometrico demonstration continues to linger in their minds, including Huygens’, who no doubt refused to publish his mechanical algorithm because he felt that, in contrast to the canonical framework for geometric demonstrations, it was filled with lacunae. In the Theoria motus abstracti (1671), Leibniz’s project is to develop a phoronomia ele­ mentalis that, like geometry, will be capable of expanding upon the laws of mechanics

The Theoria motus abstracti: “Conatus” as Motion Indivisible

31

in a rational and abstract way. In particular, this phoronomia must formulate the laws of impact in as general a way as possible, without relying on physical phenomena such as those that experience reveals. The rules of impact obtained by building on the facts of experience do not meet the standard of a model of demonstrative knowledge. For the latter necessitates that arguments be deductive and founded on geometric axioms and appropriate definitions: such definitions must not therefore imply any internal contradiction in the conception of the corresponding object.25 This approach as a whole can be inferred from the section “Usus” of the Theoria, where Leibniz conceives of the originality of his project within the context of contemporary works on mechanics. (A VI 2 273–76) What is interesting to us about Leibniz’s position is that it builds upon a certain conception of the relation between the explanation of phenomena and a priori reasons, and that this conception produces significant aporia. In the Galilean phoronomia experimentalis, reasons are, according to Leibniz, formulated on the basis of concepts signifying empirical characteristics: hence the unresolvable indeterminacies regarding the nature of the content itself, the reason behind circular motion, the principle of variation of motions in bodily interactions and impacts, and the cause of the cohesion of bodies. It is precisely with the goal of removing such difficulties that he sets out to derive from geometry the means of establishing unambiguous principles that would allow for systematizing mechanics. But two aspects of the phoronomia elementalis indicate the non-nominalist character of Leibniz’s conception of the abstract geometry of motion. On the one hand, that which will be justified by geometric reasons in the theory of abstract motion will apply ipso facto to physical reality, since the essence of physical objects only corresponds to the possibility of geometrically generating them, even if this possibility does not suffice for producing them without a sufficient reason of the order that they realize in concreto. One can in effect conceive of bodies as being generated solely by motion dividing an extension into shapes. This division corresponds to the delimitation of the cohesive parts of an extension, and the cohesion itself is the consequence of the reciprocal motions of the integrative parts, provided that analysis be pursued to the level of their “indivisible” constituents. The relation between geometry (as found in the phoronomia elementalis), physics and mechanics is adequately illustrated in the Problema generale:

25

Cf. letter to Oldenburg, 13/23 July, 1670 A II 12 95: “I have presently established some Elements of the true reasons of Motion, which are demonstrated from definitions of terms alone, by the geometrical method. In these, it seems that I have unveiled the cause of the connection, flexibility and hardness of bodies, which no one had explained in this way. The rules that the incomparable Huygens and Wren have set about motion are not primary, absolute, nor clear. They occur by accident, because of a certain state of this globe of earth, water and air (as is the case with gravity); they are not axioms nor demonstrable theorems, but experiences, phenomena, observations, however successful and remarkable (beyond all that anyone had accomplished of the sort).”

32

The Initial Scientific Project

To physically model all possible lines, figures, bodies and motions in terms of lines, simple rectilinear and equal motions, and likewise pure curvilinear motions of every type, no matter the body. This model is threefold: geometrical, that is imaginary, but exact; mechanical, that is real, but not exact; and physical, that is real and exact. The geometrical model contains the modes by which bodies can be formed, although often by God alone, provided it is understood that they do not imply any contradiction, such as forming a circle by bending a straight line at its least points (per minima); the mechanical model contains our modes; and the physical one those by which nature produces things, that is to say, by which bodies produce themselves. (A VI 2 270)

At the same time, the principle of deductively connecting the theory of abstract motion with the “exact” conception of physical reality is justified by not only relying on Cavalieri’s method of indivisibles, but also on the realist interpretation of indivisibles of space and motion. In 1670, Leibniz set out to perfect the model of indivisibles that Cavalieri had developed in his Geometria indivisibilibus continuorum (1635), then in his Exercitationes geometricæ sex (1647).26 The method for generating solids promoted by Cavalieri was founded on the principle that both cylinders and cones have the same proportions as the elements that generate them; these generating elements are determined by sections of the volume along the perpendicular axis, and visualized as equidistant planes, which are infinitely drawn together. By infinitely summing these elementary surfaces (indivisibles), one could produce a solid shape; or at least one could determine the relations of proportion between, on the one hand, the generating elements examined in their finite dimension and, on the other, the constitution of the finite shapes that result from infinitely summing the generating elements. One could forgo this infinite summation by geometrically intuiting the structures implied at the level of the generating elements and at that of the resulting shapes.27 But, as Cavalieri relies on this intuition to justify his system composed of an infinite number of indivisible elements, he cannot eliminate every analogy with real composition from the way in which he represents the artifice of summation. Hence the misguided metaphor of the Exercitationes, which Belaval cites: “It is evident that two-dimensional shapes must be conceived of as a web of parallel strings, and solids as books composed of parallel pages.”28 In short, if indivisibles appear to be useful fictions, then Cavalieri does not ultimately succeed in eliminating a presumption of reality in favor of the operational atoms that are indivisibles. The latter serve to artificially produce the extended contents of a superior order: lines,

26 27 28

Cf. (Giusti) 1980. Brunschvicg (1972), 164–65 notes, with reference to Cavalieri’s Geometria, that the method of comparison allowed him to forgo in his calculations any consideration of indivisibles, which had been used to formulate problems. Belaval (1960), 314, n.2.

The Theoria motus abstracti: “Conatus” as Motion Indivisible

33

surfaces or solids. In the end, Cavalieri never developed a notion of the infinitesimal as the fictive limit of a real quantity.29 Leibniz might have discovered this method of indiscernibles through his reading of Hobbes; at least he claims to have demonstrably taken advantage of it in the Theoria motus abstracti, declaring for instance: By it, it seems that I have established, and even demonstrated, that the parts of indivisibles lack distance, and thus, as one conatus is greater than another conatus, so is one point greater than another point; and this is the reason why I avoided the whole labyrinth of continuous composition, and even saved Cavalieri’s geometry. (Letter to Velthuysen, 5 May 1671, A II 12 163–64)30

This point is important. Leibniz maintains that he upgraded Cavalieri’s doctrine from a simple hypothesis based on a probable estimation/calculation to a demonstrable doctrine. For it would now be possible to deduce from it conclusions relating to conatus.31 In short, the definition that Leibniz ascribes to the indivisible of space would grant a sort of axiomatic status to the theory that he derives from it. This definition is precisely the one that Leibniz presents to Arnauld as one of his scientific achievements: In geometry, I have demonstrated certain fundamental propositions on which depends a geometry of indivisibles, that is, a source of discoveries and demonstration. These include the fact that any point is a space less than any given space; that a point has parts, though these are dense (indistantes); that Euclid was not wrong in speaking of parts of extension; that there are no indivisibles, yet there are non-extended things; that one point is greater than another point, but in a relation less than any which can be expressed, or incomparable to any sensible difference; that an angle is the quantity of a point. (A II 12 278; L 149)

Leibniz misinterprets Cavalieri’s indivisibles, which are ultimately solved via a trick that allows one to compound finite relations into an indefinite summation; he aims to transform them into infinitely small, actual elements that would permit him to reconstruct what is extended by summing the non-extended elements.32 However, beyond the insurmountable paradoxes of compounding the continuums that remain, Leibniz envisions understanding dynamical extension in terms of motions conceived of

29 30 31 32



Cf. Ibid., 315–16. Cf. a similar statement in a letter to Carcavi on 22 June 1671, A II 12 209. Cf. letter to Carcavi on 17 August 1671, A II 12 236. One can easily rewrite Cavalieri’s integration formula in the style of infinitesimal calculus. Proposition 24 of Book II of the Geometria focuses on a parallelogram divided into triangles by a diagonal line; these triangles are created by summing the parallel lines representing the bases of the two tria angles. Cavalieri’s equation can be formulated in the following way: ∫ x 2 dx = 1a 3 . This formula 3 0 n+1 a n can be generalized as: ∫ x dx = a . n +1 0 Cf. Brunschvicg (1972), 162–63; Belaval (1960), 317–18; Boyer (1985), 362–63.

34

The Initial Scientific Project

as points with potential speeds. The conatus theory built around pseudo-Cavalierian indivisibles probably constitutes the inventive dimension of the Leibnizian approach. This invention largely owes to what is borrowed from Hobbes, or rather to reinterpreted Hobbesian theses. In De corpore (1655), Hobbes proposed in effect a certain number of theses that Leibniz takes advantage of. Hobbes’s ambition was to describe phenomena solely in terms of a concept of moving matter. He derived the elements of his natural philosophy from an applied geometry of the axiomatic-deductive sort. Furthermore, in order to explain the sensory qualities of bodies, Hobbes set out to build speculative ad hoc models. As Howard Bernstein points out,33 one can in a sense interpret the Leibnizian distinction between “abstract theory” and “concrete theory of motion” as a reflection of the two sides of De corpore. The starting point for abstractly assembling and reconstructing the elements of nature is the notion of space as an infinitely divisible continuum;34 the displacement of bodies in this space gives one the notion that time represents the before and after in motion.35 Bodies are conceived of as existing in this space independently of the subject that conceives of them, while the accidents or properties of these bodies are reducible to the various ways in which one conceives of them. Extensive size and shape constitute actual accidents, as opposed to other types of accidents, of which one can ask: to what extent can these be reduced to motions that are inherent to or associated with bodies? Motion is change of place continuous across time,36 and appears as a continuous process even in an instant, wherein it appears radically different from rest, which is at most a fiction. Motion, insofar as it generates a certain duration across time, constitutes speed (velocitas).37 Force, as Hobbes defines it, corresponds to quantity of motion conceived of as m | v |: Motions are said to be simply equal to one another, when the swiftness of one, computed in every part of its magnitude, is equal to the swiftness of the other computed also in every part of its magnitude: and greater than one another, when the swiftness of one computed as above, is greater than the swiftness of the other so computed; and less, when less.38

The mechanist vision of moving matter is completed by formulating the principle of inertia.39 Furthermore, the model of causality that Hobbes adopts is significant. The full cause of an effect is comprised of a host of requisites, that is to say, accidents (i. e., the inherent and associated conditions) of the subject as agent and as patient, from

33 34 35 36 37 38 39

Bernstein (1980). De corpore, II, vii, § 13, OL I 89; EW I 100–101. Ibid., II, vii, § 3, OL I 83–84; EW I 94–95. Ibid., II, viii, § 10, OL I 97; EW I 109. Ibid., II, viii, § 15, OL I 100; EW I 113. Ibid., II viii, § 18, OL I 102: EW I, 115. Ibid., II, viii, § 19–20 and xv, § 1, OL I 102–103, 177; EW I 115–16, 205–206.

The Theoria motus abstracti: “Conatus” as Motion Indivisible

35

which the effect necessarily follows. An entire cause is therefore always sufficient.40 The requisites of the agent make up the efficient cause, while those of the patient the material cause. In principle, Hobbes reduces the formal cause to the efficient cause. This involves connecting the requisites of the agent to those of the patient via an intermediary, the intellection of ideas as abstract signs.41 To know the essence of a thing allows one to know that thing; the “form” is thus an efficient cause of the knowledge of the effect. The same analysis can be made of final causation, if one is dealing with bodily realities endowed with sensibility and volition. In short, the whole causal structure of natural realities boils down to the requisites of the full cause: when the latter is given, the effect is produced in the same instant. That does not mean that the process must be strictly instantaneous, that is, absolutely without any duration whatsoever. To the contrary, the process by which the effect is produced forms a “certain continuous progress”,42 even if this occurs in an instant. It is here that one finds the first, intuitive primer for the notion of conatus. At the same time, the efficient cause is exclusively equated with action when there is direct contact, the agent being contiguous and in motion relative to the patient. And, in an anti-Cartesian analysis, which Leibniz will notice, Hobbes rejects the opposition of motion to rest, since they are only contrary to one another as abstract terms; according to the rules of causation, real opposition can only consist in contrary motions.43 Nonetheless, nothing indicates how one can reconcile this conception with the notion of force via the model m | v | , wherein mass appears as a factor accounting for resistance to contrary motion. When Hobbes returns to his doctrine of force in De corpore, III, xv, § 1, he modifies his definition by seemingly ignoring the factor of mass. But he immediately reestablishes a relation between quantitas motionis and the size of bodies endowed with the same speed. In fact, Hobbes tends to associate the dynamic element with the velocity of the minimum part of matter, and to approach the summation of these parts by using artificial integrals as Cavalieri had done. This thesis is confirmed when Hobbes declares: Change is therefore motion, whether of the agent’s or patient’s parts. This is what needed to be demonstrated. From this it follows that rest cannot be the cause of anything, [and that no action can proceed from it], since nothing moves nor changes because of it.44

The Leibnizian theory of conatus will also ignore the resistance that mass provides.

40 41 42 43 44

Ibid., II, ix, § 5, OL I 108; EW I 122. Ibid., II, x, § 7, OL I 117; EW I 131–32. Ibid., II, ix, § 6, OL I 109–110; EW I 123–124. Ibid., II, ix, § 7, OL I 110–11; EW I 124–125. Ibid., II, ix, § 9, OL I 112; EW I 126.

36

The Initial Scientific Project

The analysis of causality is reproduced in that of the relation of power to action.45 Active power is reducible to all the requirement that agent needs to satisfy to determine a future effect. As the efficient cause is characterized by the presence of a contiguous body in motion acting on another body, active force is a form of instantaneous motile inclination. To active power one opposes passive power, which describes the action that will be produced by the cause’s motion when it ceases to merely be an inclination. Understood as such, passing from power to action implies a relation of necessity. Contingency then merely represents an apparent lack of determinative reasons, born of our own ignorance of the way motions are inter-connected. Chapter III, xv of De corpore contains the heart of the theory of conatus, for which it proposes the following definition: I define endeavour (conatus) as motion made in less space and time than can be given; that is, less than can be determined or assigned by exposition or number; that is, motion made through the length of a point, and in an instant or point of time.46

To establish this definition, Hobbes rejects the definition of a point as being dimensionless and thus without quantity, which implies indivisibility contrary to all extensive magnitudes. Rather than adopting Euclid’s definition, the author of De corpore proposes the idea of an extended point, but whose quantity and parts remain beyond consideration. By ignoring these properties, an extended point could be utilized in demonstrative combinations “so that a point is not to be taken for an indivisible, but for an undivided thing; just as an instant is to be taken for an undivided, and not for an indivisible time.”47 Conatus is an elementary motion, in the sense that its duration and length are negligible and indeterminate; but this elementary motion possesses a magnitude proportional to that of the effect that it produces in space and time. Hence, the various conatus can be calculated in relation to one another by determining equalities and inequalities. To this potential motion, Hobbes ascribes an intensive magnitude, that of the speed that determines it instantaneously. This degree of speed is defined as impetus: I define impetus, or quickness of motion, as the swiftness or velocity of the body moved, but considered in the several points of that time in which it is moved. In which sense impe­ tus is nothing else but the quantity or velocity of endeavour.48

The combination of impetus and duration comprise the dynamical factor of a motive effect. If impetus varies across time, the measure of the series of impetus expressed in motion will account for the mean value of this dynamical factor. Likewise, one can use the calculation of impetus to describe accelerating and decelerating motions. 45 Ibid., II, x, OL I 113–117; EW I 127–132. 46 Ibid., III, xv, § 2, OL I 177; EW I 206. 47 Id. 48 Ibid., III, xv, § 2, OL I 178; EW I 207.

The Theoria motus abstracti: “Conatus” as Motion Indivisible

37

Hobbes therefore defines resistance as the opposition of conatus to one another. But, given this state of affairs, how can one account for the effect of resistance via calculus? One must return to the elementary impetus present together in the span of an instant, and propose a sort of summation: I define force as the impetus or quickness of motion multiplied either by itself or by the magnitude of the movent, by means of which the said movent acts more or less upon the body that resists it.49

But this proposition breeds an equivocal interpretation. Hobbes in effect maintains that a body at rest is necessarily affected by the impetus, no matter how small, of the body impacting it. And he ascribes no efficacy whatsoever to rest. But to what extent does he deny that velocity is multiplied by the mass of the body? In fact, this thesis is not firmly established, because he alludes to an impetus multiplied by itself. Undoubtedly, one must devise, as Cavalieri had done, a means for integrating the minimal elements of motion that comprise a moving body of a given speed. Given these conditions, mass is understood as corresponding to a sum of relations of conatus that combine so as to form an active power. By virtue of all these tricks, Hobbes hopes to analyze dynamical effects via kinematic transpositions of the relations of conatus and impetus that determine them. The calculations therefore seem to be able to be carried out according to the composition and decomposition of conatus in mechanical exchanges. Potential speed is ultimately the factor that would make such calculations possible. If Leibniz is inspired by this paradoxical doctrine in Theoria motus abstracti, he suggests from the outset tweaking certain definitions.50 With respect to the dimensions of a point, Hobbes and Leibniz take issue with Euclid’s definition, but they differ quite significantly in their positive definitions. For Hobbes, a point is an extended element, the parts and quantity of which one can ignore. By contrast, Leibniz will reject the thesis that a point is extended, as well as the sleight of hand by which Hobbes ignores its parts. For him, a point lacks extension but comprises indistant parts. (A VI 2 265) As such, Leibniz allows for the possibility of relating conatus to an atomic mens. But more fundamentally, this enables him to conceive of a plurality of internal determinations within the same conatus, and of a combination of speeds via such antagonistic determinations in the unextended element. Hence, the Leibnizian conatus will also distinguish itself from the Hobbesian conatus. Hobbes’ conatus is in effect an inclination to motion determined by an impetus measured instantaneously by velocity. Leibniz will integrate the property of impetus into the conatus itself, as a sort of internal law of instantaneous causality. This law is designed to create infinite variations in the interaction of bodies, but in accordance with the integrative determinations of the conatus, which either

49 50

Ibid., III, xv, § 2, OL I 179; EW I 212. Cf. Bernstein (1980), 26–29.

38

The Initial Scientific Project

combine within a point (conatus componibiles) or not (conatus incomponibiles). Finally, a sequential law of the elements of conatus componibiles can be ascribed to a physical point: at least the possibility of this capacity deriving from the mens is not excluded, as it would be if conatus were reducible to an extended atom subject to being determined by motions. The scope of the methodological shift that Leibniz’s position represents with respect to the Hobbesian model is summarized well in the letter to Arnauld in early November 1671: “I saw that geometry, or the philosophy of position (de loco), was a stepping stone for the philosophy of motion and bodies, and that the philosophy of motion was a stepping stone for the science of the mind.” (A II 12 278; L 148) As it quickly becomes apparent, the Leibnizian interpretation relies on the principle of sufficient reason, which makes it possible to pass from the abstract theory of conatus to a system of actualized essences.51 The section Fundamenta prædemonstrabilia precisely explicates those abstract concepts on which the ontological relevance of the phoronomia elementalis depends. Though inspired by Hobbes, the theory of conatus that Leibniz advances there includes aspects that contrast epistemologically with the theses of De corpore. In particular, the geometric analysis of extensive continuum in terms of conatus was interpreted by Hobbes as a purely abstract codification of the system of sense-perceived appearances: by virtue of such a codification, one could perform general calculations relating to phenomena while respecting the geometric rules established from the outset. For his part, Leibniz will interpret conatus as the constituents of reality, while also using them as symbolic, conceptual instruments for analyzing possible modifications in motions. In order to understand motion in its initial and final states, one must presuppose what space, time and motion are ultimately composed of. Space is comprised of an “unextended” thing that has, in distinction with other unextended things infinite in number, the power to generate finite extensions. Because the extension of this physical point is incommensurate with all other determinable magnitudes, but also possesses the ontological property of every realized extension (i. e., being comprised of juxtaposed parts), the parts comprising it are “indistant”. “Indistance” represents the limit of what is divisible, and allows us to obtain such points. Points neither lack parts, nor are their parts inconceivable. Rather, they lack extension, that is, their parts are indistant, having a magnitude so small as to be inconceivable, unassignable, and inferior to every other magnitude that can be expressed through a proportion – unless infinite – to another sensible magnitude. This is the foundation of Cavalieri’s method, which evidently demonstrates the truth of this, and thereby makes it possible to think of certain rudiments or beginnings of lines smaller than any other that might be supplied. (A VI 2 265)

51

For an initial version of this analysis, cf. Duchesneau (1985).

The Theoria motus abstracti: “Conatus” as Motion Indivisible

39

Defining a point as an unextended element of extension helps establish the definition of conatus, which is invoked to play an analogous role in analyzing motion. Hence proposition 10: “Conatus is to motion what a point is to space, or unity to infinity, it is in effect the beginning and the end of motion.” (A VI 2 265) This analogical definition of conatus finds itself dependent on certain preconditions in the text. The essential difference between motion and rest is illustrated by the relation of 1 to 0. The consequence of this radical distinction is that one cannot reconstruct the continuity of a motion from discrete elements of motion separated by intervals of rest; hence the criticisms addressed to Gassendi’s theory on how motions combine, and his attempt to use relative inertia to explain resistance to motion and the possibility of reaction in impacts. In place of this imaginative hypothesis, Leibniz formulates an axiom of inertia that cements the spatial disparity between motion and rest: [Prop. 8] Once a thing has come to rest, unless a new cause of motion occurs, it will rest forever. [Prop. 9]. On the contrary, what has once moved, insofar as it depends on itself, will move forever with the same speed and in the same direction. (A VI 2 265)

This version of inertia, combined with the conatus theory, entails the known consequences of the first mechanical doctrine of Leibniz, such as the motion of an object, no matter how small, infinitely propagating itself through space. This occurs no matter how great the obstacles are, insofar as conatus as initial action begins to move all of these obstacles; and it does not matter if the resulting effect of conatus, as an instantaneous action, is hindered. A plurality of opposing conatus can simultaneously coexist in the same body, resulting in a motion that corresponds to the mutual compensation of the conatus – here, the state of rest of the impacted body is not factored into the calculation of the result. In the timespan of a conatus, supposing that the indivisibles are homogenous, the space traversed by a point will increase in proportion to the strength of the elementary conatus, and in any event, will prove incommensurable with the unextended space of the point, i. e., a simple extended indivisible. In fact, as we will soon show, this simple extended indivisible only possesses physical reality as a linear displacement generated instantaneously by conatus. For our understanding, the emergence of elementary mass would be subordinate to the condition sine qua non of the inclination to motion, which no doubt amounts to relegating the specific concept of mass to indeterminacy owing to the mutual, instantaneous compensation of the conatus. Moreover, the coexistence of opposing conatus in a material point across the indivisible homogeneity of time is no doubt the most adequate analogy for mass, as propositions 15 and 16 attest: [Prop. 15] In the timespan of an impulse, an impact, an encounter, the two extremities or points of the bodies either penetrate each other or are in the same point of space: as one of the impacting bodies strives for the other’s place, it will begin to occupy it, that is, it will start penetrating and uniting with the other. For a conatus is a beginning, a penetration, a

40

The Initial Scientific Project

union; the bodies are therefore in the beginning of union, that is to say, their limits are one in the same. [Prop. 16] bodies that press upon and impel one another form a whole, since they share the same limits; and because these limits are one in the same, forming a whole, they are continuous or coherent according to Aristotle’s very definition, for if two things occupy the same place, one cannot be impelled without the other. (A VI 2 266)52

But conatus do not coexist for longer than an instant; owing to the principle of inertia, only an unimpeded conatus generates motion that lasts beyond an instant. Coexisting conatus combine their effects by means of algebraic addition/subtraction. In the case of conatus servabiles or componibiles, the effect is a combination of motions that at most equal the motions that would respectively be generated by the conatus involved. In the case of conatus incomponibiles, the resulting motion is less than the potential effect of the independent conatus: this diminishment corresponds to the algebraic difference between the conatus, with the direction of the motion being determined by the strongest. The typical case of conatus servabiles is that of a circle rolling along a straight line on a plane. A point drawn on the circumference indicates a cycloidal trajectory by combining circular motion and rectilinear motion, the two motions being so to speak intermingled “per minima seu conatus”. (Prop. 19, A VI 2 267) The conditions for conserving conatus beyond an instant in the continuation of a non-rectilinear motion present a major difficulty for the abstract theory of motion. The motion generated by the instantaneous conatus is uniform, rectilinear and possesses a constant speed when unimpeded. Maintaining the curvilinear motion implies a sort of causality other than the one implied by the conception of conatus as indivisibles comprising the elements of bodies. A similar problem seems to arise when accounting for accelerated motion. Within the same physical point (defined in terms of conatus), a conatus as cause intervenes to generate a simultaneous conatus, which, as effect, integrates the causal property of the previous conatus and makes it part of its own causal determination, which will in turn be exerted upon the next embryonic conatus. This cumulative process builds up the whole effect of acceleration. And the same reasoning conversely applies to deceleration. A conatus is a “sign” of what is to come, for within the span of an instant, one conatus is instantaneously the cause and the other the effect; but at the same time, the latter is likewise a sign of the causal power of the one preceding it, the series of signs being ampliative ad infinitum. “In accelerated motion, which augments with each instant and almost immediately after it begins, augmentation presupposes a before and after; in any given instant, one sign precedes another.” (Prop. 18, A VI 2 266) In signs, or the indistant nominal moments that occur instantaneously and generate uniform series,

52

The reference is to Aristotle’s definition of the continuous as “that whose extremities are a single thing” (Physics, Ζ 1 231a19–20).

The Theoria motus abstracti: “Conatus” as Motion Indivisible

41

embryonic conatus are thus presented as originating in the same initial conatus, and these presuppose a form of integration that the essential form of conatus (according to the theory of abstract motion) does not entail. Leibniz will return to this subject at the end of the Problemata specialia to insist that abstract reasons do not suffice for adequately representing acceleration/deceleration. To be sure, he imagines resolving these two types of problems – the generation of curvilinear motions and the process whereby effects are integrated into uniform accelerated motions – through models that conform to the principles of the phoronomia elementalis, even if the justification or sufficient reason of these constructions cannot be deduced from the abstract theory. Regarding the first type of problem, the phoronomic model derives from the hypothesis of the “third way”, if one might use this expression. Examining impacts between equal conatus incomponibiles, Leibniz suggests the idea: “If incomponibiles co­ natus are equal, the combined directive effect (plaga) is canceled, or rather a third intermediate effect is chosen if one can be found, the speed of the conatus being maintained.” (Prop. 23, A VI 2 268) We will see what kind of intermediary conforming to the methodology of abstract phoronomic representation this involves. For the moment however, let us draw attention to the following explanation, where we find the idea that the justification of the abstract model surpasses any purely geometric symbolization: Here, rationality achieves its apex in motion: something is accomplished not merely by subtracting equal sums, but rather by means of a third, more favorable position, by a type of incredible but necessary wisdom that one does not easily find in geometry and phoronomy. Since all other propositions depend on the principle that the whole is greater than the part, Euclid indicates at the beginning of the Elements that everything can be resolved by addition and subtraction alone. Thus, from this idea alone, along with the fundamental Prop. 20 [the moved body transfers to another, without its own motion diminishing, everything that the other can receive salvo motu priore], which itself depends on the supremely valuable Prop. 24 [nothing is without reason], the following ideas follow: that the minimum amount of change should occur, that a middle term should be adopted when faced with contraries, that something should be added to one in order to prevent a subtraction from the other; and many other propositions that are equally dominant in civil science. (A VI 2 268)

This minor adjustment would prevent motile determinations of equivalent but directionally diametrically opposed geometric conatus from cancelling one another out; it belongs to an architectonic design for physics. A similar argument is evoked in a seemingly less apparent case, where the transmission of velocity in an impact depends on the quantity of speed that the second moving body can receive without exceeding the speed of the first. In this case, the sum of the speeds would be greater and thus contrary to what is rationally determined. Or, according to a more positive formula, the quantity of the cause (conatus A) finds itself within the effect (instantly combining conatus A and B), as if the effect were sub-

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The Initial Scientific Project

ject to the restrictions of a self-preserving system, or in other words, as if the effects themselves represented this system. What process then can save the conatus from reciprocally cancelling out one another? The geometric representation of the solution describes the collision of bodies as angular rather than rectilinear. But one must no doubt determine here what these constantly repeating deviations are, as they prove to be necessary for saving the conatus from mutual destruction. Theorem 7 provides some elucidation in this regard. (A VI 2 268–69) Leibniz specifies the circumstances where two equivalent, impacting bodies follow the same rectilinear trajectory after colliding, thus bisecting the angle of impact. In this way, the theory of abstract motion would imply that angular deviations diminish more or less rapidly rather than repeating themselves. Leibniz mentions hypothetical cases where this diminishment would not apply. This is the case (1) when at the instant of the impact, one of two equivalent conatus is subject to a finite scalar measure, and the other to a “differential” vectorial measure, e. g., when the motion of one of the moving bodies is uniform and the other accelerated, in which case parabolas and other types of lines are generated (as Hobbes showed). This is also the case (2) when one is dealing with conatus servabiles: the two conatus can therefore coexist in a physical point, but also virtually maintain the difference between their respective effects while producing a combined, continuous effect: “the two conatus can mutually add to one another, just as the rectilinear and circular motions of a spiral, the speed of each being conserved.” (A VI 2 269) It soon becomes evident that the two cases are one and the same, because the only way to prevent motive determinations from being homogenized, and thereby ensure the repetition of the angular deviation, presupposes that non-rectilinear motions are actualized in conformity with the various conic sections. As such, we return to the main question regarding the possibility, and thus reality, of circular motion. It is evident that for Leibniz the method of indivisibles allows one to assimilate the indecomposable elements of curves into the infinitely numerous elements of the polygonal perimeter of a circle’s quadrant. This is evinced in a passage in the Præde­ monstrabilia where Leibniz, following Hobbes, outlines a theory of angles founded on the method of indivisibles, and defines an angle as a quantity of a convergence point (punctus concursus): “The whole doctrine of angles concerns quantities of what is not extended. [To illustrate this:] An arc smaller than any arc that can be conceived of is certainly larger than its chord, even though the latter is smaller than any chord that can be demonstrated, as it consists in a point.” (A VI 2 267) The justification for the quadrature of the circle is therefore provided by the possibility of postulating an extensive equivalence in terms of infinite indivisibles, which involves abstracting from the unactualizable equivalency in terms of finite quantities. However, if one can technically represent conatus with circular motion as “unextended” analogues of conatus with rectilinear motion, does this equally apply to the inertial effects of conatus, that is, to the motions that result when no obstacles are present? The paradox is described in precise terms (A VI 2 267).

The Theoria motus abstracti: “Conatus” as Motion Indivisible

43

Unraveling this paradox implies that one distinguish between unextended quantities of motion – from this perspective, one can identify conatus of circular motions and rectilinear motions – and the infinite repetition of these conatus in extended space, all obstacles being removed. Rectilinear motion is generated by instantaneous conatus and is not determined by other factors. Circular motion depends on the motile disposition to constantly and repeatedly deviate, that is, on a law of inflection that the initial conatus cannot sustain beyond an instant. The indefinite motile expression of this co­ natus coincides with the tangent to the curve at the unextended point it occupies, i. e., with the indeterminable angle of the secant and the tangent at this point. The geometric expression that makes motion intelligible in the phoronomia ele­ mentalis prevents us from conceiving of the effects of conatus in other terms than tangential, rectilinear expansion. One can surely reconstruct a circular motion as an infinite series of successive rectilinear expressions reducible to an indeterminate extension, i. e., to physical points having the form of indivisibles. But in this case, it is the law of inflection represented by this infinite series that becomes inconceivable when one tries to understand it as inherent in some element of the series. The law of inflexion transcends the conatus insofar as the latter are capable of being geometrically expressed. Given the circumstances, how could there be no need to introduce something like the principle of indiscernibles as an architectonic solution to that kind of problems in physical theory? Leibniz’s solution to the problems that we have identified remains insufficient in the formal mechanics offered by the Theoria motus abstracti. Nonetheless, one already finds there substitutes for the principle of indiscernibles, or rather nascent formulations of postulates that texts relating to dynamics will develop. Section 17 of the Fun­ damenta suggests that the ontology required for keeping the system “alive” shall be completed by adding to conatus, which are understood as physical points, the elements of another order that are capable of representing a law beyond the span of an instant. These “psychic” elements would ensure that the inflexion law correlates with motion, where motion is the mode for expressing conatus as physical points: No conatus lasts beyond an instant without motion, except in minds. For conatus, within the span of an instant, is bodily movement across time. […] Thus, every body is an instantaneous mind (mens momentanea), i. e., a mind that lacks memory since it does not maintain, beyond an instant, both its own conatus and another opposed to it (for both action and reaction, or comparison and therefore harmony, are required for sensation (ad sensum), and for pleasure and pain, without which there is no sensation); a body thus lacks memory, a sense of its actions and passions, and thought. (A VI 2 266)

In short, the solution appears to be the following. There is a continuous gradation between conatus, as the elements that ultimately express physical reality, and selves, as the elements of substantive individuality that express themselves through the modalities of phenomenal self-consciousness. As an unconstrained effect of conatus, motion can

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The Initial Scientific Project

only be rectilinear, uniform and constant. Circular motion, as well as acceleration-deceleration in rectilinear motions, requires a law in order to account for its serial and progressive states. The application of this law to instantaneous conatus can only be conceived of via an analogy with conscious representation, which spans an indefinite sum of temporal indivisibles; that is, an analogy is made with conscious reminiscence. But the implications of the continuous gradation underpinning the analogy go beyond this. Is conatus itself not presented as mens momentanea, or instantaneous mind? With this metaphor, a real problem arises in Leibniz’s mechanics, that of the origin of cona­ tus. This problem could lead one down a metaphysical path, which would not appear strange given that Leibniz’s thought will subsequently trend toward monadology, the doctrine of monads. At the moment however, the problem takes on a more epistemological shape. Co­ natus, in the geometric sense, is only an “indivisible” of motion. The inertial effect, or the uniform rectilinear motion that it produces beyond an instant, expresses a “power of expansion” that cannot in any way be related to a concept of mass, which is absent from the phoronomia elementalis. The cohesion of bodies itself can only be assured by the mutual convergence of “constantly regenerating” conatus that together resist the pressure of their surroundings. The “stable” system of endogenous conatus thus forms a counterpart to the sum of exogenous conatus. As a last resort, the problem can be reduced to adequately conceiving of conatus as mutually and instantaneously penetrating one another, with each constituting one of the indistant terms of an indivisible representing the whole system of these incepting motions. But the integrated conatus tend to destroy themselves beyond an instant as a result of algebraic subtraction. Maintaining the whole system of converging conatus, as indistant antagonistic terms in the same un-extended indivisible, presupposes a kind of constant actualization amounting to regeneration across time, that is, across the indefinite sum of indivisibles of duration. This case likewise requires one to apply some law of serial expression (or expansion) to physical points; this would yield the antitypy of a point, as an individuated system of endogenous conatus resisting the sum of exogenous impacts. These requisites, being more implicit than explicit, seem to suggest that the mechanical theory must overcome what would be indiscernible in the postulated physical elements. This indiscernibility would surely emerge for a host of reasons: (1) opposite motile determinations being exhausted at impact; (2) antitypy being suppressed beyond an instant; (3) angular deviations in the inertial effect of conatus cancelling out. Each and every one of these factors would bring about the destruction of an actualized system of conatus; and these would equally be reasons for invalidating any attempt to conceive of physical nature as conforming to self-sufficient laws. To compensate for such systematic lacunae, phoronomia elementalis needs to supplement the determination of conatus. Either metaphysical sufficient reason will procure this additional determination, or auxiliary postulates or hypotheses will be required to directly connect the theory of abstract motion with phoronomia physica.

The Theoria motus abstracti: “Conatus” as Motion Indivisible

45

Let us first examine the metaphysical compensation strategy.53 This strategy would be designed as a compromise: according to Leibniz, the mind is invoked to substitute for the role of geometrical determinations since it remembers the various opposing conatus beyond the instant in which they converge. To cite a fragment of the same period as the Hypothesis: “if bodies lacked mind, motion could not be eternal (it would be endlessly diminishing).” (A VI 2 280) To account for circular motions, mind conserves so to speak the law of curvature; for body, reduced to a physical point, would cor­ respond to a motile determination along the tangent. In order to constantly maintain the total quantity of velocity in the universe, an intervening spiritual entity must once more be postulated. Since the deficiencies in the system of conatus manifest in every instantaneous interaction, one must assume that the action of the mind is immanent to the very structures of physical reality – that it is an “enduring” indivisible (psychic in nature) that must underpin and determine the instantaneous indivisible (purely phoronomic in nature). A panpsychic metaphysics would therefore be invoked to fill in the gaps of the geometrical conception of the phenomenal world, and to override the mere intelligibility of the concept of abstract motion in the analysis of the combinations of conatus.54 It is likewise paradoxical to postulate that conatus are conserved, and then apply an algebraic summation (subtraction) of velocities in impacts, which causes the original differences between these conatus to mutually destroy one another. The non-paradoxical “conservation” of conatus can only occur on a level other than geometry, namely on the level of continuous mental representations, to which there pertains the laws that are presumed to be immanent within the order of conatus. The irreparable discontinuity in the effects would be compensated for by metaphysically postulating a “parallel” continuity of conatus endowed with psychic characteristics. What we have witnessed here is a compensatory strategy: Leibniz postulates psychic entities and governing metaphysical principles to account for the laws that determine the operations of such entities; this world of soul-like principles and actions provides substantive grounding to the insufficient system of physical conatus. How­ ever, based on that overarching system of reasons, it must be possible to search, on the level of physical theory, for models that might offer some improvement to the abstract mechanics. This role is assumed by what we have identified as “auxiliary postulates”. Firstly, there is the requirement of representing the physical system in terms of circular motion. One of the most explicit passages in this regard can be found in the fragment already cited:

53 54

Gueroult (1967), 8–20 does a good job of analyzing this option, and has guided our attempt here to bring together corresponding arguments. In this instance, Gueroult (1967), 17 concludes that “given its limited resources, [Leibniz’s 1671] mechanism is incapable of providing the principle (substituting for mass) by virtue of which the positive element of nature (by which one means speed, motion and force) would neither diminish nor increase.”

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The Initial Scientific Project

If primary matter is moved in one way, that is, along parallel lines, then it comes to rest and thus ceases to be. There is a plenum, since primary matter and space are one and the same. All motion is therefore circular, or composed of circular motions, or at least returning upon itself. Many circulations [of matter] impede or act upon one another. Many circulations [of matter] come together to become one; otherwise, all bodies would come to rest, that is, be annihilated. (A VI 2 280).

Because space and matter endowed with impenetrability are equivocated (theory of plenum), the diffusion of motion through space entails the circular interaction of the parts of matter, and consequently, physical motion must conform to a closed trajectory that is circular in nature. Such a conception of physical reality likewise implies that circular motions are nested within one another down to the level of the physical point, that is, to the primary structure of the conatus. Indeed, even assuming that circular motion derives from a law immanent to the conatus, this motion tends to be exhausted in collisions involving the motion of opposing conatus. But at least this model suggests a geometrical pattern that satisfies the requirement that physical points be understood as autonomous sources of motion.55 It is remarkable that the immanent determinations (i. e., conatus) of physical points are expressed in the curvilinear expansion of motion. As we have pointed out, curvilinear expansion requires that a formal determination be present in each element of the trajectory as a body moves across it. In a concrete physics that conforms with phoronomia elementalis, spherical structures (bubbles) nested within one another ad infinitum and capable of being modified by the impact of subtle bodies that penetrate them at various angles, will represent the formal modalities of equilibrium and interaction – the structural conditions for conserving functional dispositions – in a system of phenomena subject to a geometrico-mechanical order. By making bodies in circular motion interact, one would generate precisely the sort of empirical effects that constitute the regular changes seen as being in conformity with an orderly law. However, the possibility that such dispositions to circular motion are maintained in nature with sufficient constancy requires exogenous conatus to converge ad infinitum in series of micro-impacts that form circles. The constancy of dispositions to circular motion implies “as a corollary” that conatus as nascent motions work in unison so as to determine orderly deviations in such a way that circular collisions can be fully implemented. In this case, one would be dealing with various types of constantly repeating deviations, which represent indivisibles of tendency, successively generating themselves following a law of summation that transcends the instantaneous determination to move. In a certain way, the law of summation must be located within the conatus, or rather, since the

55

Hannequin (1908), II, 162 in particular highlighted the importance of this auxiliary postulate when it comes to designing a system of physical conatus that corresponds to an ordered universe.

The Theoria motus abstracti: “Conatus” as Motion Indivisible

47

latter are destroyed in an instant, except for their breeding inertial motions that would develop indefinitely, this law is made dependent on a “substrate” of the conatus that can maintain the motile tendency. At this stage, the law of summation evidently lacks a “physical” foundation, something which the dynamics will provide later on by revisiting the notion of conatus with a view to making it capable of expressing how impetus is generated. In this way, cona­ tus will constitute so to speak the differential expressions of vis viva. Hence, points of force will be the physical source of all variation within the extended order. Conatus, as motion differentials, will then be abstract and uniform elements in our account of physical action. They will become, so to speak, the basic symbols of a general system for expressing motile tendencies. The symbols will refer to concrete elements that are internally irreducibly diverse. The original diversity of the various tendencies to motion will radiate outward from these concrete elements. To the understanding capable of developing the analysis of the “contents” of physical points, they would appear as intrinsically “discernable”. In Leibniz’s mature philosophy, an analogical expression will connect the strictly individualized monads and the centers of force, which are responsible for the hetero­ geneity of motile dispositions. In 1671, the theory does not yet explain how a law determining the serial development of conatus inheres in the physical point. The infinite circular envelopment of motive determinations, and thus the effects produced by conatus, anticipate the latter theory to an extent. At the same time, the inadequacy of the concepts at play reveals the necessity of introducing a system of physical postulates that the phoronomia elementalis pretended to do away with. An adequate system of such postulates should overcome the functional indiscernibility that arises when representing equivalent but opposed conatus eliminated by algebraic subtraction. Under these circumstances, the progressive elimination of homogenous conatus proved incompatible with the conception of nature’s enduring power to generate action. In a similar way, the Leibnizian mechanics of 1671 lacks a notion of mass that is even slightly different from the notion of shaped extension to which the delimitation of bodies is reduced. One of the consequences of this situation would be the total dispersion of rectilinear motion in the plenum. Among the supporting postulates that Leibniz relies on as corrective measures, the theory of ether plays a special role by helping to account for the elasticity of bodies. Indeed, in passing from abstract mechanics to the organized system of physical phenomena, one must presuppose a divine arrangement that maintains the most orderly and convenient relation between effects. For instance, according to the abstract theory, in all cases of oblique incidence, reflection conforms to a different angular ratio depending on the magnitude of the angle of incidence: the smaller angle is complementary (supplementum ad rectum) to the larger one multiplied by two; the angle of incidence and that of reflection only equal one another in a limit case, when the angle of incidence measures 30 degrees. But experience shows that the angles of incidence and reflection equal one another as a general rule, which forms the

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The Initial Scientific Project

basis of explaining phenomena, e. g., optical phenomena, via a teleological principle of sufficient reason: the principle of the most determined path. This principle fits well with a mathematical assessment of the best forms (de formis optimis).56 Moreover, making this type of correction to the abstract laws of motion ensures that the same angular deviation in motions is self-repeating, which is one of the conditions for curvilinear motion maintaining its trajectory. However, from the abstract laws of motion one cannot derive elasticity, which yields this arrangement of effects. We are thus forced to invent an imaginary solution to the problem, a “unique hypothesis that permits us to discover the divine solution”.57 The development of this hypothesis is the main objective of the Theoria motus con­ creti or the Hypothesis physica nova. The hypothesis itself is a response to the 11th special problem of the Theoria motus abstracti, which concerns the mutual reverberation of bodies in impacts. Among the theorems that are instrumental in finding a solution to the problem is proposition 21: “A discontiguous body resists more than a contiguous body.” (A VI 2 270) The concept of a discontiguous body is a provisional equivalent of mass. If there are countless discontiguous parts inside a solid, then any exogenous conatus acting on the solid will generate series of impacts within it. And owing to the countless discontiguous elements moving in each direction, the motile action could in a sense dissipate inside the solid without displacing the whole in the slightest. To be sure, the theorem cannot be demonstrated within the framework of the abstract theory. Perhaps one might even refer to it as a paralogism with regard to the requirements of that theory, which are as follows. The corpuscles situated along the line of impact can only act in the direction opposed to the exogenous impact if they exert countercurrent conatus, no matter how small they might be. Discontinuity as a geometric arrangement of the components of bodies is not a sufficient condition for making sense of this antagonistic conatus determination. Furthermore, discontinuity itself seems to entail another difficulty regarding the interval between parts. Would it not be necessary to postulate an inter-corpuscular vacuum in order to guarantee this structural discontinuity? The vacuum hypothesis must however be excluded. The only way to overcome this impasse will be admitting: (1) that the discontinuity of the internal parts owes to the motions that segment the apparent extensive continuum; and (2) that those motions are curvilinear and closed. (Theorem 22, A VI 2 270) The ether theory will additionally express the conditions of the discontiguity of extensive elements in actualized nature. The main property of this subtle fluid is that its parts do not obstruct one another; it is as if one were dealing with elements with a constant conatus that only intervened to “react” to the motile expansions of some extensive element of another kind. In this 56 57

Leibniz will develop this teleological approach to demonstrating the Snell-Descartes law in dioptrics in his Unicum opticæ, dioptricæ et catoptricæ principium (1682). For analyses of these later arguments, cf. Duchesneau (1993), 262–283; McDonough (2009), (2010) and (2016). Hannequin (1908), II, 72.

The Theoria motus abstracti: “Conatus” as Motion Indivisible

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context, “reacting” means reverting or inflecting the conatus of this element without any decrease in speed. This is the true reason that the circular motion of the extensive parts is maintained. This is also the sufficient condition for elasticity, as a property of extensive parts. Hence the wording of the 11th special problem: Producing a mutual repercussion: this will occur if the two bodies are carried by a discontiguous liquid that, according to theorem 21, is so subtle that the greatest part moves back and forth between the poles of the two bodies, without being impeded by the impacts; then, each body will reciprocally transfer its impetus to the opposite body, not only generating percussions, but also permutations in the directions and speeds. (A VI 2 271)

Taking this supporting hypothesis as his starting point, Leibniz will attempt to explain “all observable motions” (phoronomia physicalis), and more generally, all real-world phenomena, particularly those relating to light. The equivalence of kinetic hypotheses (as demonstrated by Descartes58) will help Leibniz conceive of a moving ether and bodies that are moved and modified by the action of this subtle fluid. This is the proper goal of the Hypothesis physica nova. For our purposes, it is sufficient to note that certain mechanical phenomena that would prove paradoxical in the context of phoronomia elementalis tend to support this theory when it is buttressed by the supporting hypothesis. Leibniz therefore suggests that it will be possible to annex mechanical conceptions foreign to the abstract system; such conceptions include Wren’s and Huygens’ laws of impact, Descartes’ conservation of the quantity of motion, elasticity as the basic state of bodies, the possibility of simple circular motion (according to Hobbes), material cohesion, etc. In fact, ether, this subtle, penetrating fluid, complements a system reduced to geometrically analyzable properties to ensure: (1) that the conatus of various physical points conform to a law of alternating expansion and compression (allowing for elasticity); and (2) that they generate circular effects, the centrifugal motion issuing from the conatus being constrained by the ether along a regular deflection. More precisely, the initial determination of this deflection is related to the extensive structure of the particles of the ambient fluid. Hence, the reactive role of ether is to circumscribe the sphere of action intrinsic to physical points. As a theoretical entity, its importance lies in this compensatory role. Indeed, the difficulties of phoronomia elementalis are not overcome by merely recurring to the auxiliary hypothesis. However, this recourse shows that the conatus theory is subject to an architectonic rule, the radical discernability of physical points. As the centers from which motion radiates, physical points must constantly serve as physically causal agents. To achieve this, they must not be subject to the destruction that threatens conatus in their rectilinear and uniform motion. If inter-related instantane-

58 Cf. Principia philosophiæ, III, § 25 and § 44, AT VIII-1, 84–85, 99.

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The Initial Scientific Project

ous conatus incomponibiles are inclined to lose their effects in an algebraic summation, then conatus componibiles or servabiles demonstrate the irreducibility of their distinct determinations, which persist in the global phenomenal effect. Evidently, the supporting ether hypothesis allows one to attribute the characteristic of servabiles to all mutually related conatus. It is in this way that Leibniz attributes an individual causal function to actualized physical points across the continuous series of indivisibles of time, and across the ordered series of varying conatus that depend on it. In the revised mechanics, the relationship between primitive force and derivative force, combined at each level with the internal opposition between active and passive force, will compensate for the arbitrariness of the auxiliary hypothesis by ensuring the perfect individuation of centers of force. The role that ether plays in the phoronomic system will be replaced by the notion that the dynamical disposition of each center of force is internally limited. This limitation is expressed at the phenomenal level by the respective conditions of elasticity between finite masses that mutually communicate motion. At the substantive level, it is expressed by the law of expansion and compression, which can be applied to physical points since it is possible to integrate the diverse conatus that result from them. The replacement of the auxiliary postulate will be carried out in a remarkable way however. In 1671, it seemed that a bridge between the physical hypothesis and phoro­ nomia elementalis had only been established by some deus ex machina, since the two distinct levels of analysis were somehow merely juxtaposed. The ether hypothesis will be retained in Leibniz’s later physics: he will never really renounce the arguments that he makes in the Theoria motus concreti, whereas the Theoria motus abstracti will be replaced by the principles of the dynamics. By contrast, the new theoretical postulates, which define the level of the mechanical analysis, will cease (at least ideally) to integrate propositions that would have been derived a posteriori from the physical representation of phenomena. The ether theory will no longer be invoked in order to compensate for the inadequacy of the principles of mechanics. It will only be employed as a sort of model for relating the formal dynamical concept of elasticity to concepts that express phenomenal effects of elasticity: gravity, magnetic impulse, reflection and refraction. 3. The Hypothesis physica nova: Unifying the Models The Hypothesis physica nova (1671), which was dedicated to the Royal Society, is typically regarded as a mere appendix to the Theoria motus abstracti that Leibniz sent to the Académie des sciences in Paris that very same year.59 Both texts were intended to

59

Hannequin (1908) and Gueroult (1967), 8–20, subscribe to this interpretation.

The Hypothesis physica nova: Unifying the Models

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demonstrate the scientific acumen of the young German philosopher before his diplo­ matic mission to Paris. When Henry Oldenburg, the secretary of the Royal Society, received the Hypothesis, he requested Leibniz to send him a copy of the Theoria, which was key to understanding the scientific theses advanced there.60 And John Wallis, who undertook the analysis of these texts for the Royal Society, both sensed that the Theo­ ria was the true foundation of the Hypothesis and agreed with its author that physics could not be analyzed without bringing geometrical reasoning into play. Did Leibniz himself not associate the physical model, which is real but exact, with the geometrical model, which is imaginary and exact? (A VI 2 270) And was he not convinced that everything in the physical world obeys the laws of phoronomia elementalis? That being said, we have remarked the inherent insufficiencies of these abstract laws when it comes to understanding how circular motions are determined by material elements, and explaining the effects of mass. Does the Hypothesis function then to “correct” the inconsistent results of an abstract geometry of conatus? If so, this would make it one vast ad hoc hypothesis for “saving phenomena”.61 It indeed appears necessary to ground the physical model in a complex set of theoretical postulates. But at least a portion of these would fall outside the scope of the geometry of conatus, even though they would not undermine its exactitude. If the geometry of conatus conforms to the principles of the physical models, these nevertheless have their own intelligibility. What is generally missing from previous accounts of the Hypothesis is an assessment of the epistemological features of Leibniz’s physical models.62 Our goal here is to provide this analysis. The letter to Honoré Fabri in early 1677 contains some interesting post facto information in this regard. (A II 12 441–66) In his correspondence with Ignace-Gaston Pardies, Fabri had previously attempted to distance his work from corpuscular theses, which he hoped to achieve by criticizing the systems of Gassendi and Descartes. In an additional attack on Hobbes’ De corpore, he objected to Leibniz’s treatise. Leibniz prepared his defense by first showing what aspects of Aristotelean science he preserved, how he differed from the atomists and particularly Gassendi, and what prevented him from adopting Descartes’ scientific methodology. He admires Aristotle’s contributions to 60 61 62

Cf. Letter of Oldenburg to Leibniz, 14/24 April 1671, A II 12 150. For an initial version of this interpretation of the Hypothesis physica nova, cf. Duchesneau (1989). An exception is Beeley (1996). In particular, Beeley (ibid., 212–27) shows that the 1671 Hypothesis relies on economic principles for harmonizing observation-based inferences about the “system” of phenomena with the mathematical framework that consists in the abstract laws of motion. In so doing, Leibniz aimed at providing unifying explanations of the mechanisms underpinning such major physical properties as elasticity, gravity, magnetism, etc. Leibniz then intended to deploy these mechanisms to account for a host of specific phenomena. For instance, he would write Hermann Conring in early May 1671, A II 12 152–53: “You will find that my hypothesis accounts with remarkable concision and harmony for gravity, elastic force, magnetic orientation towards the poles, the pendulum, fermentation, all chemical reactions, the cohesion of bodies, the motions of tides and winds, and all that, because of the motion of light, i. e., ether around the Earth.”

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The Initial Scientific Project

“civil science” and the natural history of animals, and is ready to incorporate and reinterpret certain explanations present in the Physics concerning principles, motion, continuum and the soul. He objects to Democritean natural philosophy on the grounds that the postulation of incorporeal substances is necessary, that it is impossible to derive motion from the nature of bodies, that naturally indivisible corpuscles are inconceivable, and that any theory of simulacra is purely speculative. He is interested in the fact that Gassendi makes a significant distinction between extension and the essence of physical material realities – the first step in phenomenalizing extensum. His position regarding Descartes is far more radical. Leibniz denies having had adequate knowledge of the Cartesian theory when writing the Hypothesis. He had not yet mastered Cartesian mathematical techniques, and contrary to Descartes, his intention was not to rely on hypotheses that only indicated possible causes, but failed to realize anything certain and real in physics. The methodological approach that Leibniz favors consists in “deducing” more complex phenomena from simpler ones that have already been investigated. This allows one to weed fictitious hypotheses (fictitiæ hypotheses) out of natural philosophy. (A II 12 448) The transition from causal models to those based on the resulting effects is made possible by laws derived from mechanics, or rather geometrical phoronomia. Causal models refer to the simplest and most general phenomena that appear in nature; hence, their capacity for deductively generating more specific content is considerable. Leibniz confirms these various points: I thought that one should attempt to see if more difficult natural phenomena could be deduced from evident and known phenomena. For, in so doing, it was evident that one vainly assumed possible causes to be true causes, since the true and certain causes were themselves inaccessible […]. I [therefore] sought to examine the consequences of these very powerful and far reaching causes [of the motions of celestial bodies and light] by relying solely on the laws of mechanics. (A II 12 447–48)

Two demonstrative paths for moving from the effects to the causes are possible. Firstly, in certain cases it suffices to simply examine a set of effects in order to identify the causal process from which they result. This Baconian approach encounters difficulties however, when the effects can be explained by a plethora of possible causes. Secondly, another demonstrative path is secured when the causes are directly expressed in the field of observable phenomena, or are derived therefrom by unproblematic analyses and inferences. If physics must be built on demonstrative certainties, this can only be achieved by explaining specific phenomena by more general ones; the latter are intended to represent the simplest elements of the phenomenal world. This type of explanation depends on the availability of conceptual analytic tools. According to the perspective developed in 1671, such tools would include the geometrical conatus theory and the rules of algebraic summation of motion indivisibles. In virtue of this

The Hypothesis physica nova: Unifying the Models

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twofold phenomenal-analytic approach to hypotheses, the Hypotheses would appear to present an original research program: I believed that the most ingenious minds might be inspired by this example to pursue natural philosophy, as far as possible, by excluding fictitious hypotheses and instead adopting causes whose existence in nature is evident and observable. (A II 12 448)

How can science be developed within these confines? Experiments are the cornerstone of science, whether they have already been carried out or because analysis suggests us to undertake subsequent ones. The proper function of analysis is to link known facts or empirical elements together in a chain of sufficient reasons. The conceptual basis of physical analysis consists, as we have shown, of the abstract elements provided by geometry, particularly motion indivisibles or conatus. In 1676, following his stay in Paris, to these Leibniz adds the laws of physical mechanics proper, such as the laws of impact proposed by Wren and Huygens. These laws formalize the sensible data of motion in the phenomenal world in a more direct way. At the same time however, Leibniz does not yet question the possibility of reducing the empirical laws of impact to a mathematical system by algebraically summing conatus. In any event, the chain of sufficient reasons can be made to demonstratively explain specific phenomena, provided that the required empirical surveys and verifications have been carried out. The phenomena that are consequently derived from the principles of scientific knowledge undoubtedly help us to uncover the order in which phenomena are combined. In this way, one can discover “what the nature of the elements consists in and by what combinations everything is formed”. (A II 12 460) Preoccupied with the logic of discovery, in 1676 Leibniz opposes the strategy – adopted by Descartes among others – of constructing arbitrary hypotheses. He proposes instead an approach that mirrors the practice of mathematical analysis. This involves taking inventory of the available data to find the solution to problems and designing experiments to demonstratively make sense of phenomena, the goal being certainty rather than mere probability. Leibniz’s views in 1676 are the result of his growing confidence in applying the analytical methodology characteristic of mathematics, this methodology having been the focus of his scientific reflections since his stay in Paris. This is shown by the following passage, which captures the essence of the project: From these few elements, I think one can appreciate the open field that lays before us, where philosophy can be conducted with exactness unaided by hypotheses. This is true, at least, if from now on experiments (that have been diligently documented, and which analysis demands that we undertake) are connected with geometrical elements and mechanical laws; and I do not doubt that with due caution one can demonstrate something certain regarding the system of things and the basic motions that occur about us. Afterwards, we shall derive the explanation of various phenomena from particular things, and our dominion over nature will be secured. (A II 12 460)

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But the same methodological trend had been sketched in the Hypothesis. The phenomenalist and analytic method that Leibniz adopts there is undoubtedly what distinguishes the Hypothesis from the work of Descartes the most. Indeed, Leibniz will become increasingly critical of Cartesian methodology and physics in the period during which he first dedicates himself to reforming the mechanics (1676–1686). That being said, one must appreciate the originality of the methodological advancements made in the 1671 project. If Hobbes’ influence on the conatus theory is evident, the blueprint laid out in the Hypothesis for physics represents a revised empiricist strategy, better structured and more capable of constructing demonstrative chains of sufficient reasons. In his letter to Hobbes on 13/23 July 1670, (A II 12 90–94) Leibniz himself had expressed his reservations about the Hobbesian theory of cohesion, and had opposed reducing the causality of motion to purely material reasons. Understanding conspiring motions as the source of discrete masses in extension, but also of the global economy of phenomena, exceeded Hobbes’ theoretical conceptions. In the Hypothesis, Leibniz particularly focused on these requisites for grounding his empirico-analytical models. We shall now attempt to describe and analyze this approach. While refusing any preliminary recourse to a corpuscular theory, Leibniz postulates a minimum of prerequisites in order to conceive of how physical phenomena are generated by different types of motions in homogenous space. With regard to our planetary system, a simple model is put forth, comprising the Sun, the Earth and the “mass at rest” that ether constitutes as a fluid occupying the intermediary space. (Prop. 1, A VI 2 223) The parts of the Sun and the Earth rotate around the axis of their respective spheres. These celestial bodies are produced by the differentiation of the ether, which is caused by various circular motions combining together. Conatus within the ether would completely permeate everything were it not for the sui generis motive inclination that causes celestial bodies to dissociate themselves from the ether. Apart from the gyratory motion of solar particles caused by axial rotation, one must suppose a specific motion inhering in these parts that produces the emission of light and the resulting mechanical effects. Torricelli and Hobbes wrongly attributed these effects to gyration itself: were this the case, then projection along the tangent would exhaust the solar matter, thus preventing the particles from adequately differentiating themselves. This is why Leibniz imagines a great number of circular motions within the Sun: depending on the angle at which these moving parts collide, the resulting trajectories would follow the external bisectors. (Prop. 7, A VI 2 225) Motions in ether are modified accordingly, creating a phenomenon that corresponds to the emission of light. Based on our visual experience of celestial bodies, the possibility of transmitting light can be attributed to every perceivable point, and one can thus identify a fluid matter within the heavens where light conatus expand unimpeded. Owing to the analytic implications of the notion of plenum, one must admit that all of these circular motions correlate with one another; however, one must also assume that the various motive determinations responsible for the differences among the masses in the diaphanous ether are functionally heterogenous.

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Leibniz no doubt shares with Descartes the idea of deriving the principal mechanical effects of the physical universe from the nature and behavior of light. But his theory deliberately excludes the corpuscular models that Descartes advanced in the Principia philosophiæ to account for cosmic phenomena. Leibniz supposes that the Earth’s motion is rotational, and that it is a sphere comprised of a matter with a density somewhere between that of the elements of air and earth, that is, a density like that of water. Light emissions impact the fluid mass and generate various important phenomena. Ex hypothesi, the Earth moves on its axis from west to east; being subject to the motion of light, a subtle ether places pressure on the globe; the motion of this pressure is westward, and as such, we can interpret it as a real motion if we believe that the Earth is motionless on its axis. Such is the universal motion from which all explanations of phenomena can be drawn. (Prop. 10, A VI 2 226) Hannequin is right to point out that Leibniz proposes a hypothesis for the formation of the terrestrial globe, when he writes in the letter to Fabri: “Fluids surrounded by heterogenous fluids join together to form a round droplet.” (A II 12 453) Owing to the various internal motions of fluids, when heterogenous fluids converge, the overall equilibrium is disturbed. The densest fluid naturally acquires a volume and shape that offers the greatest resistance to external impacts. Hannequin does not perceive anything essential in this addition, which would fail to offer any sufficient reason for the functional heterogeneity of fluids and the globular form that results therefrom.63 This critique misses the mark. After all, Leibniz attempts to account for the structural conditions of gravity when it exerts its effect at every point of the sphere. If one considers axial rotation alone, the conatus of light emissions should exert pressure as if it were acting on a cylindrical surface rather than a spherical one. This problem, which can equally be found in Descartes’ vortex theory, is in Leibniz’s view adequately solved via the bubble theory, which forms a central component of the 1671 physical theory. The letter to Fabri explicates the argument that justifies this theory, which likewise occupies a central role in the Hypothesis. The bubble theory is grounded on empirical analogies and analytic requirements derived from the conatus theory. The phenomena that inspire it are boiling and fusion, whether caused by heat or the mechanical action of light emissions. The conatus model implies that the spherical structure of the parts of the densest fluids be attributed to conspiring inner motions. Subtle fluids that penetrate the dense structures increase the endogenous resistance that their “hard crust” offers to external fluids. Glass blowing suggests an experimental model for grasping this mechanism: the glass bubble results from the circular motion of the fire and the rectilinear motion of the

63

Cf. Hannequin (1908), II, 111.

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air blown through the pipe, which act simultaneously on the vitrescible fluid matter. “Similarly, from the circular motion of the Earth and the rectilinear motion of light, bubbles are created.” (Prop. 11, A VI 1 226) We can understand this analogy by inverting the hypotheses. Let us suppose that the Earth is immobile and composed of a vitrescible fluid acted upon by the ether, which gyrates circularly and is subject to impacts of light perpendicular to the axial diameter. Leibniz seems to have acknowledged the necessity of attributing a spherical shape to it. This attribute would depend on the motions that, within a mass of fluid parts, would be occasioned by the multidirectional impacts of the surrounding fluid. The hollowing out of these bubbles, in addition to the increased likelihood that their endogenous conatus react, would itself result from a combination of circular and rectilinear motions. In any case, one must appreciate the composite nature of the material elements here, since heterogeneous conatus produce the bubble structure. Leibniz discovers the key to theorizing about physics in the notion of the bubble: These bubbles are the seeds of things, the fabric of species, the containers of ether, the cause of consistency, the foundation of the great variety we observe in things, and the source of an impetus so great that we find it in motions. If these bubbles were absent, then everything would be like sand without lime, and the ether, expelled by the gyration of dense bodies, would fly off, leaving our Earth dead and condemned. However, owing to the bubbles reinforced by the gyrating motion around their centers, everything is solid and maintains its form. Spherical objects benefit from an admirable solidity; hence balls of glass demonstrate greater resistance in elasticity experiments, while objects endowed with every other form will sooner break. (Prop. 12, A VI 2 226)

If we suppose that the original state of the Earth is fluid, then owing to the axial rotation that guarantees its cohesion, the terrestrial particles will each be subject to a dominant confluent motion along the equator, which is reproduced at each successive parallel on the globe. The cohesion of the parts situated on the same parallel is facilitated by these confluent rotative motions. The cohesion between parallels will be weak, if not non-existent. Hence, the ether particles of light emissions penetrate the sphere between the parallels along the lines of inferior resistance and combine with the parts of the generic terrestrial fluid to form bubbles. As a result, a number of individualized elements of various sizes and densities comprise the successive layers of the sphere, there being an infinite gradation in the materials that constitute it. The Leibnizian bubble has nothing in common with the Democritean atom. For in the first place, the bubble can be transformed in a number of ways by surrounding motions; and secondly, bubbles contain infinite and immeasurable series of progressively smaller bubbles within themselves. As such, one can represent bubbles as individualized and complex elements determined by their nested structures. Interestingly enough, Leibniz associates this concept with the observations of microscopic organizations compiled by micrographs like Athanasius Kircher and Robert Hooke:

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One must appreciate, as the illustrious microscopists Kircher and Hooke have observed,64 that most things we perceive via our senses in larger bodies could be apprehended in smaller ones – on a proportionate scale – by a subject endowed with especially acute vision. If one proceeds in this way to infinity, which is certainly possible since what is continuous is infinitely divisible, every atom would be a world with infinite species, and one would find worlds within worlds ad infinitum. (Prop. 43, A VI 2 241)

To be sure, in the correspondence with Johann Friedrich of Hanover, this theory of nested bubbles representing worlds within worlds will provide a material basis for drawing an analogy between the circle of influence of the mens and the points of convergence of motions in the phenomenal world.65 From the perspective of constructing scientific models, the characteristic feature of this theory is the coherence of its explanatory system, which is restricted to four elements: ether, air, water and earth. Owing to the geometric properties of conatus, ether is the cause of the functional diversification of the “masses” of bubbles forming the other elements. Leibniz discards the hypothesis of an even more subtle ether that would constitute a fifth element. (Prop. 29, A VI 2 243) On the one hand, no phenomenon suggests the need to appeal to such a postulate; on the other, the structural conditions of the physical universe require a type of element that can generate gravity and other analogous properties of bodies without relying on properties similar to the ones it explains. Indeed, the structure of the ether and its action are the outcome of an abstract theoretical model. If the latter is coherent and suffices to explain phenomenal effects, then why adopt additional hypotheses in contradiction to Occam’s principle? To assess the fruitfulness of this model, it will be appropriate to focus on the masses of the other elements and the infinite variety of their components, as well as on how ether acts on the material parts that constitute bubbles. In line with this theory, Leibniz attempts to develop a model to explain the phenomenon of weight (gravitas) and analyze its effects. The model would be a blueprint for constructing a system of phenomenal representations capable of giving more detailed explanations. (Prop. 15, A VI 2 227) The proposed model requires that hypotheses concerning motion and rest apply equivalently to the Earth and ether. (Prop. 2, A VI 2 223) Let us once more suppose that the Earth is immobile and surrounded by an ocean of ether moving from east to west. Not only does this ocean surround the globe, but through an infinite network of conduits, it also exerts its motive action within the masses and the bubbles that comprise these masses. Any heterogeneous body that obstructs the free circulation of the subtle ether particles causes the following reaction: the subtle particles of the ether push toward the center of the Earth any body or bodily 64 65

Cf. A. Kircher, Scrutinium physico-medicum contagiosæ luis, qua dicitur pestis (1659); R. Hooke, Mi­ crographia or Philosophical Description of Minute Bodies (1665). Cf. Letter to Johann Friedrich, 21 May 1671, A II 12 174.

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part whose mass cannot be dissociated or dispersed by attaining a degree of subtlety akin to that of ether. The centripetal motion of heavy bodies seems to depend on the spherical surface of the ether layer that they disturb. Leibniz thus proposes a false but logical law according to which the increase in gravity is proportional to two times the diameter of the ether’s sphere of circulation. Moreover, this law seems to account for the accumulation of successive impacts in falling bodies: As the ether disturbs their circulation, they are expelled, not upwards, for then they would create an ever greater disturbance (because the ratio of the expansion of the superficial surfaces to the diameters is 2:1 and not merely 1:1, resulting in a greater inequality between the sections acting on the same body), but instead downwards, meaning they descend. Thus, in addition, there is an increase in impetus owing to the renewed impressions throughout the descent in any part of the ether, which is free or freer than what the ratio of this thing to it implies. (Prop. 18, A VI 2 228)

As such, the rate of speed decreases in function of the distance traversed by the falling body. But the effect is perfectly consistent with the notions of the Theoria motus, since the effect of mass is ignored and one only takes into account the conatus, which figure into the algebraic addition of the impact speeds, but decrease in proportion to two times the diameters. Gravity measures the effort of ether required to restore the falling body’s motion when it is disturbed by bodies of various densities. The letter to Fabri will stipulate that the same cause – the combined motions of the inner parts being impacted by the surrounding ones – is responsible for the spherical shape of fluids contained in less dense media. (Prop. 10, A II 12 454) The explanation for elasticity follows a similar course, this being a general phenomenon caused by the disruptive and expansive action of the ether when it reduces the heterogeneous material parts within its circular flow to a degree of subtlety akin to its own.66 Here, Leibniz sets out to analyze phenomena from a physical perspective. For instance, he relates the physical laws of reflection and refraction – whether applied to light or mechanical effects – to elasticity, which implies a flow of ether into more or less cohesive bodies. Wren’s and Huygens’ laws of impact, which the Theoria mo­ tus abstracti could not account for, seem to Leibniz compatible with the fundamental mechanical thesis once the law of elastic action and reaction is established. (Prop. 23, A VI 2 231) In the same way, one can theoretically represent the causes behind the oscillatory motions of pendula and their isochronism, the catoptrical (reflection) and dioptrical (refraction) effects, and an infinity of more specific physical effects, such as the accelerated and decelerated motions of projectiles in the atmosphere. Air is in effect the basis of elastic motion in the sensible world. It is important to point out the

66

For Leibniz, the heaviest bodies never lack elasticity, and the most elastic bodies, which approach the subtly of ether, continue to a degree to descend toward the center of the globe.

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discontinuous structure of material parts, which alone justifies the effects of mass and the reaction of bodies to the impulsions of the parts of ether. Discontinuity is achieved by the conspiring gyratory motions of bubbles, which represent the “hard” elements in the physical world; and reactivity is a function of the dispersive power of the ether particles that disturb endogenous gyration. All bodies perceived by the senses are hard owing to an inner self-returning motion; all are discontinuous; hence, everything else being equal, the effect of mass is greater; all are elastic, that is to say, if compressed and then left to themselves, they return to their previous state owing to the gyration of ether. (Prop. 22, A VI 2 230)

With these basic theoretical concepts in hand, Leibniz intends to couple fluid statics with a coherent representation of the elementary structure of bodies interacting in virtue of gravity and elasticity. (Prop. 23–29, A VI 2 231–35) This system would then be extended to various categories of phenomena. These can be analyzed as featuring asso­ ciations of sensory qualities, which refer to subtle but imperceptible motions. Leib­ niz’s goal however is to link these phenomena to gravity and elasticity by constructing a general model that goes beyond sensible appearances and connects them with modes of motion. This particularly applies to so-called sympathetic and antipathetic motions, that is, to magnetism and attractions on the one hand, and to chemical reactions on the other. The mechanism behind magnetic attraction is explained by a type of gravity. (Prop. 33– 34, A VI 2 237–39) The pressure of ether on the parts of the globe repels solid corpuscles towards the center of the Earth. But owing to the globe’s axial rotation, when the centripetal motion of these corpuscles is impeded by the impact with denser parts, “magnetic” matter is deflected toward the poles along the meridian lines. In short, in the same way that the letter to Fabri will clarify the nature of the spherical consolidation of denser fluids, (Prop. 6–8, A II 12 453) magnetic declination occurs through the convergence of centripetal conatus, which have a drop-like structure, and gyrating conatus along the parallel lines. Following the same logic, magnetism is explained by the porosity of iron and the flow of ether particles along meridian lines. Similarly, electrical attraction is understood in terms of the dynamics of exchanges between fluid bodies striving to achieve equilibrium within the circulation of the ether. The general model conceived of by Leibniz not only combines electromagnetic and gravitational effects, which is interesting from a programmatic perspective, but also relies on the disruptive flow of ether to understand chemical phenomena in terms of elasticity. An infinite variety of effects is derived from elasticity as a result of the different relations between the containers and their contents, and ultimately between the bubbles impacted by ether particles and their endogenous motions. A framework for drawing all the different chemical effects under the influence of the analytic concepts of the geometry of conatus is thus established. Nothing is more significant in this regard than article 46 of the Hypothesis. There, Leibniz justifies his model in virtue of its

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general ability to reconcile the abstract laws of motion with an integrated economy of natural processes: In order that the reader, who might be struck by a seeming contradiction, does not instantly reject the harmony altogether, since most of the time, as I showed in the case of motion, experiences are clearly at odds with the inner principles of things at first and cannot be made to agree with them save by the manifold artifice of an all-encompassing economy, the admirable wisdom of the Creator englobing the genesis of things, one must briefly show a priori that our hypothesis is a little more than a hypothesis. For, without bubbles and containers, the most subtle corpuscles cannot be retained. Two general types of bodies are thus needed: containers and contents, that is very small ones (and I would not deny that some escape from bubbles, even though they may themselves consist in smaller bubbles), which are solids and liquids, bubbles and masses. (Prop. 46, A VI 2 242–43)

The different types of masses, or principles of composition, correspond to the primary elements – ether, air, water and earth – considered purely in terms of extensive quantity. The self-returning motions within these masses produce the spherical structures that are modified and thereby acquire their dynamical properties when impacted by ether particles.67 Elements of mass possessing a bubble structure thus define the content of these “vases” or “containers” depending on the motion and density of the surrounding external fluids. The relation of inner motion and density to external motion and density will be “ordinary” when the effects compensate for one another; this state tends to persist, restoring itself when disturbed. However, major changes can occur, destroying the equilibrium between the mechanical actions and reactions that were previously in the range of low amplitudes. When conatus in the surrounding fluids encounter those in the parts of the bubbles, sudden changes in the gyrations of the ether occur. The phenomenal effect that inspires this explanatory analogy is the permutation of the states of density and rarefaction. Specifically, Leibniz attempts to incorporate various concepts from chemistry into such a model, including: the Ancients’ notions of “deflagration” and fermentation, notions of sulfur and niter inherited from Paracelsus, as well as those implied in the various principles invoked by the Moderns, such as acids and alkalis. In order to advance a physical model of the agents and processes of chemical reaction, attempts such as Boyle’s, which were simultaneously empiricist and mechanist, seem to have developed along lines akin to those of Leibniz’s program, as the reference to the English natural philosopher’s Sceptical Chemist (1665) attests. (Prop. 38, A VI 2 240) Leibniz suggests that we attempt to understand the system of reactions by conceiving of a general mech-

67

Hannequin (1908), II, 127, n.3, notes that, in this case, Leibniz’s statements are regarding a “system” conceived of so as to represent the composition of the elementary structures and the dynamic equilibria that bubbles tend to produce.

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anism. After invoking density and rarefaction to account for motions that establish equilibrium, Leibniz concludes: The same thing can be said about all other reactions, the diversity of which can be found in their origins, being solely attributable to the size and number of bubbles, their position and shape, and their degree of rarefaction and compression. (Prop. 40, A VI 2 241)

Rather extensive developments in the Hypothesis on the variety of modifications undergone by bubbles generate a table of possible combinations in which the system of elementary properties is represented.

Fig. 2 Table of a combination of the physical properties of bubbles and the phenomenal equivalents of these properties in Hypothesis physica nova (GP IV 205)

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This is a significant attempt at establishing a theoretical key for deciphering the specific processes that experience alone allows us to grasp. This theoretical framework conforms to an architectonic principle, which implies that the diversity of effects be generated by a significantly limited system of causes, factors, or elements functioning as general sufficient reasons. Indeed, such a fiction must be justified a posteriori via empirical analysis. But would the experience of phenomena not remain incomprehensible without this type of invention, which is consistent with the general analogy of facts? This hypothetical table yields a possible schematic representation framed by an architectonic idea: Whoever frames his hypothesis here must no doubt give pause, lest he be too reckless. For a more precise application depends on experience. I always believed that the admirable wisdom of the Creator had organized things such that few principles would govern many. Thus, If I dreamed, I would say […]. (Prop. 51, A VI 2 244)

The resulting framework is a table of chemical elements conceived of as consistent with the general physical hypothesis. And this was undoubtedly the programmatic objective of Leibniz’s approach. Was he not essentially interested in conceiving of a set of articulate concepts that would connect physical theory with experience by reconciling the methodologically acceptable principles furnished by partially antagonist conceptions? The framework arrived at thereby would surely permit others to progressively expand the scope of scientific knowledge and its application: It suffices to have advanced a sufficient cause for explaining all motions: taking the simplest, most evident and comprehensible principles as one’s starting point, it suffices to have deduced high-flown theories down to the level of experience, theories that otherwise would have been alien to life and practical analysis; it suffices to have brought to the fore what all sects can admit regardless of their particular opinions. (Prop. 54, A VI 2 246)

It is in the critique of corpuscular physics that Leibniz’s framework particularly reveals its “theoretical functionality”. That being said, Leibniz certainly does not question the maxim of the Moderns, according to which all physical phenomena must be mechanically explained. In fact, he agrees that explaining phenomena via geometrico-mechanical reasons is perfectly consistent with building “machines” that bring the laws of motion into play in cases involving specific material structures, for example clocks constructed with the properties of gravity and elasticity in mind. (Prop. 58, A VI 2 249–50) But the uniqueness of the Leibnizian model lies in its capacity to produce sufficient theoretical reasons for the various mechanisms. There are in fact three possible reasons for which a body’s power will increase, for instance, so that less force will be required to lift weights: (1) the distance of the bodies from the Earth’s center of gravity, (2) the impetus produced by falling, and finally (3) the nisus that certain types of things possess owing to their inner arrangement. In the last case, this inner arrangement determines encounters of motions that produce dynamical effects; such is

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the case, for example, with the explosive force produced by gunpowder or the motive force of animals’ muscles. Reasons (1) and (2) are directly explained by the concepts of weight and elasticity, while (3) should be explicable via a more indirect analysis that focuses on the complex interplay of “microsystems”. According to the Leibnizian approach however, one tends to ignore Democritean properties like antitypy or primary density, or the elementary shape of such and such solid particles or atoms. Vacuum is also ignored in order to establish a relative ratio between density (densum) and rarity (exhaustum). This ratio is solely derived from the original principle that motion is able to produce differentiated, discontinuous structures. Leibniz thus builds on the perfect geometrical consistency of his basic hypothesis, which avoids postulating physical properties that cannot be analyzed in terms of conatus. At the same time however, justifying the adopted theoretical concepts hinges on the critique of corpuscular theory, which is incapable of “ultimately” accounting for empirical properties. Leibniz mentions the case of Boyle, who imagined helixes in order to explain why the elastic force of air is greater than that of water. It is easy to discard Boyle’s model68 if one postulates that the greater agitation of air bubbles owes to the dispersive impacts of ether particles, which produce gyratory motions that make it more difficult for air to penetrate narrow tubes than water particles; denser and less compressible, the latter have greater difficulty restoring themselves to the specific form of their “mass” in its cohesive disposition. Leibniz thus demonstrates that he possesses a model that Boyle would have lacked for studying capillarity in narrow tubes. The followers of Descartes and Gassendi, and this equally applies to Boyle, thus have a false conception of density as the cause of cohesion. This conception led them to imagine shapes specific to the elementary corpuscles, whether these were understood as atoms or not. For their part, the Cartesians found themselves obliged to adopt a hypothesis whereby the categories of elements are determined by size and velocity.69 For Leibniz, the supposition of these differently shaped corpuscles – just like that of the vortex motions of solid particles of various sizes – is tantamount to inventing pure fictions deprived of any explanatory power, given their inadmissibility in a coherent and systematic analysis of phenomena. On the contrary, one must attempt to combine the resources of a uniform and coherent geometrico-mechanical model that is applicable to all phenomena, with specific inferences based on observations of those phenomena that seem to express the inner structure of bodies (bubbles for instance). Formulated in this way, the Leibnizian hypothesis should not fail to grasp the sufficient reason behind natural realities and processes. On the one hand, a rule of simplicity must be respected when formulating postulates. On the other, a network of explanations modeled after specific phenomena must be furnished without recurring to

68 69

Cf. R. Boyle, Nova experimenta physico-mechanica de vi aeris elastica et eiusdem effectibus (1669). Cf. Descartes, Principia philosophiæ, III, § 46–52, AT VIII-1, 100–105.

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superfluous ad hoc hypotheses. The contrast between the corpuscular methodologies and those of Leibniz is clearly delineated: I always believed that what is said about variously shaped atoms, vortices, branchings, hooks, globules, and other such configurations resembles a mind game more than nature’s simplicity and specifically our experiences, this game being so sterile that it manifestly has no relation to phenomena. The present hypothesis, on the other hand, unites the diffuse and unstable corpuscles by means of bubbles; it deduces the motions and effects of bubbles, and ultimately of all species, solely from the universal motion of the universe’s system; and thus having its origin in primary and abstract principles, being grounded in the most essential experiments of chemists, and explaining the very simple phenomena of gravity and elasticity with a view to our globe in its entirety, it unites mechanical theories and observations in a superbly clear and harmonious way. (Prop. 57, A VI 2 248)

Ultimately, what can one infer from the empirico-scientific model of theorization that Leibniz lays out in 1671 in the Hypothesis physica nova? Indeed, the Theoria motus ab­ stracti and the first iterations of the conatus theory that it presents will be superseded when the mechanics is reformed and dynamics emerges as a science. Furthermore, as our analysis set out to show, the Hypothesis fulfilled an essentially programmatic role, suggesting only a handful of prescriptive ideas for constructing a system of scientific explanations from an indefinite inventory of phenomena. The specific models that Leibniz had proposed in connection with the bubble analogy, for instance, in no way constituted a real improvement with respect to the models of corpuscular philosophy, which he had rejected because of their sterile ad hoc nature. In our view, the Hypothesis was nonetheless worth reexamining to the extent that it embodies an interesting idea: scientific theorization depends on an analytic process, which seeks to systematize the inventory of specific phenomena by combining models. These unified models must be formulated and progressively revised so as to satisfy architectonic norms. Such norms demonstrate the necessity of synchronizing abstract mathematical or quasi-mathematical explanatory models with the goal of creating a systematic network that would connect the various types of phenomena. From the perspective of the Hypothesis physica nova, observational and experimental data were to be regarded as more or less provisional expressions of potential conceptual connections that reflect a complex system of sufficient reasons. 4. Conclusion The initial scientific project, realized in 1671 in the Theoria motus abstracti and Hypo­ thesis physica nova, met with several important problems. Leibniz himself will recognize these, and contrast them with his subsequent theories to emphasize the latter’s scope and originality. At this first stage of his career, Leibniz had relied on particularly

Conclusion

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insufficient mathematical models. If he discovered Cavalieri’s geometry of indivisibles, then his interpretation deviated substantially from it, being much more akin to that of Hobbes. It is during his stay in Paris from 1672 to 1676 that Leibniz, under the influence and direction of Huygens, will truly discover the mathematics of the Moderns; soon thereafter, he will lay the foundations for his infinitesimal calculus. Regarding the natural sciences, if Leibniz had closely adhered to the mechanism of the Moderns, then he did not fail to explore different methodological avenues by attempting, for better or worse, to synthesize them. Two dominant trends clearly had a more direct influence on him. On the one hand, there is the experimental philosophy, inspired by Bacon, which employed empirical methodologies such as Boyle’s. This explains several of Leibniz’s inquiries into the analysis of phenomena and the combinatorial models applicable to them. On the other hand, Leibniz is impressed by the demonstrative reconstructions whereby the abstract elements of a system of nature could be mobilized more geometrico. In this respect, Hobbes’ De corpore (1655) no doubt serves as a model for Leibniz’s first physics. Nonetheless, without fully adhering to the philoso­ phia reformata project and its goal of reconciling Aristotelean physics with modern mechanism, Leibniz focuses on the limits of mechanist conceptions when it comes to explaining the formal dimension of physical realities and translating, via analysis, their ratio essendi. At this relatively indeterminate stage, Leibniz outlines a unique conception of the process for constructing theoretical models. This imperfect and in certain respects paradoxical conception is an indispensable historical key for understanding how Leibniz studies the scientific method, and applies it to mechanical theory. Leibniz’s first philosophy, which is inspired by mechanism, evolves to the extent that motion is increasingly grasped both as the causal source of various phenomena, and as a substantial modality that cannot essentially be reduced to extension and its modes; hence the necessity of relying on a concept of mens as the foundation of the dynamical and causal element that gets expressed in and through the alterations of bodily realities. We have highlighted the fact that these reflections are accompanied by epistemological analyses. The Nova methodus discendæ docendæque jurisprudentiæ (1668) advances a three-tiered structure of scientific explanation that is (1) historical, which involves the particular description of phenomena; (2) observational, which focuses on inductive generalizations and the formulation of both contingent and general laws; and (3) theorematic, consisting of universal propositions and expressing the demonstrative structure required for deducing the reasons behind phenomena. If in physics the meaning of basic terms refers to the sensory or reflexive experience of qualities, then some of these “empirical” qualities yield combinatorial models, which help us to analyze phenomenal realities into structural and causal determinative reasons. A prime example is the search for an explanation of the solidity of bodies. If Leibniz adopts Hobbes’ principle of inertia and his notion of conatus as the tendency to motion, then he goes a step further by attempting to connect the cohesion of bodies with the idea

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of constantly opposing conatus; we are thereby obliged, as a last resort, to accept a not entirely phenomenal foundation for the system of actions and reactions. This analysis goes hand in hand with overcoming Hobbes’ “hyper-nominalism” when it comes to constructing theoretical models. Principles of inference that can presumably be derived from the reflexive experience of the mind make it possible to overcome inductive generalizations by codifying, in a combinatorial fashion, the empirical facts and by postulating anticipatory explanatory reasons that nominally represent the phenomenal world; the convergence between such nominal expressions is cited as evidence to establish their relation to a real foundation. In his correspondence with Jacob Thomasius in 1669 and in the Preface to Nizolius (1670), Leibniz shows what dependance on the mechanistic principles can mean from an epistemological perspective. Theoretical concepts must strictly correlate with simple terms that justify the geometrization of phenomena. But an adequate representation of the causality that underpins the constellation of motions assuring the cohesion and regular interaction of bodies is often referenced in connection with the analogy of the mens to subjects that transcend the physical world. When Leibniz publishes the Theoria motus abstracti (1671), Huygens, Wallis and Wren had just formulated the empirical laws of impact, which contradict the abstract laws defined by Descartes in his Principia philosophiæ (1644). Henceforth, it seemed that only geometric models conforming to experience would be acceptable. But Leibniz seems to reject this tendency. He intends to advance a phoronomia elementalis that, drawing on a priori definitions of abstract concepts, could produce a demonstrative mechanics by taking a combinatorial approach. By adopting such a geometric model, the physical model will then be invoked to substitute the indeterminate speculations of experimental mechanics with exact hypotheses. Leibnizian theory is based on the notion of conatus inherited from Hobbes, which Leibniz interprets in light of Cavalieri’s model of indivisibles. But there is a significant difference: Leibniz conceives of points as inextensive but including indistant parts; conatus is an indivisible of motion that can also contain a plurality of determinations within an instant; hence, there is a possibility of combining these determinations in certain cases, i. e., the possibility of embryonic series of determinations, if a conatus is at least connected with a mens. This definition dovetails with axioms such as inertia, but also the radical difference between motion and rest: in effect, bodies would not possess any intrinsic factor that resists motion. Regarding the effect of mass, this can be reduced solely to the instantaneous opposition of conatus to one another. From this system would follow a series of theories on the combination of conatus involved in collisions between bodies. Conatus are susceptible to algebraic addition and subtraction. Indeed, in the case of conatus that join together, their determinations coexist without being negated. In the case of conatus that do not join together, the motive effect is less than that of the initial conatus, which get subtracted. The model cannot however account for the determining factors of circular motion, nor the processes of acceleration and deceleration. In these two

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cases, another argument is required to ensure the temporal continuation of sequences whose raison d’être is not present in the instantaneous conatus, but requires their progressive integration under the same “form”, which yields a temporal law. Even the relative stability of bodies would require a structure for combining endogenous cona­ tus, one that would endure despite the impacts of exogenous conatus. This points to a sort of compensatory metaphysical strategy that Leibniz will mobilize to maintain his geometric model. But one must appreciate that this strategy is substituted by various auxiliary postulates when it comes to physics. To begin with, these include postulates like that of an extensive plenum, which accounts for the fundamental role played by the determining factors of circular motion, expressed through infinite series of tangentially orientated impacts. It is therefore necessary to admit indivisibles that involve potential effects and intervene in the physical world through connected series of states; as a result, minds (mentes) tend to serve as the “physical” foundation for the laws ruling over these states. One is already a step closer to postulating points of force that are intrinsically discernable by their serial physical effects. To account for cohesion and elasticity, Leibniz also finds himself obliged to develop a theory of ether as an auxiliary postulate. Generally speaking, rather than relying on a purely metaphysical solution to reconcile phoronomia elementalis with a model of the real physical world, Leibniz turns to a system of postulates that could be reduced to a “single hypothesis” capable of explaining the effective order of this world. It is this compensatory strategy that the Hypothesis physica nova (1671) attempts to implement in order to integrate the elements of the physical corpus that the Theoria motus abstracti seemed to repudiate, such as: the Cartesian principle of conservation of the quantity of motion, the empirical laws of impact, the laws of angular reflection, and above all, the preservation of the relative cohesion of bodies and its influence on circular motions. In fact, Leibniz does not intend to tack ad hoc hypotheses onto the deductive model of the Theoria; rather, he hopes to conceive of an adequate supporting framework. Analyzing the simplest phenomena would provide a means to accurately account for more complex phenomena within a system of physical reasons, with the abstract conatus model helping to sufficiently understand how causes operate. As opposed to the Cartesian-type hypotheses that Leibniz regards as fictive, the working methodology must analyze phenomena by the determinative reasons that emerge at this very level, reasons that explain the occurrence of complex phenomena. The Hypothesis illustrates this type of empirico-analytic model. From the outset, Leibniz excludes corpuscular models from his system in order to understand the diversity of bodies in terms of the conspiring circular motions that interact and combine with one another in various ways. Numerous categories of fundamental phenomena – the transmission of light, the effects of gravity and magnetic forces, the elasticity of bodies – are the object of models designed according to the methodological demands of analyzing diverse micro-motions. To understand phenomena associated with specific material structures, Leibniz develops his theory of bubbles according to which internal, con-

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spiring motions create spherical formations of varying sizes and densities, certain ones contained within others, and capable of being variously modified by interactions. Thus conceived, these bubbles would generate and explain the different dynamical effects that observable phenomena produce. Furthermore, it would be possible to conceive of combinations of the elementary physical properties of such bubbles, and these combinations would serve as an analytic tool for explaining complex chemical or biological phenomena. It is quite remarkable that what emerges in this Leibnizian theory of nature is a systematic representation of the structural and functional micro-properties of the elementary bubbles. It is equally remarkable that Leibniz had envisioned studying, in a combinatorial fashion, the typology of the elementary structures of material reality so as to derive an explanation for all of the apparently heterogeneous phenomena. However, beyond the suggestive images that this model provides for representing material elementary structures and their power of action, an important epistemological fact must be noted. For Leibniz, formulating hypotheses is an integral part of constructing theoretical models; and these hypotheses must make it possible to advance analogical models that are both consistent with geometric algorithms and the combinatorial characteristics of the physical properties that they are intended to explain. The deficiencies of the Hypothesis physica nova in no way discredit the methodological scope of the model advanced there.

Chapter II Reforming Mechanics The methodological model of the Theoria motus abstracti and Hypothesis physica nova will be critically reevaluated as Leibniz progresses toward establishing his own dynamics in reaction to Cartesian mechanics. Tracing this evolution is complicated because one must follow two separate developmental trajectories that overlap in various ways. In the first place, developing the reformed mechanics, and then the dynamics, will require establishing theoretical foundations for the principles and laws entailed by this new doctrine. This research will comprise, on the one hand, an epistemological analysis of the processes involved in working out the concepts and demonstrative inferences required for founding the new science, and on the other hand, a metaphysical analysis that guarantees the compatibility of the theory with a system of nature that expresses its ratio essendi. In the second place, Leibniz will set out to implement the approach for establishing dynamics as a science that conforms to the revised methodology for the natural sciences. The vast empirico-analytical model of the first iteration of Leibnizian science will, as a result, be subject to major adjustments and improvements. In order to begin examining the theoretical foundations of the reformed mechanics and dynamics, the initial steps of this Leibnizian “invention” must be traced. The analyses of the past dedicated to understanding Leibnizian science came to a great impasse in this respect. The interpreters almost entirely ignored Leibniz’s works that preceded the first “public unveiling” of the new theory, which took the critical and allusive form of the Brevis demonstratio erroris memorabilis Cartesii (1686). Michel Fichant’s research in particular has since made such an omission inadmissible.1 The principal text to which the initial invention of dynamics as a reformed mechanics can be attributed is De corporum concursu (1678). Various analyses during the period of 1676–1678 prefigure this text. They will shed light on our interpretation of the reformatio; likewise, studying De corporum concursu will permit us to better define the context in which the Brevis demonstratio emerges.

1

Cf. Fichant (1974); Fichant (1990).

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1. The First Milestones of the Reformatio Our plan is to first locate the blueprint for the sort of questions that seem to have inspired the De corporum concursu project. To this end, we will rely on several of Fichant’s findings. But the principal objective of the study can be nothing less than discovering the method that underpins the reformatio of January 1678. From this perspective, the development of the reformed mechanics would appear as its own body of theory. At the beginning of 1678, it is quite manifest that Leibniz is confronted with a problem of magnitude regarding the laws of motion. The letter to Conring on January 3 attests to this: Concerning the laws of motion, I am embarrassed and still cannot explain what happens when two hard bodies of unequal size meet, for neither Descartes, nor Huygens satisfy me in this regard, despite surpassing all others on this type of issue. (A II 12 580)

A manuscript of the same month, cited by Fichant, corroborates this fact.2 Moreover, it suffices to return to certain testimonies of the preceding year to appreciate that this inquiry is bound up with what had been for Leibniz a major theoretical challenge since his stay in Paris (1676). A letter to Berthet in 1677 confirms this: When I was on my way back to Germany, I was testing my knowledge of motion. I am sure that Mr. Descartes’ laws, but also all those that have been published until now and which I am aware of, are partly false. I see a way of resolving this issue demonstratively, but it should first be necessary to perform certain fundamental experiments that I have planned. This is my way of preparing a catalogue of experiments to be undertaken whenever I examine some issue in physics. Usually, the experiments entailed by such a list guarantee the discovery of the cause or rule at play, allowing us to demonstrate it rather than merely form a hypothesis about it. (A II 12 572)

This text successfully highlights the relationship between the main elements of Leib­ niz’s reflections on the physics of motion when he is drafting De corporum concursu, which contains in particular the catalogue of fundamental experiments on which the demonstration of an adequate system of the laws of motion is founded. The “exercise

2

Cf. Fichant (1974), 195: “The difficulty raised by Leibniz here is addressed elsewhere in the manuscript LH XXXVII, 5, f. 193 – precisely dated January 1678: ‘It is believable that when two equal, hard bodies collide, what those gentlemen mentioned occurs. For the same force and resulting direction of motion are conserved, with each body taking the speed and direction of the other. With unequal bodies however the same rule does not apply.’ The rest of the manuscript shows Leibniz’s frustration with establishing a general rule to solve this problem, ‘which is quite intricate because of the diversity of cases.’ If the rule of the conservation of the sum of forces (represented by mv) is ‘infallible,’ Leibniz neither succeeds in bringing it into conformity with the rule of the conservation of motion of the center of gravity, nor with that of the conservation of the same distance between two bodies before and after their impact.”

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regarding motion” on which it focuses seems to correspond moreover to the Pacidius Philalethi (1676), where one finds Leibniz’s most fundamental critique of the possibility of conceiving of motion as an intelligible “form” of bodies that would provide a causal explanation of phenomena. The Pacidius Philalethi itself only develops upon the inquiries laid out in the Elementa philosophiæ arcanæ, de summa rerum, whose sections date from the end of his stay in Paris. In sum, before the beginning of 1676, it does not seem that Leibniz was too far removed from the epistemological presuppositions that, since 1671, had framed the Hypothesis physica nova and the abstract model of conatus at work in the Theoria motus abstracti. These are the theses that the Propositiones quædam physicæ of 1672 systematized and synthesized. Indeed, from the beginning, the stay in Paris proves itself an occasion for reconsidering the epistemological models on which Leibniz had previously relied. The gradual discovery of the techniques of infinitesimal calculus made Cavalieri’s model of indivisibles obsolete, which Leibniz had relied on in 1671 to conceive of conatus as an element of motion. Huygens’ mentorship guides Leib­ niz in the discovery of a neo-Cartesian form of physics, where geometric expressions are employed in order to rationally account for the facts of experience; however, the latter can only be assigned a value if their geometric conversion ensures that they can be integrated into demonstrative structures. Under the influence of Huygens, whose explanatory strategy develops analysis in physics relatively independently of metaphysical postulates, Leibniz can only hope to assemble a host of mechanical analytic models that lack any unifying hypothesis, such as the one he proposed in 1671. During this period of mentorship, the architectonic requirements for developing a substitute system of physics surely did not present themselves as a dominant concern. Even by the time he had begun working on the Hypothesis, Leibniz had noted in De rationibus motus (1669) the rules of motion that Huygens had presented to the Royal Society for publication in the Philosophical Transactions. (A VI 2 157–59) These laws, like those of Wren, had been excluded from the general theory of abstract motion, and reappeared only in the Hypothesis in the context where ether is presupposed. (Prop. 23, A VI 2 231– 32) However, during the stay in Paris, Huygens’ so to speak “positive” doctrine of the laws of impact could have only been a source of renewed interest for Leibniz. Moreover, and no doubt in connection with this, in 1675 Leibniz will read and annotate Wallis and Mariotte. The Traité de la percussion ou choc des corps (1673) by Mariotte, who built upon Huygens’ analyses, particularly attracted his attention; and during the same period (1675–1676), he undertakes a critical reading of Descartes’ Principia phi­ losophiæ. (A VI 2 213–17)3 One must note that at this stage, Leibniz’s thought had less to do with his commitment to the rational rules that underpin the empirical calculus of impact as conceived of by Huygens and Mariotte – Leibniz having distanced himself more completely from Wallis’ formulas – and more to do with his reasons for

3

Cf. Belaval (1976), 57–85.

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Reforming Mechanics

being dissatisfied with interpreting phenomena in this way. These reasons no doubt convinced him to reconsider the deductive system that Descartes had tried to establish, and which the experiments interpreted by Huygens and Mariotte radically rejected. With respect to analyzing phenomena, Leibniz can only agree with the methods employed by Huygens and Mariotte. But from the point of view of causally sufficient reasons, this type of analysis raises significant difficulties for physical theory, and Leib­ niz certainly could not content himself with the ungrounded presuppositions of the “positive” doctrine, even if experimentation seemed to confirm it. He therefore looks for a way of extending geometric representations to include the entirety of mechanics; in so doing, physical theory would be given a solid foundation. This also involves making sure that this model does not conflict with the rational interpretation of experience as conceived of by Huygens and Mariotte. By 1676, it becomes obvious that Leibniz thinks himself in possession of the right tool for constructing this theoretical model. As Fichant notes, this is suggested by his correspondence with Perrault at the beginning of 1676,4 and with Mariotte in July of that year;5 Leibniz also makes similar and even more precise statements in a letter to Oldenburg on 27 August 1676: I have resolved, as soon as I find the time, to reduce all of mechanics to pure geometry, to define problems that nobody has touched yet, concerning elasticity, fluids, pendula, projectiles, the resistance of solids, frictions, etc. I believe that now all this is in our power. Hence my satisfaction with the quite perfect demonstrations of the laws of motion, and with the fact that nothing is left to be desired here. But the whole matter, surprisingly, depends on a very beautiful axiom of metaphysics, which is of no less importance regarding motion than this one: with respect to magnitude, the whole is greater than the part. (A III 1 586)

It is easy to see how this fundamental axiom for modeling a physical theory more geo­ metrico is a unique variant of the principle of sufficient reason, it being a principle of equivalence between the entire effect and the full cause, whose emergence in the programmatic texts of 1676 is particularly clear. Several significant statements must be

4

5

Letter to Claude Perrault, first half of 1676, A II 12 418: “But one would need a long series of perfectly precise geometric arguments to rightly speak about this. I believe however that this is in our power, and that we have sufficient phenomena to deduce the true nature of the sublunary world by a demonstrative analysis (analyse nécessaire). A man who devotes himself to this task will discover that in the end all difficulties would most often be reduced to certain purely geometrical problems. For I am presently convinced that I can account for the laws of motion by geometrical demonstrations, without any reliance on suppositions, nor principles of experience, and that moving forward there will only be res calculi et geometriæ.” Cf. Fichant (1978), 228. Letter to Mariotte, July 1676, A II 12 423: “I even believe that there are natural effects whose ultimate cause can be found, and this is when a physical truth entirely depends on a metaphysical or geometrical truth, as it principally happens in my opinion regarding the laws of motion.”

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highlighted here. In Catena mirabilium demonstrationum de summa rerum, dated 2/12 December 1676, Leibniz invokes this particular expression of the principle of sufficient reason: Nothing is without cause, for nothing is without all the requisites for its existence. The entire effect is equivalent to the full cause, for there must be equality between the cause and the effect, which secures the transition from the one to the other. But it consists in this equivalence (æquipotentia), and no other measure could be found for it. (A VI 3 584)

According to Fichant, Leibniz was surely inspired here by Wallis, who, in his Mechani­ ca, had advanced the following proposition: “the effects are proportional to their adequate causes”. Wallis had underlined this fundamental premise as the linchpin of the passage from mathematics to physics,6 something that Leibniz had noted in terms that he himself will reproduce in the Pacidius Philalethi, where he underscores the strategic importance of securing the transition from geometry to physics.7 Indeed, the most noteworthy modification of Wallis’ formula concerns proportionality, which is replaced with equivalence by Leibniz. Undoubtedly, the transition from one to the other consists in passing from a geometric model of fixed proportions to a model inspired by the theory of algebraic functions, and by the recent discovery of the symbolic potential of mathematical analyses based on infinite series. Even if, contrary to what he said in his correspondences, Leibniz had not resolved the theoretical problems posed by the laws of motion by 1676, which had been proven empirically but remained irreconcilable with the principle of the conservation of quantity of motion, he nonetheless thought that he possessed the key to the solution, i. e., the principle of equivalence between the entire effect and the full cause. This principle receives remarkable clarification at this exact moment. If we refer to the fragment Principia mechanica, dated 1673–1676, we discover the blueprint so to speak for this clarification, which becomes apparent in De arcanis motus et mechanica ad puram geo­ metriam reducenda in the summer of 1676. In the first text, Leibniz maintains that mechanics as a science of motion must be founded on a small number of determined principles that offer sufficient explanations. These principles must serve to reduce mechanical problems to geometric interpretation alone. We cannot proceed inductively from the characteristics of phenomena, because then we would be unable to define the system of reasons for explaining them. 6

7

Cf. Fichant (1978), 229, who cites Wallis, Opera mathematica, I, Oxford, 1695, 584: “I also thought that this universal proposition should be put first, since it opens the way that leads from pure mathematical speculation to physics, or rather connects one with the other.” In his manuscript notes on Wallis’ Mechanica, Leibniz writes: “Effects are proportional to their adequate causes. He says that this proposition opens the passage from mathematics to physics.” (A VIII 2 65) A VI 3 531: “[…] I would have asserted that transitioning from Geometry to Physics is difficult and that we need a science of motion that would connect matter with forms, speculation with practice […].”

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By virtue of an analysis grounded in geometric principles, Leibniz hopes to surpass the statics of the Ancients and forge a path for solving the sort of problems that he lays out in a letter to Oldenburg on 27 August 1676, including: the laws of impact and colliding bodies, of resisting solids and media, and of the elasticity and flow of water. From this perspective, the intended goal is “to make perfectly intelligible mechanical principles or laws of motion”. (A VI 3 102) In order to do this, Leibniz shows that it is impossible to ignore the relativity of local motion if one wishes to reach to the efficient causes beyond the many possible representations of kinematic equivalents. By the end of this inquiry, which does not seem to say anything original with respect to Cartesian science, “it appears evident that from phenomenal situs changes alone, no scientific certainty will be achieved regarding absolute motion and rest.” (A VI 3 110) However, Leibniz considers substituting the concept of absolute motion, which in fact is nothing but an ens rationis, with the concept of a real element that would be abstractly derived from the analysis of relative motions, and would reflect the system of their respective relations. Indeed, as no kinematic proposition can be demonstrated with certainty, all hypotheses that we might formulate are in principle false. However, as Leibniz affirms, “where this is justified, we shall be obliged to choose a sufficiently simple means of explanation that entails considering a cause from which we can derive all the other changes easily enough.” (A VI 3 111) Leibniz’s strategy is therefore entirely consistent with the analysis of relative displacement. In order to understand the system of causal reasons behind the various events expressed by the laws of impact, one must conceive of a system of geometric conditions that represent the ordered constants of all the particular instances described. This idea is certainly related to that of a combinatorial representation of the conditions underpinning the different kinds of bodily collisions being analyzed. Taking into account the totality of these conditions would provide for an adequate analytic representation of all conceivable possibilities. One would therefore be dealing with a system for representing the situs changes that would be equal to the complete expression of causal factors. Should this combinatorial approach not be achieved, one must undoubtedly analyze the parameters relating to the antecedent and consequent phenomena with a view to discovering a form of equivalence between the causal state and the effective state. This analytic decomposition would be relatively consistent with the combinatorial norm. The final note of the Principia mechanica fragment seems to suggest an application for the principle of equivalence between the entire effect and full cause when studying the transmission of motion in impacts: When two bodies bounce back from one another after an impact, motion must be in one or both of them after the impact. One should then examine if, by introducing various conditions, something different does not occur following the impact. One assumes that if nothing distinct can be identified outside of the impact, then nothing distinct can be identified following it, for then different full causes could produce the same full effect. (A VI 3 111)

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The question evidently depends entirely on knowing how to sufficiently discriminate between the conditions in the antecedent and consequent phenomena, so as to be able to establish an equivalence between the combinatorial expressions of the cause and the effect. It is at this point that De arcanis motus et mechanica ad puram geometriam reducen­ da (February to September 1676) specifies the methodological approach that will be adopted, one that will define the elements of a true mechanical science. In Leibniz’s mind, this entails unifying under a single principle the various rules of motion conceived of to account for empirically described phenomena. Once this principle is discovered, it should be possible to analytically derive equations from it that suffice for representing all empirically describable events. In order to achieve this, Archimedes’ proposed model of statics is followed: a principle is defined whose role is analogous to the principle of the steepest descent of the common centers of gravity in the equilibrium between bodies. Thus, statics includes systems of equations by which consequent phenomena are analytically derived from antecedent phenomena. The new mechanics, inspired by Galileo, must overcome the obstacle of transitioning from a system of equations representing dead force (vis mortua) phenomena to an analogous system capable of representing causal sequences in constrained and non-instantaneous motions. This is what Leibniz means by impetus vivus et validus, (A VIII 2 134) which acts as a link between this type of action and phenomena of impact, acceleration, oscillation, and ballistic effects. He also mentions the exceptional contributions of Galileo, and then Huygens, to this new field of research.8 Leibniz reviews contemporary attempts at developing this system of equations and formulating a mechanics that would meet the demands of a unified science. As a starting point for developing a model, one can take the Galilean principle that all bodies acquire the same speed when they fall from the same height, regardless of the slope of the inclined plane on which they descend. As for the principles that might serve as the premises of the system, Leibniz seems to suggest reconciling two different types. The first are inherited from statics, such as the principle for finding barycentres. The second must have a specifically dynamical character, and include: the principle that the same force is required to lift objects whose weight is inversely proportional to their height of lifting, and the principle according to which the size and speed of interacting bodies must compensate for one another. Ambiguity results from the fact that speed must be measured with respect to vectorial orientation, which is a relative parameter from the point of view of kinematics, while the quantity of mv conserved in equations must, at the same time, represent an actual efficient cause whose nature is absolute: “that the 8

Leibniz professes to have made a fundamental contribution to Galilean sciences by extending the principle of the composition of motion beyond abstract scenarios of equilibrium, and by accounting for the complex generation of dynamic effects; the latter supposes the seamless integration of the “ingredients” of motion into physical nature.

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efficient cause of phenomena must truly consist in the size of bodies and their speed, that was easy to determine.” (A VIII 2 134) In this regard, Leibniz exposes the gap that emerges between Descartes and the champions of the empirical laws of motion. For the latter, the principle of mechanical equivalence merely appears to be a hypothesis that is inductively justified by successful experiments; hence the risk that the explanation for this hypothesis does not accord with the real causal order, given that it does not rationally ground the laws whereby it explains phenomena. In the case of Cartesian laws, the problem is the inadequacy of the hypothesis to explain the empirical motions of colliding bodies. This suggests that, even if the principle of conservation referred to by Descartes could appear abstractly justified, the concepts at play have not adequately been explained in analytic terms. Here, the ratiocinandi filum required for developing concepts that represent efficient causality in a system of analytic equations is missing.9 Significantly, Leibniz praises contemporary authors, including Huygens and Mariotte no doubt, for having constructed workable systems for representing efficient causality by grounding them in a coherent analysis of the phenomena. In sum, it would suffice to complement these systems with a theoretical grounding. The conditions of this new approach are clear: establish the principles of mechanics, and derive theories that analytically account for the empirical laws of impact already systematized by Huygens and his successors. Leibniz thus states: In our times, men have become quite occupied with experiments, from which they have derived not merely a few observations. They surely could have inferred these, if, having established a general and true principle, they had treated all other propositions via geo­ metric arguments. Achieving this task more distinctly and advancing science by establishing new theorems before undertaking experiments are indeed worthwhile objectives. (A VIII 2 135)

What is this line of reasoning that is absent in Descartes, despite his seeming to have identified the factors that express the conservation force, that is mass and speed? De arcanis motus presents as a main axiom the principle of equivalence between the full cause and the entire effect, or more precisely, the principle that “with respect to the full cause and the entire effect the power (potentia) is the same.” (A VIII 2 135) This axiom should be obtainable by exploiting the definitions of the concepts of cause, effect and power. We would thereby obtain a metaphysical demonstration. This principle would have a status analogous to that of the totality axiom in geometry, that the whole is equal to all its parts. In the latter case, the concepts from which the definitions must be derived are the whole, the part and equality. But can all of the concepts of the mechan9

Cf. A VIII 2 134–135: “This happened to Descartes himself when he set out to furnish the laws of impact, for, had he followed this ratiocinandi filum, he could have furnished them, even laws such as those that we now have, which agree with phenomena, and he would have thereby avoided placing blame on matter and external objects.”

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ical axiom be defined a priori in the same way as those of the geometric axiom? In the text that we are examining, Leibniz forgoes this metaphysical investigation. He instead contents himself with a geometric model founded on presumably sufficient analogies. In effect, mechanical equivalence (æquipotentia) cannot be defined a priori despite assertions to the contrary;10 rather, it is an analogical model that represents causality as something that would, in principle, be defined a priori by pure concepts. Such a preoccupation with a priori demonstration will furnish, much later on, the quasi-axiomatic models of the Dynamica de potentia, as well as the conceptual (and metaphysical) distinctions relating to vis viva in the Specimen dynamicum. In 1676, the equivalent of an a priori model is produced by analogy; the model remains relative and nominal, responding to the epistemic requirements of hypotheses. The progression from the full cause to the entire effect must be one of implied necessity. The full cause corresponds to a complete set of conditions that, in action, bring about the effect. These conditions must be equivalent to one another in the cause-state and the effect-state, excepting the temporal distinction between before and after. Thus, this involves grasping the necessary connections underpinning the full cause and the entire effect, while abstracting from the accidental circumstances that one must separate from the analysis in order to render it functional. Above all, it is the identity of the definitional conditions on both sides that this functional analysis must discover. If one thereby succeeds in identifying and defining the same structure that is integral both to the cause and the effect, one will equally grasp the dynamical connection between this effect and some subsequent one in which the same structure is present. Hence the normative thesis that from the effect, one must be able to derive the formula for generating the equivalent of the initial cause. How can this be conceived of? The sequential entire states are reciprocally related in a way that can be represented by a compelling analogy. Leibniz conceives of this model by proposing to measure the entire force deployed, which equals per analogiam the power conserved from one state to another in an indefinite sequence: [Equivalent states] conform (conveniunt) with one another because the cause as well as the effect have a certain power, that is, the capacity to produce another effect; they only differ in application and situs, in the same way that the length of a line remains the same despite being curved. From this it follows that the cause can necessarily achieve as much as the effect, and vice versa. (A VIII 2 136)

The equivalence between states must be determined according to their capacity for expressing a power that is equivalent to whatever can totally recreate the initial state. To 10

Cf. A VIII 2 137: “As geometry depends on metaphysical [concepts] concerning whole and part, so mechanics depends on metaphysical ones about cause and effect. This principle of mechanics is a priori true: the effect is equivalent to the full cause, that is to say, the cause will produce the same thing, neither more nor less, provided it is neither aided nor impeded.”

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measure this, Leibniz suggests using the force required to lift a weight to a given height. In this model, the important thing is showing that the effect cannot surpass the power of the cause nor be inferior to it. If this occurs, then the demonstrative structure of the laws of motion collapses. Leibniz conceives of a total exertion of power that would enable measuring the constant constituents of causal sequences. This schema, on which the theorem for the conservation of living force will be established, would seem to appear for the first time here: the context implies that a system of analytic equations must represent the conservation of mv, assuming that a consistent measurement is in place.11 Thus the cause can be analytically converted into the effect once equivalent definitional requirements have been set for the cause and effect. Let us establish this law: the power of the full cause and the entire effect are the same. (Circumstances being constant, power is the state from which an effect of a specific (de­ terminatæ) magnitude results.) Hence the full effect can reproduce the entire cause. (A VIII 2 136)

With the help of this rule, can Leibniz overcome the difficulty posed by expressing speed as a vector when accounting for the conservation of relative speed in impacts, and the conservation of the quantity of progress when the barycentre is displaced before or after the impact? In De arcanis motus, Leibniz contents himself with demonstrating that certain contemporary theories, Huygens’ and Mariotte’s for example, are founded on the conservation of these relative quantities, but do not assign a true cause to the phenomena of motion. Such an analytic system can only be derived from experience. But there must be a means of grounding this relative system on a true “metaphysics” that explains the causal determinations at play. Thus, Leibniz affirms: I confess however that [the conservation of the respective speeds in impacts] can only be demonstrated by reference to experience. Indeed, the ultimate and true argument for these theorems depends on the primary metaphysical elements relating to cause and effect. (A VIII 2 137)

The relativist model à la Huygens translates the causes into a geometric language that expresses the relative situs changes; it implies that we abstain from all metaphysical research into the presumed substantial ground of these causes. Leibniz seems to recognize the analytic validity of such a model, but is determined at the same time to explore metaphysical reasons for it. Beyond the apparent ambiguity of such an approach, if Leibniz had not proceeded in this way, we would not have witnessed the reformatio observed in De corporum concursu. To clarify the reasons for this approach, we must

11

Cf. A VIII 2 136: “If a body or a combination of bodies is in such a state that, while exerting its whole action freely, it can lift a given weight at a given height, it will never be able to produce another effect that could achieve more; thus, all applications to this thing would be useless.”

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examine how the critique of kinematic categories in the dialogue Pacidius Philalethi (1676) prepares the way for the reformatio. Pacidius Philalethi aims to develop an analytic conception of motion that will contribute to establishing a mechanical science. The latter must be able to furnish sufficient principles for explaining impact, the collision of bodies and all mechanically identifiable phenomena, and nothing should prevent the theory thusly conceived of from extending by analogy to other types of phenomena.12 The key to this operation is formulating the analysis of phenomena in a way that mirrors analytic proofs in geo­ metry. One of the protagonists of the dialogue, Charinus, draws attention to the specific problem of motion. The impossibility of geometrically representing the reasons and causes of dynamic effects owes to the fact that the initial moment of motion entails the consecutive chain of phenomena, which seems to elude more geometrico representation. The sensory and imagined conceptualization of this moment as a point never expresses the underlying order. Hence the necessity of recurring to the facts of experience alone. But then the hypothetical causes to which one has recourse, and which one extends by analogy to similar phenomena, can breed errors since one lacks the criteria for determining an adequate methodological process. Pacidius, who speaks for Leibniz, confirms this by emphasizing that “forces and motions are not something subject to imagination and that this observation is of great importance to true philosophy.” (A VI 3 532) The hierarchical order goes from logic to geometry and from the latter to the “science of change and motion relative to time, force, action.” (A VI 3 532–33) Just as geometry can be interpreted as a mathematical logic, so can, following Galileo’s lesson, phoronomy aspire to the status of a physical logic. As a result of this, we understand that the basic concepts and argumentative structure of physics must presuppose geometric principles or axioms from the outset of the analysis of motion and the dynamical states that these motions imply. However, one cannot doubt that motion is conceivable in a rational way as a change of situs in time; that being said, a momentary state of change is unintelligible. Indeed, all change implies the transition from one state to another, and as a result, an infinite regressive juxtaposition of such successive moments: representing juxtaposed points in space, or similar moments in time, as a continuum leads us into the labyrinth of accounting for continuity in the composition of motions. Assigning a determinative reason to these series of discrete elements merging is impossible no matter how far we push the analysis. The solution comes, to a certain extent, from the thesis that bodies are perfectly flexible or elastic, which does not suppose perfect fluidity, but a status relative to the division and coherence of the parts; these parts would have their own status, and so would the parts of 12

A VI 3 530: “He complained that the foundations of general science were not established, but also that sufficiently certain precepts were not provided for percussion and impact, the increasing and decreasing of forces, the resistance of media, friction, the tension of arcs and so-called elasticity, the flowing and waving of liquids and other familiar topics of that kind.”

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these parts, ad infinitum, thus excluding the presence of any real minima. In contrast with a pile of sand made up of grains, the metaphor chosen for representing this conception is that of a leaf or tunic folded in various ways, which in turn are comprised of other folds, and so on to infinity, so that we never arrive at the final points.13 To each fold there corresponds therefore a particular motion that distinguishes it from adjacent folds, without the relative units being separate or disassociated. The notion of a point or elementary motion then corresponds only to limits, which always have a relative status when analyzing the constituents of a motion. In this context, changing of shape of the parts does not entail their actual dissociation, but only the distribution of moments of motion (momenta motus) and the change in situs that follows. Significantly, Leibniz alludes to infinitesimal calculus, which is in its early stages at that time, as a means of representing the elements of time, space and motion, but from a critical point of view: these elements are reduced to nothing other than relations to limits or extrema. The implementation of such a model leads, at least temporarily, to rejecting the presumption that punctuated moments are the basic ingredients of motion: It will be worthwhile to consider the harmony of matter, time, and motion. Accordingly, I am of the following opinion: there is no portion of matter that is not actually divided into further parts; so that there is no body so small that there is not a world of infinitary creatures in it. Similarly there is no part of time in which some change or motion does not happen to any part or point of a body. And so no motion stays the same through any space or time however small: thus both space and time will be actually subdivided to infinity, just as a body is. Nor is there any moment of time that is not actually assigned, or at which change does not occur, that is, which is not the end of an old or beginning of a new state in some body. This does not mean, however, either that a body or space is divided into points, or time into moments, because indivisibles are not parts but extrema of parts. And this is why, even though all things are subdivided, they are still not resolved all the way down into minima. (A VI 3 565–66; Leibniz, 2001, 211)

An essential consequence follows from this conception when it involves representing the causes of motions. Bodies in motion do not act as such, since at every moment of displacement, there is a transition from one state to another without the state of change itself being discernible in an instant. The sufficient reason for change must therefore belong to another reality than the one subject to situs change:

13

A VI 3 555; Leibniz 2001, 185: “If a perfectly fluid body is assumed, the finest division, i. e. a division into minima, cannot be denied; but even a body that is everywhere flexible, but not without a certain and everywhere unequal resistance, still has cohering parts, although these are opened up and folded together in various ways. Accordingly, the division of the continuum must not be considered to be like the division of sand into grains, but like that of a sheet of paper or tunic into folds. And so although there occur some folds smaller than others infinite in number, a body is never thereby dissolved into points or minima.”

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Thus action in a body cannot be conceived except through a kind of detour. If you really cut to the quick and inspect every single moment, there is no action. Hence it follows that proper and momentary actions belong to those things which by acting do not change. (A VI 3 566; Leibniz, 2001, 211)

Later on, Leibniz explicates this notion of the causal entity that underpins motions, which, in turn, he presents as being akin to phenomena or artifacts of our representations of things without any proper causal rationality. No cause can be conceived for why a thing that has ceased to exist in one state should begin to exist in another (inasmuch as the transition has been eliminated), except a kind of permanent substance that has both destroyed the first state and produced the new one, since the succeeding state does not necessarily follow from the preceding one. (A VI 3 567; Leibniz 2001, 215)

There therefore seem to be momentary transcreations, and in the Pacidius, Leibniz associates them directly with divine action; hence the implicit thesis that the conservation of a constant relation in mechanical interactions escapes kinematic representation and arises from a cause of another kind. The radical but tentative metaphysical solution, which consists in establishing a form of occasionalism for understanding change in nature, must not however obscure the methodological importance of this Leibnizian critique for the scientific mechanics that Leibniz set out to establish. Henceforth, one must invent ways of avoiding the influence of the imaginary representations of a so to speak empirical geometry. One must, in particular, develop a means of deducing the sequences of phenomena without falling victim to the aporia entailed by their extensive representation. Finally and significantly, the Pacidius abstractly understands force as intervening in an instant. This points to a framework in which force presumably exists in the “metaphysical” background of the laws of motion. Indeed, at the time of the Pacidius, Leibniz is not significantly opposed to geometrically framing mechanics as a science. And metaphysical recourse has yet to result in a conception of force capable of supporting the foundations of physical theory. Situated perfectly between the Pacidius and De corporum concursu are those analyses that we have already mentioned, which lead to the introduction of the principle of equivalence between the full cause and entire effect as a means of constructing a theoretical framework for a rationally grounded mechanics. During this period of trial and error and “empirico-theoretical” drafts, Leibniz continues to closely follow the methodological norms that Huygens and Mariotte conceived of for mechanics. The major difference is that Leibniz does not give up on reconciling and rationally integrating laws of motion a priori. He probably thinks that it is the only way to found a veritable mechanics. From the beginning, Huygens and Mariotte were stuck in a relativist position: their goal was reducible, above all, to conceiving of geometric models capable of articulating various empirical laws in kinematics. The rational system of laws that Leibniz sought to create

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would pursue another goal, since it involved constructing this system and validating it by enumerating the combination of causal reasons that would explain it. 2. The Reformatio The manuscript of De corporum concursu marks the arrival of Leibnizian dynamics as such.14 In this text, for the first time, Leibniz effectively substitutes the principle of conservation of power measured as the product of mv2 for the Cartesian principle of conservation of quantity of motion. The first nine sheets of De corporum concursu, to which the page dividers scheda secondo-secunda and scheda secondo-sexta can be added, are dated January 1678. The tenth and last scheda is, for its part, dated January-February 1678.15 Few studies have analyzed this strategic but particularly complex text.16 The complexity that characterizes it owes in effect to its twofold development. In the initial version, the first seven schedæ attempt to deduce laws of impact that conform to the principle of conservation of motion. The exercise culminates in the comparison, represented as a table, of theoretical calculations with the results of experiments that rely upon a pendulum for measuring the respective effects of displacement caused by the collision of unequal objects. Recorded in the scheda secundo-sexta, this comparison leads to replacing the Cartesian principle of conservation with a new one that alone makes it possible to explain all of the cases under consideration in a coherent way. As a result of this strategic substitution, the previous analyses must be reviewed and corrected. To the initial version, Leibniz adds commentaries and makes adjustments inspired by his reformatio of the system, in virtue of which he now hopes to establish the rationality of the phenomena of motion. For their part, the eight, ninth and tenth schedæ directly follow from the revised strategy of analysis; consequently, they are not the focus of the critical remarks of the reformatio, and are not therefore modified. The challenge of this text for today’s reader is grasping how a “logic of error” transforms into a “logic of truth”. For Leibniz’s attempt to rationalize and systematize analytic arguments for impact via an erroneous principle precipitated the emergence and justification of a new one, which could not have otherwise been proven more geometrico at the outset. The presumption of rationality that depends on Cartesian evidence could have served up to a certain point for justifying the 14 15

16

The manuscript of De corporum concursu has been for the first time edited by Michel Fichant, with a commentary and translations into French, in Leibniz (1994). The text is now available in A VIII 3 527–660. According to Bodemann’s classification, the first seven schedæ were listed under LH XXXV (Mathematics), 9, No 23, f.1 to 22, and the final three under LH XXXVII, 5 (Mechanics, Dynamics), f.86 to 91. Michel Fichant sent us his transcription of De corporum concursu. It is this transcription that we relied on in Duchesneau (1994). We wish here to recognize the magnitude of work that Fichant undertook and express our gratitude to him for the access he initially gave us to the text. Cf. Fichant (1974) and (1990), and this author’s analyses in Leibniz (1994); Robinet (1986), 211–15.

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principle of conservation of quantity of motion; and the difficulty of adopting the new principle could have owed to its lack of “intuitive” validity. From the outset, Leibniz’s rational and deductive investigation leads him to supply another, more architectonic, sort of justification for the new principle. At the same time, the reformatio promotes a methodologically re-orientated approach to mechanics as science. Throughout 1677, Leibniz had scrupulously revisited Huygens’ and Mariotte’s theories with the goal of formulating the principles that would suffice for organizing the system of reasons governing mechanics. We can build on Michel Fichant’s analysis of the manuscripts associated with this project.17 Leibniz became interested here in Mari­ otte’s third proposition (or second principle of experimentation), according to which percussion remains the same so long as the respective speed of approaching bodies remains the same.18 Percussion is therefore measured according to this relative speed, not according to the actual speed of impacting bodies, which we cannot empirically determine. Leibniz is without a doubt preoccupied with moving beyond this relativist rule to calculating real speeds. This concern is consistent with two fragments, dated 10 June 1677, where Leibniz questions the universality of the boat model, and therefore the principle of the composition of motion, on which Huygens based his calculations of all occurrences of impact in bodies. Remarkably, these fragments imply the conservation of the quantity of motion across kinematic equations, which must be revised when they prove irre­ concilable with maintaining the inherent factors of speed, which would in principle be discernable from those of relative speed. In these notes,19 Leibniz employs the architectonic criterion of an artificial perpetual motion that would lack a sufficient reason. This criterion allows him to reject the composition and decomposition of motions that would seem to imply an increase in mv (since in Descartes’ mind, v is understood as a scalar quantity). Abandoning the relativity of motion by adopting a system of equations that reflect the potential combination of cases seems to be Leibniz’s strategy for founding a mechanics based on conserved power. It is in this context that one uncovers the attempt reported in De corporum concursu. One may rightly think that De corporum concursu responds to the project of reconciling the principle of the conservation of quantity of (absolute) motion with the rel-

17 18

19

Fichant (1974), 199–201. E. Mariotte, Traité de la percussion ou chocq des corps (Mariotte 1673), 25–26: “When two bodies directly impact one another, the force or power of the impact to affect either is the same. This is true whether they are moving toward one another at the same or different speeds, whether only one of them is in motion, or whether both are moving in the same direction, provided that the speed of each is uniform according to the first supposition, and that being at the same distance when they start moving, they take equal time to meet, that is, provided that their respective speeds always remain the same.” These manuscript notes (LH XXXVII 5, 157–58 and 191–92) are edited in A VIII 3 468–73 and 460–67, and analysed by Fichant in Leibniz (1994), 368–75.

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ative principles introduced by Huygens and Mariotte (conservation of relative speed and conservation of the displacement of the common centre of gravity). The principle is affirmed and analyzed from the outset, and will only be replaced after the series of experiments reflecting the combinatorial order of events will have revealed a more adequate measure of absolute power. Consequently, the universal validity of the relative principles will be confirmed and their compatibility guaranteed. In short, one will witness, post reformationem, a modification of the system of equations that achieves perfect continuity across the apparent disparity of cases. Once the “mathematico-experimental protocol serving as the main reference point” is established, everything happens as if all of the theoretical elements perfectly lined up following their initial discordance.20 The goal here is not to investigate the details of the analysis conducted in each case. In striving to exhaust the possible combinations of variables, Leibniz constructs a table of cases that we cannot reproduce here. It seems more interesting to discern the strategy of the analysis that gets implemented. This strategy can be described as having two principal elements: the principles for formulating and managing equations on one hand, and a classification of empirical cases on the other. The first principle at play is that of the conservation of force or quantity of effect, this quantity being measured by multiplying mass by speed. This principle appears at the top of the scheda prima, (A VIII 3 530) and will be accompanied post reformationem by the remark “Error: this does not follow therefrom in our system.” (A VIII 3 530) The model of the boat also provides a basic principle to start from. As Leibniz states: This supposition must be admitted because appearances remain the same, and moreover, neither are the forces increased nor diminished according to this supposition. And thus, following the impact, the same appearances would occur as in the boat model. (A VIII 3 532–33)

Secondly, however, he adds a significant commentary that suggests the non-fictional character of the model, and justifies its elevation to the status of a principle: The same thing can be explained without fiction by thinking that bodies, insofar as they are transferred with equal speed in the same direction, do not act on one another, and therefore this equal speed is neither increased nor diminished. (A VIII 3 532 [note in the margin])

Indeed, it then appears that this principle cannot measure the real forces at play, but only their enduring equilibrium, as is evident in the version relating to the enduring progression of the center of gravity. Then, starting with the scheda secunda, lemmas are introduced to combine the cases into a synthetic model. The beginning of the scheda in question reveals this synthetic

20

Robinet (1986), 212.

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combination in progressive steps, the realization of which is desired insofar as possible. The first lemma points to a conversion of the state following the impact, such that the antecedent state can be reproduced: If two bodies moving away from one another after the impact are conceived of as returning to the point of impact with the same speed that they moved away from one another, then following this subsequent impact they would return to the state that preceded the first impact. (A VIII 3 542)

This lemma is consistent with the empirical analysis that has already been conducted in certain cases. But its meaning surpasses pragmatic considerations because it is consistent with reason to postulate the equivalence, and therefore reciprocal conservation of cause and effect. Leibniz reinforces this presumption of rationality by introducing the principle of sufficient reason: since moving from the effect to the cause can only create an effect closely resembling the cause, there is no justification for presuming an effect other than the one that corresponds to the initial cause.21 Regarding this lemma, Fichant points out how Leibniz progressively adjusts the cases.22 In the scheda prima, Leibniz established that when a larger and faster body strikes a smaller and slower one, the speed of the larger one is diminished, while that of the smaller is increased: hence the lemma should allow for inversion in this case. But Leibniz will not concede that the smaller body is not then reflected and that it transfers this much speed to the larger one. However, post reformationem, he confirms the universality of the lemma contrary to the exceptions that he previously made: “one should rather say that in these two cases it must never be reflected. I think that this lemma is correct. I think that by this lemma alone everything can be resolved.” (A VIII 3 544 [note in the margin]) The entirety of the scheda secundo-secunda is dedicated to resolving this apparent aporia in cases where the respective speed of colliding bodies and the displacement of the common center of gravity do not seem to be conserved after the small body presumably rebounds in violation of the lemmatical rule. Countering this initial error, the universality of the principle will be reestablished and confirmed soon thereafter, once the Cartesian principle of conservation is rejected:

21

22

Recourse to this type of lemma is implied in the analyses of Huygens and Mariotte: cf. Proposition 20 of the Traité de la percussion and proposition 5 of De motu corporum ex percussione Cf. Mariotte (1673), 122: “If two equal or unequal elastic bodies directly impacted one another, whether both have moved or only one, and they impact a second time with the speeds they had acquired following the first impact, they will return after the second impact to the same speed or rest that each had before the first impact.” Cf. Huygens (1967), XVI, 47: “If two bodies subsequently collide at the same speeds at which they were respectively repelled from one another following the impact (impulsum), then after the second impact they will acquire the same speeds at which they were initially impelled toward one another.” Fichant (1974), 204.

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This piece of paper rightly concludes that the same displacement of the center of gravity or the same distance cannot be conserved if one postulates that the same quantity of motion is maintained. In truth, it falsely concludes that the distance and direction of the center are not conserved, for the Cartesian hypothesis that the quantity of motion must be conserved is false. (A VIII 3 553)

The scheda tertia introduces the perspective of continuous series to the analysis, whether it is the larger body impacting the smaller or vice versa, by taking either body as the fixed basis for the calculation. The following lemmas imply recourse to the principle of continuity (even before any reformatio). The role of this principle seems essential for harmonizing the series of varying cases. The principle of continuity also elegantly expresses the equivalence between antecedent and consequent events, which provides an intelligible ground for all conservation in cases of impact: This lemma is useful when a given case or hypothesis infinitely approximates another hypothesis to the point of merging with it; one event thus continuously approaches another event that conforms with the hypothesis until it completely coincides with it; and here there cannot be a leap [between the two] such that, when any change occurs in the case that is too small to be assigned, a great and notable change would occur in the resulting event. (A VIII 3 563–64).

The lemmas of the scheda tertia attempt to systematize the cases where speed or the difference of size tends to vanish and where, at the limit, they merge into a case where the parameters are discrete and the effects well demonstrated. This prefigures the critique of Descartes’ rules as it will appear in Animadversiones in partem generalem Principio­ rum Cartesianorum (1692), along with the idea of graphically representing continuous series of cases so as to illustrate their being governed by a dominant harmonious order. This is confirmed by a note in the margin: “utility of progressions and figures for encompassing a plurality.” (A VIII 3 566) The great organizing principle of the analysis is that of equivalence between the entire effect and full cause. It is in the scheda septima, just before the reformatio takes place, that this principle is evoked as a means of integrating all of the previously conducted analyses: “it serves to basically reorder the whole thing as if a new light had appeared.” (A VIII 3 627) To be sure, as Leibniz highlights in a note added post reformationem, (A VIII 3 537) the continuity in expressions of laws that the new architectonic principle entails will include a revised version of curves that may be drawn for representing the progression of cases. In particular, this will result in non-rectilinear paths in a certain number of cases and certain graphical representations will entail hyperbolic inflexions in curves as a consequence of substituting mv2 for mv for measuring the conservation of force. The development of the principle of causal equivalence directly refers to the analysis conducted in De arcanis motus (1676). However, analytic refinement consists here

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in first explicating how cause and effect are formally assimilated as successive moments of a continuous series. This is why causality so described is subsumed under the principle of continuity: the identification of a law of series, hence a formal order, underpins these changes and determines their intelligibility in a necessary way – indeed a necessity ex hypothesi. Effectively, the effect differs from the full cause less than it differs from any given quantity. This means that the formal structure of the cause can be infinitesimally extended to include that of the effect; inversely, one can return to the cause by thus extending the effect analytically. For example, Leibniz reports that the present state of the world differs as little as possible from its previous state, which is its complete cause. Another example: from the isosceles triangle ABC, one can constructively derive the figure BDCE, which does not differ from it in size (same area):

Fig. 3 Isosceles triangle and equal squared area (Duchesneau 1994, 119).

The previous state of a machine generates the next state by a simple displacement of the situs of the forces that define its power: from one state to another, the progression is reduced to the extension of equivalent structures across situs changes. The first part of the formulation of the principle highlights how formal structures are equivalently transformed at the limit: The entire effect is considered as similar as possible to the full cause. For the entire effect is but a change to the full cause, and the smallest one possible. […] Indeed, the effect and the cause differ only by a certain particular formal difference: in sum, they correspond to each other (conveniunt). […] The entire effect originates from the entire cause; and the concept of the effect originates from the concept of the cause, insofar as it entails at the same time the necessity of the change. The change must always be conceived of as the least possible. (A VIII 3 627–28)

The second part of the statement is presented as a corollary of the first. One would need in effect to conceive of a sufficient reason for varying the power or force in the background of the processes. However, it would clearly appear that here, this involves a formal element par excellence that cannot be identified with the changing of the situs, and therefore with motion alone. To the contrary, this element must remain constant across the transformations of the situs of bodies. Once again, the law for generating series can serve as a model: each element appears here as an adequate expression of the law, and furthermore, indicates the next expression by progressive variation. Clear-

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ly, the full cause embodied by the law should, as a last resort, refer to an entity that is distinct from phenomenal bodies. Regarding the physics of 1678, the entity referred to is still understood according to a form of occasionalism that tends to make divine will the only true cause. Evidently, the prevailing model at the level of secondary causes is based on the variables of serial equations that represent systems of bodies, or “machines”; the entirety of the physical universe finds itself endowed with this status. Thus, as Leibniz expresses: Hence, the entire effect is equivalent to the full cause, that is to say, it has the same power. This is a corollary of the preceding proposition because it is not necessary to change the power, even if it is necessary to change the situs. Note that in strict metaphysics the preceding state of the world, or the machine, is not the cause of the following one, but rather God, although the preceding state is a sure sign (indicium) of the one that follows or is destined to follow. But we speak here of physics [physice], and no mistake can arise therefrom owing to the very fact that this is a sure sign. The same quantity of force subsists in the same machine or aggregate of any number of bodies cohering in action and passion. An external body is excluded or is at least certainly not considered. There is always the same quantity of force in the world because the entire world is but one machine. There always subsists therefore the same quantity of motion in the world. [Later correction] This is doubtful in the case of the quantity of motion, but true regarding the quantity of force. This is doubtful when it comes to the quantity of motion, true about quantity of force. (A VIII 3 628)

This notion of a closed system in which causal sequences could correspond to a serial generation of states is what the concept of natura irresistibilis emphasizes after the reform. By this, Leibniz understands the correspondence between, on the one hand, the relative laws of displacement of the common center of gravity and of conservation of respective speed, and on the other, the principle of conservation of quantity of force measured by the quantity of effect. The necessity of formal correspondence is in the background of the causal analysis, which is reduced to its algebraic intelligibility (through combining equations). Similarly, Leibniz tends to reinterpret the model of the boat as providing the combinatorial link between equations from the point of view of relative displacements. In a way, this would express the rule of equivalence between complete causes and entire effects in its application to changes in situs, that is to the conservation of respective speeds before and after impact.23 Likewise, what will be qualified as the principle of conservation of the quantity of progress, must be inter-

23 Cf. Scheda nona, A VIII 3 650: “Certainly all bodies are always understood as being linked by the center of gravity, and thus as a single total aggregate acted on by the cause known as gravity. But there is another cause in nature that acts on bodies such that they always approach and recede from one another at the same speed, or such that the same relation is always conserved between bodies. And these two forces comprise conatus: it is therefore not surprising that all things, as I have said, can be explained by the boat.”

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preted as the symbolic expression of a combinatorial sequence involving the constant displacement of the common center of gravity of a system, which is governed by the principle of causal equivalence. An architectonic rule therefore governs the models by which the order of phenomena is represented: This axiom, that nature as a whole is irresistible (irresistibilem), entails that forces remain the same and that the direction of the center of gravity remains the same. For two or more bodies, supposing that they are isolated and ignoring the action of the other bodies on them, constitute a nature unto itself, that is, so to say, a separate world. And the whole machine conserves its forces, for nothing diminishes them, and it retains its overall direction insofar as everything has the same overall direction. (A VIII 3 655)

This mode of combinative expression allows one to distinguish between vis ascen­dendi, which corresponds to the power consumed in the effect, and the vis agenda in se in­ vicem, seu sese propellendi, the force of impulse properly speaking, which constitutes the residue of vis ascendendi after the increase in speed has been reduced to its initial instantaneous effect. When this order is reversed, we get a model that prefigures the integration of elementary conatus into an impetus, which generates an effect capable of measuring the living force. However, the analysis of 1678 only skims the surface of this generative process. Leibniz contents himself with postulating the formal reconciliation of the principles of conservation. The various equations that express them seem to be adjusted to one another. Up to a certain point, we find in De corporum concursu only the formal skeleton of the future dynamics. The exact meaning of potentia or vis measured by the quantity of effect remains to be developed. A semantic analysis of the concept of vis viva is evidently lacking, but will be at the heart of the theory when we come to the Specimen dynamicum (1695) and to other post reformationem canonical texts. However, from the beginning, Leibniz’s analysis raised the question of providing a metaphysical foundation for the new principle. While not a dynamical theory, it surely seeks to link motion to its grounding principle. Already outlined in the Pacidius Philalethi (1676), this problematic becomes the main priority of the project entitled De motu tractationis conspectus, dated February 1678. (Gerland 114–15) According to this plan, studying the subject of motion in book I (metaphysi­ cus) would lead to conceiving of its cause and true motor, God, whether motion is understood as relative or absolute. Hence, as we point out, the persistence of a sort of fundamental occasionalism at this stage of Leibniz’s thinking. Book II (geometricus) would study the fundamentals of kinematics, while book III (organicus) would focus on displacements as arising from the system of situs of bodies, and on the latter’s impenetrability. Roughly speaking, this involves the relation between motion and the equilibrium of masses, and thus the convergence of statics and kinematics. It is only in book IV (physicus), separated from metaphysics by the whole field of extension-related considerations, that Leibniz would address the question of the interaction of forces and take into account the sequences of motions resulting from solids resisting pene-

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tration. Beyond the questions of space and formal impenetrability, it is therefore the intelligible structure of the agent of change that requires explication. A note in the margin essentially describes the theory of equivalence and reciprocity of forces, while connecting it with the principle of conservation of respective speeds.24 Book V (De machinis seu mechanicis) would rebuild the analytic description of natural and artificial mechanical phenomena on this foundation. To properly evaluate this transition to a proto-dynamics, one must return to the De corporum concursu where the critical experiment that marks the beginning of the reformatio can be found.25 Leibniz envisions a synthesis of the different types of bodily collisions that involve increasingly more complex factors. Basically, however, the first levels of the analysis concern the collision of bodies when one is at rest, and then when both are moving in the same direction. From the scheda sexta and after, the analysis focuses in particular on cases of impact proper, that is, when the colliding bodies were moving in the opposite direction before the impact. It is in this context that one finds the outline of the crucial experiment:

Fig. 4 Schema of the experiment in De corporum concursu (A VIII 3 621, by permission of De Gruyter).

The experiment is set up in the following way. Two bodies (two hard wood balls) a and b are suspended, forming pendulums with the same half-diameters CA and DB. 24

Cf. Gerland, 114, n.1: “Nature always tends toward en end, and once attained, returns (recedit) with the same force. This is necessary for preserving the variety of things. Otherwise, everything would surely be reduced to stillness. Hence a body returns back with the same speed that it moved forward.” 25 Cf. C. Santi & P. Rubini, “Experimenta percussionis domini Regnault. On Leibniz’s use of second-hand experimental data in De corporum concursu (1678) and their previously unknown source” (fortcoming), with reference to F. Regnault, Lettre à B. de Monconys, 21 décembre 1655, in B. de Monconys, Journal des voyages, Lyon, 1666, III, 52–56.

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At the perpendicular, the suspended, resting bodies touch. The lines AC and BD are divided into homologous parts representing the descending squares, from 100 to 1. At each point a parallel line intersects the quadrants, and the square root of each squared number is represented along the curves, from 10 to 1. Following Galileo’s law for falling bodies, bodies that fall from a height akin to 100 acquire a speed akin to 10, etc., and have the capacity to be elevated back to the same height, at least when we control for external friction and more generally for the resistance encountered by objects in motion. Suppose that the body a is akin to either 16, 8, 4, 2 or 1 units in relation to body b, whose value is akin to 1 and is at rest. We can then introduce the axioms of calculus for percussive force, as they are postulated at the beginning of the scheda sexta: Two bodies that collide with impact (cum percussione) withdraw from one another after the impact. This proposition is a fact of experience (experimentum); or if we say that hard or elastic bodies collide with impact, this will serve as a definition. If the impacting body is increased, then the impact will also be increased, everything else being equal. If the impacted body is increased, everything else being equal, the impact will also be increased. If the speed of the impacting body is increased, then the impact, i. e., the conatus of the bodies to recede from one another will be increased. If two bodies collide and the power of each is nearly equal to the other, then the impact is greater. If two bodies with equal force collide, then the impact is as great as the total force. In all other cases, the impact is less than the total force. (A VIII 3 589)

The general formula for calculating the force of impact for bodies a and b whose speeds are respectively e and 0 would be: v 2 =  ab e a +b After the impact, the residual force is noted:

v 2− p 2 = ae − ab e a +b This quantity divided by a + b will yield the speed of displacement, once the force of the impact is omitted. The result is thereby noted:

j = 

a2 e  a + ab + b

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The force of a will be equivalent to: 2 j a =  a ae a +b

And the force of b equivalent to: 2 j b =  a be a +b

But the body a has two opposing conatus: the forward one represented by j, and the backward one represented by half of the force of impact. In the case where e = 1, the formula for the speed of displacement of a is measured by the function:

ε = 

2a 2 − ab − b 2 e 2 2 (a + b )

In this case a > b. In the inverse case, the numerator should be written as:

2a2 + ab + b2 In the same way, we measure the speed of displacement of body b, which is represented as:

v = 

5a 2 + 3ab e 2a 2 + 4ab + b 2

From there one can reconstruct the total force measured by mv by the formula: 2a 3 + 4a 2 + 2ab 2 e = ae 2a 2 + 4ab + 2b 2 Based on this system of equations, the tables that accompany the scheda secundo-sexta become easily understandable. The first section of the first table represents the calculation for the speed of displacement of a after impact (calculated from the square roots of the heights) where a = 16, 8, 4, 2, or 1, and b = 1 and is presumed to be at rest. The second section represents the calculation of the speed of displacement of b (calculated from the square roots of the heights) after a has impacted it. The third section represents the combination of factors corresponding to the total product mv. Moving on to the second table, the first section reports the speeds calculated according to the preceding formulas for the continued displacement of a and the rise of b mentioned above. Then, in parallel, the speeds are calculated based on the actual experiment. The disparity between the two calculations represents the force that has been lost (vis perdita). The next sections of the table list the results obtained by calculating from values other than 1 on the quadrants, e. g., 2, 3, 4, 5, etc.

The Reformatio

Fig. 5 First table in De corporum concursu (A VIII 3 618–19, by permission of De Gruyter).

93

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Fig. 6 Second Table of De corporum concursu (A VIII 3 620, by permission of De Gruyter).

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In light of these disparities, Leibniz then radically revises the fundamental equation of conservation. He recognizes the unavoidable contradiction between the presumed conservation of quantity of motion and the systems of Huygens, Wren, Wallis and Mariotte. His first reflex is to overthrow these systems.26 But one can imagine that the shape of the quadrants had provided him with a potentially compatible solution, since the values that appear there represent the speeds as the square roots of their corresponding perpendicular heights (according to Galileo’s law). Secondly, he formulates a solution, which can be found here and surely merits being cited in extenso: I now see where the error is here. The force in a body should not be calculated in terms of the speed and size of the body, but according to the height from which it falls. And the heights from which bodies fall are as the squares of the speeds in question. In this way, the forces will also be such, supposing that the bodies are the same. Generally then, forces are in a compound ratio of the bodies and the speeds squared. Hence, two bodies have equal forces, not as we typically think, when the speeds are proportional to the bodies, but when the squares of the speeds are proportional to the bodies. From which it becomes evident that the same quantity of motion is not conserved, but only the same force. Also, we shall call the squared speeds momenta, so that moments will be to force what speed is to the quantity of motion. In our system, moments must be squared speeds, because the effect is equivalent to the height to which the body can be elevated; but heights are proportional to the squared speeds. Perhaps in another world system, where speeds have a different relation to heights, one should also make a different calculation of forces. From this it follows that, once impetus has been generated, bodies do not move by themselves, for how could they remember the height from which they fell, or understand in which system they are moved. But they must be perpetually moved by a general mover (which is hardly satisfying, since bodes must still possess a force of their own that would combine with the general one), or instead they must be perpetually pushed by a very wise cause that remembers everything and cannot err, in which case the laws of motion would be nothing other than reasons of the divine will, which likens effects to causes, inasmuch as reason permits. (A VIII 3 623–24)

These remarks lead to a third table where, significantly, the results of the calculation and the measurements agree as a result of the corrections suggested by the adoption of the formula mv2. If these remarkable passages initiate the reform of mechanics, to which Leibniz had previously dedicated himself, the reformatio of 1678 entails its limits. This is particularly true of the theoretical concepts to which Leibniz has recourse. 26

A VIII 3 622: “One force or another will always be lost in the experiment; but to be sure, the quantity of motion will never be increased. Therefore, these experiments overturn the systems of Huygens, Wren, Wallis and Mariotte.”

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Fig. 7 Third Table in De corporum concursu (A VIII 3 625, by permission of De Gruyter).

Henceforth, the system of equations must be based on the fact that the conserved force is directly proportional to the height of the fall from which the bodies acquire it; and this force is proportional to the square of the speed generated thereby. It is therefore to this formula that the measure of force owes itself. Hence the wording of the law: generally, forces belong to a compound ratio of bodies simply and the speeds squared. One notices the usage of the adverb generally, which indicates that this solution appears both universal and contingent to Leibniz. This perspective is revived by Leibniz’s insistence on the necessary relation of this law to the particular structure of our universe (“in nostro systemate”). In any other system of the world, one could conceive of another

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relation of speed to the height of the fall. One can therefore hardly be certain that by 1678 Leibniz had thoroughly considered the overarching justification of this principle as an optimal principle. Furthermore, it is incontestable that prima facie this discovery jeopardizes geometric intelligibility, since force becomes more abstractly symbolic and can therefore be more indirectly represented. Without a doubt, Leibniz insists on the relation of v2 to force as analogous to that of v to quantity of motion. He thus wishes to highlight that algebraic translation of this relation is possible; in this way, the new principle could be integrated into an analytic mechanics, or one analogically consistent with the ideal of geometry. Moreover, bodies seem to be endowed with autonomous forces that are irreducible to kinematic relations. This raises the question of how these elements of force can be introduced into the system of nature. Pure occasionalism appears under attack since it seems that bodies possess their own force, which makes it necessary to conceive of how those forces are combined at the primordial level of the mechanical system of our world. Leibniz does not yet allude to the existence of multiple centers of force in harmony with one another, nor does he invoke theoretical concepts and architectonic principles to account for them. His position in 1678 seems closer to a partial occasionalism: the laws of nature continuously express the reasons belonging to divine will, which link effects to causes. On the one hand, physics must be fashioned according to the principle of equivalence between the entire effect and the full cause, and to this end, it must mobilize the resources of infinitesimally continuous expressions and transitions to the limit for representing series of phenomenal states that form a system. On the other hand, the theoretical reference in the background of this type of analysis is a form of metaphysical presupposition: continuous “transcreation”, due to a supreme rational agent, would secure the lawful relation of the things of our world beyond the geometric labyrinth of abstract continuities. Is this not the lesson of the Pacidius Philalethi that endures in De corporum concursu and demonstrates its influence on the initial outline for a dynamical theory?27

27

One should point out here the remarks made in the conclusion of Fichant (1990), 65–68. In the first place, in subsequent presentations of the dynamics, Leibniz will indeed erase the traces of his initial steps. The analysis of 1678, which focuses on the systematization of the rules of impact of bodies, will be withdrawn in favor of more direct demonstrations of the principle of conservation of living force (by a posteriori means), and of the principle of conservation of motive action (by a priori means). In the second place, post reformationem, Leibniz initially tends to consider the principle of conservation of living force as belonging to a particular physical system. He will first presume that the revised principle of the quantity of motion (according to the relation to time) would continue to be universally applicable. In the third place, furthering the reform will in particular entail generalizing the formula of conservation mv2 and the concept corresponding to force, so that it can serve as the foundation of a physical theory relating to the particular structure of our universe. Finally, Fichant points out – and thereby rediscovers one of Gueroult’s theses – that if Leibniz uses Huygens’s formulas, he gives them a new “philosophical identity” that goes far beyond the models or mathematical expressions of which they formerly consisted.

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3. The Brevis demonstratio: Living Force as a Model Leibniz abandoned Descartes’ principle of conservation of motion from the beginning of 1678. He connected the empirical laws of impact with a new principle of conservation, which measures moving force by the product of mass and speed squared. Consequently, Leibniz must justify his rejection of the Cartesian principle. He must also justify his being forced to appeal to a principle of conservation founded on variables that escape the norms of geometric representations. In attempting to establish a new principle, Leibniz will have to reconsider the epistemological criteria that underpin and govern theoretical physics. This preoccupation reveals itself in the first public presentation of the main principle of the reformed mechanics, some eight years after his having written De corporum concursu. The Brevis demonstratio erroris memorabilis Cartesii et aliorum circa legem naturalem, secundum quam volunt a Deo eandem quantitatem motus conservari, qua et in re mechani­ ca abuntur (A Brief Demonstration of a Notable Error of Descartes and Others concerning a Natural Law, according to which God is Said to Conserve the Same Quantity of Motion; a Law which They also Misuse in Mechanics) appears in the Acta eruditorum in March 1686. (A VI 4 2027–30) The argument in this text reappears in the unpublished Discours de métaphysique that same year. (§ 17–18, A VI 4 1556–59) It marks the beginning of an analytic justification of the reformed mechanics, along with the structural implications that this process brings about in physical theory. The problem that interests Leibniz in this context is the establishment of “subordinate maxims” or “laws of nature.”28 The notion of a subordinate maxim refers to the subordinate status of these laws. The metaphysics to which Leibniz refers renders them in effect dependent on the general order governing the created world. This general order includes all of the reasons that prevail upon and determine the development of its diverse finite substances. Consequently, it accounts for the expression of these substances at the phenomenal level and the apparent interaction of bodies such that it manifests itself in experience. The rules, which express the order at the phenomenal level, are therefore subordinate in relation to the divine plan for a system of nature. Their relation to such a system determines the specific form that they must take. They are also subordinate in yet another way: our finite understanding accesses them via a more or less systematic conception of the facts of experience. Essentially a posteriori, this process of unveiling presupposes that the determining reasons for those facts are beyond the sphere of empirical knowledge; these reasons would only be translated in the laws themselves, which could only represent them insofar as they are reflected by sequences of phenomena.

28

Cf. for instance Discours de métaphysique, § 7, A VI 4 1538–39; L 367.

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As an example of such a law of nature, article 17 of the Discours de métaphysique chooses to examine the Cartesian principle of conservation of quantity of motion. This principle is formulated on two levels, which the Brevis demonstratio attests to just as much as the Discours. On the one hand, we are dealing with the conservation of a theoretical entity, or potentia motrix: “it is consistent with the reason that the same quantity of motive force is conserved in nature and does not diminish […] nor augment.” (A VI 4 2027) On the other hand, this theoretical entity is interpreted following a model, that of the quantity of motion, which is represented by the product mv, where mass, taken as the dense volume of a body, and speed are represented. Passing from the theoretical entity to the model, Leibniz points out, is conceived of by Descartes as a substitution of equivalents: “Descartes, who took motive force and the quantity of motion to be equivalent things, declared that God conserved the same quantity of motion in the world.” (A VI 4 2028) The principal justification for this model, according to Leibniz, owes to the prevalence of notions of statics in the conception of simple machines, such as the angle, lever, wheel, pully, and screw. When we are dealing with these, the effects of speed and mass are cancelled out by one another, as in states of equilibrium. This Archimedean model would produce the paradigm for the Cartesian model. Regarding the basic theoretical proposition, one must interpret its conformity with reason as signifying the analogical relation of specific phenomena to a sufficient reason. In effect, it is by referring to phenomena that Leibniz separates out every principle that would authorize the augmentation or diminishment of the total entity that determines exchanges of impetus in nature. According to this reconstructed Cartesian argument, experience would refute every hypothesis implying perpetual mechanical motion, and one would rightly observe that the force of a body is not diminished unless a part of it is ceded to another adjacent body, that is, so long as this force does not dissipate in the interior of the same body.29 In the Principia philosophiæ (1664), Descartes had presented his argument in a significantly different way. Making God the cause of motion in the universe, Descartes deduced therefrom the continuous conservation of the same overall quantity of motion and rest. The experience of change suggests to us that this quantity is universally maintained in the mechanical interaction of bodies. Consequently, one must formulate a rule of proportionality that adapts speed to the geometric mass of bodies in order to ensure that the motive power remains constantly the same. Hence the principle of conservation of motion.30 Justification for the principle comes, on the one hand, from God’s being a perfect cause, and on the other, from a conception of the modes of corporeal reality that reduce them to pure relations of extension. From this perspective, theoretical entities can be represented a priori without having to recur to analogies based on 29 30

It is in virtue of arguments of the latter kind that Leibniz will account for the apparent dissipation of living force in bodies that are not perfectly elastic. Principia philosophiæ, II, § 36, AT VIII-1, 61.

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experience. Moreover, the Cartesian version dispenses with presupposing two levels of the principle, depending on whether theoretical entities or interpretive models are being dealt with. At least this seems to be Descartes’ epistemological stance. In reality, it is not certain whether the twofold structure suggested by Leibniz is entirely absent from Descartes’ conception of physical theory per se. Whatever the case, Descartes will be obliged to return to the twofold structure each time that he analyzes specific phenomena. However, linking Descartes’ arguments with the statics model emphasizes the ideal of a unitary explanation. This model is attractive primarily because it is conducive to pure geometric expression where the modes of a substance endowed with motive force are concerned. The same geometrization of these modes requires Descartes to interpret the factor v as a quantitative parameter that must be treated as scalar when it comes to the laws of impact founded on his principle of conservation.31 When Huygens, Wallis, Wren and Mariotte bring the empirical rules of impact to light, these only seem to be compatible with a constant summation of the factors mv before and after the impact if one treats speed as a vector quantity; but this means that “negative quantities of motion” come into play, or can come into play, depending on the vectorial orientation chosen as a reference. As a result, the theoretical principle of conservation comes under attack: no lawful relationship between phenomenal and theoretical entities can be determined. This system of contingent rules destroys in fact the Cartesian framework in its attempt to account for the world of phenomena. Whoever rejects the methodological norm of a fixed relationship between the theoretical and phenomenal planes, while also holding onto the idea of a metaphysical basis for the laws of nature, is forced to retreat to a priori geometric representation. Otherwise, one must come to terms with phenomenalism and the mere nominal codification of relations between colliding bodies. This strategy is, for example, Huygens’ own. From the outset, Leibniz prefers a system comprised of models and theoretical principles. This certainly does not exclude the possibility of subsequent complications arising when these two standards are decoupled. At this point, we can turn to Leibniz’s first public demonstration. The desired conclusion is the following: There is thus a big difference between motive force and quantity of motion, and the one cannot be calculated by the other as we undertook to show. It seems from this that force is rather to be estimated from the quantity of the effect it can produce; for example, from the height to which it can lift a heavy body of a given magnitude and kind, but not from the velocity which it can impress upon the body. (A VI 4 2029; L 297)

The negative goal is no doubt the destruction of the Cartesian principle. But what could the positive goal be? Carolyn Iltis previously pointed out the extent to which the Brevis demonstratio seemed insufficient for demonstrating the principle of conser-

31

Ibid., II, §§ 40–42, AT VIII-1, 65–70.

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vation of living force.32 A new measure for establishing motive force must indeed be established! Establishing the theoretical principle would require one to presuppose the possibility of mechanisms that transfer force without diminishing or significantly changing it, beyond the limit case of collisions between perfectly elastic bodies. However, Leibniz is unable to provide empirical evidence in such cases. But Iltis’ position is truly untenable because one cannot hope for the thorough empirical demonstration of any principle of conservation. What is precisely at issue here are theoretical principles that govern the interpretation of facts. Hence, one cannot be expected to abandon them except by means of a systematic refutation.33 Gregory Brown also leveled a critique of Iltis’ thesis from another point of view. According to him, the Brevis demonstratio did not aim to destroy one principle of conservation and replace it with another, but simply to reject the identification of motive force with quantity of motion by instead identifying it with living force.34 For Leibniz, the existence of a theoretical entity representing motive force and capable of conserving itself went without saying. The question was one of finding a concept capable of filling the role of this unknown theorical entity. Given the evolution of the question concerning the laws of impact since Huygens’, Wren’s and Wallis’ contributions in 1668–1669, the principle of conservation of quantity of motion had remained problematic. Convincing the Cartesians of the necessity of a reform only required dissociating quantity of motion and motive force, and furnishing a new concept that represented motive force. This perspective seems fair. From the outset, Leibniz in effect distinguishes between the abstract theoretical principle of conserved motive force and the empirical-theoretical concepts that warrant the correlation between the abstract principle and its models. Without a doubt, one might rightly affirm that the immediate objective consists in substituting one model for another. But is it not disingenuous to suppose that the argument stems from the necessity of resorting to a principle of conservation? For Huygens, for instance, the propensity to “phenomenalize” the laws of impact was not a nominal tendency. Leibniz’ argument must show that measuring motive force in terms of absolute

32

33

34

Iltis (1971), 27: “Leibniz presented important mathematical arguments that mv2 and not m | v |  was a correct measure of something conserved in nature. He did not however present convincing arguments that this measure of force was also conserved in the physical instances he claimed for it, with the exception of elastic collision. In many of his other arguments Leibniz does not adequately specify a closed conservative system, since the mechanics for transferring ‘force’ among the parts of the system are not specified.” This position is a bedfellow of the one developed by Gale (1973), 206, n.24: “As to the claim that metaphysical grounds are not sufficient for Leibniz’ establishing a search for a conservation principle, it is questionable that there are in fact any other sorts of sufficient grounds. Meyerson’s extended research and argument has conclusively shown that the conservation principles present in physical science cannot be said to have been originated and confirmed as either a priori demonstrative, or a posteriori demonstrative. Their status can only be what might be called metaphysically ‘plausible’.” Brown (1984), 122–37.

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power allows for reestablishing the theoretical significance of the principle of conservation despite more phenomenalist versions of the laws of nature. After all, the model that Leibniz favors will deeply affect the interpretation of the principle of conservation, and its epistemological status in the framework of a new mechanics. From this point of view, Brown’s perspective would seem too reductionist. The way in which the demonstration is developed goes to the heart of what Gue­ roult designates the a posteriori method.35 It will become a constant of the Leibnizian approach in its analytic phase. First, Leibniz adopts as premises two principles that the Cartesians accepted without issue. The first presupposes that a body falling from a certain height acquires from its fall, excluding external hindrances, the force to return to its original position. The second presupposes that as much force is required to lift a body A weighing one pound the height CD of four yards as is required to lift a body B weighing four pounds the height EF of one yard. The first, implied by Galileo’s mechanics, is explicitly formulated by Huygens in the Règles du mouvement dans la rencontre des corps (1669), and then applied to the pendulum in Horologium oscilla­ torium (1673).36 The other principle had been stated by Descartes in the small treatise on mechanics attached to the letter to Constantin Huygens on 15 October 163737 and in his correspondence with Morin and Mersenne;38 it had also been formulated by Pascal in his Traité de l’équilibre des liqueurs (1663).39 The combination of these two principles allows us to treat the forces obtained by elevating A to C and B to E as equal and proportional to the heights of the fall CD and EF.

Fig. 8 Figure in Brevis demonstratio erroris memorabilis Cartesii (GM VI, fig. 11).

35 36 37 38 39

Cf. Gueroult (1967), 28–30. Cf. Huygens, Horologium oscillatorium, part 4, hypothesis 2. AT I, 435–36. Letter to Morin on 13 July 1638, AT II 229; letters to Mersenne on 12 September 1638, AT II 353, and 15 November 1638, AT II 432. Pascal (1963), 237–39.

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With a view to showing that the Cartesian model for motive force could not possibly work, Leibniz relies on the law of falling bodies that Galileo uncovered in the Discorsi e dimonstrazioni matematiche (1638). With regard to freefall, the space is proportional to the square of the time taken to traverse it, while the speed is proportional to the time itself. It follows that the speed is proportional to the square root of the space, that is, of the height of the fall. Following the trajectory CD, the body A acquired a speed proportional to 2, while traversing the trajectory EF permitted B to acquire a speed proportional to 1. The application of the product mv to each results in a quantity of motion proportional to 2 in the first case, and 4 in the second. Hence, there is no conservation of motive force according to the model of quantity of motion. Calculations based on the model of the resulting effect assume that the return of the object to its starting point consumes all of the force generated by the fall itself. This is an application of the rule of equivalence between the full cause and the entire effect: force must be calculated based on the quantity of effect that it can produce. This rule is the linchpin of the demonstration that links the model to the theoretical principle of conservation, which the model empirically translated. If we assume that the heights of the falls are reciprocally proportional to the masses of the objects (by interpreting mass as the dense volume of bodies in a way similar to Descartes’), and if we assume that the product of mass and height represents the entirety of the force consumed in the effect, we obtain the product mv2 as the measure of this force. In simple machines, to which the approach of Archimedean statics applies, equi­ librium is achieved by the mutual compensation of the quantities m and v, which follows from the nature of machines. After all, given the preservation of the state of equilibrium, forces cannot generate effects that exhaust them, as they are limited to their virtual instantaneous expression. In this case alone it happens per accidens that force is conserved, and perhaps calculable by virtue of the equivalence between the implied quantities mv. The Brevis demonstratio concludes with a significant statement: It must be said, therefore, that forces are proportional jointly, to bodies (of the same specific gravity or solidity) and to the heights which produce their velocity or from which their velocities can be acquired. More generally, since no velocities may actually be produced, the forces are proportional to the heights which might be produced by these velocities. They are not generally proportional to their own velocities, though this may seem plausible at first view and has in fact usually been held. (A VI 4 2030; L 298)

The majority of the previous theoreticians of mechanics had been victims of this presumptuous illusion of intelligibility in favor of calculations based on mv. Therefrom resulted the difficulties that arise when interpreting phenomena relating to gravity, impact, and pendular oscillation. Leibniz presents the new theoretical principle as one that should be capable of systematizing the phenomena under a productive organizing principle. This aspect is clearly alluded to in article 18 of the Discours de métaphysique, which insists on the heuristic and regulatory role of the new theory of conservation:

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This consideration, in which force is distinguished from quantity of motion, is of importance not only in physics and mechanics in finding the true laws of nature and the rules of motion, and even in correcting many errors in practice which have slipped into the writings of a number of able mathematicians, but also in metaphysics for the better understanding of the principles. (A VI 4 1558–59; L 315)

From this perspective, relying on the theoretical principle requires us to conceive of the analyses of phenomena as rightfully derivable from a system of causal reasons that surpass the limits of geometric representation. Indeed, the very order of phenomena must be able to express itself in mathematical or mechanical relations. Regarding the phenomena of motion, which constitute the object of geometric representation, the relations cannot lead to assigning cause and effect, since the chosen reference system arbitrarily determines the sequence of the series. The insufficiency of this system for determining relations of motive force reveals the necessity of developing a deeper level of active and passive elements. The concept of body is modified so as to account for such a structure on two levels. Nonetheless, there remains the problem of finding an adequate way of representing the “profound” structure corresponding to force or the “immediate cause of these changes.” (A VI 4 1559; L 315) From a Leibnizian point of view, the question seems all the more central given that force is more real than the phenomena through which it expresses itself, and that it represents in particular the substantial element in which the effects of motion inhere. Leibniz thus claims that force defines the real activity of bodies in their interaction with one another, and that “there is a sufficient ground for ascribing it to one body rather than to another.” (A VI 4 1559; L 315) This previous claim must be interpreted in conjunction with two other assertions that emerge. On the one hand, if the concept of body entails more than extension and its modes, this means that the element of form is reintroduced by analogy with the Aristotelean concept of substance. On the other hand, if one resorts to such an element, it can only be at the level of a general explanation concerning the order and underlying principles of physical phenomena. It follows from this combination of theses that causes cannot be assigned at the level of the phenomena themselves, but instead at that of their relation to one another as a whole. It is by virtue of architectonic arguments rising from reasons based on continuity and harmony that one can allocate the relative properties of activity and passivity to phenomenal bodies. Ian Hacking alludes to this problem. His thesis seems to be that Leibniz considers the theorem of the conservation of living force as representing a system of real and therefore causal references, in contrast with kinematic frames of reference that only have a relative and phenomenal meaning.40 40 Cf. Hacking (1985), 142: “Leibniz’s insight is that there is a fundamental difference between motion (represented by a vector v in some system) and vis viva (represented by mv2 in some system). The laws concerning vis viva are ‘real laws’, namely conservation laws, invariant laws. The laws con-

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Evidently, the problem is that measuring mv2 in phenomena does not permit us to empirically differentiate the subjects of causal activity from those that undergo the effects. As a result of this, Hacking points to a metaphysical level of explanation, which is distinct from the empirical level. To this dualistic interpretation we can oppose the thesis that a theorem like that of the conservation of living force serves as a regulative norm and allows us to privilege, as causally significant, certain kinematic analyses that would represent a structurally organized image of natural phenomena. The serial integration of the given cases would permit us to grasp that they form some continuous chain and consequently to label certain phenomenal entities as active and others as passive. The overall distribution of the system dictates, it seems, how causal responsibility should be presumably shared. Indeed, the quantification of living force represents the constant governing the system. And this constant secures the connection between the geometric reasons, which express the relationship of motions to one another, and the metaphysical reasons, which point to the combination of underlying forms that go beyond the relativity of motion.41 How is this connection possible? In fact, the new theorem of conservation serves as a basic theoretical postulate for discursively integrating the specific laws of nature; at the same time, it expresses these architectonic reasons that form the requisites of a metaphysical system of nature. Such is the way, at least, for untangling the apparent ambiguity of the physical principles that lie somewhere between the models for mechanical representation and metaphysical concepts: And although all particular phenomena of nature can be explained mathematically or mechanically by those who understand them, it becomes more and more apparent that the general principles of corporeal nature and of mechanics themselves are nevertheless metaphysical rather than geometrical and pertain to certain forms and indivisible natures as the causes of what appears rather than to the corporeal or extended mass. (A VI 4 1559; L 315)

The difficulty of determining the epistemological status of principles of this nature will reveal itself in the successive controversies to which Leibniz’s reform of mechanics will give rise, starting with his confrontation with the Cartesians. It is in order to circumvent, if not resolve, this problem that Leibniz will formulate the synthetic approach found in the Dynamica de potentia (1689–90). So begins the series of theoretical models that Leibniz will develop in order to articulate dynamics as a fundamental discipline.

41

cerning position, its first vector derivative, velocity, and its second vector derivative, acceleration, are merely phenomenal laws, none of which can be regarded as literally true.” Cf. Bernstein (1984), 102: “For Leibniz, true motion is essentially a metaphysical notion dependent on the absoluteness of an indwelling, non-geometrical force concept. In spite of the equivalence of hypotheses, there is a real internal difference between the intensity of force residing in body which is phenomenally displayed in (scalar) vis viva – the ‘live force’ of the body in motion – and mere geometrical change of relations.”

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4. Conclusion Leibniz’s first scientific synthesis, that of 1671, will be radically revised when Leibniz “invents” the reformed mechanics. This invention, which Leibniz presents as a refor­ matio, takes place at the beginning of 1678, while he is writing De corporum concursu. During his stay in Paris, in 1676, Leibniz studied the problems raised by the laws of motion and the impact of bodies in depth. His ambition is to surpass the relativist perspective embodied in Huygens’ and Mariotte’s presentations by bringing mechanics back to the realm of geometry. Leibniz believes that he possesses the key to possibly reconciling empirical laws with an a priori principle of conservation of the quantity of motion analogous to that of Descartes. This key would be the principle of equivalence between the full cause and entire effect. Leibniz plans to establish a system of definitional equivalents for cause and effect that provide a uniform measure of motive force in all cases. He presumes that this measure will be consistent with the product of mass and speed. In the Pacidius Philalethi (1676), he had noticeably divorced motive forces, conceived of as momenta motus, from the geometrical representations that they occasioned. But Leibniz does not then understand how to avoid the thesis according to which a continuous form of “transcreation” would underpin the chain of physical changes. In opposition to this form of “occasionalism”, he does not yet possess genuine physical categories for representing the permanence of the causal order in the agents of phenomenal interactions. De corporum concursu is an epistemological text unique for its kind. There, Leibniz undertakes a systematic “deduction” of the laws of impact with a view to establishing their consistency with the Cartesian principle of conservation of quantity of motion. He compares the theoretic calculations of the deduction with the results of an experiment based on the properties of pendulums and designed to measure the effects of relative displacement following the impact of unequal objects – the impact of equal objects is considered as the initial element of a graduated series of unequal impacts. In this experiment, one of the bodies is at rest and the motion of the other is generated by its fall from different heights. Leibniz then notes the disparity between the inferred results and the actual facts, which prove admissible in virtue of the theory of the experiment itself. He then reformulates or “reforms” the deduction by grounding it on the presumed conservation of the product of the mass and the falling heights of the bodies before impact; and this time, harmony is achieved between the different cases following a continuous gradation. Thus, precisely in the case of our physical universe, the system of nature must be organized such that this type of equivalence function between cause-occurrences and effect-occurrences be produced and maintained across physical changes. However, if the explanation is formulated architectonically following the equivalence postulate between the full cause and the entire effect, and is in conformity with a methodological rule of continuity, then the theory still functions without truly conceptualizing the underlying forces. Thus, the “metaphysical” level of the

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theory fails in a way, with the exception of a form of occasionalism that is close enough to the one suggested by the Pacidius Philalethi. For this reason, we are forced to notice that Gueroult’s theory regarding the metaphysical genesis of dynamics does not correspond to the model of 1678. Indeed, Leibniz will improve upon and perfect the reformed mechanics by developing a system of arguments and theoretical concepts, but one must understand that this attempt will be essentially driven by the search for a scientific formula capable of maximizing the coherence and functionality of this new mechanics. Evidently, and in good part, this theoretical refinement will bring about subtle conceptual distinctions at the border between science and metaphysics. Moreover, it is when he is able to develop a comprehensive metaphysical theory of his own that Leibniz officially gives birth to the reformed mechanics. This occurs in the short article Brevis demonstratio erroris memorabilis Cartesii et aliorum circa legem naturalem (1686); similar arguments are integrated into the Dis­ cours de métaphysique, which is written the same year, but remains unpublished. The context relates to the laws of nature. In opposition to the tenets of a mechanics reduced to empirical laws alone, Leibniz agrees with Descartes in recognizing that a principle of conservation of motive power must be a primordial component of the physical explication. But Leibniz relies on Galileo’s empirical law of falling bodies and axioms of equivalence accepted by the Cartesians to demonstrate the disparity between the Cartesian measure of the quantity of motion and the requisites of the concept of motive power. In order to respect the implications of Galileo’s law, and therefore the lesson learned from the experiment, he proposes a measure of conserved power based on the product of mass and the square of speed. At this stage of the analysis, Leibniz has not yet fully mobilized the true framework of his dynamics. Rather allusively, he suggests that that theory must bring models consistent with the experimental facts into agreement with the substrate of theoretical entities embodying the very order of the system of nature. It is henceforth recognized that a theoretical entity must be conserved; this entity is called potentia motrix, and its definition eludes the intelligibility of strictly geometric concepts. For the moment, the new principle of conservation is presumed to possess a regulative role in systematizing various empirical laws. And Leibniz holds that it can serve as Ariadne’s thread in all attempts at achieving the level of theoretical explanations. The characteristics of the theories targeted remain to be specified, but it is henceforth admitted that the causal reasons that will be thereby articulated will surely exceed the limits of a mere kinematic formalization of phenomena.

Chapter III The Structure of Dynamics In challenging the foundations of Cartesian mechanics in De corporum concursu (1678), Leibniz had certainly subscribed to the view that physical explanations must formulate determinative reasons consistent with the findings of empirical laws, namely the laws of impact. Consistency with the results of experimentation is a fundamental requirement of physical theory. At the same time however, the theoretical construction had resulted from the application of methodological principles to the analysis of empirical data. More precisely, Leibniz had relied on the postulate of equivalence between the full cause and the entire effect, and on the architectonic principle of continuity. Lastly, the basic propositions of the theory, and particularly the new principle of conservation, had seemed to exceed the conceptual norms of purely geometrical intelligibility. The problem to which Leibniz will dedicate a large portion of the remainder of his scientific career is epistemological in nature, and thus also methodological. It involved adequately conceiving of the way in which a mechanical theory based on the concept of force could be formulated, both with respect to organizing principles and inferences, and to justifying concepts that provide sufficient reasons for its explanation. Even if the issue of the synergy between metaphysics and science is constantly encountered in Leibniz’s approach – something which the majority of commentators have noticed – his successive attempts at establishing what will become known as the dynamics will be centered around the problem of internally organizing its theoretical structure. From this perspective, Leibnizian science constantly entails examining the modalities of its own constitution. From 1686 to 1690, or from the Brevis demonstratio to when the Dynamica de potentia is drafted, which includes the Phoranomus situated between the two, this problem will be the focus of many controversies. Beyond the systematic development effected by the Dynamica, it will also continue to influence subsequent conceptions of dynamics, including those of the Animadversiones in partem generalem Principiorum Cartesianorum (1692), the Specimen dynamicum (1695) and the later Essay de dynamique (c.1700). The theorem of conservation of living forces is henceforth considered a constitutive as well as regulatory principle regarding the laws of nature; as a result, many subsequent analyses will focus on the formative criteria of the physical theory to be constructed.

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Given this host of analyses, we must order the texts so as to distill their epistemological implications. A major feature of Leibniz’s thinking is that it examines problems step by step. The argument advanced by certain versions of reformed mechanics and dynamics is analytic in nature, the Brevis demonstratio being a prime example of this, in addition to the later Specimen dynamicum and Essay de dynamique. It is not insignificant that among these analytic attempts, one historically finds a synthetic and systematic presentation of the arguments, the presentation of the Phoranomus being a first attempt at this, and that of the Dynamica de potentia being a model for it. It is as if the analytic phase had reached the point at which a synthetic construction became necessary. Then, this synthetic phase seems to have required further analytic developments. It is true that these various texts possess significantly different statuses. Analytic texts would respond to the need for reaching a greater audience; by contrast, the Dynamica de potentia would be part of a projected demonstration aimed at specialists initiated in the ways of more geometrico expositions in physics.1 It could even be viewed as a sort of counterpart to Newton’s Philosophiæ naturalis principia mathematica (1687). The subsequent return to the analytic method could be explained by the requirement that the so to speak metaphysical and methodological foundations of dynamics as a science be further explored. One should not however strictly identify the synthetic phase with the search for a priori demonstrations. Part of the subsequent correspondence with Johann Bernouilli, Burchard De Volder, Christian Wolff and others will focus on the possibility of such proofs, which had been drafted in the Phoranomus and Dynamica, and which analytic versions refrain from resorting to. But this debate, otherwise quite fundamental, takes a backseat to the argumentative layout of the main texts in which Leibniz reveals his dynamics. We will reserve our interpretation of this controversy for later, when we examine in particular the epistemological status of causes and laws for Leibniz. Our interest here is retracing the argumentative phases where Leibniz’s treatment of dynamics after the schematic and allusive presentation of the Brevis demon­ stratio can be found. We will therefore trace the limits of the framework underpinning attempts to formulate a priori proofs, which will appear to the majority of commentators as being more fragile and problematic. In a published article, De primæ philosophiæ emendatione et notione substantiæ (1694), Leibniz mentions that he dedicated a special science to the notion of force, i. e., “dynamics,” and that this methodological development elucidates the metaphysics of substance.2 This assertion causes a problem to a certain extent. Leibniz’s previously

1 The Phoranomus, where Leibniz first outlined his synthetic argument for the dynamics was, although exoteric in style, aimed at an audience familiar with the methods of geometrical physics. 2 Cf. GP IV 469; L 433: “To give a foretaste of this, I will say for the present that the concept of forces or powers, which the German call Kraft and the French la force, and for whose explanation I have set up a distinct science of dynamics, brings the strongest light to bear upon our understanding of the true concept of substance.”

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published scientific texts do not support the claim of having founded dynamics as a science. If the unedited texts show that, since De corporum concursu (1678), Leibniz had been in possession of his theorem of conservation of living force, then the first official attack on Cartesian mechanics came in 1686 in the Brevis demonstratio erroris memorabilis Cartesii, the argument of which reappears in the Discours de métaphysique. Indeed, since the Brevis demonstratio, Leibniz worked to demonstratively invalidate the Cartesian principle of conservation of quantity of motion. He proposed in its place a new measure of motive force that is conserved in mechanical exchanges, one based on estimating the absolute power of living force (expressed as the product of mv2). Afterwards, the quarrel with the Cartesians exploded over the possibility of “deducing” this new principle presenting itself as the foundation of the system of the laws of nature.3 On what grounds could Leibniz have pretended to have founded dynamics as a science? For Leibniz, there was no question that a genuine science entails a system of demonstrative arguments that makes synthetic exposition possible.4 The analytic procedures implemented as a posteriori proofs of a new principle of conservation would suggest the possibility of establishing a system of demonstrations and conceptual connections as the theoretical basis for explaining phenomena. But this objective seemed attainable only if Leibniz had showed that the analysis resulting in the principle of conservation of living force was constructed such that it could, in turn, give rise to a synthetic exposition capable of illustrating the new principle’s value as the lynchpin of physical theory. It is with this objective in mind that Leibniz began drafting the Phoranomus (1689), and then the Dynamica de potentia (1689–1690)5 in the period immediately preceding the statement that we extracted from De primæ philosophiæ emendatione. From these two texts a canonical version of dynamics as a science will emerge. How are we to understand and interpret, with regard to its genesis, purposes, and argumentative structures, this more geometrico theoretical framework in which the dynamics project unfolds? Once this framework has been restored, we shall reexamine the way in which it influenced the subsequent analytic expositions, those principally introduced in the Specimen dynamicum (1695) and later Essay de dynamique (c.1700). How did Leibniz reconfigure and adjust the arguments of dynamics after having attempted to formulate them in keeping with the demonstrative requirements of the synthetic scientific approach?

3 4 5

Regarding the significance of the vis viva controversy, cf. Laudan (1968); Iltis (1971); Gale (1973); Papineau (1981). Cf. Duchesneau (1989); Duchesneau (1993), 86–101. Regarding the relationship between the two texts, cf. Robinet (1989), 171–186; Robinet (1988), 81–95.

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1. The Phoranomus: A Turning Point A decisive step in the emergence of dynamics as a science occurs in the Phoranomus seu De potentia et legibus naturæ, a complicated and unfinished draft undertaken during the summer of 1689 during Leibniz’s stay in Rome.6 The Phoranomus seeks to found a science capable of integrating and complementing the reformed mechanics, which was conceived of in 1678, and had been developed ever since. Owing to the theoretical discussion it sparks, it marks the transition that underlies the emergence of dynamics proper.7 The Phoranomus is presented, like the Pacidius Philalethi in 1676, as a dialogue. But this new dialogue has strong realist connotations, since it seems to closely reflect the valuable discussions that Leibniz potentially had with Roman scientists at the Accademia Fisicomatematica where he regularly visited. Indeed, in two successive sessions, illustrated respectively by Dialogus I and Dialogus II, the Phoranomus introduces Leibniz and certain members of this academy exchanging on several of the central themes of Leibniz’s reformed mechanics and developing dynamics. An introduction written in the form of a letter that Leibniz had presumably addressed to a French correspondent, Melchisédec Thévenot, a learned member of the Académie des Sciences in Paris, sets the stage with a critical evaluation of Cartesian philosophy and science. As Simon Foucher, himself a critic of Cartesianism, would have informed Leibniz, Pierre-Daniel Huet had recently published Censura philosophiæ cartesianæ (1689). Reacting to this news, Leibniz invokes the critique of Descartes’ physical hypotheses that he had previously leveled, and states his belief in the necessity of reforming and updating Cartesianism.8 The errors made by Descartes regarding the establishment of the laws of nature motivated Leibniz’s research, the result of which is presented in the Phoranomus.9 In this regard, there is no doubt that the work takes up the task of the Brevis demonstratio of 1686 and, going even further back in time, that of the De corporum concursu of 1678. But the specific objective remains to be determined with greater precision. 6

Regarding the stay in Rome, which occurs from mid-May to 20 November 1689, cf. Robinet (1988), 41–192. 7 A. Robinet reconstructed this text from manuscripts: cf. Dialogus I: LH XXXV, IX, 1, f.1r–6v; LH XXXV, X, 11, f.89r, LH XXXV, IX, 1 f.7r–18v (+ copy of the biggest part: LH XXXV, IX, 1, 19r–40v); Dialogus II: LH XXXV, X, f.8–86; LH XXXV, IX, 41r–54r. And he provided this in an annotated edition: cf. Leibniz 1991a and Leibniz 1991b. An improved edition of the Latin text with an Italian translation has been provided by G. Mormino: Leibniz (2007), 679–930. 8 This critique appears, for instance, in his correspondence with Hermann Conring in 1678. Cf. Leib­ niz’s letters on 3/13 January, 19/29 March, and June 1678, A II 12 578–84; 597–607; 630–34. 9 Cf. Phoranomus I, § 1, Leibniz (2007), 682: “Certainly, because he made errors in founding the laws of nature, [Descartes] gave me the opportunity to establish the real ones, as it will appear from this dissertation.”

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Incomplete and imperfect, the Phoranomus is essentially a rough draft that introduces the mechanical theses regarding force that the Dynamica de potentia will develop and articulate as part of an entire system immediately following the works of the summer of 1689. Up until now, as De corporum concursu and the Brevis demonstratio attest, Leibniz had founded his recognition of a new principle of conservation of motive power, represented as the product of mv2, on a demonstration that was at least partially a posteriori. This demonstration depended in effect on Galileo’s empirical law of falling bodies and required a universe with gravity – a phenomenon for whose explanation Leibniz and his contemporaries relied solely on hypothetical models. Indeed, Leibniz also relied on axioms derived from Archimedean statics that had been reformulated by mechanists in the seventeenth century. Above all, he based himself on an architectonic principle that required a sufficient reason for motion in the phenomenal universe, the principle according to which the entire effect and the full cause must be equivalent in mechanical interactions. It is because the motive power generated by the fall of a body from a given height is completely expressed or consumed by the elevation of this object to its original height that the product of the mass and distance traversed, which is proportional to the square of the speed, measures the force that is preserved across the series of processes wherein force is generated and destroyed. But does such conservation not require a sufficient reason that would operate within the boundaries of a particular system of phenomena? And do the characteristics of this system not depend on an empirically founded model? If such is the case, and we must admit as much, then Leibniz’s reformed mechanics is justified by a demonstrative system constructed and founded a posteriori, at least with respect to a significant portion of its premises. The challenge that Leibniz focuses on in the Phoranomus is universalizing the measure of motive power for all phenomena involving motion. The project requires identifying the same underlying causal influence, whether it involves a violent effect, such as in impacts that exhaust motive power, or an unconstrained effect, such as the one that occurs in a uniform translational motion that is neither facilitated nor hindered by the intervention of an external force. It is this that Leibniz qualifies as “uniform motion such that it would be entirely moved by itself ” (motus æquabilis qualis per se est omnis). (II §G, Leibniz 2007, 817)10 Attempting to calculate a priori the force involved in this unconstrained motion, Leibniz will lean toward a more a priori system of arguments in which empirical premises would only serve as confirmation. At the same time, he

10

Robinet (Leibniz 1991b, 821) uses the expression “essential motion” in this context. The expression is a little confusing since Leibniz could not have possibly interpreted motion, which is always relative, as the essence of some reality. Extension itself, which for the Cartesians is the essence of corporeal substance, and the basis for the modes of motion, belongs, according to Leibniz, to the phenomenal order as a non-essential property, but one founded on finite realities that are ultimately immaterial.

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will present himself as having established dynamics as a science after having unveiled its theoretical foundations. A significant clue in this respect is the fact that the term itself “dynamics” begins to appear in Leibniz’s science in the wake of the Phoranomus,11 where it serves to characterize a new science that the Dynamica de potentia intends to found. Thus, while the Phoranomus took a rather exploratory approach to the new science of force and effect (nova de potentia et effectu scientia), (II, §G, Leibniz 2007, 824) the great work that follows it will set out to demonstrate this new science regarding force and action (nova de potentia et actione scientia), which is identified by the term and concept of dynamics.12 In reference to the birth of dynamics properly speaking, Robinet’s interpretation, which is based on a reconstruction and analysis of the manuscripts, merits some critical attention. Three major elements seem to characterize it: (1) a certain conception of the authors and theses to which Leibniz responded critically in his essay on the universalization of reformed mechanics; (2) a radical view of the paradoxes that emerged from a theory of force constructed a priori on the basis of the unconstrained effects of motion; (3) a so to speak “causal” representation linking the “failure” of the Phorano­ mus to the birth of the Dynamica de potentia and the emergence of the dynamical science. This text and the specific status given to dynamics would result from relying on an a priori demonstration. But this time the process would refer to a concept of action that surpasses the simple concept of formal effect, and would thereby avoid the paradoxes that Leibniz encountered in his attempts at systematization during the summer of 1689. In preparing the Dynamica de potentia, Leibniz, it seems, adopted the term “dynamics” to designate a new science for the laws of nature. Also, and in this regard, the gaps in the Phoranomus would have created the necessary pretext for transitioning from the reformed mechanics to dynamics properly speaking. These three fundamental points can be appreciated for various reasons. Let us first consider the theoretical context in which Leibniz’s analysis appears. Previous interpreters of the dynamics, beginning with Gueroult, misunderstood and neglected the

11

If it is still not used in the former work, it appears, as Robinet points out, (Leibniz 1991a, 432) in the correspondence with Rudolf Christian von Bodenhausen, and thus in direct connection with the writing of the Dynamica de potentia. 12 Cf. Specimen præliminare, GM VI 287: “Thus, I thought it worthwhile to grasp the force of my argument by most evident demonstrations, so as to establish little by little the true elements of a new science of power and action that may be called dynamics [Dynamicen]. I have added some initial presentation of these to a special treatise from which I wished to cite the present excerpt and thereby excite minds to enquire after the truth and receive genuine laws of nature in place of imaginary ones.” Also, at the end of the first section of the first part, GM VI 464: “Thus it seems that I have uncovered sources for the new dynamic science on the nature of power and action that had not been sufficiently explored yet.” In a preliminary piece entitled Conspectus operis, (GM VI 284–85) Leibniz already uses the term dynamica in a context both involving abstract analysis and concrete analysis, regarding action and force respectively, and the active causes and effects at play in the “system of things”.

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Phoranomus, and instead readily attributed the Dynamica de potentia to the desire to construct a deductive system capable of competing with Newton’s mechanics, which is explicitly justified a posteriori. And it is true that, at the moment when he set out to found the dynamics, Leibniz had recently become acquainted with the Philosophiæ naturalis principia mathematica (1687), and was already working toward defending his hypothesis of harmonic circulation against the gravitational theory of Newton and his disciples. However, reading the Phoranomus reveals quite another ambition. Robinet rightly highlights the strategic importance of the scientific elements of the Galilean tradition in the two dialogues. These elements clearly serve as the same point of reference for Lubinianus’ interlocutors in the dialogues, regarding mechanics and mathematical physics. Leibniz would therefore attempt to show the superiority of his demonstrative models in comparison to those that Galileo and his disciples were able to advance where, as a last resort, one had to content oneself with empirical validity, having failed to provide perfectly rigorous geometric demonstrations. Unquestionably, the text attempts to critically resume and overcome the Galilean tradition, along with the Archimedean statics on which it was partly based. However, at the same time, a dominant concern of the text is continuing the critical refutation of Cartesian mechanics and its flaws regarding the conservation of quantity of motion. Indeed, this two-pronged attempt to overcome Galileo and Descartes in no way escapes those who contrast it with the tenets of a direct and nearly exclusive focus on Newtonian mechanics. In our view, two complementary remarks must nonetheless be made here. On the one hand, the evolving theoretical framework of Leibniz’s science in the Phoranomus is clearly sui generis. It distinguishes itself in an interesting way with respect to the hypothetico-deductive model that Cartesian physics seemed to offer. More so than in any other text perhaps, Leibniz’s very unique project for developing a system of demonstration based on a priori principles and proofs reveals itself here. Furthermore, can Leibniz’s immediate reasons for distancing himself from Cartesian mechanics not be attributed to the recent controversy with Catelan and Malebranche, which followed the publication of the Brevis demonstratio, whose highpoint was undoubtedly the publication of the letter on the principle of continuity and its role in the theorization of mechanics?13 Through the characters Auzout and Ciampini, Leibniz introduces in particular the Cartesians’ main objection, namely, that time should be taken into account when interpreting the implied relations of motive force in the elevation of bodies in which power has accumulated from the preceding fall. (Leibniz 2007, 846–48) If one subjects phenomena to the same measure of time, does the conservation of the quantity of motion not prevail over that of the quantity of living force? 13 Cf. 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 server de réplique à la réponse du R. P. D. Malebranche, GP III 51–55, and my analysis of the principle of continuity according to Leibniz (Duchesneau 1993, 311–74, Duchesneau 1994).

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Such is the case with the summation of the impetus when calculated in the first instant without subsequent integration of the conatus. The same occurs in statics when bodies are attached to the extremities of the arms of a balance according to a ratio of distances inverse to that of their respective masses. There, the time it takes for each constrained vertical displacement to occur corresponds to the conservation of mv, rather than of mv2. Making use of the character Baldigiani, Leibniz already had the opportunity to expose the paradox encountered by the Cartesians with regard to the Galilean model of falling bodies. Given an empirically established equivalence between the full cause and the entire effect in cases of falling bodies, the relation mv2 is implied as a constant. Why, under these circumstances, does one tend to relate the elevation of bodies to the respective speeds of the moving objects? The reasons for this error can be traced to a principle in statics: two bodies are in equilibrium when their respective situs is such that if one begins to descend, the motions created thereby imply speeds that are proportional to the respective masses of the two bodies. Hence the presumptuous tendency of inferring, as if the principle were evident, that “powers are equal where speeds are inversely proportional to bodies”. (Leibniz 2007, 810) Here Leibniz argues that statics presents us with a unique case in this sense, insofar as the heights are proportional to the speeds, whereas the dynamical account of cases in which the expression of force is allowed to develop implies the proportionality of masses with the square of the speeds. In any event, the powers are equal when the masses are inversely proportional to the heights to which they can raise themselves. The error of the Cartesians lied in assuming that the laws governing connected and therefore constrained bodies also applied, more generally, to unconstrained bodies and the motive force that they generated and consumed. 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 proportional to the heights to which they can be lifted. […] This was indeed known to the learned, but as they had only envisioned lifting bodies that are bound together rather than freely ascending, and therefore had not examined bodily motion in this form, they were deprived of the excellent fruit of a very general truth, which sheds the most light on the communication of motion and the interaction of bodies. (Leibniz 2007, 832–34)

However, Leibniz’s true counterargument in the Phoranomus to this specious objection of the Cartesians consists in demonstrating that the temporal limitation imposed on the integral expression of motive powers proves incompatible with the architectonic rule of continuity. In effect, the application of this criterion reveals the apagogical consequences resulting from the incompatibility of the temporal clause with the rule according to which the common center of gravity of a system of bodies cannot, whether by gravitational force or some other cause, be raised higher in the effects than in

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the causes.14 In the first place, the demonstration directly depends on the empirically corroborated hypothesis of the conservation of the common center of gravity. Secondly, in order to raise the supporting principle to an a priori epistemological status, Leibniz relies on the counterfactual supposition that, if there were a divergence from the conservation of the common center of gravity, the implied consequence would be an admission of perpetual mechanical motion. The same type of apagogical demonstration allows for eliminating the divergences that would presumably originate from bodies taking a more or less oblique trajectory, which would imply different lengths of time for the same total effect. Eventually, Leibniz works to systematically overturn the objection formulated by Catelan and the Cartesians by asserting that, in violent motions, the space traversed, i. e., the height achieved by the total exhaustion of power, implies that time is already included in the measure of speed squared with respect to this height. Hence the redundancy of factoring in time across the space traversed. This obviously entails, as we shall see, the opposite problem of reducing motion to a formal effect lacking any external causal restraint. (Leibniz 2007, 866–68) In short, it seems that the demonstrative formulation of the arguments against the Cartesians, which descend from the arguments of the polemic with Catelan and Malebranche, determines first and foremost the demonstrative construction that Leibniz is attempting here. The second point to be highlighted concerns the paradoxes of an a priori model of force based on the analysis of the effects of unconstrained motion. The model is presented as follows. Its architectonic premise consists in the principle of equivalence between the full cause and the entire effect, but its applicability extends beyond the specific case of our system of things. In this system, phenomena of motion in colliding bodies are in effect interpreted with the help of an empirically supported secondary hypothesis, i. e., the Galilean law for falling bodies, which applies where impetus is generated by gravity. According to the system of constraints in colliding bodies, motive force is measured by the violent motions that it produces, and by virtue of which it is completely consumed. The sought-after demonstrative approach involves, by contrast, abstractly conceiving of the measure of force on the basis of the resulting unconstrained motion, as if the latter expressed such an intrinsic causality in and through its uniform progress alone. However, in this new case, which abstracts from the hic et nunc conditions of the system of things, the same fundamental premise, the same “thread of Ariadne” identified previously, that is, the architectonic principle of equivalence between the full cause and the entire effect, must be applicable. Even if this principle

14

Cf. Leibniz (2007), 848: “I shall therefore furnish demonstrations that are clearer than any that mechanical mathesis has ever seen. And indeed I shall obtain the same result by various means. To begin with, I think you will grant me this hypothesis, which resembles those that the learned have occasionally used: namely that neither by the force of gravity, nor by the force between bodies resulting solely therefrom, does one find the common center of gravity of bodies higher in the end than it was in the beginning.”

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seems to be more general, it is initially applied in the context of Archimedean statics,15 and in that of the phoronomy of moving bodies (in cases of violent motion).16 In fact, the argument Leibniz tries to develop in the a priori mode in the Phoranomus stems from Archimedean statics. He assumes that his interlocutor, in this case Charinus, relies on the equivalence between the inverse proportions of mass and speed for unequal bodies in equilibrium and applies it to account for free translation on a horizontal surface. This transition from statics to kinematics introduces an aporia between the purely modal status of speed, and the “substantial” and therefore real status of bodies, complicating the calculation of motive force.17 If a body with two units of mass can be assigned twice the force of a body with one unit of mass moving at the same speed, then the same would not apply to a body being displaced at a speed of two in relation to a body of the same mass being displaced at a speed of one. Indeed, the power of the first body does not, according to the parameter mv, equal twice the power of the second body when the speeds achieved in a given time are geometrically represented as traversed space. As Lubinianus-Leibniz highlights, it is only by proposing a proportionality based on the inverse ratio of the times of traversal and the direct ratio of the products of masses and spaces traversed that one can reestablish a system of equivalences regarding the motive forces that are involved. Given the same bodies and the same spaces, the powers will be inversely proportional to the times it takes to traverse them. And from this it follows that in a given unit of time, the powers will be measured by the product of msv, or, owing to the equivalence of s and v regarding displacements measured in terms of a given unit of time, they will be measured by the product mv2. But this argument can easily appear fallacious, having hardly more validity than the fallacy according to which a body with a mass of one and a speed of two equals a body with a mass of two and a speed of one. The flaw of such an equation lies in the lack of congruence between the cases (casus), for the cases in question would combine states (status) and things (res) and could not possibly generate homologous relations. “The question is not what the moving body can do, but […] how a proposed case is estimated, given the time and space, that is to say, how the given case is resolved into two cases equivalent to each other and from which it is compounded.” (Leibniz 2007, 810) The thesis that Leibniz affirms a priori rather than proves postulates that if the spaces traversed and the inverse of the traversal times are considered together, then the two cases may be seen as

15 16

17

Cf. I, § 18, Leibniz (2007), 742. Cf. II, §D, Leibniz (2007), 802: “To finally escape this labyrinth, the only thread of Ariadne that I required was a calculation that assumed this metaphysical principle, that the entire effect is always equal to the full cause. As I admitted that it perfectly accords with the experiences and resolves all doubts […].” This is apparent in very difficult passages in the Phoranomus, cf. Appendix II, Leibniz (2007), 880– 84.

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congruent. The double spatio-temporal status combined with the res would designate a measure for the motive force that gets expressed in unconstrained uniform motion: Thus, through the resolution of bodies into parts, the speed, or space and time, being conserved, we inferred and demonstrated that, given the same speeds, the powers were proportional to the bodies. Similarly, we demonstrated, which is paradoxical but absolutely true, that, the body being conserved, and time and space being resolved jointly (for otherwise the given case could not be divided in several cases congruent with, but different from, one another), given the same bodies, the powers are proportional to the square of speeds. (Leibniz 2007, 882)

The justification provided for this system of equivalence between freely expressed powers owes therefore to a combination of relations involving modal parameters and signifying, via homologous combinations, the respective effects of these forces. On the one hand, it seems that no other element of motions can be accounted for in the synthetic representation of the cases. On the other hand, this combination of modal elements suffices for producing an analytic equivalent of the powers whose effects uniform and unconstrained motions express; following, by contrast, the Cartesian calculation modeled after Archimedean statics, speed alone could not suffice to express this effect of motive forces, no more than the space traversed could be considered as the only essential parameter.18 Ultimately, Leibniz’s thesis, as Baldigiani summarizes it, essentially affirms that the equivalent effect of powers in unconstrained uniform motions (motus æquabilis qualis per se est omnis) is determined by a combination of two “requisites”: space traversed and the speed of displacement. (II, Leibniz 2007, 814) Leibniz cannot justify his measure for the effect other than by combining the effect considered in the unit of time as a sort of equivalent effect that accompanies the actual effect as it gets entirely consumed. The suspected flaw of this analysis may lie in inadequately distinguishing the extensive and intensive components mixed together in the effect, which is constantly regenerated by a living force that remains intact throughout the motion. More than anything else, Leibniz still lacks a causal model for “metaphysically,” that is abstractly, accounting for this combination of intensive and extensive factors. Owing to this deficiency, the model for congruent casus undoubtedly remains aporetic, and ultimately is only secured by an analogy borrowed from the a posteriori demonstration of the principle of conservation of living force. The complex writing borne out in the passages concerning an a priori demonstration in the Phoranomus are no doubt consistent with an

18 Cf. Phoranomus II, Leibniz 2007, 812: “Therefore the estimation of powers will be the same when the cases that we raised are nothing but the effects of powers. And the extent to which this conforms to reason appears from the fact that in uniform motion, speed is not determined by a given space traversed, or the contrary. This is why both reasons must be conjoined, which in the case of equal times increases [the speed] twofold.”

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insufficiently developed model for justifying the combination of parameters regarding action in its essential expression. Despite the shortcomings of this notion of effect, the methodological model invoked in the Phoranomus seems to suggest a demonstrative pattern for generalizing the principle of conservation of motive power so as to include cases of unconstrained motion. In the Phoranomus, Leibniz likens this project to Galileo’s own for extending, as far as possible, the demonstrative structure of geometry to natural phenomena. The protagonists of the dialogue emphasize the necessity of combining experiments with formal models by having recourse to mathematical analysis and the various techniques that it offers. But invoking this demonstrative structure also means having recourse to principles. However, beyond the Archimedean inspired statics, it does not seem that Leibniz has yet found the principles for grounding and securing a system of demonstrative inferences for the phoronomy, i. e., the part of mechanics that deals with the communication of motion between colliding bodies.19 It is this challenge that Leibniz intends to address in the Phoranomus. To attempt this demonstrative expansion, Leibniz uses a key that a mathesis mechani­ ca in development is supposed to provide.20 This mathesis combines several conceptual instruments that Leibniz seems to have employed in a more or less unconnected way in his previous works concerning the reformed mechanics, including: (1) a principle that we will qualify as “architectonic,” the principle of equivalence between the full cause and entire effect, which underpins the arguments previously described in the reformed mechanics items; (2) models directly borrowed from infinitesimal geometry that serve to represent static and phoronomic phenomena involving transition to a limit; (3) models invoked to represent the combined ingredients of theoretical concepts relating to the order of causes and effects; and finally, (4) abstract definitions of these conceptual ingredients. These definitions are in a manner of speaking “metaphysical” because they go beyond that which is geometrically representable, and as a result could not possibly be based on some imaginative schematization. In an undoubtedly less analytic manner, this methodological program is announced in the introductory remarks that Baldigiani addresses to Lubinianus-Leibniz. Just as geometry is subject to an analytic calculus owing to the equality between the whole and all its parts, in mechanics, owing to the equality between the effect and the entirety of its causes, or between the cause and the entirety of its effects, certain so-called equations and a sort of mechanical algebra are attained by the use of this axiom. Thus, to your great

19 20

Cf. regarding this question, Phoranomus, I, § 5, Leibniz (2007), 694–96. The expression is used by Leibniz in the passage where he intends to respond to the objections like those that Catelan and the Cartesians had raised. These objections seem at first glance to depend on the recourse to models that are more purely geometric than those that Leibniz intended to develop. Cf. Phoranomus, II, Leibniz (2007), 848: “I shall therefore furnish demonstrations that are clearer than any that mechanical mathesis has ever seen.”

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advantage, you shall furnish a most vitally useful science, should you illuminate the great darkness we inhabit, and impose laws not only on statics, which Archimedes held captive, but also on universal phoronomy and the explanation of moving forces. (I § 6, Leibniz 2007, 696–98)

Charinus’ arguments for advancing the discussion indicate the same methodological goal of developing a mathesis mechanica that would reach to the level of abstract intelligibility required for the new theory of force. In geometry and numbers I observe evident principles of unavoidable necessity. Everything gets explained by parts of the same magnitude being arranged in different ways. But the moving forces seem to me to possess an unknown incorporeal something that is not at all subject to the imagination. Therefore, every time I would conceive of the powers of machines, I was confronted with something unexplored and not admitting of any image. (Phoranomus, I § 8, Leibniz 2007, 702–704)

We can now examine the various elements introduced in the mathesis mechanica, but above all, determine the extent to which they are successfully combined into an adequate demonstrative strategy. The formulation of the principle of equivalence between the full cause and the entire effect is the leitmotiv of the arguments in the Phoranomus. As this principle had already furnished the demonstrations of De corporum concursu and Brevis demonstratio, one could not argue that it plays an entirely original role here. But here, Leibniz sets out to reinterpret the entirety of Archimedean inspired statics in light of this principle, and in so doing, broadens so to speak the sphere of what can be geometrically represented so as to include the causal perspective of the underlying motive forces. The universalizing role of the principle with respect to its previous and necessarily limited uses should therefore be noted. An architectonic principle essentially helps formulate the theoretical explanation by providing it with a heuristic framework. This heuristic function is also twofold, since, on the one hand, it eliminates erroneous models, and on the other hand, imposes a method for formulating models that best satisfy the presumption of sufficient reason.21 Consistent with this pattern, the principle of causal equivalence plays a determinative role in extending the model from statics to phoronomy, that is, beyond cases of relative equilibrium between moving bodies, or cases of conservation of the common center of gravity among interacting bodies. (I § 18, Leibniz 2007, 742) Although static phenomena are generally susceptible to geometric representation, which would allow us to restrict the study of causes to extension parameters, the transition to the theory of forces assumes that one is now looking for

21

Cf. regarding this, Duchesneau (1993), 374–79.

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a system that may represent causes of changes arising from the collision of moving bodies.22 The abstract representation of these determinative causes by adequate models derived from infinitesimal mathematics and from certain definitions of theoretical concepts ensures that the principle of equivalence applies to this new domain. In the passage that immediately precedes the a priori argument concerning unconstrained uniform motion, Leibniz notes that this “metaphysical” principle seemed to him to provide an Ariadne’s thread for estimating forces. Moreover, insofar as it also allows inferences consistent and conformable with experimental data, this same principle yields a system of causes irreducible to “some blind property of nature,” and reflective of an “intelligent order” that makes itself known via “metaphysical reasons.” (II, Leibniz 2007, 802) Here, we understand that the development of models, particularly theoretical models relating to the intelligible representation of force as a cause, made the application of the principle of equivalence in the context of phoronomy possible. Hence the critical importance of such models when it comes to conceiving of mathesis mechanica. These models, as we have seen, aim on the one hand to promote a geometric analysis inspired by infinitesimal calculus, and on the other, to develop a system of concepts capable of representing the causal and formal elements behind the phenomena relating to motion. We must now consider the reference to infinitesimal geometric models. Even when he continues to develop Archimedean statics, particularly the theory of the centers of gravity, Leibniz takes advantage of the techniques of the transition to the limit for representing infinitesimal ratios. But models of this type appear more significant when one turns to representing acceleration and deceleration as constrained effects of the centrifugal or gravitational force. Similarly, the models put forth for understanding phenomena of resistance will be grounded on the same logic of analysis that relies on series of infinitesimal ratios. Leibniz starts with his conceptual distinction between vis viva and vis mortua. He stipulates that the relation between “dead” and “living force” is analogous to that of the finite to the infinite, or, to that of a point, as the beginning of a line, to the line itself. Conatus corresponds to dead force, while impetus to living force. (I § 18, Leibniz 2007, 742–44) Such are the first distinctions of the reformed mechanics that Leibniz will refine in the texts that characterize the dynamics. Impetus is conceived of as being generated by the continuous accumulation of conatus in the moving body. One therefore

22

Even if Leibniz draws a connection between his use of the term “phoronomy” and the Phoronomica of Joachim Jungius, he insists, contrary to Jungius, on not limiting himself to purely geometric and kinematic considerations. Cf. I, § 23, Leibniz (2007), 755: “but living forces, that is, impetus, belong to the phoranomica.” What Jungius did not take into consideration were “the very laws of nature that motive forces observe in the communications of motion”.

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passes from a latent (or embryonic) stage of motive force to a developed one. In this regard, Leibniz recalls that Galileo and Giovanni Alfonso Borelli had likewise contrasted the force of percussion with gravitational force, as the infinite to the finite. All bodies in motion thus possess an impetus, and this represents the existence of a living force generated by the summation of conatus. It is noteworthy that while the proposed model adequately identifies the relation of conatus to impetus, it does not clearly show how the mathematical expression of impetus distinguishes itself from that of living force. However, as with his more definitive theses, Leibniz relates inertia to the action of conatus, as the resistance of bodies to motion would imply inverse summative processes. If all motion, no matter how small, and belonging to any body whatsoever, can act upon another body, no matter how large, then the speed communicated to the second body and relating to the impetus will be the smaller as the body affected will be larger, according to the ratio between their respective impetus. “But if the surrounding bodies do not interfere, it is certain that […] a body, no matter how big, can be moved by a body, no matter how small. And the inertia of matter does not consist in absolutely resisting being moved, but rather in receiving less speed when the receiving matter is greater.” (I § 20, Leibniz 2007, 746–48) However, with reference to the explanatory reasons derived from the infinite accumulation of elementary conatus, one must, more precisely, adequately represent the effect of gravity, which would allow for understanding the acceleration of motion in falling bodies and, indirectly, the force conserved in bodily impacts. The method for explaining gravity relies on an analogy with the summation of elementary moments of centrifugal force, such as it had been defined by Huygens. But Leibniz interprets this analogy on the basis of his own algorithm for the summation of infinitesimal quantities, so that, in centrifugal force, he distinguishes the vis impressa from its cumulative effect in the impetus centrifugus. (I § 21, Leibniz 2007, 750–52) But how can we transition from this to a model for the force of gravity? Leibniz insists on the fact that gravity is a physical phenomenon the causes of which remain obscure and controversial. He clearly refuses to exclusively explain it as a form of attraction with which the divinity could have endowed matter. Rejecting Newtonian attraction – without naming Newton himself – on the grounds that this would contradict the requirement for a sufficient mechanical reason, the author of the Phorano­ mus, following in the footsteps of Kepler, Descartes and Huygens, imagines that, by virtue of causes similar to centrifugal impressions and their summation in the form of impetus, highly dense but imperceptible bodies tend to move from the center and push bodies with less density toward it; as a result of this, the bodies are less likely to move from the center. (I § 21, Leibniz 2007, 748–50) Leibniz even designs an experiment for the purpose of directly comparing centrifugal force with gravitational force.23 This

23

Cf. I § 22, Leibniz (2007), 752. This experiment will be taken up again in the Dynamica de potentia, GM VI 452.

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experiment involves an obliquely rotating tube, the shorter end of which is submerged in a body of water. Owing to the centrifugal force resulting from its rotation, the water can rise in the tube to a certain height. This effect offsets the corresponding effect of gravity and can therefore serve to measure it. Thereafter, Leibniz recasts Galileo’s account of the empirical law of falling bodies according to his own model of infinitesimal summation. Further on, the same type of model can be applied for establishing laws of transition to account for changes of states owing to the elasticity and resistance of bodies.24 The third aspect of mathesis mechanica relates to the combination of conceptual elements that serve as the framework for hybrid models. Hence one finds, against the backdrop of the a priori argument, a more or less explicit combinative conception of the notion of “effect” with respect to unconstrained uniform motion. It is, one could say, owing to the analytic requirement of a similar model that the Dynamica de potentia will substitute this provisional notion of effect with the notion of formal action, which combines intensive and extensive factors such that the formal effect is understood in terms of the velocity with which it is achieved. To advance his analysis of effect in the Phoranomus, an analysis that is certainly imperfect, Leibniz tries to combine various relations that would together represent the equivalent of a determinative reason for the continuous effects of the force involved. A series of laws regarding uniform motion (motus æquabilis) are thereby obtained. (II, Leibniz 2007, 814–24) These laws purport to relate the dynamical factors of mass and speed with the properly kinetic factors of time, speed and distance traversed. The series in question will be reconsidered in the Dynamica, where it will be revised so that it conforms with the analytic components of formal action, which are expressions of a power that perpetually regenerates itself. The demonstrative value of the conceptually combinative models is highlighted by the type of validation that they allow. In this respect, Leibniz appeals to a scholastic precept, which also mirrors a particular method for apagogic proofs in mathematical physics, starting with Archimedean statics: “The conclusion is true, and one finds no other possible reason [than this one], thus this reason is true.” (II, Leibniz 2007, 814) The truth of the conclusion in the case that interests us would be established a pos­ teriori. But the explanatory reason would be provided a priori by the combination of concepts. And this self-consistent formal model would allow for eliminating every other hypothetical formulation of sufficient reason for the given phenomena. The demonstrative link depends on the condition that the contrary is deemed impossible, while an a posteriori argument seems to have the advantage of being a nearly direct empirical inference. However, in this latter case, if validity is immediately established, then it is circumscribed by a particular state of the system of nature, such as the one uncovered in 24

Robinet, in Leibniz (1991a), 538, suggests that Leibniz returns here in his text to issues that he had recently raised in the Schediasma de resistentia medii et motu projectorum gravium in medio resistente, Acta eruditorum, January 1689, 38–47.

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the experiment, and raising the inferential conclusion to the status of a general law presupposes, once again, recourse to a reductio ad absurdum that would exclude other combinations of possible reasons owing to the contradictions that they entail. Thus, while he reproduces in the Phoranomus the a posteriori arguments that had formed the basics of the reformed mechanics, Leibniz tends to support them with apagogical demonstrations whereby he concludes that every other alternative approach to giving explanatory reasons would lead to adopting perpetual mechanical motion. The combinative model prevails at the heart of the two styles of argument, but in the first case, or that of the a priori approach, the very nature of things would seem to reveal it; and in the second, or that of the a posteriori approach, it would be indirectly expressed through a hypothesis. The conjunction of the two approaches attests to this dual aspect of the model: Thus I advance with haste to considerations more agreeable and satisfactory to the imagination, which although assumed a posteriori, have the advantage of gently persuading the mind, while these a priori demonstrations (demonstrations natura priores) force a reluctant mind; and perhaps I would not have even suspected and thus attained the more profound ones […], had I not first discovered the truth in virtue of those that I now make known exoterically. (II, Leibniz 2007, 826)

However, applying this model in compliance with the principle of equivalence between the full cause and the entire effect and the analogies of infinitesimal geometry requires one to bring into play a system of abstract definitions supporting a coherent theoretical model. If, on the a posteriori side, these definitions explain the underlying reasons of the phenomenal order, then, on the a priori side, they simultaneously represent the objects of the system of theoretical concepts. Hence one discovers in Dialogus I the strategic distinction between two types of force, vis mortua and vis viva; and this distinction is developed alongside a parallel distinction between conatus and impetus, which respectively signify dead force and living force. Between conatus and impetus, summation builds up from the finite to the infinite, just as a point, taken as the beginning of a line, progresses through the line that it creates in so doing. (I § 18, Leibniz 2007, 876) Nonetheless, it is noteworthy that impetus corresponds to a measure of living force expressed in an instant, and not in its extended and so to speak consumed effect. At the same time, impetus is negatively measured by the inertia of the body whose mass absorbs the summation of the instantaneous conatus. This system of distinctions will be clearly developed and refined in subsequent texts, particularly the Specimen dynamicum. In the Phoranomus, Leibniz contents himself with making these distinctions in a context where he aims to going beyond purely geometric concepts in order to pin down the causes or reasons for the transfer of motion and the expression of motive force. The reality of force refers to “ideal principles of metaphysics”. (I § 9, Leibniz 2007, 704) The representation of the underlying causes exceeds the limits of geometric imagination and leads to theoretical entities that require us to go beyond the scope of geometric conceptualization to conceive of the nature of bodies.

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According to the opening passage of Dialogus II, in order to understand the natural inertia involved in the transition from conatus to impetus, the doctrine already requires us to go beyond the Cartesian and Democritian (i. e., atomist) conceptions of the essence of bodies (i. e., as simple extension, or as both impenetrability and extension). In one of the most significant passages where he is critical of his first mechanics, which he had developed in the Theoria motus abstracti (1671), Leibniz explains that he has gradually distanced himself from the fictional laws of a pure phoronomy, and searched elsewhere for an intelligible causal foundation for inertia and motive force, a foundation that could, furthermore, guarantee the preservation of the total quantity of motive power in the universe. (II, Leibniz 2007, 800–802) But this theoretical foundation cannot be furnished by some entity measured by motion itself, since the latter is essentially relative. Leibniz’s solution, a common feature of both a priori and a pos­ teriori approaches, will also consist in proposing essential sufficient reasons capable of satisfying the requirements of the architectonic principle of the equivalence between the full cause and the entire effect. This involves, above all, “declaring” which theoretical entities are consistent with this principle.25 However, to represent and express this theoretical hypothesis, one must be able to establish models that satisfy the requirements of extended geometric intelligibility. This geometric intelligibility requires the resources of infinitesimal analysis just as much as those of a combination of forms that go beyond mere quantitative analogies. In return, Leibniz will take issue with every attempt at recurring by one means or another to the occult qualities of the scholastics. Eventually, what the approach of the Phoranomus reveals despite its imperfection is the essential search for a correspondence between the proposed “geometric” models and the abstract definitions employed in conceiving of theoretically adequate hypotheses. The method to achieve this relationship relies on the use of architectonic principles in this case, that of the equivalence between cause and effect. It is at this cost that one can attempt to develop a mathesis mechanica that will translate the system of forces into a body of demonstrative arguments. Ultimately, before the Phoranomus, the reformed mechanics presented itself as a demonstrative system that was at least partially a posteriori, insofar as an empirical law had been introduced among its premises, that is, Galileo’s law of falling bodies. Under these conditions, the axioms of statics and architectonic principles, beginning with the principle of sufficient reason in the form of the postulated equivalence between the full cause and the entire effect, and the law of continuity, seemed only to apply to a factual state of the physical universe. Being an attempt to overcome Galilean and above all Cartesian science, the Phoranomus immediately sets out to apply a priori premises, particularly the principle of equivalence between cause and effect, to the analysis 25 Cf. Phoranomus II, Leibniz (2007), 824: “[BA] You have astonished me, Lubianus, with such unexpected and admirable decrees of the new science of power and effect, by which laws are prescribed to the universe itself.”

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of uniform and unconstrained motion, which would involve constant self-reproduction. In fact, Leibniz proposes a strictly equivalent relation between the effect of this motion – which is measured as directly proportional to mass and displacement, and inversely proportional to the speeds of displacement – and the measure of force in cases where the impact completely exhausts it. The aporia of this proposition consists in the fact that Leibniz has not yet succeeded in combining conceptual elements so as to guarantee this equivalence, since “substantial” as well as “modal” elements are combined here, giving the res a double spatio-temporal status. Leibniz also lacks an adequate model for representing action in terms of both its formal aspects of intensity and extension. Moreover, the justification that he provides is mainly apagogical, being founded on the sufficient character of the proposed model, which is related to the absence of a more adequate alternative. Could this signify that drastic change had to take place to bring about the theoretical inventions of the Dynamica de potentia? The fact is that the Phoranomus spells out the methodological requirements of a mathesis mechanica that will be fully operational in the Dynamica. This mathesis combines mathematical models that appeal to the resources of infinitesimal geometry, with abstract conceptual models of a more metaphysical kind supported by 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. Hence a relatively continuous transition from the Phoranomus to the Dynamica de potentia is assured, with the term “dynamics” emerging in this context to designate this new and improved science. 2. The Dynamica de potentia: Implementing a Theory Leibniz began writing the Dynamica de potentia et legibus naturæ corporeæ in September of 1689, just after finishing the Phoranomus. The successive phases of writing this new synthetic text comprise the end of the Leibniz’s stay in Rome and his stay in Florence (22 November – 22 December 1689). These ended with a whole series of adjustments that Leibniz sent, up until 18 March, 1690, from Venice to the preceptor of Cosimo III’s son, the Baron Rudolf Christian von Bodenhausen, to whom he had already given a copy of the text during his stay in Florence.26 Leibniz’s project consisted in effect in publishing, under the auspices of the Grand Duke, a version of the Dynamica transcribed by Bodenhausen.27 This project will never be completed because Leibniz will put off writing and sending the final part of the work indefinitely. He will make excuses, such as the desire to integrate a host of new considerations, but also the fear of an

26 27

Regarding the course of this writing, cf. Robinet (1988), 83, 101, 200, et passim. The dedication to the Grand Duke Cosimo III attests to this, GM VI 283–84.

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unfair reception of the theses put forth.28 Following the death of Bodenhausen on 9 April 1698, Leibniz will focus only on recuperating the bundles of documents comprising the Dynamica project that had been in the possession of his former correspondent. In the Dynamica de potentia, Leibniz undertakes to systematize his principles for mechanics so as to frame a general physical theory that can subsequently be applied to the whole set of phenomena that characterize bodies in motion. The argumentative structure of this treatise conforms to a series of more geometrico demonstrations. Leibniz’s intention seems to have been to deduce, through combinative synthesis, the laws governing mechanical phenomena. And all analysis would reveal here that project for synthetically deducing mechanical laws founded on immanent force, or vis insita.29 Indeed, this major text will not see the light of day until 1860 in Gerhardt’s edition of Mathematische Schriften,30 but one must take note of its strategic position in the series of Leibniz’s works on dynamics. Leibniz will, it seems, formulate all of his subsequent analyses by relying on this synthesis. It is in this sense that the Specimen dynamicum (1695) constitutes if not a part of, then at least a complement to the Dynamica, one that will bring the metaphysical component of Leibniz’s theses to light. Other outgrowths of this project include the Animadversiones in partem generalem Principiorum Cartesianorum and the first Essay de dynamique (1692),31 which was clearly written to demonstrate the superiority of Leibniz’s positions over those of Descartes and the Cartesians following the living forces quarrel. The combination of equations relating to the formulation of an absolute equation for the conservation of the quantity of action – which no doubt constitutes the most remarkable aspect of the later Essay de

28

Several accounts discuss this matter, cf. the letter to Foucher in January 1692, A II 3 494; the letter to the Marquis de l’Hospital on 15/25 January 1696, A III 6 617; and the letter to Johann Bernoulli on 8/18 March 1696, A III 6 707–708: “While I was in Rome in the year 1689 and was discussing these issues with Auzout, a very learned Frenchman, I penned a small book that would set my meditations in order, and in which all is demonstrated concerning absolute as well as directive force, conservation of progress of the center of gravity, and other matters not inferior to these. Passing through Florence I left it with a friend, learned in mathematics, who wanted to publish it. He carefully edited everything, but, as the end of the book was still lacking, something which I had meant to provide, I take responsibility for the fact that the edition was not produced, for I did not complete it, partly because many new things emerged that deserved to be added, partly because I did not dare to impose such beautiful truths on those whom I saw would not have received my views appropriately.” 29 Cf. De ipsa natura sive de vi insita actionibusque creaturarum, pro Dynamicis suis confirmandis illus­ trandisque (1698) § 4, GP IV 505–506; L 499: “One important indication of this, among others, is provided by the foundation of the laws of nature, which is not to be found in the principle that the same quantity of motion is conserved, which is a commonly held view, but rather in the principle that the same quantity of active power is necessarily conserved, and indeed – a thing which I discovered happens for a most beautiful reason – so is the same quantity of motive action, the value of which is far different from what the Cartesians think of as the quantity of motion.” 30 GM VI 281–514. A critical edition is currently being published: Dynamica de potentia et legibus naturæ corporeæ tentamen scientiæ novæ. Édition, présentation, traduction et annotation par A. Costa, M. Fichant et E. Pasini. Hildesheim: Olms, 2023. 31 Cf. regarding this point, Costabel (1960).

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dynamique (c.1700) – results as well from the deductive chains formulated in the Dy­ namica. The question of what can and should constitute a synthetic demonstration upon which a unique science might be founded requires us to juxtapose a posteriori and a priori reasons, or integrate them into the structure of Leibniz’s dynamics. Generally, commentators have tended to avoid this question by concentrating instead on the essentially analytic and a posteriori approaches evinced in the texts of 1686 and those of the initial controversy with the Cartesians, as well as in the Specimen dynamicum of 1695. Gueroult is an exception to this insofar as he consecrates an entire chapter of his Leibniz. Dynamique et métaphysique to the “a priori method (by motive action).”32 Gueroult builds his interpretation on the distinction between two types of demonstrations at work in the dynamics. The a posteriori type relies on the principle of equivalence between the entire effect and full cause to derive, from an empirical law, Galileo’s law of falling bodies, a theory of the conservation of living force which itself requires a meta-empirical interpretation. On the other hand, Leibniz combines two equations of relative conservation (the conservation of relative speed before and after the impact, and the conservation of the same quantity of progress) from which he develops an equation for the absolute conservation of living force. To a certain extent, the justification seems to owe to analogous truths of fact that can be established from experience. Thus, the theorem is consistent with the empirical laws of motion established by Huygens, Wallis, Wren and Mariotte. At the same time however, the objective meaning that Leibniz attributes to it seems to depend on a system of truly metaphysical concepts. From the start we may wonder whether such a description of the a posteriori approach leaves enough room for hypotheses that can be constructed with reference to empirical data interpreted via mathematical models, and by means of regulative principles. Moreover, can one maintain that interpretative frameworks that combine equations, such as those adopted in the Dynamica and exemplified by the late Essay de dynamique, have nothing in common with the a priori approach at the heart of the analytic, and more generally, a posteriori framework. How is the a priori approach presented to us? The living forces quarrel inspired Leibniz to combat the Cartesians on their own ground by adopting the strategy of a priori demonstration displayed in geometric phoronomy. Thus, he takes abstract conceptions of space and time as his only starting points. Gueroult considers this scientific approach a dead end. Leibniz could only hope to avoid the paralogisms that underpin the premises of his syllogisms by synthetically interpreting theoretical concepts with the help of frameworks belonging to his metaphysics of finite substances. This harsh verdict follows at least in part from the analytic strategies that Gueroult adopted. After having conceptualized the a priori approach on the basis of the theorems in the Dy­

32

Cf. Gueroult (1967), 110–53.

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namica de potentia, he essentially relies on the syllogistic codification of the demonstration as it is formulated in the correspondence with Johann Bernoulli and Burchard De Volder, and on the model that Wolff proposed to Leibniz, to which the latter reacted. In our view, the “a priori rational” character of Leibniz’s thought process ought to be more adequately described. Should we not account for the possible differences between Leibniz and his Cartesian interlocutors regarding what may be counted as an a priori justification for theorems in physics? Excluding the possibility that his a priori approach is merely an artificial façade, our analysis shall focus on clarifying the structure that Leibniz intends to give to theoretical physics. One must identify the “empirical” arguments that underpin the two types of Leibnizian demonstration, and then show how Leibniz’s theories are conceived of with the help of architectonic principles. It seems arbitrary to associate these principles purely and simply with metaphysics. The dynamics belongs in effect to a category of knowledge that Leibniz refers to as mixed science,33 the science of ex hypothesi necessary truths.34 This conception can no doubt clarify the structure of the seemingly divergent demonstrations between which dynamics is shared. In our view, the Dynamica de potentia exemplifies the systematization of a priori and a posteriori demonstrations despite the tensions and difficulties that Gueroult detects among the different approaches taken by Leibniz. Indeed, for polemical reasons, Leibniz often unduly emphasizes the apodictic character of certain arguments relating to the so-called a priori method. This happens in particular with the definitions of the theory’s basic concepts, and we must determine to what extent these definitions are real or simply nominal. Such has been the focus of discussions on the a priori approach. But in the first place, it is important to reconstruct the methods by which Leibniz proposes to establish a physical theory based on the dynamics. The a priori method is supposed to advance principles respecting the norms of a theory of motion (phoronomia) constructed solely from abstract notions relating to space and time – note the apparent similarity with the methodological ideal that had prevailed when Leibniz wrote the Theoria motus abstracti. However, as we well know, the logic that Leibniz argues applies to natural sciences, starting with physics, is one of inferences grounded in hypothetically necessary truths at best, that is, truths of fact that are assumed ex hypothesi to be equivalent to truths of reason. This thesis is, for instance, well expressed in the following passage of the Nouveaux Essais sur l’entendement humain: I agree that the whole of physics will never be purely a science for us; but still we shall be able to have some physical science, and indeed we have some examples of it already.

33 34

Cf. the statements relating to the mixed propositions in the Nouveaux Essais sur l’entendement hu­ main, 4.11.14, A VI 6 446–47, and those concerning the status of physics at 4.12.10, A VI 6 453–54. Cf. Ishiguro (1982), 90–102; Duchesneau (1982).

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For instance, magnetology can be regarded as such a science. For from a few assumptions grounded in experience we can demonstrate by rigorous inference a large number of phenomena which do in fact occur in the way we see as being implied by reason. (NE 4.12.10, in A VI 6 453)

It is within such a formal framework of hypothetically assumed necessary truths that Leibniz, in the first part of the Dynamica devoted to the simple elements of the abstractly conceived dynamics, inserts a deductive procedure based on definitions. And this procedure is noticeably similar to Hobbes’ a priori synthetic approach to definitions. One axiom is postulated: “that the same quantity of matter being moved across the same distance in less time produces greater action.” (GM VI 349) As for the definitions, their nature is not specified, but rather depends on a rational model wherein their objects depend on the criterion of possibility, that is, logical consistency. These objects ultimately boil down to the actions of corporeal parts changing place (situs) within a given spatio-temporal frame of reference, and therefore to motions reduced to their geometrical features. All of the definitions only have a provisionally real status. Leibniz derives them from an abstract reformulation of the phenomenal data of phoronomy, that is, from the rational analysis of the displacement of material parts in space, where displacement is reduced to geometric properties. Nonetheless, at the very end of the first part, Leibniz composes a remarkable chapter dedicated to analytic calculus, the Pro phorometria dynamica. (GM VI 425–31) As the concepts instrumental to infini­ tesimal calculus exceed the possibilities of the geometric representation of relations, it thus becomes evident that abstraction is no longer restricted to the conceptualization of phenomenal relations in terms of finite proportions between points, lines and various figures. Moreover, in the simple phoronomy that relates to uniform motion, Leibniz extensively refers to equivalent infinite series of proportions that would cor­ respond to finite ones at the limit. Certain propositions with an ostensibly paradoxical character can be explained by presupposing that such serial relations between factors are implicitly equivalent. The justification for doing so is the raison d’être of the chapter De ductibus seu de æstimationum compositione. (GM VI 307–19) It is clear that Leibniz will, relying on this method of representation inherited from mathematical analysis, propose a conception of abstract states (status) that would provide sufficient reasons for the phoronomically representable empirical processes. By means of this geometric transposition, these states would afford analytic equivalents for the mechanical properties elicited by the interaction of bodies. As such, Galileo’s legacy, that is, the conception of the effects of gravity, might be radically clarified: [Galileo] brought the acceleration of motions produced by gravity to light. Encouraged by these early advances, we discovered the true criterion for estimating the power of bodies, from which some rather remarkable general laws of things arise and the hitherto scarcely known nature of body and motion becomes evident. (GM VI 283)

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From the way the demonstration unfolds in the Dynamica, it is clearly the case that applying empirical laws in such a way depends on the abstract framework of concepts and relations that, conforming to mathematical models, constitutes the theoretical basis for the new physics and for the principle of conservation of living force governing the reformed mechanics. In the Specimen præliminare. De lege naturæ circa corporum potentiam, Leibniz presents the different possible ways in which the principle of conservation of living force can be demonstrated. The first three, which correspond to the a posteriori approach, depend on a common lemma and three distinct axioms. From the outset, one might suspect these axioms of having the same status as the only axiom introduced in the first part of the work on simple dynamics. These axioms are each stated in turn: The same power that lifts four pounds one foot can lift one pound four feet. There is no perpetual mechanical motion. It cannot happen that the center of gravity of the bodies is raised by the force of gravity alone. (GM VI 288–90)

The lemma, for its part, further develops upon Galileo’s law by means of a hypothetical formula, the conception of which draws upon a doctrine of motion and weight. The justification for this lemma is based partly on the possibility of a geometric representation, and partly on experimental control: The perpendicular heights of heavy bodies are as the square of the speeds that they can acquire by descending from these heights, or by means of which they can raise themselves to the same heights. This is a proposition of Galileo, demonstrated from the nature of the uniformly accelerated motion of heavy bodies, accepted by mathematicians and corroborated by many experiences. (GM VI 288)

The fourth demonstration is supposedly founded on reasons for motion that abstract “from sensible matter” (a materia sensibili). It involves deducing propositions for measuring and conserving motive action, which will be established on the axiom concerning the quantification of formal action. It is the implementation of this deduction that will, for instance, become the focus of the discussions with Bernoulli and De Volder, and which Gueroult characterizes as a degenerative process in Leibniz’s theory. The scholium that accompanies the summary of this a priori demonstration specifies the latter’s methodological relevance: Although this last demonstration may not be to the taste or within the reach of all, it should nonetheless please those who seek to clearly perceive truths. If it was the last to be discovered, then it certainly seems to be the first in dignity, since it proceeds a priori and is born from the consideration of space and time alone without any presumption of gravity or hypotheses posterior by nature. […] Following the example of the previous lemma, this is exactly what one can independently conclude from the fourth demonstration. (GM VI 292)

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Leibniz thus intends to give a new a priori demonstration of the principle of conservation of force solely on the basis of phoronomic considerations, and consequently by means of abstract concepts of motion for understanding displacement in terms of the general parameters of space and time. However, it is not evident that such a proof can be derived from “totally abstract” conceptions of space and time. Are we not rather dealing with types of theoretical entities that are represented abstractly as objects that can be analyzed via quantitative models and the methods of geometry and infinitesimal calculus? And because these concepts are conceived of in connection with architectonic principles, do they not retain some objective relation to the system of nature that they are invoked to represent? Section III, De actione et potentia, where the “a priori” demonstration can be found, is preceded by a section, De motu et velocitate, where Leibniz combines notions to formulate the abstract representation of the modes attributable to body in motion. He provides the following definition: “A is moving or in motion. Everything in it [A] that is homogenous and comparable to B has a point E, which cannot occupy the same spatial point in any moment in time.” (GM VI 230) When this first definition is advanced, Leibniz mentions that, formulated in this way, motion also allows us to represent rest as a motion whose velocity is so small as to be indeterminate; this indeterminately small velocity can be applied to a group of bodies, both moving and at rest. The proposed definition has the advantage of jointly dealing with what moves and what is moved, because an active substrate of homogenous relations of displacement is implied by the notion. This substrate can be represented by phoronomic relations that apply globally to a given group. Leibniz insists on the functional role of the definition for any study that applies to the interrelated elements of motion and rest, even inside a point taken as a unity without parts, but accountable for the homogenous relations it implies. The second definition holds that: “The time of motion is such that one cannot presume the existence of any part in which some point of a presumably homogenous element in the moving body would not be displaced.” (GM VI 323) This definition aims to connect the measure of time with the relations of motion between homogenous and heterogeneous elements down to the infinitesimal relations that they entail. Consequently, time is not a neutral element, but corresponds to the intensio of motion in its successive moments. The definitions that follow concern uniform motion, equally distributed motion, in which speed and length are equally distributed, simply simple motion, and simply simple motion that is rectilinear, in which there is a constant distance between a body’s parts (consentiens), that is equidistributed, and that is equidirectional. These definitions frame a conception that combines the various formal elements that characterize the notion of motion. A good example of this is the manner in which speed in equally distributed motions is interpreted such that the existence of implied ratios would be justified: Speed in equidistributed motion is a modification (affectio) of the moving body (a formal modification found in it owing to motion alone) that is proportional to the line that a

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point of the moving body would represent if this motion continued for a time of a given magnitude, this very property of the mover body being conserved. The speed remains the same if the same point represents equal spaces in equal times. (GM VI 323)

Here, speed (velocitas) has a twofold status. It is both a modification (affectio) of the moving body and an abstract measure for the variable of spatial displacement over a given period of time that can be as short as possible to the point of being indeterminate. At the limit, this abstract measure returns to the original relation between the homogenous and heterogeneous elements that characterizes the moving body’s inten­ sio. Consequently, the property is defined as formal or inherent to the moving body owing to its displacement alone (and even its virtual displacement). At first glance, it seems that we find here an equivalent to conatus that would serve as a basic element in dynamics. But this property, which is related to actualizable motion, is expressed in terms of proportional relations. All definitions relating to simply simple motion are based on an analogical method, but aim to reduce, by means of abstraction, the essential heterogeneity of the elements of motion. The simple is therefore a symbolic expression for a diversity of elements that are infinitesimally synthesized by being represented as a geometric continuity. It is the congruency between the successive moments that accounts for their integration. The principal definition states: Motion is simply simple when the motions of the points of a given mobile agree [conveni­ ent] with and among themselves in such a way that they could not agree more. (GM VI 341)

And the explanation provided shows the precise role played by homology in representing motion as ultimately homogenous for any given concrete moving body, that is, as abstractly simple: This happens when, examining their motions alone, one can neither distinguish one state of a point from another, nor one point itself from another, before or after. These motions always being similar and similarly placed with respect to one another, there is no principle whereby one can distinguish their points – even if their points are differentiated by virtue of the different bodies they occupy – by their situs in bodies, something which cannot be avoided. And here I ignore those qualities in virtue of which one distinguishes between the parts of bodies, the reason for this being inapplicable in the present case. It suffices here to consider the motions of points alone to understand the degree to which they accord with one another. (GM VI 341–42)

This implies that the law of continuity helps frame an abstract representation of motion such that the aporias entailed by what is completely indiscernible can be overcome. Does Leibniz not suggest that the physical elements of simply simple motion are essentially different? At the same time, does the geometric analogy formulated with reference to continuous series, and therefore serial homogenization of these ele-

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ments, suffice for expressing the constant relation governing phenomenal variations? And does it not then suffice for grounding a general phoronomy? The shift to action and power in the following section is characterized, in our view, by inversing the steps of the analogy. It is there, however, that the quest for an a priori demonstration of the fundamental theorem of dynamics resides, or rather its equivalent, the principle of conservation of the quantity of motive action. The presuppositions of the demonstration rely on the definitions of quantity of formal effect and quantity of formal action: The measure of the quantity of formal effect in motion is determined by the fact that a certain quantity of matter (motion being equally distributed) is moved a certain distance. (GM VI 345) The measure of the quantity of formal action in motion is determined by the fact that a certain quantity of matter is moved a certain distance (motion being uniformly equidistributed) within a certain time. (GM VI 345)

These definitions introduce a remarkable condition. In La Dynamique de Leibniz,35 I interpreted these definitions as nominal and falling short of the goal to construct the demonstration on a deductive a priori foundation. According to this interpretation, Leibniz would have deviated from the a priori approach by proposing hypothetical models that depended on a more analytic than synthetic strategy. I will now correct this interpretation. These definitions bear in fact on a priori determined measures that can apply to the modes of motion. They match a specific requirement: they have to represent effects arising from actualized motion. On the other hand, they are conceived of by abstracting from other reasons that would refer to the particular structure of bodies, beyond geometrically or rather analytically intelligible modes of motion. At the same time, these modal effects may not rightly be defined without reference to a metaphysical reason for their correlation, a reason that functions as an architectonic requirement. This requirement is presented in the remark that follows the definitions: I have qualified the effect as well as the action as formal because they are, according to the definition given here, essential to motion; they thus differ from the other effects or actions that originate from some obstacle, such as the force of gravity that pushes bodies toward the centre of the earth, or the resistance caused by the medium or some contraction, or the elasticity of a body to overcome, or similar accidents of concrete matter. If one hardly supports the use of a metaphysical term in a mathematical topic, let him reflect that no other more convenient ones came to my mind and that once the definition was given, all ambiguity was removed. (GM VI 346)

35

Duchesneau (1994), 183.

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The difference between the two types of modal effects owes to the fact that the former would be directly observable in unconstrained motion, while the latter would be characterized by the body’s resistance to undergoing change and transforming its state, and would express an action proportional to this inherent potential for resistance. Hence, the latter would be abstract, while the former would refer to features of corporeal reality, which are manifest to sensory experience. But even this abstraction results in purely a priori conception, and as in the case of mathematical concepts, it simply corresponds to a geometric interpretation of concrete realities whose inner relations are adequately expressed by these abstract models. The abstractness of the concepts thus defined is not purely mathematical: if they are framed to geometrically represent corporeal modes, and more specifically modes of motion, they imply relations of sufficient reason that underpin the causal role attributed to these modes. The reason for the causal relations implied pertains to principles of architectonic order that govern a priori the combination of these concepts. In the 1684 Meditationes de cognitione, veritate et ideis, Leibniz argued that the test for determining the distinctness of a concept consists in perceiving the rational consistency of its representative contents, which may be done in either of two ways: either by a logical analysis that can be achieved through conceptual models, or by perceiving the existence of the corresponding objects.36 In the present case, Leibniz’s strategy consists in a conceptual model adapted to meet architectonic requirements, as if one were dealing with a priori hypotheses. But the validity of this model would ultimately be tested a posteriori through experiences that demonstrate that the implied ­consequences of the hypothesis match features of the corresponding, well-founded phenomena. The intention of the second part of the Dynamica was to provide instances that would offer such confirmation. This will also be the case in the 1695 Specimen dynamicum, which only alludes to the a priori process, and compensates for its shortcomings by postulating constant primitive active and passive forces underpinning the phenomenal relation between derivative forces. If formal effect and formal action are distinct and intrinsically distinguishable notions, then it is important to grasp the architectonic connection between these two notions, beyond the quantitative calculations to which they give rise. It might even be important to assume that this connection will reveal itself through the analysis of these notions. The adopted strategy involves relying on definitions to formulate converging propo­ sitions that relate both to motive effect and motive action. Take for example propo­

36

Cf. A VI 4 589; L 293: “We know [the possibility of a thing] either a priori or a posteriori. We know it a priori when we reduce the concept to its necessary elements or to other concepts whose possibility is known, and we know that there is nothing incompatible in them. This happens, for instance, when we understand the method by which the thing can be produced; hence, causal definitions are more useful than others. We know an idea a posteriori when we experience the actual existence of the thing, for what actually exists or has existed is in any case possible.”

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sition 3 in the chapter on formal action of motion and its effect: “The formal effects of motion and, supposing an equal speed, even the formal actions of motions, are in a compound ratio to the moving bodies and lengths.” (GM VI 348) Next, this involves applying an axiom that links this system of relations with the measure of duration: “The fact that the same quantity of matter is moved across the same length in less time indicates greater action.” (GM VI 359) Consequently, there is a compound ratio between the power to act and the quantity of matter and spatial displacement, and an inverse ratio between the power to act and the time required for the action. And this power to act translates into a series of relations that are proportional to the quantity of motive action depending on the speed at which various sequences of formal effects occur. An additional definition distinguishes between the perspectives of spatial displacement and power to act: The diffusion of action in motion or its extension (extensio) is the quantity of formal effect in motion. The intensity (intensio) of the same action is the quantity of speed with which the effect is produced or by which matter is carried over a distance. (GM VI 355)

But these distinct viewpoints give way to a new combination of concepts centered around a joint sufficient reason. Thus proposition 11, “the formal actions of motions are in a compound ratio of diffusions and intensities,” (GM VI 355) points to the joint articulation of the two viewpoints as a product of intensio and extensio factors. Consequently, the a priori argument renders the expression of the formal action ratios subject to an architectonic requisite while preserving the geometrical and combinative style of the undertaken demonstration. How is this integration justified? A simple but likely insufficient response presents itself. On the one hand, are formal motive actions not the product of a compound ratio of the effects and speeds of translation? On the other hand, does this additional definition not relate the extensiones or diffusiones to the effects, and the intensiones to the speeds (velocitates), in keeping with the axiom that postulates a fundamental relation regarding the power to act? Indeed it does! But that does not prevent combining a latent instantaneous effect, represented by v, with an effect carried out across the translational space, represented by s, assuming that the latent effect intervenes across the entire path. Here, motive action is understood as a sort of causal element that combines the propensity for action and motive effect. As all inertial resistance is bracketed, it is presumed that this propensity is entirely conserved across the motive effect, such that one could add it to this effect as a permanent gain in virtual translation. The equation for formal motive effect postulates a relationship between the moving bodies and the translational spaces, or, equivalently, “a compound ratio of the moving bodies, speeds and times of motion.” (Proposition 16, GM VI 356). Thus, it is easy to advance propositions that combine the motive effect with the conserved virtual effect in order to create a discursive analogue of the power to act. This leads for example to Proposition 17: “The formal actions of motions comprise a compound ratio formed of the times and moving bodies simply and

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of the speeds doubly.” (GM VI 356) And it also leads to the propositions that isolate the various ingredients of proportionality, particularly when one assumes that the times are equal, and that the actions are determined by virtue of a relation that considers moving bodies and the square of their speed together (cf. the scholium of Proposition 20).37 This formula itself depends on a previous formula where, furthermore, the quantities of matter were assumed to be equal (Proposition 18).38 Once this system of mixed arguments is postulated, Leibniz formulates the a priori abstract proof that also figures in the Dynamica as the fourth demonstration of the Specimen præliminare. This order is not random, as the deduction largely depends on a preliminary analysis of the fundamental phenomena, and this analysis is based on so to speak “formalizing analogies”. Returning to Proposition 4, which is governed by the additional axiom, Leibniz suggests two possible parallel equations. (1) If the same length is successively traversed at four different speeds, and among these BC is to EG as LM is to EG, then likewise the formal action corresponding to BC will be to the action EG as the action LM is to the action NG. Hence, according to Proposition 7: if the length traversed remains the same, then the actions are proportional to the speed, whether it remains the same, is multiplied, or divided. Lacking a sufficient reason to conceive of a factor that would provide for multiplication or division, the relation is restricted to simple speed. Then, according to Proposition 18, (which itself depends on Definition 4), the time being the same (that is, equal to one unit of time), action is reconceived of in proportion to the square of speeds (this occurs, in our view, by inscribing the latent speed in the power to act conserved across the action, that is, across the translation along each unit of length). (2) The second possible formulation of the equation involves starting out, this time, with the hypothesis of the successive speeds exercised in a unit of time. Indeed, the same equation of actions is conserved, and in this way, the actions can be said to be proportional to the simple speeds:

BC =  LM EG NG Henceforth, it suffices to introduce the relation of action, thus constituted, to equal lengths, in order to arrive at a new equation for representing the composition of the intensio and diffusio of action. In the so-called a priori inference that follows, which syn-

37

38

GM VI 359: “Calculating actions involves calculating effects, i. e., lengths, and speeds, but lengths are in a compound ratio of times and speeds; actions therefore are in compound ratio of the times once and of the speeds twice. Thus, if the times are equal, the actions are in a duplicate ratio of speeds.” GM VI 357: “If the quantities of matter are equal, and the times of actions are equal, then the formal actions of the motions will be in compound ratio of speeds and lengths of motions.”

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thesizes the two approaches, everything revolves around two considerations: on the one hand, the possibility of representing extensive and intensive factors in view of the relations that speed has with lengths of space and time respectively; and on the other hand, the analogy between this symbolic comparison of speed with itself and a “physical” interpretation of the power to act (expressed by the verb facere), the postulation of which the additional axiom and Definition 4 make possible: Until now everything matches up; the only problem is this: supposing that the length is the same, the ratios cannot be compounded; and thus the actions will be proportional to the speeds, or reciprocally proportional to the times; but supposing that the times are the same, the ratios can be compounded; and thus one cannot say that the times being the same, the actions will be proportional to the speeds. I will therefore show when there is an occasion for composition. Suppose three actions: A achieves twice as much in one unit of time; B achieves twice as much in two units of time, and finally C achieves one unit [of action] in one unit of time. The ratio of A to C, which includes two actions with different speeds but the same times, is compounded from the ratio of A to B, which includes actions with different speeds but the same lengths, and the ratio of B to C, which relates two units of action to one unit. Hence it is evident that the ratio of A to C is two times greater, which deserved to be demonstrated separately by a specific proposition, and thus what we said about the ratio of B to C needed to be introduced first. (Scholium after Proposition 20, GM VI 358)

The problem would involve restricting ourselves to the phoronomic relation of A to B, within which a simpler relation cannot be established. For this reason, we would find ourselves strictly limited to the proportionality of actions to speeds, or, where the times are equal, to the continuous effects that result therefrom. This is the end of the Cartesian variety of phoronomy, which was incapable of geometrically representing the physical reasons required for justifying the combinative approach. From the outset, Leibniz’s version takes a mixed approach. To the arguments limited to the analysis of the phoronomic relation between A and B, Leibniz opposes a synthetic argument that he summarizes as follows: Calculating actions involves calculating effects, i. e., lengths, and speeds, but lengths form a compound ratio of times and speeds; actions therefore involve a simple ratio between the times and duplicate ratio between the speeds. Thus, if the times are equal, the speeds of the actions are in a duplicate ratio. (GM VI 359)

In order to account for the implications of this mixed, “analogical” a priori approach, it remains to examine the developments that Leibniz derives from it in chapter II: De potentia motrice absoluta demonstrata a priori. The a priori nature of the demonstration stems here entirely from a new definition linked to the axiom in the previous chapter, and to the analogically a priori demonstrations relating to the diffusio and intensio that combine in formal motive action. This is the definition of absolute force:

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The absolute power of what is moved is this property (affectus) of this thing that is proportional to the quantity of action resulting from the state of what is in motion and remains in motion for a given time, or that is proportional to the quantity of formal action the moving body would exert if it maintained this motion uniformly for a time of given magnitude. This is why, the times of the action being equal and the formal actions being uniform, the absolute motive powers are proportional to the formal actions. (Definition, GM VI 359)

Significantly, Leibniz pursues here the inverse of the approach of geometric abstraction characteristic of phoronomy. Dynamical representations based on the combination of factors in motive action help him reconstruct a sufficient reason for the power to act in moving bodies, and this sufficient reason is evoked as a property (affectus) of the model. The quantity of formal action, translated by squaring the relative factor v, will allow for expressing the absolute factor of living force by virtue of relations of proportions. This evidently involves the latent living force that is conserved within the given body, and not the living force that gets translated into exhausting effects, as in the case of free fall, and even in respective effects, as in the case of percussion, spring tension and pendular oscillation. It is in this conceptual context that Leibniz demonstrates Proposition 4: “If the moving bodies possess an equal quantity of matter, the absolute motive powers pre­sent a duplicate ratio of the speeds, or are proportional to the square of the speed.” (GM VI 362) This proposition derives from the definition of absolute force and Propo­sition 18 of the preceding chapter, bearing in mind a limiting condition: the equal quantity of matter in the moving bodies. The methodological process underlying the demonstration is clearly identified with the conservation of the power of action across the formal action that expresses it. This is why speed is compounded with itself, which allows for assigning to mass an absolute force that is both actualized in motion and preserved intact throughout the motion in a given unit of time. The first aspect constitutes the phenomenal and extensive expression of the second by virtue of the fact that the parameter for velocity is both intensive and diffusive: “In power, that which is uniformly diffused in action across time is instantaneous.” (GM VI 364) The whole demonstration is based on studying bodies in motion that are neither constrained nor acted on by gravity. It goes without saying that its application can be extended to actions and forces expressed through various motions, such that we are able to analyze them in terms of uniform and equally distributed combined motions. An interesting case emerges regarding translational motions that result from accelerated and delayed motions, caused by gravity for example. Linking absolute force with the effect of gravity is done in correlation with the principle of conservation of power, represented as the quantity of formal motive action, but in such a way that a posteriori arguments appear as analytic requisites that can be attached to the synthetic exposition. In the perpendicular descent, characteristic of the effect of gravity on equal quantities of matter, the forces are proportional to the heights of fall, and as it turns out, the heights of fall are proportional

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to the square of the speeds. But the latter relation here is part of the concessa, and for this reason, must be verified a posteriori: “And as this is taken for granted, it follows that our demonstrations are verified a posteriori.” (GM VI 367) At this stage, Leibniz only makes further use of this argument to prevent the dynamical phenomenon from being subjected to the Cartesian principle of conservation of quantity of motion, which would lack proof in the given case. But this principle is equally liable to create paralogisms in all cases where one must grasp how latent action is implied in the generation of formal action. More precisely, the concept of force, analogically expressed by the integration of latent action into formal action, has the ability to disentangle such paralogisms. This disentangling is achieved by an equation that has an absolute value, and establishes the conservation of the total quantity of motive force in whatever time (in tempore quocun­ que). Nonetheless, to expose the foundation of this principle in concreto, Leibniz must necessarily integrate an a posteriori derivation of analogical equivalents of potentia absolu­ ta into the analysis. The case of uniformly accelerating and decelerating motions, like the effects of gravity on an initially constrained moving body, will allow for generalizing the analysis. Failing this, potentia absoluta risks remaining an abstract model, akin to an algebraic unknown. Indeed, this concept is grounded in formal analogies representing, in an increasingly adequate way, the power to act in concrete moving bodies; however, without being connected to the a posteriori approach, the a priori approach could not possibly constitute a coherent mixed science of the fundamental phenomena of dynamics. At this point, we must pause our analysis and give thought to what the elements of a provisory conclusion might be. We have only considered a part of the Dynamica de potentia, and precisely the part that seemed prima facie to justify the thesis involving an a priori demonstration that Leibniz could have pursued in order to establish a theorem of dynamical conservation. One should immediately notice that the theorem wherein the principle is stated refers to a theoretical entity, absolute power (potentia absoluta), that is represented by the quantity of formal action across a given unit of time that is so small as to be unquantifiable. We are dealing with a strategy of geometrizing phoronomic relations and extending them to infinitesimal ratios – which differs by the way from Leibniz’s resorting to conatus as mere indivisibles of motions in his early mechanics. However, and this constitutes the second deviation from the initial method, Leibniz builds on these relations of formal effect and action a system of references to theoretical entities of a different order. He thus aims to represent the power to act in bodies by means of both intensive and extensive effects. Relying on a demonstrative framework of axioms and definitions, he builds models that do not refer to an order of geometrical representations alone, but that are symbolically linked to concepts representing a metaphysical foundation that suffices for justifying harmony between the various phoronomic relations.39 In the Dynamica, it seems as if the more geome­

39

I will not develop this point now, but the transition to a metaphysical explanation to ground the

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trico propositions would also incorporate concepts of that type. The principal phoronomic theorems exemplify a mounting progression through combinative relations toward theoretical causal notions. But this is all the more true when Leibniz derives the principle of conservation of formal action from these theorems: then the axioms and definitions, insofar as the they refer to an absolute power to act, tend to express the architectonic reasons behind the causal system articulating the various modes of motion. As a result, the system of equations derived from the theorems synthesized in this way would presumably apply to a possible phenomenal world in which general harmony is achieved. By virtue of the axioms and definitions relating to the power to act, the theoretical concepts representing sufficient physical reasons tend henceforth to regulate the formulation of systems of equations and the theorems whereby such systems are founded. This tendency undoubtedly becomes apparent when Leibniz, at the beginning of the second part of the Dynamica, establishes the definitions and the axiom for reconstructing the a posteriori demonstration of the theorem of conservation of living force. Leibniz conceives of the second part of the Dynamica de potentia as analytically extending his theses regarding the “formal” characteristics of motion. These theses suggested a conceptual formulation characterized by the idea of a latent power to act at every instant of unconstrained homogenous motion. Diffusio conceived of as the length of instantaneous traversal does some justice to this power, as long as it is correlated with a coefficient of instantaneous speed representing the intensio of motion. This appears, for example, when Leibniz lays the foundation for a phoronomic theory of acceleration and deceleration. He then redefines the notion of quantity of motion and provides a functional equivalent to it by appealing to a combination of two concepts: the quantity of conatus measured in its instantaneous connection with mass and the quantity of translation, which is a function of the generated conatus summed in their temporal effects: From this one understands that impetus is the quantity of motion, but only momentary motion, and that the term quantity of conatus better captures it; but truth be told, as motion takes time, one should rather interpret quantity of motion as that which gets produced by the aggregate of conatus present for the entire duration; and this is what we refer to as the quantity of translation. (GM VI 399)

a priori quasi-geometrical argument is well exemplified in the correspondence with De Volder, in particular when the interpretation of the notion of præstantia of action is invoked, a term which P. Lodge translates as the “productivity” of action, but which for Leibniz means an efficiency achieved that leads to the combination of the power involved in the action performed with the intensity with which it is accomplished. Cf. letter to De Volder on 9/20 January 1700, GP II 201–206: “I agree if by the efficiency of action you mean this intensity […] that along with extension in time compounds quantity of action, i. e. if you mean velocity.” Cf. Leibniz (2013), 150.

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Leibniz adds that he saves the name “quantity of motion” by reducing it to an intensive effect, that of impetus, or the instantaneous summation of conatus in a mass; herein lies the basis of inertial motion. At the same time, one can by abstraction discern the quantity of conatus that therefore inheres in the mass; this quantity is represented as an intensive factor v. This is what is meant by the following definition: “The intensity (intensio) of the motion or the energy (vigor) in a moving body is the speed of a body that is uniformly moved, and it is equal to the mass and impetus of this body; thus, its speed multiplied by its mass yields its impetus.” (Definition 2, GM VI 399) Evidently, the formal effect results from the expression of impetus, which is assumed to be constant across time. Suppose that there is a motion in which the repetitive addition of conatus amplifies the effect of impetus as the motion unfolds. The intensio thus reconstructs the impetus that defines the formal effect as a constant progression. In abstract phoronomy, it is not yet a question of conceiving of the force by which conatus are progressively integrated. The study of gravity and the determination of centrifugal force will go hand and hand with the transition to the dynamical study of physical effects. Already however, Leibniz must consider, on an abstract level, the distinction between the factors mv on the one hand, and v (intensive) on the other. This distinction allows us to represent a variation of the product mvv in terms of acceleration g, a measure that refers to the instantaneous summation of conatus that comprise impetus (mv), as long as we conceive of it as being accomplished in an instant. The analysis of the Dynamica de potentia reveals that relations such as summations of conatus in impetus, abstract summations of instantaneous conatus, and virtual accelerations in the effect of mass, are inferred from abstract definitions and do not depend on taking into account effects that express the exhaustion of the power to act in bodies. For instance, Leibniz introduces an abstract measure for the quantity of translation or formal effect of motions in the accounts he gives of uniform acceleration/deceleration. This measure comprises two elements of formal effect: on the one hand, the summation of instantaneous conatus that are progressively integrated across time, and on the other, impetus across its constant development. Proposition 6 of chapter 2 expresses this, offering a definition relating to the average conatus and impetus achieved by the temporal elements. If the intensities of motions apply to time in an orderly way, the product obtained is a length of translation, and if the impetus apply to time in an orderly way, the product obtained is as the quantity of translation or the formal effect, and thus the impetus (applied in an orderly way to the element of time) is an element of the effect. (Proposition 6, GM VI 410) The average energy (vigor) used or average speed attained (over a duration of time) is such that, multiplied by the amount of time elapsed, it yields the length of translation; or the speed attained is that in virtue of which the constantly moving body would have traversed, in an equal amount of time, as much actual distance as it now has. And the average impetus,

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which is obtained by multiplying the average speed attained by the mass, or by the time, gives the actual space, that is, the quantity of translation. (Definition 3, GM VI 411)

By the end of this reformed phoronomic approach, the definitions that served to support the deductive formulation of a mixed science are related to the abstract methods of the infinitesimal analysis of quantities. The techniques of calculus are called upon to symbolize relations in the phoronomy of uniform motions, as well as motions that are uniformly accelerated in gradually increasing series. Evidently, in the latter cases, the resources of calculus allow us to generalize the meaning of an analysis intended to ground the extensive via an intensive relation inherent in the power to act. However, given the chosen system of definitions, all of these phoronomic considerations at the limits of what is geometrically representable form a network of necessary hypothetical inferences. The effort to ground this system of abstract representations in the phenomenal world is secured when one proceeds to the first section of the second part of De causa et effectu activis. (GM VI 435) The network of necessary hypothetical inferences is expanded upon here. At the same time, we witness the direct intervention of a hypothesis that supposes a causal agent beyond the modal effects that the phoronomy accounts for. The very last phrase of the section describes the way in which Leibniz perceives the theoretical formulation that he elaborated in the more concrete portion of the analysis: And thus it seems that I have discovered the sources of the dynamic science with regards to the nature of power and action, which had not been explored enough until now. For I have removed all ambiguities by establishing the most simple and general principle of equality between cause and effect, from whence we shall deduce all of nature’s other wonders in a specific way. (GM VI 464)

The program that finally emerges is that of a series of physical explanations governed by the axiom of equivalence between the full cause and entire effect, and arrived at by extending the laws of dynamical science to other more specific phenomena. At the beginning of the second part, Leibniz addresses causality from a twofold perspective in the preliminary definitions and the axiom that he adds to them. On the one hand, the definitions have the same abstract character as those that one finds in the first part, the only difference being that they represent themata or rerum status, the referents of which are what one could call real phenomena. According to definitions 1 and 2, the active state or the state endowed with force (potentia præditum) is the one from which physical change results, while all of the concomitant states are assumed to be inert. If, from an active state there results an active effect without our having to assume the existence of a third productive force, and if, moreover, the context allows us to assume that the other bodies are inert, then the first and second states constitute the cause and its absolute effect. Furthermore, if no other effect can result from the cause, then a full cause and an entire effect are involved. (GM VI 435–37) From there Leibniz proceeds to recall

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the axiom of equivalence between the entire effect and full cause. From this follows the impossibility of conceiving of an effect capable of producing more than its cause – hence the rejection of perpetual mechanical motion – but also the impossibility of a diminishment in power in the transition from the full cause to the entire effect. This relation allows one to define the quantity of force as a measure of the quantity of the entire active state capable of undergoing change, from the first “theme” to the second “theme,” or from the full cause to the entire effect. Indeed, Leibniz affirms the metaphysical origin of the axiom, but at the same time, turns it into a law of physical nature that proves to be indispensable for every estimation of force, that is, when considering every conception that aims to identify a sufficient causal reason for the effects of motion. From this physical point of view, the justification is therefore pragmatic in nature. However, as a metaphysical proposition, the axiom does not represent a strictly arbitrary rule derived from a purely nominal game, but rather a form governing our apprehension of truths of fact, which can be inferred from our reflexive experience regarding the order of things: The equivalence between the entire effect and the full cause is a proposition belonging to a sublime metaphysics that does not content itself with mere words, but instead concerns what is universal in things. Nature observes this law with utmost constancy, and truth of it can be inferred from the fact that, if suppressed, there would remain no means for estimating powers nor for determining the magnitude of effects from their causes. (GM VI 437)

In the explanations that accompany the definitions, Leibniz underlines the relation to experience that one can establish by applying the concept of substitutability between cause and effect. It is in this way that he identifies, in the absence of obstacles, the role that the power to act plays in producing dynamical changes, for instance, the diffusion of a contained, pressurized gas when a valve is opened, or when a body falls after the rope suspending it is cut. In distinction to the formal effect of motion, the active or absolute effect assumes the expenditure of a body’s previously accumulated force, as is the case for a body raised to a certain height before the rope suspending it is cut. Conversely, the expenditure of accumulated force is compensated by the dynamical capacity the body acquires in the physical state that results. Consider, for instance, a ball in motion displaced across a frictionless flat surface that impacts a series of elastic bodies, which, because of the way they are tied together, are constrained to reverberate the impetus once the motion of the impacting body is exhausted. The total displacements obtained thereby represent the entirety of the productive cause of the elastic shocks. Assuming that the cause of friction is in the displacement of the ball, for example, in the resistance offered by the fibers of a rug, the active effect is merely spread out across a greater number of bodies that reproduce but also partially absorb it owing to their more or less “porous” internal structure. In the propositions that follow, Leibniz will steadily piece together in this way both a theoretical perspective in which the principle of conservation is directly applied, and an empirical perspective in which the principle sets the norm for equivalent explanato-

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ry accounts. Regarding this norm, the apparent disparities with regard to conservation in phenomena can be justifiably reduced by virtue of the attending circumstances that mark the transition from one dynamical state to another. The fundamental theoretical idea is that of a closed system of communication between forces, where the physical universe is presumed to be such a system: “The power is always the same in systems of bodies that do not communicate with others.” (Propo­ sition 7, GM VI 440) But this idea must be interpreted in terms that can quantify the expression of the forces conserved in dynamical substitutions. Leibniz suggests that this expression must correspond to the total quantity of action of the system over equal times. The flaw in conceptions based on the conservation of quantity of motion consists in that impetus corresponds only to an instantaneous latent effect, without accounting for an actualized dynamical expression. The major difference evidently consists in that the quantity of action represents a power that is constantly conserved, and in which the latent dimension of impetus is abstractly represented as an integrative element in the product of mass and speed squared (which combines an intensive element and an extensive one). To dynamically represent the causality of phenomena, a direct integral expression for the product mv2 must be demonstrated. The series of propositions that aim to establish this direct integral expression of force is not inconsequential, as it translates a mounting analytic reconstruction of sufficient reasons. From the proportionality of full causes to entire effects, Leibniz moves on to a measure of power based on the relation between weight and perpendicular height, then to reciprocal relations between the weight of a body and the height from which it falls in case of dynamical equivalence between states. One thereby arrives, for example, at proposition 17: An equal power can lift one pound two feet or two pounds one foot; this generally means that if the heights of elevation are reciprocally proportional to the weights, the powers lifting them are equal. (GM VI 445)

In a series of propositions governed by the idea of the conservation of equilibrium between the forces at play, Leibniz argues for a measure of compensated forces determined by the product of mass, or weight, or dense volume, and space. The examples are borrowed from statics, hydrostatics, and dilatation and condensation under the effects of heat. Consequently however, these are cases of actualized constraint, where the dynamical effect endures without the cause being used up. In order to ensure the transition to a direct evaluation of force via its depletion in the cause, Leibniz offers new nominal definitions in Propositions 27 and 28, those of centrifugal conatus and of the generation of living force based on dead force (which is initially comprised of centrifugal conatus): Whatever is moved along a circular line is forced to recede from the center of the circle; and the force of receding is called centrifugal conatus. (Proposition 27, definition 4, GM VI 451)

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When the centrifugal conatus continues for a certain time, it acquires the speed to recede from the center; and the centrifugal conatus is to this speed to recede, which is generated by the continuing conatus, what the infinite is to the finite. Force that relates to speed in this way, or which is infinitely less than the speed, I call dead; but force that has speed or is comparable to its speed I call living. (Proposition 28, definition 5, GM VI 451)

The experiment on which the distinction between dead and living force is based is that of a body at rest in a tube rotating circularly around a center that is at one of its extremities. Assuming that the rotation is a uniform motion, the moving body within the tube has an initial conatus that recedes from the center, which is established by the circular motion and the initial conatus of motion along the tangent. Leibniz’s thesis is that centrifugal conatus is an infinitesimal of displacement that requires us to sum an infinite number of such elements to progressively generate the impetus of the mass. Thus, it is by this means alone that the effect of change occurs, i. e., a displacement whose quantity is commensurate with that of the resulting effect of rotation. Hence it is the relation of the finite to the infinite that characterizes the centrifugal conatus in relation to the impetus generated by the summation of successive conatus, even if impetus itself is an instantaneous measure of the power to act. Regarding this subject, Leibniz recalls Galileo’s assertion that the force resulting from percussion would be infinite in degree compared to the instantaneous expression of the force of gravity.40 For Leibniz, such is the relation that a generated impetus expressive of a causal state has with a conatus representing the still virtual tendency in the descending motion of a heavy body. According to Leibniz’s analysis, centrifugal conatus and impetus are privileged in relation to the elements of vis percussionis and gravity owing to their perfectly seamless homogeneity. Another reason is the fact that the effect of gravity seems to result from an “occult” cause for which it can be difficult to substitute strictly mechanical reasons – and such had also been the case of elasticity since the Hypothesis physica nova. By contrast, centrifugal force seems to derive from the tangential determinations of mobile bodies that are circularly displaced by rotation. In fact, Leibniz immediately applies this model to an explication of gravity and elasticity, as propositions 29 and 30 show. He thus ignores the contingent material conditions that could alter the geometric interpretation, or more precisely, they are subsumed under the rule for generating impetus by summing conatus according to the quantity of mass: 40

This analogy is well developed in the letter to Varignon on 12 August 1707, GM IV 159: “To say that resistances are infinitely small, one must stipulate what they are infinitely small in comparison to; they are infinitely small in comparison to a moving body’s speeds, which I sometime also call impetus (impétuosités) to distinguish them from these imperfect or latent (embryonnées) speeds, such as those that a heavy body has at the first instant of its fall and receives at every moment thereafter. This is why Galileo inverted the comparison and taking weight to be something ordinary, said that impetus, and so also percussion, was infinite, while, taking speed to be an ordinary magnitude, weight and also instantaneous resistance which is homogenous with it, would be infinitely small.”

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These propositions assume however that gravity acts continuously in accordance with mathematical reasons. And they do not even take into account sensory features (sensibi­ lia) where force can be so small that the elevation of a great weight may not be noticeable or the impetus exerted is absorbed by intermediaries such as fibers, strings and similar things. However, the dead forces are augmented by the continuation and generate impetus or speed. (GM VI 453)

The demonstration of the theorem of conservation takes a mixed approach, whereby Galileo’s law is integrated as an a posteriori argument into a system structured with help of the theorem of conservation of motive action and the axiom of equivalence between the full cause and the entire effect. Consider a proposition analogous to the law advanced by Galileo: “The perpendicular heights from which heavy bodies fall are in a duplicate ratio comprised of the speeds acquired thereby.” (Proposition 53, GM VI 453) The proposed demonstration is formulated as follows. As premises, we adopt the axiom of equivalence, and measure the absolute motive action of the bodies by placing them in a compound ratio where their weights are multiplied by the square of their speeds. The entire effect resulting from the motive power is proportional to the height to which a body raises itself by virtue of its motive action. The total consumption of motive action is measured in direct relation to this vertical distance. Hence, there is the implication that this height, for any given moving body, is directly proportional to the square of the speed of the moving body at the moment when its direction changes from horizontal to vertical. At the same time, the power acquired by a body falling from this height can be entirely transferred into unconstrained motive action. This effect of active power is itself independent of the weight of the body, but directly proportional to the vertical space traversed by which it is measured. Thus, one arrives at an absolute measure of the latent conatus and its law of summation. Henceforth, Leibniz can invoke Galileo’s law by asserting that if the heights are directly proportional to the square of the speeds, then the increases in height can be accounted for by a compound ratio of the speeds and their rates of acceleration. Considering the direct relation of speed to time, one comes to a conclusion that returns to Galileo’s empirically established formula: “And thus, to conclude, the science of motion is liberated from hypotheses.” (GM VI 453) This means that, by proposing a complex phoronomic measure of motive action, the science successfully furnishes a system of representation in which Galileo’s law is incorporated as an empirical model. But we are not then presented with a purely a priori demonstration, because the equivalence between force and the power to act expressed in terms of motive action is the result of an analogy drawn from experience, i. e., the dynamical effect of pendular oscillation which yields the conservation of conatus that are successively transformed into impetus and vice versa. At the same time, the effect of gravity incorporated into this analogy indicates that the height of the fall generating the maximum impetus measures the dynamical causality of the phenomena. The essence of the theoretical framework provided by Leibniz is to work

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out the correlation between impetus and absolute power. But the measure for these quantities is not identical, and Leibniz establishes that quantity of motion is merely a formal element, and is insufficient for characterizing the motive power of the effect that entirely expresses it. The measure of absolute power can only be obtained by an exhaustive effect of motive action, such as the one revealed by work that compensates for the cumulative effect of gravity. In relation to this novel systematic approach, Leibniz attempts to distinguish the effects relating to the quantity of motion from those relating to absolute power, both in the global conservation of mechanical effects and in bodily impacts. Regarding the global conservation of mechanical effects, he invokes Galileo’s law or rather the more theoretical version in which he couches it. He thereby distinguishes the conservation of living force from that of impetus, given the impossibility of perpetual mechanical motion. Indeed, if one had accepted a quantity of absolute action equal to impetus, then the action corresponding to the compensation of the effect of gravity in free fall would result in a significant disequilibrium in the relation of the full cause to the entire effect and vice versa. This is one of the most frequent arguments that Leibniz opposes to the Cartesians. Nonetheless, it is out of the question to deny a certain conservation of impetus in bodily impacts, which will be demonstrated by the analysis of displacements of the center of gravity of bodies before and after impact. This will give rise to the second relative equation in the Essay de dynamique, which concerns the conservation of the quantity of direction (quantitas nisus, seu conatus ad certam directionem sive vim directricem). For the time being, the issue concerns the role played by vis respectiva in the causal relation between absolute powers and their effects. Definition 7 provides the nominal rule for this relation: Respective force is that in virtue of which two bodies act upon one another; and when two colliding bodies stop one another, one will say that each has the same respective force. One can also call it the force of impact (ictus) or the force of percussion. (GM VI 462)

Leibniz therefore conceives of the relation of equilibrium between impacting bodies in such a way that the resistance between each is exactly proportional. The following proposition 41 establishes this equilibrium in terms of impetus, that is, in terms of the quantity of motion in colliding bodies, supposing that they are equal in all other respects (the cæteris paribus clause).41 The demonstration of this proposition specifies that within a unit of time, constrained effects are proportional to impetus; consequently, their mutual compensation is responsible for the state of equilibrium. In effect, a unit of time can be reduced to an instant wherein the latent conatus are related to mass,

41

Cf. GM VI 462: “If two moving bodies directly or perfectly collide with the same quantity of motion, they will have the same respective force, that is, they will mutually stop their progression.”

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without absolute force being expressed in effects that incorporate motive action in both its intensive and extensive dimensions. Then, Leibniz proposes, without further justification, the possibility of deriving relative force from absolute force, as one of its elements. The Essay de dynamique will develop upon this point by formulating the equation of absolute conservation through combining the relative equations. These relative equations can be derived from one another once the equation of absolute conservation is established. It turns out that in cases of equilibrium, and in all cases where action is limited to the interplay of compensating forces, the latter can be substituted salva veritate for the relation of absolute powers or living forces, which cannot then be concretely generated and deployed. The compensating impetus make it impossible for such effects of living forces to take place in an instant. This fact explains the error of the Cartesians who generalized compensation by equilibrium for all dynamical interactions, despite its being a border case. In this way, they mistakenly interpreted the conservation of quantity of motion as a fundamental law of nature. Leibniz’s law has the systematic advantage of including the Cartesian law among all the relations between the factors at play. From a physical point of view, this can be represented as follows: the entire effect of power is a combination of the diffusion of im­ petus and intensive speed. At the beginning and end of a motion, diffusion either does not yet occur or has already been reabsorbed. The result is an effect of mass combined with latent conatus alone. In bodily collisions, the diffusion of impetus is reabsorbed at the moment of impact, and everything returns to a state wherein dynamical effects compensate for one another, owing to mass and the related elementary conatus. But living force can be regenerated given the elastic nature that prevails in the structure of the concerned bodies. This is why, aside from the strict equilibrium between impetus, in all of the other cases, living force reappears after the impact in the effects that exceed the phase of latent motions that are constrained to remain as such. In the case of equilibrium, rest is only an appearance representing an indefinite sum of virtual motions, something which Leibniz excellently demonstrates with his theory of impetus. To complete the analysis of the dynamical “system,” we must note that section II of the second part, entitled De centro gravitatis et directione motus, (GM VI 464) is connected with a new series of propositions that introduce relative equations. Thus, for example, Leibniz develops his rule for combining nisus in terms of the diagonal of the parallelogram of forces.42And he also conceives of all curvilinear motion as being the 42

Cf. Proposition 2 and Definition, GM VI 470–71: “If two rectilinear motions, uniform or endowed with proportional speeds combine such that the moving body placed at an angle scours during the same time one side of the angle with a given speed, and the other side with another speed, the compounded motion will develop along the diagonal of the parallelogram based on those sides and will scour it during the same time; and it will be similarly uniform or increasing or decreasing proportionally with the prior motions. Hence, if the speed and direction of the compounding motions are represented by the sides of the parallelogram, the speed and direction of the compounded motion will be represented by the diagonal. We shall call nisus the direction along with

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result of two linear motions, where one is uniform or uniformly accelerated/decelerated.43 But more fundamentally, he accounts for the displacement of the center of gravity of a system when the elementary quantities of progress of the masses comprising it will compensate for one another and maintain the same overall configuration. All of these considerations help establish the systems of equations that represent the conatus of material points when circumstances prevent the absolute power from fully developing in the material masses in which these conatus are inscribed. Thus, this involves an abstract phoronomy representing the phenomenal constants of motive actions. And the hypotheses that aim to account for this constitute alternatives extrinsic denominations. The choice between these hypotheses can only respond to pragmatic criteria. The task of formulating hypotheses becomes radically different when living forces are introduced, for one is then establishing a physical reference founded on what is most real in phenomena. Nonetheless, the criterion of simplicity constitutes an important index when it comes to choosing “objective” nominal representations for relative phenomena: This is why nothing prevents any one of the hypotheses from being true if we search only for what is mathematically possible. We surely understand that which is purely mathematical in motion and can alone be accounted for, namely positions (situs) and their changes. But the physical part, namely the subjects of the acting cause and of forces, and the explanations capable of making sense of them, belong to another sort of thinking. Whether the same phenomena can be saved by any of the proposed hypotheses when it comes to bodies impacting one another is something we shall determine below. If there were several hypothe­ ses, and one hypothesis held that bodies moved uniformly and another did not, and no external causes were present, then the hypothesis of uniform motion should be preferred, since all motion is in itself so. This then is how things shall be when causes of change are lacking. However, in rectilinear motions, several uniform motions can produce the same effect, whichever subject is considered to be at rest or in motion. (GM VI 485–86)

This system of hypothetical representations relates to reality, on the one hand, owing to the physical causes that determine the modifications that express the forces at work in the chain of motions, and on the other, owing to the analogical models that make it possible to rationally generate the most complex phenomena of displacement. It is in keeping with the spirit of mixed science to formulate abstract representations that serve as necessary ex hypothesi derivations. At the same time, this system can express

43

the speed correlated to the quantity of matter, but when we designate at the same time a flexion, we shall call it conatus, a more general term. Hence all nisus are rectilinear, while conatus are rectilinear as well as circular, or otherwise curvilinear.“ Cf. Proposition 5, GM VI 472: “Every curvilinear motion on a plane can be understood as being composed of two rectilinear motions, one of which is uniform or has a speed that increases or decreases according to a given law. For motion in a solid is comprised of three rectilinear motions.”

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real phenomenal causalities essentially because it reconciles a posteriori and a priori inferences about forces and actions. Section III of the second part, De concursu corporum, is the last one in the manu­ script. (GM VI 488 sq.) There, Leibniz develops theorems relating to the laws of impact for bodies, beginning with the fundamental theorem of the conservation of absolute power. This power is equally divided between bodies before and after impact, according to the complete effects that correspond to the exhaustion of related forces. In view of a general hypothesis of elasticity, one can, with the help of this theorem, generate rules for combining mass, speed and direction, from which one will obtain a continuous representation of the real possibilities of mechanical effects. The general hypothesis of elasticity depends on the architectonic principle of continuity. In this regard, Leibniz explicitly refers to the article of 1687 that appeared in the Nouvelles de la République des Lettres.44 But the argument that he presents here is typical, as it generalizes the analogical concept of gradual transition from one phenomenal state to another by requiring a sufficient reason that can justify whatever leap one might imagine. “But, in nature, there can be no attributable or recognizable instantaneous change; and therefore, one cannot pass from one degree of speed to another without intermediary degrees.” (GM VI 491) This negative argument is equally one that allows us to disqualify the hypothesis of atoms. If the argument has metaphysical roots, it is employed here in such a way as to analogically make sense of a general condition for representing the ordering of phenomena. Furthermore, Leibniz does not hesitate to invoke empirical observations to justify the inherent flexibility of impacting bodies. Take, for instance, a boat crew pushing off the shore with its oars. This example illustrates the force of elasticity or internal motion by which a body rebounds from another. Another significant example involves balls impacting a door that is ajar and passing through it rather than causing it to close, for if more force is required to breach it than to close it, however penetration is instantaneously facilitated by elasticity. This account of elasticity, which is at least partly empirical, serves to establish the principle of conservation of vis respectiva in impacts. This vis respectiva constitutes the portion of potentia absoluta generated by attributing to impacting bodies speeds that are reciprocally proportional to their masses: “The respective force by which two bodies can act on one other is this part of the absolute force that occurs when attributing speeds to the bodies that are reciprocally proportional to these bodies; and these speeds are such that they maintain the same respective speed following the impact.” (Proposition 9, GM VI 493) It is therefore the case that directly opposed impetus compensate for each another instantaneously, such that the motion of the bodies is gradually brought to a halt by interpenetration; the presumed intrinsic elasticity of bodies 44 Cf. 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 server de réplique à la réponse du R. P. D. Malebranche (1687), GP III 51–55; L 351–54.

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allows us to conceive of a summing of conatus that generates an inverse impetus equivalent to the initial impetus. One thereby arrives at the following proposition: Two directly impacting bodies with speeds reciprocally proportional to the bodies exert their total force on one another and are reflected at their initial speed, at least if they are elastic enough and the force of percussion is not absorbed by the parts of the body itself. (Proposition 8, GM VI 493)

Furthermore, the relation between the respective forces implies a constant ratio between their speeds throughout the impact. This ratio consists in a relation between relative speeds, parallel and thus analogically similar to the tension of the elastic body (tensio elastri). (Proposition 10, GM VI 494) Then, Leibniz adds the conservation of vis progressiva, which is combined with vis respective to form potentia absoluta.45 Significantly, a semi-concrete, semi-abstract model serves to represent this combination of partial causal determinations. Suppose, for example, that there is a group of bodies at rest or in motion connected along (perfectly elastic) rigid lines. These lines between bodies will maintain perfect reciprocity between their intensive nisus, that is, between their relations of impetus. At the same time, supposing that all of the bodies comprising such a system are endowed with a motion that causes displacement, a vector applied to the center of gravity will express this directional tendency. On the basis of this model, if one assumes the existence of different impacts within the system, not only is the total sum of quantity of motion maintained, so too is the directional displacement of the center of gravity for the entire system. In bodies that gradually absorb the effects of impacts, the total result will seem to indicate, as experimentation attests, a reduction in absolute power or living force due to the impetus involved infinitely compensating for one another. But directive force will remain in its entirety, even if the force of impact (vis ictus) is no longer manifest. In keeping with his goal, Leibniz goes on systematizing the various combinations of dynamical relations. Quantity of motion and quantity of progress depend on the exercise of living force under conditions of constraint. The types of constraint that allow for establishing equations of relative conservation are related to elastic impact considered at its inception. The respective force can be distributed among the two impacting bodies in accordance with their respective impetus. In this regard, Leibniz underlines the fact that once the equation of conservation of living force is postulated, the relative equations are implied: “It is evident that the same quantity always subsists, to which one must relate the power present in the force of impact as well as in the directional force.” (GM VI 499)

45

Cf. Proposition 11, GM VI 495: “The absolute power of an aggregate of several bodies which results from their motion is comprised of their respective force for acting on each other and of the force of progression (for acting on a third one) along the way of one of them, that is, by the force of direction.” Proposition 12, GM VI, 496–99, extends this theorem to cases involving a plurality of bodies.

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These principles of relative conservation yield particular expressions of it, whether they illustrate the conditions of dynamical equilibrium between impetus that owe to elasticity, or they express the compensation of impetus in the constant and free displacement of a system, which constitutes its quantity of progress. If the quantity of motion remains intrinsically the same when bodies interact, its dynamical application can either cause a constant quantity of directional impetus to remain, or an equal distribution on both sides, or a partial expansion in one direction, accompanied by an equivalent subtraction on the opposite side. It might be the case that what had caused an addition before the impact also causes a subtraction afterwards, or vice versa. From this it follows that one might be dealing with relations of quantity of motion that compensate for each other within the given system. As the following demonstration indicates: The same quantity of progress remains before and after the impact; and if it is established in one and the other state by simply adding the progress of the two bodies, then surely the quantity of absolute motion will remain the same; insofar as a subtraction has to be made in either one or the other or in both states, the entire progress will be the difference between those quantities of motion traced along parallel lines. (GM VI 500)

Ultimately, these principles of conservation have something in common with the relativity of hypotheses concerning the reciprocal displacement of moving bodies. But Leibniz draws a remarkable conclusion from them regarding the convergence of phoronomic hypotheses in relation to the principle governing living forces. In effect, it is possible to break down phoronomic phenomena into combinations of various motions from which these phenomena would result, and conversely, to combine these various motions into one. This is the flexibility that the relativity of hypotheses grants. If, however, one cannot establish a strict relationship between the summation of living forces on the one hand, and the summation of relative and directional forces on the other, the laws of relative conservation can, with one exception, be combined in such a way that one infers the conservation of absolute powers that are calculated on the basis of the phoronomic combinations involved.46 The exception is motion comprised of two motions that form a right angle. The motion that results is diagonal according to the rule of the parallelogram of forces, and there is, in this case, proportionality between the relative equations and the absolute equations by virtue of the phoronomic hypothesis itself.47 In all other cases, Leibniz indicates, the relativity of phoronomic 46

47

Although he neither takes into account the Dynamica de potentia nor the problem of combining a priori and a posteriori approaches, M. Spector (1975) analyzes the system for integrating equations on the basis of the later Essay de dynamique. Thus, he simultaneously discovers a reason for emphasizing the universal aspect of Leibnizian dynamics from a theoretical perspective, as opposed to the Cartesian analyses, whose conditional and so to speak “localized” validity Leibniz exposes. Cf. Proposition 23, GM VI 512: “In a motion comprised of two motions forming a right angle, the absolute power along the diagonal is the same as in the two motions taken from the sides of the rectangle.”

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hypotheses never poses a problem to interpreting any system of motion in light of the principles of conservation of respective force and of quantity of progress. The demonstration extends, for example, to the analysis of gyratory phenomena, such as those that determine the combination of conatus along the tangent and conatus that constitute the pressure of the surroundings.48 It also extends to analyzing the cohesion of material parts in virtue of antagonistic conatus.49 But Leibniz envisions its extension to physical phenomena that assume the existence of a law determining the order and relationship between the “themes” or states implied in causal activity, for instance, the equality between the angles of incidence and reflection when a body conserves in an oblique impact all of its force of percussion owing to perfect elasticity.50 According to the rational model suggested by the facts observed, this involves a so to speak normative property of harmony for all physical bodies. Ultimately, Leibniz implies in the Dynamica de potentia that, behind the often complex phoronomic analysis, there looms the possibility of breaking down phenomena into rectilinear and uniform motions capable of forming a system. Hypotheses based on extrinsic and thus relative “denominations” represent this system more geometrico. Then, to account for the production of these systems of ordered motions, the dynamics sets about constructing adequate and sufficient causal models. Such models rely on abstract concepts and regulatory principles, such as those of continuity and equivalence between the full cause and the entire effect: and this is how the principle of conservation of formal action is established a priori. On the other hand, according to what is suggested by experimentation and the principle of causal equivalence, Leibniz proposes a theorem of conservation of living force and absolute power that can be considered as being congruent with, or equivalent to, the a priori demonstrated principle salva veritate. The analysis of combined phoronomic factors representing the power to act, then serves to formulate the theorems of relative conservation of respective force and quantity of progress. These theorems are based on the elementary physical element of force, impetus, which is itself representable as an instantaneous summation of the conatus in the mass. However, Leibniz’s theory synthesizes these different moments, and, to this end, establishes and identifies an architectonic connection capable of integrating the various components of these phoronomic models. Indeed, these propositions can, to a certain extent, be situated in a framework of deductive inference.51 Furthermore, one can analytically establish the same inference by retracing, in the reverse order, the combination of theorems of relative conservation

48 49 50 51

Cf. Proposition 18, GM VI 503–507. Cf. Proposition 20, GM VI 508–511. Cf. Proposition 25, GM VI 513–14. This is something that Gueroult successfully demonstrated with regard to deriving the absolute equation from the relative equations by following the presentation of the later Essay de dynamique. See Gueroult (1967), 50–51, particularly, n.3.

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in such a way as to produce an analogue of the causal theorem concerning the order of absolute powers. But Leibniz is primarily concerned with bringing to light the fact that the proposed inferential model illustrates the organic unity of phenomena, that it is inscribed within the harmony of the various mechanical laws. These are evoked in correspondence with one another in the theoretical structure. Thus, for instance, Leibniz provides arguments for justifying that tendencies to circular motion in the physical universe, when analyzed in phoronomic terms, perfectly accord with the laws of dynamical conservation. The concordance between the theoretical representations is the fundamental justification for this methodologically mixed approach: But since the impressions made by the surrounded bodies are cancelled out by the receding conatus in the gyrating bodies, and therefore nothing, owing to the inherent forces of the bodies themselves, deviates from the rectilinear motion of what is moved in the vortex, the same forces as before, absolute and respective, will persist. In this regard, I have declared that the constancy and harmony of Nature encountered in the observation of these laws merited admiration, and that one has yet to sufficiently consider them to this date. (GM VI 506)

In the wake of this argument, Leibniz challenges the fact that Descartes had separated the force measured as quantity of motion from the direction of the implied motion, thus rendering the latter arbitrary and bereft of any causal foundation in the laws of nature. Indeed, Descartes made use of this presumed absence of sufficient reason regarding the application of force as a means of conceiving of how souls can determine the orientation of bodily motions. For Leibniz, the hypothesis of there being a regulated concomitance between the actions of souls and those of bodies can alone secure the theoretical coherence and independence of an explanation of the reasons behind physical phenomena. However, from the point of view of the phenomena of motion themselves, Leibniz insists that an inherent architectonic reason governs the combination of the fundamental mechanical determinations of directive (relative) force and living (absolute) force: “Because nature does not strive, with less attention, to conserve the sum of directive forces than that of absolute forces (which this philosopher [Descartes] confused with the quantity of motion).” (GM VI 506–507) In sum, on the one hand, Leibniz resorted to abstract models, including some borrowed from infinitesimal mathematics, to represent the causal generation of motive effects, but, on the other hand, he subordinated to formal requirements the combination of conceptual ingredients that were invoked to represent the dynamical processes, including architectonic principles that governed a priori the formulation of physical hypotheses. This methodology was systematically implemented, notably in the second part of the Dynamica, when Leibniz put forth definitions and axioms that made it possible to recast the a posteriori demonstration of the theorem of conservation of vis viva within the theoretical framework of reasons provided by the principle of conservation of formal action. Recall his declarations in the Specimen præliminare that, owing to the a priori method of demonstration:

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Not only does a remarkable harmony between the truths become manifest, but so too does a new way open for proving Galileo’s propositions concerning the motion of heavy bodies without the hypotheses to which he resorted, namely that, in the uniformly accelerated motion of heavy bodies, equal increases in speed are generated in equal times. (GM VI 292)

It is probably useful to recall here that grounding Galileo’s physics on the a priori principles of mechanics counted among the objectives that Hobbes had wished to pursue in the De corpore. Not surprisingly, a similar objective can be found at the outset of Leibniz’s Dynamica, one that had also appeared in the Phoranomus a few months earlier. But Leibniz’s was a deeply revised program, for it implied that kinematic concepts were to be established in virtue of higher level reasons, which included architectonic principles and metaphysical requisites that would guide the formulation of physical hypotheses. Finally, one might argue that the two separate parts of Hobbes’ method in physics (i. e., causally deducing natural effects from the concepts defining the modes of motion a priori, and analytically proceeding from phenomena to their causes by means of hypotheses) are still reflected in the architecture of the Dynamica at the heart of Leibniz’s mature physics, but that the distinction between the two ways has been radically revised. For Leibniz, a metaphysical approach to the true system of nature underpins the abstract combination of concepts relating to the modes of motion. And these modes are now represented through systems of integrated relations that express architectonic requirements. For Leibniz, in nature, the order of causes and the order of ends pertain to the same unifying system of reasons. In his mature physics, Leibniz surely preserved what he could of the twofold strategy of the early mechanics, but he would revise, complete and even surpass it so that it could conform to a higher level of rational requirements. Though still inspired by his former methodological approach, he had somewhat shifted away from it by constructing all a priori geometrical models of the laws of motion in function of theoretical propositions that, involving metaphysical and architectonic concepts, were to guide the formation of adequate hypotheses. 3. The Specimen dynamicum: Presuppositions The Specimen dynamicum, the first part of which appears in the Acta eruditorum in 1695, constitutes, nine years later, the necessary follow-up to the Brevis demonstratio. Here, emphasis is placed not only on the principle of conservation of living force and the a posteriori argument the justifies it, but also on reforming the foundations of the physical theory that underpins the dynamics. Indeed, the Specimen is also a prolongation of the Dynamica de potentia of 1689–1690, this more geometrico formulation remaining in-

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complete and unpublished. Leibniz can hardly ignore the synthetic presentation that gave rise to an a priori argument, and which led to the emergence of a new principle in relation to the theorem of living forces, that is, to the principle of conservation of formal action. In certain respects, one can consider the Specimen dynamicum as the reflection and expression of the theoretical improvements of the preceding years. Without a doubt, it also provided an alternative to a theory that was, at once, more knowledgeable, more rigorous and more axiomatic, but also more “superficial” so to speak. For it was restricted to the level of mathematical symbolization and to the representation of a priori models, rather than tackling head-on the relationship between models and theoretical entities and the causal order underlying the series of mechanical states represented in geometric models and their corresponding equations. However, in return, this new text adopts a style and direction that is more programmatic than systematic. The more geometrico demonstration is refracted into an analytic format that allows one to formulate the methodological and metaphysical presuppositions of the physical theory being developed with greater precision. But the problem with constructing mathematical models and determining their ontological meaning is, like the problem with the a priori reformulation of arguments, left somewhat aside. It is as if epistemological reflection had yet to attain, amongst these considerations, a sufficiently coherent status. If there is no final achievement in this regard in the Specimen dynamicum, new frameworks capable of influencing the subsequent analysis might nonetheless emerge from the text. Masterful but elliptical, the Specimen points to the complex nature of the ingredients comprising the physical theory being formulated by Leibniz, and goes beyond the so to speak provisional and relative harmony between representations of dynamical effects articulated with the help of mathematical models. In fact, the Dynamica de potentia had presented a marvelous framework of definitions. But how can one prove that this framework corresponds to the causal order underlying well-founded phenomena? Evidently, from a Leibnizian perspective, the more a combination of definitions proves essential, the more it may legitimately pretend to symbolize the real order of things. The Dynamica had taken for granted the fact that the abstractly-defined could be translated into the real within the theoretical constructions of the synthesized models. This methodological “hypothesis” required a search for justifications, which Leibniz undertakes in the Specimen dynamicum by associating the fundamental demonstration of dynamics with the analysis of its presuppositions in a system of architectonic principles and theoretical concepts. As the Specimen dynamicum has been the object of numerous analyses in the literature,52 we shall limit ourselves to searching the text for those elements that will allow us 52

It is without a doubt that this text, among others, is at the heart of Gueroult’s interpretation, to the extent that it represents the methodological prevalence of the a posteriori approach as being incontestable (cf. Gueroult 1967). The same can be said of the interpretations proposed in Garber (2009), Tho (2017), Garber and Tho (2018).

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to formulate a thesis regarding the epistemological status of the principles and theoretical postulates of dynamics. The context is established from the outset, since it involves justifying the foundations of dynamics as a new science.53 The analytic search for determinative reasons requires examining the foundation of corporeal reality beyond extensive properties and motion. Extensive properties by themselves could not possibly represent the sufficient reason for the power to act that manifests itself in nature, nor satisfy the rational requirements for unifying phenomena into substantial relations. Nonetheless, these arguments remain implicit while, by contrast, the inadequacy of motion as concept capable of describing a real foundation is underlined. Indeed, neither could this possibly entail renouncing the achievements of the new physics, which made motion the foundation of bodily action, nor contesting the fact that all physical processes belong to a sequential chain of motions. But the only real ingredient of motion lies in the instantaneous disposition to motion: this generates the series of the successive moments of motion, which can be dissociated and identified as such only by abstraction. This “force striving toward change” (vis ad mutationem nitens) (GM VI 235; AG 118) refers to metaphysical equivalents analogous to Aristotelean forms or entelechies. We must compare these initial suggestions with the developments in the Système nouveau de la nature et de la communication des sub­ stances, (GP IV 477–87) which appeared the same year. The characteristic trait of the latter text lies in that it assimilates force and substantial form into the foundation of the general theory of entelechies. Leibniz even goes so far as to consider perception, revealed through reflective experience, as an analogue of force. As a result, the metaphysical order undergoes a sort of physicalizing, whereby the ultimate units of nature are presented as formal atoms, or substrata wherein the fundamental capacities of dynamics are contained. This conception of the system still lacks the deep, groundbreaking perspective that the monadic model will furnish. It is only with the introduction of the monad that force will appear as a well-established phenomenon, one that attests to the essential property of perception-appetition in the simple, finite substances called monads. For the moment, the search for the reasons behind the physical order is still situated within a two-tiered hierarchy. Leibniz conceives of the power to act at the foundation of physical reality as a disposition that spontaneously expresses itself. He refers to this disposition by the terms nisus or conatus. These terms, particularly conatus, have a history in the development of Leibniz’s thinking. As it has been shown, they derive from the influence of Hobbes and Cavalieri in the days of the Hypothesis physica nova. From the beginning, they had signified indivisibles of motion that, combined together, cause extensive masses to be displaced. If the term conatus has lost the connotations that inadequate mathematical 53

As we pointed out at the beginning of the chapter, this notion of dynamics as a new science found itself officially advertised in 1694 in De primæ philosophiæ emendatione, et de notione substantiæ, GP IV 469; L 433.

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and physical models had previously conferred to it, then it reemerges in the context of dynamics as the ingredient of potential force that expresses itself in all physical beings that begin to act. This model translates the immanence of dynamical dispositions into units comprising phenomenal realities. At the same time, it suggests that extensive properties themselves result from the diffusion or continuation of these elementary tendencies to action, counterbalancing one another by virtue of the mutual resistance of bodies. Typology is of crucial importance here. Leibniz presents the physical theory suggested by this metaphysical conception of nisus or conatus by outlining the distinction between active force and passive force, and by further categorizing each term in virtue of a new criteria, that of being either primitive or derivative. We thereby arrive at four terms: vis activa primitiva, vis activa derivata, vis passiva primitiva and vis passiva deriva­ ta. (GM VI 236–37; AG 119–20) This central typology has been reinterpreted time and again; however, being somewhat like the analogy of the line in book VI of Plato’s Re­ public, it is somewhat difficult to differentiate between the various methods by which it might be formulated. The text is presented as follows: Active force (which some might with reason call power) has two versions: as primitive, it inheres in every corporeal substance per se (as it would be contrary to nature, I believe, for a body to be wholly at rest); as derivative, it derives from collisions between bodies and exerts itself in different ways depending on the limits of primitive force […]. Likewise, there are two versions of passive force, primitive and derivative. And the primitive force that is acted upon and resists [other forces] constitutes, if one properly understands it, exactly what the scholastics call primary matter: it is the reason why one body does not penetrate another but instead impedes it, and why this body is also endowed with a sort of laziness, that is, opposition to motion, and therefore does not allow itself to be moved by a body without diminishing some of the latter’s force. Hence, the derivative force of being acted upon appears in various ways in secondary matter. (GM VI 236–37; AG 119–20)

In this two-variable table, the level of derivative forces and resistances corresponds to the phenomenal effects, while that of primitive forces and resistances represents the determinative reasons behind such effects. Relying on these distinctions, George Gale insisted on the necessity of presuming the existence of an additional level of analysis. The direct consequence of this would mean that the level of primitive forces would become the middle one, while the indirect consequence of this would mean that this level would constitute a special case in the regressive analysis with which the Specimen dynamicum seems to be concerned.54 To this level there would correspond the postulation of theoretical entities that cannot be reduced, on the one hand, to simple empirical descriptions, and on the other, to a monadological ontology.

54

Cf. Gale (1970), (1974), (1988).

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According to Gale, the Leibnizian model would consist in three levels: (1) the ob­ servational level, where derivative active and passive forces can be found, and where one is dealing with bodies as phenomenal entities; (2) the explanatory level, which concerns active primitive forces (i. e., substantial forms) and passive primitive forces (i. e., primary matter); the substrate referred to at this level would be that of corporeal substance per se, which is needed as a true ground for phenomena; and (3) the strictly metaphysical level, which relates to monads and to their distinct and confused perceptions. The two first levels are qualified as phenomenal, while only the third surpasses the limitations of the phenomenal order. At the developmental stage that the Specimen dynamicum represents, the devices mobilized by this interpretation do not quite cor­respond to the direct meaning of Leibniz’s distinctions. Thus, an additional level is out of the question. Active primitive forces are considered equivalent to primary entelechies, to which substantial forms and souls correspond. One is a long way from a system of strictly phenomenal sufficient reasons. If one might give some credit to Gale’s hypothesis, then it owes to the fact that Leibniz associates the extension, impenetrability and inertia found in corporeal reality with second level explanatory characteristics. Gale relies on Leibniz’s position here to reconstruct a system of parallel concepts on the side of active force. In so doing, he relies on a suggestion taken from the letter to De Volder on 3 April 1699, where force is likened to a law that persists across the series of states that it generates.55 This reconstruction might represent a subsequent stage in the evolution of Leibniz’s epistemology. However, and above all, this stage will not involve what Gale understands to be a phenomenal reference point. Let us consider the supposed denotation of passive primitive force. From the outset, extension, which is a phenomenal characteristic par excellence, does not find itself on the “explanatory” level. By contrast, impenetrability and inertia do, but as abstract characteristics, owing to the intrinsic limitation of the entelechies under consideration. At the phenomenal level, impenetrability and inertia are related to the extension of bodies, from which one cannot truly detach them. Hence, from the point of view of the science to be constructed, the explanatory level cannot be that of properly phenomenal entities. Instead, at that level, we are presented with theoretical entities that are abstractly defined and conform to architectonic principles. It seems doubtful to us that Leibniz cultivates, at this presumed intermediary level of scientific intelligibility, a plain realism of phenome55

GP II 171; Leibniz (2007), 74–76: “And so the resistance of matter contains two things, impenetrability, i. e., antitypy, and sluggishness, i. e., inertia. And since they are equal throughout bodies or proportional to their extension, I locate the nature of the passive principle, i. e., matter, in these, just as I recognize primitive entelechy in the active force exercising itself in various ways through motion and, in a word, in something analogous to the soul, whose nature consists in a certain perpetual law of the same series of changes, which it undergoes unhindered one by one. We cannot do without this active principle, i. e., the ground of activity, for accidental or changeable active forces, and motions themselves, are certain modifications of some substantial thing. But forces and actions cannot be modifications of a merely passive thing.”

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nal entities.56 Likewise, a fortiori, active primitive force appears to us as a theoretical framework founded on the use of general regulatory principles. One therefore finds oneself relegated to an abstract level beyond all manner of empirical systematization.57 Besides, with respect to the distinctions of the Specimen, the analogical relationship of primitive active and passive forces to form and matter that gets introduced might allow us to identify, in a certain sense, the governing concepts of the theory with the theoretical entities that constitute the system. Hence, and by implication, the relative inseparability of the level that Gale seeks to radically divide into the explanatory and the metaphysical. In the Specimen dynamicum, the approach that Leibniz takes to elaborating dynamical concepts is essentially analytical, that is, guided by the search for sufficient reasons capable of justifying the relation and harmony of phenomena at the non-phenomenal level. Such methods of analysis can belong to various orders, some more metaphysical (focusing on the properties of entelechies) and others more epistemological (focusing on regulatory and architectonic principles), without posing a problem to the rationality of the explanatory process. At its heart, is Leibniz’s metaphysics of nature not a system of conceptual models for justifying regular and thus well-founded series of observed phenomena?58

Fig. 9 Illustration of the demonstration of the theorem of living forces in Specimen dynamicum (GM VI, fig. 25)

56 57

58

Gale continues to defend this position in Gale (1984), 70, where he argues that Leibniz identifies forces as entities that have a specific ontological status as the substrate of both actual and future effects. In Gale (1988), Gale leans toward a more nuanced position. He shows in effect that the postulates of dynamics serve as physical analogies for a metaphysical conception of the system of nature built on the notion of harmony. However, generally speaking, Gale seems to ignore the role that architectonic principles play in establishing the foundational postulates of physical theory, and in defining the concept of force as a theoretical construct capable of explicating phenomena. We must mention that the historical analysis at the root of this work remains unsatisfactory insofar as it fails to take into account the reformatio of 1678. Our interpretation equally diverges from that of Okruhlik (1985). For the basic concepts of Leibniz’s physical theory do not seem to relate to metaphysical entities that would be identified by the term “ghost,” and would radically distinguish themselves from the abstract and regulatory formulations by which Leibniz seeks in fact to explain phenomena.

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The demonstration of living forces that the Specimen dynamicum proposes is analogous to that of the Brevis demonstratio, with the minor exception that the experimental set-up entails pendular displacements. Nonetheless, Leibniz refines the methodological formulation of the a posteriori argument. First, he highlights the option between the a priori and a posteriori approaches for dynamics, while postponing in a way the presentation of the a priori demonstration based solely on the concepts of space, time and action. Then he specifies the methods for applying the principles of equivalence between cause and effect. It is necessary that the effect be equal to the total force expended, which obliges us to conceive of processes that would produce a violent effect, in contrast to an innocuus effect, which occurs when the action produced does not diminish the force accumulated in the cause. It is equally necessary that one be able to establish homogeneity between the quantified factors. Thus, in calculating force as the product of mass and speed, the Cartesians had assumed that motive force is expressed by the inverse ratio between the two parameters. This was to take “modalities” for realities. Change in the modalities of size and speed does not necessarily imply an equivalent change in the reality that this product of the modalities aims to translate. One must discover the analytic means for comparing homogenous terms, something which, in phenomena entailing elasticity, can cause significant difficulties. Finally, it is the principle of equivalence that secures the passage from a homogenous measure of the parameters of motive force to the adoption of the principle of conservation of living forces: Moreover, I take it to be certain that nature never substitutes things unequal in their forces for one another, but that the entire effect is always equal to the full cause. And in turn, we can safely substitute things equal in force for one another in our calculations with complete freedom, just as if we had made that substitution in actuality, with no fear of perpetual mechanical motion arising as a consequence. (GM VI 245; AG 129)

Owing to this methodological rule, and under the guise of the regulatory principle of equivalence, Leibniz permits himself to assume, by a sort of Gedankenexperiment, the substitution of motive forces between different bodies, which are susceptible to quantitatively identical effects.59 He consequently responds to the objection of those like the Cartesian Papin, but also Huygens, who contested the transition from calculating living force to the universality of a principle of conservation, on the grounds that it would be impossible to empirically verify the transferal of this quantity in mechanical exchanges. The same methodological device destroys the basic Cartesian laws of motion by demonstrating the absurdity of the hypothesis according to which the increasing effects in relation to the causes would generate perpetual mechanical motion. 59

Cf. GM VI 245–46; AG 129: “It does not matter whether this substitution can actually be brought about through the laws of motion, for we are able to safely substitute things equal in power for one another, even mentally.”

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Identified as a posteriori, the inequality of effects would permit us to assign, by means of a rational criterion, the non-equivalence of causes. The a priori process would consist here in substituting one cause for another on the basis of quantitative effects that can be substituted for one another.60 The method of a posteriori demonstration thus established, the first part of the Speci­men aims to ground the epistemological preambles of dynamical science in a conceptual system with two levels, the phenomenal and “metaphysical.” It is this system that the doctrine of force illustrates. The notions of active primitive force and passive primitive force reflect both the substantial status of the entelechies that underpin phenomenal bodies, and the role of architectonic principles involved in the formulation of the theoretical postulates of dynamics. These components certainly have not been completely dissociated from one another, and significant tensions arise in the theoretical model that Leibniz proposes. In this regard, three elements of its presentation merit our attention: 1) the analytical development of the notions of active and passive derivative forces; 2) the explanation’s connection to the metaphysical level; and 3) the identification of the basic principles of the physical theory that dynamics inspires, this third element constituting, above all, the object of the unpublished second part of the Specimen dynamicum.61 Empirical concepts and rationally derived principles together account for derivative forces. The primitive forces postulated surpass all empirical references. Indeed, Leibniz recognized that the notion of nisus provides for an analogical transition from the empirical plane to the rational one. To these nisus there correspond laws of action “which are understood not only by reason, but are also corroborated by the senses themselves through phenomena.” (GM VI 236; AG 120) Nisus can be empirically related to motions to which the analysis of phenomena can be reduced. It is by analytically breaking down motion into its generative factors that Leibniz hopes to formulate a theory of nisus that elucidates the real order of derivative forces. At the same time, he presents this approach as an attempt to nominally define the terms the combination of which would adequately represent the forces at work in phenomenal nature. This representation could serve to abstractly symbolize more geometrico the real relations derived from the more basic metaphysical structure. This is literally the model that Leibniz wants to construct, and he states as much:

60 GM VI 245; AG 129: “I have also used his method repeatedly to define a posteriori two states of unequal power and, at the same time, to find a sure mark for distinguishing greater power from less. For, when perpetual mechanical motion or an effect that is greater than its cause arises from substituting one thing for another, these states are hardly of equal power. Rather, that which was substituted for the other was more powerful since it caused something greater to appear.” 61 The second part of the Specimen dynamicum appeared for the first time in 1860 in Gerhardt’s edition of Leibniz’s Mathematische Schriften.

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I would not want to claim on these grounds that these mathematical entities are really found in nature, but I only wish to advance them for making careful calculations through mental abstraction. (GM VI 238; AG 121)

Let us begin with motion conceived of as change in place over time. If one considers the displacement of the moving body in a unit of time, which can be reduced to an instant, one obtains its speed, which varies more or less according to the space that it could traverse in the unit of time. Speed reduced to its instantaneous expression appears as an intensive tendency. In fact, the elements of kinetic analysis are concentrated in the notion of this intensive tendency that, together with the direction of the moving body, represents conatus. The product of the mass of a body and the speed it achieves in its first moment is defined as impetus. Impetus is what the Cartesians called quantity of motion, but with the qualification that it involves a quantity generated in what can be considered a unit of time (quantitas motionis), and as a result, must be distinguished from a quantity of motion that would assume effective duration (quantitas motus). This quantity of motion would follow from integrating impetus in time through summation, either of equal quantities, or of unequal quantities (when there is acceleration or deceleration). We must also note that the initial impetus also represents the instantaneous and infinitesimal integration of conatus into the elements of mass. It is the case of centrifugal force or gravity that instantaneously manifests itself in contrast with the effects resulting from an already accomplished summation that goes beyond the instant. The cumulative effect manifests itself then by the summation of the impetus themselves, which is, for example, the case with the actualized force of percussion. In this way, in the words of Leibniz, the apparently enigmatic statement of Galileo could be explained when he affirmed percussive force to be infinite in contrast to the simple tendency of gravitational effect.62 This problem is illustrated by the experiment with a tube rotating in a uniform motion around a point on a horizontal plane. Assume, for instance, that in the tube is a ball attached to a rope that gets cut. At this very instant, the centrifugal tendency of the ball is initially infinitely less than the impetus that it already possesses from the rotation of the tube. Hence the distinction between the elementary nisus corresponding to conatus, which is a sort of sollicitatio, an original dynamical tendency, and the nisus formatus, which is a sort of tendency that materializes through the continuation and repetition of the elementary nisus, even if this tendency toward summation is itself considered as instantaneous.

62

This point already appeared in the Phoranomus, I, § 18, Leibniz (2007), 744, and in the Dynamica, GM VI 451.

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Fig. 10 Illustration of the distinction between conatus and impetus in Specimen dynamicum (GM VI, fig. 24).

From this nominal reconstruction, Leibniz, as he had begun to do in the Phoranomus, infers a system comprised of two forms of integration, that of dead force (vis mortua) and that of living force (vis viva). Dead force represents an integration of the cona­ tus without its being actualized in actual motion, that is, without time determining in some manner the result of the integration. This is the case of the ball in the tube and that of the stone in a drawn sling that has not yet launched it into motion. Living force is expressed by an actualized motion that causes a summation at another level of integration. If the force of gravity, and if the force of elastic tension, for example, have already been exerted for some time, then one obtains a continuous succession of impressions caused by dead force, a succession that translates into living force: From this it is obvious that the nisus is twofold, that is, elementary or infinitely small, which I also call solicitation, and that which is formed from the continuation or repetition of elementary nisus, that is, impetus itself. […] From this it follows that force is twofold. One force is elementary, which I also call dead force, since motion does not yet exist in it, but only a solicitation to motion […] The other force is ordinary force, joined with actual motion, which I call living force. (GM VI 218; AG 121)

One possible symbolic expression of conatus (given that it is a vector quantity) would be: conatus = dv = gdt And that of impetus, reduced to an instantaneous quantity of motion (i. e., quantitas motionis), in contrast with Descartes’ m | v |, would be: t

impetus = ∫ 0 gdt = mv That of the temporal effect of impetus: t

t

temporal   summation  of  impetus = m ∫ 0 gtdt = m ∫ 0 vdt

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That of dead force: t

vis   mortua = m ∫ 0 gdt = mv That of living force: t

t

vis  viva = m ∫ 0 gtdt = m ∫ 0 vdt = ms = mv 2 The expression of impetus integrated in time connects therefore with that of living force, as the expression of conatus integrated in the form of instantaneous impetus had connected with that of dead force.63 Following Gueroult,64 one can thus justify the final two terms of the equation for living force. Let us then formulate the equation as: gt 2 t space  traversed = s = ∫ 0 vdt 2 Since v = gt, one can write:

v 2 = g2t2 = g ∙ gt2 = 2gs Hence the proportionality of speed squared to space, and the justification for the equation of living forces: t vis  viva = m ∫ 0 ds dt = ms = 1 mv 2 ∝ mv 2 2 dt As Leibniz will explain to De Volder, this whole system of definitions aims to reconcile static relations with the representation of dynamical effects that forces generate by their causal activity. The algorithmic model of infinitesimal calculus permits us here to represent the process for generating an effective nisus as doubly integrated, whereby impetus represents the intermediary level between conatus as elementary solicitation, and vis viva as the summation of the effects of impetus in time; the mediatory role played by the quantity of motion is thereby adequately translated. Hence the analogies drawn from geometry and infinitesimal calculus of which Leibniz makes use for representing the relations at play.65 Conatus are equivalent to the speed differentials dx, where speed is represented by x, while living forces are equivalent to the integrals of the product of the speeds and the speed differentials: xdx . Or, when geometrically transcribed as x2, they develop according to the square of the speeds.66 63 64 65 66

We are inspired by the formulas proposed by L. Loemker, with some corrections (cf. L 451). These formulas are quite close to those furnished by Gueroult (cf. Gueroult 1967, 38–39). Cf. Gueroult (1967), 39 and n.3. Cf. Leibniz’s letter to De Volder on 27 December 1698, GP II 156; Leibniz (2013), 28. As E. Grosholtz mentions in Grosholtz (1984), 207: “In a number of passages, Leibniz claims that the transition from the infinitesimal level of vis mortua to the finite level of vis viva is to be effected by integration, though his account of what precise form this integration will take is ambiguous.”

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To conclude his account, Leibniz aims to expose every other combination of conceptual elements capable of figuring into the analysis of derivative force. In this way, he breaks down the total living force of a system of interacting bodies into partial living forces, which are accounted for by means of an abstract interpretation of relative force and directive force. Here, he relies on conatus, which relate to the various material elements of a system, and impetus, which are assigned to the entire system according to the displacement of the common center of gravity, with the impetus of each element being subject to a vectorial measure. As when developing the corresponding theorems of the Dynamica de potentia, this obviously signals the system of linear, plane and solid equations that will be found highlighted in the later Essay de dynamique. It is thus possible to conceive of the interaction of living forces as the result of relations of conatus and impetus along the coordinates postulated in the analysis. Integrating these analytic relations could suffice for representing the dynamical interaction of bodies by adequate symbolic models. Everything here comes down to formulating theoretical sufficient reasons that may be represented by geometric relations involving progression to infinity. The Specimen seems to insist, on the one hand, on the infinite summation of impetus as dead or elementary force in the living force resulting from actual motion, and on the other hand, on the infinite summation of elementary nisus (conatus) in the generation of instantaneous impetus. Again, the strategy consists in creating an analytic system of symbolic relations that, one presumes, sufficiently express the underlying causal order represented by the typology of primitive forces. From a Leibnizian perspective, a theoretical foundation is required to guarantee these symbolic equivalences. Indeed, Leibniz brings together such and such expressions in order to establish operational equivalences for assessing the actions and interactions of bodies. But this procedure must be justified by sufficient reasons of another kind, rather than by simply reconciling the nominal definitions at play. The question thus arises: what connects the mathematical models of the components of force to the theoretical entities and principles from which the empirical laws governing phenomena derive? Leibniz’s explanatory system, we believe, comprises heterogenous elements that analogical reasoning is alone able to reconcile and render compatible by relying on architectonic principles and abstract models that account for the ultimate reasons behind phenomena. One must note the formal role played by such principles in the analogical frameworks introduced. In the Specimen, the The ambiguity arises mainly from statements where the first integration of conatus into mass as quantity of motion is not explicitly indicated. Cf. for example, the letter to the Marquis de L’Hospital on 4/14 December 1696, A III 7 214: “Dead forces, such as weight, elasticity and centrifugal tendency, do not have an assignable speed, but only an infinitely small speed, which I call solicitation, and they are but an embryo of the living force that the continuation of solicitations brings forth. They observe the laws of equilibrium, that is, mass and speed compensating for one another, in the way we can conceive of it with quantity of motion, whereas I find that living force, that is, the force that gets conserved, would not observe them.”

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metaphysical level comprises two components: on the one hand, active and passive primitive forces as fundamental theoretical entities, and on the other hand, the architectonic principles introduced to construct the physical theory, that is, to articulate the mathematical models together and connect them to the higher-level theoretical concepts. These various types of components are complementary and inform one another, but without truly becoming one and the same. From the moment when one conceives of the Leibnizian approach as the combination of various analytic methods that seek to harmonize the plurality of models and connect them with the metaphysical entities that serve as sufficient reasons for grounding the phenomenal order of mechanical effect, the blueprint of the dynamic theory appears both complex and thoroughly unified. Leibniz’s position on the structure of the theory that he proposes is stated with precision in the following passage: But after I examined all of this more deeply, I saw what a systematic explanation of things consists in, and noticed that my earlier hypothesis about the notion of body was imperfect. I also noticed, through other arguments as well as this one, that one can establish that something should be posited in bodies over and above size and impenetrability, something from which the consideration of forces arises, and that by adding the metaphysical laws of this something to the laws of extension, the laws of motion that I called systematic arise, namely, that all change comes about by stages, that all action has a reaction, that a new force is not produced unless an earlier one is diminished, and therefore that a body that carries another off with it is always slowed by the one it carries off, and that there is neither more or less power in an effect than there is in its cause. Since this law does not derive from the notion of bulk, it is necessary that it follow from something else inherent in bodies, indeed from force itself, which always maintains its same quantity, even if it is realized in different bodies. (GM VI 241; AG 124–25)

The concept of systematic explanation refers to the failure of non-systematic explanations in the topical sense that Leibniz gives to this expression. The analysis of these failures is revealing. In the first place, it focuses on the inherent flaws of Cartesian mechanics concerning its reduction of conserved motive force to the quantity mv, and its unjustified application of the laws of dead force to the analysis of the phenomena of living force. With regard to falling bodies, the increase in descending motion demonstrates, as Galileo began to show, the intrinsic accumulation of force in the moving body. However, at the first moment of the fall, and thus at the limit of the state of equilibrium, the moving body possesses a dynamical determination reduced to impe­ tus alone. It is therefore not the case that static concepts are incapable of symbolically representing the inception of the effects of living force; rather, this symbolization must be linked to a series of successive integrations, such as those that the Specimen tracks. A causal principle must underpin the integration process from which formal expressions are produced.

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Another major flaw of Cartesian mechanics lies in its failure to conceive of the factors of speed and direction as they combine in accounting for the real laws of impact. Indeed, by postulating the conservation of mv, Descartes conceived of speed as a scalar quantity, not as a vectorial quantity. As Leibniz shows in the Animadversiones in partem generalem Principiorum Cartesianorum (1692),67 this results in major discontinuities in the series of cases covered by the laws of motion bound to such a principle of conservation. Leibniz seems to level two criticisms in connection with this matter. On the one hand, in disassociating speed from direction, Descartes could only formulate an artificial system for upholding the absolute conservation of the quantity mv in mechanical exchanges. Instead, he attempted this by upholding an architectonic principle according to which, in collisions between moving bodies, dynamical determinations cause as little overall change as possible. This thesis prefigures in a certain sense Leibniz’s principle of equivalence between cause and effect in conjunction with the principle of continuity. However, according to Descartes’ principle of coherence, at certain times, speed and direction would be considered in unison, while at other times separately from one another. Above all, the author of the Principia philosophiæ treated modal properties (corresponding to extrinsic denominations) abstractly, as if they were real properties, and in vain sought to introduce them under an architectonic norm. Following Huygens’, Wallis’ and Wren’s formulations of the empirical laws of impact, the Cartesians went in the opposite direction, rejecting the architectonic norm and the reference to the real for a purely nominal system of determinations. This was, for example, the case for Huygens and Malebranche. Hence the shift in neo-Cartesian science away from a conception of force as a real cause of phenomenal changes. The relativity of kinematic relations tends to directly overlap with a theory of occasional causes, wherein the theoretical entities and agents between God and phenomena seem to be lacking. Thus, contrary to the spirit of Cartesian philosophy, relativist rules are intertwined with a purely meta-empirical principle of conservation that even escapes broader geometric intelligibility.68 This analytic strategy is to a given point analogous to the one that Leibniz had himself introduced in Theoria motus abstracti and in Hypothesis physi­ ca nova. He had thus proposed an inadequate conception of conatus and of their purely algebraic combination. The first physics had reduced bodies to their simple geometric properties, without proposing an inherent cause for the effects of inertia and the resistance to motion. This theory of corporeal nature lacked a true dynamical dimension. Nothing was presented that could justify the equivalent of a relation comprising active

67 68

II, ad §§ 46–53, GP IV 375–84; L 398–403. Cf. GM VI 240; AG 123: “Nor has anyone before us explained the notion of force. These matters have hitherto troubled the Cartesians and others who could not even grasp that the totality of motion or impetus (which they take to be quantity of force) might be different after a collision than it was before, because they believed that if that were to happen, the quantity of force would change as well.”

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and passive derivative forces. Leibniz therefore mentions that the deficiencies of such an abstract theory needed to be compensated for by a providential system constantly maintained by God. This artificial sort of solution thus connected the first physics with a sort of neo-Cartesian model. And this is precisely what the recourse to architectonic principles aims to eliminate from the updated system developed in Specimen dynami­ cum. The problem posed by recourse to such principles has to do with their epistemological interpretation.69 According to Leibniz, it is necessary to conceive of theoretical entities whose essential properties go beyond the extensive magnitude and impenetrability of bodies. However, these theoretical entities must be able to account for phenomenal properties that are more or less geometrically expressible. To these theoretical entities, which are sufficient reasons of an ontological and causal nature, there apply “metaphysical laws” that must be added to the “laws of extension.” From the combination of these two types of laws “systematic laws of motion” emerge. (GM VI 241; AG 124) These metaphysical laws comprise the principles of continuity, the equivalence between action and reaction, the equivalence between the full cause and the entire effect, and the conservation of the efficiency of force across time without increase or destruction. In the Specimen, force is, as we know, analyzed according to the twofold distinction between the primitive and the derivative, and the active and the passive. Active and passive derivative forces express the activity of primitive forces on the phenomenal level. But the fundamental role of primitive forces seems to be justifying the insertion of architectonic principles into the theory. And these principles in turn serve to construct models for derivative forces that are as adequate as possible. Locating living force beyond the extensive characteristics of bodies involves justifying the reason for applying truths other than those based on logical and geometrical axioms relating to “large and small, whole and part, shape and position (situs),” (GM VI 241; AG 125) and thus to physical phenomena. These truths or principles concern the relation of causality, and the relation of equivalence between action and passion, and serve to rationally explain the “order of things”. (GM VI 241; AG 125) It is in this context that Leibniz’s rejection of the two types of metaphysical theses, those pertaining to occasionalism and those pertaining to “animism”, must be situated.

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M. Wilson (1981) shows how Leibniz in dynamics excludes the possibility of reducing the laws of nature to the status of necessary, geometric truths and rejects theoretical concepts that represent corporeal realities exclusively in terms of extensive properties. But her analysis does not establish how architectonic principles structure and demarcate physical theory in contrast with a system of strictly metaphysical concepts. Wilson points out, without emphasizing the importance of this point, that Leibnizian science recurs to a certain extent to geometric models and that, according to Leibniz, explanations via efficient causes can in principle be achieved. She does not show how these aspects of the analysis work together with the teleological arguments and the postulation of non-geometric theoretical entities. To the extent that a truncated picture of dynamics is given, Leibniz seems to have defended epistemologically discordant theses.

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In opposition to the first type, Leibniz supports the existence of an autonomous cause of motive force that underpins phenomena, and does so without having to rely on a deus ex machina that transcends that natural order. The supposition of this sort of immanent causation responds to the demand of founding a system of sufficient reasons in conformity with architectonic principles in the order of things. Under the label of animism are listed all the doctrines that attribute a direct causal role in phenomena to formal principles, like Van Helmont’s archei and Henry More’s hylarchic principles. This involves causes whose intervention could not be explained by a geometric model. According to Leibniz, the efficient causes at work in the physical universe are all ultimately representable by virtue of their geometrico-mechanical effects. More precisely, analyzing these effects must adequately reveal how they are determined by causes congruent to the geometrico-mechanical characteristics that express the intelligibility of phenomena. Leibniz excludes all recourse to faculties, occult qualities and sympathies of the Aristotelian-scholastic philosophy of nature from his physics. A legitimate scientific explanation of mechanics is erected on two levels. On the level of particular phenomena, this involves producing, by building on geometrically representable properties, models for accounting for mechanical processes. But constructing these models supposes arguments of a higher order. From a general and unspecific point of view, reasons that are metaphysical in nature, particularly the requirement for attributing sufficient reasons, dictate that these models be fashioned according to directives yielded by architectonic principles. One can indeed imagine such metaphysical reasons taking the form of a divine plan, but the most important thing is conceiving of and developing the reasons that determine the rationality of the models. Thus, when this involves establishing such models, and then evaluating their functionality in an explanatory system of phenomena, one is obliged to account for general reasons. Architectonic principles serve to express the methodological conditions of this operation. And the concepts that refer to metaphysical theoretical entities outline a system of causes whose operative processes would be expressed by the principles.70 Since theoretical entities are expressed architectonically in the resulting phenomena, and these are geometrically representable, it seems legitimate to acknowledge that in the system of phenomena, there is an inherent teleological relationhip that one can establish beyond the analysis carried out in terms of mechanical causality. Hence the classically Leibnizian assertion:

70

Cf. According to McGuire (1976), Leibniz’s philosophy of nature would strive to integrate concepts appertaining to the ideal, the actual and the phenomenal respectively. A principle, such as that of continuity, would secure the analogical – and often problematic – articulation of the different elements of the system of representation.

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In general, we must hold that everything in the world can be explained in two ways: through the kingdom of power, that is, through efficient causes, and through the kingdom of wisdom, that is, through final causes. (GM VI 243; AG 126)

We may abstain here from considering the metaphysical meaning ascribed to the kingdom of ends in the monadological system. From a more epistemological perspective, and on an empirico-theoretical level, the efficient causes behind phenomena are identified with derivative forces; and on a purely theoretic level, they are identified with primitive forces, which are connected with the intervention of autonomous entelechies. As such, derivative forces could not possibly be likened to final causes, which would be to covertly insert physical intentionality into the mechanical chain of phenomena. Expressing the substantial reality of bodies, the related phenomena are perfectly capable of being analyzed in geometrico-mechanical terms. However, the interaction of derivative forces responds to models that should be constructed by proposing analogical sufficient reasons; hence the possibility of relying on the strategy of teleological analysis at this level. But what assurance do we have that this analytic approach effectively responds to the structure and arrangement of material reality beyond the phenomenal plane? Can recurring to theoretical entities provide us with the type of assurance that we are looking for? Defined with reference to entelechies that combine active and passive dynamical states, primitive forces embody laws of serial development that conform with the overall architectonic design. At this theoretical level, architectonic principles intervene as well; however, rather than serving to construct analogical models, they appear as formulas that express how the distribution of dynamical states in and between the various entelechies is regulated. And, as a consequence, theoretical entities arranged in this way underpin the explanation for the causal regulation that phenomena seem to imply. This is why Leibniz, on the one hand, relegates final causes to the plane of general and metaphysical reasons, but on the other, assigns teleological arguments a role in discovering empirical laws.71 When one cannot recur to causal deduction as a rule, but is limited to a set of often very complex hypotheses, then the analogical search for ends can guide the formulation of adequate models. For example, Leibniz refers here to his demonstration of the sine law in dioptrics, which he demonstrated with the help of architectonic principles, as the Unicum opticæ, dioptricæ et catoptricæ principium (1682) attests. Moreover, he relies on the endorsement that William Molyneux had given regarding this analytic approach in the Dioptrica nova (1692).72

71 72

This is attested to, for example, in the Discours de métaphysique, § 22, A VI 4, 1564–66; L 317–18. Cf. GM VI 243; AG 126: “Indeed, one can even bring final causes to bear from time to time with great profit in particular cases in physics (as I showed with the clearly remarkable example of an optical principle, which the most celebrated Molyneux greatly applauded in his Dioptrics), not only the better to admire the most beautiful works of the Supreme Author, but also in order that

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In short, Leibniz seems to suggest that the theoretical level of explanation entails relying both on the concepts of entities and on principles. Entelechies and their dynamical properties thus serve to establish a causal order that is reflected in an architectonic arrangement of effects. And architectonic principles serve to give shape to a translational or expressive system for bringing theoretical entities within the scope of phenomena. The recourse to entities and principles justifies conceiving of the general laws of nature as teleological. In the process of formulating hypotheses, and progressing analytically to exposing these laws, one can assume the teleological arrangement of phenomena. This methodological presumption must nonetheless conform with an overall appraisal of the order of, and connection between, phenomena. Moreover, causal sequences must be determined by a model that conforms with geometric intelligibility, and experience must corroborate such constructions. The epistemological inventory of the Specimen dynamicum would be incomplete if the analysis of the second part, unedited in Leibniz’s lifetime, were not integrated. Two points dominate these analyses with respect to the laws governing physical nature, and it is important to grasp their connection. Leibniz emphasizes, on the one hand, the necessity of developing concepts for theoretical entities that account for the production of causal sequences. From this point of view, the concepts on which a Cartesian sort of science rests appear relative. Concepts of space, time and motion seem here to reflect a strictly nominal order of phenomena; if they do represent causal modalities of physical reality, they could not possibly designate the efficient causes that articulate the system of phenomena along with the changes that manifest in it. If physics must take for its object a reality endowed with action and passion, it cannot hope to understand it by reducing it to the modes of intelligible extension alone. However, on the other hand, this argument is related to the positive thesis according to which it would be possible to represent the geometric characteristics of corporeal reality by deriving them from the concept of a subject of inherence, the required substrate for founding powers of action and resistance. By refusing this theoretically foundational requirement, the Cartesians arrived at occasionalism and relegated the faculty for generating the causal order of phenomena to a transcendent subject. Leibniz’s solution, by contrast, aims to ground this faculty in the heart of physical nature. If geometric formulations can abstractly symbolize the phenomenal world, then one must descend into the sui generis causal background to establish the reality of these models. At first glance, the strategy adopted for understanding phenomena seems to involve restoring Aristotelian substances? But one can discard this interpretation even if it appears credible.73 Leibniz’s strategy is different from the outset, since it builds a great deal

73

we might sometimes discover things by that method that are either less evident or follow only hypothetically on the method of efficient causes.” In this respect, we would like to distance ourselves from a position like the one Daniel Garber takes. Cf. for example, Garber (1985), 99: “When understood properly, Leibniz’s mechanical expla-

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upon the geometric representation of phenomena. This representation requires a sufficient reason that is both suited to the geometric order that renders phenomena representable, and to assigning causes for the system of phenomena. This is why Leibniz proposes a theoretical conception of primitive force capable of accounting for the extensive expression of force, through which phenomenal properties become analyzable. In the analysis of effects, this is evinced for example by the fact that the respective displacements of bodies in motion provide an occasion for implementing the rules of conservation of relative speed before and after impact. Evidently, the notion of relative speed allows for assigning purely arbitrary and infinitely variable modes of motion and rest. Hence the equivalence of every conceivable hypothesis falling under this general rule of conservation. This relativity of kinetic hypotheses imposes normative constraints on the theoretical explanation on which one must rely in the end. In other words, recourse to theoretical entities must yield sufficient reasons that are adequate and therefore accord with the geometric intelligibility whereby phenomena are translated: It also follows from the relative nature of motion that the mutual action or impact of bodies on one another is the same, provided that they approach one another with the same speed. That is, if we keep the appearances in the given phenomena constant, then whatever the true hypothesis might finally be, to whichever body we might in the end truly ascribe motion or rest, the same outcome would be found in the phenomena in question, that is, the same outcome would be found in the resulting phenomena, even as regards the action of bodies on one another. (GM VI 248; AG 131)

In conformity with the geometric intelligibility of the level of effects, this means that, at the level of causes, where force is something real or absolute, one is working with theoretical concepts that can only be adequately conceived of by including the requirements of geometry. And this means that Aristotelean concepts such as entelechies, instead of being fully restored, are overstepped when it comes to representing the entities of theoretical physics. On the basis of this requirement for an adequate sufficient reason, recourse to active and passive primitive forces is molded by the principle of continuity, which stipulates the method whereby it is achieved: It also follows from our notions about bodies and forces that what happens in a substance can be understood to happen of that substance’s own accord, and in an orderly way. Connected to this is the fact that no change happens through a leap. (GM VI 248; AG 131)

nations do not replace Scholastic explanations, but are grounded in them; mechanical explanations are a schematic and partial way of describing what goes on in the form, what all forms have in common. It is what we have to fall back on in our ignorance of what specifically God programmed in.” Garber’s analysis unduly omits the role of architectonic principles and mathematical models in Leibniz’s conception of physical theory, which cannot be reduced to the postulates of a scholastic metaphysics of nature.

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Connecting the principle of continuity with the notions of primitive force constitutes the foundation of the theoretical structure as illustrated by the Specimen. On this foundation one can construct a system of consequences that constitute physical theory proper. These consequences are developed in a schematic but rigorous fashion. One must first reject the notion of atoms, for absolute solidity would render the effects of impact incoherent, given that a continuous transition between the successive states in the chain of antecedents and consequents would not be achieved. By contrast, the law of continuity requires that we conceive of all the interacting material elements in impacts as being fundamentally endowed with the property of elasticity. Supposing that we admit, according to the subtle ether model, that elasticity owes to bodies being permeated by such a fluid, one would also have to admit that we will never arrive, by analytic regression, at some ultimate subtle form of matter.74 There is another consequence of this: the propulsion of a body, whether directly or by reflection, following an impact with another body, implies that the smallest elements beneath the body’s surface are deformed, only to then be reflected in a process that potentially results in the subsequent reestablishment of the body’s state of equilibrium, depending on the degree of primitive forces that characterize it.

Fig. 11 Illustration of the collision of bodies endowed with the fundamental property of elasticity, in Specimen dynamicum (GM VI, fig. 26-27).

The law of continuity also explains the fact that rest must be understood as a progressively diminishing motion, just as equality can only be the limit of a disappearing inequality: “[so…] that there is no need for special laws such that particular rules are not required for equal bodies and bodies at rest. Rather, these laws arise per se from the laws of [un]equal bodies and motions.” (GM VI 249; AG 133) If there were different rules for motion and rest, then harmony would be lost; hence the methodological requirement that the rules necessarily accord with one another. It is, for example, on this basis that Leibniz criticizes the Cartesian laws of impact in his Animadversiones in partem generalem Principiorum Cartesianorum, and as evidence, graphs the solutions

74

Cf. Breger (1984) on the development of this notion as a theoretical concept.

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for continuity that these laws imply regarding the transition between one case and another infinitesimally distinct from it.75 If we examine the other consequences of the physical theory that Leibniz points out, it is easy to notice that the same principle of continuity exercises at the very least a regulatory function here. Thus, in collisions, no matter the adopted kinematic hypothesis, one can attribute half of the effect to each of the two moving bodies, which supposes a strict correlation between their passivity and their inherent power to act. Compression and the degree of elasticity are equally distributed among the two moving bodies according to their respective speeds. In sum, from the intrinsic passivity translated by the impact one can infer an equivalent action in the same bodies themselves.76 Applying to the motion of each body in relation to the common motion of their centers of gravity, which together represent the phenomenal order, the continuity of active and passive forces in the same bodies and between bodies yields the law of equality and opposition between action and reaction: “the action of bodies is never without reaction, and both are equal to one another, and directed in opposite directions.” (GM VI 252; AG 135) In this case as well, one must understand equality as the limit of a diminishing inequality and interpret the vectorial reflection that reaction represents with respect to action as the result of a continuous transition: the elementary nisus are in equilibrium with and oppose one another in a sequence of interpenetrations that the corresponding impetus measure, until the generation of the diminishing impetus is superseded by the generation of the impetus growing in the opposite vectorial direction. Likewise, the explanation of solidity as a phenomenal disposition rests on the continuous and gradual chain of motive determinations. From force and the instantaneous n­ isus that represents its differential ingredient at the source of motion, there derives the tendency of all physical motion for rectilinear expansion; thereafter, the motions can only be combined by the effect of these original nisus expanding rectilinearly. In circular motions, projection along the tangent corresponds to the expression of this type of determination. The pressure of the surroundings on particular bodies whose motion is determined at the tangent generates a centripetal tendency, and this combines with the former to produce seemingly simple and absolute circular motion. The pressure of the surroundings

75

76

Cf. GM VI 250: AG 133–34: “When I examine the Cartesian laws of motion with respect to this touchstone, which I transposed from geometry into physics, it happens, much to my surprise, that a certain gap or leap, entirely abhorrent to the nature of things, displays itself. For representing quantities by lines, and taking the motion of B before the collision as the given case, represented in the abscissa, and this motion after the collision as the outcome sought, represented in the ordinate, and extending a line from one end of the ordinate to the other in accordance with Descartes’s laws, this line was not a single continuous line, but was something wondrously gaping and leaping in an absurd and incomprehensible way.” Cf. GM VI 251; AG 135: “[…] that the repercussion and bursting apart arises from the elasticity it contains, that is, from the motion of the aethereal matter permeating it, and thus it arises from an internal force or a force existing within itself.“

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is equal to a magnetic device placed at the center, and appears as a true analogue for the presumed force of attraction. In sum, solidity results from the convergence of rectilinear motions that are interlinked such that the interaction of the nisus prevents them from separating from one another. Consequently, solidity appears as the result of the continuously integrated nisus; and this relative property must also be evaluated in terms of an infinite gradation of elementary forces in the background of extension and the inertial characteristics of bodies. Geometrically symbolizing the “elementary” determinations of such complex properties is accomplished via the equivalence of hypotheses. This allows above all for indefinitely combining and deconstructing the determinations of motion. However, underpinning this analytic use of equivalent hypotheses of the geometric and kinetic sort is the idea of integrating nisus into a continuous process, which would lead to the possibility of expressing the causal rationality at work in phenomenal reality. A methodological condition governs this symbolization via hypothetical equivalents: it must account for the greatest possible number of facts through which the phenomena can be analyzed. Across the geometric models constructed in this way, a serial chain of hypotheses that responds to the requirements of continuity and expresses a maximum of architectonic requirements, takes shape. In this respect, we can reconcile two passages from the second part of the Specimen dynamicum. On the one hand, Leibniz affirms: Since rotation also arises only from a combination of rectilinear motions, it follows that if the equivalence of hypotheses is preserved in rectilinear motions, however they might be placed in things, then it will also be preserved in curvilinear motions. (GM VI 253; AG 136–37)

On the other hand, Leibniz imposes an additional condition for methodologically justifying the appropriate system of geometrico-hypothetical representations: Whenever we are dealing with the equivalence of hypotheses, we must take into account everything relevant to the phenomena. From these things we also understand that we can safely apply the composition of motions, or the resolution of one motion into two or however many more motions […]. For the matter warrants a proof in any case, and it cannot be assumed as if it were self-evident, as many have done, (GM VI 254; AG 137–38)

By these somewhat arcane statements with which the Specimen dynamicum concludes, Leibniz indicates that geometrical hypotheses must translate the analytic data representing the phenomenal order in the most complete way. On the other hand, a compro­ batio is required that can justify the explanatory character of hypotheses of this type. But this comprobatio was paradigmatically sketched in Leibniz’s dynamics and in the physical theory that he articulated there. It sufficed there to submit the evaluation of hypotheses to the architectonic principle of continuity and to that of the equivalence between the full cause and the entire effect. These principles served as a cornerstone of the explanatory strategy and as a special means for articulating models for analyzing phenomena and formulating theories with the goal of causally interpreting the laws governing the system of phenomena.

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4. The Essay de dynamique: Combining Principles The manuscript in question, entitled Essay de dynamique sur les lois du mouvement,77 sets out to produce a systematic version of the dynamics. André Robinet, who has worked on dating this text, concluded that it could be placed between 1699 and 1701. Leibniz mentions in effect that Malebranche abandoned the principle of conservation of quantity of motion, which had only been confirmed by a letter that Malebranche had addressed to him on 13 December 1698.78 It is in 1700 that Malebranche will make the radical revisions to his theoretical position public in a reprint of his treatise Des lois générales de la communication des mouvements. Moreover, it is easy to recognize references to the contemporary discussion on the a priori or a posteriori status of demonstrations relating to the conservation of motive action, as developed in the correspondence with Johann Bernoulli and De Volder. Leibniz seems to construct his formulation of this topic on the basis of a previous critical reflection. Nonetheless, the Essay de dynamique merits attention more than anything because it proposes an original methodological conception of the relation between the various principles of conservation at the foundation of dynamics as a science, and elucidates the role of architectonic principles in the structure of the theory. In the Essay de dynamique, Leibniz unites so to speak two types of arguments: one takes an analytic approach to the dynamical theorems that the Dynamica de potentia had introduced; the other is connected with the epistemological considerations that were the focus of the Specimen dynamicum, and is more concerned with the architectonic reasons governing physical theory. Thus, in the epistemological vein, Leibniz emphasizes that a principle of conservation of some absolute thing seems to be a prerequisite of all theories that are intended to be physical rather than merely mathematical. “Mathematicians” can content themselves with equations that represent the conservation of relative quantities or factors, and, in this regard, the theoretical entities to which they refer can only be the object of nominal definitions that express the apparent compatibility of the abstract and symbolic elements, but are unable to ensure real compatibility with the concrete essences in the behind phenomena. Malebranche bears the brunt of this critique here because he rejected the Cartesian principle of conservation of quantity of motion without replacing it with another principle of physical conservation involving a true causal order. In fact, from 1668 and thereafter, when the discovery of the empirical laws of impact by Huygens, Wallis, Wren and Mariotte challenged the status of the Cartesian principle as an effective physical law – in view of the empirical inconsistencies that 77 78

The full title is Essay de dynamique sur les loix du mouvement, où il est monstré, qu’il ne se conserve pas la même quantité de mouvement, mais la même force absolue ou bien la même quantité d’action motrice, GM VI 215–31. Cf. Robinet (1986), 266–67.

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it produced – Cartesians thinkers adopted a twofold strategy: on the one hand, they rejected the notion that phenomena could be analyzed via the conservation of a causal agent responsible for the constancy and order of mechanical changes; on the other, they argued that acknowledging the actual ordering of phenomena would suffice in accounting for all mechanical changes in geometric terms. If Malebranche, obliged by Leibniz’s demonstrations, henceforth rejects quantity of motion as a permanent causal agent in the background of phenomena, then he gladly falls back on the self-sufficiency of the product mv for algebraically representing the criterion of intelligibility in the analysis of mechanical phenomena. Apparently, it suffices to manipulate the signs applied to the symbolized quantities in the equation to arrive at a coherent representation of these phenomena on the kinematic and geometric level. However, must we not then reject the idea that the equations’ symbolic representations can be physically grounded as constant laws of nature? And from a Leibnizian point of view, would this not be equivalent to giving up on the search for an order of sufficient reasons? We now begin to disabuse ourselves [of the Cartesian principle], especially since this view has been abandoned by some of the most longstanding, capable and noteworthy of its supporters, and particularly by the author himself of the Recherche de la vérité. But if there has been a drawback to this, it is that we have gone too far in the opposite direction and no longer recognize the conservation of something absolute, which would replace the quantity of motion. However, this is what the mind expects, and this is why I notice that philoso­ phers, who do not engage in the profound discussions of mathematicians, are reluctant to abandon an axiom such as the quantity of conserved motion without being offered another that they might adopt. (GM VI 216)

Faced with the relativity of mechanical laws, which the neo-Cartesians take advantage of, Leibniz underscores two functions of principles of real and physical conservation: (1) satisfying the demand for a sufficient causal reason at the physical level without recurring to a providential metaphysical order, which evidently does mean that the foundations of physical causality can be conceived of without any metaphysical hypothesis; and (2) assuring the systematic integration of the laws of relative conservation, which have the analytic status of nominal definitions. From a technical point of view, the Essay de dynamique is essentially comprised of two sets of demonstrations. The first involve revealing the subordinate relation of the principles of relative conservation to some absolute principle of conservation that guarantees its intelligibility in a system of physical explanations. The second entail establishing a principle of conservation of quantity of action that expresses in geometric terms – and therefore, according to the Cartesians, acceptable terms – a system of sufficient causal reasons that govern the analysis for the whole order of mechanical exchanges. The methodological ideal that Leibniz pursues consists in “satisfying both the rigor of the mathematicians and the expectations of the philosophers, that is, both the experiences and the reasons derived from various principles.” (GM VI 228) The “rigor of the mathematicians” refers, it

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seems, to the requirement that phenomena be geometrically transcribed into series of equations that can be rendered analytically equivalent to one another via substitution of terms salva veritate. The “expectations of the philosophers” no doubt boil down to the search for an ordered system of sufficient causal reasons in which the well-founded and thus real phenomena that experience presents us can be grounded.79 And the recourse to principles can only appear as the cornerstone of a physical theory that responds simultaneously to these two methodological requirements. The Essay de dynamique examines these principles by introducing some noteworthy elements to interpret them. The first of these elements stems from assimilating the fundamental theorems of absolute conservation to the architectonic principles as such, the function of which appears to be regulatory and heuristic above all else. Among the laws that potentially govern the creative action of God, Leibniz considers essential: [T]hese two laws of nature that I was the first to reveal, the first being the law of conservation of absolute force or motive action in the universe, with some other absolute conservations that depend on it that I shall one day explain, and the second being the law of continuity, according to which, among other consequences, every change must happen by unassignable transitions and never by leaps. (GM VI 229)

Indeed, the principle of continuity is introduced here to justify the application of theorems of absolute conservation to the world of phenomena – to the extent that elasticity is a fundamental property of all bodies and underpins the phenomenal properties of hardness and relative flexibility. The same type of argument had dominated the second part of the Specimen dynamicum, which illustrated the regulatory function that architectonic reasons and relations play in elaborating theoretical concepts in physics. In the Essay de dynamique, Leibniz resorts for instance to the principle of continuity to reject corpuscular hypotheses, as the laws of impact that one assumed to be in accordance with hypotheses of this sort would no longer be derivable from any principle of absolute conservation. Representing the collision of non-elastic, hard bodies requires that one admit the existence of discontinuities in the passage from rest to motion and motion to rest. In addition, the conservation of relative speed is no longer guaranteed in duration, since the capacity to restore dynamical states is no longer guaranteed to be present in colliding bodies. In short, reference to the principle of continuity allows one to underscore why it is necessary to rely on graduated series of states that underpin the interaction of bodies if one wants to preserve a causal order without gaps. The architectonic requirement expressed by this principle thus guides the reinterpretation of those empirical situations where no real dynamical factor seems to be conserved. These situations are represented in a model: a sack of small, hard, elastic balls touching one another. According to this model, living force continuously regenerates itself in

79 Cf. De modo distinguendi phænomena realia ab imaginariis, GP VII 319–22; L 363–65.

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the interaction of infinitely smaller and smaller parts, while the dynamical capacity as a whole diminishes within the container, which seems to absorb force without restoring it, as the phenomena indicate. The diminishment of force in such cases may or may not overturn the laws of relative conservation. In impacts with soft bodies, it would thus seem that relative speed before and after the impact could not possibly be conserved, while the quantity of progress of the common center of interacting bodies remains unaffected. But then we lack an overall formula for simultaneously accounting for the variation in the different factors. In effect, only abstractly representing phenomenal interactions can justify the equations of relative conservation when one does not consider them as being subordinate to theorems of absolute conservation, and conceives of them outside the scope of a physics that seeks to provide causally determinative reasons. Hence, a partial justification in terms of the empirical references that the equations are able to symbolize is given. For Leibniz, there is no doubt that the theory demands a higher degree of intelligibility and organic justification. To satisfy this requirement, are we not obliged to recur to principles that provide an interpretive framework and assure the coordination of relative laws and principles beyond the symbolization of phenomena in a mere empirical framework?80 It is, without a doubt, in view of this goal of coordinating explanatory principles that the demonstrative argument of the Essay de dynamique must be interpreted. It then appears that the model for coordinating these principles is combinative. The first part of the demonstration aims to establish the principle of conservation of quantity of action as an equivalent of the principle of conservation of living force. In Leibnizian terms, such an equivalence means that it is possible to reduce the two principles to substitutable combinations of definitional requisites. Proving the analytical equivalence of these principles can go hand in hand with the revelation of an analogical role in the explanation. In fact, Leibniz will even tend to take this functional analogy, which characterizes the system of determinative reasons for phenomena, as confirmation of a stricto sensu analytic equivalence, at least when the latter is not directly explicit. The arguments supporting the conservation of living force are in no way distinct from those to be found in the previous discourses, particularly the Brevis demonstratio and Specimen dynamicum, with the exception that the Cartesian principle of conservation of quantity of motion is advanced here as a limit case of the new dynamical principle, since it only applies where there is equilibrium and the exercise of living force

80

In light of this we can interpret the apparent refutation of the theorem of conservation of living force in collisions between bodies that are imperfectly elastic owing to their composition, cf. GM VI 230–31: “However, this loss of total force or failure of the third equation [of the conservation of total absolute force or motive action] in no way detracts from the inviolable truth of the law that the same amount of force is conserved in the world. For what is absorbed by the small parts is not absolutely lost to the universe, even though it is lost in regards to the total force of the colliding bodies.”

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is suspended by mutual compensation in the initial impetus. Everything is therefore conceived of as if the bodies impacting one another interact instantaneously, and by virtue of the ratio of the respective factors mv of each body, as if this process was bound to prolong itself by preventing force from expressing itself in time. In cases of enduring equilibrium, summation of the instantaneous conatus in the mass is prevented beyond the immediate impetus, and consequently no further integration of the impetus themselves occurs. This case is reflected in the laws of statics: For we find in statics that two bodies are in equilibrium when, because of their situation, their speeds are reciprocal to their masses or weights, or when they have the same quantity of motion. (GM VI 218)

One can symbolically draw a connection between this situation and that of two bodies colliding with equal quantities of motion. Even if the living force of the bodies present is unequal in the instant of impact, the interaction is measured according to “the laws of dead force or statics.” (GM VI 219) Insofar as one assumes that bodies are elastic, they successively compress one another according to the differential of living force, that is, according to the instantaneous elements of impetus. Such elements, into which impetus are broken down, reciprocally compensate for one another in a continuous series until all of the impetus of the two touching bodies is used up. Once this happens, impetus is restored in the opposite direction following the same continuous series. Leibniz does not hesitate to liken this case to that of equilibrium without a previous or resulting motion, since at the instant of impact the parameters of the equation are equal on both sides: But the quantity of dead force is calculated, according to the laws of equilibrium, by the quantity of motion, which in truth is infinitely small, but whose continuous repetition finally exhausts the total quantity of motion of both bodies; assuming that this is equal in both bodies, the quantity of motion of each is exhausted at the same time, and as a result, both bodies are simultaneously reduced to rest by the pressure of their springs, which, returning to their original states, put things back into motion. (GM VI 219)

If, when it comes to combining equations, the case of equilibrium can be likened to that of the collision of two bodies with the same impetus, then Leibniz tends to emphasize the fact that this only occurs in particular cases, which themselves can be subsumed under the general principle of conservation of living force. In order to achieve this, he first proposes, as if it were a hypothesis, to measure force as a violent effect, one that entirely uses up its cause; in the end, the concepts on which this operation relies are subject to nominal definitions. Thus, if one measures living force by the weight of moving bodies multiplied by the height they can achieve by their own motion, then one is able to compare cases where the height is calculated according to the square of the speed (e. g., in cases of falling bodies or where a cumulative effect of force is generated) with those where it is calculated as simple speed (e. g., where there is equilibrium

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or mutual compensation between the impetus).81 The result is yet another combination of equations. Nonetheless, at this stage of the demonstration, the fundamental theorem of dynamics continues to rest on an essentially empirical justification. Indeed, Leibniz professes that the proof is both rational and empirical: Reason and experience discover that absolute living force, which is calculated by the violent effect it can produce, is what is conserved, not the quantity of motion. (GM VI 219)

But the element of rational proof negatively consists in excluding from the Cartesian principle a number of consequences that would contradict the implications of the principle of sufficient reason. It would thus be justified to eliminate hypotheses that assume perpetual mechanical motion by supposing that the forces at work in nature are amplified, or conversely, hypotheses that would require one to compensate, by some extraordinary means, for the exhaustion of the factor representing the constant of physical efficacy in the background of phenomena. Here, this involves regulatory rational criteria for hypotheses that have already been formulated in other ways. But these principles obviously could not possibly dictate the formulation itself of a hypothesis capable of serving as a fundamental principle of physics. The other part of the demonstration must therefore take the form of an a posteriori justification that references Galileo’s law of falling bodies, as this empirical law allows for measuring violent effect and thus the entire equivalent cause. In the rest of the Essay de dynamique, Leibniz works to build up the rational proofs that might ground some manner of a priori argument. This attempt consists of two phases. On the one hand, Leibniz develops the proof for a principle of conservation equivalent to that of living force, but capable of being geometrically represented in a more adequate way: this involves the principle of conservation of quantity of action. Furthermore, he demonstrates the integration of principles of relative conservation into the principle of absolute conservation, which itself henceforth possesses two equivalent analytic sides. We have already described the demonstration of the principle of conservation of quantity of motive action in the Dynamica de potentia. For the most part, the Essay de dynamique just repeats it. Leibniz relies on stipulative definitions, and first, that of for81

Leibniz criticizes Descartes for not having grasped the fact that the two cases are analogous. Cf. the handwritten remark in the margins of the text, GM VI 218: “It is therefore surprising that Mr. Descartes avoided with such skill the pitfall of interpreting speed as force in his small treatise on statics or dead force, where doing so would not have caused a problem since, everything being reduced to weights and heights, it would have made no difference, while he abandoned heights for speeds where it was necessary to do the opposite, that is, where percussions or living forces that must be measured by weights and heights were involved.” Leibniz is referencing Descartes’ Explication des engins par l’aide desquels on peut avec une petite force lever un fardeau fort pesant, which was joined to his letter to Constantin Huygens, on 5 October 1637 (AT I 435–48).

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mal effect, which expresses the product of mass and the length of the space traversed. Formal effects are distinguished from violent effects insofar as they express force by conserving it (in conformity with the principle of inertia) rather than exhausting it. Motive action for its part is defined by the product of formal effect and the speed at which it is accomplished. In a unit of time, it is evident that the effect is equal to the product mv, and motive action is equal to the product mv2. One could indeed maintain that these definitions can be empirically validated where experimental verification guarantees the reality of the object being represented, that is, at least its status as a well-founded phenomenon. Leibniz himself argues that one can legitimately speak of a true a priori justification: The justification for this definition of motive action is a priori enough [assez a priori] since it is obvious that in a purely formal action considered by itself, such as that of the moving body being considered by itself here, there are two points to examine: the formal effect or what is changed, and the rapidness of the change, for it is quite obvious that what produces the same formal effect in less time accomplishes more. (GM VI 221)

But what does he understand by an a priori justification, or rather one that is a priori enough (assez a priori)? Later on in the text, Leibniz speaks of a general verification of the conservation of motive action that would rest on the axiom of conservation of force without specifying the measure that expresses it; hence, such an axiom would no doubt have a status close enough to the principle of sufficient reason applied in a general way for identifying a constant cause underlying mechanical changes. The other aspect of the justification would owe to the fact that “in the end the exercise of force or force carried out in time is action, the abstract nature of force consisting in nothing more than that.” (GM VI 222) This element of proof will have to be reconsidered in the context of the discussion with De Volder and Johann Bernoulli. But it seems from the outset that the proof would result from a reflective apperception whose object is the concept of power. This therefore entails a priori reflective experience rather than analytically deployed conceptual requisites. It is true that, from reflection to intellection of the truths of reason, the Leibnizian theory of knowledge tends to conceive of a continuous gradation in cognitive functioning and in the propositional content where it is expressed.82 If the definition of motive action can be said to be a priori, such is the case insofar as the abstract model elaborated by Leibniz is part of the extension of a primordial truth concerning the reflective meaning of the concept of power or force. In any event, it is difficult to see in the nascent a priori argument presented to us any-

82

NE, 1.1.11, A VI 6 81: “The intellectual ideas that are the source of necessary truths do not come from the senses; and you acknowledge that there are ideas that owe to the reflection of the mind when it reflects on itself.” NE, 2.7.1, A VI 6 129: “It seems that the senses could not convince us of the existence of sensible things without the help of reason. I would thus say that the thought of existence comes from reflection, that those of power and unity come from the same source […].”

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thing more than a hint of the reflective intelligibility requirement that must be adopted when conceiving of the fundamental principles of dynamics. The demonstrative strategy of the Essay de dynamique consists in effect in acknowledging the concepts that ground fundamental principles by means of abstract definitions. Once this a priori hypothesis is in place, one might ascertain its validity by invoking a posteriori arguments. Thus, at least one can begin by interpreting statements like these: But if someone should insist on disputing this definition of motive action with me, it would suffice to say that my decision to call it motive action is an arbitrary one and to repeat what I have just said, so long as nature justifies the reality of my definition afterwards. (GM VI 222) But I intend to verify it [the principle of conservation of the quantity of motive action] with the help of those laws of motion that are established by experience and commonly held. (GM VI 222)

Nonetheless, there is no certainty that these seemingly a posteriori arguments would satisfy the requirements for theory building if severed from the more fundamental elements of a priori validation. For Leibniz does not fail to emphasize that demonstrations of this empirical sort are capable of being presented as copies of a more general demonstration that depends on the conception of the abstract nature of force, once the requirement of providing a principle of absolute conservation of the causally efficient factor is accepted.83 However, as we have pointed out, this requirement constitutes a presumption of sufficient reason, which determines so to speak the a priori physical explanation of mechanical phenomena. On the other hand, does the abstract nature of force not refer to a concept produced by reflective apperception? And would it not therefore be a concept incapable of being derived from the empirical cases that it is supposed to represent by analogy? The abstract hypothetical character of a priori arguments and the reliance on a pos­ teriori demonstrations as ectypical equivalents do not jeopardize, it would seem, the presentation of dynamics as a unified theory. Of course, Leibniz presumes that the empirical arguments on which such demonstrations rely could be explained by real a priori definitions that might rightly be formulated. At least one comes as close as possible to achieving this end by establishing the coherence of the combined equations, and thus that of the definitions that these equations express. Establishing coherence of this sort can nonetheless only be achieved by subordinating the formulation of equa83

Cf. GM VI 222: “Now I mean to verify the conservation of motive action. I can give the general demonstration of it in a few words because I have already proven elsewhere that the same force is conserved, and because the exercise of force or force carried out in time is action, the abstract nature of force consisting only in this.”

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tions to the quest for formulating a sufficient causal reason. One can indeed adopt formulations of problems that only rely on the postulated empirical laws of impact, formulations that conform for example with Huygens’ model of the boat. One additional postulate might be sufficient for inferring the architectonic order of principles and laws that comprise dynamics.84 The minimal supposition that this postulate entails no doubt illustrates the necessity of extending the determinative reasons of conservation to the level of the underlying physical entities: these are invoked to explain the geometric relations that symbolize mechanical phenomena. The demonstration we are given links this theoretical maneuver to a combination of equations and at the same time to physical reasons. The system comprises three equations, linear, plane and solid, that in effect encapsulate the structure of arguments on which dynamics as a science depends. They schematically represent the hierarchic combination of cases that both forms and justifies the theoretical model. Consider the abstract model of two bodies impacting one another at their respective centers. The speed of a before and after the impact is noted as v and x respectively; the speed of b is noted as y and z respectively. One says that the speeds “conspire” with one another when they are both orientated in the direction of the displacement of the common center of gravity of the two bodies. Supposing that v is orientated in such a way, for every other speed orientated in another direction, the negative sign “–” will be used to modify the value of the quantity signified. The linear equation expresses the relative conservation of the respective speed before and after the impact in cases where perfectly elastic bodies are involved. It states: v − y = z − x In the case of negative speeds, the equation is modified according to the algebraic rule of signs. The plane equation is based on the quantity of motion before and after the impact, which are noted as positive values in the direction of the displacement of the common center of gravity, and as negative values in the opposite direction, aside from the conventional attribution of a positive value to v. The algebraic treatment of quantities will translate the conservation of the total quantity of progress of the two bodies before and after the impact. One thus obtains the equation: av +by = ax + bz

84

Cf. GM VI 226: “But here I will show that this is proven by the very rules of percussion that experience has justified, the reason behind which can be given by the method of the boat, as Mr. Huygens did, and by many other means, even though we are always obliged to assume something non-mathematical that comes from above.”

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Leibniz presents the solid equation as adequately expressing the total conservation of force or motive action. It is formulated as: av2 +by2 = ax2 + bz2 Squaring speeds allows for eliminating the relation to the direction of displacement of the common center of gravity. If one finds in the parameters of the equation a combination of the factors of relative speed and progress in a certain direction, then the resulting combination expresses an absolute quantity in contrast to the relative algebraic quantities. This third equation has the advantage of presenting itself as resulting from the combination of the first two, while also symbolizing an integration in terms of causal sufficient reasons, since it suggests an order that underlies the theoretic entities, whose endurance assures a conservation that is not relative to geometric, and therefore to phenomenal and abstract, parameters. This theoretical role of the solid equation as the lynchpin of an integrative hierarchy of relative theorems is based on the reciprocity of analytic and synthetic operations that make it possible to combine the distinct equations that the most integrated expression comprises.85 Everything is carried out according to the often-invoked model for decomposing concepts with a view to ultimately exposing the internal compatibility of their elements. The only difference is that one is dealing with propositions rather than concepts here, and that these propositions illustrate quantitative relations that are irreducible to the strictly conceptual analysis since they relate to the extensivity and relativity of phenomenal realities. But at least the combination of propositions analogically represents a combination of concepts that one cannot easily decompose, given that the concepts involved are expressed at different levels. Among these concepts, the first are abstractions representing the order of phenomena, such as relative speed and the quantity of progress. These cannot be completely broken down analytically such that their intrinsic meaning as ultimate ingredients of reality would be revealed. They only have a relative value in the geometric representation of empirically observable effects. The other concepts at play almost provide an adequate and symbolic expression of the 85

Cf. GM VI 228: “Although I bring together these three equations for the sake of beauty and harmony, two would nonetheless meet the demands of necessity. For taking any two of these equations, one could infer the remaining one. In this way then the first and second yield the third. The first yields v + x = y + z, the second yields a(v − x) = b(z − y), and multiplying each side of one equation by the corresponding side of the other yields a(v − x)( v + x) = b(z − y)( z + y), which gives avv − axx = bzz − byy, or the third equation. Likewise, the first and the third yield the second, for a(vv − xx) − b(zz − yy), which is the third divided by the first v + x = y + z yields side by side a(vv − xx): (v + x) = b(zz − yy): (x + y), which gives a(v − x) = b(z − y), that is, the second equation. Finally, the second and the third yield the first, for the third a(vv − xx) = b(zz − yy) divided by b zz − yy ) the second, that is, by a(v − x) = b(z − y) gives: A ( vv − xx ) = ( which in accordance with a (v − x ) b ( z − y) the first equation gives: v + x = y + z.”

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sufficient reasons underlying phenomena, such as living force and motive action, but their status remains twofold since their contents belong in a way to a metaphysics that relies on categories of reflective apperception, as well as postulates sufficient causal reasons invoked to explain the empirico-geometric theorems of mechanics. The passage from the former to the latter means moving from what is more quantitative and relative to what is more qualitative, while appreciating that one is still dealing with quantitative expressions at the level of dynamical principles – even if they do have real foundations beyond the relative symbolization of phenomena. This relationship can be clarified by referring for example to the definitions of quantity and quality in the Initia rerum mathematicarum metaphysica (after 1714): Quantity or magnitude is that in things which can be known only through their simultaneous compresence – or by their simultaneous perception. Thus, it is impossible for us to know what a foot or yard is unless we actually have something to serve as a measure, which afterwards can be applied to other objects. A foot or a yard can therefore not be explained adequately by a definition: that is, by one which does not include something similar to the thing defined. For though we may say that a foot is twelve inches, the same question arises concerning the inch, and we gain no greater insight, for we cannot say whether the notion of the inch or of the foot is prior in nature, since it is entirely arbitrary which we wish to assume as basic. Quality, on the other hand, is what can be known in things when they are observed singly, without requiring any compresence. Such are the attributes which can be explained by a definition or through the various modes that they involve. (GM VII 18–19; L 667 modified)

We can thus understand why Leibniz orders his argument in the Essay de dynamique the way he does. It is impossible to reduce the concepts of dynamics to purely analytical terms, since they do not meet the standard of concepts that can be entirely broken down by analysis. What is quantitative is indeed ipso facto relative. Thus, one should not hope to completely disentangle the relations of similitude and difference between the forms underpinning phenomena. The interplay between forces is expressed in terms of motions, and therefore in a relative way. However, the limitation of our resources concerning a priori conceptual analysis is offset by our capacity to construct a system of interlocking symbolic expressions that represent the causal sufficient reasons behind the mechanical phenomena that one seeks to explain. At the top of the model and ensuring its integration are symbolic expressions that coordinate and cover the relative expressions of a lower level. At the same time, these formulas can legitimately claim to represent the activity of the real physical causes. For, responding to the architectonic norms of a so to speak optimal theoretical model, they project a constant system of determinative reasons onto phenomenal effects, and guarantee the coherent formulation of empirical laws.

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5. Conclusion From a methodological perspective unique to Leibniz’s science, the principle of conservation of living force, which is essentially arrived at analytically via empirical laws and axioms of equivalence, must be justified by its ability to deductively explain the laws of mechanics, that is, to generate a theoretically-founded system of equations for motion. This project first gives rise to the Phoranomus (1689). There, Leibniz focuses on formulating a science on powers and effects. This involves overcoming the element of empirical relativity that is introduced by analyzing impact phenomena by means of Galileo’s law of falling bodies. From a theoretical point of view, this also involves overcoming the limits imposed by relying exclusively on the Archimedean statics models in Galilean as well as Cartesian mechanics. The Phoranomus attempts to apply the principle of equivalence between the full cause and the entire effect to the analysis of unconstrained, constant and uniform motion, in which vis motrix neither increases nor exhausts itself. The effect in this case, measured in terms of the direct relation between mass and the distance traversed, and in terms of the inverse relation between the traversal times, is postulated as being equivalent to the measure of living force in impacts. The combinative model advanced for calculating this effect nonetheless remains problematic, since it directly combines “substantial” elements with “modal” ones. But it provides a definitional framework capable of representing the system of causes that underpin both violent and unconstrained motions. The methodological model of this mathesis mechanica inspires the search for reasons that are inherent to formal action, and their accordance with those that represent the analytical equivalent of living force. Dynamics as a science and its conceptual identity emerge when the methodology of this combinative analysis is mobilized in order to overcome the insufficiencies of the Phoranomus. The Dynamica de potentia (1689–1690) presents this new theory synthetically. The a priori and a posteriori approaches have been artificially opposed to one another up until this point in dynamics. In any event, Leibniz conceives of the theory as a framework built upon heuristic principles, and as capable of justifying the formulation of phenomenal relations. The definitions, which are provided a priori, imply that conceptual elements be combined in accordance with the requirements of architectonic principles. These definitions essentially serve to provide models that allow for the application of geometrical analysis and infinitesimal calculus. Formulated step by step, these synthetic representations of the formal elements of motion combine relations of proportionality. At the heart of a priori geometric representation, the definitions for the quantity of formal effect and for the quantity of formal action necessarily refer to an analysis that, owing to the convergence of ratios, can analogically reconstruct the power of bodies to act in motion, which is a power that both implies intensive and extensive dimensions. This system of arguments, which results from “formalizing analogies”, serves as a premise of the a priori proof of the theorem of motive action. It is

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as if the underlying dynamical representations and their agreement with one another ensure that factors can be transposed into a system of parameters, which meets the requirements of a deduction ex hypothesi. In the second part of the Dynamica de potentia, Leibniz sets out to validate the theorem of conservation of living force via the same process of abstract postulates. Thus, he adopts a priori models that combine conatus and impetus to account for the generation of effects representing living force. This involves establishing a network of necessary hypothetical references based on such conceptual postulates. Leibniz also adopts definitions for representing the dynamical properties that underpin physical changes and express themselves in the relationship between cause and effect. These definitions lead to the notion of quantity of force when they are interpreted via the axiom of equivalence between the full cause and the entire effect. The analysis progresses when the combination of formal factors capable of providing the equivalent of phenomenal properties is unveiled. In this way, Leibniz can take advantage of the model for summing infinitesimals to account for the integration of conatus and impetus, and to translate the effect of the forces that are actualized in material, and not merely formal, effects. By constructing complex theoretical analogues, Leibniz is in fact attempting to provide an adequate theoretical representation of the combination of factors that emerge in the empirical laws inspired by Galileo’s law of falling bodies. The following is a characteristic of the analytical strategy that the theoretical formulation deploys: if the concept of force includes the required combination of analytical parameters, then it becomes possible to formulate equations for the conservation of vis respectiva and vis progressiva on the basis of the theorem of the absolute conservation of force (vis abso­ luta) by, for example, interpreting cases of equilibrium as borderline cases where the dynamical effect is suspended. From this point of view, theorems relating to dead force are implied by the principle of conservation of living force. Ultimately, the Dynamica de potentia appears to combine definitions that are presumed to be real, and which are articulated in accordance with the regulatory principles that ensure the theory’s coherent formulation. The validity of this far-reaching hypothesis owes to its analytical coherence, and to the fact that it can provide an adequate and sufficient analogue for phenomenal properties and their various relations. The Specimen dynamicum, the first part of which appears in the Acta eruditorum of 1695, seems to conform to the a posteriori development of the argument, as if Leibniz had renounced the geometric structure of the Dynamica in this public presentation. In fact, the text focuses on the system of architectonic principles and theoretical entities that underpin the empirico-analytic model comprising the theoretical laws and models. To account for the substantive element in dynamic effects, Leibniz resumes the concept of conatus, and reinterprets the combinative relations involving conatus by means of a model that conforms to infinitesimal calculus. On the other hand, the explanatory strategy gets its theoretical grounding in a two-tiered typology comprised of four terms: vis activa primitiva, vis passiva primitiva, vis activa derivata, and vis passiva

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derivata. In opposition to Gale’s interpretation, it must be acknowledged that, if the level of derivative forces concerns empirical laws and the models that represent them, the level of primitive forces does not correspond to a mere level comprised of theoretical entities representing bodies. For Leibniz, primitive forces seem both to incorporate a system of architectonic principles and of concepts whose meaning could be given via different analogies with the metaphysical order of forms and entelechies. By reviving the a posteriori argument for establishing the theorem of conservation of living force, Leibniz models concepts that can be combined to account for the generation and correlation of derivative forces. This modelling seeks to combine various levels of expression of force (conatus, impetus, and vis viva), and thus symbolize physical properties by formulas amenable to the infinitesimal algorithm. But such formulas require sufficient reasons of another kind, ones that represent the very structures at the foundation of phenomena. To this end, Leibniz mobilizes a strategy for superimposing analogical relations that combine different analytic approaches. The norm of intelligibility of this formulation owes to “metaphysical laws” that are architectonic principles, by virtue of which a system of justifications comprised of concepts relating to primitive force can be proposed. Moreover, these principles specify the meaning to be given to theoretical entities, borrowed from the metaphysics of entelechies, which serve the ends of an explanatory system of phenomena. This mainly entails conceiving of the order that originates in the causal series underpinning mechanical interactions. This is what Leibniz works to achieve by rejecting any recourse to occasionalism and conceiving of a physics where the reasons that represent theoretical entities will be integrated in a systematic way. The Essay de dynamique (c. 1700) has the advantage of being composed toward the end of the discussions on a priori and a posteriori proofs in dynamics. Moreover, it brings together previous approaches, specifically the analytic treatment of theorems and the attempt to deepen our understanding of the foundation of the theory. Leibniz’s strategy consists here in algebraically integrating the principles of relative conservation – relative speed before and after impacts and quantity of progress in the same direction before and after the impact – into a principle of absolute conservation, itself comprising two equivalent analytical modalities: the conservation of living force and the conservation of quantity of action. The combination of principles analogically expresses the necessity of relying on sufficient causal reasons for phenomena, sufficient reasons that themselves entail a complex integration of parameters. The integration of these models mirrors the system of architectonic principles, suggests the nature of theoretical entities, and determines how hypotheses based on the fundamental structures and elementary properties of physical realities can be critically analyzed. Ultimately, it seems that physical theory can be thus charaterized: (1) it combines models whereby equivalences and integrations can be accounted for at the level of laws; (2) it introduces architectonic principles that determine the formulation itself of the theory, whether (a) positively, by establishing the type of relational order that must prevail in the

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description of phenomena as well as in the formulation of theoretically explanatory reasons, or (b) negatively, by regularly eliminating inadequate hypotheses; and finally (3) it postulates, regarding the underlying forms of phenomena, theoretical entities for understanding, by analogy, the causal sequences expressed in extensive relations. Metaphysical representations indeed appear on the horizon; however, within the scientific discourse, the system of sufficient reasons is limited, it seems, to the principles required for accounting for the laws of power and action, which govern so to speak the rules of motion and, by implication, all empirical laws. Dynamics would thus constitute the core of a research program for progressively explaining the facts of experience.

Chapter IV The a priori Analytic Model On several occasions after publishing the Specimen dynamicum (1695), Leibniz will attempt to define dynamics as a science. He does so, in particular, when discussing the fundamental principles of this science and the types of arguments for proving them with Johann Bernoulli, Denis Papin, Burchard De Volder and Christian Wolff among others. In fact, the focal point of the debate lies in the status to be attributed to a priori arguments for establishing the principle of conservation of quantity of action. Generally speaking, these arguments were developed in the Dynamica de potentia (1689–1690), but they remained mostly unpublished. Leibniz will eventually invoke them outside of the theoretical context of this treatise. Consequently, he will have to direct his attention toward an analytic formulation of his arguments. Such a formulation will highlight the importance of architectonic principles, which guarantee the objective meaning of the theoretical concepts that ground the system purportedly founded on a priori reasons. These principles, and primarily the law of continuity, provide a framework that is sufficiently intelligible for speculatively anticipating the laws of nature. By adopting such an approach, which Gueroult regarded as a regressive aspect of Leibniz’s science, is Leibniz not revealing the theoretical mode of construction that characterizes for him the science at hand, and above all, its method of invention? To ground this interpretation, which differs significantly from Gueroult’s, it should be necessary to trace the arguments developed throughout the main texts of the controversy regarding the a priori approach. Our interpretation should stress that the dynamics requires a framework of theoretical intelligibility, and that the axioms of the physical theory represent the basic structure of this framework. In general, these speculative considerations seem concomitant with the transition to the final version of Leibniz’s metaphysics that we know as the monadology. Can these two developments be linked? The question is all the more interesting given that interpreters have grasped the relationship between dynamics as a science and previous versions of the metaphysics, such as can be found in the Discours de métaphysique (1686) and Système nouveau de la nature et de la communication des substances (1695). In both cases, the a posteriori approach to justifying the theorem of conservation of living force prevailed, producing a model that sought to conceive of the transition from the

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phenomenal order to that of the underling individual substances. But the monadological model implies specific characteristics insofar as it invokes analogies of another sort – essentially organic and psychological ones – to account for the whole system of phenomenal realities. How does the a priori, analytic model for justifying the conservation of formal action get articulated in this new representation of the ordered system of nature? Analyzing these discussions on the a priori approach can help clarify Leibniz’s most developed conceptions of the method for constructing theoretical physics and its accompanying proof structure. 1. Unveiling a priori Arguments to Johann Bernoulli Johann Bernoulli’s (1667–1748) investigation of the theses of the Specimen dynamicum primarily focused, like the objections advanced by the Cartesians, on interpreting the summation of speeds in terms of time according to Galileo’s law of falling bodies. The presumption behind these qualifications owes to the fact that, when the times are equal, it might seem that the quantity conserved can be measured in terms of mv rather than mv2: “the times being proportional to the number of impulsions” leads to “the powers being proportional to the speeds themselves, and not to their squares”1. The entire effects – which correspond to the cause being wholly consumed in the elevation of a moving body or in the tensioning of a spring – in virtue of which Leibniz conceived of living force, did not account for the duration of the exercise of force, but only for the total displacement in space. The main response that Leibniz offers his correspondent focuses on making the distinction between summing real quantities and summing modal quantities. Indeed, at any moment, the quantity of impulsion generated by gravity remains constant, but not the force found within the body, which varies according to its previous state. If the impulsion corresponds to a constant degree of speed, this is not true of the force generated. If body 2A with speed e equals twice body A with speed e, the same is not true of body A with speed 2e in relation to body A with speed e. To truly double the real unit, the thing itself must be doubled and not simply one of its modalities. When measuring force, the factor of mass, where force is intensively present, cannot be ignored in the calculation, which must also be based on an extensively realized modality. The measure of power can therefore only be determined by the repetition of real units and not merely modalities, even if these real units consequently imply some relative internal complexity compared with modalities that are nominally conceived of, such as speed.2 1 2

Letter from Johann Bernoulli to Leibniz, 24 August/3 September 1695, A III 6 486; GM III-1 215. Letter to Johann Bernoulli, 20/30 October 1695, A III 6 526–27; GM III-1 220–21: “For measuring, the real production or destruction of a certain power needs to be repeated. I mean that a power including its own subject, which is a real power, is repeated: for instance, a degree of speed is repeat-

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Indeed, the power of Leibniz’s arguments owes to the combination of two factors: calculating force assumes that one can aggregate and separate it into basic units without introducing discontinuity into its mathematical expression – whether one pays a talent or its equivalent in obols, it is all the same; on the other hand, depending on the extensive features – volumic mass and height of elevation – that one chooses, it appears that the basic units can be aggregated or separated in a way that is consistent with experience. Hence, there is this pragmatic, corroborating component that Leibniz likens to the agreement between rational arguments and experiments when solving problems that had hitherto been put on hold by other theoretical models. This inspires, for example, his relying on the impossibility of perpetual mechanical motion to ground his theory on the correlative aggregation and separation of effects. Such a principle cannot be imposed apodictically,3 but must be illustrated in a way that reflects the equivalence between the full cause and the entire effect, a principle introduced by Leibniz to play a regulatory role. To be sure, relying on the impossibility of perpetual mechanical motion to expose the aporias of the Cartesian principle of conservation of quantity of motion and confirm the rationality of the new principle nonetheless represents a limit case of demonstrative argumentation. Thus, once Bernoulli embraced the dynamics, he looked for a more straightforward version. And from this perspective, the previous controversy between Leibniz and Denis Papin (1647–1714), which he discovered in the Fasciculus dissertationum de novis quibusdam machinis (1695), seemed to offer some paths to explore. Bernoulli therefore attempts to construct an a priori model of the gravitational effect revealed in Galileo’s law of falling bodies. Given, as Papin states, that the speed of the gravitational matter acting on a body at rest is infinite in relation to the speed of the latter, it follows that the effect of gravity does not differ depending on whether the body is at rest or in motion. Like all the Cartesians, Papin inferred from this that the effect of gravity must be strictly measured in terms of time, not the height of the fall. For his part, Bernoulli postulates that, depending on its speed, the body is subjected to the effects of a somewhat considerable number of particles of gravitational matter, whereby the effect of gravity, which is uniform in terms of time, ceases to be so, owing to the speed acquired by the moving body. Following this abstract model, the

3

ed without the body being repeated, or several degrees of speed are supposed to exist at the same time in the same body. And you will perceive from experience that, when using a real repetition, everything accords under the various suppositions; when using a modal one, this is not the case: for in a modal repetition, not everything is found equally repeated.” Cf. Letter to Johann Bernoulli, 23 December 1695/2 January 1696, A III 6 600; GM III-1 227–28: “Your opinion that […] the argument leading to perpetual motion, which I often use, could be rejected by a rigorous opponent, would be, I think, excessive in this science; and I judge this hypothesis to be safely assumed, as it is agreeable to reason and corroborated by infinitely many experiences. Whosoever holds perpetual mechanical motion to be possible, will also judge it possible for water to spontaneously ascend a mountain.”

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increased vis insita of the body in question gets attributed to the space traversed in the descent, thereby confirming the Leibnizian principle. Unfortunately, Leibniz interprets this model as deficient, insofar as gravity’s action can only be uniform across time, just as the pressure of wind is uniform across a sail. (Letter to Johann Bernoulli, 28 January/7 February 1696, A III 6 646; GM III-1 239) By contrast, he rejects the premise that the effect of gravity could not be different, whether the affected body is moving or at rest. In fact, the particles of gravitational matter could possibly be infinitely small and, as a result, their infinite speed would produce percussive effects commensurate with the various states of macroscopic bodies. In effect, Leibniz tends to present the cumulative summation of nisus generated by gravity in bodies as not being commensurable with the uniform flow of time, but as expressible in terms of the space traversed. However, when Bernoulli proposes another abstract model, it receives Leibniz’s support this time around. This model relies on the theorem of the composition of motions to prove a priori that force is conserved when squaring speeds.

Fig. 12 Illustration of the demonstration proposed by Johann Bernoulli to Leibniz in his letter on 18/28 January 1696 (GM III-1, fig. 50).

Take two equal bodies A and B, where A obliquely impacts B, which is at rest. Suppose that the angle is such that the speed AB is decomposed into AC + CB (two equal vector segments). Following the impact, A is displaced at a speed expressed by the vector BF = AC. And B is displaced at the speed BE = CB. In this case, where A’s speed before the impact is equal to twice B’s after the impact, which is equal to 1, the ratio of powers corresponds to the ratio of (AB)2 to (CB)2, that is, to the ratio of speed A squared to speed B squared. With the help of an additional construction, Bernoulli establishes that the Cartesian formula would imply a false equation. Supposing in effect that only a part of the speed CB is transferred to B, GB for example, it would follow according to the conservation of the quantity of motion that A(AB) = A(AG) + B(BG), and since A = B, then AB = AG + BG, which is absurd. From this interesting approach involving a geometric model for representing the relation of forces in terms of speeds, it becomes apparent that one must discover a deductive, apodictic demonstration that would al-

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low for dispensing with the Cartesian explanatory model that relies on indirect validation by reductio ad absurdum.

Fig. 13 Illustration of the demonstration proposed by Leibniz to Johann Bernoulli in his letter from 18/28 January 1696 (GM III-1, fig. 51).

Leibniz happily welcomed this type of model. Accordingly, he proposed a variant of the argument involving the composition of motions. Take three equal spheres, A, B and C. A moves towards and collides with B and C simultaneously, such that their centers of gravity at the moment of impact form a right triangle at A. Before the impact, the speed of A is represented by the vector 1A2A. Following the impact, the body A ceases to move, and the bodies B and C are displaced respectively along the sides of the right angle at speeds 2B3B and 2C3C. The ratio of these vectors to the vector of A’s displacement before the impact corresponds to the ratio of the sides of the square to its diagonal. This illustrates once more a ratio of powers involving the square of the speeds. Leibniz stipulates that this a priori model had not been presented to Papin owing to his insistence upon rejecting the principles of dynamics. (A III 6 648; GM III-1 240) Clearly, such models only make sense for those who from the outset adopt a number of theoretical concepts, especially those capable of articulating a deductive model. It is in this context that Leibniz proposes to Bernoulli a series of arguments that summarize his conception of the schema of the a priori demonstration. Here is the argument: 1. An action achieving twice as much in once the time is (virtually) double an action achieving this in twice the time; that is, running two miles in one hour (virtually) requires twice as much as running two miles in two hours. 2. An action achieving twice as much in twice the time is (formally) double an action achieving once the effect in once the time; that is, running two miles in two hours (formally) requires twice as much as running one mile in one hour. 3. Thus, an action achieving twice as much in once the time is quadruple an action achieving once the action in once the time; that is, running two miles in two hours requires quadruple as much as running one mile in one hour. 4. If we replace twice with thrice, quadruple, quintuple, etc., then the resulting action would be nine, sixteen, and twenty five times greater; and it would generally

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appear that the uniform equi-temporal actions of equal bodies are proportional to the square of the speeds, or to put the matter another way, when a body is identical or equal, the forces are in a duplicate ratio. (A III 6 648; GM III-1 240) Leibniz adds some important points to this argument. The argument can only be convincing to those who have already heard the a posteriori one, which relied on the properties of heavy objects and Galileo’s law of falling bodies. The structure of the argument is based on notions of time, space and action, which are the most fundamental and abstract principles whereby one can form a model for calculating force.4 Leibniz is therefore able to give the most appropriate response to the Cartesians who criticized him for not taking time into account when determining what gets conserved in the generation of mechanical effects. By far, the most significant aspect of this epistemological formulation is the idea of providing the most abstract and fundamental model possible, one that comes as close as possible to the primary elements of a combination of concepts, that is, of relations expressing the reality behind well-founded phenomena. Dynamics rests on the principles of a mathesis, or more precisely a physico-mathesis, whose application is vast and includes the entire phenomenal order. The main goal of such a system based on analytic/synthetic arguments is establishing the connection between force, which is presumed to be real, and the geometric models designed to make phenomena intelligible; this assures the transition from the conceptual to the imaginable, that is, the compatibility of the theoretical foundations with the descriptive models.5 The combination of principles includes such relative laws as that of the quantity of progress. At the very moment when he introduces the a priori argument, Leibniz thus imagines connecting it to the models that essentially serve to represent empirical laws. The combinatorial nature of the physico-mathesis allows one to interpret these arguments of different levels as belonging to the same sort of relational order and therefore having the same causal determination. Leibniz expects that his well-intentioned interlocutor will accept the harmonic relationship between the terms involved in the demonstration. It is probably because this way of conceiving of the argumentative structure of the physical theory was foreign to the Cartesians that Leibniz demonstrated such reticence in presenting the a priori approach to them. Even with Bernoulli, will Leibniz not be forced to debate the matter in epistemological terms that can only seem outdated to him? And will he not be forced to justify himself by relying on a demonstrative model that has nothing to do with an architectonic conception of

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A III 6 648; GM III-1 240: “It is taken from principles most primitive and abstract, namely the notions of time, space and action.” A III 6 651; GM III-1 243: “You easily understand how important it is to rightly establish the principles of this largely evident mathesis, or rather physico-mathesis, that adds the consideration of forces, which is not subject to imagination, [and thus to] to Geometry, which is the universal science of images.”

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the science of nature? Bernoulli immediately raises the objection that virtual action and formal action are heterogenous, which would prevent them from being combined together. From this perspective, the a priori argument would be a solipsism founded on the equivocation of one of the terms (i. e., action) which would be serious breach of the axiom of equality. As a result, the Leibnizian argument might be corrected by proposing two distinct syllogisms, according to which one takes into account either virtual or formal actions: 1. An action achieving twice as much in once the time is virtually double an action producing the same thing in twice the time. 2. An action achieving twice as much in twice the time is virtually equivalent to an action achieving once the action in once the time. 3. Thus, an action achieving twice as much in once the time is double an action achieving once [the action] in once the time. Or alternatively: 1. An action achieving twice as much in once the time is formally equivalent to an action achieving the same in twice the time. 2. An action achieving twice as much in twice the time is formally double an action achieving once as much in once the time. 3. Thus, etc.6 In both cases, it would seem impossible to conclude that an action with twice the effect could be calculated as having quadruple the power of another action with a “simple effect” conducted in the same amount of time. Leibniz responds with two arguments. In the first place, he makes the distinction between the virtual and formal a relative one when dealing with motive action. Both cases involve calculating values, but only in the latter case, when comparing formal actions, is the calculation expressed as a ratio of extensive congruence. The example provided deals with money values: a ducat is worth twice a thaler, which itself is worth twice a half-thaler. But the first relation is merely a reflection of buying power; by contrast, the second, which is also a reflection of buying power, expresses a relation of numerical value, and in this case a fraction that can be represented extensively. Evidently, the analogy with money could not possibly hide the problem of identifying the basic unit of calculation and verifying its unambiguous place within the twofold relation between the virtual and the formal. Leibniz seems unable to directly respond to this challenge by manipulating the general and abstract factors of time and space. This is why his response assumes that speed has a causal significance, which apparently contradicts its purely modal significance:

6

Letter from Johann Bernoulli to Leibniz, 22 February/3 March 1696, A III 6 672; GM III-1, 250.

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Just as traversing two miles in two hours cannot possibly be formally equivalent to traversing two miles in one hour, neither can there be any virtual or calculable equivalence; indeed, who would not doubt that there is more reality to accomplishing the same thing more rapidly than slowly. (Letter to Johann Bernoulli, 8/18 March 1696, A III 6 706; GM III-1 258)

This point is illustrated by the metaphor of the runner: would one not exert twice the energy to traverse twice the distance in twice the time? And would one not expand twice this amount of energy to traverse the same distance, but in once the time? Evidently, the metaphor once again does not give us direct access to the theoretical underpinnings of the basic concepts of dynamics. It only suggests that virtual action causally precedes formal action, which itself is interpreted as expressing the causal element illustrated in and through virtual action. The other argument advanced by Leibniz is based on a definition of the concept of motive action. This argument, which Gueroult identifies as a posteriori given that it rests on an essentially nominal definition and thereby must be empirically proven, would appear from the outset to be the more satisfying from the point of view of the theoretical model: The (uniform) motive actions of the same moving body are in a compound ratio composed of the immediate effects, or the distances traversed and the speeds. However, the (uniformly traversed) distances are in a compound ratio composed of the times and the speeds. The motive actions are therefore in a compound ratio composed of a simple ratio of the times and duplicate [ratio] of the speeds; as such, given the same times or elements of time, the motive actions of the same moving body involve a duplicate ratio of the speeds, or if one is dealing with multiple moving bodies, a compound ratio involving the simple ratio of the bodies and the duplicate [ratio] of the speeds. Our principles thus permit us to show with greater truth and exactitude that the same quantity of motive action is conserved in the universe at any moment, that is, given equal times […]. Powers, if neither impeded nor added, and exercised in the same amount of time, are proportional to actions. (A III 6 707; GM III-1 259)

One must understand in which sense this demonstration is and is not a priori. For Leibniz, it is another version of the same argument that he previously formulated as a syllogism by relying on assumed a priori premises. Both forms have the same foundation, but the second one is more tailored to the accepted methods of demonstration in physics. However, what does this model share with Cartesian physics? Two points are advanced: first, the hypothesis meets the requirement of giving an explanation that allows for some order to be preserved across phoronomic changes; second, it adequately illustrates the principle of efficiency in both the production and conservation of phenomena. From this point of view, the quantity of action could easily look like an expression rigorously determined by what might be the causa sive ratio of the physical

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order. And, insofar as it continuously underpins the unfolding of phenomena without ever being exhausted, it would appear to better represent the constant laws of nature than living force. In its derivative form, living force is subject to the effective temporal variations of each concrete object; this holds true even if, in the phenomenal interaction of moving bodies, it refers to a law of universal conservation, and even if, at the metaphysical level of primitive forces, the constancy of the causal principle underpinning living force presents itself as analogous to the law according to which predicates inhere in their substantive subject. Descartes’ hypothetico-deductive model at the foundation of mechanics might have seemed to better satisfy these two requirements. In fact, however, it proved insufficient since it failed to produce a uniform and universal quantitative equivalent for the motive action that was presumably conserved in the universe. The aporias that emerge from applying the principle of conservation of quantity of motion revealed a paralogical application of Descartes’ hypothetico-deductive model when accounting for causal efficiency and its basic laws. From this perspective, the Leibnizian principle of conservation of the quantity of action might be seen as meeting the requirements of a demonstrative approach, the intelligibility of which would be confirmed in an essentially analytic way: above all, this involved establishing that the principles and premises used would either meet the requirement of an a priori geometric proof, or would appear to be the most distinct expressions of a sufficient architectonic reason. In both modes of a priori demonstration, Leibniz oscillates between the geometric conception and the metaphysical or architectonic conception of principles. In fact, he seems to propose interpreting geometric statements as expressions founded on nature, which implies that one interpret their terms by attributing them the role of theoretical entities, and therefore a partially metaphysical meaning. This is what passages like the following reveal: By this twofold demonstration of mine it has been established that the quantity of motive action involves a compound ratio composed of the simple ratio of the masses and the duplicate ratio of the speeds. This is why it is evident a priori that God would not act by virtue of perfectly wise laws if he conserved, in accordance with the opinions of the Cartesians, the same quantity of motion, as they believe; for this very reason, he would not prove to be consistent. (Letter to Johann Bernoulli, 15/25 May 1696, A III 6 758; GM III-1 275)

A subsidiary argument that accords with the first form of demonstration seems to support our interpretation. To counter Bernoulli’s first objection, Leibniz thus suggests that one should “divide” the relation between an action producing twice the effect in once the time and another producing once the effect in once the time. This division, or rather this search for a median, leads to introducing an intermediary term, an action producing twice the effect in twice the time. (Letter to Johann Bernoulli, 8/18 March 1696, A III 6 706; GM III-1 258) Everything seems as if one were looking among related forms for intermediary ones as similar as possible to the two initial forms. One

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will recall that one of the characteristics of the combinatorial art involves seeking for a more fundamental relation than the congruency between quantities, i. e., the similarity of forms that generate qualitative congruencies along progressive series. In short, the analogy between similar forms allows one to go beyond the quantitative comparison of terms and move toward laws of similarity that express the substantial and causal foundation of extensive and modal changes. The generation of phoronomic sequences thus seems to require an underlying explanation of the causal reasons that extend the same fundamental similitude to phenomenal effects. Given the situation, Bernoulli comes to accept the Leibnizian argument without resistance. Indeed, his response is significant. He henceforth recognizes the elegance of the a priori approach and its potential impact on the Cartesian milieu, but above all, sees in it a means to reinforcing the a posteriori approach.7 He sees in it a supporting analytic model capable of contributing to justifying the hypothetico-deductive path. It is in virtue of this strategic assessment that he recognizes the inadequacy of his former objections. Such is his explanation. He had interpreted virtual action as force itself or intrinsic dynamical disposition. In this case, formal action could only seem to be the resulting effect of this action. Thus, there was no opportunity for interpreting the speed of displacement as an element of the resulting action. To appreciate the Leibnizian argument, it is necessary to grasp the concept of action as a concept belonging to a theoretical construction representing the constancy of the expression of force. Force exerts itself without exhausting itself in the effect whose duration, represented as a unit of time, measures its intensity. To arrive at this relation, motive action must clearly be generated continuously and instantaneously as it extensively unfolds. This requires a type of a theoretical model to account for action as a concept that is not strictly reducible to its phenomenal characteristics. The exchange between Leibniz and Johann Bernouilli will continue in this direction and lead to several important clarifications regarding the combinatorial nature of the concept of motive action. As we know, Bernoulli is more receptive to the second, hypothetico-deductive form of the a priori argument.8 He therefore calls upon Leibniz to provide an axiom and justify the relationship that he establishes – on the basis of the modal nature of speed – between two actions producing the same effect in a given time and in twice that time. Thus, by virtue of which principle can one draw an analogy between a relation of speed and a relation of power (potentia)? Could one not simply admit that twice the speed means more power without having to stipulate that the power also doubles? For Bernoulli, it is possible and legitimate to admit such an ex hypothesi relation and deduce from it an analytic framework to account for the laws expressed through phenomena. But can one not go further than that, as the first version of the a priori proof seems to suggest?

7 8

Cf. Letter from Johann Bernoulli to Leibniz, 7/17 April 1696, A III 6 733 (GM III-1 266). Cf, Letter from Johann Bernoulli to Leibniz, 9/19 June 1696, A III 6 787, (GM III-1 281).

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Leibniz’s response is particularly instructive, since it identifies the basic axiom of the a priori approach and justifies it as such in an epistemic context, where a subsequent procedure demonstrating the axiom itself would be possible. Thus, we know that on multiple occasions Leibniz defended the utility and fruitfulness of reducing axioms as much as possible, and that with respect to necessary truths, this reduction must ultimately lead to the principle of identity applied to simple definitions. Indeed, in the present case, the geometric character of the principles is merely analogical, and the premises of the argument necessitate reliance on the principle of sufficient reason. This type of reduction cannot then be truly carried out. Leibniz falls back however on the technique of analytic regression that so often characterizes his scientific approach. This involves accounting for a complex and derivative axiom by showing how it implies a more general and basic axiom, for which it represents a more specific application. The derivative axiom is validated by showing how the conceptual connections in the two statements resemble one another, and by moving from this resemblance to demonstrating how both are functionally the same when it comes to articulating an analytic framework. The most systematic example of this process is provided by the many instances where the principle of continuity is invoked as a fundamental requirement of sufficient reason when it comes to explaining how phenomena are connected. In the present case, the axiom at play is the one that underpins the transition from the proposition “That which achieves the same action in less time is greater” to the thesis that power and the measure of speed are equivalent to one another per unit of time. From the first proposition, one derives the assertion that action A is greater than action B; thus, the action increases, while the effect remains the same and time decreases. However, to then establish the reciprocal relation between the times and the actions, one must invoke the axiom “Whenever one does not find a reason for a compound ratio, one must admit that the ratio is simple”. (Letter to Johann Bernoulli, 16/26 June 1696, A III 6 798; GM III-1, 286) It is evident for us, if it was not so for Bernoulli, that this principle is a brilliant and almost direct consequence of a technical application of the principle of sufficient reason. But Bernoulli does not seem to have admitted the logic of this analytic regression, since he asks for this obscure principle to be clarified, and interprets what was merely a premise of the analytic argument as being more evident and better meeting the requirements of a veritable axiom.9 Leibniz’s response is noteworthy. He is ready to concede that the aforementioned proposition serves as an axiom, so long as it can be demonstrated, as in the case of every derivative axiom. Supposing that the axiom can be analytically derived, one would undoubtedly arrive at

9

Letter from Johann Bernoulli to Leibniz, 21/31 July 1696, A III 7 49; GM III-1 297–98: “This principle: whenever one does not find a reason for a compound ratio, one must admit that the ratio is simple, seems obscure to me, and it does not prove what it should, but I would assume as an axiom that other proposition: what achieves the same action in less time is greater, from which you say we should start.”

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the obscure proposition, which would thereby become clear. It would above all appear evident that, without the more fundamental axiom that requires a sufficient reason for every compound proportion, it would be impossible to make several demonstrations in physics. And either way, the present case essentially involves the principle that governs how the theoretical concepts of power and action are identified.10 Once more confronted with the need to conceive of an analytic chain of reasoning as Leibniz does, Bernoulli will make a revealing misinterpretation. Even though Leibniz founded his usage of the theoretical concepts on a regulatory axiom derived from the principle of sufficient reason, and thus on a non-primitive principle, Bernouilli claims that the concept of action must be derived from some absolute axiom, since he fears that Leibniz might admit a form of infinite regression in the principles. In addition, he does not understand why one should not retain simple nominal definitions of the basic theoretical principles, given that one can use them to articulate a model apodictically founded on geometric axioms.11 But it is precisely this type of hybrid conception that Leibniz hopes to surpass. On the one hand, there seems to be some misunderstanding about the nature of the basic axioms of physics: these are capable of being derived a priori, but on the basis of the principle of sufficient reason, and by establishing relevant equivalences. Hence, they can only represent an ex hypothesi and non-absolute necessity, contrary to what can be the case with geometric principles. On the other hand, Leibniz relies on such principles precisely to establish the meaning of theoretical concepts in a way that accords them more reality than Bernoulli would grant. At this stage however, Leibniz is satisfied with raising the possibility of a demonstrative articulation of the axioms that are mistakenly thought to be irreducible; unfortunately, his models are limited to the geometric sphere, and he apparently sets aside the undoubtedly more complex and problematic case of physico-mathematical axioms. However, rather than tracing an unfinished epistemological discussion on physical theory, let us return to Leibniz’s use of his regulative axiom when it comes to elaborating the concepts of action and power. Consider the possible objection that arises from the apparent distinction between action and power. This distinction owes to the fact that the measure of power and that of action are not based on the same relations. Consider the case of A achieving exactly what B achieves in twice the time. The action

10

11

Cf. Letter to Johann Bernoulli, 31 July/10 August 1696, A III 7 74–75; GM III-1 312: “I profess that this proposition, that the action producing the same in less time is greater, can be assumed as an axiom, but note that for me every axiom is in need of a demonstration, otherwise science would be imperfect. Whoever demonstrates this axiom, I believe, will open the way for that other one which is in need of a demonstration. However, I do not disapprove at all if someone assumes such propositions without demonstration. There are, I believe, certain propositions that will lack demonstration without this axiom. I do not think it is obscure, if correctly understood. However, it will suffice to our purpose that I assume it for the sake of calculating action and thus also power, until there is a rigorous demonstration of it.” Cf. Letter from Johann Bernoulli to Leibniz, 15/25 August 1696, A III 7 102; GM III-1 216.

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of A is twice that of B, and its power quadruple. This paradoxical result would seem to deny the correlation of the two principles of conservation, motive action and living force. But this is not the case if one manages to reduce everything to a simple temporal ratio. Action is in effect a compound ratio of power and the time it takes to be enacted. It suffices to reduce the measure of action to a given amount of time to connect it to the measure of power. In the same way, one can combine two calculations of the quantity of action, one based on both power and time, and the other on the extensive/material effect and the intensive/formal effect, which perfectly captures the approach taken in the first iteration of the a priori demonstration.12 In any case, Leibniz refers to the synthetic project of the Dynamica de potentia, which he interprets as correlating different analytic approaches under the aegis of architectonic principles, such as excluding all combinations of factors when a sufficient reason cannot seem to justify such a complex relation. In sum, determining the theoretical concepts of dynamics seems to occur a priori insofar as it involves architectonically explaining the system of factors whereby phoronomic phenomena can be measured. But this system of correlations cannot possibly be considered purely nominal, insofar as it derives from the a priori regulatory application of the principle of sufficient reason in its various derivative forms. 2. The Parameters of Action: The De Volder Correspondence In his correspondence with Burchard De Volder (1643–1709), Leibniz is no doubt close to reaching the final stage of his epistemological reflections on dynamics. De Volder was a professor of philosophy, physics and mathematics at the University of Leiden. At the time when Leibniz was corresponding with him, the main project of this Dutch academic was publishing Huygens’ posthumous works, of which he was the executor. It is through an intermediary, Johann Bernoulli, then professor of mathematics at the University of Groningen, that De Volder establishes contact with Leibniz; moreover, the correspondence between Bernoulli and Leibniz will on numerous occasions produce commentaries that interpret the ongoing discussion with De Volder. The focus of Leibniz’s and De Volder’s exchanges is twofold. On the one hand, it is through his replies to De Volder that Leibniz will mostly develop his justification for his system of a priori proofs. In addition, as De Volder attempts to understand how dynamics can fit into a mechanistic theory of nature, Leibniz finds himself forced to elaborate the system of theoretical presuppositions of the new science. It is in this way

12

Cf. Letter from Johann Bernoulli, 16/26 June 1696, A III 6 797–98; GM III-1 286.

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that the monadological model is officially introduced as the framework for a physical theory whose basic structure is reconceived. The debate begins in De Volder’s correspondence with Johann Bernoulli. In his letter to Bernoulli on 21 November 1698,13 De Volder summarizes the problems that stand out to him in Leibniz’s dynamics, which he knows from the few relevant publications and the details of the controversy with Papin. De Volder is interested in the soundness of axioms that Leibniz used to construct his a posteriori demonstration, particularly the axiom inherited from Descartes (and Pascal), according to which the same force is required to lift a body weighing four pounds one foot as is required to lift a body weighing one pound four feet. He doubts that this is universally true independently of time, that is, of the speed with which the body is lifted or dropped. It thus suffices to reiterate Papin’s hypothesis on the role of gravity: the speed of the gravitational matter being somewhat infinite in relation to that of the body on which it acts, everything occurs as if the impact of gravity were strictly a function of time and inversely related to the resistance of the body, this resistance itself being measured by the speed that the body acquires. Relying on this model leads to assuming that, in a given unit of time, the effect of force would be measured in terms of speed multiplied by mass. This inevitably contradicts the presumed axiom. At the same time, De Volder seeks to understand Leibniz’s criticism of Papin for his inability to replicate the purely modal qualities, i. e., the speeds in the reciprocal relationship between mass and speed. The situation would be different if we were dealing with mass multiplied by height, since the product would represent a real and therefore complete quantity, and not merely a modal and thus incomplete one.14 However, in a more general sense, De Volder wants Leibniz to justify reconciling the theorems of relative conservation with that of the conservation of force as an absolute quantity.15 Thus, he quite rightfully derives the principle of conservation of the quantity of progress from rule 5 of Huygens’ laws of impact. But to him, this hardly seems to justify maintaining an absolute quantity of force beyond the algebraic addition and subtraction of quantities of motion. Moreover, the relativity of the principles is confirmed when one considers the fact that Huygens assumes that bodies are perfectly hard and capable of being reflected in impacts. This hypothesis was formulated without taking into account how either hard, non-elastic bodies or soft bodies respond to impact. If Leibniz follows in Huygens’ footsteps by insisting that force is universally conserved across the relative changes represented by the incomplete and abstract parameters v and mv, then De Volder suspects that he must subscribe to the

13 14 15

A II 3 476–82; GP II 148–152; Leibniz 2013, 13–25. Cf. Letter from Leibniz to De Volder, 27 December 1698, A II 3 501–502; GP II 157; Leibniz 2013, 37: “But if a new degree of speed is given to the first body alone, the repetition is not of a complete thing but only of something from which a measure cannot be taken safely.” Cf. Letter from De Volder to Johann Bernoulli, 21 November 1698, A II 3 479–80; GP II 150; Leibniz 2013, 19.

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universal elasticity of bodies. But this conception would appear to him all the more speculative given that it seems to contradict the geometrico-mechanical conception of the nature of bodies. De Volder challenges Leibniz to produce a theory of “active” substance. In the Acta eruditorum of 1694, he found the essay De primæ philosophiæ emen­ datione and eagerly accepted Leibniz’s critique of occasionalism. But can one replace occasionalism with an a priori conception that establishes the intrinsic dynamism of substances? The problem is that sensory experience gives the idea that matter is active. But a purely conceptual analysis of the concepts requires us to recognize that this idea contains no intelligible notion of causality. This is why Leibniz’s theory is so attractive, but demands further elucidation. Leibniz must successfully produce the analytic system that will transform his dynamics into a legitimate natural philosophy. He must elaborate what we would consider today to be the metaphysical presuppositions of his theory. De Volder therefore seems to set the bar high: For if we had an a priori demonstration that every substance is active, I might easily persuade myself that from this most fruitful source of truth follows not only the resolution of my worries but also of those difficulties that have burdened every natural philosopher up until now. Indeed, due to the ignorance of this fact they have been compelled to summon God ex machina for any collision of bodies. For since they learned from experience that matter is active but did not observe this force to flow from the nature of body, they had nothing else in which to take refuge. But if it is established that the nature of corporeal substance involves action, which remains the same as long as there are still bodies, the matter is resolved. And unless I am mistaken, it will also be possible with this to explain those things about the cause of the continuation of motion that seems completely confused to many people. It will be clear what striving is that persists in a body even if it has no effect because of obstacles (something that seems extremely difficult to comprehend to me), in what way contrary forces may follow from the same active force, and how the forces nonetheless remain the same even if the effects cancel each other out. Finally, lest I pile on too many things perhaps it will be clear what that thing is that must be ascribed to matter besides extension. I confess I have no concept of this so far. (A II 3 482; GP II 151–52; Leib­ niz 2013, 21–23)

Leibniz responds to this challenge in two significantly different ways. On the one hand, he wants to justify the “axiomatic” structure of the a posteriori demonstration, that is, to show how it is analytically sufficient. Then he will turn to the metaphysical assumptions relating to active substance, but in the context of an analysis that he seeks to squarely situate within the realm of physical theory. He will first argue in favor of the axiom of the Cartesians: the same amount of force is required to lift a four-pound object one foot as is required to lift a one-pound object four feet. His justification is based on the reductio ad absurdum that would follow from denying this: by presuming the conservation of the quantity of motion, one is obliged to admit effects that are greater than their causes, which would imply perpetual mechanical motion. However,

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by accepting the axiom, force remains within the limits of the notion that causes are physically equivalent to their effects. De Volder perceived the power of this argument, but recognized an unsurmountable difficulty: the relationship between forces in equilibrium satisfies, according to Archimedes’ law, the Cartesian principle of conservation, but only cases where forces are completely exhausted would demonstrate the axiom and, by implication, prove the validity of Leibniz’s principle. Above all, Leibniz must establish that the cases of equilibrium and those where force is completely exhausted belong to the same, coherent system of laws. In a beautiful passage, Leibniz draws together the theoretical concepts that allow for subsuming cases of equilibrium under those where force is causally expressed. These concepts are respectively conatus and impetus, and vis mortua and vis viva. The instantaneous conatus are elements of dead force, just as the impetus are introduced to generate the quantity of living force. As we know, a twofold framework is introduced here, one that the Specimen dynamicum had sought to analyze. In his reply to De Volder, Leibniz insists on reconciling the theoretical propositions or laws relating to the phenomenal characteristics with which the established explanatory concepts must agree. And this reconciliation, by which “everything comes together very beautifully”, (Letter to De Volder, 27 December 1698, A II 3 497; GP III 154; Leibniz 2013, 29) must directly connect with the means of expression that the infinitesimal algorithm furnishes. Impetus therefore results from infinitely summing the elementary conatus as a function of mass, but at the same time, one goes from dead force to living force when impetus, initially reduced to a virtual effect, unfold across time and thereby initiate the progressive integration of conatus. In the case of equilibrium, which Archimedes’ law takes into account, only a virtual effect is produced, represented by mv. When a body falls, and unconstrained impetus is reproduced across time by the summation of an infinite series of instantaneous conatus, the phenomenon must be expressed mathematically by mv2. To conceive of equilibrium as a limit case of the causal expression of force, it suffices to rely on the notion of a continuous progression or regression involving a force differential. This would constitute an ultimate nominal or relative value whose justification would depend on the mathematical fiction that allows for representing it to the limit. From here we also see that nature reconciles most elegantly the law of equilibrium of colliding [bodies], which is relative, with the law of equivalence of causes and effects, which is absolute, through the mediation of the law of transition, which is gradual, avoiding all leaps. (A II 3 497; GP II 154; Leibniz 2013, 31)

To support this point, Leibniz symbolically represents the terms and allows one to analogically grasp their relations. The tendencies or basic solicitations would be represented as quantities dx, while the speeds would be represented as x and the forces as xx or even, to indicate their law of production, as xdx . Furthermore, he cites the intrinsic elasticity of bodies to explain the virtual effect of impetus in cases of

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equilibrium. Therefore, the tension placed on the system of the bodies in equilibrium instantly dissipates the force without the latter expressing itself in some spatially measurable effect. This implies, according to Leibniz, that the parts of the system interact instantaneously owing to an elasticity that affects the tiniest of their infinitely divisible elements. But action cannot then be expressed by any displacement in space whatsoever. Rather, the effect of this action translates to a virtual displacement. If the Cartesian principle of the conservation of the quantity of motion illustrates the case of equilibrium, that is, of force constrained instantaneously, then the expression of forces beyond an instant would not be captured by it. And, as a result, the Cartesian principle must be subsumed under a general principle, of which it would represent a limit case: According to the law of equilibrium, the measure proportional to the product of the [size of] a body and the speed has a restricted (peculiarem) necessity, and should not be applied to absolute living force. Indeed, as I once noted in the Acta eruditorum, the general rule for the evaluation of force through its effects applies no less in the case of the law of equilibrium (which concerns dead force) than in the case of the law of equivalence (which concerns living force). For in both cases, the measure of force is proportional of [the size] of the body and the distance, i. e., the descent. (A II 3 504; GP II 158; Leibniz 2013, 39)

Subsuming the law of equilibrium under that of equipollence reveals the nature of the flaw in Papin’s axiom, on which De Volder also relied. The supposed equality between the number of degrees of gravity overcome by the body during its ascent and the number of degrees of force thereby acquired is the product of two errors. On the one hand, this argument involves reducing an effect to the limit case of equilibrium, while the effect is realized in time and cannot be conceived of as a simple additive succession of moments of speed without progressively integrating the conatus into the impetus. On the other hand, everything is undertaken as if the influence of force were added or subtracted across time without either being entirely exhausted or accumulated. It is the causal reality, as it were, of gravity’s effect that escapes the framework put forth by Papin and the Cartesians. But is this ontological dimension of the power of action in phenomenal realities not precisely the reason for which De Volder wants Leibniz to establish an a priori system? Is this not the crux of eliminating Cartesian mechanical models? The combined efforts of Leibniz and Bernoulli had already led to various demonstrations via geometric models. And while Papin first gave his approval of them, he later retracted it. These models involved one body obliquely colliding with two others at the bisection of a right angle; when the three bodies are equal, at the moment of impact, their centers form an isosceles right triangle. (A II 3 505; GP II 159; Leibniz 2013, 41) In a given unit of time, the distances travelled and thus the speeds are inserted into an equation where the effects are measured by the masses multiplied by the square of the speeds.

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Fig. 14 Illustration of the demonstration proposed by Leibniz to De Volder in the first letter of their direct correspondence (GP II 159).

In this geometric diagram, body R collides with bodies S and T at rest, which are both equal to one another and to R; R passes from motion to rest in conformity with the laws of motion, and the impact creates the displacements S2S3 and T2T3 following the law of the composition of motions. If one calculates the force of R as being directly proportional to the square of R1R2 before the impact, then the forces S and T must add up to the equivalent of this value after the impact. Thus, by transferring R’s diagonal motion into two motions that comprise the sides of the square, and applying the Pythagorean theorem, we are able in particular to formulate the equation: (R1R2)2 = (V1T)2 + (W1S)2 = (T2T3)2 + (S2S3)2 If, however, one calculates force as proportional to the speed represented by the simple vector R1R2, then the principle of conservation conflicts both with the law of the composition/decomposition of motions and the Pythagorean theorem. Leibniz presents this argument as a simple geometric analogy. The properties of the Pythagorean theorem are verified a priori in dynamics so long as we understand bodily displacements in terms of the theorem of the composition of motions and the properties of right triangles. But can this model be extended to other situations where the parameters vary. Leibniz must be conscious of the limits of this type of geometric demonstration. Furthermore, to what extent can such abstract models account for the force that inheres in the agent’s own structure? It is from this perspective that De Volder formulates his request, asking for an a priori conceptual exposition of the fundamental principles of dynamics, that is, a theory of phenomenal reality that adequately accounts for the empirical laws of dynamics. Obviously, Leibniz postpones the moment when he will satisfy this request. He claims that the metaphysical part of the theory relating to the nature of phenomenal reality remains unfinished in comparison to the mathematical part of the dynamics:

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Would that I could explain my metaphysical meditations concerning the nature of substance and the things depending on it just as clearly, or that I might have the issues set out just like the mathematical part of the dynamics. (A II 3 510; GP II 162; Leibniz 2013, 47)

At the same time however, he refers to partial expressions of the theory, in particular, in his articles published in academic journals. There, the system is mostly presented through analogies rather than in a systematic way. These are provisional attempts that would in theory lead to a deductive account. At least several fundamental notions of the theory crop up here. Leibniz proposes the elasticity of bodies as an analytic requirement, situating it beyond the realm of phenomenal properties. The ether hypothesis, which Descartes and Huygens both endorsed, proves to be an insufficient explanation in this context. Meanwhile, Leibniz searches for a concept that agrees with the architectonic law of continuity and explains the infinite regression that one encounters whenever relying on purely physical concepts inferred from the perceptible properties of bodies. Because the phenomenal universe reveals itself as being comprised ad infinitum of worlds within worlds, the best we can do is assume a type of order or architectonic reason governing how phenomena unfold. Adequate representations of this order could not possibly be derived inductively from sensory experience data; instead, they must undoubtedly be drawn from the rational requirements that the human understanding encounters when framing representations of the sufficient reason whereby empirical correlations can be accounted for. However, this order must be corroborated by the most analytically adequate hypothesis that that can be advanced on the basis of our relative understanding of phenomena. For the time being, Leibniz contents himself with postulating that the concept of an active principle in material substances is necessarily required for justifying the dynamics’ equations. This principle is represented in terms of nisus, which always undergo an effect, even if it seems to be halted by material obstacles. And even the inertia of bodies, or the power to resist the exercise of force, must be conceived of as deriving from the nisus belonging to the substances that it is part of. The dominant idea behind this notion of substantial active principle is a continuous series of effects that would express it. For Leibniz, this is a blueprint of the a priori framework of the theory that dynamics requires. De Volder’s request shapes the meaning of the a priori demonstration that Leibniz reformulates this time around, the demonstration being modeled after the descriptions that he had previously sent to Bernoulli and Papin. According to De Volder, Leibniz’s dynamics must be demonstrated a priori to be justified; that is to say, by analyzing its concepts from the perspective of their determined meaning, one must show that the notion of material substance necessarily implies a principle of activity. In short, one must apply the Cartesian criteria of clear and distinct ideas when conceiving of bodies to show that the concept of res extensa must be reformed without prejudicing geometric intelligibility, which is the basis of the new physics. One must therefore arrive at the cause of those phenomena of force described by the empirical laws of impact and

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explained by the principle of conservation of living force that Leibniz invented. But this operation must be carried out by exposing the intrinsic intelligibility of a notion of material substance distinct from the Cartesian notion.16 Leibniz accepts the challenge by first underscoring the limits of the demonstrative process. (Letter to De Volder, 24 March/3 April 1699, A II 3 544–51; GP II 168–75; Leibniz 2103, 69–83) Failing rigorous demonstrations, one must content oneself with hypotheses that do not prove deficient when analytically explaining phenomena, that is, hypotheses that can legitimately be regarded as being intrinsically intelligible and corresponding to the facts.17 One must therefore be convinced that they meet the conditions of certain knowledge. Such is the case for the law of continuity itself, which is justified both because it meets the rational requirement for order and because experience never fails to confirm it.18 By experience, Leibniz means here validation by a growing number of diverse phenomena. By the rational principle or requirement for order, he means the regulatory idea of an order that becomes increasingly detailed as our analysis of objects becomes more refined. This argument is all the more important because the law of continuity is increasingly invoked when conceiving of the facts of experience in accordance with theoretical models.19 In any case, the preferred methodology for judging such models is not reducible to the mathematical rigor of a priori demonstrations, but to the development of a set of sufficient reasons: these are valuable because the abstract statements that express them are found to be consistent with one another, and because they prove to relate to a seemingly significant number of phenomena. Take, for example, Leibniz’s thesis regarding the universal elasticity of bodies. This theoretical concept can account for a number of empirical cases. But more than anything, it ensures the consilience between the laws of dead force and 16

17

18 19

Cf. Letter from De Volder to Leibniz, 18/28 February 1699, A II 3 530; GP II 166; Leibniz 2013, 59–61: “And so, most renowned Sir, if you would like us to agree with you without any scruples, I think that it will be necessary to descend to the notion of substance and to demonstrate that it is necessarily active from its nature, or certainly that the nature of corporeal nature is such that it is necessary that its forces are always conserved.” Cf. A II 3 548–49; GP II 172; Leibniz 2013, 77: “For even though I cannot easily present everything in such a way that it is demonstrated a priori with geometrical rigor or explained deeply, even when I see the reasons, nonetheless, I venture to promise that no objection can be raised that I would not hope to answer. I think that this is something that should not be disdained in matters so remote from the senses, especially since the agreement of doctrines, both with the phenomena and among themselves, is among the most powerful marks of their truth.” Cf. A II 3 547; GP II 170; Leibniz 2013, 75: “And so two things on which I always rely here, success in experience and the principle of order […].” Cf. A II 3 545; GP II 168; Leibniz 2013, 71: “For since everything happens through the perpetual production of God and, as they say, by a continuous creation, why could he not have, so to speak, transcreated a body from one place to another at a distance from it leaving behind a gap either in time or in space, e. g. by producing the body at A and immediately thereafter at B, etc.? Experience teaches that this does not happen, but the principle of order, which makes it the case that the more things are analyzed the more they satisfy the intellect, establishes the same thing. This does not happen with leaps, where in the end the analysis leads us, so to speak, to mysteries.”

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living force, and between the relations of inertia and motive force in bodies, as well as making it possible to conceive of the hierarchical integration of the absolute and relative principles of conversation. What was missing in theories like the one Huygens proposed regarding the fundamental laws of impact was a concept guaranteeing the passage from perfectly hard and elastic bodies to ones with the characteristics of relative softness and hardness. Hence the relative inconsistency of Huygens’ doctrine when it comes to articulating the ontology required by the science of dynamical and kinetic relations. The limit of Huygens-type physical science is a reluctance to rely on analytic hypotheses that are ultimately grounded in architectonic principles. With respect to the theoretical foundations of the system of nature, Leibniz’s methodological position involves developing such analytic hypotheses. This is the context in which interest is rekindled in the syllogistic argument that was previously proposed to Bernoulli, and which had come to De Volder’s attention. In the first place, Leibniz suggests the possible assimilation of the virtual and formal components of motive action, which had proved difficult for Bernoulli. The example cited in the present case is a ducat being (virtually) worth two thalers, and a thaler being (formally) worth two half-thalers. In terms of purchasing power or market value, the ducat can be exchanged for four half-thalers if both the virtual and formal are jointly converted into a total power to intervene in exchanges. The syllogism that had already been presented to Bernoulli reappears, but formulated with the supporting hypotheses that underpin its interpretation: In a uniform motion of the same body, (1) “An action bringing about a double effect in two units of time is twice an action bringing about a simple effect in a single unit of time.” For example, the action of covering two leagues in two hours is twice the action of covering one league in one hour. For the first action formally contains, i. e. exactly repeats, the second one twice, since it traverses one league in one hour twice. (2) “An action bringing about a single effect in a single unit of time is twice an action bringing about a single effect in twice the time.” For example, the action of covering one league in one hour is twice the action of covering one league in two hours. Clearly, that which produces the same effect more quickly does more. I also assume that actions producing the same effect are proportional to the speeds, or inversely proportional to the times. And so an action that traverses a distance at twice the speed is twice as efficacious as an action traversing the same distance at one unit speed, i. e. which amounts to the same thing, the latter is contained virtually in the former twice. From here the conclusion now follows, namely, (3) “An action bringing about a double effect in two units of time is four times an action bringing about a single effect in the same double amount of time.” For example, the action of covering two leagues in two hours is four times the action of covering one league in two hours. […] And generally actions lasting the same time are proportional to the square of the speeds. (A II 3 549–50; GP II 173; Leibniz 2013, 79–81)

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De Volder sees this argument as a paralogism. And he focuses more precisely on the minor premise and the axiom that supports it: the agent’s force increases to the extent that it accomplishes its task more quickly. It seems to him that, by virtue of the same Leibnizian methodological axiom according to which the entire effect equals the whole cause, the measure of force must derive from the effect. Speed would merely constitute a modal property that in no way affects the reality of the effect into which the force is translated: in this case, the effective translation of the moving body.20 Since Leibniz replies by defending the certitude of the axiom, which is both confirmed by experience and perfectly satisfies the intellect, De Volder reiterates his rebuttal by reinterpreting the meaning that his interlocutor attributes to speed: far from agreeing that an action expresses more reality and therefore greater perfection the more quickly it is achieved, De Volder will assume that the more or less lengthy duration of an action can “compensate for” the inverse ratio of speed.21 The effect of translation alone, no matter the duration nor speed at which it is accomplished, can express the true causality that force represents. And we know that, in this case, it is far from evident that the principle of conservation of the quantity of action measured by msv/t, that is mv2, indicates some type of theoretical entity with a real referent. As we know, across a given unit of time, it is the relative ratio of quantities of progress that seems to account for phenomenal occurrences. Leibniz is therefore forced to insist on the power relation that underpins the relative speed at which motive action is accomplished by linking the latter to the spatially measured effect. But the only argument that he is able to advance derives from the strategic role this hypothesis plays in organizing the facts of experience, which are explained by the principles of conservation of force and motive action. Without this hypothesis, one could not possibly hope to deduce phenomena in a systematic way. At least, it seems impossible to otherwise justify the analysis of phenomena in distinct and therefore adequate and precise terms:

20

21

Cf. Letter from De Volder to Leibniz, 13/23 May 1699, A II 3 564; GP II 180–81; Leibniz 2013, 95: “Indeed, since the whole nature of action consists in producing an effect, and the force of an action, given that it is the force of the cause, is equal to the effect, it seems clear to me that actions are proportional to effects. Consequently, the second proposition is assumed incorrectly, with the result that “actions producing the same effect are proportional to the speeds, or inversely proportional to the times.” Nor do I admit that “that which produces the same effect more quickly produces more force.” For whether it takes a shorter or longer time, it will use precisely this force, from which the same effect follows. Therefore, in both cases it seems to be neither more nor less than the effect, but equal to it.” Cf. Letter from De Volder to Leibniz, 1 August 1699, A II 3 586; GP II 188; Leibniz 2013, 113: “I would think that what is more real in the quicker motion is compensated by the longer duration of the slower one.”

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I assumed that “it is more to produce the same thing more speedily,” and that there is also a great profit in the nature of time, which every experience indeed confirms. And so I thought this postulate was reasonable. If it is rejected, I admit that my most recent demonstration, and almost every measure of the forces in nature, is useless. In the meantime, it may be enough that my measure is demonstrated from this hypothesis and that all the phenomena are derived from it. (Letter to De Volder, 23 June 1699, A II 3 574; GP II 185; Leibniz 2013, 111)

From this second perspective, the axiom presumably refers to the simplest and most evident notions that one can introduce. It is interesting to note that the unsent version of the letter contains a more relativist and pragmatic understanding than the sent version, which contents itself with positively affirming the axiom. In Leibniz’s next letter, (Letter to De Volder, 1/11 September 1699, A II 3 592–602; GP II 189–95; Leibniz 2013, 117–31) the argument is presented analytically. In addition to affirming the criteria that the axiom satisfies, Leibniz introduces formulations of notions and distinctions that he considers crucial. Regarding the purely modal role that time would play in the expression of force, he introduces the distinction between force’s formal effect and real effect. In the latter case, where a force is entirely consumed in the motive effect, the spatial displacement that the moving body undergoes expresses the force in toto. Therefore, one does not account for time when calculating the motive cause; indeed, it is apparent that the greater or lesser speed at which a motive action or dynamic effect is accomplished owes to the circumstances being more or less favorable to the exercise of the total intrinsic cause, which Leibniz illustrates by way of a moving body being obliquely displaced on a vertical plane along the curve AB and then the tangent BC. Thus, since the circumstances in no way present an obstacle, the action is accomplished as fast as possible.

Fig. 15 Illustration from Leibniz’s letter to De Volder in response to the latter’s letter of 1 August 1699 (GP II 192).

However, in the second example force is exerted without being consumed, that is, without translating into a motive effect that exhausts it. Force therefore remains intact, while the effect that expresses it without being altered is measured by the intensity of the action across time. Hence the necessity of examining the speed with which a moving body is displaced in terms of the effect of its intrinsic force, and in the absence of all external constraint that might hinder it. Owing to the purely modal

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nature of time, one can no doubt disqualify the idea that it effectively figures into the theoretical model of formal action. Leibniz skirts this difficulty by describing the measure of time as a simple index for quantity of action and thus for the underlying causal structure of a moving body. The true reality of formal action lies in the perfection or intrinsic constitution of the agent that reveals itself in a more or less rapid action. From this point of view, quantity of time may be interpreted as “a posteriori” conceived to express what is analogically constructed, and as “a priori” conceived to represent the internal form of the physical agent, and in this case, the moving body as the subject of action. (A II 3 594–95; GP II 189–90; Leibniz 2013, 119) In fact, Leibniz will go so far as to point out that taking into account both power and the inverse of time does not work in the case of violent or real action. The latter exists where other bodies create resistance in a process that depends on highly contingent circumstances, even if the resulting effects are modeled after a universal law that conforms to the order represented by the principle of the conservation of living force. However, the free exercise of force translates into a constant motive effect whose factors are derived from representing the resulting action in an intelligible, geometric or, even more precisely, combinatorial way. The process of abstract conceptualization that allows one to represent such a theoretical entity extends even to the limits of the possibilities of experience, which include the conditions under which bodies causally interact with one another, as well as the interactions of force/resistance in the phenomena that express them. This is why Leibniz proposes his analytic model of motive action based on the formal effect as an “a priori calculation”.22 De Volder’s argument unduly assimilates the process by which force is conserved and transferred. In the controversy with Catelan following the publication of the Brevis demonstratio, Leibniz already had the opportunity to argue how taking time into consideration would prove his system. But it seemed that doing so would also serve the interests of the Cartesian theory, for with the addition of time, the theorem of living forces is reduced to a relative principle of the conservation of quantity of progress, which is measured algebraically by mv. The consideration of time acquires its true theoretical, and not merely empirical-modal, meaning when Leibniz is able to build the system of the principles of motive action – a fact that he cautiously shares with only a few correspondents. An additional analytical argument follows from the fact that Leibniz combines conceptual elements in an original way to show that, in the case of motive action, where the effect is formal, it is nonetheless possible to establish that a certain real effect is involved.

22

Cf. A II 3 594; GP II 190; Leibniz 2013, 119: “Even if there is no action in nature that is actually free from obstruction, nevertheless, that which belongs to an object intrinsically may be separated through mental abstraction from what is added in by accidents, especially since the former is measured from the latter, as if a priori.”

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Conversely, in the case of violent action, where time has a strictly modal meaning, it is possible to imagine that, on the whole, time is preserved within the metaphysical sufficient reasons. Both cases involve analogical arguments. To begin, suppose that a freely moving body expresses a formal effect, as per the calculation mentioned above. In this case, one must conceive of a sort of causal transfer of motive force from one moment to the next, in virtue of which the speed of displacement of the moving body is reproduced sequentially. This is effectively possible only if one maintains that there is an intensive dimension of action reproduced across every instant of its uniform transfer: In this sense, the axiom of the equality of the full cause and the entire effect is verified even in formal, i. e., free, action. But in this case, it is rather that the previous force is conserved than that a new one is produced. It is also clear from this that a new force cannot be produced unless the previous force is used up at the same time. Otherwise the force in the universe would increase. (A II 3 596; GP II 191; Leibniz 2013, 123)

Now let us examine the equivalent model for the violent effect, according to which the force within the cause expresses itself by consuming itself. Leibniz emphasizes that, from the point of view of a finite understanding, time has no real meaning here. However, one can conceive of a system of final causes comprehensible to the infinite understanding of the Creator. According to this system, in the smallest interval of time, the maximum real motive effect compatible with the rule of conservation of action would be produced from one moment to the next, and this would occur across the inestimable diversity of conditions affecting the interaction of bodies. The argument that Leibniz seems to favor above all others involves combining analytic models so as to include both the quantity of action and living force in the mathematical formula. The model for this type of argument dominated how this science was conceived of going back to the Dynamica de potentia, and it is wonderfully exemplified, as the Essai de dynamique will show, by the convergence of absolute and relative principles. But the main point of convergence is the one that can be affirmed by bringing to light the harmony between principles of absolute conservation: A remarkable harmony is seen there, in that the same measure emerges whether the moving thing acts by conserving its force while acting and producing nothing else – except what I would call its own formal effect, or operation, namely, change of place – or whether it produces some other effect, by violent action, thus consuming the force of that which is acting, as when it raises a weight or stretches a spring. Either way, it follows that the forces are calculated as the squares of the speeds. (A II 3 595; GP II 190–91; Leibniz 2013, 121)

As the rest of the correspondence reveals, De Volder concedes to Leibniz the minor premise of the syllogism for the a priori proof of the theory of conservation of quantity of action. The basic axiom indeed seems admissible to him. According to the axiom, the agent’s greater perfection would lead to the action being accomplished more quickly; it follows then that the action’s perfection is measured by the speed at which

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it is accomplished, and thus in terms of time.23 But the true value of the action, its formal characteristic, is independent of the speed of accomplishment since it remains identical across the various moments of the time it takes to be accomplished. Given the circumstances, De Volder suggests revising the major premise of the syllogism and considering space and time as not to be combined, but rather treating them as factors that cancel one another out. A faster action traversing a shorter distance can also be equivalent to a slower action traversing a greater distance given their reciprocal proportions. But the præstantia of the action would be measured in terms of its speed, to which the power of the agent would correspond. It would therefore seem that quantity of action is proportional to speed rather than its square. Leibniz endeavors to resolve what he considers to be De Volder’s new paralogism by separating the now admitted premises from the still inadequate theoretical hypotheses. He notes that, from now on, one can accept the principle that actions are composed of compound ratios of power and time, that is, of intensiones and extensiones. In fact, only the first part of this assertion wins De Volder’s approval for the time being, as he argues that the præstantia of the action for an agent is proportional to its intrinsic power to act, and thus its speed. But at the same time, in order to determine the quantity of action, one must take into account the diffusion of the præstantia, which for De Volder means understanding time and displacement as reciprocally related, and thereby invalidating the combination of intensive and extensive dimensions. In contrast, Leibniz asserts that the diffusion must be accounted for in two ways: Actions by which the same distance is traversed are calculated as the reciprocals of the expended times, but the powers by which the same distance is traversed are calculated as the reciprocals of the squares of the expended times. (Letter to De Volder, A II 3 621; GP II 201; Leibniz 2013, 147)

This new construction implies that what De Volder had believed to be force itself is now merely understood as a modal ingredient – the intensity of action – that one can and must strictly identify as complementing another ingredient, diffusio, which is understood in terms of space, and not merely the times relating to the manifestation of intensity. To illustrate this point, Leibniz introduces a twofold framework to analyze action, that is, two analytic approaches to conceiving of the basic theorem. One can conceive of action as either proportional to the combination of power and time, or the combination of the effect/distance travelled and the speed. De Volder’s problem consists in completely reducing power to the intensity of action, which is measured by speed. One must distinguish two possible conceptualizations of extension, one understood in terms of time, and another in terms of place, that is, in terms of spatial

23

Cf. Letter from De Volder to Leibniz, 12/22 November 1699; A II 3 658; GP II 196; Leibniz 2013, 135–36.

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perspective. In the first case, intensio = potentia. In the other, intensio = velocitas. One cannot interpret intensio as being entirely representative of force and measurable solely in terms of instantaneous speed.24 Leibniz proposes restoring the two hypotheses by reducing them to their intelligible terms, which best approximates the a priori aspect of the concepts with regards to their implied meanings. This yields two possible relations: actions are the product of power and time; actions are the product of speed and distance traversed. One can analyze the second relation by considering the fact that space is measured in terms of the product of speed and time. Actions are therefore conceived of as the product of speed squared and time. Supposing that the propositions apply for the duration of a single unit of time, it would appear that both possible calculations of the value of action basically involve equating power with speed squared. Once again, this is the sign of an analytic construction that responds to the demands of the combinatorial architectonic: “In this way you again see how beautifully everything fits together and is connected by an undeniable relationship.” (A II 3 623; GP II 203; Leibniz 2013, 149) At the same time, the conclusion to the a priori argument is meaningful. Leibniz returns to the conceptual distinctions that underpinned the argument. He makes a distinction between free action and violent action, and with respect to free action, a distinction between speed intensity and power intensity, which respectively comprise extension understood in terms of time and extension in terms of location. Two parallel equations for the quantity of action are thus provided. The principle of conservation that these equations express brings together the theorems established a posteriori in cases where action implies violence and the a priori proof structure of the dynamics. What is both problematic and fruitful about this strategy for explaining phenomena is weaving together the network of inferences by including more empirical or more metaphysical arguments within the same rational structure, that is, concepts that are more or less abstract in relation to the facts of experience and the related phenomenal order: In fact, in this whole business, the account is balanced and everything adds up so that, whether you measure free actions more metaphysically or violent actions more physically, no objection (of which I am now aware) could be raised to which I could not guarantee a definite answer. (A II 3 625; GP II 205; Leibniz 2013, 155)

Obviously, the epistemological structure of the theory does not entail any clear distinction between a priori reasoning and a posteriori reasoning. But the fact remains

24

Cf. A II 3 624; GP II 204; Leibniz 2013, 151–53: “But I do not agree with what you propose if by the productivity of the action you mean what we meant above, namely, the power. And to add here what has been said before, I agree if by the productivity of the action you mean the intension that comprises quantity of action together with extension over distance, i. e., if you mean speed. But I do not agree if by the productivity of the action you mean the intension that we assumed initially that comprises quantity of action together with extension through time, i. e., if you mean power.”

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that the dominant framework of this theory is a complex process involving various combinatorial analytic models. The incompleteness of every analytic model would not seem to be a critical obstacle when it comes to establishing the demonstrative validity of the theory. This essentially owes to the complementary nature of the relations that it expresses, and to their functional organization as a whole. From this perspective, one must trace the emergence of new analytical developments capable of demonstrating the richness of the theory. After De Volder embraces Leibniz’s analysis of the distinction between power and speed,25 Leibniz once more sets out to clarify the meaning of his a priori demonstration. De Volder’s conversion had above all been motivated by arguments relating to the theorem for composing and breaking down oblique motions. We find for example this type of argument advanced by Leibniz in his correspondence with Papin, but also by Johann Bernoulli in his correspondence with Leibniz. It entails, in Leibniz’s view, imperfectly a priori arguments compared with the demonstration of the theorem of motive action, since the former invoke auxiliary hypotheses, just as the a posteriori arguments underpinned the theorem of conservation of living force. However, these various arguments are brought together under a single general hypothesis capable of covering a considerable array of increasingly complicated cases.26 By contrast, the theorem of conservation of motive action saves the greatest number of supporting hypotheses, since one contents oneself there with a combination of principles that respond both to the norms of mathematical reasoning and “metaphysical” reasoning. From an epistemological point of view, the proof on which the above analytic model is based owes to its “conforming the most with reason”, that is, best satisfying our desire to depict the rational creation of phenomena. Leibniz’s intention here is to emphasize: That the principles of nature are no less metaphysical than mathematical, or rather, that the causes of things lie hidden in a certain metaphysical mathematics, which measures perfections, i. e., degrees of reality. (A II 3 656; GP II 213; Leibniz 2013, 181)

The conservation of motive action in the system of nature for a given amount of time figures among the epistemologically adequate representations that meet this requirement for a metaphysical mathesis.

25 26

Cf. Letter from De Volder to Leibniz, 5 April 1700, A II 3 630–31; GP II 207–208; Leibniz 2013, 163–65. Cf. Letter to De Volder, 6 September 1700, A II 3 655; GP II 212; Leibniz 2013, 179: “And I would certainly dare to contend that nothing other than our hypothesis (which is the one true hypothesis) will satisfy the various more or less complicated cases that I have tried out in the past in many ways, involving oblique collisions and other things.”

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3. Justifying the a priori Way for Papin, Bayle, Jacob Bernoulli, Wolff, and Hermann Ultimately, as Leibniz had declared to De Volder, the principles governing all mechanical phenomena in nature belong to mathematics and metaphysics together: the hypothetical-deductive structure of the explanatory reasons presented gets expressed by means of analytic models of a geometrical sort, but in accordance with architectonic principles that connect them with the conception of a harmonic system of causes and effects. Some other correspondences, in particular those with Denis Papin, Jacob Bernoulli and Jacob Hermann, provide relevant illustrations of this presumed a priori demonstration way, This orientation of Leibniz’s analytic strategy has been noted by Alberto Ranea in his study of the controversy with Denis Papin.27 The latter, as he points out, went through two phases. The first was based on the discussion of the theorem of conservation of living force and developed from 1689 to 1691 through a series of articles. These are the texts that interested commentators, most of all when they sought to identify the use of similar arguments with those advanced by Catelan and the Cartesians, to which Leibniz seems to have formulated a significant response.28 From 1692 up until 1700, the correspondence was direct and focused on the argument for motive action. Ranea brings these texts to light and offers us a very interesting analysis of the Leibnizian concept of quantity of action. Moreover, these texts prove all the more revealing since Papin, as an interlocutor, was infinitely more contentious and eager to prevail over Leibniz than Johann Bernoulli or even De Volder were. If the latter ultimately rejects the Leibnizian system, this seems to be for reasons concerning the metaphysics of the monad, rather than physics as such. With Papin, the polemic, which is very heated, concentrates on the physical world to the exclusion of any ontological underpinning. Moreover, from this point of view, Leibniz will prove himself reluctant to reveal the heart of his theory of dynamics to his adversary – something he plainly communicates to Johann Bernoulli. He will therefore complain about Papin’s rigid obstinacy and his strategic retractions, which he sees as motivated by an unwillingness to cede ground at any costs. Having explained what the controversy analyzed by Ranea tells us about the a priori demonstration, we shall turn to the remaining correspondences, such as those with Bayle, Jacob Bernoulli, Wolff and Hermann. Can these exchanges verify the assertions found in the correspondence with Johann Bernoulli, De Volder and Papin? And can our interpretative framework, as we have conceived of it, ultimately make sense of the project to abstractly demonstrate the conservation of quantity of action, as this pro-

27 28

Cf. Ranea (1989), 42–68. Cf. Stammel (1982), 168–80.

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ject had appeared from the outset in 1689–1690, in the Phoranomus and Dynamica de potentia? We can no doubt ignore the first part of the controversy between Papin and Leibniz: from the moment that Leibniz adopted a style of demonstration that invoked Galileo’s law of falling bodies, this part revolved around the role of time in calculating living force. The aporia caused by the relation to time in Galileo’s theory of falling bodies leads Leibniz to assert that he can overcome the difficulty by establishing the truth of his calculation of force through abstracting from physical factors such as gravity and the elasticity of bodies. (Letter to Papin, March 1696, A III 6 699) But Leibniz generally adopts a prudent strategy of sharing his a priori argument as a basic principle only with those who already accept the new theory of living forces.29 Thus, Leibniz first presents his interlocutor with a simple request by asking him to accept a proposition in which the minor premise of the syllogism is already present, but without advancing the technical concept of motive action.30 Papin will concede that one can then probably speak of action, but that this only refers to the fact of persisting in the same state: any exertion of force presupposes overcoming resistance, and, in the present case, the motion unfolds uniformly without either losing or gaining force, thus making it impossible to measure force and even to suppose that some degree of force is exercised. Leibniz finds himself obliged to respond after Papin invents a troubling Gedankenexperiment in his letter on 27 December 1697. Suppose that in a void there are freely moving bodies whose state can only be modified by some impact. Two bodies with, respectively, masses 1 and 2 and speeds 2 and 1 collide. They possess an equal quantity of motion and thus stop one another. One can infer from this that the force exerted by each is measured by the quantity of motion. Suppose that, in another region of space, there is a fine matter formed from imperceptible particles. For no apparent reason, certain bodies encountering this fine matter are reflected, and the effect apparently conforms to a different measure of force, mv2. The choice is between two modes of analysis: according to the first, one must admit that two laws are involved; the other solution consists in extending the application of the principle established for the first case to the second one by supposing that the imperceptible particles of some fine material are at play. Unless Leibniz can establish that, in the case of free uniform motion, it is the quantity mv2 that gets expressed, the problem of coherence compels him to accept the Cartesian principle of conservation in all cases.

29 30

Cf. Letter to Papin, 16 January 1698, A III 7 724: “[…] with those in whom my other reasons have found favor.” Cf. Letter to Papin, 20 August 1696, A III 7 96: “Here is one: traversing one league in one hour is like accomplishing double (triple, etc.) with respect to what the same moving body would accomplish in traversing one league in two (three, etc.) hours. I assume that each motion is uniform, and that the space as well as the moving body is the same, so that the difference is only in the length of time. I am also thinking of a body without weight or impediments (empêchements).”

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Leibniz reveals his a priori argument in two stages. First, he presents the demonstration founded on oblique impact, which invokes the principle of composition of motions. It is interesting to note that this proof is presented by Leibniz and accepted by Papin as a priori only insofar as it ignores gravity. On other occasions, such as the correspondence with De Volder (and Bernoulli), Leibniz considers it insufficiently a priori in comparison to the syllogistic demonstration involved in calculating the quantity of action. Apriorism indeed seems to be a relative feature of arguments. With ­Papin having been seemingly persuaded by the “kinetic” argument, Leibniz sends him a more radical proof: Take the uniform motions of one and the same body where the times are represented by t, speeds by v, spaces by s, actions by a, powers by p. We shall have: (1) s is proportional to tv, or the spaces traversed are in a compound ratio composed of the times expended and the speeds. (2) a is proportional to sv, or the actions are in a compound ratio composed of the spaces traversed and the speeds at which they are traversed. (3) Therefore, (if we take (2) and substitute according to (1) tv for s) a is proportional to tvv. Or rather: the actions are in a compound ratio composed of the simple ratio of the times and the duplicate ratio of the speeds. (Letter to Papin, 14/24 April 1698, A III 7 756–57)

This form of argument will frequently be relied upon in later exchanges. It has the merit of articulating certain combinatorial factors at play without actually supplying real definitions – as opposed to mere nominal definitions. Papin objects to the fact that the second premise implies that force does not depend on the resistance of the medium. Leibniz then suggests a new syllogistic version that mirrors the classical versions found in the Dynamica de potentia as well as in the De Volder correspondence. However, what interests us in this text is the preamble to the proof: Perhaps this way (assuming [the principle], or ex hypothesi), will please you as much as the other manner owing to its simplicity, for it comes more easily to all sorts of persons than the method of combining reasons. (Letter to Papin, 8 August 1698, A III 7 863)

In this respect, there is a certain ambiguity in the reading proposed by Ranea, who seems to argue that the proof formulated here is more profound from Leibniz’s perspective, and that Leibniz did not previously formulate it in such a direct manner, for example in August 1696, because he had lacked the proof for the minor premise. Ranea cites a passage from the correspondence with Johann Bernoulli to support this claim.31 But Ranea has surely gone astray, for the new proof is clearly an inferior version of the one that Leibniz had already presented in a combinatorial form. Achieving demon31

Cf. Letter to Johann Bernoulli, 16 June 1696, A III 6 797; GM III-1 286: “I have not yet found a way of demonstrating this proposition a priori via congruence; nor even this one, that an action achieving the same in less time is greater, from which one should start.”

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strative superiority would have required a combination of arguments that conformed to a model that went beyond the geometric order, and drew from the similarity and difference of forms symbolized by algebraic parameters. From this point of view, the argument of April 1698 was clearly superior in quality. One must understand the epistemological conclusion of July 1698 as a retreat from the deeper Leibnizian position rather than as an affirmation of it: However, even if this ever simple and reasonable axiom or principle, which states that actions are in a compound ratio comprising the spaces and speeds, amounts only to a hypothesis, its great success and ability to reconcile everything despite protestations, and some studies by which we submit it to experience and reason, ought to legitimize it. And if you include its agreement with this other demonstration regarding the composition of motions, then Sir, I can hardly believe that, having remarked all this, you might hide your shock and refuse to recognize it with your usual sincerity. (Letter to Papin, 8 August 1698, A III 7 865)

However, Ranea is right to point out that the foundation of the a priori argument lies in a combination of reasons: “We can conclude that the several numerical versions of the a priori proof are merely different ‘interpretations’ of the true formal or rational proof based on a combination of reasons.”32 In our opinion, Ranea advances an essentially nominal version of this combination of reasons, and from this perspective, his analysis hardly differs from Gueroult’s. For him, Leibniz’s definition of action as the product of space and speed is comparatively more suitable than any other definition that one might think of; by suitable, he means that it synthesizes the greatest number of relations without inconsistencies, and that it agrees with the empirical inferences. Nonetheless, Ranea points us in the direction of an interesting theory, since he affirms that Leibniz bases an “order of justification” on an “order of discovery”. By this he means that Leibniz interprets the syllogistic argument as a mode of expression capable of deriving its source exclusively from the axiom of motive action, which is justified methodologically by its ability to describe the essential structure of phenomena independently of adventitious circumstances. The letter to Papin on 8 August 1698 seems to express this: Reflecting on my axiom, one clearly sees that it is reasonable. For, by setting aside every external factor, is it not true that every relevant effect, impediment or resistance, is directly and immediately determined by space and speed? (A III 7 864)

Given the circumstances, the most recent syllogism, whose major premise quantifies action in terms of space, and whose minor premise in terms of speed, seems to him the most advanced since it distinguishes between the two elements of action once the axiom is pragmatically justified. Following this strategy of combining nominal argu-

32

Ranea (1989), 55.

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ments, Ranea is surprised that Leibniz returns to invoking the concepts of extensio and intensio to analyze the two modal elements. As we have noted, there is nothing accidental about this approach, which is at the very heart of the exchange with De Volder and, in our view, represents a significant step in the method of using analytic models to describe the causal aspect of force, in contrast to a properly kinetic approach à la Descartes. Ranea sees in this the reemergence of the Calculatores’ physics of the fourteenth century. It would seem as though Leibniz substantialized, or ascribed an ontological dimension to, the quantitative factors that only have objective meaning when considered as extensive modes.33 At the same time however, Ranea could not stop there. Indeed, one could not possibly admit that Leibniz’s recourse to the dimension of intensity lacked grounding. For, at the very least, this factor has its own reality, one that is irreducible to any extensive modality given that the combination of the factors space and time, which presupposes the intensive dimension, is confirmed by experience. We also believe that this is an analytic model designed to account for the causal functioning underpinning the phenomenal order that the kinetic factors fail to represent a priori. From this point of view, Ranea identifies a metaphysical fiction, or at least an indicator of a transphenomenal entity, where one should instead see the need for a causal system that is adequately expressed by phenomena. Take for example Leibniz’s assertion here to Papin on 24 June 1699: By the way, I shall only say that the quicker motion is also intrinsically and essentially the more perfect, since it does not fail to act more on what externally and accidentally offers resistance, whatever it might be. (A III 8 166)

This is an illustration of a moving body to which the quantity of action, as the intrinsic and undissipated power to act, can be formally assigned. If we suppose that a free uniform motion on a horizontal plane is transformed into vertical motion, this power to act can be measured by the highest elevation it can attain given the resistance of gravity; but it is still necessary that it be readily present in some virtual sense. The combination of reasons is therefore intended to represent a theoretical entity endowed with causal capacities and not simply an amalgamation of kinetic factors. In the end, the approach that Ranea takes leads him to resurrect a certain understanding of a priori argumentation that allows one to overcome the seemingly radical 33

Cf. Ranea (1989), 57: “Leibniz’s inconsequent stratagem is of great importance, however. It 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 [extension and intension] 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 motion. 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 the Fourteenth Century’s Physics when he states that a faster motion is essentially more perfect than a slower one.”

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dichotomy between a strictly formal demonstration relying on the logical evidence of premises and a system of empirically grounded inferences. Ranea rightly identifies Leibniz’s preoccupation with abstracting his theory from accidental empirical conditions. He argues, like Leibniz himself admits, that this project can only be achieved by deriving mechanical properties revealed by experience from principles, that is, by means of concepts representing efficient causes. Leibniz therefore thinks that he can convince Papin: “Thus you claim that it would be difficult for you to […] accept [these arguments] until I prove my hypothesis, not only by final causes, but still a priori by efficient ones.” (Letter to Papin 2/12 December 1697, A III 7 660) He therefore presents his approach to calculating force in terms of the quantity of motive action as: more profound and a priori, as everything should be calculated in its source; and the source of the power capable of producing actions of the second type [i. e. violent effects, in which the force is measured from exhaustion] is the faculty of producing formal actions of the first type [i. e. free actions expressed through uniform motions without exhaustion of force]. (Letter to Papin, 28 August/7 September 1698, A III 7 891)

But if this analysis largely clarifies the particular meaning of a priori argumentation in the dynamics, it remains defective because it fails to grasp how the principles function in an analytical construct aimed at symbolizing the order of efficient causes behind the phenomena of free action. Furthermore, Leibniz can only realize this project by making use of architectonic principles that guide the combination of reasons. These must signify formal properties capable of generating both the intensive and extensive properties of mechanical phenomena and ensuring that the latter can be explained by the former. Lastly, a methodological requirement is imposed upon this process: the analysis must conform to the system of reasons governing the calculation of living force, which finds itself guaranteed by a system of a posteriori inferences, equally governed by being subordinated to architectonic principles. At this stage, we should turn to the final formulations of the a priori argument to verify the interpretation that our analyses of the correspondences with Johann Bernouilli and De Volder have already suggested. Directed at Bayle, Leibniz’s remarks are cursory and in line with the syllogistic demonstration. The most significant aspect of dynamics thus seems to be the convergence of demonstrative arguments. The case of horizontal displacements in which force is not consumed through its effects is raised from the outset as the initial framework on which Descartes had based his principle of conservation of motive force. Understanding the transformation of such motion into violent effects, for example, when a body is elevated against the force of gravity or when there is a percussive impact, allowed one to reject the erroneous Cartesian attempt to relate the principle of conservation of force to the measure of the quantity of motion. However, by reconsidering unconstrained uniform motions and establishing the formula for motive action by means of axioms, it is possible to verify the conformity of the general law in relation

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to the theorem of living force. This abstract model, reserved for those who understand this sort of approach, allows one to establish that “action is nothing other than the exercise of force, and [that it] is reducible to the produce of force and time”. (Letter to Bayle, 6/16 April 1699, A II 3 556; GP III 60) This framework underpins the combinatorial agreement of the terms in virtue of which one represents motive force, whether one begins with the effect that exhausts it or the neutralized effect wherein it is continuously reproduced. The exchange with Jacob Bernoulli on the principles of dynamics is also rather brief; but its orientation is entirely different since, from the outset, the discussion focuses on the theory’s so to speak epistemological meaning. The elder Bernoulli brother confirms his commitment to the principle of living force, but by reducing its scope to the systematization of empirical laws of impact, such as those established by Wallis, Huygens and Mariotte. The principle of the conservation of quantity of motion is empirically invalidated. For elastic bodies, the conservation of living forces applies. One must admit that the power conserved in mechanical exchanges therefore requires force to be understood as being essential to bodies.34 We will see all of Bernoulli’s attention focused on the problem of how the center of gravity of every mechanical system is maintained. By virtue of this principle, he could easily overcome Papin’s objections, whom he considered a shoddy mathematician. However, he is far from being prepared to engage with Leibniz’s properly theoretical concerns. Leibniz hopes to rectify the interpretation that reduces his dynamics to the empirical laws of impact, as if dynamics were a mere nominal synthesis. Hence his epistemological declaration: “I have neither discovered nor deduced [my dynamics] from phenomena like those that Mariotte and others have produced, but from the causes themselves, with which the phenomena must subsequently agree.” (Letter to Jacob Bernoulli, April 1703, GM III-1 68) A profound remark if one takes into account the determinative role played by the principles of equivalence and continuity in the initial structure of dynamics. After all, it was always possible to adhere to empirical laws of impact by following an approach like Huygens’, refusing to base one’s analysis of phenomena on a higher-level theory that implied a law of conservation with an essential and causal meaning. Leibniz’s strategy regarding Jacob Bernoulli consists in presenting him the formal framework, or the system of equations that expresses the combinatory of the theoretical concepts. He begins with living force by presenting the system of three equations – linear, planar and solid – from the Essay de dynamique. Thus, one notes the correlation between the laws of conservation of relative speed and quantity of progress, and the law of absolute conservation of living force. Leibniz insists on the combinatorial relationship of the three laws and on the fact that they apply universally to all bodies, since

34

Cf. Letter from Jacob Bernoulli to Leibniz, 15 November 1702, GM III-1 64.

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they primarily concern the impacts of elastic bodies, and since all other cases can be related to these if one considers the intrinsic elasticity of bodies that, empirically, seem to be endowed with relative inelasticity. However, to complete the theory, one must conceive of a model that gives us access to a more fundamental dimension of the system. This will appear under the guise of definitions abstracted from the specific conditions of gravity and elasticity. This approach can be regarded as purely a priori insofar as it consists of analytic constructions guaranteed by principles.35 The definitions must be introduced in a way that allows for a system of deductions that express the intelligible causal order. This yields, via a combination of arguments, what is now considered the classical demonstration. The terms are space traversed, time elapsed, speed, body, power, effect and action. Among the series of equations, one recognizes the fundamental proposition according to which action is the product of the effect and speed; the formal effect itself, disregarding all constraint and exhaustion, is measured as the product of the body and the space traversed. However, the latter relation is the one that ensures that the others are discursively integrated: action is the product of power and time. Leibniz thus explains the meaning of this term: “Power is made intelligible by its fruits, that is, by action, if one maintains that action is the exercise of power, that is, results from the replication of power, or its multiplication by time.” (GM III-1 70) There is no doubt that if one interprets this meaning as a priori, it can only be in a mathematical sense. The apriorism it involves seems only to derive from the analytic projection of a representation of the cause behind the unconstrained, uniform displacement. The conclusion to be drawn from the inference does not at first seem to capture this analytic path, which is comprised of expressions that conform to principles. One instead seems to be dealing with an algebraic solution: Hence our calculation of power is thus established: tp such as a, but a such as ev, and ev such as cs, therefore tp such as csv. But s as vt; therefore tp as cvvt or p as cvv, i. e., powers are in compound ratio of bodies once and speeds twice, which had to be demonstrated. (GM III-1 70)

Evidently, the combinatorial usage of the terms continues to depend on the meanings of the concepts that they represent, and the possibility of integrating the latter into the same analytic process.

35

GM III-1 69: “However, in my opinion, this doctrine was in need of something more sublime and profound, namely, that our calculation of power should be conducted entirely a priori, without concern for observations about heavy or elastic bodies, from definitions of power, effect and action alone, while imagining at the same time the body in uniform motion, all things considered formally and in themselves according to the essence of motion and all violence and accidental factors having been removed in every body moved. This is what I successfully accomplished and do not easily communicate except to those who know how to appreciate principles.”

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Jacob Bernoulli is, in the end, hardly sympathetic to this type of argument, which seems to him fraught with inappropriate metaphysical references that neither calculus nor experience can justify. He particularly takes issue with the overdetermination of key concepts: “I do not dare to interject my judgement, since metaphysical notions of action, effect, etc., in a mathematical discourse lack enough evidence for me.”36 To illustrate this, he confronts Leibniz with the fact that action is equal to the product of the effect and speed, and that the effect itself would already be measured as the product of the body and space, or similarly as the product of the body, time and speed. Replicating the factor of speed places the terms of the equation on a higher level, but fails to justify doing so with relevant mathematical evidence. Leibniz’s response successfully clarifies the combination of factors at work here. Take the notion of effect. In the concept of uniform motion, an additional connotation is introduced concerning the rate at which the effect is accomplished. Consider then a combination of three factors determining the notion of motive action. This combination is sufficient to account for the dynamical properties that underpin the body’s displacement, but at the same time, it is somewhat inescapable insofar as one cannot conceive of an additional factor.37 For Leibniz, the meaning of these concepts goes beyond the limits of mathematical evidence, but can one thereby deny them every fundamental characteristic of intelligibility? “Their evidence is no less metaphysical than mathematical if one treats them adequately (recte).”38 This last condition is significant. From the concepts at play, one derives a theorem that experience confirms, as well as hypothetical inferences resulting from taking into account elasticity and gravity. However, more fundamentally, this involves a so to speak sufficient and quasi-necessary expression of the causal process when force is generated constantly rather than being used up in a so-called accidental effect. The combination of conceptual reasons shapes the whole system of determinations, which corresponds to this essential nature of force. Lacking what is required of an essential analytical manifestion of force, these “accidental” variations would be without grounding. What characterizes the analytic procedure in this case, connecting it to the mathematical analysis and its requirements of exactitude and efficacy?39 The response seems to lie in the possibility of constructing an abstract model that responds to the data of the problem to be solved; in the case of dynamics, this problem concerns the requisites of the conservation and generation of force underpinning violent and non-violent effects. The generality of the principles of dynamics permeates the construction of this integrative model.

36 37 38 39

Letter from Jacob Bernoulli to Leibniz, 7 October 1703, GM III-1 76. Cf. Letter to Jacob Bernoulli, 3 December 1703, GM III-1 83: “[…] I call this the action itself, for in it nothing else has to be taken account of, which might contribute to the calculation.” Ibid., GM III-1 83. Cf. ibid., GM III-1 83: “I am personally satisfied with this demonstration, for in these rather metaphysical matters, scarcely has something been achieved with the rigor and success of mathematics.”

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Can examining later exchanges with partisans of Leibniz’s science help support such an interpretation? The discussion between Leibniz and Christian Wolff on dynamics focusses on the model of a science that develops beyond the limits of geometrical patterns. Wolff concludes that Leibniz claims to go beyond mathematics by conceiving of a more powerful and efficacious science wherein reasons would be intelligible in themselves, a science that abstracts from the facts of experience and constructions of the imagination. He links this program with the notion that, even though everything is generated mechanically in nature, “the principles of mechanism derive from a higher principle by means of final reasons (per rationes finales)”.40 Leibniz will, to a certain extent, endorse this view that relates the foundations of physics to those of a science of formal reasons. He insists that combinatory allows one to create a theory of forms founded on relations of similitude, beyond quantitative congruencies. Algebra itself, he argues, is subordinated to the requirements of this science of forms when it relies on combinatorial formulas. In the same breath, he proposes to illustrate mechanics’ dependence on higher-level principles, i. e., the axiom of equivalence between the full cause and the entire effect. This axiom is labeled as “metaphysical” (Letter to Wolff, Briefwechsel, 129): in the spirit of the text, this means that it possesses an intrinsic rational certainty, independent of all empirical inferences, even though it proves indispensable in accounting for the phenomenal world. One cannot however defend the idea that this axiom, which governs the construction of a physical theory, serves as a basic foundation: ontologically speaking, it makes sense to look for the reason behind these principles in the design of divine wisdom.

Fig. 16 Illustration of Christian Wolff ’s letter to Leibniz on 31 December 1710, and Leibniz’s response (Briefwechsel, fig., 11).

Along these lines, Wolff proposes directly deducing the theorem of living forces from the axiom of equivalence.41 In so doing, he violates the purely regulatory role that this axiom played for the a posteriori inference that classically led to this theorem. Suppose that there are two equal bodies A and B, and that the latter’s speed is twice the former’s. In the same span of time, A traverses CD and B traverses EF, which is twice the distance of CD. Suppose that EG is half of EF and EL half of this. Each impetus of B will be twice that of A. Across EL, which equals ½ CD, B will generate as much motion as A across

40 41

Letter from Wolff to Leibniz, 8 November 1710, Briefwechsel, 128. Cf. Letter from Wolff to Leibniz, 31 December 1710, ibid., 130.

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CD. Hence the conclusion that B’s force is quadruple that of A. Leibniz responds that the conclusion would hold for the quantity of action and not motion as such, which is calculated by the degree of speed and therefore in function of time. In fact, Wolff employs a strategy that would favor those who take quantity of motion as the measure of force. To show Wolff his error, Leibniz gives the following example: suppose a line TP representing time with the moment M on it. The impetus of the two bodies, measured by their respective speeds, are represented by the perpendiculars at M, with MB being double of MA. All the impetus of A across time TP are represented by the rectangle TQ, while those of B by the rectangle TR, which is twice as big as the first. Let us return to Wolff ’s figure. When EL is traversed, the sum of the impetus corresponds to the rectangle TN, which is not equivalent to the rectangle TQ, but rather to half of it. If Wolff had replied that applying impetus to time does not work, and that impetus can only be applied to the distance traversed, the conclusion might be different. But Wolff must then respond to the argument that we have already encountered in Papin and Jacob Bernoulli, according to which there is no formal reason for multiplying the space traversed by speed, since the space traversed is already represented by vt. It is not evident, Leibniz adds, that the product of these factors (vt) follows from the axiom of equivalence itself, since this axiom concerns the entire effect, to which the speed of accomplishment and therefore the impetus as understood here do not seem pertinent. At this stage, Leibniz introduces his theorem of motive action, the demonstration of which he presents syllogistically, and then via the combinatorial calculus of factors. These demonstrations do not reveal anything that our analysis has not already explored. We wish only to highlight that Leibniz rejects Wolff ’s paralogism, which was based on calculating force by the product of speed and time. No adequate model of the causal reason behind the pure motive effect can be found here. Things are different when determining the effect of uniform and unconstrained translation via impetus or degree of speed, because one is relying a combinatorial representation of the conditions for the continuous production and reproduction of force in terms of time. An abstract model of the causal order at the level of forces inherent and specific to the displaced bodies is thus given. Leibniz confirms the combinatorial nature of the proof of the theory by pointing out that one can derive increasingly complex theories from the model when examining non-uniform motions: it suffices to formulate these theoretical propositions on the basis of calculations that combine factors in virtue of the most analytically complete and fruitful basic model.42 Is this model established a priori? Leibniz himself refers to it as the product of an abstract combinatory, which is justified as an intelligible model of the causal order that 42

Cf. Letter to Wolff, ibid., 133: “In non-uniform motion, this thing also occurs. However, the factors that form the compound ratios must be respectively analyzed in their succession, and the summing up of what is found in these elementary components provides the whole evaluation. And in these elements is contained the part of my dynamics that is most abstracted from sensible things.”

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presumably underpins mechanical phenomena. In his Principia dynamica (1728), Wolff will not refrain from representing Leibniz’s science as being deduced from principles.43 In so doing, he will undoubtedly reinforce a mode of analytic deduction that Leibniz conceived of by combinatorial means instead. The exchange with Jacob Hermann, just after the criticism of Wolff ’s model, interestingly confirms the role assigned to this combinatorial conception of abstract factors in the theory. It is while working on hydrodynamics that Hermann reexamines the principles of Leibniz’s theory: his goal is to combine the theorem of living forces with the lemmas of statics and derive therefrom principles sufficient for establishing theorems relating to hydrodynamics; in particular, this involves extending the scope of Leibniz’s theorem to hypotheses for measuring gravity, as well as to a general theory of centers of oscillation.44 Contesting Leibniz’s science is not therefore the goal here, but rather integrating, in a deeper sense, different theoretical fields under the guise of Leibnizian principles. Hermann revisits Johann Bernoulli’s model for demonstrating the theory of living forces via the composition of motions. He does not interpret it as an a priori proof, but rather as an elegant a posteriori procedure,45 which more directly expresses the calculation of forces via mv2. Thus, he himself proposes such a direct expression, which he arrives at by means of a mechanical theorem. This theorem holds that the areas under the curves, which represent the solicitations, are proportional to the squares of the speed ordinates of the figure, these being produced by the succession of solicitations. This leads to conceiving of force (vis) as the effect of integrating the speeds into the mass via the series of speed differentials in the uniformly accelerated motion: force would then be calculated by multiplying mass, which is presumably constant, by speed multiplied by the speed differentials (vdv). Take the following figure.

Fig. 17 Illustration of the demonstration that Jacob Hermann proposed to Leibniz in his letter on 1 June 1712 (GM IV, fig. 65).

43 44 45

Cf. Wolff (1728). Cf. Letter from Hermann to Leibniz, 2 June 1711, GM IV 366–67. Cf. Letter from Hermann to Leibniz, 1 June 1712, GM IV 368.

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The ordinates EF and GK express the speeds of the moving body A being displaced toward Q at points E and G. The ordinates A2A, B2B, C2C, etc., represent the elementary solicitations. Force (nisus) would be measured successively by the areas A2A2EE, A2A2GG. However, as Newton demonstrated, these areas are proportional to the squares of the speed ordinates. Hence the search for a direct expression of Leibniz’s theorem of living forces. Leibniz is not opposed to this expression, so long as it relates to the already established definition of living force. In effect, one would first have to establish that the force differential is proportional to the product of the displacement differential and the speed differential, while assuming that the time differential remains constant. Failing this, Hermann’s symbolism would be useless. How can one provide a priori justification for this model? Would Hermann have a response to this problem? Leibniz asserts that he had recently provided Johann Bernoulli with the “metaphysical principles” that can be regarded as “the true sources of this doctrine”. (Letter to Hermann, GM IV 372) The abstract model that Hermann then constructs proves unsatisfying insofar as it implies a petitio principii: it assumes that the basic solicitation, measurable by the quotient of the speed differential and the time differential, is a product of the factors multiplied by the space differential. But this is precisely what should be established from the beginning. Leibniz’s solution consists in presenting metaphysical arguments that would allow one to overcome this aporia. Leibniz initially returns to the so-called a posteriori approach and suggests that the empirical elements of the theory of gravity would only serve to corroborate a model based on abstract concepts; the latter would be justified by the order that they project onto the expression of phenomena. Among the basic elements of this theoretical model, one must no doubt include principles such as the equivalence of cause and effect and continuity, which frame such a regulated expression.46 Nevertheless, the model found in the public version seems fragile because it draws upon the supporting thesis of gravity, and because measuring force by its violent effect requires accounting for accelerated and decelerated motions. But to analyze such motions, one must rely on mathematical models comprised of fictitious limits, or infinitesimals. Hence suspicions that some trickery or ad hoc hypothesis is involved, which in turn haunt the theorem of living forces. To overcome this aporia, Leibniz introduces the elements of his demonstration of the theorem of motive action in such a way as to highlight the interplay of conceptual reasons that come together to account for motive action. Leibniz does not seem to be preoccupied with formally verifying if this model corresponds to a strategy of adopting either a nominal or real definition. The linchpin of the formulation of reasons is

46

Cf. Letter to Hermann, 9 September 1712, GM IV 378: “Although the physical hypothesis of gravity and experiments seem to be evoked here, in reality, experiments serve only for confirmation, for this demonstration proceeds from a hypothesis itself mentally abstracted from experiments.”

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certainly provided by calculating action in terms of the product of the pure (or nonviolent) effect and the speed at which it is accomplished. If there is metaphysical justification for this “theoretical” hypothesis, it consists in the link established between the requirement of force and the notion of action. Leibniz affirms: The notion of power is such that power multiplied by the time during which it is exerted produces action, that is, the temporal exertion of power is action, for power cannot be known except from action. (Letter to Hermann, 9 September 1712, GM IV 379)

The meaning of this proposition can be explained as follows: the causal agent required to account for mechanical phenomena must be conceived of; this role is satisfied by the concept of force; the stable state of force, where force is maintained without being intrinsically modified, must be captured by an abstract model. Only the combination of concepts capable of expressing this state and the resulting effects is derived from the conception of action, understood as the product of the formal effect and the speed of accomplishment. This is an abstract analytic model that meets the requirements for a model of force that captures unconstrained uniform motions. With respect to this analytic model, Hermann’s objections are precisely combinatorial in nature. In effect, he easily notices that the demonstration of the theorem of motive action depends on three “analogies” combined together.47 He sees no problem in admitting the first: that that effect is expressed by the product of the body and the distance traversed (“quod sit e ut cl”). The third seems equally acceptable to him: that action be expressed in terms of the product of power and time (“quod a ut pt”). It is the pivotal proposition that seems to entail an unnecessary redundancy: that action be expressed as the product of distance and speed (“quod a ut ev”).48 We have already encountered a similar objection in Papin and Jacob Bernoulli, but with the subtle distinction that Hermann, from the outset, adopts the demonstrative procedure involving a combination of factors. He happily uses it himself to defend his own model against the Leibnizian criticism of petitio principii. The major difference between this ingenious construction and Leibniz’s owes to the fact that, according to Hermann, the basic combinatorial relation is the result of multiplying the elementary solicitations by the elements of space.49 It is remarkable that Hermann relies, like

47 48 49

Cf. Letter from Hermann to Leibniz, 22 December 1712, GM IV 385. Ibid., GM IV 385. Ibid., GM IV 384: “The acting cause that inserts into a moving body m, during time dt, an element of speed dc, will give the ratio mdc : dt. However, the power (dP) relevant to the element of speed (dc) will be equivalent to the acting cause and the extension joined together. For me, the extension of action is the space or extension in which the cause ceaselessly and continuously acts, i. e., in whose singular points it operates. Thus, dP gives mdcdl : dt; hence, since dl : dt is as the speed (c), it follows that dP is equivalent to mcdc, and P to mcc. I had presumed that by this reasoning I had obtained an a priori demonstration.”

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Leibniz, on such an abstractly constructed model to satisfy the requirement of causally representing the force behind unconstrained, uniform motions. From a pragmatic perspective, the expression that he chooses better satisfies the theory of momenta of speed in all accelerated and decelerated motion – in fact, it derives from such a theory. Furthermore, it has the merit of overcoming the shortcomings of a model that would understand degrees of speed as real properties. Relying on infinitesimals guards the internal coherency of the model against the contradictions of any scalar conception of these degrees. But at the same time, Hermann falls somewhat victim to his ability to circumvent the problem. Reliance on the artifices of infinitesimal calculus renders his combinatory purely nominal, depriving it of the metaphysical significance that Leibniz hopes to attach to his analytical causal model. Leibniz responds by showing that the concept of solicitation merely yields another expression for instantaneous acceleration. Consequently, with regard to the elements, one reverts to the cliché of equating the space differential and the product of the speed differential and the time differential. Even if he approves of the idea of symbolizing the causal agent underpinning uniform motions by some term, Leibniz has his concerns about the proposed model, which invokes a concept of solicitation that is both conceived of within and beyond power because, on the one hand, it would be measured by the speed differential, and on the other hand, it would extensively generate a summation of vdv: such a concept, symbolized by the letter S is not analytically clear and Hermann only succeeds in providing it as an ens rationis. Leibniz’s main critique can be summarized as follows: I do not accept that the acting cause that gives speed dc to a moving body m in duration dt should form the ratio mdc : dt. And I do not see how one can prove this unless one assumes it as a definition. But then I do not understand or see how this notion combined with space constitutes power, and why another might not say with equal reason that the acting cause forms the ratio mdl : dt, or something else. On the other hand, with the simplest Elements like those here, the question is not what the acting cause produces in something else, but what it produces in itself as a cause. (Letter to Hermann, 1 February 1713, GM IV 388)

Under these conditions, It is obvious that Hermann constructed his model by reversing, so to speak, the combinatorial order; hence the aporias that he encounters. The motions he refers to appear to him in the form of accelerations and decelerations regarding the generation of speed. To represent them, he must artificially construct basic infinitesimal quantities; furthermore, in such motions, the factor of time does not determine how the effect is measured. In turn, Leibniz argues that his formulas are justified expressions because they correspond to distinct notions. The latter are such that it is impossible to conceive of ones that are more determinate; intrinsically, they provide the basics of an analytic model that combines reasons in an easy and intelligible way. In this respect, these definitions provide a precise layout of the theoretical entities, one that is sufficient

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for understanding dynamical phenomena in general.50 In effect, Leibniz presents this system of concepts as the sufficient reason of the physical system of force in all those cases where it is perpetually regenerated or exhausted through the violent effect by which it is measured. Regarding the objection that he is redundantly relying on speed to calculate action, Leibniz sets out to dispel this objection by insisting on the analytic distinction between the formal effect and the speed at which an action is accomplished. Indeed, one can conceive of the same effect regardless of whatever the moving body’s translational speed might be, and, on the other hand, this formal effect, which is calculated independently, can be added to a calculation of speed where the parameters have significantly different meanings. Herman will be especially sensitive to this aspect of reciprocal distinction and adequate determination that Leibniz invokes in support of his model. He will recognize it in the remarks accompanying his final acceptance of Leibniz’s arguments.51 These parameters refer to distinct relations that express the combination of “forms” in the idea of the underlying cause of mechanical effects. From this point of view, Leibniz would no doubt happily admit that his analytic model of motive action cannot be conceived of independently of a theory of force whose basic concepts are necessarily overdetermined in relation to every representation limited to empirical facts. This overdetermination owes to an architectonic combinatory according to which explanatory concepts must be interwoven together and satisfy the requirements of order embodied in the architectonic principles. Is the model for motive action not labeled a priori because, in the end, Leibniz cannot conceive of adequate rules for constructing this model a posteriori? The overdetermination of theoretical concepts in relation to what they refer to in experience therefore seems to follow mainly from the architectonic principles that articulate the combination of reasons that these concepts signify. For Leibniz, this level is a “metaphysical” one that is sufficient for establishing theories that explain physical phenomena.

50

51

Cf. ibid. GM IV 388: “The three ratios that I have composed, namely that e is proportional to cl, a to ev, a to pt, are nothing but definitions. That is, according to me, cl defines the effect, ev defines the action, and a : t defines the power, or rather a is proportional to pt. Indeed, I define power (which is best known from its exertion) as that which, through its exertion, is conveyed in time, and thus produces action, by which power acts as much as possible. Once you have reflected enough on those matters, you will probably discover that these notions can be distinguished and accepted easily enough, and that no ratios more precise can be found.” Cf. Letter from Hermann to Leibniz, 2 March 1713, GM IV 390.

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4. Conclusion Debates around Leibniz’s elaboration of an a priori system of proofs for dynamics are developed in and through important correspondences. They influence the way that Leibniz ultimately conceives of the epistemological status of this new science and the justifications he gives in support of his theoretical models. In the exchange with Johann Bernoulli, Leibniz initially relies on indirect arguments relating to the formal characteristic of analysis: the distinction between modal units and real units when breaking vis motrix down into factors; and the problem of perpetual mechanical motion, which the Cartesian principle of conservation of quantity of motion would imply. Once in agreement with the Leibnizian arguments, Bernoulli endeavors to elaborate proofs capable of challenging the Cartesians on their own terrain, i. e., in terms of geometric intelligibility. Once the basic theoretical concepts derived via a posteriori demonstration are admitted, the two correspondents first envision models that invoke geometric properties. Leibniz ends up proposing a syllogistic argument inspired by the Dynamica de potentia. This argument combines the virtual and formal dimensions of action to conclude that forces are proportional to masses and the squares of speeds. In the spirit of Leibniz, a physico-mathesis approach is taken, one that combines the resources for geometrically representing parameters with a theoretical model for their integration. To counter the objection of equivocation between the formal and virtual dimensions of actions, Leibniz introduces metaphors illustrating the causal element that makes it possible to integrate the two dimensions. In fact, within the Cartesian hypothetico-deductive framework, Leibniz assigns both a geometric and (theoretical) causal meaning to the concepts whose analysis is reflected in the premises of the arguments. The apriorism of the approach ultimately belongs to the regulatory role played by the axioms that express the sufficient reason requirement, and uphold the combinatory of factors representing force in motive actions. Beginning in 1696, De Volder gets involved in the discussion on the foundations of dynamics. In the exchange between De Volder and Leibniz, the search for an a priori proof is justified by the need to establish an analytic groundwork through concepts perfectly suited to a theory capable of accounting for the causality behind the kinematic arrangement of phenomena. This is also why De Volder will ultimately coax Leibniz into entering a discussion on the notion of active substance. From the outset, Leibniz shows that the laws of dynamics and those of statics can be reconciled through a synthesis that, relying on the resources of infinitesimal models, represents the generation of vis viva by the summation of basic conatus into impetus; here, the equilibrium signified by the conservation of vis mortua only constitutes a moment of virtual effect of the impetus. The dominant idea in Leibniz’s account is an active principle governing the determination of serial effects; the laws of dynamics would adequately express this principle, something which an analytic study of these laws should bring to light. Leibniz’s goal is above all to advance a hypothesis that can analytically accommodate all the

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reasons required for explaining the diversity of mechanical phenomena. It is this line of reasoning that must be elaborated by a sufficient system of conceptual equivalences and analogical transpositions. The interpretation of the syllogism establishing a priori the principle of the measure of the quantity of action must be understood within this framework. This argument is valid only if one agrees to invoke the modality of time in the free expression of force: the reasons for this concession are found in the causal determinations that presumably underpin the accomplishment of the formal effect. The apriorism essentially seems to be justified by the convergence of analytic models, starting with those that respectively establish the theorem of conservation of motive action and the theorem of conservation of living force. In opposition to the hypothesis proposed by De Volder, whereby the præstantia of action would be constituted by its speed of accomplishment alone, Leibniz establishes that this præstantia can only result from the combination of intensio and extensio: this combination, moreover, also appears when force is calculated in terms of time. The a priori argument seems to affirm the substitutability of formulations of laws when considering the appropriate system of conceptual equivalences; the basic concepts entailed by this combinatorial interplay of equivalences must therefore entail an adequate reference to an underlying causal order capable of describing the entire system of effects, since the latter are transformed into geometric models. Thus, the order of a priori reasons is subjected to the architectonic principles of continuity and the equivalence between the full cause and the entire effect. We set out to verify this interpretation by examining the second phase of the controversy that set Leibniz and Papin against one another, a phase that A. G. Ranea notably reconstructed. The a priori argument must in effect be understood by avoiding the mistake of interpreting it as a strictly formal demonstration in virtue of the logical evidence; indeed, opposing it to every hypothetico-deductive approach would be at odds with reality. The dynamics as a theory appears to be based on an analytic model in which the explanatory concepts ultimately derive their meaning from the architectonic and combinatorial expressions that they confer upon relations of cause and effect. From this perspective, subsequent accounts of the so-called a priori approach – in the correspondence with Bayle, Jacob Bernoulli, Wolff and Hermann – confirm this epistemic structure of the arguments involved. In this respect, the concepts and laws of the theory overdetermine the observed phenomenal connections by translating them into combinatorial relations governed by architectonic principles.

General Conclusion Our analysis of Leibniz’s dynamics responded to a number of concerns. The science advanced by Leibniz is no doubt epistemologically interesting. Is this science not one of the most thoroughgoing examples of the modern science? Does Leibniz not bring to fruition in some way the mechanics elaborated by Galileo, Descartes, Hobbes, Huygens and others? Can this refined mechanics not be compared to Newton’s, his immediate contemporary? Among the possible approaches to reforming mechanics lurks the future of physics as a general science of nature. Regarding how such a reform should be conducted, contrasting visions of scientific methodology surely present themselves. In this respect, Leibniz appeared to be one of the major protagonists of the debate. Indeed, undergoing various changes, the Newtonian methodological paradigm has dominated the sciences until the previous century. But on the one hand, the Leibnizian alternative never lost its effectively underground influence. In addition, contemporary science has numerous affinities with a program that constructed “geometric” models under the direction of regulative principles; and this program sought to derive, from sufficient theoretical reasons, a system that could combine hypothetico-deductive explanations. The causal order underpinning phenomena would be reached through combinatorial analyses of the factors at play, and by relying on the resources for analogically and symbolically transcribing phenomenal data. Our goal was to show how this program characterized one of Leibniz’s most fundamental scientific projects, that is, the dynamics to be found at the heart of the reformed physics. Historical epistemology required that we incorporate both the genesis and structure of Leibniz’s dynamics within the same analysis. The lucid picture of the development of dynamics that we have today allows us to understand the subject in a new light. Henceforth, we can read the texts and decipher the themes of Leibniz’s oeuvre without the distorting effect of positivist or neo-Kantian conceptions of scientific knowledge. From this point of view, it was incumbent upon us to reinterpret the relations between theoretical entities and mathematical models, as well as between these theoretical structures and the analysis of phenomena. If there is a metaphysical level in the system of scientific explanations according to Leibniz, then one must, we believe, distinguish it from the metaphysical doctrine as such and identify it with a group of architectonic

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principles established for performing hypothetical deductions and making analogical inferences capable of yielding theoretical reasons. Moreover, the whole vision of Leibniz’s dynamics considerably evolved since the remarkable synthesis that Martial Gueroult envisioned. Regarding the origins of dynamics, contemporary works, like those of Michel Fichant, require us to review the initial phases of the program’s establishment. With respect to the structure of dynamics, we must take into account the network of contrasting arguments that Leibniz employs throughout his accounts and responses to objections: the challenge was grasping the flexible way in which these various inferential strategies are incorporated. The procedures unveiled by the analysis of the dynamics make it possible to understand this method of invention and demonstration, the main epistemological springboard of Leib­niz’s science. The conception of the theoretical model that crops up in Leibniz’s first scientific forays provides us, from the outset, the key to a number of subsequent analyses, or improvements. Even before 1670, Leibniz, who had adhered to the “mechanism of the Moderns”, had attempted to conceive of a science endowed with a demonstrative structure, a science capable of taking advantage of empiricist methodologies of observation, classification and inductive inference. This science was therefore based on a geometrization of phenomena of motion and on a combinatory of perceptible qualities – with the addition of an irreducible causal agent that is necessarily likened to a soul, a mens. However, in 1671, the goal of the science seems to be split between two different projects. In the Theoria motus abstracti, Leibniz develops an abstract mechanics based on the notion of conatus that he borrows from Hobbes, and uses a geometrical model for various states of conatus, a model that integrated a notion of indivisible inspired by Cavalieri’s. However, this phoronomia elementalis conflicts with the empirical laws of impact and implies aporias within the combinative processes that are hypothesized to account for circular motions and accelerations/decelerations. A system of supporting hypotheses is thus required to form a veritable physical theory. In the Hypothesis physica nova, Leibniz therefore advances some postulates: extensive plenum, subtle ether, and the mens that grounds series of conatus. These postulates buttress the desired explanation of phenomena. An explanation of this type involves constructing certain micro-mechanical models – such as the theory of nested, interacting “bubbles” – that imply the determinative reasons behind the various systems of physical motions. Throughout the speculative fiction, there emerges the methodological ideal of a theory based on interrelated hypotheses and accounting for phenomena via analytic arguments. For Leibniz’s science, the critical turning point comes after the end of his stay in Paris. As of 1676, Leibniz returned to the problem of the laws of motion and impact with the goal of systematically deducing empirical laws, such as those formulated by Huygens and Mariotte. These deductions would bring them under a principle of conservation analogous to the Cartesian principle of conservation of quantity of motion. In the texts

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preceding De corporum concursu (1678), there emerges a significant conceptual disparity between the intrinsic features of motion and their kinematic transcriptions. Without adequate physical categories to account for the causal substrate of motion, Leibniz invokes a form of occasionalism. However, at the same time, he relies on the principle of equivalence between the full cause and the entire effect in an attempt to establish a complete system of equations that would lend intelligibility to the empirical laws of impact. The De corporum concursu illustrates the “reformatio.” Leibniz finds himself forced to conceive of a new principle of conservation, one based on the product of the body’s mass and the height of its fall before impact. In this way, he preserves the uniform architecture of the equations representing the effects in the various cases. This text did not fully reveal the metaphysical reasons that would later justify the dynamics. However, a minimal theoretical structure, justified by a seemingly occasionalist epistemological argument, underpinned the model that governed the new principle. The official publication of this new principle in the Brevis demonstratio erroris memo­ rabilis Cartesii (1686) undoubtedly places Leibniz’s reformed mechanics in opposition to Cartesian science and to its argumentative structure. Meanwhile, the first canonical formulation of the metaphysical doctrine will appear in the Discours de métaphysique (1686), demarcating itself from every form of epistemological occasionalism. How­ ever, if the new principle is supposed to reflect a system of theoretical entities, then these entities, as well as the way in which the empirical laws will be theoretically structured, remain unspecified. The turn toward dynamics begins in the unpublished Phoranomus (1689). Leibniz sets out there to apply the principle of equivalence between the full cause and the entire effect to the formal effect. The latter notion applies to unconstrained motion, which is constantly and uniformly maintained without either generating or exhausting force. To conceive of the formal effect as resulting from the underlying force, Leibniz first proposes an imperfect combinatorial model that introduces substantial states and purely modal ones under the same homology. But this model is paradigmatic, since it “simulates” the assumed parallel between the causes of formal effects and the explanatory reasons for living force. This analysis initiates the revision and correction of the previous theoretical frameworks. From a phoronomic science of power and effect, a transition occurs toward a dynamic science of action and power. The structure of dynamics as a science is revealed in its entirety in the similarly unpublished Dynamica de potentia (1689–1690). In this vast demonstrative synthesis, definitions and heuristic principles make it possible to conceive of series of theoretical models capable of accounting for empirical laws. As such, Leibniz proposes a so-called a priori approach to demonstrating the theorem of conservation of motive action. Relying on “formalizing” analogies and models derived from geometric analysis, a system of convergent arguments develops; this system allows one to deduce ex hypothesi both the intensive and extensive characteristics of a body’s power to act. Leibniz draws upon the same methodological procedure for

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establishing the combination of formal factors that play a role in the conservation of absolute force. Both with respect to theoretical concepts and the formulation of laws, the analogical framework expresses the absolute causal order underpinning phoronomic correlations. Among the resources required to achieve this theory, an ingenious system of analytic equivalences secures the combination of theoretical factors and corresponding equations. In 1695, the Specimen dynamicum will reveal, through another aspect, the theoretical method of construction at work in the dynamics. By all appearances, the demonstrations of this text take an a posteriori approach. In fact, here Leibniz proposes a research program for conceiving of sufficient reasons. This program would be based on theoretical reasons and would analyze the empirical world through a set of architectonic principles. The typology of primitive and derivative forces and the models for integrating the formal ingredients of force (conatus, impetus, vis viva) are part of the blueprint for establishing analogical relations that reference the underlying causal order. In the later Essay de dynamique (c. 1700), one witnesses the last phase of structuring of dynamics. There, Leibniz subsumes the relative principles of conservation under the absolute principles of conservation. The synthesis of the various models whereby dynamical phenomena are represented is achieved with the help of architectonic principles. The latter govern the combinatorial interplay of factors and theoretical concepts, highlight poorly formulated hypotheses that need to be eliminated, and symbolically connect the postulation of theoretical entities to the expression of continuous causal series. During the decades 1690–1700 and 1700–1710, the question of a system of a priori proofs for Leibniz’s dynamics is further examined. In the correspondence with Johann Bernoulli, Leibniz defends an approach that consists in combining the geometric representation of the factors of force and the theoretical representation of their mode of integration. The apriorism thus owes to the governing principles that make this analogical combination of instances possible, and which underpin the premises of the syllogistic demonstration of the conservation of the quantity of action. In the cor­ respondence with De Volder, there dominates the idea of merging analytic models in such a way that they can be expressed through an integrative system of conceptual equivalents. This system would represent, presumably a priori, the causal determination of force both in its formal effect and in the violent effect that would consume it. For Leibniz, the a priori conceptual analysis of the factors of force always presupposes that one refer to an underlying causal order, which can only be partially translated by geometric analogues; hence the necessity of synthetically combining such factors, so to speak. Furthermore, combining factors would not fall outside the jurisdiction of architectonic principles, namely, the equivalence between the full cause and the entire effect and the continuity of causal series expressed by antecedents and consequents. As one can establish in light of Leibniz’s various exchanges concerning the “a priori approach”, particularly with Papin, Bayle, Jacob Bernoulli, Wolff and Hermann, it would

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not be a question of demonstrations composed of propositions in their purely logical form. To the contrary, the approach remains hypothetico-deductive, apart from the fact that the theoretical concepts, grounded in architectonic principles, reveal their power to symbolically anticipate the causal connections implied in the phenomenal sequences of motion; they would “geometrically” transcribe the determinative reasons behind phenomena of force. Regarding phenomena of force, the procedures of Leibniz’s methodology only apply insofar as they provide a causal explanation for these. Although it must necessarily have a hypothetico-deductive structure, an explanation of this type must respond to “logical” criteria that imply the analytical adequacy of “geometric” models in relation to the objects represented, and in relation to the compatibility between analytic and synthetic paths, and between a posteriori and a priori methods of proof. But the desired explanation must also satisfy empirical and pragmatic criteria: the theory is presented as a research program capable of achieving an understanding that progressively becomes more complete and covers more phenomena. Hence, the theory must be developed by analogically extending explicative reasons, while also respecting the fundamental requirement of empirically verifying hypotheses more and more. Any analysis of Leibniz’s methodology, such as the one that we have previously conducted in Leibniz et la méthode de la science,1 must necessarily be articulated in terms of the dominant theoretical models of Leibniz’s science, along with the epistemological meaning that he himself attributed to them. Analyzing the nature of science depends on understanding science in action, but science evolves by revising its own methodological structure. This fact is easily demonstrated by examining the successive phases of the development of dynamics. Elucidated by this study, Leibniz’s method reveals its various epistemological innerworkings. The laws of nature surely belong to the category of contingent truths. Among these truths, one must locate the complementary processes of discovery and demonstration, which build on the combinatorial conceptualization of phenomenal factors, and on their expression via “geometric” models. An analysis of this type serves as the basis for systematically explaining how phenomena are produced. The search for sufficient causal reasons that follows therefrom constitutes the theoretical level of science. Indeed, there are theoretical approaches that are more a priori and others that are more a posteriori. However, generally speaking, the achieved theoretical framework must adequately capture the complexity and “organicity” of phenomena. In Leibniz’s science, theorization can and must incorporate hypothetical inferences that respond both to norms of empirical adequation and analytic coherence. But that does not suffice, for these hypothetical inferences must still embody the equivalent of an architectonic blueprint that allows us to “reconstruct at the theoretical level” phenomenal reality from its structures and fundamental causes.

1

Cf. Duchesneau (1993) and (2022).

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To our finite understandings, architectonic principles such as the principle of continuity or the equivalence between the full cause and the entire effect define the operating rules of this architectonic blueprint. In the end, Leibniz’s dynamics is an original example of a science constructed from and transformed by principles, a science that leads to the combinatorial integration of a plurality of analytic models.

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Index Académie des sciences  50, 111 Accademia Fisicomathematica  111 acceleration  40–41, 44, 66, 75, 105, 121–22, 130, 141–42, 147, 164, 235 action (conservation of quantity of formal action)  113–15, 119, 123, 126–51, 154–55, 157, 160, 162, 178–79, 181–85, 188–89, 191–93, 197–238 a priori (demonstration)  10–13, 77, 101, 108–09, 112–25, 128–41, 148, 151–57, 162–63, 178, 183–91, 192–238, 241–43 Archimedes  99, 103, 112, 117–23, 189 Architectonic principle  49, 62, 71, 83, 86, 89, 97, 104–05, 108, 112, 115–17, 119–20, 125, 129, 132, 134–36, 141, 151, 154–57, 161–63, 167–80, 189–93, 198, 205, 213, 221, 226, 236–38, 242–44 Aristotle  17, 24, 27, 40, 51–52, 60, 158, 171, 174 Arthur (Richard)  12 atomism 20 attraction  59, 122, 177 axiom  72, 76–77, 89, 125, 130–31, 136, 138, 141, 143–44, 147, 190, 199, 202–08, 214–17, 224, 230–31 Bacon (Francis)  18, 52, 65 Bayle (Pierre)  221, 226–27, 238, 242 Beeley (Phillip)  51 Belaval (Yvon)  11, 32–33 Bernoulli ( Jacob)  221, 227–29, 234, 242 Bernoulli ( Johann)  109, 127, 129, 131, 178, 184, 193–206, 209, 211, 213, 221, 223, 232–33, 237, 242 Bernstein (Howard)  34, 37, 105 Berthet ( Jean)  70 boat (model of the boat)  83–84, 88, 186 Bodenhausen (Rudolf Christian)  126–27

Bodéüs (Richard)  25 Borelli (Giovanni Alfonso)  122 Bouquiaux (Laurence)  12 Boyer (Carl E.)  33 Boyle (Robert)  60, 63 Breger (Herbert)  9 Brown (Gregory)  101–02 Brunschvicg (Léon)  32–33 Burkhardt (Hans)  11 capillarity 63 Carcavi (Pierre de)  33 Catelan (François)  116, 119, 216, 221 Cavalieri (Bonaventura)  32–35, 65–66, 71, 158, 240 centrifugal motion  22, 49, 121–23, 142, 145–46, 164, 167 chemistry 60 combinatory  227, 230–31, 235–37, 240 Conatus  17, 19, 21–23, 29–60, 63–67, 71, 88– 92, 115, 121–25, 133, 140–55, 158–59, 164–69, 190–91, 208–09, 237, 240, 242 Congruence  117, 199, 223 Conring (Hermann)  51, 70, 111 continuity (principle of continuity)  22, 86–87, 108, 114, 151, 174–77, 180, 203, 244 Cosimo III of Tuscany  126 Costabel (Pierre)  127 Couturat (Louis)  11, 15, 18 dead force (vis mortua)  75, 121, 124, 145, 165–68, 182–83, 196, 208–09, 212 Democritus  15, 52, 56, 63, 125 density  55, 60–63, 122 De Raey ( Jan)  25 derivative force  50, 135, 159, 167, 170–72, 191, 242

254

Index

Descartes (René)  15, 29–30, 35, 49, 51, 53–55, 63, 66–67, 69–72, 74, 76, 78, 82–85, 98–103, 106–10, 114–19, 122, 125, 127, 138, 140, 153, 162, 168–70, 173–76, 181, 206, 211, 226, 239, 241 De Volder (Burchard)  109, 129, 131, 141, 160, 166, 178, 183–84, 189, 193, 201, 205–21, 223, 237–38, 241–42 Digby (Kenelm)  25 dioptrics  48, 172 Du Hamel ( Jean-Baptiste)  25 elasticity  47–51, 56–67, 72, 74, 79–80, 207– 12, 222, 228–29 entelechy 160 equivalence (between full cause and entire effect)  72–77, 81, 86, 88, 97, 103, 106, 108, 115–20, 124–28, 143–47, 154, 162, 169–170, 178, 189–90, 195, 230, 238, 241–44 Euclide  30, 37 extensio  136, 225, 238 Fabri (Honoré)  14, 51, 55 Fichant (Michel)  10–12, 69–73, 82–83, 85, 97, 240 Foucher (Simon)  111 Gale (George)  101, 110, 159–61, 191 Galilei (Galileo)  31, 75, 91, 95, 103, 107, 114– 16, 119, 122–23, 125, 130, 147–48, 156, 183, 189, 193, 195, 222, 239 Garber (Daniel)  12, 17, 157, 173–74 Gassendi (Pierre)  15–16, 20, 39, 51–52, 63 God  17, 21, 32, 88–89, 98–99, 169–70, 174, 180, 201, 207, 212 Grosholtz (Emily)  166 Gueroult (Martial)  11–12, 15–16, 45, 97, 102, 107, 113, 128–29, 131, 157, 166, 193, 200, 240 Hacking (Ian)  104–05 Hannequin (Arthur)  16, 20, 46, 48, 60 Hermann ( Jacob)  221, 232–236, 238, 242 Hobbes (Thomas)  15, 17, 20–26, 29–30, 33–38, 42, 49, 51, 54, 65–66, 130–31, 156, 158, 239–40 Hooke (Robert)  26, 56 Hall (Francis)  18 Hofmann ( Joseph E.)  11 Huet (Pierre-Daniel)  112 Huygens (Christiaan)  10, 14, 29–31, 49, 53, 58, 65–66, 70–72, 76, 81–85, 95, 100–01, 106,

122, 128, 162, 169, 178, 186, 205–06, 211, 227, 239–40 identity (principle of identity)  203 Iltis (Carolyn)  100–01, 110 impenetrability  10, 46, 89–90, 125, 160, 168, 170 impetus  36–37, 47, 49, 56–58, 62, 75, 89, 95, 99, 115–16, 121–25, 141–49, 151–54, 164–69, 176, 182–83, 190–91, 208–09, 230–31, 237, 242 indiscernible  13, 33, 43–44, 133 indivisible  29–49, 52, 66, 105, 240 inertia  22–23, 29, 34, 39–47, 65–66, 122–25, 136, 142, 160, 169, 177, 184, 211–13 infinitesimal calculus  9, 14, 33, 65, 71, 80, 119, 121–27, 130, 132, 140, 143, 146, 155, 164–66, 189, 190–91, 208, 235 intensio  132–33, 136–38, 141–42, 219, 225, 238 Ishiguro (Hidé)  11, 15 light  19, 23, 26, 49, 51–58, 67 living force (vis viva) (principle of conservation of quantity of vis viva)  105, 110, 114, 118, 121–24, 127–28, 131, 139, 141, 145–57, 161–70, 179–83, 188–94, 201, 205, 208–09, 212–13, 217–22, 226–33, 238, 241 Johann Friedrich of Hanover  57 Kabitz (Willy)  16 Kant (Immanuel)  12, 239 Kauppi (Raili)  11, 15 Kepler ( Johannes)  122 kinematics  37, 74–75, 79, 81, 83, 89, 97, 104– 07, 117, 121, 156, 169, 176, 179, 241 Kircher (Athanasius)  56–57 Knobloch (Eberhard)  11 Laudan (Larry)  110 L’Hospital (Guillaume de)  127, 167 magnetism, 51, 59 Malebranche (Nicolas)  116, 151, 169, 178–79 Mariotte (Edme)  10, 71–72, 76, 78, 81–85, 95, 100, 106, 128, 178, 227, 240 mass  26, 35–37, 39, 44–48, 51, 54–60, 66, 76, 84, 98–99, 103, 105–07, 112, 117, 123–26, 139, 141–51, 154, 162, 164, 167, 182, 184, 189, 194–95, 206, 208, 232, 241 mathesis mechanica  119–126, 189 McDonough ( Jeffrey)  48 McGuire ( J. E.)  171

Index

mens  17, 21, 23, 25, 27–28, 37–38, 43–44, 57, 65–66, 240 Mercer (Christia)  25 mixed science  129, 140, 143, 150 Moll (Konrad)  16 Molyneux (William)  172 More (Henry)  171 motion (laws of motion)  29, 48, 51, 60, 62, 70, 72–74, 76, 78, 81, 95, 106, 128, 156, 162, 168–70, 176, 185, 210, 240 motion (conservation of quantity of motion)  10–11, 29, 34, 49, 67, 73, 82–88, 95–108, 110, 114, 117, 140–42, 145, 148–49, 152–55, 164–67, 178–83, 186, 196, 201, 207– 09, 222, 226–27, 231, 237, 240 Newton (Isaac)  9, 12, 109, 122, 239 nisus  62, 148–150, 152, 158–59, 163–67, 176–77, 196, 211, 233 nominalism  24, 66 occasionalism  81, 88–89, 97, 106–07, 170, 173, 191, 207, 241 Okruhlik (Kathleen)  161 Oldenburg (Henry)  31, 51, 72, 74 Papin (Denis)  162, 193, 195, 197, 206, 209, 221–26, 234, 238, 242 Papineau (David)  110 Pardies (Ignace-Gaston)  51 Pascal (Blaise)  102, 206 percussion  83, 85, 122, 139, 146, 148, 152, 154, 164, 186 perpetual motion  83, 195 Perrault (Claude)  72 Pérez-Ramos (Antonio)  18 phoronomy  41, 79, 115, 117, 119–21, 125, 128, 130, 134, 138–39, 142–43, 150 Plato  16, 159 Plotinus 16 primary matter (materia prima)  26, 46, 159–60 primitive force  50, 159–63, 174–75, 191

255

progress (quantity of progress)  29, 78, 88, 128, 152–54, 181, 186–87, 191, 198, 206, 216, 227 Pythagoras 210 Ranea (Alberto Guillermo)  221–25, 238 reflection  21, 29, 47, 50, 58, 67, 154, 157, 175–76 Regnault (François)  90 Robinet (André)  90, 110–14, 123 Rubini (Paolo)  90 Santi (Carlo)  90 Serfati (Michel)  11 situs  19, 74–75, 77–80, 87–89, 115, 130, 133, 159, 170 Spector (Marshall)  153 speed (conservation of respective)  83, 85, 88, 151, 186 Stammel (Hans)  221 statics  59, 74–75, 89, 182–83, 189, 232, 237 substantial form  16, 26–28, 160 Thévenot (Melchisédec)  111 Tho (Tzuchien)  12, 157 Thomasius ( Jacob)  17, 22, 25–28 time  75, 79–80, 83, 97, 103, 114–18, 123, 128– 46, 148–49, 162, 164–66, 170, 173, 184–85, 194–38 Torricelli (Evangelista)  54 Van Helmont ( Jan Baptist)  171 Varignon (Pierre)  146 vis impressa 122 vis insita  127, 196 Wallis ( John)  10, 29, 51, 66, 71, 73, 95, 100–01, 128, 169, 178, 227 Weigel (Erhard)  16, 25 White (Thomas)  25 Wilson (Margaret)  170 Wolff (Christian)  109, 129, 193, 230–32, 238, 242 Wren (Christopher)  10, 29, 31, 49, 53, 58, 66, 71, 95, 100–01, 128, 169,178

Gottfried Wilhelm Leibniz (1646– 1716) launched a new science dedicated to the theory of force. His “dynamics” is an important episode in the history of the scientific revolution. Its starting point has been a particular theory of the combination of motions placed within the framework of a mechanistic natural philosophy. Its turning point was Leibniz’s discovery in 1678 of a new principle later known as principle of conservation of live force, which he proposed in 1686 as replacement for Descartes’ principle of conservation of quantity of motion. In Dynamica de po-

ISBN 978-3-515-13520-7

9

78 3 5 15 13 5 2 0 7

tentia (1689–90), Specimen dynamicum (1695), Essay de dynamique (ca. 1700), and various pieces of correspondence, Leibniz applied his creative method­ ology, fostered original scientific models and hypotheses, and refined demonstrative arguments in support of his theory. As a result, the dynamics would comprise the conceptual and architectonic fundamental elements for a revised “system of nature”. Our endeavor has been to unveil the genesis and explain the makeup of this Leib­ nizian scientific and philosophical achievement.

www.steiner-verlag.de Franz Steiner Verlag