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The Economy of Nature – Maupertuis and Euler on the Principle of Least Action

Ansgar Lyssy, LMU Munich

This paper has appeared in French: “L’Économie de la Nature – Maupertuis et Euler sur le Principe de Moindre Action”, in: Philosophiques, Vol. 42.1, 2015, pp. 31-51.

Please do not quote this version.

1. Introduction: The Principle of Efficiency

Since antiquity, philosophers have assumed that the world follows some kind of economic principle. One may express this as “nature does nothing in vain”, or as the fact that all means have to be accounted for in regard to their aims. Even though this idea is not always discussed by itself, it is often the underlying foundation of many fundamental concepts of ontology, natural philosophy and rationality alike. It’s a methodological premise for the way we do philosophy and what we account for as rational. Take the following passage from Leibniz as an example:

As for the simplicity of the ways of God, this is shown especially in the means which he uses, whereas the variety, opulence, and abundance appears in regard to the ends or results. The one ought thus to be in equilibrium with the other, just as the funds intended for a building should be proportional to the size and beauty one requires in it. It is true that nothing costs God anything, even less than it costs a philosopher to build the fabric of his imaginary world out of hypotheses, since God has only to make his decrees in order to create a real world. But where wisdom is concerned, decrees or hypotheses are comparable to expenditures, in the degree to which they are independent of each other, for reason demands that we avoid multiplying hypotheses or principles, somewhat as the simplest system is always preferred in astronomy.1

Here, the same principle holds just as much for human as divine rationality and thus also for creation itself. Leibniz relates the ‘price’ of complexity and multitude, ontological as well as epistemic, to the ‘values’ ascribed to creation and to our possibility of understanding the phenomenal world. It is clearly not one mere philosophical idea amongst many others, but rather one that has a transformative and regulative function for metaphysics, natural philosophy and epistemology alike. This idea both serves as an ontological criterion for what there is and as a foundational epistemic criterion to which other theorems have to conform. I will call this idea the “Principle of Efficiency” (hereinafter abbreviated as PE). For the 1 Discourse on Metaphysics, § V, translation following Leibniz, Gottfried Wilhelm: Philosophical Papers and Letters, ed. and translated by Leroy E. Loemker, 2nd edition, Dordrecht: Kluwer, 1989, pp. 305 f.

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purposes of this paper, I will only give it a working definition. A comprehensive study of the history of philosophy and science would be needed for a more historically accurate definition. This working definition is: the entities involved in a cause or foundation are minimized in regard to the respective thing or effect produced by them. By saying that nature is ‘saving’ its forces or expenses, we attribute a certain economy to it and thereby conceive it as teleological, since this act of ‘saving’ is directed at a specific outcome. We understand what has been ‘saved’, what has been ‘spent’ and what has been ‘earned’, only in regard to this outcome and thus only a posteriori. This kind of economical understanding of nature thus exhibits a two-fold aspect: Either we regard the end or outcome of an event and notice that it was caused by a minimum of forces or effort; or we regard the means or forces involved in something that is happening and we notice that these forces cause the maximum of change that was possible.2 Thus stated, the PE may appear as a principle of epistemology and philosophy of nature, but we also see how it determines what kind of questions philosophers can ask, how questions and answers are to be structured and what they have to appeal to in order to be valid.

In this paper, I want to show that and how this philosophical principle was transformed and formalized into the Principle of Least Action (abbreviated as PLA) in the writings of Pierre-Louis Moreau de Maupertuis and Leonhard Euler. The implication is that since then the PE has lost a great deal of its tremendous philosophical importance and has become a major principle of science. We can see how the formalization of one rather vague philosophical idea into a precise physical formula has lead to the ‘relocating’ of certain parts of our knowledge and some interpretative schemes from one discipline into another and even their mutual restructuring. Of course, I do not dare to give a detailed reconstruction of the arguments that established the PLA. This would require a book-length treatise, and has already been done before in a brilliant manner.4 Instead, I merely want to highlight some aspects of the philosophical background relevant for the deduction of the PLA and the philosophical problems regarding its generalization from a principle of optics to a general principle of physics. In advance it should be noted that, in this context of (meta-)philosophical principles some researchers have argued for a Principle of Simplicity.5 This was often articulated as some form of Natura simplicitatem amat and for which we can give another working definition: Nature will always prefer the simple thing, way or outcome over the more complicated alternatives – and so should we as rational, epistemic subjects. It seems to me, though, that such a Principle of Simplicity can be grounded in or derived from the PE, if we apply it to our concept of nature: After all, simplicity is favored by nature because it’s ‘cheap’ and usually deemed ‘efficient’. The PE, however, excludes the possibility of a thing that is simple but superfluous within the greater scheme of things, which is probably something most philosophers had in mind, even though it is technically not stated in any assumed ‘Principle of Simplicity’. The PE also has

2 I follow Petzoldt, J.: “Maxima, Minima und Oekonomie”, in: Vierteljahresschrift für wissenschaftliche Philosophie, vol.14 (1890), pp. 206-239, pp. 354-366 and pp. 417-442, here: p. 363

4 See Pulte, Helmut: Das Prinzip der kleinsten Wirkung und die Kraftkonzeptionen der rationalen Mechanik, Stuttgart: Steiner 1989 (Studia Leibnitiana, Sonderheft 19); Panza, Marco: “De la nature épargnante aux forces généreuses: le principe de moindre action entre mathématiques et métaphysique. Maupertuis et Euler, 1740-1751”, Revue d’histoire des sciences, vol. 48.4 (1995) pp. 435-520. Also worth mentioning: Suisky, Dieter: “Über eine Differenz in der Begründung des Wirkungsprinzips bei Maupertuis und Euler”, in: Hecht, Hartmut (ed.): Pierre Louis Moreau de Maupertuis – Eine Bilanz nach 300 Jahren, Berlin: Arno Spitz Verlag, 1999, pp. 293-320; Schramm, Matthias: “Zur Entstehung des Prinzips der kleinsten Aktion”, in: ibid., pp. 321-340.

5 See, for example, Yourgrau, Wolfgang / Mandelstam, Stanley: Variational Principles in Dynamics and Quantum Theory, New York: Pitman, 1960 (2nd edition), p. 4

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an advantage over the Principle of Simplicity to the extent it can account for the fact that the world is not simple or uniform, but rather contains multitudes and heterogeneities, because this is the ‘price’ that exacted for the coming into being of something of value (order, harmony, praise for God, intelligent beings, salvation, etc.) can be. The creator ‘pays’ for this good by giving up simplicity.From the PE we can derive such diverse ideas as there can be no cause without an effect, the impossibility of the vacuum, and in some ways even the idea of the regularity of nature.The PE is even directly or indirectly related to other philosophical theorems and ideas, such as the principle of parsimony often called ‘Ockham’s Razor’, and to multiple scientific discoveries, such as the laws of reflection and refraction of light or Carl von Linné’s idea of an economy of nature in a rather ecological sense. It was implicitly involved in the early modern debates concerning occult qualities. We find it in several theologians, such as Augustine or Aquinas: “If a thing can be done adequately by means of one, it is superfluous to do it by means of several; for we observe that nature does not employ two instruments where one suffices.”6 One can also argue that this was the main reason to favor the Copernican worldview over the epicyclic model.7 Its importance is also to be tracked in a historical trajectory that passes through various different paths in the major camps in Christian philosophy, spanning from Paul to Schelling, where it takes the form of an economy of grace.8 While the genealogical search for origins in philosophy will usually be fruitless, given the enormous impact the principle of economy had on Western thinking, one might be tempted to think of it as one of the major historical foundations of Western philosophy itself. Historically, the perhaps most influential expression of this principle is found in Aristotle: “Nature does nothing in vain”.9 This means basically two things: First, nature does not omit anything essential; and, second, nothing inessential can be found in the world. We find several examples of this in Aristotle. Stars and snakes have no feet because they do not need them for their movement. The principle of efficiency governs movements in nature, for example the circular movement of stars.10 Here it is evident that this principle is different from the principle of simplicity: Doing nothing in vain means that everything has to be accounted for. This can be understood in an economical sense: Everything that exists is needed for something else, for otherwise it wouldn’t exist at all. In this way, we can also catch glimpse of how the PE is closely connected to the idea that nature does what’s ‘the best’. This, however, is highly ambiguous, to say the least. It needs to be expressed within a certain framework of values that are intimately connected not to human actions or intentions, but instead to the world as whole. In Aristotle, while ‘the best’ is relative to multiple possibilities (‘a is better than b, c, d, ...’), it is mostly meant relative to living beings, i.e. what is best for an animal. For Aristotle, understanding living nature can serve as a model for understanding nature as a whole. It thus serves as a principle of scientific inquiry (especialy in biology) inasmuch as it serves as a criterion for affirming or discarding the possibility of some ideas in favor of others. It also can be understood as an individuating principle, since by means of it we understand animal organs as being perfectly matched to 6 “Quod potest fieri sufficienter per unum, superfluum est si per multa fiat: videmus enim quod natura non facit per duo instrumenta quod potest facere per unum.” Summa Contra Gentiles, Book III., Chap. 70.

7 See, for example, the classic text by Burtt, Edwin A.: The Metaphysical Foundations of Modern Physical Science, New York: Harcourt, 1925, chapter 2.

8 See Agamben, Giorgio: The Kingdom and the Glory: For a Theological Genealogy of Economy and Government, Stanford: Stanford University Press, 2011; Richter, Gerhard: Oikonomia. Der Gebrauch des Wortes Oikonomia im Neuen Testament, bei den Kirchenvätern und in der theologischen Literatur bis ins 20. Jahrhundert, Berlin / New York: De Gruyter, 2005.

9 De Incessu Animalium, IA 8, 708a9-20. For a more nuanced interpretation see Leunissen, Mariska: Explanation and Teleology in Aristotle’s Science of Nature, Cambridge: Cambridge University Press, 2010.

10 De Caelo, I. 4, 271a22-33.

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their function.

Another influential proponent of the PE was Newton. Take, for example, his First Rule of Reasoning:

We are to admit no more causes of natural things, than such as are both true and sufficient to explain their appearances. To this purpose the philosophers say, that Nature does nothing in vain, and more is in vain, when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.11

Simplicity alone is a subordinate principle of economy and serves the overall efficiency of our explanations. However, one might assume that his famous phrase: “I don’t feign any hypothesis” accounts for the inefficiency of metaphysical speculation within a scientific context: It does not yield any desirable outcome and thus has to be abandoned.

As mentioned in the beginning, the PE had a great impact on early modern optics. Here, however, the problems of applying this kind of rather vague metaphysical proposition to specific, individual problems within the physical world inevitably become apparent. Imagine the problem of refraction. A ray of light follows a certain path s1 from any given point A in a medium1 to another given point B in a medium2, wherein both media are determined by their own density:

Why exactly is path s1 more efficient than any other path s2, s3, …? Why should nature prefer the shortest way over the fastest or vice versa? The PE provides us with a reason why light ‘chooses’ the most efficient path, at least within the framework of a necessarily teleological interpretation. It was Fermat who postulated this in a mathematical framework, which showed that light always choses the fastest way.

Gottfried Wilhelm Leibniz took on this problem and argued that light does not take the shortest or fastest route. It will always take the “easiest” path12, which is the path of least resistance. This connects the difference between the densities of the media to velocity insofar as density determines resistance and velocity is relative to the resistance the light faces passing through it. In this way, the efficiency of the pathway s1 can be articulated in more precise mathematical terms. The concept of action, which originates in this debate, then allows us to bring this optical principle into dynamics. (There is, however, some scholarly disputation about whether Leibniz had already introduced the concept of action. I will avoid this ongoing debate by rather focusing on the writings of Maupertuis and Euler, whose work on the PLA was much more influential than Leibniz’s).

11 Newton, Isaac: Principia, Berkeley and Los Angeles: University of California Press, 1966, p. 398.

12 “Lumen a puncto radiante ad punctum illustrandum pervenit via omnium facillima […].” Leibniz, Gottfried Wilhelm : Unicum opticae (1682), in: Die philosophischen Schriften von Gottfried Wilhelm Leibniz, ed. by C. I. Gerhardt, vol. VII, Berlin, Weidmannsche Buchhandlung, 1890, p. 273; see Pulte, Das Prinzip der kleinsten Wirkung, p. 51.

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2. Maupertuis

The French mathematician and philosopher Pierre Louis Maupertuis was the first one to elaborate a formalization of the PE. He introduced the PLA in a paper called Accord between different laws of Nature that seemed incompatible (1744) and, reacting to Leonhard Euler’s take on the PLA,13 he further developed his own account in Derivation of the laws of motion and equilibrium from a metaphysical principle (1746). He was a profoundly inspired by Newton’s works, but also by Leibniz’s metaphysical idea of an economy of creation, both of which play an important role in Maupertuis’ deduction of the PLA.

Maupertuis arrives at the PLA by criticising and expanding on Fermat’s theorem. While he deduces the PLA in the context of optics, he then extends it into the domain of mechanics. He criticises Fermat for not giving a reason for why light should take the shortest or the fastest way. He makes into a metaphysical and teleological principle of optics the claim that “Nature, in the production of its effects, acts always by the simplest means.14 One might object that a “blind“ nature, merely driven by efficient causes, does not “know” which way the most efficient one is. Indeed, Maupertuis was open for teleological arguments in optics, like those proposed by Robert Boyle, who wanted to prove God’s causal efficacy in the world. Boyle had renewed the idea of a teleological nature and was backed up by Newton. So to propose a teleological interpretation of something usually described in corpuscular mechanics was not completely unheard of at that time. Consequently, Maupertuis’ saw it fit to substitute time as a variable in Fermat’s approach with a new variable, namely action:

After meditating deeply on this topic, it occurred to me that light, upon passing from one medium to another, has to make a choice, whether to follow the path of shortest distance (the straight line) or the path of least time. But why should it prefer time over space? Light cannot travel both paths at once, yet how does it decide to take one path over another? Rather than taking either of these paths per se, light takes the path that offers a real advantage: light takes the path that minimizes its action.15

The abstract concept of action was known before, mostly as a vague mechanical concept in the distinction actio-reactio.16 The above quoted idea that nature works in an efficient way implies such a teleological interpretation of natural events because the efficiency of something can only be understood by taking into account the entire event from start to finish. Maupertuis gives action the precise mathematical expression ∫vds:

Now I have to define what I mean by ‘action’. When a material body is transported from one point to another, it involves an action that depends on the

13 “M. le Professeur Euler donna à la fin de la même année son excellent Livre: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Dans le Supplement qui y avoit été ajouté, cet illustre Géomêtre démontre ; Que dans les trajéctoires, que des corps décrivent par des forces centrales, la vîtesse multipliée par l’elément de la courbe, fait toujours un minimum.” Histoire de l’Académie Royale des Sciences et des Belles Lettres (1746), p. 267

14 “La Nature, dans la production de ses effets, agit toujours par les moyens les plus simples.” Accord de différentes loix (1746), in: Oeuvres, vol. IV, Lyon, Jean-Marie Bruyset, 1768, p. 12.

15 “En méditant profondément sur cette matière, écrit-il, j’ai pensé que la lumière, lorsqu’elle passe d’un milieu dans un autre, abandonnant déjà le chemin le plus court [...], pouvoit bien aussi ne pas suivre celui du temps le plus prompt: en effet, quelle préférence devroit-il y avoir ici du temps sur l’espace? la lumière ne pouvant plus aller tout-à-la fois par le chemin le plus court, et par celui du temps le plus prompt, pour quoi iroit-elle plutôt par un de ces chemins que par l’autre? aussi ne suit-elle aucun des deux, elle prend une route qui a un avantage plus réel: le chemin qu’elle tient est celui par lequel la quantité d’action est la moindre.” Ibid. p. 16. Translations for this text follow the translation available online at: http://en.wikisource.org/wiki/Translation:Accord_between_different_laws_of_Nature_that_seemed_incompatible

16 See Pulte: Das Prinzip der kleinsten Wirkung, pp. 56 ff.

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speed of the body and on the distance it travels. However, the action is neither the speed nor the distance taken separately; rather, it is proportional to the sum of the distances travelled multiplied each by the speed at which they were travelled. Hence, the action increases linearly with the speed of the body and with the distance travelled.17

However, his arguments for the PLA are obscure. They have the air of being more of a metaphysical postulate designed to fit empirical observations rather than being rigorous, scientific modelling. He simply postulates that there is some kind of variable that depends on velocity and distance. Some interpreters point out that, even though he found out a scientific ‘truth’, he came to it by pure guesswork.18 This suggests that instead of reconstructing a proper, well-formulated argument, we would have better luck if we were to trace the course that lead Maupertuis to his discovery.

Maupertuis’ follows the Cartesian measure of movement as vds, but refrains from using Leibniz’ way of calculating forces as mv2. He also makes use of Newton’s idea that the pathway of light is the way that light-corpuscles have traveled and thus generalizes this optical formula by making it applicable to all physical bodies. He adds: “This action is the true expense of Nature, which she manages to make as small as possible in the motion of light.”19 Or, as he similarly puts it: “Until some change happens in nature, the quantity of action necessary for such a change is the smallest possible.”20 What is especially to note here is how Maupertuis has made action the true expense of nature in general and has thus transferred it from optics to all natural events.

The metaphysical background to this reasoning is derived from Leibniz’s philosophy.21 Maupertuis highlights the organizational structure of the universe, which can be understood as governed by divine wisdom and causal efficiency alike:

One cannot doubt that everything is governed by a supreme Being who has imposed forces on material objects, forces that show his power, just as he has fated those objects to execute actions that demonstrate his wisdom. The harmony between these two attributes is so perfect, that undoubtedly all the effects of Nature could be derived from each one taken separately. A blind and deterministic mechanics follows the plans of a perfectly clear and free Intellect. If our spirits were sufficiently vast, we would also see the causes of all physical effects, either by studying the properties of material bodies or by studying what would most suitable for them to do.22

17 “II faut maintenant expliquer ce que j’entends par la quantité d’action. Lorsqu’un corps est porté d’un point à un autre, il faut pour cela une certaine action: cette action dépend de la vitesse qu’a le corps et de l’espace qu’il parcourt, mais elle n’est ni la vitesse ni l’espace pris séparé ment. La quantité d’action est d’autant plus grande que la vitesse du corps est plus grande, et que le chemin qu’il parcourt est plus long, elle est proportionnelle à la somme des espaces multipliez chacun par la vitesse avec laquelle le corps les parcourt.” Accord de différentes loix, p. 17.

18 von Helmholtz, Hermann: “Rede über die Entdeckungsgeschichte des Princips der kleinsten Action”, in: Harnack, Adolf v. (ed.): Geschichte der Königlich-Preußischen Akademie der Wissenschaften zu Berlin, vol. 2, Berlin 1900, p. 294.

19 “C’est cela, c’est cette quantité d’action qui est ici la vraie dépense de la Nature, et ce qu’elle ménage le plus qu’il est possible dans le mouvement de la lumière.” Accord de différentes loix (1746), in: Oeuvres, vol. IV, Lyon, Jean-Marie Bruyset, 1768, p. 16.

20 “Lorsqu’il arrive quelque changement dans la Nature, la quantité d’action nécessaire pour ce changement est la plus petite qu’il soit possible.”Recherche des lois du mouvement (1746), in: Oeuvres IV, p. 36. Translation A. L.

21 For a more detailed account see Pulte, Helmut: Das Prinzip der Kleinsten Wirkung; and Hecht, Hartmut: “Gemeinsame Denkmotive bei Leibniz und Maupertuis,” in: ibid. (ed.): Pierre Louis Moreau de Maupertuis – Eine Bilanz nach 300 Jahren, Berlin, Arno Spitz Verlag, 1999, pp. 207-224.

22 “On ne peut pas douter que toutes choses ne soient réglées par un Etre suprême qui, pendant qu’il a imprimé à la matière des forces qui dénotent sa puissance, l’a destinée à exécuter des effets qui marquent sa sagesse; & l’harmonie de ces deux attributs est si parfaite, que sans doute tous les effets de la Nature se pourroient déduire de chacun pris séparément. Une

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Here the Leibnizian idea that events can be explained by causal just as much as final explanations is repeated. This is important as it allows him to apply a teleological explanation first and then assume that there must be an explanation by efficient causes. Of course, the PE or the economy of action derives from the divine wisdom by which creation itself is formed. In this case, action is a means for establishing a certain natural order as an end: “Nature, in the production of its effects, always acts by the most simple means.”23 This is the reason why we never find any indication that Maupertuis might have found it problematic to assume that a teleological principle derived from optics holds for all mechanics or dynamics: He adheres to Leibniz’s idea of a teleologically directed universe, following the plan of the creator. He makes it clear that this is an economical principle in which means are related in the most economical manner to their ends:

But this ‘budget’, this expense of action that Nature minimizes in the refraction of light, is it also minimized in the direct propagation and reflection of light? Yes, it always has the smallest possible value.”24

This is, of course, a sign of the divine intelligence by which God has created the world. All events follow his wisdom. God has directed things according to his plan by forces and Maupertuis just iterates Leibniz’ postulate of a harmony between final and efficient causes. There is a determined quantity of action in the world and nature spends as little of this as possible. In the case of optics, nature ‘saves’ action in reflection and linear propagation so that light always takes the shortest and fastest path. In this way, Fermat’s law is not a proper law by itself, but rather a consequence or expression of a more fundamental and universal law, namely the PLA. Maupertuis here takes it for granted that the laws of optics are basically just a special case of mechanics. In 1746, Maupertuis presents a paper called Derivation of the laws of motion and equilibrium from the attributes of God at the Berlin Academy of Science, which already points towards the foundation of mechanics in ideas of a rather theological nature. The published paper bears the more modest title: Derivation of the laws of motion and equilibrium from a metaphysical principle (1744). Maupertuis applies the PLA to the direct impact of two bodies. He seems to be inspired by a certain understanding of Leibniz’s concept of forces and the vis-viva controversy. He adapts Leibniz’s definition of force as mv2 and, assuming that in any interaction of two bodies Σmv2 is constant, he derives the change of velocity in both bodies after collision as m(v0 – v1)2 + m‘(v‘1 – v‘0)2. This is what he calls the quantity of action that is involved. He conceives, as it was common at his time, the nature of light not as a physically special thing, but rather as the result of the movement of very small bodies. As such, it was natural to suppose that the laws of reflection and refraction were supposed to be part of the mechanics of rigid bodies. Consequently, it is no leap for Maupertuis to assume on metaphysical grounds that the PLA holds for all physical events, but nonetheless he cannot show that it is valid in dynamical systems where one particle is subject to multiple forces. Once again, the PE is a metaphysical principle based a specific understanding of nature as creation. It is one of the universal regularities by which we can discover the relationship

méchanique aveugle & nécessaire suit les desseins de l’Intelligence la plus éclairée & la plus libre, & si notre esprit étoit assez vaste, il verroit également les causes des effets Physiques, soit en calculant les proprietés des corps, soit en recherchant ce qu’il y avoit de plus convenable à leur faire exécuter.” Accord de différentes loix de la nature, in: Oeuvres IV, p. 21.

23 “La Nature, dans la production de ses effets, agit toujours par les moyens les plus simples.” Ibid., p. 12. Translation A. L.

24 “Mais ce fonds, cette quantité d’action que la Nature épargne dans le mouvement de la lumière à travers différens milieux, le ménage-t-elle également lorsqu’elle est réfléchie par des corps opaques et dans sa simple propagation? Oui, cette quantité est toûjours la plus petite qu’il est possible.” Accord de différentes loix de la nature, en: Oeuvres IV, p. 19.

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between God and nature itself.25 This means that, on the one hand, Maupertuis can deduce the PLA from the PE and, on the other hand, simultaneously claim that the universal laws and regularities of the world prove the existence of God and the fact that the world is created by a supreme intelligence. There’s the “need to search for the proofs of God’s existence in the general laws of nature. Especially the laws governing motion’s conservation, distribution and destruction are based on the attributes of a supreme intelligence.”26 In short, in Maupertuis’ writings we find the earliest – or one of the earliest – occurrences of the PLA, which here serves as a regulative principle for scientific inquiry and metaphysical speculation alike: We understand physical effects and divine intentions alike in regard to the PLA, but the ideas that support it derive from the metaphysical idea of the PE.

3. Euler The mathematician and physicist Leonhard Euler was convinced that Maupertuis’ account of the PLA was basically correct. Euler brought Maupertuis’ empiricist stance to an extreme by demanding that the “indubitable” mechanics and empirical science should be the true source and foundation of “imaginary” metaphysics: “It would be absurd to think that mere imaginations could serve as foundations for the real principles of mechanics.”27 Mathematics and experience are the only true foundations of mechanics and by rejecting experience as a valid truth criterion, metaphysical thought is subject to phantastical speculation devoid of any relation to the actual world. However, despite what may appear to us to be a proto-positivist position contra traditional rationalist metaphysics, Euler in fact combines a mathematical approach with some metaphysical presuppositions in his own deduction of the PLA. While Maupertuis was searching for universal principles that were obvious in optics, but equally valid in other scientific disciplines, Euler was rather looking for mathematical extrema.28 In his treatise A method for finding curved lines enjoying properties of maximum or minimum (1744), Euler had developed a calculus of variations, i.e. a geometrical method to find curves (or other geometrical figures) in which a certain variable had a maximized or minimized value. Amongst such problems was the question: Which curve a moving body would pass through, given a constant velocity, but subject to a constant force, which in this case was given by mass or gravitational force?

25 Pulte: Das Prinzip der Kleinsten Wirkung, p. 43.

26 “Il faut chercher les preuves de l’existence de Dieu dans les Loix génerales de la Nature. Que les Loix selon lequelles le Mouvement se conserve, se distribue et se détruit, sont fondées sur les attributs d’une suprême Intelligence.” Les Loix du mouvement et du repos, p. 277. Translation A. L.

27 See Reflexions sur l’espace et le tems [E149], in: Memoires de l’Academie des Sciences de Berlin, 4, 1750, pp. 324-333. Translations of Euler’s writings usually follow those accessible at http://eulerarchive.maa.org/.

28 Boudri, J. Christiaan: What was Mechanical about Mechanics? The Concept of Force between Metaphysics and Mechanics from Newton to Lagrange, Dordrecht, Kluwer, 2002, p. 156.

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The question then naturally arises: Why does the body follow the path s1 instead of any other possible path s2, s3, …? The calculus of variations offers an answer. To wit, the curve of a projectile can be determined by finding the property of maximum or minimum:

Since all the effects of Nature obey some law of maximum or minimum, it cannot be denied that the curves described by projectiles under the influence of some forces will enjoy the same property of maximum or minimum. It seems less easy to define, a priori, using metaphysical principles, what this property is. But since, with the necessary application, it is possible to determine these curves by the direct method, it may be decided which is a maximum or a minimum.29

It’s the curve s1 that is determined by such a minimum: Let the mass of the projectile be M, and let its speed be v while being moved over an infinitesimal distance ds. The body will have a momentum Mv that, when multiplied by the distance ds, will give Mvds, the momentum of the body integrated over the distance ds. Now I assert that the curve thus described by the body to be the curve (from among all other curves connecting the same endpoints) that minimizes ∫Mvds or, provided that M is constant along the path M∫vds.”30

Euler realized that Maupertuis’ momentary action is valid only for infinitesimal moments and that consequently the action along an arbitrary infinitesimal section of s has a minimal value. One needs to integrate this momentary action over the entire path of the particle. Action, thus, is not a property of things or events, but rather a property of the geometrical model that

29 “Quoniam omnes naturae effectus sequuntur quandam maximi minimive legem; dubium est nullum, quin in lineis curvis, quas corpora projecta, si a viribus quibuscunque sollicitentur, describunt, quaepiam maximi minimive proprietas locum habeat. Quaenam autem sit ista proprietas, ex principiis metaphysicis a priori definire non tam facile videtur; cum autem has ipsa curvas, ope Methodi directae, determinare liceat; hinc, debita adhibita attentione, id ipsum, quod in istis curvis est maximum vel minimum, concludi poterit.” Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Additamentum 2: De motu projectorum in medio non resistente [E065], Lausanne / Geneva, Marcus M. Bousquet, 1744, p. 311. Translation follows Dugas, René: A History of Mechanics, Mineola, Dover, 2012, p. 273.

30 “Sit massa corporis projecti = M, ejusque, dum spatiolum = ds emetitur, celeritas debita altitudini v; erit quantitas motus corporis in hoc loco = Mv; quae per ipsum spatiolum ds multiplicata, dabit Mdsv motum corporis collectivum per spatiolum ds. Jam dico lineam a corpore descriptam ita fore comparatam, ut, inter omnes alias lineas iisdem terminis contentas, sit ∫Mdsv, seu, ob M constans, ∫dsv minimum.” Ibid., pp. 311-12. Translation follows the one available at http://en.wikisource.org/wiki/Methodus_inveniendi/Additamentum_II. Euler writes √𝑣 for velocity, but I have abridged that according to the modern notation.

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describes the unfolding of an event over time. Euler’s account differs from Maupertuis’ in the way that it is deduced in a proper mathematical manner.31 The resulting curve sab is determined by the PLA as the sum of all infinitesimal sections and has a minimal value in comparison to all other curves s’ab. As action is formalized as I = ∫mvds and since v = ds/dt can also be written as I = ∫mv2dt. Here we encounter again the Leibnizian force (vis viva) f = mv2, which is here conceived as the causal effect of forces in regard to velocity in a certain point. With this in mind, we can also say that the resulting curve sab=s1 is determined by the PLA as the sum of all momentary expressions of force over time and is minimized in comparison to all other curves s’ab=s2, s3 etc. Of course, Euler presupposes not only that curves and all types of change are continuous, but also that all possible curves form a spectrum of continuity.

He further recognized that this was the same PLA as formulated by Maupertuis. But it instead expressed the movement of a body under forces expressed by geometrical means, while Maupertuis was restricted to massless bodies without forces. Consequently, Euler had to add another condition to make his formula valid, namely that the forces involved had to be constant – expressing it in a modern way: That the energy involved remained constant. This new definition of action makes the PLA directly relevant for our understanding of causal processes and allows for an interpretation in terms of causal relations: “His [Maupertuis] principle is, that of the least possible action; by which he means, that in all the changes which happen in nature, the cause which produces them is the least that can be.”32 The curve s1 can be found with a direct and indirect method. The so-called ‘direct’ approach consists in relating the different moments of a movement to the forces involved, explaining every consequent change by Newtonian forces. The indirect approach is a geometrical approach: It is based on the calculus of variations and it defines the movement over the course of time in regard to a variable, here action, which is minimized over the entire course. It is considered a teleological principle, as here the entire course of events is considered, but deemed effective as a determining principle from the very onset on. Both methods can be equally employed, depending on the variables given in regard to a specific situation:

From this one can see that there has to be a double method for resolving the problems of mechanics; one is the direct method, which is grounded in the laws of equilibrium, or movement; but the other is that which I am going to speak about, where the solution can be found by the means of the methods de maximis et minimis, if you know the formula, which has to be a maximum or a minimum. The first one provides the solution in determining the effect of efficient causes; and the other one is regarding the final causes.33

Here we see that Euler implicitly refers to Leibniz’ doctrine that every causal relation can be deduced from efficient and final causes alike. Despite his attack on metaphysics, especially on Leibnizian metaphysics, Euler does not distinguish in clear terms between metaphysics and physics, which makes his arguments shift seamlessly between the one and the other. He criticizes Leibniz for deriving his physical forces from metaphysical forces,34 yet he clearly 31 See Yourgrau / Mandelstam : Variational Principles in Dynamics and Quantum Theory; Pulte : Das Prinzip der kleinsten Wirkung.

32 Lettres à une Princess d’Allemagne, Part II [E344], Lettre X, Paris, Charpentier, 1843, p. 203.

33 “Par là on voit qu’il doit y avoir une double méthode de resoudre les problèmes de Mécanique; l’une est la méthode directe, qui est fondée sur les loix de l’équilibre, ou du mouvement; mais l’autre est celle don’t je viens de parler, où sachant la formule, qui doit être un maximum, ou un minimum, la solution se fait par le moyen de la méthode de maximis et minimis. La première fournit la solution en déterminant l’effet par les causes efficientes; or l’autre a en vue les causes finales.” Recherches sur les plus grands et plus petits qui se trouvent dans les actions des forces [E 145], dans: Mémoires de l’académie des sciences de Berlin 4, 1750, p. 151. Translation A.L.

34 See Gedancken von den Elementen der Cörper (1746 [E81]), in: Opera Omnia: Series 3, vol. 2, pp. 347-366.

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accepts that the notion of force needs a metaphysical background to be understandable.

Euler decides to ignore speculations about the nature of forces in favor of inquiring into their effects. The idea guiding his research is that bodies obey forces as little as possible.35 Forces are real, but derivative entities. They result from underlying substances. This also explains, again in an obviously Leibnizian manner, that there are final causes embedded into natural forces. In one of his later writings, Euler makes it clear that final causes are real and embedded in the metaphysical or substantial nature of bodies: The success of the inquiry into movements by final causes is “founded in the nature of bodies themselves”.36 There is quite a lot of textual evidence that Euler’s justification of the PLA draws from two sources or ‘background’ ideas: First, from the mathematical attempts to determine the unequivocal minimum of a curve, say, a chain fixated on two points; and, second, from a metaphysical background easily identifiable as Leibniz’s philosophy. This consists in the same believe in the two reigns of final and efficient causes as has been shown in regard to Maupertuis, which in Euler’s writings connect the economy of the world to theological ideas. This is evident in one of the Letters to a German Princess:

“If we consider the order and economy of the world in this point of view [in regard of the non-arbitrary decree of God], nothing can be more perfectly disposed for the attainent of this great end [human felicity]. All events, the calamities themselves which we undergo, are the most suitable means for conducting us to true happiness; and in this respect it may be with truth affirmed, that this is indeed the best world possible, as every thing in it concurs to promote our eternal salvation. When I reflect that nothing befalls me by chance; but that every event is directed by Providence.”37

Here it becomes apparent that Euler, even though an adherent of mechanism and the explanation of all events by mathematical science, nonetheless advances the thesis that there is a moral, even normative aspect to be found in all events. Another piece of evidence points us directly to the economy of nature as proposed by Aristotle:

The earliest philosophers had already recognized that nature does nothing in vain and this relates perfectly to [the question of] least expense. For if nature were to employ superfluous expense, it would clearly involve nature doing something in vain. Aristotle already makes references to this doctrine, and it seems that he has taken this idea from those who preceded him rather than envisioning it himself. The proposition has subsequently had such enormous influence in the Schools that it was regarded as one of the first precepts of philosophy until Descartes ultimately dared to reject it.38

35 Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Additamentum 2: De motu projectorum in medio non resistente [E065], Lausanne et Geneva, Marcus M. Bousquet, 1744, p. 320; See Dugas, René: A History of Mechanics, Mineola, Dover, 2012, p. 274.

36 See Lettres à une Princess d’Allemagne, Part II [E343], Letter VI, Paris, Charpentier, 1843, pp. 192 ff.

37 “Si nous regardons sur ce pied l’arrangement et l’administration de ce monde, tout ne saurait être mieux disposé pour ce grand but [la félicité des hommes]. Tous les événements et même les adversités que nous éprouvons, sont les moyens les plus propres pour nous conduire à notre vrai bonheur: et à cet égard on peut dire que ce monde est effectivement le meilleur, puisque tout y concourt à opérer notre salut. Quand je réfléchis qu'il ne m’arrive rien dans ce monde par hasard, et que tous les événements en sont dirigés par une providence.” Lettres à une Princess d’Allemagne, Part II [E343], Lettre XLV, Paris, Charpentier, 1843, p. 293. Translations for this text follow the translation by Henry Hunter, London, Murray and Highley, 1802.

38 “Car les plus anciens Philosophes avoient déjà reconnu, que la nature ne faisoit rien en vain, ce qui s'accorde parfaitement avec la moindre dépense ; car si la Nature employoit des dépenses superfluës, il n’y a pas de doute qu’elle ne fit quelque chose en vain. Aristote fait déjà souvent mention de ce dogme, et paroit l’avoir plutôt pris de ceux qui l’avoient précedé, que l’avoir imaginé lui-même. La proposition a fait ensuite un si grand progrés dans les Ecoles, qu’on l’a regardée comme un des premiers préceptes de la Philosophie, jusqu’à ce qu’enfin Descartes a osé la rejetter.” Sur le principe de la moindre action, in: Memoires de l'academie des sciences de Berlin 7, 1753, pp. 199-218, here p. 199. Translation follows the one made by Michael Saclolo and Peter Wake available here: http://eulerarchive.maa.org/pages/E198.html

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Euler here correctly assumes that the PE has been a main principle of philosophy from Aristotle to Descartes and he relates this philosophical idea directly to his idea of least expense. For all the anti-philosophical attitude that he occasionally displayed, metaphysical speculation about the overall economy of the world does nonetheless help Euler as an epistemic guide for his scientific research.

The economy of nature is articulated in the more concrete striving of nature for a minimum, or, depending on which unities or aspects the scientist focuses on, a maximum. As Joseph Petzoldt has claimed,39 the PLA provides Euler a mathematical access for understanding nature as determined. We even find such a statement in Euler’s own writing:

For since the plan of the universe is the most perfect possible and the work of the wisest possible creator, nothing happens which has not some maximal or minimal property, and therefore there is no possible doubt that all effects in the world can be determined equally well from final causes using the method of maxima and minima, or from the efficient causes themselves.40

Here Leibniz’s influence on Euler is, once again, clearly discernable. Leibniz has made it a core theme of his philosophy that each event can be equally understood and explained by mechanistic as by final causes. This, of course, is due to the fact that the world is a creation by God who has outline a divine plan for everything that has happened or will ever happen. For Leibniz, though, the determining principle is the Principle of Sufficient Reason, which postulates that nothing happens without a determining reason, a reason that also be understood as a cause. Yet, the mathematical approach was more favorable to Euler thanks to his anti-metaphysical tendencies. He used this advantage of a purely mathematical approach to criticize Maupertuis‘s attempt to relate the principle of least action to the actions of forces: “the determination of the quantity of action bears no relation to any force whatsoever”.41 We can formulate action as I = ∫mvds and mass, velocity and path are here the only relevant factors – but his metaphysical idea of the equivalence of explanations by efficient and final causes leads him to assume that this would correspond to Newton’s forces. The advantage of the PLA is that Maupertuis’ unresolved question of how much force is required to give a body a certain velocity can be simply ignored, at least within the mathematical approach. The determination of movement by forces (e.g. vis viva) requires a metaphysical background Euler does not want to deal with.

There are other presuppositions at play when we understand curve s1 as ‘determined’. Any possible curve that would not satisfy the PLA would have a “twin”: a symmetrical and corresponding trajectory with the same values. The PLA helps to determine the one trajectory that is unequivocally determined, i.e. that has no such “twin.” However, this only holds for idealized curves. As soon as friction is taken into account, the actual pathway of a projectile within a gravitational field would be an asymmetrical parable, something that rather looks like the curve s3 in the graphic 2 displayed above. For such an asymmetrical parable, there’s a symmetrical “twin” for which action (I = ∫mvds) has the same value.

39 See Petzoldt: “Maxima, Minima und Oekonomie”, chap. I.

40 “Cum enim Mundi universi fabrica sit perfectissima atque a Creatore sapientissimo absoluta, nihil omnino in mundo contingit, in quo non maximi minimive ratio quaepiam eluceat; quamobrem dubium prorsus est nullum, quin omnes Mundi effectus ex causis finalibus ope Methodi maximorum et minimorum aeque feliciter determinari queant, atque ex ipsis causis efficientibus.” Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Additamentum I. De curvis elasticis, [E065] 1744, p. 245. Emphasis mine. Translation follows Philip E. B. Jourdain: “Maupertuis and the Principle of Least Action”, in: The Monist 22 (1912), p. 435.

41 “La determination de la quantité d’action n’a egard à aucune force.” Euler’s letter to Maupertuis, 24 May 1746, in: Opera Omnia, Series IV.a, vol. 6, p. 64. Translation A. L.

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The same holds for all other non-conservative forces. As such, Euler’s mathematical version of the PLA only holds for moving projectiles subject to none or only one constant force and then only under idealized conditions, as it would allow for two possible solutions. The direct and non-teleological approach would unequivocally determine this as the only possible solution. Of course, it should be noted, Euler is well aware of the problem that his deduction of the PLA holds only for conservative forces and he assumes that a further elaboration of metaphysics will lead to a greater clarity of his solution.42 Consequently, he rather proposes to rework the distinction between conservative and non-conservative forces instead of restraining the general validity of the PLA, which he assumes on purely metaphysical grounds. The major difference between Maupertuis’ and Euler is that the latter extended the PLA to dynamics and supplies it with a theory of forces with which he intends to show how all types of forces can be reduced to one (see below), which is clearly meant to justify the generalization of the PLA. This is not a minor detail. It is supposed to support the validity of the PLA for all kinds of physical events: Since there is only one type of force, all kinds of action can be modeled after the trajectory of one body, the collision of two bodies and the parallelogram of forces.

Leibniz had postulated that there are two fundamental properties of matter, namely inertia and impenetrability. Both correspond two different types of forces, that is, derivative and primitive forces. Even though Leibniz makes it sufficiently clear that only one of these possesses substantial reality and the other one, the derivative force, is only a phenomenon, Euler argues against Leibniz by maintaining that there is only one the fundamental property of matter: impenetrability. Euler then goes on to derive extension, mobility and inertia from impenetrability and thus to ground the different physical aspects of matter in one single property. This property is then to be conceived as an irreducible and thus substantial property of matter itself and can consequently be understood as a metaphysical concept.43 To summarize the argument very briefly, it goes as follows: Without extension, impenetrability is meaningless, for a thing that has no extension cannot occupy a place. Thus, an impenetrable object is necessarily extended.44 Due to its impenetrability an object occupies a place and thus can potentially change place – in other words, impenetrability implies mobility. If an object is mobile, then also inertia has to be attributed to it, for otherwise any change would occur without a sufficient reason.

In the impact between bodies, their impenetrability produces only the minimum force required to protect them from penetration (...) and no doubt it is this fact that is the basis of the general principle that all changes in the world are produced at the least expense possible, or with the smallest forces capable of producing the effect concerned.”45

42 “Cuius rationii vis, etiamsi nondum satis perspiciatur, tamen, quia cum veritate congruit, non dubito, quin ope principiorum sanioris Metaphysicae ad maiorem evidentiam evehi queat; quod negotium aliis, qui Metaphysicam profitentur, relinquo.” Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Additamentum 2: De motu projectorum in medio non resistente [E065], Lausanne / Geneva, Marcus M. Bousquet, 1744, p. 320.

43 See Pulte: Das Prinzip der kleinsten Wirkung, pp. 23ff. and 110ff.

44 See Anleitung zur Naturlehre [E842], Opera Omnia: Series 3, Volume 1, pp. 16-180, §§35-39, 75; and Theoria motus corporum solidorum seu rigidorum [E289], Opera Omnia: Series 2, Volume 3, §§131-135. See also Suisky, Dieter: “Euler’s Mechanics as a Foundation of Quantum Mechanics”, in: Bradley, Robert (eds.): Leonhard Euler. Life, Work and Legacy, Amsterdam, Elsevier, 2007, p. 516; also Gaukroger, Stephen: “The metaphysics of impenetrability: Euler’s conception of Force”, in: British Journal for the History of Science, 15, 1982, pp. 132-154.

45 “Ainsi dans le choc des corps leur impénétrabilité ne fournit toujours que la plus petite force, qui es capable de les garantir de la pénétration; et c’est sans doute sur cette circonstance, qu’est fondé ce principe si général, que tous les changemens au monde sont produits aux moindres dépens qu’il est possible, ou avec les plus petites forces, qui sont capables de cet effet.” Recherches sur l‘origine des forces [E181], in: Memoires de l'academie des sciences de Berlin 6, 1752, pp. 419-447, here pp.

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Both force and action are subject to the same minimum principle. All forces and relevant physical properties are effects of impenetrability. This should be best understood as an attempt to eliminate the unwelcome metaphysics of force from his physics. Euler outlined an order or hierarchy of the characteristics of physical bodies, and did so expressly in his later Anleitung zur Naturlehre (published posthumously in 1862, but most likely written between 1744 and 1748): “The impenetrability does by itself already include extension and mobility and consequently also inertia in itself. Thus, if one ascribes impenetrability to a body, one also has to ascribe him the other attributes too.”46 However, this has important consequences for the interpretation of the PLA within the context of Euler’s physics: All force can only be attributed to bodies insofar as they collide or could potentially collide. Euler accepts no action-at-a-distance forces, so all action results from contact. Thus the theory of contact forces is supposed to unite all physical phenomena. In the end, all motion results from uniform motion and all kinds of bodies are composed of bodies of infinitesimal magnitude.47 This allows him to use collision as paradigm for any type of change48 and therefore to understand forces by means of the well-defined mechanical account of collision. J. Christiaan Boudri highlights the importance of this reduction:

Euler‘s foundation of the force of impact shows even more strongly than Maupertuis‘s argument that the final cause and efficient cause have the same metaphysical basis. Where for Maupertuis this basis was the oneness of God, for Euler the reference to God is as it were reflected onto matter itself. […] What is concealed in his foundation […] is that the minimality of the integral of action does not say anything about the separate material substances, but about their structure. In the end, the forces of impact are not derived from the characteristics of the material substances, but from a principle of material structures: that the interactive effects are as small and as direct as possible.49

Despite this metaphysical dimension of Euler’s concept of force, he understands the PLA as justified not by an a priori deduction, but with an a posteriori verification, since the conservation of energy or of forces had to be presupposed. Yet it was considered a basic principle, as it was to be found and confirmed in experience. For Maupertuis and Euler, action, nonetheless, is the measure of God’s own efficiency and not justifiable by scientific inquiry alone. It needs a confirmation that effectively includes experience and metaphysical speculation alike.

For both, material substance cannot be a starting point of philosophy anymore. The concept of force is based on experience and the idea of a universal regularity based on divine wisdom, which allows them to assume the equivalence of final and efficient causes within a metaphysical framework, albeit the latter is more of an outline than a developed system. The PE is here the conceptual background for the deduction of the PLA, without which it would remain a mere principle of optics (Maupertuis) or a mathematical property of idealized curves (Euler). For both, the PLA is a regularity principle to which other laws and theorems have to confirm. 431 f. Translation follows Boudri: What was so Mechanical about Mechanics, p. 168.

46 “Die Undurchdringlichkeit [impenetrabilité] schliesset für sich schon die Ausdehnung und Beweglichkeit und folglich auch die Standhaftigkeit [inertia] in sich. Wenn man also dem Körper die Undurchdringlichkeit zueignet [ascribe, attribute], so muss man ihm auch die übrigen Eigenschaften zuschreiben.” Opera Postuma 2, ed. by Paulus Henricus Fuss and Nicolaus Fuss, Saint-Pétersbourg1862, vol. II, p. 472. Translation A. L. See also: “So nehmen alle Kräfte, welche die Veränderungen in der Welt hervorbringen aus der Undurchdringlichkeit der Körper ihren Ursprung, und daher sind in der körperlichen Welt keine anderen Kräfte anzutreffen, als die, welche aus der Undruchdringlichkeit der Körper entstehen.” Ibid., p. 474.

47 Mechanica sive motus scientia analytice exposita, part I (1736 [E15]), Propositio III; see Suisky, p. 134.

48 See Lettres à une Princess d’Allemagne, Part II [E343], Letter X.

49 Boudri: What was so Mechanical about Mechanics, p. 169 f.

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It has become a global principle independent from any specific understanding of forces. It’s not force anymore that defines what is physically possible or not, but in a great extent the PLA itself. In this sense, the PLA does not substitute the PE, but gives it a precise and applicable physical interpretation. As a result, the PE has been transformed from a metaphysical concept of force to a way of conceiving of force by means of its quantifiable effect. The idea that force has to be grounded in a material substance was transformed into a project of grounding force in the structure of matter. Geometrical deduction has taken the place of former attempts at giving force a metaphysical foundation.