E´milie du Chaˆtelet Between Leibniz and Newton edited by Ruth Hagengruber LONDON: SPRINGER-VERLAG, 2012, 256 PP., US $179.00, ISBN 978-94007-2074-9 REVIEWED BY ROBYN ARIANRHOD

he aim of E´milie du Chaˆtelet between Leibniz and Newton is ‘‘to make the intellectual strength of E´milie du Chaˆtelet apparent and to facilitate a better understanding of the singularity of her work.’’ The book focuses on her contribution to the philosophy of science, so Ruth Hagengruber’s editorial introduction begins by asking, ‘‘What is knowledge and how do we achieve confident knowledge?’’ It is a perennial topic of debate, but the purpose here is to explore du Chaˆtelet’s approach to the problem: the book’s eight contributors were participants in a conference celebrating her three-hundredth birthday, at Potsdam in 2006, and most of their contributions focus on aspects of her attempt to provide a Leibnizian-style metaphysical foundation for Newton’s revolutionary mathematical theory of gravity. Du Chaˆtelet was working at a crucial time in scientific history: Newton’s theory was not yet well tested nor universally accepted, and the separation between philosophy and physics was still relatively tentative; indeed, theoretical physics was known as ‘‘natural philosophy’’ well into the nineteenth century. Conflict between supporters of Newton and Leibniz still lingered: Johann Bernoulli I—who had been a friend and advocate of Leibniz—was still active, and du Chaˆtelet was connected with his circle; so was her friend and occasional mathematics teacher Maupertuis, who generally favoured Newton over Leibniz, especially when it came to explaining the motion of the planets around the sun. Leibniz had supported the Cartesian hypothesis, according to which the planets were pushed along in their obits not by a disembodied force of gravitational ‘‘attraction’’ but by the direct impulsion of massive ethereal vortices. Unlike Leibniz and many of his followers, du Chaˆtelet did not allow metaphysical objections to Newton’s theory—in particular, that it ‘‘failed’’ to supply a causal mechanism for gravitational attraction itself—to get in the way of her acceptance of the theory as an astonishingly powerful model of reality. But unlike many of Newton’s awestruck admirers, she did question whether or not his model was the truth about reality. Her path ‘‘between Leibniz and Newton’’ was unusual, and it brought acclaim from many Leibnizians whilst perplexing many Newtonians, notably Voltaire: like many scholars in more recent times, he could not see why du Chaˆtelet would bother trying to integrate the precise, predictive theory of gravity into an

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imprecise Leibnizian philosophical framework. Emilie du Chaˆtelet between Leibniz and Newton helps shed some light on the matter. Philosophy is something that perhaps most working mathematicians and scientists don’t pay enough attention to, so this book offers not only an important contribution to the literature on E´milie du Chaˆtelet, but also insight into the philosophy of science. Nevertheless, in this review-essay for mathematical readers, I will add some Newtonian context. This will necessarily entail a critical reading of some parts of the book—a reading that might seem at odds with its stated intention to showcase du Chaˆtelet’s intellectual strength. But I do not mean to undermine du Chaˆtelet; on the contrary, I agree with the contributors that her intellect was truly remarkable, as was her determination to understand the intricacies of science and philosophy in such an extraordinary era—a post-Newtonian, post-Leibnizian era of transition from ‘‘natural philosophy’’ to modern mathematical physics, which Dieter Suisky likens to the transition between the birth of quantum theory in 1900 and the development of quantum mechanics in 1925. The book’s first chapter is by Hagengruber; then follow chapters by Hartmut Hecht, Sarah Hutton, Fritz Nagel, Dieter Suisky, Andrea Reichenberger, and Ursula Winter, whereas Ana Rodrigues has contributed an extensive bibliography relating to du Chaˆtelet. Chapter 1 is an extensive (54-page) overview of du Chaˆtelet’s ‘‘transformation of metaphysics’’; the chapter summarizes the intellectual process that culminated in du Chaˆtelet’s attempt to synthesize Newton and Leibniz in her book, Institutions de physique, which was first published in 1740; it was revised and published as Institutions physiques in 1742, and it was also published in Italian and German. Hagengruber begins by showing that du Chaˆtelet was highly regarded by Continental intellectuals, including Wolff (Leibniz’s leading exponent), Euler, Maupertuis, Clairaut, and Diderot—a point made and illustrated by most of the other contributors, too. For Hagengruber, this starting point is a necessary counterpoint to the subsequent lack of attention to du Chaˆtelet’s work, which ‘‘was neglected after her death and almost sank into oblivion.’’ Her portrait of du Chaˆtelet-asphilosopher is that of an independent thinker who rebelled against ‘‘blind devotion to Newton’’; a thinker who also rejected parts of the empiricist philosophy of John Locke, who was a friend of Newton and a favourite among lumie`res such as Voltaire. At the same time, du Chaˆtelet reexamined the unfashionable rationalist views of Descartes and the scholastics; her goal was thus ‘‘to establish a new metaphysics, which satisfies the demands of rationality as well as the standards of the experientially dependent contents’’ of a theory. It was certainly an impressive intellectual endeavour, and it is heartening to see her efforts receiving the attention they deserve, and her work set out in accessible detail by Hagengruber (and also by the other contributors). This is an expository chapter, however, not an analytical one, so du Chaˆtelet’s views are not criticized in any way here. But analysis is not Hagengruber’s purpose: she is focussed on tracing the evolution of du Chaˆtelet’s thought,

and showing how it differed, very early on, from that of her peers—including Maupertuis and Voltaire, whose work has generally been held to be the source of most of her ideas. I heartily applaud this goal, and Hagengruber does an excellent job of using du Chaˆtelet’s writings and correspondence to prove her point. But independent thinking is not necessarily correct thinking. I’m in awe of du Chaˆtelet’s achievements, but I think her ‘‘rebuffing’’ of Locke is based on a misreading of his Essay on Human Understanding; after all, her critique appeared in one of her very first works of scholarship (Fable of the Bees). As for her critique of Newton’s theory, this, too, was a relatively early work: she wrote her Institutions de physiques in the late 1730s, and revised it in the early 1740s, before she had thoroughly studied Newton’s Principia (a point made by Hutton). In the main, the philosophical import of Newton’s paradigm isn’t discussed in this book; the emphasis is du Chaˆtelet’s Leibnizian philosophy, so the key work discussed throughout the book is Institutions, with very little on her later translation and commentary on Principia. But I think it is important to note that the Cartesian/Leibnizian approach to scientific theory emphasized the role of causality: it required an intelligible ‘‘sufficient reason’’ or mechanism to explain physical phenomena—such as direct impulsion, as in the hypothetical ethereal vortex model of celestial motion. The key word is ‘‘intelligible,’’ not ‘‘testable.’’ Newton deliberately reversed this approach, emphasizing measurable physical effects rather than metaphysical or mechanical causes, and expressing these effects in quantitative, predictive (testable) mathematical analogies. Even Euler seems to have missed this in the early stages of his work on Newtonian mechanics: Suisky shows that Euler complimented du Chaˆtelet on her clear exposition of the role of hypotheses, which, he agreed, are the only way ‘‘to a certain knowledge of physical causes’’ (my emphasis). Because Leibniz emphasized metaphysical causes, whereas Newton emphasized measurable effects, du Chaˆtelet’s program of trying to unify these two approaches in her Institutions was doomed to fail. Few contributors offer such a direct assessment, preferring simply to outline the details of her arguments, and rightly to praise the intellectual strength and independence behind them. But Hecht sums it up nicely: ‘‘Her scientific methodology proved highly successful in pinpointing philosophical implications of scientific problems in an exemplary way, yet it generally failed when it came to solving particular scientific dilemmas.’’ (Some of these successes and failures are discussed throughout this review.) Winter, however, offers an interesting alternative view, suggesting that the purpose of Institutions was not so much to synthesize Newton and Leibniz as to present two complementary approaches to different aspects of nature: Newton’s system for astronomy, and Leibniz’s view of the universe as ‘‘energetic.’’ She begins her discussion of these complementary approaches by suggesting that in the Enlightenment, Leibniz and Newton offered ‘‘two theories of equal value,’’ and she seems to suggest that du Chaˆtelet believed Leibniz actually had the better methodology. I felt this part of the chapter was weakened by a lack of analysis. Leibnizian metaphysics may have influenced some of

the work of some Enlightenment physicists—including Maupertuis and Euler, according to Hecht and Suisky—and it certainly guided Leibniz himself to his fertile (and fledgling!) idea of conservation of ‘‘living force’’ (kinetic energy). But not even this achievement can compare in complexity, rigour, or influence with Newton’s theory of gravity and its predictive, experimental/mathematical paradigm. I had a similar reaction to parts of several chapters in this book: without adequate analysis of du Chaˆtelet’s later work on Principia, and of the way physics actually developed in the eighteenth century and beyond, the emphasis on du Chaˆtelet’s Institutions can read as if some of the contributors are concerned with repositioning not just du Chaˆtelet but also Leibniz himself. Let me offer another example. Trying to present Leibniz as a thinker comparable to Newton is an understandable but very fraught exercise because it generally entails comparing apples with oranges: Leibniz was the most brilliant and versatile philosopher of his era, whereas Newton was simply incomparable when it came to mathematical physics. But Hagengruber notes, without comment, that du Chaˆtelet claimed Newton and his theory were ‘‘not incomparable,’’ because he built on previous work by Kepler, Galileo, and so on. Winter also makes this point, but in this context she and Hagengruber are focused on Institutions. At the beginning of her later commentary on Principia, du Chaˆtelet expressed a very different view—one that did, indeed, single out Newton. As all the contributors make clear, however, there is still much to be learned from du Chaˆtelet’s Institutions: her intellectual motivations in trying to synthesize Newton and Leibniz provide a marvellous insight into early eighteenthcentury ways of thinking about these two very different approaches to understanding the physical world. And in general, her understanding of the nature of scientific hypotheses was truly modern. In fact, Hagengruber points out that she directly influenced Kant, who ‘‘wrote his first reflections on natural philosophy on the occasion of the publication of her pamphlet on the living forces.’’ Suisky and Winter also discuss du Chaˆtelet’s influence on Kant, and Suisky considers her a forerunner of Popper because of her analysis of the role of hypotheses in scientific theories. Unfortunately, her starting point in this analysis was Newton’s infamous statement ‘‘I do not feign hypotheses,’’ ‘‘Hypotheses non fingo,’’ from the penultimate paragraph of (his second and third editions of) Principia. Hagengruber quotes from Institutions, wherein du Chaˆtelet said, ‘‘M. Newton, and above all his disciples, … rose up against hypotheses.’’ Hagengruber adds that in making his famous statement, Newton ‘‘interdicts further reasoning about the primary cause of gravitation.’’ I’m not sure if du Chaˆtelet herself believed this when she wrote Institutions, but it is certainly a misreading of Newton’s passage. He did not say that it was unnecessary to search for the cause of gravity; rather, he said he hadn’t yet been able to deduce this cause from physical evidence, which was the only way to proceed in his new paradigm for doing theoretical science because ‘‘I do not feign hypotheses.’’ I believe he was referring here to unfalsifiable mechanical or metaphysical causal hypotheses, not to testable hypotheses built on observation and induction,  2016 Springer Science+Business Media New York, Volume 38, Number 4, 2016

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such as those that underlie the theory of gravity itself. Indeed, du Chaˆtelet recognized this, too: in Hagengruber’s words, ‘‘Newton used hypotheses up to the point where the hypothetical connections appeared to be plausible and probable.’’ Hutton also gives specific instances of Newton’s use of hypotheses in his mathematical derivations in Principia. But it is Suisky who points out that the prevailing mechanism of the time gave Newton good reason to be cautious about being seen to ‘‘feign hypotheses’’ about the cause or nature of gravity: Newton wanted to make it clear that all he was offering was a purely mathematical law that described the effects of this mysterious force. Du Chaˆtelet, too, made this clear in her commentary on Principia, in which she, ‘‘like M. Newton himself,’’ distinguished between the gravitational ‘‘force’’ of attraction and the ‘‘cause’’ of that attraction. (Winter actually quotes this passage, but in a different context: she doesn’t link it to any new insight on Newton’s ‘‘not feigning hypotheses’’ since du Chaˆtelet wrote Institutions.) Hutton emphasizes that du Chaˆtelet’s criticisms in Institutions ‘‘are directed not so much at Newton as at his followers, such as Keill and Freind,’’ and that generally ‘‘her criticisms are in line with the kind of criticisms that were being advanced in both France and Germany at that time.’’ It is true that many early Newtonians were responding not so much to Newton’s words as to the preface by his first scientific editor, Roger Cotes: in fact, on the penultimate page of his 1873 Treatise on Electricity and Magnetism, James Clerk Maxwell specifically differentiated between Newton’s ‘‘definite law of attraction at a distance’’ and the ‘‘dogma of Cotes, that action at a distance is one of the primary properties of matter, and that no explanation can be more intelligible than this fact.’’ Physical action-at-a-distance is something that Newton never claimed—as Hecht points out—and something that du Chaˆtelet tried to disprove using Leibnizian metaphysics. Maxwell replaced electromagnetic action-at-a-distance with a mathematical ‘‘field’’ of force, which Einstein adapted in his field theory of gravity. Two centuries earlier, du Chaˆtelet was intellectually courageous to support Newton’s theory of gravity whilst challenging Cotes’s dogma about gravitational action-at-a-distance. She was also right to clarify the role that hypotheses must play during the building of scientific theories. Hagengruber outlines du Chaˆtelet’s program for making ‘‘good’’ hypotheses, as opposed to the ‘‘bad’’ (i.e., causal) hypotheses of Cartesians: (1) hypotheses cannot contradict the (Leibnizian) ‘‘principle of sufficient reason’’; (2) they must be built on the ‘‘certain knowledge of the facts within our reach, and [on] all circumstances attendant upon the phenomena to be explained’’ (to avoid hypotheses being overthrown by new facts); and (3) ‘‘one experiment is not enough for a hypothesis to be accepted, but a single one suffices to reject it when it is contrary to it.’’ She spoke about ‘‘a science of hypotheses, not a science of final truths.’’ She even believed that numbers are hypothetical constructs, in contrast to Maupertuis, who was ‘‘convinced mathematical laws demonstrate truth.’’ Most important, she pointed out that the greater the number of experiments that confirm the logically deduced consequences of a hypothesis, the greater its probability of being correct. This, together with rule (3) (and the essence of rule 80

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(2)), constitutes an insightful and very modern-sounding assessment. As for du Chaˆtelet’s first rule, it obviously misses the experimental, nonmetaphysical point of Newton’s paradigm—as does her use of other Leibnizian hypotheses, including the ‘‘laws’’ of contradiction and continuity, which are briefly discussed by Hagengruber, Hecht, and Winter. Interestingly, Reichenberger shows that even in a philosophical context, du Chaˆtelet’s use of Leibniz’s ‘‘principle of sufficient reason’’ led her into error, specifically in testing the hypothesis of conservation of ‘‘living force.’’ Nevertheless, du Chaˆtelet’s exposition of the role of hypotheses is lucid and important, especially in light of Newton’s brief and apparently confusing comment on ‘‘not feigning hypotheses’’: in Hagengruber’s paraphrasing of du Chaˆtelet, with my emphasis, ‘‘Hypotheses are the basis for preliminary scientific explanations, the scaffolding needed for any form of construction! And, as is generally the case with scaffoldings for buildings, they also supersede their initial use and lose overall significance once the building is finished.’’ Hutton also highlights this analogy, and rightly praises du Chaˆtelet’s analysis of the ‘‘provisional’’ nature of hypothetical theories, until they become more certain through the results of experiments. I don’t think Newton would have disagreed! Certainly Maxwell spelled out clearly, from a working scientist’s point of view, the role of hypotheses in the construction of a modern physical theory, and it is a tribute to du Chaˆtelet’s perspicacity that he wrote in a somewhat similar vein about hypotheses as scaffolding, and about the difference between mathematical analogies and reality. Du Chaˆtelet was also on the mark in her defence of Leibniz’s vis viva or ‘‘living force.’’ The debate between Cartesians (and some Newtonians, including Voltaire) on one hand, and Leibnizians on the other, concerned which formula was the true measure of ‘‘force’’: the ‘‘Cartesian’’ mv or the ‘‘Leibnizian’’ mv2. I use inverted commas because in this context, Descartes and Leibniz generally expressed their equations in words, and it was Newton who first articulated the concept of mass (m). And of course his second law of motion gave a general definition of force, which was not the same as either of the above formulae: rather Newton’s force was defined as a change in the quantity that we now call momentum (mv), but which he, following Descartes, called ‘‘quantity of motion.’’ Du Chaˆtelet played an important role in re-invigorating this debate in the 1740s, and her work on this topic is discussed in several chapters in this book. Hagengruber points out that what was at stake for du Chaˆtelet in this debate was Leibniz’s concept of conservation of ‘‘living force’’ (or, in modern terms, conservation of kinetic energy), as opposed to Descartes’s conservation of ‘‘quantity of motion’’ (or momentum). Furthermore, what would such a law mean for free will? As du Chaˆtelet wrote to Maupertuis, if the ‘‘quantity of force [is] always the same in the universe,’’ then do we, as individuals, have the ‘‘power to set something in motion, or are we not free?’’ Such a question gives a fascinating insight into eighteenthcentury thinking! Hagengruber uses the question of free

will to emphasize the impact of du Chaˆtelet’s questions on Maupertuis’s development of his principle of least action. (She does not discuss the principle of least action itself, but Hecht and Reichenberger give a little more detail on it). In his chapter on du Chaˆtelet and Euler, Suisky offers a brief summary of the ‘‘living forces’’ controversy, and notes that Euler was able to resolve it by distinguishing between finite forces and masses and infinitesimal changes in ‘‘dead force’’ (momentum) and ‘‘living force’’ (kinetic energy). The proper understanding of infinitesimals allows calculus to be used—as both Newton and Leibniz knew. Presumably because of the novelty of calculus, and its then-shaky foundation (in the centuries before limit theory was adequately developed), Newton chose to present most of the proofs in Principia in geometrical form, and Suisky notes that Leibniz, too, deliberately obscured the foundation of his own version of calculus. Euler played the key role in applying calculus to Newtonian mechanics, thereby giving us the modern version. For instance, Suisky gives Euler’s integral equations showing the difference between force, ‘‘living force,’’ and ‘‘dead force’’: integrating force with respect to time gives ‘‘dead force,’’ whereas integrating the same force with respect to distance gives ‘‘living force.’’ Consequently, Suisky claims that as early as 1736, Euler had ended the debate (in a then-unpublished paper). Reichenberger, however, points out that a mathematical understanding of the situation did not equate to an ontological understanding. This is certainly true even from a working scientific point of view, because terms such as energy, work, and momentum were not commonly used and understood until the mid-nineteenth century. Nevertheless, the mathematical understanding was used advantageously in eighteenth-century mathematical physics, not only by Euler but also by Laplace and many others. Reichenberger’s chapter is entirely devoted to ‘‘Emilie du Chaˆtelet’s Institutions in the context of the Vis Viva Controversy,’’ and she gives some historical background to the dispute, as well as a detailed summary of du Chaˆtelet’s arguments in support of ‘‘living force.’’ She also shows how du Chaˆtelet included some Leibnizian metaphysics in her rendering (in Institutions) of Newton’s first and second laws of motion: in particular, in the first law she replaced Newton’s word ‘‘force’’ by ‘‘cause,’’ and in the second law she added a clause noting that the connection between force and change in (quantity of) motion offered a ‘‘sufficient reason’’ for such a change. Although many contributors seem to use du Chaˆtelet’s metaphysical ‘‘amendments’’ or critiques of Newton as evidence of her intellectual independence and of the strength of Leibniz’s approach, Hutton offers a different view: she suggests that the emphasis on Leibnizian metaphysics in Institutions was educational (because du Chaˆtelet addressed the book to her 12-year-old son)—so that the metaphysical emphasis was a way to reach ‘‘readers without an advanced level of mathematical expertise.’’ Hutton adds that du Chaˆtelet’s own knowledge of mathematics and of Principia was limited at that time. In her brief discussion of the ‘‘living force’’ controversy, Hutton argues that du Chaˆtelet tried to reconcile Leibniz and Newton through this debate, by suggesting Leibniz’s

‘‘living force’’ solves a problem identified by Newton in a query in his Opticks. Hutton doesn’t specify the problem— the purpose of her chapter is to summarize the links between du Chaˆtelet’s philosophy and that of Newton’s disciple, Samuel Clarke, about whom she gives some interesting biographical information. But she does say it was a problem ‘‘for which [Newton] had not produced an answer, beyond proposing that force must be conserved by divine intervention to re-calibrate the universe.’’ Reichenberger gives a similarly brief statement, as does Winter. I presume they are referring primarily to Query 31 at the end of Opticks—a query that bothered du Chaˆtelet. In this query, Newton noted that, compared with the idealized Keplerian/Newtonian ellipses, the observed orbits of the planets contained ‘‘some inconsiderable irregularities, [which] may have risen from the mutual [gravitational] actions of comets and planets upon one another, and which will be apt to increase, till this System wants a Reformation.’’ He added that the system ‘‘will continue by the laws [of gravity] for many Ages’’ before it needs such a ‘‘Reformation.’’ Leibniz was scathing about this, saying the Newtonians assume ‘‘God Almighty needs to wind up his watch from time to time… He had not, it seems, sufficient foresight to make it a perpetual motion.’’ As Reichenberger points out, du Chaˆtelet, too, was concerned that Newton seemed willing to envisage a physics that allowed God not just to create the world, but also to intervene and tweak it when necessary. As it turned out, however, Newton was correct in that the solar system does appear to be inherently unstable— although by current estimates it will continue, without disastrous distortions of planetary orbits, for billions more years. Back in 1740, du Chaˆtelet’s concern that Newton had left this ‘‘Reformation’’ up to God was pertinent. Reichenberger notes that du Chaˆtelet (following Leibniz) assumed the law of conservation of ‘‘living force’’ actually ‘‘proved’’ that the universe would not run down, so that there was no need for God to intervene to keep the system stable. It is certainly interesting to see how questions about God and metaphysics helped pioneer such important laws as conservation of energy. But the law of conservation of total energy is not sufficient to explain these planetary distortions, as Laplace would later show. Even at the time, however, it was clear that the law of conservation of ‘‘living force’’ (kinetic energy) does not hold in every situation—it fails, for example, in inelastic collisions, which Leibniz eventually realized. Reichenberger and Hecht suggest this lack of universality was a motivating factor in Maupertuis’s and Euler’s development of the principle of least action. Hagengruber suggests this, too, and says du Chaˆtelet’s questions directly helped Maupertuis in this endeavour. Hagengruber devotes two pages to du Chaˆtelet’s discussions with Maupertuis about elastic and inelastic (solid) bodies—it is exciting to see her intellectual struggle to work out whether or not ‘‘atoms’’ exist, two centuries before the development of our modern view, and Hagengruber shows the depth of du Chaˆtelet’s thought about such fundamental issues. Hagengruber also suggests that the problem of inelastic collisions meant du Chaˆtelet was not uncritical of the ‘‘living force’’ hypothesis—whereas  2016 Springer Science+Business Media New York, Volume 38, Number 4, 2016

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Reichenberger says that in her Institutions, du Chaˆtelet chose to question the experimental results of inelastic collisions rather than the law of conservation of ‘‘living force’’ itself. It is true that not all experiments are valid, but perhaps du Chaˆtelet was so concerned with keeping God out of physics that she was willing to suspend her own rule on the importance of experiments. Either way, this secular goal was an important motivation behind du Chaˆtelet’s attempt to synthesize Newton and Leibniz in her Institutions. In addition to the concerns discussed previously, she worried that because the cause of gravity was not addressed in Newton’s otherwise brilliant theory, then the theory itself must assume that God is the cause. A number of contributors discuss this and related issues: Hagengruber, Hutton, Winter, and Hecht, who adds that du Chaˆtelet drew support from Leibniz’s ‘‘principle of sufficient reason,’’ because she believed even God must have a reason for creating gravitational attraction, and this reason should be the basis of any theory involving gravitational attraction. Hutton, however, points out that theological objections were never a primary feature of Institutions, and that there were even fewer objections in her revised 1742 version; Hutton says this suggests ‘‘that as far as Newtonianism was concerned, theological issues were no longer so pressing.’’ This is in contrast to the emphasis of the other authors mentioned, whose apparent purpose is to present du Chaˆtelet’s position on God in Newton’s theory as evidence of her intellectual independence. From a scientific perspective, du Chaˆtelet’s theological concern about the cause of gravity obviously misses Newton’s point that in physics, it is important to address only what we can reliably know, while at the same time acknowledging there is much more that we do not know. It is true Newton was tempted on occasion to leave what we don’t know to God—notably in some of the queries appended to Opticks, and in the final appendix (or ‘‘general scholium’’) that was added to later editions of Principia, in an apparent attempt to satisfy metaphysically-minded critics who wanted him to declare the source or cause of gravity. It’s also true that in any scientific theory, such an expedient—a ‘‘God of the Gaps,’’ as it has been called in recent times—needs to be critiqued, especially from a philosophical point of view. But making too much of Newton’s occasional comments on God misses the point that he didn’t want to have to prematurely bring a cause into his theory. As he said at the end of the penultimate paragraph of his final ‘‘scholium’’ in Principia: ‘‘It is enough that gravity really exists and acts according to the laws we have set forth, and is sufficient to explain all the motions of the heavenly bodies and of our sea.’’ Enough indeed: for most applications within the solar system, Newton’s theory is accurate at least to one part in ten million. One of the most important aspects of du Chaˆtelet’s attempt to synthesize Leibniz and Newton in her Institutions was her critique of the Newtonian concepts of absolute space and time. She was right to follow Leibniz in suggesting that space and time are merely conventions for the relative order of one body with respect to another, and one event with respect to another. By contrast, in Principia Newton had to define motion in the simplest possible way 82

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so that it could be analysed mathematically—and this meant he had been forced to assume that mathematically, space and time were absolute, the same for everyone no matter where they are in the universe or how they are moving. Einstein supported Newton, saying he had used the only ‘‘fruitful’’ choice open to him at the time; but Einstein added that Leibniz’s (and Huygens’s) objections were ‘‘intuitively well-founded,’’ even though they were supported by ‘‘inadequate arguments.’’ Indeed, du Chaˆtelet’s and Leibniz’s critiques of absolute space and time included theological arguments, as Hutton and Winter show. Suisky adds other metaphysical arguments, and also discusses the related issue of relative motion. He includes Euler’s critiques, too, and mentions his transformation of ‘‘Leibniz’s metaphysical model of monads into the methodology of observation. The metaphysical Leibnizian pattern is transformed into a mechanical model. The perspectives of observers … become part of the theory [of relative motion].’’ Winter says that ‘‘renowned scholars have stated an affinity of [Leibniz’s] theories on space and time with Einstein’s space and time theorem’’ (I presume she means the theory of relativity), but Suisky says it was Euler, not Leibniz or du Chaˆtelet and others, who effectively modelled the observer’s motion analytically ‘‘by means of thought experiments performed with frames of reference that had been later called inertial systems.’’ In other words, unlike Leibniz and du Chaˆtelet, ‘‘Euler replaced the [Newtonian] absolute space with a frame of reference made up of bodies.’’ Hecht, too, briefly mentions Euler’s scientific interpretation of Leibniz’s philosophy of monads, but he adds that du Chaˆtelet’s Institutions—translated into German in 1743, three years before Euler’s work on monads— ‘‘had already presented a much more detailed and more extensive contribution to the field.’’ The purpose of Hecht’s chapter is to compare and contrast Maupertuis’s and du Chaˆtelet’s interpretations of Leibnizian ideas. He describes Leibniz as the source of the metaphysical aspects of Maupertuis’s work on comets and the principle of least action. I found particularly fascinating Hecht’s detailed discussion (and critique) of du Chaˆtelet’s attempt to use the metaphysical principle of continuity in mathematics and physics (including the reflection of light, and falling motion), and of Maupertuis’s resolution of the problem of inelastic bodies and Leibniz’s ‘‘living force’’ via his principle of least action. There is much more in this book, beyond the major themes that I have briefly touched on so far. For instance, Winter outlines the response by du Chaˆtelet’s peers to her commentary on Newton, and she also gives an interesting account of various drafts and editions of Institutions. She uses this to show something important for du Chaˆtelet’s reputation: du Chaˆtelet’s one-time tutor, Samuel Ko¨nig, accused her of plagiarizing her work on Leibniz from Ko¨nig’s notes, but Winter shows that letters and drafts of Institutions prove that du Chaˆtelet was familiar with much of Leibniz before she had even met Ko¨nig. Nagel’s chapter is based on his own thrilling discovery—in 2006—of a complete version of du Chaˆtelet’s Essai sur l’Optique, which until then had been presumed lost. Nagel found the essay in Basel, where Johann I Bernoulli

and his son, Johann II, had lived, and he begins his chapter with a discussion of the relationship between du Chaˆtelet and the famous Bernoulli family. He also discusses the souring of du Chaˆtelet’s relationship with Maupertuis. Du Chaˆtelet’s essay is based on Newton’s Opticks, and Nagel gives a detailed account of its contents, in particular her attempt to grapple with the cause of reflection and refraction. She believed it was some form of attraction—possibly electrical in origin, with a different law from the law of gravity. Again, it is fascinating to see how she grappled with fundamental concepts, and Nagel does an excellent job of explaining and critiquing her efforts. Finally, let me suggest that from a mathematician’s point of view, the most impressive evidence of the strength of du Chaˆtelet’s intellect is her French commentary and translation of Principia: what energy and intelligence it must have taken simply to read and understand the whole magnum opus! The first 110 pages of her appended commentary summarize Newton’s development of the theory of gravity, and discuss such applications as the shape of the earth, the

tides, and the precession of the equinoxes; this summary includes developments by Clairaut, Maupertuis, and Daniel Bernoulli. But the final seventy-odd pages of the commentary contain du Chaˆtelet’s own proofs of many of the theorems in Principia. I mentioned earlier that for various reasons, Newton chose to present his theorems in terms of geometrical propositions and proofs, rather than using his new calculus; du Chaˆtelet reworked a number of these in terms of calculus, using Leibniz’s superior symbolism. This mathematical ‘‘synthesis’’ of Newton and Leibniz is not a feature of Emilie du Chaˆtelet between Leibniz and Newton, but I emphasize it here in support of the aim of this book: restoring du Chaˆtelet to her rightful place in the history of science. School of Mathematical Sciences Monash University Victoria, 3800 Australia e-mail: [email protected]

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