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Niccolò Guicciardini ( ISA AC NEWTON ON MATHEMATICAL CERTAINTY AND METHOD "+ "* ") & Isaac Newton on Mathematical Certainty and Method Transformations: Studies in the History of Science and Technology Jed Z. Buchwald, general editor Dolores L. Augustine, Red Prometheus: Engineering and Dictatorship in East Germany, 1945–1990 Lawrence Badash, A Nuclear Winter's Tale: Science and Politics in the 1980s Mordechai Feingold, editor, Jesuit Science and the Republic of Letters Larrie D. Ferreiro, Ships and Science: The Birth of Naval Architecture in the Scientific Revolution, 1600–1800 Sander Gliboff, H.G. 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Isaac Newton on mathematical certainty and method / Niccolò Guicciardini. p. cm. - (Transformations : studies in the history of science and technology) Includes bibliographical references and index. isbn 978-0-262-01317-8 (hardcover : alk. paper) 1. Newton, Isaac, Sir, 1642–1727—Knowledge-Mathematics. 2. Mathematical analysis. 3. Mathematics-History. I. Title. QA29.N4 G85 2009 510-dc22 2008053211 10 9 8 7 6 5 4 3 2 1 Nam Geometriae vis & laus omnis in certitudine rerum, certitudo in demonstrationibus luculenter compositis constabat. For the force of geometry and its every merit laid in the utter certainty of its matters, and that certainty in its splendidly composed demonstrations. —Isaac Newton, late 1710s (MP, 8, pp. 452–3) Contents Preface xiii Abbreviations and Conventions I Preliminaries 1 xxi 1 Newton on Mathematical Method 3 1.1 Early Influences 3 1.2 First Steps 1.3 Plane Curves 1.4 Fluxions 1.5 In the Wake of the Anni Mirabiles 11 1.6 Maturity 6 7 10 14 2 Newton on Certainty in Optical Lectures 19 3 Descartes on Method and Certainty in the Géométrie 31 3.1 Analysis and Synthesis in Pappus 31 3.2 Analysis and Synthesis in Descartes' Mathematical Canon 3.3 An Example of Analysis and Synthesis of a Determinate Problem 3.4 An Example of Analysis and Synthesis of an Indeterminate Problem 3.5 The Limitations of Descartes' Mathematical Canon 56 38 49 53 viii Contents II Against Cartesian Analysis and Synthesis 4 5 6 59 Against Descartes on Determinate Problems 4.1 Lucasian Lectures on Algebra 61 4.2 Demarcation and Simplicity in the Construction of Equations 4.3 Neusis Constructions 4.4 Conic Constructions 4.5 Against Cartesian Synthesis 4.6 Against Cartesian Analysis 64 68 72 74 76 Against Descartes on Indeterminate Problems 79 5.1 In Search of Ancient Analysis 79 5.2 Porisms 5.3 Newton's Two-Step Approach to the Pappus Problem 5.4 Organic Description of Conics 5.5 Tensions 81 89 93 106 Beyond the Cartesian Canon: The Enumeration of Cubics 6.1 Studies on Cubics 6.2 Absence of Demonstrations in the Enumeratio 109 6.3 Common Analysis in the Enumeratio 6.4 Projective Geometry in the Enumeratio 6.5 New Analysis in the Enumeratio 130 III New Analysis and the Synthetic Method 7 61 The Method of Series 117 137 139 7.1 Wallis's Arithmetica Infinitorum 139 7.2 Criticism Leveled at Wallis 7.3 The Binomial Series 7.4 Infinite Series and Quadratures 7.5 Resolution of Affected Equations 7.6 New and Common Analyses 144 148 154 164 158 121 113 109 Contents 8 9 ix The Analytical Method of Fluxions Barrow 8.2 Preliminaries to the Method of Fluxions 8.3 The Direct Method of Fluxions 8.4 The Inverse Method of Fluxions 8.5 The Inverse Method in De Quadratura 202 8.6 Methodus Differentialis 210 8.7 A Question of Style 193 211 213 213 9.2 "Geometria Curvilinea" 9.3 First and Ultimate Ratios in the Principia 219 9.4 Lemma 2, Book 2, of the Principia 223 217 9.5 Limits in De Quadratura 226 9.6 A Method Worthy of Public Utterance 230 233 The Principia 235 10.1 12 179 186 Synthetic Quadratures in De Methodis IV Natural Philosophy 11 169 The Synthetic Method of Fluxions 9.1 10 169 8.1 Genesis of the Principia 235 10.2 An Overview of the Principia 10.3 Did Newton Use the Calculus in the Principia? Hidden Common Analysis 238 252 259 11.1 Proposition 30, Book 1, of the Principia 259 11.2 Newton's Synthetic Construction for Proposition 30, Book 1 11.3 Descartes' Construction of Third-Degree Equations 11.4 The Analysis behind Proposition 30, Book 1 11.5 Concluding Remarks Hidden New Analysis 262 266 267 12.1 Concessis curvilinearum figurarum quadraturis 267 12.2 The Inverse Problem of Central Forces 268 262 261 x Contents 12.3 Attraction of Solids of Revolution 12.4 Concluding Remarks V Ancients and Moderns 13 14 279 288 291 Geometry and Mechanics 293 13.1 The Preface to the Principia 293 13.2 Mathematical Manuscripts Related to the Preface 299 13.3 Mechanical Curves in the Study of Planetary Motion Analysis and Synthesis 309 14.1 Analysis and Synthesis in "Geometriae Libri Duo" 14.2 Realism and Constructivism 14.3 Analysis and Synthesis in Natural Philosophy VI Against Leibniz 313 The Quarrel with Leibniz: A Brief Overview 16 Scribal Publication, 1672–1699 Newton's Reluctance to Publish 16.2 Manuscript Circulation 16.3 Correspondence with Collins and Leibniz 16.4 Disclosures, 1684–1699 361 Fluxions in Print, 1700–1715 365 In the Public Sphere 17.2 Commercium Epistolicum 17.3 The "Account" 385 339 346 17.1 381 331 339 16.1 Conclusion 315 329 15 17 305 365 372 351 309 Contents xi A Brief Chronology of Newton's Mathematical Work References Index 413 391 389 Preface There is no doubt that Isaac Newton is one of the most researched giants of the scientific revolution. It is triviality to say that he was a towering mathematician, comparable to Archimedes and Carl Friedrich Gauss. Historians of mathematics have devoted great attention to his work on algebra, series, fluxions, quadratures, and geometry. Most notably, after the publication of the eight volumes of Mathematical Papers (1967–1981), edited by D. T. Whiteside, any interested reader can have access to Newton's multifaceted contributions to mathematics. This book has not been written with the purpose of challenging such a treasure of scholarship and information.1 Rather, I focus on one aspect of Newton's mathematical work that has so far been overlooked, namely, what one could call Newton's philosophy of mathematics.2 The basic questions that motivate this book are What was mathematics for Newton? and What did being a mathematician mean to him? It is well known that Newton aimed at injecting certainty into natural philosophy by deploying mathematical reasoning. It seems probable that he entitled his main work The Mathematical Principles of Natural Philosophy (1687) in order to state concisely and openly what constituted the superiority of his approach over the Cartesian Principles of Philosophy (1644). The role of this program in Newton's philosophical agenda cannot be overestimated. Little research has been devoted, however, to Newton's views on mathematical certainty and method, views that are obviously relevant to his program, for if mathematics is to endow philosophy with certainty, it must be practiced according to criteria that guarantee the certainty of its methods. But the new algebraic methods that Newton mastered so well, and that are the salient characteristic of seventeenth-century mathematical development, appeared to many far from rigorous. Newton participated in the debate on the certainty of mathematical method and elaborated his own answer. However, his views on mathematical method have attracted scant attention from historians, 1 So writes Whiteside, the doyen of Newtonian studies, about his edition of Newton's mathematical papers: "Let it be enough that the autograph manuscripts now reproduced are no mere resurrected historical curiosities fit only once more to gather dust in some forgotten corner, but will require the rewriting of more than one page in the historical textbook." MP, 7, pp. vii–viii. I dare to hope that my book will in part answer Whiteside's desideratum; it would have been impossible to write without a many-years-long full immersion in his eight green volumes. 2 This is a term that Newton would not have understood. xiv Preface whereas we know a great deal about, say, René Descartes' or Gottfried Wilhelm Leibniz's philosophies of mathematics.3 In order to implement and divulge his innovative approach to a mathematized natural philosophy, Newton tackled a series of questions that have been overlooked or misunderstood by historians, such as When are mathematical methods endowed with certainty? How can one relate the common and new algebraic analyses of the moderns to the venerated methods of the ancients?4 When is a geometrical construction exact and elegant? What guarantees the applicability of geometry to mechanics? In tackling these issues Newton mobilized deeply felt convictions concerning his role as a philosopher and as a mathematician. He positioned himself against the probabilism endorsed as a moral value by most of the virtuosi of the Royal Society, such as Robert Hooke and Robert Boyle, but he also held a polemical position against two great giants in the common and new analyses, Descartes and Leibniz. On the other hand, Newton found affinities with the thought of Isaac Barrow, Christiaan Huygens, and more obliquely Thomas Hobbes. He reconstructed the development of mathematics from ancient times so as to depict himself as a legitimate heir of the Greek tradition while distancing himself from the moderns. Newton's antimodernism had important consequences for his policy of publication, for the general outlook of his foundational thought, for the mathematical structure of his Principia, and for the negotiations he set afoot in order to trade his mathematics in the milieu of British mathematicians and to affirm the superiority of his method over Leibniz's. In this book I depart from a tendency that has prevailed in the literature devoted to the history of seventeenth-century mathematical methodology. Generally speaking, historians of mathematics have tended to concentrate on questions related to the definitions of basic terms such as fluxion, infinitesimal, limit, and moment, that is, on questions concerning rigor. These questions, which relate to the "labyrinth of the continuum," certainly had a great importance for a deep thinker like Leibniz. Not so for Newton, who was more concerned with questions about the legitimacy, elegance, and exactness of procedures for solving geometrical problems, that is, with questions about method. Leibniz's concerns appear much closer to modern foundational issues in the philosophy of mathematics. Newton's discourse on method is much more opaque to modern readers trained in contemporary foundational literature. This is a natural consequence of scientific training in an era of mathematical 3 Just to take a telling example, in Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (1996), an informed and thorough study of the philosophy of mathematics in the seventeenth century, Newton's name occurs only three times and in passing references. 4 The terms common analysis and new analysis referred to Cartesian algebra carried on via polynomial equations and to a more general algorithm where infinitesimals, infinite series, and infinite products could be deployed. Preface xv practice shaped by the exigencies that emerged after the early-nineteenth-century rigorization of the calculus, axiomatization of geometry, and development of abstract algebra. Since then, definitions of basic terms have become a prominent feature of the foundations of mathematics. Newton did devote attention to, say, the nature of infinitesimals, but he was much more concerned with methodological issues related to the analysis (or resolution) and synthesis (or composition) of geometrical problems. In this he showed himself to be influenced by Descartes, who was deeply involved in the program of redefining his algebraic method of analysis as an alternative to the analysis of the ancients ("Analysis Veterum").5 Descartes also traced the boundaries between legitimate (exact) and illegitimate (not exact) means of geometrical construction and classed them on the basis of criteria of simplicity that broke with the ancient tradition conveyed by Pappus's Collectio. Newton fiercely rejected Descartes' canon of problem resolution and composition, and proposed an alternative that looked to ancient exemplars as models superior to the mathematics practiced by the "men of recent times." But Newton's admiration for the ancients—arguably intertwined with many facets of his philosophy of nature and religion—opened a gap between his views on mathematical method and his mathematical practice. Several hitherto unexplained aspects of Newton's mathematical work are related to this condition of stress and strain characterizing his thoughts on mathematical method. Why did Newton fail to print his method of series and fluxions before the inception of the priority dispute with Leibniz? Did he use his new analysis in the Principia, as he claimed in retrospect? And if so, why did he hide his competence in his new methods of quadrature when writing the Principia? Why does his Arithmetica Universalis, a work devoted to algebra, end with pointed criticisms of the use of algebraic criteria in the construction of geometrical problems? Why did Newton, a master of symbolic manipulation, express deep disparagement of the algebraists (whom he, according to hearsay, labeled the "bunglers of mathematics")?6 Which strategies did he adopt in order to maintain the superiority of his method compared to Leibniz's extraordinarily efficient algorithm? Why did he engineer Commercium Epistolicum (1713), which was meant to prove his priority in the discovery of the calculus, using documents that proved everything he had achieved on series and quadrature but revealing nothing about the rules of the calculus? Why did he—contra Leibniz—attribute these rules to Isaac Barrow rather than to himself? Was not therefore his policy during the priority dispute lacunose and contradictory? Historians have found these questions embarrassing. It is not unusual to encounter attempts to formulate answers in terms of psychological motivations, a 5 6 Bos, Redefining Geometrical Exactness (2001). Hiscock, David Gregory, Isaac Newton and Their Circle (1937), p. 42. xvi Preface sign—in my view—of the difficulty of making sense of Newton's policy and convictions. So, we often read that it was Newton's neurotic shyness, or a lack of interest in mathematics that suddenly crept into his mind, that prevented him from printing his method. Further, we are told that in the Principia he studiously tried to be obscure in order to avoid dispute and criticisms because of his obsessive feeling of being persecuted. We even encounter the thesis according to which, in the polemic against algebra and calculus, he was trying to hide his indebtedness toward an intellectual father (Descartes) whose image he found oppressive. When we move to consider the polemic with Leibniz, we find studies that highlight Newton's obsessive approach to the priority dispute, his lack of fairness, his egotism, and even his political motivations related to the Hanoverian succession. Little effort is made to try to discern in such a muddied context (and muddied and political it certainly was!) coherent methodological positions held by Newton and by his opponents. I believe we should be able to find better answers by studying more carefully Newton's writings on the nature of mathematics, which are so abundant in his manuscripts. True, the search for Newton the philosopher of mathematics is at times frustrating. What we find is a disconnected, sometimes contradictory, constellation of pronouncements scattered in the margins of mathematical manuscripts, in aborted prefaces and appendices, in letters and personal notes. They serve their purpose in a dialogical context, providing defensive grounds for mathematical practices, orienting aims, and establishing hierarchies. However, in these writings Newton reveals himself as a mathematician who—even when shattered by psychological disturbances, stymied by academic rivalries, and motivated by political interests— is able to endorse a fairly clear view of method and certainty. It is this view that prompted him to articulate his mathematical work according to codes of communication that were understood by his contemporaries, especially by his close acolytes, but that turn out to be so puzzling for modern readers. In short, even in the heated context of the priority dispute Newton has something to tell us about mathematical method. We should consider his theses, even though his stature as a philosopher of mathematics is inferior to a Leibniz or a Descartes. In this book I study Newton's methodology of mathematics by analyzing his main works, from the early treatises on series and fluxions to the writings addressed against Leibniz. Even though my book is not devoted to Newton's mathematical results, it is important to try to understand his mathematics, because his views on mathematical method interacted with his mathematical practice in a complex way. Therefore, the reader will find some pages in which I analyze Newton's mathematics at work. I have been extremely selective, and the examples I have chosen are not meant to offer an exhaustive view of Newton's mathematics; he was simply too prolific to allow condensing the wealth of his results in a single book. I have tried, however, to choose examples that are simple enough to be followed by a reader equipped with a modicum of mathematical expertise and that are representative of a method or an approach that Newton developed. The expert mathematician Preface xvii will find most of the examples far too simple. Newton was indeed a great problem solver, and his best mathematics can be admired by reading Mathematical Papers and its editor's profound commentary. In matters of notation I have avoided translating Newton's mathematics into modern terms.7 The mathematician not trained in history will find Newton's mathematical language and practice somewhat puzzling. For instance, Newton and his contemporaries did not talk about functions, but about continuously varying magnitudes. Newton did not use an equivalent of Leibniz's integral sign consistently; most often he used connected prose and referred to "the area under the curve" or "the fluent of." He was also somewhat confused about the distinction between definite and indefinite integral, and never rendered constants of integration explicit. Therefore, he often used the singular ("the fluent of"), but he was of course aware that the indefinite integral identifies a class of functions. Further, he dealt with the convergence of infinite series in very intuitive terms; their convergence was tested in cavalier terms. We have to wait for Augustin Louis Cauchy at the beginning of the nineteenth century for a modern theory of convergence. The list of oddities could continue. I have made no effort (with the exception of a few explanatory footnotes) to avoid the distinctive character of Newton's mathematics compared to modern usage. The book is divided into six parts. Part I provides some preliminaries: a survey of Newton's mathematical work and of the development of his ideas on mathematical method that began to mature just after the creative burst of the anni mirabiles (chapter 1); a comparison between his youthful program in natural philosophy with the one endorsed by influential contemporaries like Descartes, Hooke, and Boyle (chapter 2); a presentation of Descartes' ideas on analysis and synthesis as Newton found them in the Géométrie (chapter 3). In fact, Newton, who was in his mathematical practice so much a Cartesian, stood in opposition to Descartes with respect to method. Part II considers the first period of Newton's methodological thought. He began distancing himself from Cartesian method in writings that date from the 1670s in which he compared common analysis (i.e., Cartesian algebra) to ancient analysis. The occasion for these reflections was Newton's involvement in the project of revising a textbook on algebra by the Dutchman Gerard Kinckhuysen and his commitment to prepare lectures on algebra (chapter 4). It is in this context that Newton began reading Pappus with the purpose of recovering the ancient analysis (chapter 5). In this period Newton also worked on cubic curves (chapter 6); and it is in this research that the tensions between his mathematical practice and his views on method surfaced. 7 So, for instance, I avoid the integral sign and prefer to talk about the "quadrature of a curve" rather than the "integral of a function." xviii Preface Part III is devoted to Newton's attempts to provide a synthetic version of what he labeled "method of series and fluxions." Newton's early researches on series (chapter 7) and fluxions (chapter 8) were carried on in terms unacceptable to his more mature standards of validation. Consequently, he aimed at developing a synthetic method of fluxions—a "more natural approach," as he says—consonant with the practice of the ancient geometers. These researches culminated in the early 1680s in a treatise entitled "Geometria Curvilinea," in which Newton elaborated his method of first and ultimate ratios, a method that informs the most mature presentation of his method of series and fluxions offered in the Principia and De Quadratura (chapter 9). Part IV considers the mathematical methods employed by Newton in the Principia (chapter 10). I devote particular attention to the strategies he chose in order to accommodate his analytical methods in common analysis (chapter 11) and new analysis (chapter 12) in the body of a text that he presented as written according to the "ancient and good geometry." Part V concerns perhaps the most philosophically freighted texts that Newton wrote in the 1690s and early 1710s. In these decades his belief in the myth about a prisca sapientia of the ancients prospered and determined his self-portraiture as an heir to the mathematics of Euclid and Apollonius. Consequently, he wrote at length on the relations among analysis, synthesis, algebra, natural philosophy, and mechanics (chapters 13 and 14). Newton's views concerning mathematical method emerge again in the polemic with Leibniz, which occupied him especially from 1710 (chapter 15). In order to trace the rationale for Newton's polemical strategy, Part VI devotes attention to his policy of publication adopted in circulating manuscripts, in correspondence (chapter 16), and later in printing some of his tracts on the new analysis that he had composed years before (chapter 17). Acknowledgments During the long period necessary to write this book I profited from the advice of a great number of scholars. First, I should mention my debt to two giants in the Newtonian scholarship, I. Bernard Cohen and Derek T. Whiteside; without their impressive works and the generous support they gave me through correspondence, my research would not have been possible. Colleagues in Siena (Emanuela Scribano and Alessandro Linguiti), Milan (Umberto Bottazzini), and Bergamo (Andrea Bottani, Richard Davies, and Maddalena Bonelli) were of help in many ways (thank you, Ric, for rescuing me from the pitfalls of too many "Italianisms"). It is a pleasure also to thank Fabio Acerbi, Franco Giudice, Massimo Mugnai, and Michael Nauenberg for much-needed assistance on ancient geometry, optics, Leibniz, and mathematical physics, respectively. A one-year sabbatical spent as a Visiting Fellow at Clare Hall (Cambridge) allowed me to begin planning my book in a friendly Preface xix atmosphere. At Cambridge I exploited Scott Mandelbrote's kind willingness to share his expertise in Newtoniana with me. A subsequent semester as Mellon Visiting Professor at the California Institute of Technology gave me the chance of working in one of the most important centers for early-modern history of science and Newtonianism in particular. I thank the Andrew W. Mellon Foundation for generous support. At Caltech the exchanges I had with Jed Buchwald, André Carus, Moti Feingold, Kristine Haugen, Diana Kormos-Buchwald, Rose-Mary Sargent at Dabney and with Tinne Hoff Kjeldsen, Andrea Loettgers, Jesper Lützen, Tilman Sauer during seminars at the Einstein Papers Project, and a memorable visit from George Smith and Christopher Smeenk, shaped many of the theses defended in this book. During meetings of the Wissenschaftlicher Beirat of the Bernoulli-Edition in Basel I received advice from one of the greatest Leibniz scholars, Eberhard Knobloch. My former supervisor and present friend Ivor Grattan-Guinness and the distinguished physicist and writer Klaas Landsman read and extensively commented on a draft of this book. Henk Bos's detailed comments rescued me from a great number of blunders; I am grateful to him for the extreme, and undeserved, kindness with which he addressed the task of revising my script, equaled by the widely appreciated rigor of his scholarship. In the very last stage of preparation of the book for the press, Massimo Galuzzi rendered his profound knowledge of Cartesian and Newtonian mathematics available to me with a generosity exemplary of what excellency in scholarship should be; I consider it great luck to have him as one of my correspondents and friends. The competence of the librarians working in the "Angelo Mai" library in Bergamo, the library of the Institute and Museum of the History of Science in Florence, the University Library in Cambridge, the Royal Society Library in London, the Huntington Library in San Marino, and the Millikan Library at Caltech rendered this book possible. Sara Damiani of the Centro Arti Visive of the University of Bergamo generously helped me in preparing the figures with Photoshop. Silvia Maschio of Compomat s.r.l transformed my sloppy LATEX file into a magnificent camera-ready pdf. The help of Laura M. Keith and Sergio Knipe in revising the English is gratefully acknowledged. Without the patience and kindness of the staff at MIT Press, producing this book would have been much more difficult. The greatest "thank you" is for my wife, Elena, and my daughter, Pepi; they know why. Abbreviations and Conventions All dates are old style (O.S.) unless otherwise specified (N.S.). When necessary, I use January 17/27, 1712/13 for January 17, 1712 (O.S.) = January 27, 1713 (N.S.). The following abbreviations are employed: § refers to section numbers in this book. ∝ = "is proportional to." "Account" = "An Account of the Book Entituled Commercium Epistolicum Collinii & Aliorum de Analysi Promota; Published by Order of the Royal Society, in Relation to the Dispute Between Mr. Leibnitz and Dr. Keill, About the Right Invention of the Method of Fluxions, by Some Call'd the Differential Method." Philosophical Transactions 29, no. 342 (1715): 173–224. Author: Isaac Newton. Published anonymously. Facsimile in Hall, Philosophers at War (1980), pp. 263–314. Add. = Additional manuscript in the Cambridge University Library. AT = Descartes, René. Oeuvres de Descartes. Edited by C. Adam and P. Tannery. 11 vols. New ed. Paris: Vrin, 1964–67. Commercium Epistolicum = Commercium Epistolicum D. Johannis Collins, et Aliorum De Analysi Promota: Jussu Societatis Regiae In Lucem Editum. London: typis Pearsonianis, 1712 (printed late 1712, distributed January–February 1713 (N.S.)). Correspondence = Newton, Isaac. The Correspondence of Isaac Newton. Edited by H. W. Turnbull, J. F. Scott, A. R. Hall, and L. Tilling. 7 vols. Cambridge: Cambridge University Press, 1959–77. De Analysi = MS LXXXI, no. 2 (Royal Society of London). Author: Isaac Newton. Title: De Analysi per Aequationes Numero Terminorum Infinitas. Date of composition: 1669. Edited in MP, 2, pp. 206–47. First printed in Newton, Analysis per Quantitatum (1711), pp. 1–21. De Methodis = MS Add. 3960.14 (Cambridge University Library). Author: Isaac Newton. Title: untitled since the first folio is lacking, known as Tractatus de Methodis Serierum et Fluxionum. Date of composition: 1670–1671. Edited in MP, 3, pp. 38–328 (on pp. 32–7 the first missing leaf is from a transcript by William Jones). First printed in English translation as Newton, Method of Fluxions (1736). De Quadratura = Tractatus de Quadratura Curvarum. First printed in Newton, Opticks (1704), pp. 165–211. Excerpts were communicated to Wallis for inclu- xxii Abbreviations and Conventions sion in volume 2 of his Opera (1693), pp. 390–6. Date of composition: early versions in 1691–1692, revised for publication in 1703. The numerous early versions and the revisions (mostly in MSS Add. 3960.7–13, 3962.1–3, and 3965.6 (Cambridge University Library)) are edited in MP, 7, pp. 24–182, and MP, 8, pp. 92–167. Enumeratio = MS Add. 3961.2, ff.1r–14r (Cambridge University Library). Author: Isaac Newton. Title: Enumeratio Linearum Tertii Ordinis. Date of composition: 1695. Edited in MP, 7, pp. 588–645. First printed in Newton, Opticks (1704), pp. 139–62 (+ 6 Tables). Geometria = Descartes, René. Geometria, à Renato Des Cartes Anno 1637 Gallicè Edita. Amsterdam: Apud Ludovicum & Danielem Elzevirios, 1659–61. "Geometriae Libri Duo" = Edited by D. T. Whiteside mostly from Add. 3962.1, 3962.3, 3963.2, and 4004 (Cambridge University Library). Author: Isaac Newton. Title: "Geometriae Liber Primus" and "Geometriae Liber Secundus." Date of composition: mid-1690s. Edited in MP, 7, pp. 402–561. Géométrie = Descartes, René. La Géométrie. In René Descartes, Discours de la Méthode, pour Bien Conduire sa Raison et Chercher la Verité dans les Sciences: Plus la Dioptrique, Les Meteores, et la Geometrie, Qui Sont des Essais de Cete Methode, pp. 297–413. Leiden: Maire, 1637. Intended Preface = MS Add. 3968.9, f. 109 (Cambridge University Library). Author: Isaac Newton. Date of composition: late 1710s. Edited in MP, 8, pp. 452–59. Preliminary drafts in MP, 8, pp. 442–52. Cohen and Whitman provide an English translation in Newton, Principles, pp. 49–54. Lucasian Lectures on Algebra = MS Dd.9.68 (Cambridge University Library). Author: Isaac Newton. Title: untitled. Edited in MP, 5, pp. 54–491. Deposited late 1683 or early 1684. First printed as Newton, Arithmetica Universalis (1707). MP = Newton, Isaac. The Mathematical Papers of Isaac Newton. Edited by D. T. Whiteside. 8 vols. Cambridge: Cambridge University Press, 1967–81. Optical Lectures = MS Add. 4002 (Cambridge University Library). Author: Isaac Newton. Title: Lectiones Opticae. Date of composition: 1669/70—2. Edited in Newton, The Optical Papers (1984), pp. 46–279. A later version, MS Dd.9.67 (Cambridge University Library), entitled Opticae pars 1a and Opticae pars 2da , deposited in October 1674, appeared posthumously, first in an English translation of pars 1a as Optical Lectures: Read in the Publick Schools of the University of Cambridge (1728), and then the complete Latin text as Lectiones Opticae (1729). Preface = "Praefatio ad Lectorem." In Newton, Isaac. Philosophiae Naturalis Principia Mathematica (1687), pp. i–ii. Facsimile (as in third (1726) edition) in Principia, pp. 15–7. English translation in Principles, pp. 381–3. Principia = Newton, Isaac. Philosophiae Naturalis Principia Mathematica, The Third Edition (1726) With Variant Readings. Edited by A. Koyré and I. B. Cohen, assisted by A. Whitman. Cambridge: Cambridge University Press, 1972. Abbreviations and Conventions xxiii Principles = Newton, Isaac. The Principia: Mathematical Principles of Natural Philosophy . . . Preceded by a Guide to Newton's Principia by I. Bernard Cohen. Translated by I. B. Cohen and A. Whitman, assisted by J. Budenz. Berkeley: University of California Press, 1999. Isaac Newton on Mathematical Certainty and Method I Preliminaries It is useful to have a general scheme of Newton's mathematical work at the outset, before delving into the details of his works on algebra (part II), fluxions (part III), natural philosophy (part IV), geometry (part V), and the more philosophical issues that emerged in the context of the polemic with Leibniz (part VI). Therefore, in chapter 1, I give a general overview of the development of Newton's ideas on mathematical method, from his anni mirabiles down to his maturity. Chapter 2 considers a program that Newton stated as a very young Lucasian Professor in his Optical Lectures. According to this program, which never ceased to inform Newton's agenda, mathematics is the vehicle that can guarantee certainty in natural philosophy. This—he was adamant—had escaped Descartes and his followers, who had remained trapped in a qualitative and hypothetical mode of explanation of natural phenomena. When considering the ideas that Newton later developed concerning mathematical method, this agenda should always be kept in mind. Chapter 3 considers the great antagonist against whom Newton positioned himself: Descartes. What Descartes had to say in the Géométrie (1637) concerning method and certainty was rejected by Newton for reasons which become the object of study for the rest of the book. We cannot understand Newton's methodology unless we are aware of the antagonist that he always had in mind, and appreciate that his polemic against Cartesian method engendered a deep tension between his mathematical practice, which was indebted to Cartesian ideas, and his methodological views, which were at odds with it. Therefore, chapter 3 considers the basic elements of the Cartesian mathematical canon. 1 Newton on Mathematical Method: A Survey For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more then at any time since. — Isaac Newton, 1718 1.1 Early Influences Mathematics played a prominent role in Newton's intellectual career. This was not, of course, his only concern. A polymath and polyhistor, Newton devoted years of intense research to the reading of the Books of Nature and Scripture, deploying the tools of the accomplished "chymist" (at the furnace and at the desk), instrument maker (he made his own instruments, among them the first reflecting telescope), experimentalist, astronomer, biblical interpreter, and chronologist. In all these fields mathematics entered as one of the most powerful and reliable tools for prediction and problem solving, and as the language that guaranteed accuracy and certainty of deduction. Newton would not have achieved most of his results without it.1 It is no coincidence that the adjective mathematical enters into the title of his masterpiece. When Newton matriculated at Cambridge in 1661, he possessed only a modicum of mathematical training. Two years later the first Lucasian Chair of Mathematics was conferred on Isaac Barrow, a scholar of broad culture who would play an important role in Newton's intellectual life. The existence of such chairs, which provided mathematical teaching at the universities, was something of a novelty in England.2 Barrow therefore had to defend his discipline and lectured on the usefulness of mathematical learning. He did so in verbose and scholarly lectures, which Newton probably attended. Barrow patterned his peroration following the agenda set by Proclus, and he had in mind a late-sixteenth-century debate over the certainty of mathematics, which was sparked in 1547 by Alessandro Piccolomini's commentary Epigraph from MS Add. 3968.41, f. 85r. For a discussion of this memorandum see Westfall, Never at Rest (1980), p. 143, and "Newton's Marvelous Years of Discovery and Their Aftermath" (1980); Hall, Philosophers at War (1980), pp. 10–23. See also Whiteside, "Newton's Marvellous Year" (1966). The best guide to Newton's mathematical work is to be found in Whiteside's commentary to Mathematical Papers. 1 Jed Buchwald and Mordechai Feingold are currently examining Newton's work on chronology. Their research reveals the importance of new mathematical techniques in treating astronomical and historical data. 2 For the antecedents, see Feingold, The Mathematicians' Apprenticeship (1984). 4 Table 1.1 Chapter 1 Mathematical Books Annotated by Newton in the 1660s René Descartes Geometria, à Renato des Cartes, Amsterdam, 1659–61 François Viète Opera Mathematica, Leiden, 1646 Frans van Schooten Exercitationum Mathematicarum, Leiden, 1657 William Oughtred Clavis Mathematicae, 3d ed., Oxford, 1652 John Wallis Operum Mathematicorum Pars Altera, Oxford, 1656 John Wallis Commercium Epistolicum, Oxford, 1658 on pseudo-Aristotle's Problemata Mechanica.3 The rising status of mathematics was opposed by some Aristotelian philosophers like Piccolomini, who maintained that mathematics did not possess the deductive purity of syllogistic logic and was not a science because it did not reveal causal relationships. Barrow's defense of geometry as a model of reasoning and his idea that since geometrical magnitudes are generated by motion, a causal relationship can be captured in such mechanically based geometry must have impressed the young scholar. These typically Barrovian ideas remained the backbone of Newton's views about mathematics. Newton soon began to read advanced mathematical texts, possibly borrowing them from the Lucasian Professor. The mathematical books he had on his desk, which he annotated extensively, are listed in table 1.1. As is often repeated in later memoranda and hagiographic biographies, he devoted little attention to ancient geometry, which is at odds with his mature predilection for the ancients, which began to flourish in the 1670s. As far as we know, of the ancient corpus he studied only Euclid's Elements in Barrow's algebraized edition.4 He learned algebraic notation from Oughtred's Clavis Mathematicae in the third 1652 edition, and from Viète's Opera Mathematica (1646). These last two works were based on the idea that algebra is not a deductive theory, like the Elements, but rather an analytical, heuristic tool that can extend the possibility of finding solutions to problems, especially geometrical problems. The annotations to Oughtred and Viète show how interested Newton was in this promising method of discovery.5 Algebra was still a novel language in England. Oughtred had been a pioneer (his Clavis had first appeared in 1631), but in the 1660s there was still need for an updated text on algebra. In 1669, Newton became involved in the project of producing such a text3 Piccolomini, In Mechanicas Quaestiones Aristotelis (1547). On Barrow's reading of Proclus, see Stewart, "Mathematics as Philosophy: Proclus and Barrow" (2000). On the debate initiated by Piccolomini, there is a vast literature; see Jardine, "Keeping Order in the School of Padua" (1997). 4 Euclidis Elementorum Libri XV Breviter Demonstrati (1655). 5 MP, 1, pp. 25–88. Newton on Mathematical Method 5 book (see chapter 4). Algebra was interesting as a tool for practical applications (it answered the needs of cartographers, instrument makers, mechanics, accountants, land surveyors) but it was also promising for more theoretical purposes. The latter motivation was the stimulus for Newton. The seminal text in Newton's mathematical formation is a highly abstract essay: Descartes' Géométrie. He borrowed and annotated the second Latin edition (1659– 1661) by Frans van Schooten.6 Here Descartes had proposed a novel method for the solution—he claimed in the opening sentence—of all the problems of geometry. It was on this text that Newton concentrated his attention. Descartes taught how geometrical problems could be expressed in terms of algebraic equations (this process was termed the resolution or analysis of the problem). He maintained that finding the equation and determining its roots, either by finite formulas or approximations, is not the solution of the problem (see chapter 3). It was not a surprise for the contemporaries of Descartes and Newton to read that in order to reach the solution, one had to geometrically construct the required geometrical object. A geometrical problem called for a geometrical construction (a composition or synthesis), not an algebraic result. Traditionally, such constructions were carried out by means of intersecting curves. Thus, Descartes provided prescriptions to construct segments that geometrically represent the roots and are therefore the solution of the problem. By Newton's day the heuristic method proposed by Descartes was labeled common analysis. It was contrasted with a more powerful new analysis, which tackled problems about tangents and curvature of curves and about the determination of areas and volumes that cannot be reached by the finitist means envisaged by Descartes. Common analysis proceeds by "finite" equations (algebraic equations, we would say) in which the symbols are combined by a finite number of elementary operations. The new analysis instead goes beyond these limitations because it makes use of the infinite and infinitesimal. Basically, Newton and his contemporaries understood both the common analysis and the new analysis, where respectively "finite" and "infinite equations" (infinite series and infinite products) were deployed, as heuristic tools useful in discovering a solution. Analysis, however, had to be followed by synthesis, which alone, in their opinion, could provide a certain demonstration. Barrow much concerned himself with synthesis and, in his lectures defending mathematical certainty, aimed to provide synthetic demonstrations of the results reached by the heuristic techniques characteristic of the new analysis. His young protégé was making inventive forays into the new analysis. Newton was aware, however, that a synthetic construction was needed, and he later turned to Barrow for inspiration. 6 Newton worked on the second Latin edition, but he might also have encountered the smaller first Latin edition prepared by van Schooten, which appeared in 1649. A copy of the first edition (University Library (Cambridge) Adv.d.39.1) might have been in Newton's possession, but "its brief manuscript annotations are not in Newton's hand." MP, 1, p. 21. 6 Chapter 1 1.2 First Steps Newton's early notes on Descartes' Géométrie reveal how quick he was in mastering algebra applied to geometry. In 1665 he began to think about how the equation could reveal properties of the curve associated to it via a coordinate system. Actually, he began to experiment with alternative coordinate systems to orthogonal or oblique axes. He tried, for instance, what we call polar, bipolar, or pedal coordinates. He also began to work with transformation of coordinates. One line of research consisted in trying to extend algebraic treatment beyond the conic sections. In De Sectionibus Conicis, Nova Methodo Expositis Tractatus, which Newton read in the Operum Mathematicorum Pars Altera (1656), Wallis had developed an algebraic treatment of conics as graphs of second-degree equations in two unknowns. Newton began to extend the definitions of diameter, chord, axis, vertex, center, and asymptote to higher-order algebraic curves. In the late 1660s he made his first attempts to graph and classify cubic curves.7 Another line of research concerned the so-called organic description (or generation) of curves.8 This was an important topic, since in order to determine the point of intersection of curves in the construction of geometrical solutions, it was natural to think of the curves as generated by a continuous motion driven by some instrument (an oργανoν). It is the continuity of the motion generating the curves that guarantees a point of intersection can be located exactly. Descartes had devised several mechanisms for generating curves. In De Organica Conicarum Sectionum in Plano Descriptione Tractatus (1646), which Newton read in Exercitationum Mathematicarum (1657), van Schooten had presented several mechanisms for generating conic sections. This research field was connected with practical applications, for instance, lens grinding and sundial design, but it was also sanctioned by classical tradition and motivated the highly abstract needs underlined by Descartes. Newton was able to devise a mechanism for generating conics and to extend it to higher-order curves (§5.4). In 1665, Newton deployed organic descriptions in order to determine tangents to mechanical lines, that is, plane curves such as the spiral, the cycloid, and the quadratrix that Descartes had banned from his Géométrie (see chapter 3). The study of mechanical lines, curves that do not have an algebraic defining equation, was indeed a new, important research field. How to deal with them was unclear. Newton was able to determine the tangent to any curve generated by some tracing mechanism. He decomposed the motion of the tracing point P , which generates the curve, into two components and applied the parallelogram law to the instantaneous component velocities of P (see the parallelogram law on the top of the left margin 7 8 MP, 1, pp. 155–244. MP, 2, pp. 134–42 and 152–5. Newton on Mathematical Method 7 in figure 1.1). For instance, the point of intersection of two moving curves will generate a new curve whose tangent Newton was able to determine. Such a method for determining tangents without calculation pleased Newton as much as did his new techniques for the organic description of conics. This was an approach to the study of curves—alternative to the Cartesian algebraic—that Barrow had promoted and that in the 1670s Newton began to couple with ideas in projective geometry. Already in 1665 the master in the common and new algebraic analyses was experimenting with non algebraic approaches to geometrical problems. In these early researches one encounters a characteristic of Newton's mathematical practice, a deep intertwining between algebra and geometry, that eventually led to unresolved tensions in his views on mathematical certainty and method. Indeed, it is often the case that in tackling a problem Newton made recourse to a baroque repertoire of methods: one encounters in the same folios algebraic equations, geometrical infinitesimals, infinite series, diagrams constructed according to Euclidean techniques, insights in projective geometry, quadratures techniques equivalent to sophisticated integrations, curves traced via mechanical instruments, numerical approximations. Newton's mathematical toolbox was rich and fragmentary; its owner mastered every instrument it contained with versatility. But he was also a natural philosopher who envisaged a role for mathematics that did not allow him to leave the toolbox messy, albeit efficient, and open for unauthorized inspection. 1.3 Plane Curves How did the young Newton tackle a problem that was quite difficult in his day: the drawing of tangents to plane curves? Figure 1.1 shows the first folio of a manuscript dated by Newton (in retrospect?) November 8th, 1665, and entitled "How to Draw Tangents to Mechanicall Lines." In the left margin there are an Archimedean spiral, a trochoid, and a quadratrix.9 Tracing the tangent to the spiral was particularly handy. To a point b of a spiral with pole a (see figure 1.2) Newton associated a parallelogram having a vertex in b whose sides, the former bc directed along the radius vector ab and the latter bf orthogonal to it, are proportional to the radial speed and to the transverse speed of b. The diagonal bg determines the tangent at b. In other cases, the method was more difficult to implement, and Newton made a couple of blunders, which he soon corrected, in tracing the tangent to the quadratrix and to the ellipse.10 9 In modern symbols these three curves have equations r = c θ (r, θ polar coordinates, c con0 0 stant), x = c1 t − c2 sin t, y = c1 − c2 cos t (parametric equations, c2 < c1 ), and x = y cot(πy/2c3 ) (x, y, Cartesian coordinates). 10 MP, 1, pp. 379–80. 8 Chapter 1 Figure 1.1 Newton's kinematic method for drawing tangents to mechanical curves. From top to bottom of the left margin, below the illustration of the parallelogram, are the following curves. (i) The Archimedean spiral is traced by a point that slides with constant speed along a straight line that rotates with constant angular speed. (ii) The trochoid (sometimes called curtate cycloid) is traced by a point on a disk that rolls without sliding along a straight line. (iii) The quadratrix is a curve traced by the intersection of a radius and a line segment moving at corresponding rates. A square and a circle are drawn so that one corner of the square is the center of the circle, and the side of the square is the radius of the circle. A radius rotates clockwise from the side of the square to the base at a constant angular speed. At the same time, a line segment falls from the top of the square at constant vertical speed and remains parallel to the base of the square. Both start moving at the same time, and both hit the bottom at the same time. Newton also considers two "Geometricall lines," namely (iv) the ellipse, and (v) the hyperbola. Source: Add. 4004, f. 50v. Reproduced by kind permission of the Syndics of Cambridge University Library. Newton on Mathematical Method 9 Figure 1.2 Tracing the tangent to the Archimedean spiral. Source: Add. 4004, f. 50v. Reproduced by kind permission of the Syndics of Cambridge University Library. Newton applied his method for drawing tangents not only to mechanical but also to geometrical lines: the ellipse and the hyperbola.11 In his early papers, Newton intertwined the geometrical approach to tangents with the development of a new algorithm, which he called the method of series and fluxions. This method allowed the calculation of the tangent and curvature to all plane curves known in Newton's day. Later, I describe Newton's algorithm for the determination of tangents (§8.3.6) and its application to the conchoid (Cartesian equation x2 y 2 = (c1 + y)2 (c22 − y 2 )). Undoubtedly, this algorithm, referred to in modern textbooks as the calculus, is the most celebrated discovery that Newton made in the years 1664–1666. This highly symbolic and algebraized tool of problem solving is discussed in part III. It should be stressed, however, that what appears, with the benefit of hindsight, to be Newton's greatest achievement was perceived as just one among many alternative approaches to problem solving by its inventor. Infinite series allowed Newton to study the properties of mechanical curves, such as the cycloid (the curve traced by a point on the circumference of a circle that rolls along a straight line: the parametric equation of the cycloid generated by a circle with radius a is x = a(t − sin t), y = a(1 − cos t)).12 Most notably, they 11 MP, 1, pp. 369–99. See also the beginning of the "October 1666 Tract on Fluxions." MP, 1, pp. 400–1. Kirsti Andersen studied this technique and presented her analysis at a meeting in Oberwolfach (Germany) in December 2005; see Andersen, "Newton's Inventive Use of Kinematics in Developing His Method of Fluxions" (2005). 12 As Newton wrote in 1684, "To be sure, convergent equations can be found for the curved lines 10 Chapter 1 allowed him to calculate curvilinear areas, curvilinear volumes, and arc lengths; these calculations were generally called "quadrature problems." So, squaring a curve meant calculating the area of the surface bounded by it. Nowadays we would use Leibnizian terminology and speak about problems in integration. Sections §7.4 and in §8.4.5 take up Newton's calculation of the area of the surface subtended by the cycloid and by the cissoid (Cartesian equation y 2 (a − x) = x3 ). The fact that Newton's method allowed him to tackle mechanical curves and quadratures is due to a mathematical fact of which he was well aware. Using Leibnizian jargon, we can say that while differentiation of algebraic functions (accepted by Descartes) leads to algebraic functions, integration can lead to new transcendental functions. Newton referred to what are now called transcendental functions as quantities "which cannot be determined and expressed by any geometrical technique, such as the areas and lengths of curves."13 Infinite power series—in some cases fractional power series— were the tool that young Newton deployed in order to deal with these mechanical (transcendental) curves. 1.4 Fluxions A Newtonian memorandum, written more than fifty years after the momentous intellectual revolution it describes, gives an account, basically confirmed by manuscript evidence, of his early mathematical discoveries: In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon . . . . All this was in the two plague years of 1665–1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more then at any time since.14 There would be much to say to decipher Newton's words and place them in context. For instance, the task of commenting on the meaning of the term philosophy would require space and learning not at my disposal. commonly dubbed 'mechanical,' and with their assistance problems on these curves are solved no differently than in simpler curves." MP, 4, p. 559. "Quinetiam ad curvas lineas vulgo dictas Mechanicas inveniri possunt aequationes convergetes et earum beneficio problemata in his curvis non aliter solvi quam in curvis simplicioribus." MP, 4, p. 558. 13 MP, 3, p. 79. "quantitates . . . quae nullâ ratione geometricâ determinari et exprimi possunt, quales sunt areae vel longitudines curvarum." MP, 3, p. 78. 14 Add. 3968.41, f. 85r. This passage is contained in a draft (August 1718) of a letter that Newton intended for Pierre Des Maizeaux. It is discussed in Westfall, Never at Rest (1980), p. 143. Newton on Mathematical Method 11 Note three things about this memorandum. (i) The "method of approximating series" is the method of series expansion via long division and root extraction. Newton achieved also other methods for expanding y as a fractional power series in x, when the two variables are related by an algebraic equation. These methods, later generalized by Victor-Alexandre Puiseux, allowed Newton to go beyond the limitations of the common analysis, where finite equations were deployed, and express certain curves locally in terms of infinite fractional power series, which Newton called infinite equations.15 (ii) The "rule for reducing any dignity of any binomial" is now called the binomial theorem for fractional powers, which Newton attained in winter 1664 by interpolating results contained in Wallis's Arithmetica Infinitorum (included in Operum Mathematicorum Pars Altera (1656)).16 Such methods of series expansion were crucial for attaining two goals: the calculation of areas of curvilinear surfaces and the rectification of curves. (iii) Newton does not talk about discovering theorems, but rather methods and a rule. This last fact is of utmost importance because it reveals that, in his view, his results belonged to the analytical, heuristic stage of the method of problem solving. In October 1666, Newton gathered his early results in a tract whose incipit reads "To resolve Problems by Motion these following Propositions are sufficient."17 He conceived this tract as devoted to a method of resolution (i.e., "analysis") of geometrical problems, which makes use of the concept of geometrical magnitudes as generated by motion. This method, referred to in Newton's memorandum as the "direct and inverse method of fluxions," is discussed in part III. Note that the inverse method was always conceived by Newton as deeply intertwined with the method of approximating series and with the binomial rule.18 1.5 In the Wake of the Anni Mirabiles In 1669 the first challenge arrived for the young mathematician. A slim book entitled Logarithmotechnia, printed in 1668, the work of the German Nicolaus Mercator, came to his attention. What Newton saw was worrying. Mercator had used an infinite equation (in our terms, a power series expansion of y = 1/(1 + x)) in order to square the hyperbola (i.e., calculate the area of the surface subtended by the hyperbola). This result belongs both to pure mathematics and to practi15 These techniques are discussed in many treatises on algebraic curves: e.g., Brieskorn and Knörrer, Plane Algebraic Curves (1986), pp. 370ff. 16 MP, 1, pp. 89–142. 17 The "October 1666 Tract on Fluxions" is Add. 3958.3, ff. 48v–63v, and is edited in MP, 1, pp. 400–48. 18 For a recent evaluation of Newton's early work on series and fluxions in the period 1664–1666, see Panza, Newton et les Origines de l'Analyse: 1664–1666 (2005). 12 Chapter 1 cal applications. It is, in fact, useful in facilitating the calculation of logarithms, a need deeply felt by seventeenth-century practitioners in all fields from navigation to astronomy. This was one of the results that Newton had achieved via binomial expansion or long division. He was able to do much more than this and therefore summarized his results regarding infinite series applied to quadrature in a small tract entitled De Analysi per Aequationes Numero Terminorum Infinitas (1669).19 Barrow, who was well informed about Newton's discoveries, immediately sent De Analysi to a mathematical practitioner called John Collins. The choice could not have been happier. Collins was at the center of a network of British and Continental mathematicians whom he kept up to date with an intense and competent correspondence. After taking copies of De Analysi, Collins informed a number of his correspondents about Newton's discoveries. He also made Newton aware of the Scotsman James Gregory (or Gregorie), who was pursuing researches on series expansions at a level comparable to what could be found in De Analysi. Collins's correspondence was the vehicle that allowed Newton to establish his reputation as a mathematician. Collins's network overlapped with that of the Royal Society; its president, William Brouncker, the secretary, Henry Oldenburg, and Wallis were certainly interested in Newton's mathematical researches on infinite series. In 1672, Newton was elected a Fellow of the Royal Society because of the construction of the reflecting telescope, not because of his mathematics. And it was because of his ideas concerning the role of mathematics in natural philosophy that he initially found himself in a difficult relationship with the Royal Society. When he presented his 1672 paper on the nature of light, Newton made it clear that the undisputable certainty of his "new theory about light and colors" was guaranteed by mathematical reasoning. This thesis displeased the secretary, Henry Oldenburg, and the curator of experiments, Robert Hooke, who refrained from subscribing to what they perceived as a dogmatic position (see chapter 2). Newton found himself embroiled in a dispute that led him, after some years of tiresome correspondence with critics, to be reluctant about printing his philosophical ideas. Famously, in a different context, he was to complain about philosophy as an "impertinently litigious Lady."20 What is relevant here is that, in the mid-1670s, much to Collins's frustration, he withdrew from any project of printing his mathematical discoveries on series and fluxions. As I argue in Part VI, Newton's policy of publication is consistent with his selfportraiture as a natural philosopher who—contrary to the skeptical probabilism endorsed by many virtuosi of the Royal Society—could attain certainty thanks to 19 20 See chapter 7 for further information. Newton to Halley (June 20, 1686) in Correspondence, 2, p. 437. Newton on Mathematical Method 13 mathematics. Printing the algebraic, heuristic method would have exposed him to further criticisms; what he aimed at was certainty, and this was guaranteed by geometry. The algebraic analysis—as he later said to David Gregory—was "entirely unfit to consign to writing and commit to posterity."21 To appreciate Newton's views on mathematics, one should not underestimate how sharp a boundary he drew in contrast with his mathematical practice between algebra and geometry, and how strongly he believed that only geometry could provide a certain and therefore publishable demonstration. Newton's perception of the two layers of algebraic analysis and geometrical synthesis is already evident in his Tractatus de Methodis Serierum et Fluxionum, composed in 1670–1671.22 The beginning of this long treatise is occupied by a revision and expansion of De Analysi. In the remaining twelve sections (labeled as problems) Newton "methodized" his researches into fluxions that he had first laid down in the October 1666 tract.23 Here he developed the analytical method of fluxions, which was divided into two parts: (i) the direct method (mainly calculations of tangents and curvatures) and the inverse method (mainly calculations of areas and rectifications of curves). De Methodis ends with extensive tabulations of areas of surfaces subtended to curves. Newton soon developed (in an "Addendum" written in 1671) the idea that a synthetic form of the method of fluxions was required (see chapter 9). This more rigorous version, where no infinitesimals occur, was based on limit concepts and geometrical-kinematical conceptions and was systematized in a tract entitled "Geometria Curvilinea," written about 1680. The synthetic method of fluxions—the method of first and ultimate ratios—informs most of the Principia (1687). Thus, in 1671—just after the completion of De Methodis, a summa of his analytical researches on series and fluxions—Newton began to rethink the status of his early researches, which are based on heuristic analogies and the use of infinitesimals, namely, on techniques that are far from the standards of exactness that he aimed at as a natural philosopher. In the 1670s he spent great effort in systematizing them, in rethinking their foundation, and in attempting alternative approaches. Several factors contributed to the more mature phase of Newton's mathematical production that followed the creative burst of the anni mirabiles. I note a few of these factors in the next section and elaborate on them in subsequent chapters. 21 "Algebram nostram speciosam esse ad inveniendum aptam satis at literis posterisque consignandum prorsus ineptam." University Library Edinburgh MS Gregory C42, translated by D. T. Whiteside in MP, 7, p. 196. See also Correspondence, 3, p. 385. 22 See chapter 8 for further information. 23 "partly upon Dr Barrows instigation I began to new methodiz ye discourse of infinite series." Newton to Collins (July 20, 1670) in Correspondence, 1, p. 68. 14 Chapter 1 1.6 Maturity Young men should prove theorems, old men should write books.24 In 1669, Newton was elected Lucasian Professor, in succession to and thanks to the patronage of Barrow. He began preparing a first set of lectures on optics in which he claimed that certainty in natural philosophy can be guaranteed by the use of geometry (see chapter 2). A concern with certainty in mathematical method thus emerged in the context of Newton's early optical researches and remained anchored to them until maturity when, in the last Query 23/31 (1706/1717) of the Opticks, he wrote a famous peroration in favor of the use of the method of analysis and synthesis in natural philosophy. The investigation of difficult things, he claimed, could be pursued in natural philosophy only by following the steps of the mathematicians' method of enquiry. Newton wished to validate his natural philosophy mathematically, outstripping the skeptical probabilism that was rampant in his day, as he complained. Synthesis, not analysis, was the method that could guarantee the level of accuracy and certainty required for such an ambitious task. Further, as a successor of Barrow in the Lucasian Chair, Newton probably felt that his new status implied delivering mathematics in rigorous and systematic form. He began writing mathematical treatises characterized by length, maturity, and apparent uselessness (they seldom went to the press).25 Newton's involvement in preparing his next set of lectures on algebra led him to conceive the idea that analysis could also be approached differently from the way promoted by the moderns: in short, there could be a geometrical analysis, a geometrical rather than an algebraic method of discovery. Up to this point in this chapter, I have somewhat incorrectly equated analysis with algebra, and synthesis with geometry. But it is necessary to avoid such equivalences because they were not accepted by Newton and by many of his contemporaries. Not only synthesis but also analysis could be geometrical. Lucasian Lectures on Algebra stemmed from a project on which Newton had embarked since the fall of 1669, thanks to the enthusiasm of John Collins: the revision of Mercator's Latin translation of Gerard Kinckhuysen's Dutch textbook on algebra. Newton's involvement in this enterprise was an occasion to rethink the status of common analysis. He began experimenting with what he understood as ancient analysis, a geometrical method of analysis or resolution that, in his opinion, the ancients had kept hidden. In his Lucasian Lectures on Algebra, which he deposited in the University Library of Cambridge in 1684 and from which William 24 Godfrey H. Hardy, quoted by Freeman Dyson, in Albers, "Freeman Dyson: Mathematician, Physicist, and Writer" (1994), p. 2. 25 But on Newton's attitude toward print publication versus manuscript circulation, see chapter 16. Newton on Mathematical Method 15 Whiston edited the Arithmetica Universalis (1707), Newton extended Cartesian common analysis and arrived at new results in this field. But even in this eminently Cartesian text one can find traces of his fascination with the method of discovery of the ancients. The ancients, rather than using algebraic tools, were supposed to have a geometrical analysis that Newton wished to restore. This was a program shared by many in the seventeenth century. He also made it clear that synthesis, or composition, of geometrical problems had to be carried on—contra Descartes—in terms wholly independent of algebraic considerations (see chapter 4). The fascination with ancient analysis and synthesis, a better substitute, he strongly opined, for Cartesian common analysis (algebra) and synthesis (the techniques on the construction of equations prescribed by Descartes), prompted Newton to read the seventh book of Pappus's Collectio (composed in the fourth Century a.d. and printed alongside a Latin translation in 1588). He became convinced that the lost books of Euclid's Porisms, described incompletely in Pappus's synopsis, were the heart of the concealed ancient, analytical but entirely geometrical method of discovery (see chapter 5).26 Newton intertwined this myth of the ancient geometers with his growing antiCartesianism. In the 1670s he elaborated a profoundly anti-Cartesian position, motivated also by theological reasons. He began looking to the ancient past in search for a philosophy that would have been closer to divine revelation. The moderns, he was convinced, were defending a corrupt philosophy, especially those who were under Descartes' spell. Newton's opposition to Cartesian mathematics was strengthened by his dislike for Cartesian philosophy. Descartes in the Géométrie had proposed algebra as a tool that could supersede the means at the disposal of Euclid and Apollonius. Newton worked on Pappus's Collectio in order to prove that Descartes was wrong. He claimed that the geometrical analysis of the ancients was superior to the algebraic of the moderns in terms of elegance and simplicity. In this context, Newton developed many results in projective geometry and concerning the organic description of curves. His great success, achieved in a treatise entitled "Solutio Problematis Veterum de Loco Solido" (late 1670s) on the "restoration of the solid loci of the ancients," was the solution by purely geometrical means of the Pappus four-lines locus. This result, much more than the new analysis of infinite series and fluxions, pleased Newton because it was in line with his philosophical agenda (see chapter 5). The importance of projective geometry emerged also in the study of cubics, when Newton found that these algebraic curves can be subdivided into five projective classes. His interest in the classification of cubic curves dates to the 1660s, but it was only in the late 1670s that, by deploying advanced algebraic tools, he achieved 26 It seems that Newton did not know that Descartes expressed similar views in the "Responsio ad Secundas Obiectiones" in Meditationes de Prima Philosophia (1641) (AT, 7, pp. 155–6). 16 Chapter 1 the array of results that later, in the mid-1690s, were systematized in Enumeratio Linearum Tertii Ordinis, a work that first appeared in print as an appendix to the Opticks (1704) (see chapter 6). One should not forget another factor that determined Newton's option for geometry in the 1670s: the encounter with Huygens's Horologium Oscillatorium. In his masterpiece, printed in 1673, Huygens had employed proportion theory and ad absurdum limit arguments (method of exhaustion) and had spurned as far as possible the use of equations and infinitesimals (in his private papers he did employ symbolic infinitesimalist tools, but he avoided them in print).27 Huygens offered an example to Newton of how modern cutting-edge mathematization of natural philosophy could be presented in a form consonant with ancient exemplars. The Lucasian Professor immediately acknowledged the importance of Huygens's work, and one might surmise that his methodological turn of the 1670s— which in part led him to cool his relationship with Collins and avoid print publication of his youthful algebraic researches—was related not only to a reaction against Cartesianism, but also to an attraction toward Huygens's mathematical style. When Newton composed the Principia, in 1684–1686, he had a panoply of mathematical methods in his toolbox, methods that he could deploy in the study of force and motion. He gave pride of place to the synthetic method of fluxions (first elaborated in a treatise composed about 1680 and entitled "Geometria Curvilinea"), claiming in Section 1, Book 1, that this was the foundation on which the magnum opus was based. But in several instances, as a close reading of the text of the Principia makes clear, he appealed to quadrature techniques that belong to his algebraized new analysis. These quadratures were not, however, made explicit to the reader. Newton chose instead to insert in the body of the text a treatment of ancient analysis and its application to the solution of the so-called Pappus problem. In Part IV I discuss the policy of publication that led Newton to structure the text and the subtext of the Principia in ways consonant with his views on mathematical method.28 After the publication of the Principia, Newton ceased to be an isolated Cambridge professor. He had to defend and establish his rising position in the political and cultural world of the capital, where he moved in 1696 as Warden of the Mint. A first challenge, in 1691, from David Gregory (§8.5.1) on quadrature techniques induced him to work hard in the early 1690s on the composition of a treatise, Tractatus de Quadratura Curvarum, which appeared in 1704 as an appendix to the Opticks. De Quadratura opens with an introduction in which Newton claims that the method of fluxions is based on a conception of magnitudes generated by motion 27 28 Yoder, Unrolling Time (1988). See also Guicciardini, Reading the Principia (1999). Newton on Mathematical Method 17 and on limiting procedures that are consonant with the methods of the ancients (see chapter 9).29 Newton's growing fascination with the myth about the prisca sapientia, a pristine superior wisdom of the ancients, that characterizes his thought after the publication of the Principia resonates with his extensive researches on the ancient analysis that he carried on in the 1690s and early 1700s. His aim was to show that the youthful analytical method of fluxions could be reformulated in terms acceptable by ancient standards. He even explored a totally new method of discovery and proof. Newton left hundreds of manuscript pages, which culminated in an unfinished "Geometriae Libri Duo," devoted to his attempts to write a treatise on projective geometry written in a style reconstructed following the authority of Pappus (see part V). These aborted attempts are the more philosophy-laden texts belonging to Newton's mathematical Nachlass, since he made a deep effort to clarify the relations among the various sectors of his mathematical method: analysis, synthesis, algebra, geometry, mechanics, and natural philosophy. These terms have been used in this first chapter in an improperly ambiguous way. But commenting on Newton's works on method in subsequent chapters will allow me to clarify this terminology and decode Newton's somewhat arcane mode of expression. Newton encouraged his acolytes to pursue researches in ancient analysis and never missed the opportunity for praising those, such as Huygens, who resisted the prevailing taste for the symbolism of the moderns, the "bunglers in mathematics."30 When the polemic with Leibniz exploded, he could deploy his classicizing and anti-Cartesian theses against the German (see part VI). Thus, Newton's last mathematical productions, publications, and (often anonymous) polemical pieces were driven by a philosophical agenda difficult to reconcile with his mathematical practice. 29 The other appendix, Enumeratio Linearum Tertii Ordinis, was also written in the 1690s, deploying notes on cubics dating from the 1670s. See chapter 6. 30 Hiscock, David Gregory, Isaac Newton and Their Circle (1937), p. 42. 2 Newton on Certainty in Optical Lectures Newton then does stand somewhat apart not only in accomplishment but also in his philosophy of science. —Barbara Shapiro, 1983 The appointment to the Lucasian Chair (1669) led Newton to ponder over his role as a mathematician. Now he was not just a young creative protégé of Barrow but rather a professor who would soon address the Royal Society with new theories concerning the nature of light. That Newton began investigating the role of mathematics in natural philosophy is evident in his first set of lectures on optics, which he supposedly delivered between January 1670 and the end of Michaelmas term in 1672. As Lucasian Professor, Newton had to deliver one lecture each week during term and to deposit at least ten lectures every year. There thus exist a set of his lectures on optics and a set on algebra (§4.1).1 In his third lecture Newton stated a program that remained a leitmotif all his life: Thus although colors may belong to physics, the science of them must nevertheless be considered mathematical, insofar as they are treated by mathematical reasoning. Indeed, since an exact science of them seems to be one of the most difficult that philosophy is in need of, I hope to show—as it were, by example—how valuable is mathematics in natural philosophy. I therefore urge geometers to investigate nature more rigorously, and those devoted to natural science to learn geometry first. Hence the former shall not entirely spend their time in speculations of no value to human life, nor shall the latter, while working assiduously with an absurd Epigraph from Barbara Shapiro, Probability and Certainty in Seventeenth-Century England (1983), p. 58. 1 An early version of the Principia was also deposited by Newton. One should not assume that Newton actually delivered these lectures to his students (if he had any). Further, the dates on the deposited manuscripts were added in retrospect. The critical edition is in Newton, The Optical Papers of Isaac Newton: Vol.1. The Optical Lectures 1670–1672 (1984), pp. 46–279, which includes both an earlier version of the lectures and the Opticae, pars 1 and 2, which Newton deposited in October 1674 in compliance with the statutes of the Lucasian Chair. The deposited lectures appeared posthumously, first in an English translation of pars 1 as Optical Lectures: Read in the Publick Schools of the University of Cambridge (1728) and then the complete Latin text as Lectiones Opticae (1729). Both these posthumous publications (based on a collation of a transcript and the original) have a complicated story, which is discussed in Alan Shapiro's commentary in Newton, The Optical Papers (1984), pp. 20–5. A facsimile of the first version of the lectures (MS Add. 4002 (Cambridge University Library)) can be found in Newton, The Unpublished First Version of Isaac Newton's Cambridge Lectures on Optics (1973). 20 Chapter 2 method, perpetually fail to reach their goal. But truly with the help of philosophical geometers and geometrical philosophers, instead of the conjectures and probabilities that are being blazoned about everywhere, we shall finally achieve a science of nature supported by the highest evidence.2 Ideas very similar to these were to occur in the Preface to the second edition of the Principia (1713), written by Roger Cotes some forty years later. Alan Shapiro has emphasized the pervasive role of the quest for certainty in Newton's thought and has demonstrated that Newton, very early in his intellectual career, conceded that natural philosophy, even when approached mathematically, cannot reach the absolute certainty of geometry. But Newton made it clear that geometers who practice natural philosophy are able to go a long way beyond the modest aim of the experimental naturalists, namely, conjectures and probabilities. Shapiro wrote that in the last years of his life Newton changed his mind about the possibility of philosophical geometers and geometrical philosophers being able to approach absolute certainty: Although he [Newton] soon modified his excessive claims and more carefully distinguished mathematical demonstrations from less than certain experimental conclusions, he continued to prize mathematical theories and seek "truth" and "certainty," words that appear frequently in his writings. Only in the last decades of his life did he accept the probabilism of his contemporaries.3 2 Newton, The Optical Papers (1984), pp. 87, 89, and 437, 439. "Sic etiamsi colores ad Physicam pertineant, eorum tamen scientia pro Mathematica habenda est, quatenus ratione mathematica tractantur. Imo vero cum horum accurata scientia videatur ex difficillimis esse quae Philosophus desideret; spero me quasi exemplo monstraturum quantum Mathesis in Philosophia naturali valeat; et exinde ut homines Geometras ad examen Naturae strictius aggrediendum & avidos scientiae naturalis ad Geometriam prius addiscendam horter: ut ne priores suum omnino tempus in speculationibus humanae vitae nequaquam profuturis absumant, neque posteriores operam praepostera methodo usque navantes, a spe sua perpetuo decidant: Verum ut Geometris philosophantibus & Philosophis exercentibus Geometriam, pro conjecturis et probabilibus quae venditantur ubique, scientiam Naturae summis tandem evidentijs firmatam nanciscamur." Newton, The Optical Papers (1984), pp. 86, 88, and 436, 438. It should be noted that Galileo in the Dialogo, which Newton knew in Thomas Salusbury's translation (see Newton, Certain Philosophical Questions: Newton's Trinity Notebook (1983), p. 202, and Salusbury, Mathematical Collections and Translations in Two Tomes (1661–65)), referred to a "filosofo geometra" who is able to apply mathematics to the study of nature: "così, quando il filosofo geometra vuol riconoscere in concreto gli effetti dimostrati in astratto, bisogna che difalchi gli impedimenti della materia; che se ciò saprà fare, io vi assicuro che le cose si riscontreranno non meno aggiustamente che i computi aritmetici. Gli errori dunque non consistono né nell'astratto né nel concreto, né nella geometria né nella fisica, ma nel calcolatore, che non sa fare i conti giusti." Galilei, Dialogo Sopra i Due Massimi Sistemi del Mondo (1998), 2, pp. 511.3–5. Similarly in Auctoris Praefatio to the Principia, Newton stated that errors are not due to the imperfections of geometry and mechanics but rather to the artificer who applies them. See chapter 13. 3 Alan Shapiro, Fits, Passions, and Paroxysms (1993), p. 14. See also Alan Shapiro, "Newton's 'Experimental Philosophy'" (2004). Newton on Certainty in Optical Lectures 21 It is difficult to assess the reasons and influences behind the rather extremist methodological position that Newton endorsed in his youth. Barrow might have instilled in the young Newton the conviction that mathematics not only possesses certainty but can also transfer it to the fields to which it is applied. But Newton's position is passionate and idiosyncratic and goes beyond anything that Barrow might have envisaged. Newton's claims concerning mathematical certainty were perceived as alien by most of his contemporaries. In order to defend his somewhat isolated position Newton had to clarify two issues: (i) why mathematics could be considered a source of certainty, and (ii) how mathematics could transfer its certainty to natural philosophy. In this chapter I do not deal with Newton's approach to these issues. It is hard to find a justification for his program in the writings of the early 1670s. Actually, Newton's critics often complained about his lack of justification and explanation, and the dogmatism with which he defended his method. But Newton soon faced these issues in the context of his researches contra Cartesian mathematics (see part II). It is only at the end of this book that one will be in a position to appreciate the negotiations and strategies that Newton had to engineer during his long intellectual career to assess the certainty of mathematics and mathematized natural philosophy vis à vis his mathematical practices, which were based on innovative, bold, heuristic, and far from well-settled methods. In this chapter I describe whom the young Lucasian Professor was criticizing, what his models were, and the aims he set himself when proposing to inject geometry into natural philosophy in his search for "highest evidence." My aim is to locate Newton's early program in a dialogical context. His idiosyncratic philomathematism is, I believe, a critical response, still lacking elaboration and justification, against a position that was "being blazoned about" in the influential circle of the natural philosophers gathered at the Royal Society. One of Newton's polemical targets might be identified as Descartes; indeed, antiCartesian feelings pervade Newton's work. In his natural philosophy Descartes had adopted what is often called a hypothetical physics whereby hypothetical mechanical models are introduced a priori in order to deduce the observed phenomena. Descartes' explanation of the magnet has become an icon of this sort of hypothetical mechanicism (figure 2.1). Descartes conceived of the magnet as emitting screw-shaped particles that penetrate the pores of matter and thus exert a circulatory motion ultimately accounting for the attraction between magnet and iron in terms of contact action between corpuscles. This model was proposed as a hypothesis based on local mechanical interactions that, although plausible, might not occur in nature. Indeed, in Cartesian physics more than one model could be invoked in order to reach a mechanical explanation of the same phenomenon. According to Newton, Descartes' hypothetical models were not derived from observation; they were assumed from first principles. Newton disliked these characteristics of Cartesian hypotheticism because, in his opinion, Descartes' models were not derived 22 Chapter 2 Figure 2.1 The earth as a giant magnet, according to Descartes. The screw-shaped particles are emitted from the North Pole and travel toward the South Pole, generating magnetic effects by direct contact with other bodies. Source: Descartes, Principia Philosophiae (1644), p. 271. Courtesy of the Biblioteca Angelo Mai (Bergamo). from experiments ("deduced from the phenomena")4 but proposed as instances of one of many plausible causal explanations of phenomena. Newton's critical reading of Descartes can appear reductive to Cartesian scholars. Indeed, Descartes had much to say about the role of mathematics, especially geometry, in natural philosophy. Geometry was the model of certain knowledge for Descartes. Further, reference to geometry was crucial in grounding Descartes' conception of matter as extension, his rejection of atomism, his view of the phenomena of nature as understandable in terms of matter and motion. It is by starting from a geometrized natural philosophy that Descartes drew such crucial consequences as the nonexistence of a vacuum. As in the case of Descartes the mathematician, Newton approached Descartes the natural philosopher from a very restrictive and biased point of view. What Newton kept in critical view was the genre of hypothetical-deductive modeling that informs so much of Descartes' Principia Philosophiae (1644), a book that was to be read, according to its author, as 4 See, e.g., the famous passage in the Scholium generale in Newton, Principles, p. 943. See also §14.3.2. Newton on Certainty in Optical Lectures 23 a "romance."5 Indeed, Descartes made clear that the corpuscular models that he proposed as explanations of a wide variety of phenomena were to be received as plausible mechanical accounts of their functioning and genesis. Each one of these explanations, taken separately, was neither unique nor certain, but their cumulative effect, according to Descartes, delivered a powerful defense of the viability of a mechanistic view of nature.6 But when expressing his dissatisfaction towards "conjectures and probabilities" Newton might well have had in mind his contemporaries: the virtuosi of the Royal Society associated with the experimental researches of Robert Hooke and Robert Boyle. As Barbara Shapiro has argued, most of the English intellectuals associated with the Royal Society "were deeply concerned with matters empirical, and they concluded that neither syllogism nor mathematical reasoning was an appropriate vehicle for ordering the data they were collecting."7 The certainty of Aristotelian logic and Euclidean geometry was deemed by them to be extraneous to experimental philosophy, where the best one could reach was probability through a patient collection of facts. The language that Hooke and Boyle promoted was one of avoiding both dogmatic certainty and extreme skepticism. Their position can be characterized as a form of mitigated skepticism according to which it is accepted that in natural philosophy one can attain probability but not absolute certainty.8 It is also true that Boyle praised mathematics and recognized that mathematical demonstration can yield certainty.9 He valued quantitative reports of experimental results when they could be attained. Mathematics occupies a prominent place in Boyle's thought, but he expressly delimited its sphere of influence. Typically, in Hydrostatical Paradoxes (1666) and Medicina Hydrostatica (1690), he defended the idea that the "exactness" and "preciseness" of mathematics cannot be attained in "experiments where we are dealing with gross matter."10 5 As Descartes famously stated in a letter to the abbé Claude Picot (translator of the first French edition of Principia Philosophiae) that Newton might have read in Descartes, Opera Philosophica (1656), which he possessed. See Harrison, The Library of Isaac Newton (1978), no. 506. 6 The literature concerning Descartes is as vast as the one concerning Newton. Two scholarly introductions are Garber, Descartes' Metaphysical Physics (1992) and Gaukroger, Descartes' System of Natural Philosophy (2002). For a recent examination of Descartes' experimental method, see Buchwald "Descartes's Experimental Journey" (2008). 7 Barbara Shapiro, Probability and Certainty in Seventeenth-Century England (1983), p. 5. 8 On constructive or mitigated skepticism, see Popkin, The History of Scepticism (2003), pp. 112ff. 9 Boyle, Of the Usefulness of Mathematicks to Natural Philosophy (1671). This was part of Some Considerations Touching the Usefulnesse of Experimental Natural Philosophy (1671) and was reprinted in Boyle, The Works of the Honourable Robert Boyle in Five Volumes (1744), 3, pp. 392–456. 10 "[Some readers] will not like that I should offer for proofs such physical experiments, as do not always demonstrate the things, they would evince, with a mathematical certainty and accurateness. . . . In physical enquiries it is often sufficient, that our determinations come very near the matter, 24 Chapter 2 Newton positioned himself against the Baconian inductivist program in vogue at the Royal Society as much as against the Cartesian rationalist hypotheticism. From his many pronouncements on method one can infer that he considered both Bacon's bottom-up experimental procedure and Descartes' top-down method doomed to yield only probability. Bacon's proposed patient collection of numerous experiments was not sufficient to guarantee certainty. Descartes alignment of clear and distinct ideas could only deliver, as Newton wrote in his maturity, "little better then a romance."11 Newton proposed to overcome the probabilistic outcomes of both Cartesianism and Baconianism by "proceeding alternately from experiments to conclusions & from conclusions to experiments"12 (see chapter 14). As far as possible, general principles of philosophy had to be deduced, not induced, from experiments. A single well-chosen experiment (the experimentum crucis) or phenomenon (the planetary laws) could allow the geometrical philosopher to deduce the truth of a principle, or better, to initiate a process of approximation to the truth grounded on successive deductions of conclusions from experiments and experiments from conclusions. Much of the polemic between Hooke and Newton over the 1672 paper on light (in which Newton proposed his two prisms experimentum crucis) can be viewed as a clash between two different conceptions of method.13 On the one hand, Newton claimed that through a single experiment he had been able to mathematically deduce a property of light beyond doubt (white light is composite; its components exhibit different refraction coefficients). On the other hand, Hooke denied Newton's claims to necessity and certainty, and advocated a methodology based on the evaluation of a plurality of experimental tests. In a passage that Henry Oldenburg, the secretary of the Royal Society, cautiously censored, Newton went so far as to state, A naturalist would scearce expect to see ye science of those [colors] become mathematicall, & yet I dare affirm that there is as much certainty in it as in any other part of Opticks. For what I shall tell concerning them is not an Hypothesis but most rigid consequence, not conjectured by barely inferring 'tis thus because not though they fall short of a mathematical exactness." Boyle, Works (1744), 2, p. 741. "[I do not] pretend (and indeed it is not necessary) that the proportion, obtainable by our method, should have a mathematical preciseness. For in experiments where we are to deal with gross matter, and to employ about it mathematical instruments, it is sufficient to have a physical, and almost impossible to obtain (unless sometimes by accident) a mathematical exactness." Boyle, Works (1744), 5, p. 480. Also available in The Works of Robert Boyle (1999–2000). 11 "But if without deriving the properties of things from Phaenomena you feign Hypotheses & think by them to explain all nature you may make a plausible systeme of Philosophy for getting your self a name, but your systeme will be little better then a Romance." Add. 3970, f. 480v. Discussed in Alan Shapiro, "Newton's 'Experimental Philosophy'" (2004), pp. 195–6. 12 Ibid. 13 Bechler, "Newton's 1672 Optical Controversies" (1974). Newton on Certainty in Optical Lectures 25 otherwise or because it satisfies all phaenomena (the Philosophers universall Topick,) but evinced by ye mediation of experiments concluding directly & wthout any suspicion of doubt.14 Hooke, who could read this passage canceled from the paper printed in the Philosophical Transactions, remarked, Nor would I be understood to have said all this against his [Newton's] theory as it is an hypothesis, for I doe most Readily agree with him in every part thereof, and esteem it very subtill and ingenious, and capable of salving all the phaenomena of colours; but I cannot think it to be the only hypothesis; not soe certain as mathematicall Demonstrations.15 Hooke was hitting upon themes that were part of the ideology of the Royal Society, defended by influential members such as Boyle, Joseph Glanvill, and Thomas Sprat. Newton's quest for mathematical certainty could appear to them as mere arrogance and as a departure from the Baconian program that informed their agenda. Hooke explicitly invoked Bacon's authority against Newton in his letter of June 1672 to Lord Brouncker, the president of the Royal Society: I see noe reason why Mr. N. should make soe confident a conclusion that he to whome he writ did see how much it was besides the business in hand to Dispute about hypotheses. For I judge there is noething conduces soe much to the advancement of Philosophy as the examining of hypotheses by experiments & the inquiry into Experiments by hypotheses, and I have the Authority of the Incomparable Verulam to warrant me.16 There was a moral dimension to the Royal Society's initial rejection of Newton's self-enrollment in the camp of geometrical philosophers: such a program was against the precept that "true knowledge is modest and wary; 'tis ignorance that is so bold and presuming." From this viewpoint the certainty of geometry should not be traded for the experimental practice that was promoted as a cooperative effort, where different explanations could and should be confronted without any dogmatic emphasis. The probabilism and moderate skepticism that were widespread in Newton's England were part of a much larger movement toward tolerance endorsed by the Fellows of the Royal Society. Newton's geometrical philosopher could seem too much akin to what Glanvill termed a "dogmatist [who] betrays a poverty and a narrowness of spirit [and, being] too confident in opinions, [shows] ill manners and immodesty."17 14 15 16 17 Correspondence, Correspondence, Correspondence, Glanvill, Scepsis 1, pp. 96–7. 1, p. 113. 1, p. 202. Scientifica (1665), p. 195. 26 Chapter 2 Feingold showed that Newton's approach, consisting in an "uncompromising conviction concerning the primacy of mathematics in the domain of natural philosophy . . . and his condescending view of the natural history tradition," was shared by only a fraction of the Royal Society's Fellows for many years to come.18 During Newton's presidency, many of the Fellows defined themselves as naturalists and opposed the Newtonian philomaths. Feingold brought new insight into the dispute that was fought between the two camps, and he documented the tension that existed among the Fellows of the Royal Society. The quarrel over Newton's 1672 paper on the experimentum crucis can be seen as the beginning of this struggle, which was still raging the year of Newton's death (1726), in the battle between Martin Folkes and Hans Sloane for the presidency of the Royal Society. Feingold opposed the idea that the distinction between the two groups could be defined in terms of scientists versus amateurs. He warned that "by the early eighteenth century the cleavage between a group comprised primarily of mathematicians, astronomers, and physicists, on the one hand, and naturalists, physicians, and general scholars, on the other, was indicative of taste, not competence."19 A towering figure in English natural philosophy such as Boyle showed a marked disinclination to merge mechanicism and mathematization, even when he recognized that his corpuscular view of matter allowed the description of the natural processes in geometrical terms. Both Boas Hall and Shapin showed that Boyle did so not because of mathematical illiteracy but because the language he promoted had to avoid unwarranted expectations of certainty and accuracy, and had to be accessible to the scrutiny of the largest number of inquirers. Modesty in theoretical assessment and accessibility were values deeply embedded in Boyle's moral depiction of the natural philosopher.20 It would be, however, narrow-minded to portray Newton as an outsider fighting single-handedly in favor of mathematized natural philosophy against a compact group of Baconian naturalists. It is true that Boyle sanitized experimental work from interference by mathematics, but alternative views emerged as well, for instance, in the works of Hooke, Christopher Wren, John Wallis, Isaac Barrow, and later Edmond Halley. In most of his works Hooke displayed due reverence toward Boyle's mitigated skepticism. For instance, in the Preface to the Micrographia (1665) he made clear that his hypotheses, while "grounded and confirm'd" by experiment, should be taken "only as Conjectures and Quaeries" whose epistemological status was far from the mathematical certainty attained through "any Infallible Deductions or certainty 18 Feingold, "Mathematicians and Naturalists" (2001), p. 78. Ibid., p. 94. 20 Boas, Robert Boyle and Seventeenth-Century Chemistry (1958); Shapin, "Robert Boyle and Mathematics" (1988). 19 Newton on Certainty in Optical Lectures 27 of Axioms." It is evident from the Preface of the Micrographia that Hooke paid attention to deeply felt values shared by his patrons: If therefore the reader expects from me any infallible deductions, or certainty of axioms, I am to say for myself, that those stronger works of wit and imagination are above my weak abilities; or if they had not been so, I would not have made use of them in this present subject before me: Wherever he finds that I have ventur'd at any small conjectures, as the causes of things that I have observed, I beseech him to look upon them only as doubtful problems, and uncertain guesses, and not as unquestionable conclusions, or matters of unconfutable science.21 The message delivered by Hooke, in a work that has been aptly described as a Royal Society venture,22 could not be more at odds with Newton's defense of the role of geometrical philosophers. It should also be noted that in his magnum opus Hooke lost no occasion to praise Bacon for his empiricism and Descartes for his corpuscular hypothetical models. However, Hooke was also interested in mathematical modeling, especially when it came to planetary motions. Recent research illuminates the extent of Hooke's mathematization in this field.23 His employment of mathematics, however, was quite different from Newton's, both in content and role. Hooke's mathematization of the planetary orbits was preeminently graphic. It can be understood as a graphic simulation of the mechanical devices (the compound pendulum and a ball rolling on an inverted cone) that he devised in order to test his theories on central force motion. In Hooke's graphic mathematics is instantiated a characteristic of his work highlighted by both Bennett and Bertoloni Meli: an entanglement between art and nature, so that mechanical devices work as cognitive tools.24 Hooke's mathematization was extraneous to Newton's much more abstract methods. However, some of the mathematical techniques that Newton deployed, especially those related to the organic description of