Niccolò Guicciardini ( ISAAC 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 Ger- many, 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 Sci- entific Revolution, 1600–1800 Sander Gliboff, H.G. 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Rocke, Nationalizing Science: Adolphe Wurtz and the Battle for French Chemistry George Saliba, Islamic Science and the Making of the European Renaissance Nicol´asWey G´omez, The Tropics of Empire: Why Columbus Sailed South to the Indies Isaac Newton on Mathematical Certainty and Method Niccol`oGuicciardini The MIT Press Cambridge, Massachusetts London, England c 2009 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and re- trieval) without permission in writing from the publisher. For information about special quantity discounts, please email special [email protected] This book was set in Computer Modern by Compomat s.r.l., Configni (RI), Italy. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Guicciardini, Niccol`o. Isaac Newton on mathematical certainty and method / Niccol`o 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 analy- sis. 3. Mathematics-History. I. Title. QA29.N4 G85 2009 510-dc22 2008053211 10987654321 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 cer- tainty 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 xxi I Preliminaries 1 1 Newton on Mathematical Method 3 1.1 Early Influences 3 1.2 First Steps 6 1.3 Plane Curves 7 1.4 Fluxions 10 1.5 In the Wake of the Anni Mirabiles 11 1.6 Maturity 14 2 Newton on Certainty in Optical Lectures 19 3 Descartes on Method and Certainty in the G´eom´etrie 31 3.1 Analysis and Synthesis in Pappus 31 3.2 Analysis and Synthesis in Descartes’ Mathematical Canon 38 3.3 An Example of Analysis and Synthesis of a Determinate Problem 49 3.4 An Example of Analysis and Synthesis of an Indeterminate Problem 53 3.5 The Limitations of Descartes’ Mathematical Canon 56 viii Contents II Against Cartesian Analysis and Synthesis 59 4 Against Descartes on Determinate Problems 61 4.1 Lucasian Lectures on Algebra 61 4.2 Demarcation and Simplicity in the Construction of Equations 64 4.3 Neusis Constructions 68 4.4 Conic Constructions 72 4.5 Against Cartesian Synthesis 74 4.6 Against Cartesian Analysis 76 5 Against Descartes on Indeterminate Problems 79 5.1 In Search of Ancient Analysis 79 5.2 Porisms 81 5.3 Newton’s Two-Step Approach to the Pappus Problem 89 5.4 Organic Description of Conics 93 5.5 Tensions 106 6 Beyond the Cartesian Canon: The Enumeration of Cubics 109 6.1 Studies on Cubics 109 6.2 Absence of Demonstrations in the Enumeratio 113 6.3 Common Analysis in the Enumeratio 117 6.4 Projective Geometry in the Enumeratio 121 6.5 New Analysis in the Enumeratio 130 III New Analysis and the Synthetic Method 137 7 The Method of Series 139 7.1 Wallis’s Arithmetica Infinitorum 139 7.2 Criticism Leveled at Wallis 144 7.3 The Binomial Series 148 7.4 Infinite Series and Quadratures 154 7.5 Resolution of Affected Equations 158 7.6 New and Common Analyses 164 Contents ix 8 The Analytical Method of Fluxions 169 8.1 Barrow 169 8.2 Preliminaries to the Method of Fluxions 179 8.3 The Direct Method of Fluxions 186 8.4 The Inverse Method of Fluxions 193 8.5 The Inverse Method in De Quadratura 202 8.6 Methodus Differentialis 210 8.7 A Question of Style 211 9 The Synthetic Method of Fluxions 213 9.1 Synthetic Quadratures in De Methodis 213 9.2 “Geometria Curvilinea” 217 9.3 First and Ultimate Ratios in the Principia 219 9.4 Lemma 2, Book 2, of the Principia 223 9.5 Limits in De Quadratura 226 9.6 A Method Worthy of Public Utterance 230 IV Natural Philosophy 233 10 The Principia 235 10.1 Genesis of the Principia 235 10.2 An Overview of the Principia 238 10.3 Did Newton Use the Calculus in the Principia? 252 11 Hidden Common Analysis 259 11.1 Proposition 30, Book 1, of the Principia 259 11.2 Newton’s Synthetic Construction for Proposition 30, Book 1 261 11.3 Descartes’ Construction of Third-Degree Equations 262 11.4 The Analysis behind Proposition 30, Book 1 262 11.5 Concluding Remarks 266 12 Hidden New Analysis 267 12.1 Concessis curvilinearum figurarum quadraturis 267 12.2 The Inverse Problem of Central Forces 268 x Contents 12.3 Attraction of Solids of Revolution 279 12.4 Concluding Remarks 288 V Ancients and Moderns 291 13 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 305 14 Analysis and Synthesis 309 14.1 Analysis and Synthesis in “Geometriae Libri Duo” 309 14.2 Realism and Constructivism 313 14.3 Analysis and Synthesis in Natural Philosophy 315 VI Against Leibniz 329 15 The Quarrel with Leibniz: A Brief Overview 331 16 Scribal Publication, 1672–1699 339 16.1 Newton’s Reluctance to Publish 339 16.2 Manuscript Circulation 346 16.3 Correspondence with Collins and Leibniz 351 16.4 Disclosures, 1684–1699 361 17 Fluxions in Print, 1700–1715 365 17.1 In the Public Sphere 365 17.2 Commercium Epistolicum 372 17.3 The “Account” 381 Conclusion 385 Contents xi A Brief Chronology of Newton’s Mathematical Work 389 References 391 Index 413 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 Mathe- matical 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 scholar- ship and information.1 Rather, I focus on one aspect of Newton’s mathemati- cal 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 philos- ophy 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 devel- opment, 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.