Isaac Newton on mathematical certainty and method

449 Pages·2009·3.402 MB·English

by  Guicciardini, Niccolò, Newton, Isaac

Isaac Newton on Mathematical Certainty and Method
Transformations: Studies in the History of Science and Technology
Jed Z. Buchwald, general editor
DoloresL.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
SanderGliboff,H.G.Bronn, ErnstHaeckel, andtheOriginsofGermanDarwinism:
A Study in Translation and Transformation
Niccol`o Guicciardini, Isaac Newton on Mathematical Certainty and Method
Kristine Harper, Weather by the Numbers: The Genesis of Modern Meteorology
Sungook Hong, Wireless: From Marconi’s Black-Box to the Audion
Jeff Horn, The Path Not Taken: French Industrialization in the Age of Revolution,
1750–1830
Myles W. Jackson, Harmonious Triads: Physicists, Musicians, and Instrument
Makers in Nineteenth-Century Germany
Myles W. Jackson, Spectrum of Belief: Joseph von Fraunhofer and the Craft of
Precision Optics
Mi Gyung Kim, Affinity, That Elusive Dream: A Genealogy of the Chemical Revo-
lution
Ursula Klein and Wolfgang Lef`evre, Materials in Eighteenth-Century Science: A
Historical Ontology
John Krige, American Hegemony and the Postwar Reconstruction of Science in
Europe
Janis Langins, Conserving the Enlightenment: French Military Engineering from
Vauban to the Revolution
Wolfgang Lef`evre, editor, Picturing Machines 1400–1700
Staffan Mu¨ller-Wille and Hans-Jo¨rg Rheinberger, editors, Heredity Produced: At
the Crossroads of Biology, Politics, and Culture, 1500–1870
William R. Newman and Anthony Grafton, editors, Secrets of Nature: Astrology
and Alchemy in Early Modern Europe
Gianna Pomata and Nancy G. Siraisi, editors, Historia: Empiricism and Erudition
in Early Modern Europe
Alan J. Rocke, Nationalizing Science: Adolphe Wurtz and the Battle for French
Chemistry
George Saliba, Islamic Science and the Making of the European Renaissance
Nicol´as Wey G´omez, The Tropics of Empire: Why Columbus Sailed South to the
Indies
Isaac Newton on Mathematical Certainty and Method
Niccolo` Guicciardini
The MIT Press
Cambridge, Massachusetts
London, England
(cid:2)c2009 Massachusetts Institute of Technology
Allrightsreserved. Nopartofthisbookmaybereproducedinanyformbyanyelectronic
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 / Niccolo` Guicciardini.
p. cm. -(Transformations: studiesinthehistoryofscienceandtechnology)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
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 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 AnniMirabiles 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 LucasianLecturesonAlgebra 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 ArithmeticaInfinitorum 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 DeQuadratura 202
8.6 MethodusDifferentialis 210
8.7 A Question of Style 211
9 The Synthetic Method of Fluxions 213
9.1 Synthetic Quadratures in DeMethodis 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 DeQuadratura 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 Concessiscurvilinearumfigurarumquadraturis 267
12.2 The Inverse Problem of Central Forces 268