University of Plymouth PEARL https://pearl.plymouth.ac.uk 04 University of Plymouth Research Theses 01 Research Theses Main Collection 2015 The Geometrical Thought of Isaac Newton: An Examination of the Meaning of Geometry between the 16th and 18th Centuries Bloye, Nicole Victoria http://hdl.handle.net/10026.1/3273 Plymouth University All content in PEARL is protected by copyright law. Author manuscripts are made available in accordance with publisher policies. Please cite only the published version using the details provided on the item record or document. In the absence of an open licence (e.g. Creative Commons), permissions for further reuse of content should be sought from the publisher or author. This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with its author and that no quotation from the thesis and no information derived from it may be published without the author’s prior consent. The Geometrical Thought of Isaac Newton An Examination of the Meaning of Geometry between the 16th and 18th Centuries by Nicole Victoria Bloye A thesis submitted to Plymouth University in partial fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY School of Computing and Mathematics Faculty of Science and Environment Plymouth University December 2014 Abstract Our thesis explores aspects of the geometrical work and thought of Isaac Newton in order to better understand and re-evaluate his approach to geometry, and specifically his synthetic methods and the organic description of plane curves. In pursuing this research we study Newton’s geometrical work in the context of the changing view of geometry between the late 16th and early 18th centuries, a period defined by the responses of the early modern geometers to a new Latin edition of Pappus’ Collectio. By identifying some of the major challenges facing geometers of this period as they attempted to define and practice geometry we are able to contrast Newton’s own approach to geometry. The themes emerging from the geometrical thought of early modern geometers provide the mathematical context from which to understand, interpret and re-evaluate the approach taken by Newton. In particular we focus on Newton’s profound rejection of the new algebraic Cartesian methods and geometrical philosophies, and the opportunity to focus more clearly on some of his most astonishing geometrical contributions. Our research highlights Newton’s geometrical work and examines specific examples of his synthetic methods. In particular we draw attention to the significance of Newton’s organic construction and the limitations of Whiteside’s observations on this subject. We propose that Newton’s organic rulers were genuinely original. We disagree with Whiteside that they were inspired by van Schooten, except in the loosest sense. Further, we argue that Newton’s study of singular points by their resolution was new, and that it has been misunderstood by Whiteside in his interpretation of the transformation effected by the rulers. We instead emphasise that it was the standard quadratic transformation. Overall we wish to make better known the importance of geometry in Newton’s vi scientific thought, as well as highlighting the mathematical and historical importance of his organic description of curves as an example of his synthetic approach to geometry. This adds to contemporary discourse surrounding Newton’s geometry, and specifically provides a foundation for further research into the implications of Newton’s geometrical methods for his successors. Contents Abstract v List of figures ix Acknowledgements xi Author’s declaration 1 1 Introduction 3 1.1 Newton’s Geometry . 3 1.2 Context . 5 1.3 Themes and questions . 7 1.4 Literature . 11 1.5 Structure of the thesis . 13 2 Early modern geometry: two points of view 17 2.1 Pappus’ Collectio in the 16th century . 20 2.2 Viète and the new analysis ......................... 37 2.3 Kepler . 44 2.4 Into the 17th century . 49 3 The Geometry of René Descartes 53 3.1 First mathematical writings: 1619–1628 . 62 3.2 Early mathematical works . 66 3.3 The Géométrie ............................... 79 3.4 Reception of Descartes’ Géométrie ..................... 92 4 The Geometry and Geometrical Thought of Isaac Newton 97 vii Contents viii 4.1 Introduction . 97 4.2 Early influences on Newton . 100 4.3 Newton’s changing view of Cartesian geometry . 109 4.4 Pappus’s problem and the organic description of curves . 127 4.5 The enumeration of the cubics . 141 4.6 Geometry in Newton’s physics . 150 5 The reception of Newton’s geometry in the 18th century 157 5.1 The publication of Newton’s work . 159 5.2 The Newtonians . 163 5.3 Other commentaries on and later references to Newton’s work . 176 5.4 Concluding comments . 179 6 Conclusion 181 6.1 Chapter summary . 182 6.2 Themes . 186 6.3 Conclusion . 190 List of references 193 A Names and Dates 207 B Neusis 209 C Cartesian instruments 213 C.1 The “mesolabum" (Instrument 3.1.2)....................215 C.2 The “turning ruler and sliding curve" procedure (Instrument 3.3.1)....216 D Newton, the geometer 217 Figures 2.1 Hyperbola for the Archimedean neusis . 27 2.2 Pappus’ problem . 32 2.3 Hyptios porism . 35 2.4 First porism . 36 3.1 Descartes’ “new compasses” [AT, 1901, 10, p.240] (left); with curve KJLM (right) . 64 3.2 Descartes’ “mesolabum” [AT, 1901, 6, p.391] . 66 3.3 Descartes’ Solution to a five-line Pappus problem . 78 3.4 Descartes’ moving ruler [AT, 1901, 6, p.393] . 86 3.5 Descartes’ solution of Debeaune’s problem . 88 4.1 Hyptios porism . 124 4.2 Main porism - first case . 126 4.3 The four-line locus problem . 127 4.4 Lemma 17 . 129 4.5 The organic construction . 130 4.6 The organic construction . 132 4.7 Organic construction of a cubic . 133 4.8 Newton to Collins, 20 August, 1672 . 135 4.9 Schooten, p.347 Exercitationes ......................135 4.10 Projection of cubics [Add. 3961.1:2r, Cambridge University Library, Cam- bridge] . 144 4.11 Projection of cubics . 147 ix Figures x 4.12 Lemma 28 . 151 4.13 Gravitational attraction of a spherical shell . 152 B.1 Neusis . 211 B.2 Conchoid of Nicomedes . 211 B.3 Angle trisection . 212 C.1 Descartes’ “mesolabum” [AT, 1964–1974, 6, p.391] . 215 C.2 “Turning ruler and sliding curve” procedure . 216 Acknowledgements I would like to thank my supervisors Professors Stephen Huggett and Jeremy Gray for their unfaltering support and encouragement. I am grateful to my examiners June Barrow-Green and Phil Dyke for their careful reading of the first thesis and for giving me a second chance. I would especially like to thank June for her detailed notes and recommendations (much more than I deserved). I am grateful to WMS, Plymouth University, the Open University, and Thomas Hintermann at the European Mathematical Society for gainful employment at various stages throughout the past eight years (and for keeping a roof over my head). I thank James Harvey and the Finance team at WMS for understanding and always taking an interest. I apologise to my extended family for being a terrible daughter, sister, niece, cousin, and aunt for too many years. To my best friend, Alfie, I am grateful for you every day. Finally I would like to thank my partner Richard Bower, without whose love and support this project would not have been close to possible. Author’s declaration At no time during the registration for the degree of Doctor of Philosophy has the author been registered for any other University award. Relevant scientific seminars and conferences were regularly attended at which work was often presented. One paper has been accepted for publication in a refereed journal. Publications : Bloye, N. & Huggett, S. (2011) Newton, the geometer, Newsletter of the EMS, 82:19–27 Conference presentations : 2012: BSHM, Research in Progress, The Obtuse History of Conformal Geometry, Oxford. 2010: BMC, History of Mathematics splinter group, Newton, the Geometer, Edinburgh 2008: BMC, History of Mathematics splinter group, Geometry and sin, York 2008: BSHM, Research in Progress, Geometry and sin, Oxford 2007: BSHM, Research in Progress, Our Distance from Euclid, Oxford Word count for the main body of this thesis: 63731 Signed: Date: Chapter 1 Introduction 1.1 Newton’s Geometry The work of Isaac Newton has been widely studied and yet as a geometer he remains relatively unknown. For most people Newton is known for his gravitational theory. In science and academia, much attention has been paid to other areas of his work, for example, in optics and the calculus. In contrast, we seek to explore aspects of Newton’s geometrical work that remain overshadowed and, we argue, undervalued. For Newton, geometrical thought was not confined to pure geometry, it extended to other aspects of his scientific work, especially in physics, and many examples can be found in the Principia. That is certainly not to say that Newton did not use other methods. Far from it, he was a skilled algebraist. However, Newton’s use of geometric methods was conscious and selective, especially in his published works. In order to gain a thorough understanding of the development of Newton’s geometry we look at his first readings of geometrical texts from his time at Trinity College, Cambridge. We also consider Newton’s close study of the methods of the ancients. We follow the progression of Newton’s growing anti-Cartesianism through examples of his contributions to geometry. The main part of our study is set between the mid-17th and mid-18th centuries, where we focus on the geometrical work and thought of Isaac Newton, and his influence upon those who succeeded him.