THE DEVELOPMENT OF NEWTONIAN CALCULUS IN BRITAIN 1700-1800 NICCOLO GUICCIARDINI The right o/the Un;",rsiljl 0/ Cambridge 10 print (lIId sell 011 monner 0/ boob WQS granted by Henry Vlllm ISJ4. T'Iu! Uni~rsi')! Iws printed and published continuously since "84. CAMBRIDGE UNIVERSITY PRESS Cambridge New York Port Chester Melbourne Sydney PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13,28014 Madrid, Spain Dock House, The Waterfront, Cape Town 800 I, South Africa http://www.cambridge.org © Cambridge University Press 1989 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 First paperback edition 2002 A catalogue record/or this book is available/rom the British Library Library 0/ Congress Cataloguing in Publication data Guicciardini, Niccolo. The development of Newtonian calculus in Britain 1700-1800 I Niccolo Guicciardini. p. cm. Originally presented as the author's thesis (Ph. D.-Council for National Academic Awards, 1987) Bibliography: p. Includes index. ISBN 0521 364663 I. Calculus - Great Britain - History - 18th century. I. Title. QA303.G94 1989 515'.0941'09033-dc20 89-7085 CIP ISBN 0 521 36466 3 hardback ISBN 0521 524849 paperback CONTENTS Introduction vii OVERTURE: NEWTON'S PUBLISHED WORK ON THE CALCULUS OF FLUXIONS I P ART I: THE EARLY PERIOD 9 I THE DIFFUSION OF THE CALCULUS (1700-30) II 1.1 Early initiates' 12 1.2 Textbook writers 13 1.3 The teaching of the calculus in the universities 18 2 DEVELOPMENTS IN THE CALCULUS OF FLUXIONS (1714-33) 28 2. I Methods of integration 29 2.2 The Methodus differentialis 31 2.3 Geometry 36 3 THE CONTROVERSY ON THE FOUNDA TIONS OF THE CALCULUS (1734-42) 38 3.1 Berkeley's criticisms of the calculus 38 3.2 The definitions of the basic terms of the calculus in the works of the early fluxionists 41 3.3 The doctrine of prime and ultimate ratios 43 3.4 The foundations of the calculus in Maclaurin's Treatise of Fluxions 47 iii iv CONTENTS PART II: THE MIDDLE PERIOD 53 4 THE TEXTBOOKS ON FLUXIONS (1736-58) 55 4.1 Teaching the algorithm of fluxions 55 4.2 Textbooks on fluxions and the science books trade 60 5 SOME APPLICATIONS OF THE CALCULUS (1740-3) 68 5.1 Maclaurin's study of the attraction of ellipsoids 69 5.2 Simpson's study of the attraction of ellipsoids 73 5.3 Remarks on the use of partial differentials in Clairaut's Theorie de la Figure de la Terre 79 6 THE ANALYTIC ART (1755-85) 82 6.1 Simpson's methodology 83 6.2 Landen's Residual Analysis 85 6.3 Waring's work on partial fluxional equations 89 PART III: THE REFORM 93 7 SCOTLAND (1785-1809) 95 7.1 The universities and the Royal Society of Edinburgh 95 7·2 John Playfair 99 7.3 Wallace and Ivory 103 7.4 Glenie and Spence 104 8 THE MILITARY SCHOOLS (1773-1819) 108 8.1 The Royal Military Academy at Woolwich 109 8.2 Charles Hutton I I I 8.3 Olynthus Gregory and Peter Barlow 112 8.4 The Royal Military College at Sandhurst 114 8.5 The Ladies' Diary and the Mathematical Repository 115 8.6 Ivory's break-through I I 7 8.7 Other journals and the encyclopaedias 118 8.8 The Royal Naval Academy at Portsmouth 121 9 CAMBRIDGE AND DUBLIN (1790-1820) 124 9.1 Fluxions in Cambridge 124 9.2 Robert Woodhouse 126 9.3 Ireland in the eighteenth century 131 9.4 The reform of mathematics at the University of Dublin 132 9.5 The Analytical SOciety 135 CONTENTS v CONCLUSION 139 APPENDIX A TABLES OF CONTENTS OF FLUXIONARY TEXTBOOKS 143 A.I Table of contents of William Emerson's The Doctrine of Fluxions (1743), (2nd edn., 1757) 143 A.2 Table of contents of Thomas Simpson's The Doctrine and Application of Fluxions (1750c) 146 A.3 Table of contents of John Rowe's An Introduction to the Doctrine of Fluxions (1751) 147 APPENDIX B PRICE LIST OF MATHEMATICAL BOOKS PRINTED FOR JOHN NOURSE 148 APPENDIX C CHAIRS IN THE UNIVERSITIES 150 C.I The Gregory's family tree 150 C.2 University of Cambridge 150 C.3 University of Oxford 151 C.4 University of Edinburgh 152 C.5 University of Glasgow 152 C.6 University of St Andrews 153 C.7 University of Aberdeen 153 C.8 University of Dublin 155 APPENDIX D MILITARY ACADEMIES 156 D.I Royal Military Academy at Woolwich (masters and assistants of mathematics and fortification) 156 D.2 Royal Military College (Sandhurst) (masters and professors of mathematics) 157 D.3 Royal Naval Academy at Portsmouth (head-masters) 158 APPENDIX E SUBJECT INDEX OF PRIMARY LITERATURE 159 APPENDIX F MANUSCRIPT SOURCES 165 Notes Bibliography Index INTRODUCTION EIGHTEENTH-CENTURY British mathematics does not enjoy a good reputation. The eighteenth century, a 'period of indecision'I as many historians would say, is said to have witnessed 'the crisis' or the 'decline' of mathematics in the country of Newton, Wallis and Barrow. However, even a glance at the following list of names should be sufficient to refute the prevailing image of eighteenth-century British mathematics. To the imported Abraham de Moivre one can add the native Brook Taylor, James Stirling, Edmond Halley, Roger Cotes, Thomas Bayes, Colin Maclaurin, Thomas Simpson, Matthew Stewart, John Landen and Edward Waring. Through their work they contributed to several branches of mathematics: algebra, pure geometry, physical astronomy, pure and applied calculus and probability. I devote this work to a theory that all these natural philosophers knew very well: the calculus of fluxions. This was the British equivalent of the more famous continental differential and integral calculus. It is usually agreed that the calculus of fluxions was clumsy in notation and awkward in methodology: the preference given to Newton's dots and to geometrical methods engendered a period which was eventually labelled as the 'Dot Age'.2 Furthermore, the calculus of fluxions is usually indicated as the principal cause of the decadence of British mathematics: the 'Dot-Age' was the price paid for a chauvinistic attachment to Newton's theory. The origin of this depressing image of the Newtonian calculus can be easily traced back to the irreverent writings of the Cambridge Analytical Society's fellows who, at the beginning of the nineteenth century, tried to introduce into Great Britain the algebraical methods of Lagrange and Arbogast. Like all the reformers, they offered a pessimistic view of the past. 3 Since then, many historians have behaved as loyal members of the Analytical Society, and a standard account of the eighteenth-century fluxional calculus has been given in the histories of mathematics. For vii viii INTRODUCTION instance, in Koppelman (1971) we find stated that the 'quiescence' of English mathematics in the eighteenth century depended upon the isolation of English mathematicians from the continent. The reason for this isolation is attributed to the' bitterness engendered by the Newton-Leibpiz priority controversy' and to the 'insularity of the English'. The result of this situation was, according to Koppelman, that 'the Newtonian school clung to a clumsy notation and, perhaps even more important, to a reliance on geometric methods out of a misguided belief that these represented the spirit of Newton' (Koppelman (1971), PP: 155-6). The difference between Newton's and Leibniz's notation has been given too much importance. Even though there are some reasons for preferring the differential notation, it is certain that the progress of the calculus of fluxions was not dependent upon the choice between the dots and the d's. Indeed the fluxional notation is still successfully used in mechanics to express the derivatives as a function of time. Another commonplace misinterpretation is that British mathematicians used geometrical methods. It is not clear to me how the researches of Stirling on interpolation, or of Taylor on finite differences, the second book of Maclaurin's Treatise o/Fluxions (1742), the work of Simpson on physical astronomy and geodesy, the results of Landen on infinite series and elliptic integrals, and those of Waring on fluxional equations could be defined as geometrical. Many British mathematicians consciously departed from the geometrical methods of the Principia, and they did so with different motivations and different results. The current account of the decline of the calculus of fluxions also includes sociological discussions. It is maintained that the practical bent of a country dominated by the industrial revolution together with the chauvinistic isolation of British scientists caused the stagnation of mathematics. However, many British scientists cultivated a deep interest in pure research, for instance in pure geometry or cosmology, and in Great Britain there was a considerable interest in mathematics as the many 'philomaths' mathematical serials show. The existence of a chauvinistic myth for the Philosophia Britannica4 is undeniable, but this does not imply that there was a total separation between continental and British scientists. For instance, continental and British astronomers were in close contact. Furthermore, the theory of the' golden isolation' of the fluxionists does not explain why there should be so many letters from continental mathematicians in the correspondence of Stirling and Maclaurin and why there existed several translations from continental mathematical works into English using Newton's notation. 5 INTRODUCTION ix It is disappointing that the only work devoted to the eighteenth-century British calculus, Florian Cajori's A History of the Conceptions of Limits and Fluxions in Great Britain from Newton to Woodhouse (1919), restates the usual account. For instance, on p. 254 Cajori simply says that 'Newton's notation was poor and Leibniz's philosophy of the calculus was poor', a statement which historians of Leibniz's mathematics would not easily accept; while on p. 279 we find that' the doctrine of fluxions was so closely associated with geometry, to the neglect of analysis, that, apparently, certain British writers held the view that fluxions were a branch of geometry'. Furthermore, Cajori is interested only in the definitions of the term . fluxion'. Since these definitions did not change very much during a whole century and were generally unsatisfactory from a modem point of view, he takes it as an argument in favour of the thesis of the decline of the British calculus. Cajori's quotations are invariably taken from introductions and prefaces of treatises on fluxions. The reader is left without any information about the authors, the length and contents of their works, and the purposes for which they were written. Thanks to the recent works of Schneider (1968), Gowing (1983) and Feigenbaum (1985) we have acquired a very good knowledge of de Moivre, Cotes and Taylor. However, it seemed to me necessary to study the whole period from Newton's work to the reform of the calculus in the early nineteenth century. I will offer a general survey of the development of the calculus in Great Britain; I will not consider therefore the impact of the Newtonian calculus on continental mathematics. I will try to concentrate especially on aspects which are not covered in other works. Whenever it is possible, I will refer the reader to studies which cover specific subjects or authors. First of all, I will take for granted a knowledge of Newton's mathematical work, which has been extensively and masterfully studied by Whiteside in his well-known edition of Newton's mathematical papers. Other works which have been useful are: Tweedie (1922) and Krieger (1968) on Stirling; Eagles (1977a) and (1977b) on David Gregory; Clarke (1929) on Simpson; Tweedie (1915), Turnbull (1951) and Scott (1971) on Maclaurin; Grattan-Guinness (1969) and Giorello (1985) on Berkeley; Trail (1812) on Simson; Chasles (1875) on Simson, Stewart and Maclaurin; Smith (1980) on Bayes; and Bos (1974) on the differential calculus. The Dictionary of National Biography, the Dictionary of Scientific Biography and E.G.R. Taylor's Mathematical Practitioners (1954) and (1966), have been indispensable tools in this work. However, the most important source of information on the lives and works of British