Sources and Studies in the History of Mathematics and Physical Sciences Editorial Board J.Z. Buchwald J. Lützen G.J. Toomer Advisory Board P.J. Davis T. Hawkins A.E. Shapiro D. Whiteside Springer Science+Business Media, LLC Sources and Studies in the History of Mathematics and Physical Seiences K. Andersen Brook Taylor's Work on Linear Perspective H.l.M. Bos Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction 1. Cannon/S. Dostrovsky The Evolution of Dynamics: Vibration Theory from 1687 to 1742 B. ChandIerlW. Magnus The History of Combinatorial Group Theory A.I. Dale AHistory of Inverse Probability: From Thomas Bayes to Karl Pearson, Second Edition A.I. Dale Most Honourable Remembrance: The Life and Work of Thomas Bayes A.I. Dale Pierre-Simon Laplace, Philosophical Essay on Probabilities, Translated from the fifth French edition of 1825, with Notes by the Translator P. Damerow/G. FreudenthallP. McLaugWin/l. Renn Exploring the Limits of Preclassical Mechanics: A Study of Conceptual Development in Early Modem Science: Free Fall and Compounded Motion in the Work of Descartes, Galileo, and Beeckman, Second Edition P.l. Federico Descartes on Polyhedra: A Study of the De Solworum Elementis B.R. Goldstein The Astronomy of Levi ben Gerson (1288-1344) H.H. Goldstine A History of Numerical Analysis from the 16th Through the 19th Century H.H. Goldstine A History of the Calculus of Variations from the 17th Through the 19th Century G. Graßhoff The History of Ptolemy's Star Catalogue A.W. Grootendorst Jan de Witt's Eiementa Curvarum Linearum, über Primus Continued after Index The Arithmetic of Infinitesimals John Wallis 1656 Translated from Latin to English with an Introduction by Jacqueline A. Stedall Centre for the History of the Mathematical Sciences , Open University Springer Jacqueline A. Stedall The Queen's College Oxford OXI 4AW England [email protected] Sources and Studies Editor: Jesper Lützen University of Copenhagen Department of Mathematics Universitetsparken 5 DK-2100 Copenhagen Denmark Mathematics Subject Classification (2000) 01A45, 01A75, 51-03 Library of Congress Cataloging-in-Publication Data Wallis, John, 1616-1703. [Arithmetica infinitorum. English] The arithmetic of infinitesimals : John Wallis 1656/[translated from Latin to English with an introduction by] Jacqueline A. Stedall. p. cm.--(Sources in the history of mathematics and physical sciences) Includes bibliographical references and index. 1. Wallis, John, 1616-1703. Arithmetica infinitorum. 2. Curves--Rectification and quadrature--Early works to 1800. 1. Stedall, Jacqueline A. 11. Title. 111. Sources and studies in the history of mathematics and physical sciences. QA626.W252004 51O--dc22 2003070361 Printed on acid-free paper. ISBN 978-1-4419-1922-9 ISBN 978-1-4757-4312-8 (eBook) DOI 10.1007/978-1-4757-4312-8 © 2004 Springer Science+Business Media New York Softcover reprint ofthe hardcover 1st edition 2004 Originally published by Springer Verlag New York, Inc. in 2004. All right reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 SPIN 10956595 springeronline.com For June Barrow-Green and Jeremy Gray, and in memory of John Fauvel who did so much to make this and many other things possible Acknowledgements This translation was made possible by a generous grant from The Leverhulme Trust. The idea arose initially out of conversations with John Fauvel and David Fowler, and the project was eventually planned and set in motion by John Fauvel, then of the Centre for the History of the Mathematical Seiences at the Open University. Unfortunately John died less than three months after we began working on it. Jeremy Gray of the Open University then took over responsibility for the project, and it has been carried out as faithfully as possible to John's original vision. Contents Acknowledgement vii Frontispiece: Title page of the Arithmetica infinitorum 1656 . x Introduction: The Arithmetic of Infinitesimals by Jacqueline A. Stedall .................................. xi An advertisement of the forthcoming Arithmetica infinitorum, Easter 1655 xxxiv To the most Distinguished and W orthy gentleman and most Skilled Mathematician, Dr William Oughtred, Rector of the church of Aldbury in the Country of Surrey 1 To the Most Respected Gentleman Doctor William Oughtred, most widely famed amongst mathematicians, by John Wallis, Savilian Professor of Geometry at Oxford ... ... .. 9 Doctor William Oughtred: A Response to the preceding letter (after the book went to press). In which he makes it known what he thought of that method .................... 11 The Arithmetic of Infinitesimals or a New Method of Inquiring into the Quadrature of Curves, and other more dijJicult mathematical problems. .......................... 13 Glossary ...................................................... 183 Bibliography. ............................................... .. 185 Index 191 10bannis 11'alJjJii, ss, Tb. D. GEOMETRJIE PROFESSORIS S AI'1t.u» JX.. I in Celeberrirns Academia 0 XO NI EN SI, ARITHMETICA INFINITORVM, SIVE Nova Methodus [nqutrendi in Curvili.... neorum Quadraturam, äli2q; djfficl1iora Matbcfeos Problemara. OXON11 , Typil L E0 N: L 1C HF I EL D Ac.dcmiz Typographi, JQ1?tn6. THO. ROß INSON. Annt 16~6. - Title page of the Arithmetica infinitorum 1656 The Arithmetic of Infinitesimals John Wallis (1616-1703) was appointed Savilian Professor of Geometry at Oxford in 1649. He was then a relative newcomer to mathematics, and largely self-taught, but in his first few years at Oxford he produced his two most significant works: De sectionibus conicis and Arithmetica injinitorum. Both were printed in 1655, and published in 1656 in the second volume of Wal­ lis's first set of collected works, Operum math ematicorum.t In De sectionibus conicis, Wallis found algebraic formulae for the parabola, ellipse and hyper­ bola, thus liberating them, as he so aptly expressed it, from 't he embranglings of the cone' .2 His purpose in doing so was ultimately to find a general method of quadrature (or cubature) for curved spaces, a promise held out in De sectionibus conicis and taken up at length in the Arithmetica injinitorum.3 In both books Wallis drew on ideas originally developed in France, Italy, and the Netherlands: algebraic geometry and the method of indivisibles, but he handled them in his own way, and the resulting method of quadrature, based on the summation of indivisible or infinitesimal quantities," was a crucial step towards the development of a fully fledged integral calculus some ten years later. To the modern reader the Arithmetica injinitorum reveals much that is of historical and mathematical interest, not least the mid seventeenth-century tension between classical geometry on the one hand, and arithmetic and 1 For the first editions of De conicis sectionibus and Arithmetica infinitorum, see Operum mathematicorum, 1656-57, 11, 49-108 and 1-199 (separate pagination). Both works were reprinted in Wallis's second set of collected works, Opera mathematica, Wallis 1693-99, I, 291-354 and 355-478. 2 Wallis 1685, 291-292. 3 See De sectionibus conicis, Proposition 48; Arithmetica infinitorum, Proposition 45. 4 Strietly speaking an 'indivisible' has at least one of its dimensions zero, (for example, a point, line or plane), whereas an 'infinitesimal' has arbitrarily small but non-zero width or thiekness. Wallis blurred the distinction between the two and generally spoke only of 'infinitely small qu antities'. For hirn 'indivisible' and 'infinitesimal' were more usefully seen as geometrie and arithmetie categories, respectively. xii The Arithmetic of Infinitesimals algebra on the ot her. Newton was to take up Wallis's work and transform it into mathematics that has become par t of the mainstream , but in Wallis 's text we see what we think of as modern mathematics still stru ggling to emerge. It is t his sense of watehing new and significant ideas force their way slowly and sometimes painfully into existence t hat makes t he Arithmetica infinitorum such a relevant text even now for st udents and historians of math­ ematics alike. Wallis 's mathematical background Wallis was educated from the age of nine by a pri vate tutor t hen, at the age of fourteen , for a year at Felsted School in Essex, and then at Emmanuel College, Cambridge." He later claimed that he had learned little or no math­ emat ics at Cambridge (though he did study some astronomy) . Instead he tau ght hirnself elementary arit hmetic from the textbooks of a younger brother who was preparing to go int o trad e. After the brief tenure of a Fellowship at Queens' College, Cambridge, Wallis was employed as a privat e chaplain, but his mathematical bent came to the fore during t he yea rs of civil war in England (1642-1648) when he regularly decoded letters for Parli ament ." His event ual appointment as Savilian P rofessor at Oxford was no doubt at least in par t a reward for his loyalty and polit ical service to the winning side. By the time he too k up his post , indeed possibly in preparation for it , Wallis had begun t o extend his mathematical knowledge by reading William Oughtred's Clavis mathematicae, the first edition of which had been published in 1631. (Second editions appeared in English and Latin in 1647 an d 1648, respectively, but the copies owned and annotated by Wallis were first edi­ tions.") The Clavis provided Wallis with his first taste of algebraic notation and, as for ot her English readers, an elementary introd uction to t he new sub­ ject of algebraic geometry first developed by Viet e during the 1590s.