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Journey through Mathematics Enrique A. González-elasco Journey through Mathematics Creative Episodes in Its History Enrique A. González-Velasco Department of Mathematical Sciences University of Massachusetts at Lowell Lowell, MA 01854 USA [email protected] ISBN 978-0-387-92153-2 e-ISBN 978-0-387-92154-9 DOI 10.1007/978-0-387-92154-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011934482 Mathematics Subject Classification (2010): 01-01, 01A05 © Springer Science+Business Media, LLC 2011 All rights 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, 233 Spring Street, New York, NY 10013, USA), 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. Cover Image: Drawing in the first printed proof of the fundamental theorem of calculus, published by Heir of Paolo Frambotti, Padua in 1668, by James Gregory in GEOMETRIÆ PARS VNIVERSALIS (The Universal Part of Geometry). Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To my wife, Donna, who solved quite a number of riddles for me. With thanks for this, and for everything else. TABLE OF CONTENTS Preface ix 1 TRIGONOMETRY 1 1.1 The Hellenic Period 1 1.2 Ptolemy’s Table of Chords 10 1.3 The Indian Contribution 25 1.4 Trigonometry in the Islamic World 34 1.5 Trigonometry in Europe 55 1.6 Fromiète to Pitiscus 65 2 LOGARITHMS 78 2.1 Napier’s First Three Tables 78 2.2 Napier’s Logarithms 88 2.3 Briggs’ Logarithms 101 2.4 Hyperbolic Logarithms 117 2.5 Newton’s Binomial Series 122 2.6 The Logarithm According to Euler 136 3 COMPLEX NUMBERS 148 3.1 The Depressed Cubic 148 3.2 Cardano’s Contribution 150 3.3 The Birth of Complex Numbers 160 3.4 Higher-Order Roots of Complex Numbers 173 3.5 The Logarithms of Complex Numbers 181 3.6 Caspar Wessel’s Breakthrough 185 3.7 Gauss and Hamilton Have the Final Word 190 4 INFINITE SERIES 195 4.1 The Origins 195 4.2 The Summation of Series 203 4.3 The Expansion of Functions 212 4.4 The Taylor and Maclaurin Series 220 vii viii Table of Contents 5 THE CALCULUS 230 5.1 The Origins 230 5.2 Fermat’s Method of Maxima and Minima 234 5.3 Fermat’s Treatise on Quadratures 248 5.4 Gregory’s Contributions 258 5.5 Barrow’s Geometric Calculus 275 5.6 From Tangents to Quadratures 283 5.7 Newton’s Method of Infinit Series 289 5.8 Newton’s Method of Fluxions 294 5.9 Was Newton’s Tangent Method Original? 302 5.10 Newton’s First and Last Ratios 306 5.11 Newton’s Lastersion of the Calculus 312 5.12 Leibniz’ Calculus: 1673–1675 318 5.13 Leibniz’ Calculus: 1676–1680 329 5.14 The Arithmetical Quadrature 340 5.15 Leibniz’ Publications 349 5.16 The Aftermath 358 6 CONVERGENCE 368 6.1 To the Limit 368 6.2 Theibrating String Makes Waves 369 6.3 Fourier Puts on the Heat 373 6.4 The Convergence of Series 380 6.5 The Difference Quotient 394 6.6 The Derivative 401 6.7 Cauchy’s Integral Calculus 405 6.8 niform Convergence 407 BIBLIOGRAPHY 412 Index 451 PREFACE In the fall of 2000, I was assigned to teach history of mathematics on the retirement of the person who usually did it. And this with no more reason than the historical snippets that I had included in my previous book, Fourier Analysis and Boundary Value Problems. I was clearly fond of history. Initially, I was unhappy with this assignment because there were two ob- vious difficultie from the start: (i) how to condense about 6000 years of mathematical activity into a three-month semester? and (ii) how to quickly learn all the mathematics created during those 6000 years? These seemed clearly impossible tasks, until I remembered that Joseph LaSalle (chairman of the Division of Applied Mathematics at Brown during my last years as a doctoral student there) once said that the object of a course is not to cover the material but to uncover part of it. Then the solution to both problems was clear to me: select a few topics in the history of mathematics and uncover them sufficientl to make them meaningful and interesting. In the end, I loved this job and I am sorry it has come to an end. The selection of the topics was based on three criteria. First, there are always students in this course who are or are going to be high-school teachers, so my selection should be useful and interesting to them. Through the years, my original selection has varied, but eventually I applied a second criterion: that there should be a connection, a thread running through the various topics through the semester, one thing leading to another, as it were. This would give the course a cohesiveness that to me was aesthetically necessary. Finally, there is such a thing as personal taste, and I have felt free to let my own interests help in the selection. This approach solved problem (i) and minimized problem (ii), but I still had to learn what happened in the past. This brought to the surface another large set of problems. The firs time I taught the course, I started with secondary sources, either full histories of mathematics or histories of specifi topics. This proved to be largely unsatisfactory. For one thing, coverage was not extensive enough so that I could really learn the history of my chosen topics. There is also the fact that, frequently, historian A follows historian B, who in turn follows historian C, and so on. For example, I have at least four ix x Preface books in my collection that attribute the ratio test for the convergence of infinit series to Edward Waring, but without a reference. I finall traced this partial misinformation back to Moritz Cantor’s Vorlesungen über Geschichte der Mathematik.1 This is history by hearsay, and I could not fully put my trust in it. There is also the matter of unclear or insufficien references, with the additional problem that sometimes they are to other secondary sources. Finally, I had to admit that not all secondary sources offer the truth, the whole truth, and nothing but the truth (Rafael Bombelli, the discoverer of complex numbers, has particularly suffered in this respect). The long and the short of it is this: it’s a jumble out there. After my firs semester teaching the course, it was obvious to me that I had to learn the essential facts about the work of any major mathematician included here straight from the horse’s mouth. I had to fin original sources, or translations, or reprints. I enlisted the help of our own library and the Boston Library Consortium, with special thanks to MIT’s Hayden Library. Beyond this, I relied on the excellent service of our Interlibrary Loan Department. But even all this would have been insufficien and this book could not have been written in its present form. I purchased a large collection of books, mostly out of print and mostly on line, and scans of old books on CD, all of which were of invaluable help. Special thanks are also due to the Gottfried Wilhelm Leibniz Bibliothek, of Hanover, for copies of the relevant manuscripts of Leibniz on his discovery of the calculus. As for the rest, the very large rest, I went on line to several digitized book collections from around the world, too many to cite individually. It is a wonder to me that history of mathematics could be done before the existence of these valuable resources. Many of the works I have consulted are already translated into English (such as those by Ptolemy, Aryabhata, Regiomontanus, some ofiète’s, Napier, Briggs, Newton, and—to a limited and unreliable extent—Leibniz), but in most other cases the documents are available only in the language orig- inally written in or in translations into languages other than English (such as those byAl Tusi, Saintincent, Bombelli, most of Gregory’s, Fermat, Fourier, da Cunha, and Cauchy). Except by error of omission, the translations in this book that are not credited to a specifi source are my own, but I wish to thank my colleague Rida Mirie for his kind help withArabic spelling and translation. 1 ol. 4, B. G. Teubner, Leipzig, 1908, p. 275. Cantor gave the reference, but with no page number, and then he put his own misleading interpretation of Waring’s statement in quotation marks! For a more detailed explanation of Waring’s test, stronger than the one given later by Cauchy, see note 21 in Chapter 6. Preface xi As much as I believe that a text on mathematics must include as many proofs as possible at the selected level, for mathematics without proofs is just a story, I also believe that history without complete and accurate references is just a story, a frustrating one for many readers. I have endeavored to give as complete a set of references as I have been able to.

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