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Prelims-N52728.fm Page i Wednesday, November 15, 2006 1:35 PM LEONHARD EULER: LIFE, WORK AND LEGACY Prelims-N52728.fm Page ii Wednesday, November 15, 2006 1:35 PM STUDIES IN THE HISTORY AND PHILOSOPHY OF MATHEMATICS Volume 5 Cover art: “Leonhard Euler”, by Susan Petry, 2002, 26 by 33cm, cherry wood. After the 1753 portrait by Handmann. Photo by C. Edward Sandifer AMSTERDAM - BOSTON - HEIDELBERG - LONDON - NEW YORK - OXFORD - PARIS SAN DIEGO - SAN FRANCISCO - SINGAPORE - SYDNEY - TOKYO Prelims-N52728.fm Page iii Wednesday, November 15, 2006 1:35 PM LEONHARD EULER: LIFE, WORK AND LEGACY Edited by Robert E. Bradley Department of Mathematics and Computer Science Adelphi University 1 South Avenue Garden City, NY 11530, USA C. Edward Sandifer Department of Mathematics Western Connecticut State University 181 White Street Danbury, CT 06810, USA Amsterdam - Boston - Heidelberg - London - New York - Oxford - Paris San Diego - San Francisco - Singapore - Sydney - Tokyo Prelims-N52728.fm Page iv Wednesday, November 15, 2006 1:35 PM Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2007 Copyright © 2007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-444-52728-8 ISBN-10: 0-444-52728-1 For information on all Elsevier publications visit our website at books.elsevier.com Printed and bound in The Netherlands 070809101110987654321 v Foreword The articles in this volume were commissioned by the editors and have not appeared elsewhere. “The Truth About K¨onigsberg,” by Brian Hopkins and Robin J. Wilson, is the only exception. It was published in May 2004 in The College Mathematics Journal and won The Mathematical Association of America’s George P´olya Award. It is included with the kind permission of The Mathematical Association of America. Indirectly, this volume is a result of a contributed paper session on Euler organized by William Dunham and V. Frederick Rickey at the 2001 Joint Mathematics Meetings in New Orleans. This attracted Euler enthusiasts from all over North America. A few, Ronald Calinger, John Glaus and Edward Sandifer, met at a local restaurant to hatch the idea of The Euler Society. The Society had its first annual meeting in the summer of 2002. Two years later, Arjen Sevenster, an editor at Elsevier, contacted us, saying, “I just came across the announcement of the Third Annual Meeting of the Euler Society: Euler 2004. I wonder, if you would be interested in editing a volume covering the same topics of the Meeting: Euler, his work and times, aiming to give a more or less complete picture.” We soon found that The Euler Society alone could not provide the “com- plete picture” the project required. Here, our friend R¨udiger Thiele stepped in and helped introduce us to a number of his European colleagues. We owe the participation of Wolfgang Breidert, Peter Hoffmann, Teun Koetsier, Olaf Neumann, Karin Reich and Dieter Suisky to Thiele’s good efforts. He also invited Michael Raith to contribute, but sadly, Raith passed away be- fore he could contribute. R¨udiger Thiele has dedicated his own contribution to the memory of Michael Raith. Peter Hoffmann wrote his chapter in Ger- man, and R¨udiger Thiele worked to render it into English, the language of the volume. Because of all of this, and because he is in all ways such a fine friend and colleague, we dedicate this volume to R¨udiger Thiele. Without his help, this volume would be much less than it is. vi Foreword Others deserve our recognition. The staff at Elsevier, especially Andy Deelen and Simon Pepping, have been helpful whenever we have needed them, as has Henk Bos, the general editor of the series, Studies in the History and Philosophy of Mathematics. Our six-member Editorial Panel of Ronald Calinger, Lawrence D’Antonio, Stacy Langton, R¨udiger Thiele, Jeff Suzuki and Homer White, did yeoman work refereeing and editing the chapters. We also thank Pat Allaire, Ken Gittelson and Theresa Sandifer for their editorial assistance. The authors of the chapters have been patient and professional with us, and have written some wonderful essays. We also thank our wives and families. Robert E. Bradley C. Edward Sandifer Garden City, NY Newtown, CT August 2006 Contents-N52728.fm Page vii Wednesday, November 15, 2006 1:36 PM vii Contents Foreword ..................................................................................................... v Introduction .................................................................................................. 1 C. E. Sandifer, R. E. Bradley Leonhard Euler: Life and Thought............................................................... 5 R. S. Calinger Leonhard Euler and Russia........................................................................... 61 P. Hoffmann Princess Dashkova, Euler, and the Russian Academy of Sciences.............. 75 R. S. Calinger, E. N. Polyakhova Leonhard Euler and Philosophy ................................................................... 97 W. Breidert Images of Euler ............................................................................................ 109 F. Fasanelli Euler and Applications of Analytical Mathematics to Astronomy .............. 121 C. Wilson Euler and Indian Astronomy ........................................................................ 147 K. Plofker Euler and Kinematics ................................................................................... 167 T. Koetsier Euler on Rigid Bodies .................................................................................. 195 S. G. Langton Euler’s Analysis Textbooks.......................................................................... 213 V. J. Katz Contents-N52728.fm Page viii Wednesday, November 15, 2006 1:36 PM viii Contents Euler and the Calculus of Variations............................................................ 235 R. Thiele Euler, D’Alembert and the Logarithm Function .......................................... 255 R. E. Bradley Some Facets of Euler’s Work on Series....................................................... 279 C. E. Sandifer The Geometry of Leonhard Euler ................................................................ 303 H. S. White Cyclotomy: From Euler through Vandermonde to Gauss............................ 323 O. Neumann Euler and Number Theory: A Study in Mathematical Invention ................. 363 J. Suzuki Euler and Lotteries ....................................................................................... 385 D. R. Bellhouse Euler’s Science of Combinations ................................................................. 395 B. Hopkins, R. Wilson The Truth about Königsberg ........................................................................ 409 B. Hopkins, R. Wilson The Polyhedral Formula............................................................................... 421 D. Richeson On the Recognition of Euler among the French, 1790 - 1830 ..................... 441 I. Grattan-Guinness Euler’s Influence on the Birth of Vector Mechanics.................................... 459 S. Caparrini Euler’s Contribution to Differential Geometry and its Reception ............... 479 K. Reich Euler’s Mechanics as a Foundation of Quantum Mechanics ....................... 503 D. Suisky Index ............................................................................................................ 527 Leonhard Euler: Life, Work and Legacy 1 Robert E. Bradley and C. Edward Sandifer (Editors) © 2007 Elsevier B.V. All rights reserved Introduction C. Edward Sandifer a and Robert E. Bradley b aDepartment of Mathematics Western Connecticut State University Danbury, CT 06810 USA bDepartment of Mathematics and Computer Science Adelphi University Garden City, NY 11530 USA The year 2007 marks 300 years since the birth of Leonhard Euler. This gives historians of mathematics their first opportunity since 1983, 200 years after his death, to celebrate an Eulerian anniversary. Academic celebrations have three traditional forms, books, meetings, and special issues of journals. This book, a collection of chapters written by outstanding Euler scholars from seven different countries, is one such celebration. Chronology and tradition provide us four opportunities to celebrate each
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