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Pappus of Alexandria: Book 4 of the Collection Pappus of Alexandria: Book 4 of the Collection For other titles published in this series, go to http://www.springer.com/series/4142 Sources and Studies in the History of Mathematics and Physical Sciences Managing Editor J.Z. Buchwald Associate Editors J.L. Berggren and J. Lützen Advisory Board C. Fraser, T. Sauer, A. Shapiro Pappus of Alexandria: Book 4 of the Collection Edited With Translation and Commentary by Heike Sefrin-Weis Heike Sefrin-Weis Department of Philosophy University of South Carolina Columbia SC USA [email protected] Sources Managing Editor: Jed Z. Buchwald California Institute of Technology Division of the Humanities and Social Sciences MC 101–40 Pasadena, CA 91125 USA Associate Editors: J.L. Berggren Jesper Lützen Simon Fraser University University of Copenhagen Department of Mathematics Institute of Mathematics University Drive 8888 Universitetsparken 5 V5A 1S6 Burnaby, BC 2100 Koebenhaven Canada Denmark ISBN 978-1-84996-004-5 e-ISBN 978-1-84996-005-2 DOI 10.1007/978-1-84996-005-2 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009942260 Mathematics Classification Number (2010) 00A05, 00A30, 03A05, 01A05, 01A20, 01A85, 03-03, 51-03 and 97-03 © Springer-Verlag London Limited 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Dieses Buch ist für Roger, Hannah und Lisa Preface In the seventeenth century, it was a common opinion among those interested in ancient Greek mathematics that there were four great mathematicians in antiquity: Euclid, Archimedes, Apollonius – and Pappus of Alexandria. Especially the fourth and seventh book of the Collection were widely read and avidly discussed. They do indeed contain a host of interesting material on ancient Greek geometry that is not attested elsewhere. The fourth book in particular offers quite a few vignettes on “higher” Greek mathematics, treating of the classical problems of squaring the circle, doubling the cube, and trisecting the angle. Also reported is an intriguing little piece on mapping the sequence of natural numbers into a closed configuration of touching circles. Despite his importance as a historical source, Pappus has nowa- days become something like the ugly stepchild among the ancient mathematicians. The present edition is intended to provide a basis for giving the writer Pappus another hearing, to rekindle scholarly interest in him. It contains a new edition of the Greek text of Collectio IV (based on a fresh transcription from the main manu- script Vat. gr. 218), an annotated translation, and a commentary. The text offers an alternative to Hultsch’s standard edition. The commentary provides access to quite a few aspects of the work that have so far been somewhat neglected. Above all, it supports the reconstruction of a coherent plan and vision within Collectio IV. This edition is developed out of a complete revision of my 1997 German dis- sertation in the Mathematics Department (Section History of Mathematics) at Mainz University (published in microfiche form in 1998). In difference from the earlier version, a Greek text was included, a new translation into English was made, and the commentary was reformulated as well. Numerous people have supported and helped me over the years with the Pappus project. To all of them, my warmest thanks. In particular, David Rowe, my disserta- tion advisor, has been extremely generous with his time, support, and advice. Henk Bos, as second reader for the original dissertation, gave encouraging, yet incisive and helpful advice. Bob Berghout discussed Pappus with me, pointed me to Ath Treweek’s edition, and let me consult Ath Treweek’s material on Collectio IV. Siegmund Probst (Leibniz-Archiv, Hannover) introduced me to the art and trade of editing manuscripts. Jeremiah Hackett helped me gain access to the Vat. gr. 218. I should also like to thank the Bibliotheca Apostolica Vaticana for the generous hospitality and the inspiring atmosphere they provide for scholars consulting their vii viii Preface collection, and especially for allowing me to study the text and figures of Vat. gr. 218 for the present edition. Ms. Regine Becker proofread the entire Greek text. Ms. Lisa Weis improved my English in numerous places. My deepest gratitude is owed to the members of my family, who have, generously, if not always voluntarily, been living with Pappus for a number of years. That is why this book is dedicated to them. Mainz/Columbia, August 2008 Heike Sefrin-Weis Contents Part I Greek Text and Annotated Translation Introductory Remarks on Part I .................................................................... 3 Part Ia Greek Text ....................................................................................... 11 Part Ib Annotated Translation of Collectio IV .......................................... 83 Props. 1–3: Euclidean Plane Geometry: Synthetic Style .......................... 83 Props. 4–6: Plane Analysis Within Euclidean Elementary Geometry ................................................................................ 88 Props. 7–10: Analysis, Apollonian Style (Focus: Resolutio) ................... 92 Props. 11 and 12: Analysis: Extension of Confi guration/Apagoge ........... 99 Props. 13–18: Arbelos Treatise: Plane Geometry, Archimedean Style .................................................................................... 103 Props. 19–22: Archimedean Spiral ............................................................ 119 Props. 23–25: Conchoid of Nicomedes/Duplication of the Cube ............. 126 Props. 26–29: Quadratrix .......................................................................... 131 Prop. 30: Symptoma-Theorem on the Archimedean Spherical Spiral ......................................................................................... 141 Props. 31–34: Angle Trisection ................................................................. 146 Props. 35–38: Generalization of Solid Problems: Angle Division ............ 155 Props. 39–41: Constructions Based on the Rectifi cation Property of the Quadratrix ........................................................................ 159 Props. 42–44: Analysis of an Archimedean Neusis .................................. 163 Part II Commentary Introductory Remarks on Part II .................................................................. 171 II, 1 Plane Geometry, Euclidean Style .......................................................... 173 Props. 1–6: Plane Geometry, Euclidean Style .......................................... 173 Prop. 1: Generalization of the Pythagorean Theorem .......................... 173 Props. 2 and 3: Construction of Euclidean Irrationals ........................ 175 ix x Contents Props. 4–6: Plane Geometrical Analysis in the Context of Euclidean Geometry ......................................................................... 181 II, 2 Plane Geometry, Apollonian Style ......................................................... 193 Props. 7–10: Plane Geometry, Apollonian Style ....................................... 193 Overview Props. 7–10 ........................................................................... 193 Prop. 7: Determining Given Features Using the Data ......................... 196 Prop. 8: Resolutio for an Intermediate Step in the Apollonian Problem .................................................................... 199 Prop. 9: Auxiliary Lemma for Prop. 10 ................................................ 201 Prop. 10: Resolutio for a Special Case of the Apollonian Problem .................................................................... 201 II, 3 Plane Geometry, Archaic Style .............................................................. 203 Analysis-Synthesis Pre-Euclidean Style ................................................... 203 Props. 11 and 12: Chords and Angles in a Semicircle ......................... 203 Prop. 11: Representation of a Chord as Segment of the Diameter ....... 204 Prop. 12: Angle Over a Segment of the Diameter ................................. 205 II, 4 Plane Geometry, Archimedean .............................................................. 207 Props. 13–18: Arbelos (Plane Geometry, Archimedean Style) ................. 207 Observations on Props. 13–18 .............................................................. 207 Prop. 13: Preliminary Lemma ............................................................
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