Pappus of Alexandria: Book 4 of the Collection

351 Pages·2010·1.95 MB·English

by  Heike Sefrin-Weis (auth.)

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
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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
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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
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