Apollonius of Perga’s Conica Text, Context, Subtext

512 Pages·2001·42.9 MB·English

by  Michael Fried

APOLLONIUS OF PERGA'S CONICA
MNEMOSYNE
BIBLIOTHECA CLASSICA BATAVA
COLLEGERUNT
H. PINKSTER· H. W. PLEKET· Cj. RUUGH
D.M. SCHENKEVELD • P. H. SCHRIJVERS
S.R. SLINGS
BIBLIOTHECAE FASCICULOS EDENDOS CURAVIT
C.]. RUUGH, KLASSIEK SEMINARIUM, OUDE TURFMARKT 129, MfSTERDAM
SUPPLEMENTUM DUCENTESIMUM VICESIMUM SECUNDUM
M.N. FRIED
S. UNGURU
APOLLONIUS OF PERGA'S CONICA
APOLLONIUS OF PERGA'S CONICA
TEXT, CONTEXT, SUBTEXT
BY
MICHAEL N. FRIED
SABETAI UNGURU
BRILL
LEIDEN· BOSTON· KOLN
2001
This book is printed on acid-free paper.
Library of Congress Catalo~~-in-Publication Data
Fried, Michael N.
Apollonius of Perga's Conica : text, context, subtext / by Michael N.
Fried, Sabetai Unguru.
p. cm. - (Mnemosyne, bibliotheca classica Batava.
Supplementum, ISSN 0169-8958; 222)
Includes bibliographical references (p. ) and index.
ISBN 9004119779
I. Apollonius, of Perga. Konica .. 2. Mathematics, Greek. 3. Conic
sections-Early works to 1800. I. Tide. II. Series.
QA31.F75 2001
516'.15-dc21 2001035937
Die Deutsche Bibliothek -CIP-Einheitsaufnalune
Apollonius of Perga's Conica: text, context, subtext / by Michael N. Fried;
Sabetai Unguru. - Leiden ; Boston; Kaln : Brill, 2001
(Mncmosyne : Supplemcntum ; Vol. 222)
ISBN 90-04-11977-9
ISSN 0169-8958
ISBN 9004 11977 9
© Copyright 2001 by Koninklyke Brill NT; Laden, The Netherlands
All rights reserved. No part qf this publication may be reproduced, translated, stored in
a retrieval -D'stem, or transmitted in any form or by a1'!)! means, electronic,
mechanical, photocopying, recording or othenvise, without prior written
permission flom the publisher.
Authon'zation to photocopy items for internal or personal use
is granted by Brill provided that the appropriate fees are paid directly
to The Copyright Clearance Center, 222 Rosewood Drive, Suite 910
Danvers MA 01923, USA. Fees are subject to change.
PRINTED IN THE NETHERLANDS
To Michael's parents, Dan and Thelma
&
To the memory qf Sabetai's, Zeida and Chitel

CONTENTS
Preface and Acknowledgements ................................................ Xl
Introduction ....... ............ ...... ................... ..... .... ............... ...... ...... 1
Apollonius's Conica and its History .................................... 2
The Plan of the Present Work .......................................... 12
I. Geometrical Algebra .......................................................... 17
A. Its Formulation .............................................................. 17
B. 1. Its Origins .................................... ...... ...... ...... ...... ...... 20
2. Strength and Weakness of Geometrical
Algebra ...................................................................... 35
C. The Influence of "Geometrical Algebra" in the
Interpretation of the Conica ....... ...... ..... ....... ...... ..... ...... 38
Figures .................................................................................. 52
II. The Elements of Conic Sections in the Framework
of Books I-III ...................................................................... 56
Introduction ....................... ....................... .................. .......... 56
Apollonius's Introduction and the General Nature
of "Elements" .................................................................. 57
The Initial Definitions ................. ...... ...... ..... ....... ..... .... ... ... 61
The Definitions of the Conic Sections and the
fiymptomata ........................................................................ 74
A Second Look at the Algebraic Interpretation of
the Elements of Conics .. ................................................ 90
fiymptomata and Equations ........ ................. ............ ....... ...... 92
The Locus Problem ............................................................ 97
Intentiones Primae and Intentiones Secundae ......................... ..... 10 1
The Place of Diameters .......... ...... ..... ................... ...... ........ 103
Conclusion ............................................................................ 107
Figures .................................................................................. 109
III. Book IV of the Elements of Conic Sections:
How Conic Sections Meet in the Plane .......................... 117
Introduction ................ ..................... .... ............ ... .................. 117
YIn CONTENTS
The Investigation of Opposite Sections ............................ 119
How Conic Sections Meet in a Plane ...... ............ ............ 128
Digression: Descartes' Treatment of Cutting and
Touching in La Geometrie ............................. ... ................ 133
The Possibility of Different Conic Sections in
One Plane ........................................................................ 135
Conclusion ... ...................... ....... ............ ....... ...... ................... 138
Figures .................................................................................. 140
IV. Maximum and Minimum Lines: Book V of the
Conica .................................................................................... 146
Introduction .... ...................... .... ... ... ............. .............. ....... .... 146
The Conventional View: Normals as Maxima
and Minima· .................................................................... 148
The Importance of Normals .............................................. 155
The Evolute ........................................................................ 167
Maximum and Minimum Lines as Longest and
Shortest Lines .................................................................. 17 7
Neusis-like Constructions .................................................... 187
Conclusion ..... ...... ................ ...... ... .................... .......... .......... 203
Figures . ... ... ...... ... ... ...................... ............. ............................ 205
V. Equality and Similarity: Book VI of the Conica .............. 221
Introduction ................. ... ...................................................... 221
Equality............................................ ................ .................... 224
Similarity .............................................................................. 242
Equality and Similarity .... ... ............ ............................... .... 252
The Constructions ........................... .... ................ ............... 260
Conclusion ............................................................................ 270
Figures .................................................................................. 275
VI. Diorismic Theorems and the Use of Analogy:
Book VII of the Conica . ......... ................... ............ .......... ... 283
Introduction .............. ... ... ................. ............. ........................ 283
Diorismoi and Diorismic Theorems .................................. 284
A Glance at Halley's Book VIII................................ ........ 293
The Diorismic Theorems of Book VII ....................... ..... 298
Images of Diameters and Latera Recta ................................ 306
Conclusion ...................... ........................... ... ... ........ ............. 320
Figures ..... ..... ...................................................... .................. 324
CONTENTS IX
VII. Once Again, the use of Analogy in Apollonius:
Conics and Circles .......................................................... 332
Introduction ................. ...... ... ... ... ......... ...... ....................... 332
The Tangent and the Asymptote: An "Internal"
Analogy in the Ganica ......... ...... ...... ...... ....................... 335
Circles and Conic Sections ............................................ 344
Conclusion ....... ........................ ............ ............................. 355
Figures .............................................................................. 357
VIII. The Ganica as an Integral Whole: Elementary
and Non-Elementary Books ............................................ 364
Introduction .......... ............ ..... .......... .................. ........ ....... 364
Was the Ganica Written as One book? .......................... 365
Elementary and Advanced Books: The Basic
Division in the Ganica .................................................. 370
Conclusion .. ........ .................... ...... ............ .................. ...... 386
Figures .............................................................................. 389
IX. Some Circumscribed Reflections on the
Historiography of Hellenistic Mathematics 390
Appendix: English Translation of Book IV............................ 413
Translator's Preface .......................................................... 413
Lettering Scheme for Diagrams in Book IV... ............. 415
Apollonius's Ganica: Book IV.......................................... 416
Diagrams ............................... ...... ...... ...... ... ... ............ ....... 454
Bibliography ................................................................................ 487
Index ............................................................................................ 495