Geminos's Introduction to the Phenomena: A Translation and Study of a Hellenistic Survey of Astronomy 9780691187150, 9780691123394, 069112339X, 2006049372

Citation preview

Geminos's Introduction to the Phenomena A TRANSLATION A N D STUDY OF A HELLENISTIC SURVEY O F A S T R O N O M Y

James Evans and J. Lennart Berggren

P R I N C E T O N UNIVERSITY PRESS PRINCETON AND OXFORD

Copyright © 2006 by Princeton University Press Published by Princeton University Press, 41 W i l l i a m Street, Princeton, N e w Jersey 08540 In the United K i n g d o m : Princeton University Press, 3 M a r k e t Place, Woodstock, Oxfordshire O X 2 0 1SY A l l Rights Reserved Library of Congress Cataloging-in-Publication Data Geminus. [Introduction to astronomy. English] Geminos's introduction to the phenomena : a translation and study of a Hellenistic survey of astronomy / [translated and commentary by] James Evans and J. Lennart Berggren. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-691-12339-4 (alk. paper) ISBN-10: 0-691-12339-X (alk. paper) 1. Astronomy, Greek—Textbooks. 2. Astronomy—Early works to 1800. I. Evans, James, 1 9 4 8 - II. Berggren, J. L . III. Title. Q B 2 1 . G 4 5 1 3 2006 520—dc22

2006049372

British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. pup.princeton.edu Printed in the United States of America 10

9

8

7

6

5

4

3

2

1

Contents

ix

List of Illustrations

xiii

List of Tables

xv

Preface INTRODUCTION

l

1. Significance o f G e m i n o s ' s Introduction

to the Phenomena

2

2. Geminos's Other W o r k s

3

3. O n " T h e P h e n o m e n a " i n G r e e k A s t r o n o m y

4

4. T h e G r e e k G e n r e o f A s t r o n o m i c a l Surveys

8

5. G e m i n o s ' s Sources for H i s Introduction

12

6. G e m i n o s ' s C o u n t r y a n d D a t e

15

7. G e m i n o s a n d the Stoics

23

8. G e m i n o s o n A s t r o n o m i c a l Instruments a n d M o d e l s

27

9. G e m i n o s o n M a t h e m a t i c a l Genres

43

10. R e a l i t y a n d Representation i n G r e e k A s t r o n o m y

49

1 1 . H e l i a c a l R i s i n g s a n d Settings

58

12. A s t r o n o m i c a l A p p l i c a t i o n s o f A r i t h m e t i c Progressions

73

13. L u n a r a n d L u n i s o l a r Cycles

82

14. O n the Text a n d T r a n s l a t i o n

101

Introduction to the Phenomena:

TRANSLATION

AND COMMENTARY I. O n the C i r c l e o f the Signs II. Aspects o f the Z o d i a c a l Signs III. O n the C o n s t e l l a t i o n s G e m i n o s ' s Stars a n d C o n s t e l l a t i o n s : A Supplement to C h a p t e r III

113 125 137 140

vi

·

Contents IV. O n the A x i s a n d the Poles

146

V . T h e Circles o n the Sphere

149

V I . O n D a y and N i g h t

161

V I I . O n the R i s i n g s o f the 12 Signs

169

VIII. O n M o n t h s

175

I X . O n Phases of the M o o n

186

X . O n the Eclipse o f the S u n

189

X I . O n the Eclipse o f the M o o n

191

X I I . T h a t the Planets M a k e the M o v e m e n t O p p o s i t e to T h a t of the C o s m o s

195

X I I I . O n R i s i n g s a n d Settings

200

X I V . O n the Paths o f the F i x e d Stars

205

X V . C o n c e r n i n g the Z o n e s o n E a r t h

208

X V I . O n Geographical Regions

210

X V I I . O n Weather Signs f r o m the Stars

217

X V I I I . O n the Exeligmos

227

Parapëgma

231

FRAGMENTS 1 A N D 2, F R O M GEMINOS'S O T H E R WORKS Fragment 1. F r o m G e m i n o s ' s Philokalia: G e m i n o s o n the Classification of the M a t h e m a t i c a l Sciences

243

Fragment 2. F r o m G e m i n o s ' s Concise Exposition of the Meteorology of Poseidônios: G e m i n o s o n the R e l a t i o n of A s t r o n o m y to Physics

250

REFERENCE MATERIALS A p p e n d i x 1. T e x t u a l N o t e s to G e m i n o s ' s Introduction the Phenomena A p p e n d i x 2. T h e G e m i n o s Parapëgma

to 257 275

Contents

·

vii

A p p e n d i x 3. G l o s s a r y o f T e c h n i c a l Terms i n G e m i n o s ' s Introduction to the Phenomena

291

A p p e n d i x 4. Index o f Persons M e n t i o n e d by G e m i n o s

301

Bibliography

303

Index

317

Illustrations

Figures have been added by the authors to provide historical context and to illustrate astronomical concepts. They occur in our Introduction, as well as in our Commentary to the various chapters in Geminos's Introduction to the Phenomena. Diagrams are photographs (or, in one case, a drawing) of illustrations that occur in the actual medieval manuscripts. They occur in the body of Geminos's chapters i and ii, and in our commentary to his chapters χ and xi.

F I G U R E S IN O U R I N T R O D U C T I O N

1.1. A p o r t i o n o f the Celestial Teaching o f Leptinës o n a papyrus. 1.2. T h e Farnese globe. 1.3. T h e M a i n z globe. 1.4. A m o d e r n c o p y o f the M a i n z globe. 1.5. A m o s a i c o f a n a r m i l l a r y sphere i n the C a s a d i L e d a , at S o l u n t o . 1.6. A Renaissance i l l u s t r a t i o n o f a n a r m i l l a r y sphere. 1.7. T h e p r i n c i p l e o f the spherical s u n d i a l w i t h c u t a w a y south face. 1.8. A c o n i c a l s u n d i a l f o u n d near A l e x a n d r i a . 1.9. A h o r i z o n t a l - p l a n e s u n d i a l f o u n d i n A q u i l e i a . 1.10. E u c l i d ' s argument using a dioptra. 1.11. A conjectural reconstruction o f H e r o ' s dioptra. 1.12. A dioptra for measuring the angular diameters o f the Sun and M o o n . 1.13. A conjectural reconstruction o f G e m i n o s ' s equatorial dioptra. 1.14. G e m i n o s ' s branches o f the m a t h e m a t i c a l arts. 1.15. A n eccentric-circle m o d e l for the m o t i o n o f the S u n . 1.16. A concentric-deferent-and-epicycle m o d e l for the m o t i o n o f the S u n . 1.17. E q u i v a l e n c e o f the concentric-plus-epicycle m o d e l to the eccentric-circle m o d e l . 1.18. Fragment o f a stone parapëgma from Miletus.

11 29 30 31 32 33 35 36 37 39 40 41 42 43 55 56 56 60

χ

·

Illustrations

1.19. A p o r t i o n of a papyrus parapëgma, written about 300 B.c. 1.20. True star phases. 1.21. F o r a star o n the ecliptic, the true m o r n i n g r i s i n g . 1.22. T h e situation s h o w n i n F i g . 1.20, after some hours have gone by. 1.23. T h e visible m o r n i n g phases f o l l o w the true m o r n i n g phases i n time. 1.24. A u t o l y k o s ' s 1 5 ° v i s i b i l i t y rule.

62 64 64 65 67 68

F I G U R E S IN O U R C O M M E N T A R Y T O G E M I N O S ' S

VARIOUS CHAPTERS

1.1. If the Sun's circle were centered o n the E a r t h . 2 . 1 . D i v i s i o n o f the z o d i a c i n t o signs i n a system attributed to E u d o x o s . 2 . 2 . D i v i s i o n o f the z o d i a c i n t o signs by astronomers of G e m i n o s ' s time. 4 . 1 . T h e E a r t h s u r r o u n d e d by the sphere o f the cosmos. 4 . 2 . T h e latitude φ is equal to the altitude θ o f the celestial pole. 4 . 3 . T h e celestial sphere for a n observer south o f the equator. 4 . 4 . T h e t w o celestial poles for an observer at the equator. 5 . 1 . T h e celestial sphere for a n observer at a b o u t 3 0 ° latitude. 5.2. T h e arctic circle for an observer at 4 0 ° latitude. 5.3. T h e arctic circle for a n observer at 2 0 ° latitude. 6 . 1 . A " d a y " as the time elapsed f r o m one sunrise to the next. 8.1. T w o successive n e w M o o n s o c c u r r i n g a b o u t 2 9 ° apart. 9 . 1 . A n argument that the M o o n gets its light f r o m the S u n . 1 1 . 1 . Aristarchos's eclipse d i a g r a m . 1 3 . 1 . E v e n i n g risings o f a star.

121 134 135 146 147 148 148 149 150 150 161 175 186 193 202

DIAGRAMS

D i a g r a m 1. C o n j e c t u r a l r e c o n s t r u c t i o n o f G e m i n o s ' s first diagram. D i a g r a m 2 . A n i l l u s t r a t i o n o f the solar theory. V a t i c a n u s gr. 3 1 8 , f. 4. D i a g r a m 3. A n i l l u s t r a t i o n o f the a s t r o l o g i c a l aspect o f o p p o s i t i o n . M a r c i a n u s gr. 3 2 3 , f. 4 8 2 . D i a g r a m 4 . T r i n e aspect. M a r c i a n u s gr. 3 2 3 , f. 4 8 2 .

122 123 126 128

Illustrations D i a g r a m 5. Q u a r t i l e aspect. M a r c i a n u s gr. 3 2 3 , f. 4 8 3 . D i a g r a m 6. Signs i n syzygy. M a r c i a n u s gr. 3 2 3 , f. 484v. D i a g r a m 7. A n i l l u s t r a t i o n o f a solar eclipse. V a t i c a n u s gr. 3 1 8 , f. 2 8 . D i a g r a m 8. A n i l l u s t r a t i o n o f a l u n a r eclipse. V a t i c a n u s gr. 3 1 8 , f. 2 8 .

.

xi 130 133 189 191

Tables

1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. I . l 1. 1.1. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 7.1. 17.1. A2.1. A2.2. A2.3.

C o m p a r i n g 19 A t h y r a n d W i n t e r Solstice O r d e r o f True Star Phases i n the Y e a r O r d e r o f V i s i b l e Star Phases i n the Y e a r R i s i n g T i m e s a n d D a y Lengths i n System A R i s i n g T i m e s for the Klima o f 1 5 D a y Lengths for the Klima o f 1 5 D a y Lengths a c c o r d i n g to G e m i n o s ' s R u l e D a y Lengths i n G e m i n o s ' s Scheme for the Klima o f 1 5 First N e w M o o n s o f the Years 1 9 6 1 - 9 0 Y e a r a n d M o n t h Lengths i n the Octaetëris and Related Cycles Y e a r a n d M o n t h Lengths i n the 19- a n d 76-Year Periods D i v i n e a n d Secular N a m e s for the Planets Geminos's Constellations: The Z o d i a c Geminos's N o r t h e r n Constellations G e m i n o s ' s Southern C o n s t e l l a t i o n s G e m i n o s ' s Stars: Z o d i a c a l C o n s t e l l a t i o n s G e m i n o s ' s Stars: N o r t h e r n Constellations G e m i n o s ' s Stars: Southern C o n s t e l l a t i o n s R i s i n g a n d Setting T i m e s o f the Signs i n the Klima o f 1 5 M o r n i n g R i s i n g s o f Sirius, A c c o r d i n g to P t o l e m y Star Phases i n the G e m i n o s Parapëgma C o m p a r i s o n o f the Second M i l e t u s Parapëgma and the G e m i n o s Parapëgma Weather Signs o f K a l l i p p o s i n P t o l e m y a n d i n the G e m i n o s Parapëgma

19 66 69 76 77 78 81 82 83

h

h

h

h

86 89 119 140 141 142 144 144 145 174 224 280 285 287

Preface

A famous m a t h e m a t i c i a n o f the twentieth century once lamented, " T h e algebraic t o p o l o g i s t has p r a c t i c a l l y ceased to c o m m u n i c a t e w i t h the point-set t o p o l o g i s t ! " T h i s r e m a r k is characteristic o f o u r time a n d c u l ­ ture, i n w h i c h k n o w l e d g e has become fractured i n t o thousands o f spe­ cialties a n d subspecialties, a n d i n w h i c h n o one science c a n c l a i m to h o l d a privileged place. It w a s not so i n G r e e k A n t i q u i t y , w h e n a s t r o n o m y was the central science, w i t h v i t a l l i n k s to nearly every other aspect o f the culture. A s t r o n o m y h a d i m p o r t a n t relations w i t h other sciences, such as physics (or p h i l o s o p h y o f nature) a n d mathematics. A s A r i s t o t l e p o i n t e d out, the m o t i o n s o f the celestial bodies were the best clues to the physics (or essential natures) o f these bodies. B u t the methods o f investigation, as w e l l as o f d e m o n s t r a t i o n a n d a p p l i c a t i o n , i n a s t r o n o m y were so thor­ o u g h l y m a t h e m a t i c a l that a s t r o n o m y w a s often considered to be a b r a n c h o f a p p l i e d mathematics. It w a s p a r t l y for this reason that P l a t o i n c l u d e d it i n the q u a d r i v i u m o f m a t h e m a t i c a l arts r e c o m m e n d e d for the e d u c a t i o n o f the guardians of his ideal state. A s t r o n o m y also h a d l i n k s to ancient r e l i g i o n , for the planets were w i d e l y h e l d to be d i v i n e , a n d the celestial p h e n o m e n a c o m m a n d e d the attention o f the poets, w h o f r o m the time of H e s i o d h a d sung o f the celestial signs a n d o f the r e v o l v i n g year. A s t r o n o m y p r o v i d e d subject matter for craftsmen, w h o represented the heavens i n the f o r m o f ingenious globes a n d m e c h a n i s m s . A n d , fi­ nally, it was one o f the m o s t significant channels o f intellectual exchange between ancient c i v i l i z a t i o n s , most n o t a b l y between the B a b y l o n i a n s a n d the G r e e k s . G e m i n o s ' s Introduction to the Phenomena manifests a l l these c u l t u r a l affiliations o f ancient astronomy. T h i s graceful m a n u a l o f astronomy, w r i t t e n p r o b a b l y i n the first century B . C . by a m a n w h o h a d h a d some experience of teaching, remains t o d a y a n engaging i n t r o d u c ­ t i o n to the central n a t u r a l science o f A n t i q u i t y . M u c h o f ancient a s t r o n o m y requires o f the reader a n a p p r o a c h over a l o n g a n d difficult r o a d . T h i s includes Ptolemy's Almagest as w e l l as the planetary theories o f the B a b y l o n i a n scribes. A n d m u c h else is either devoted to special p r o b l e m s (such as A r i s t a r c h o s ' s treatise On the Sizes and Distances of the Sun and Moon) o r consists o f repetitive m a t e r i a l arranged i n theorems a n d proofs that s u r v i v e d because it w a s useful for teaching (such as A u t o l y k o s ' s On Risings and Settings). F i n a l l y , there is a g o o d deal o f l o w - l e v e l , n o n t e c h n i c a l m a t e r i a l w r i t t e n for ancient readers

xvi

·

Preface

w h o were not w i l l i n g to try very h a r d (such as the a s t r o n o m i c a l p o r t i o n s o f Pliny's Natural History), m a t e r i a l that c a n n o t really give a m o d e r n reader a fair a p p r e c i a t i o n o f the ancient science. G e m i n o s ' s Introduction to the Phenomena is one o f a very s m a l l n u m ­ ber o f ancient a s t r o n o m i c a l w o r k s that c a n be read w i t h a p p r e c i a t i o n a n d u n d e r s t a n d i n g by a nonspecialist, but one that offers, nevertheless, a competent a n d reasonably comprehensive account o f its subject. T h e E n g l i s h t r a n s l a t i o n o f the Introduction to the Phenomena here presented is the first complete one ever to be published. W e hope it w i l l be o f interest a n d use not o n l y to historians o f science, but also to students o f ancient c i v i l i z a t i o n , as w e l l as to scientists w h o w a n t to k n o w m o r e a b o u t the origins o f their art. T h e b o o k that the reader n o w h o l d s h a d a l o n g gestation. J E e n c o u n ­ tered G e m i n o s w h i l e c o m p l e t i n g a d o c t o r a l dissertation at the U n i v e r s i t y o f W a s h i n g t o n . W o r k i n g f r o m G e r m a i n e Aujac's relatively recent e d i t i o n o f the G r e e k text (Aujac 1 9 7 5 ) , he translated most o f G e m i n o s ' s chapter v, o n the circles o f the celestial sphere, for his students to read i n a course he w a s teaching o n the h i s t o r y o f astronomy. H e was struck by G e m i ­ nos's patience a n d clarity, a n d c h a r m e d by his frequent use o f literary ex­ amples to illustrate a p o i n t o f astronomy. G e m i n o s w a s a n excellent w r i t e r for students to read—the a s t r o n o m y was accurate a n d useful, but the priorities a n d concerns o f the ancient t h i n k e r came t h r o u g h l o u d a n d clear as w e l l . A student c o u l d read G e m i n o s w i t h scientific as w e l l as his­ t o r i c a l senses o p e n . In 1 9 8 3 - 8 4 , J E spent a year i n P a r i s , w i t h the a i d o f a F u l b r i g h t G r a n t , w o r k i n g at the Centre K o y r é under the patronage of the late R e n é T a t o n , a n d g o i n g regularly to the history o f a s t r o n o m y ses­ sions o f the Equipe Copernic ( C o p e r n i c u s team) at the Paris O b s e r v a ­ tory. H e spent most of the year w o r k i n g o n eighteenth-century physics. B u t i n his spare time, s i m p l y for pleasure a n d as a w a y o f k e e p i n g u p his G r e e k , he c o m p l e t e d a draft t r a n s l a t i o n o f the w h o l e o f G e m i n o s ' s Intro­ duction to the Phenomena. Some time later, at the I n t e r n a t i o n a l C o n ­ gress o f the H i s t o r y o f Science, h e l d at Berkeley, he h a d o p p o r t u n i t y to meet G e r m a i n e A u j a c , w h o responded generously by l e n d i n g h i m her m i c r o f i l m s o f the most i m p o r t a n t G e m i n o s m a n u s c r i p t s . J E used his t r a n s l a t i o n o f G e m i n o s for m a n y years i n teaching a course o n the his­ t o r y o f a s t r o n o m y at the U n i v e r s i t y o f Puget S o u n d . A short extract f r o m G e m i n o s ' s chapter ν appeared i n his The History and Practice of Ancient Astronomy (Evans 1 9 9 8 ) . W h i l e w o r k i n g o n other projects, he o c c a s i o n ­ ally t o o k time out for his o n g o i n g c o m m e n t a r y o n G e m i n o s . J L B a n d J E h a d k n o w n each other for a l o n g time before b e g i n n i n g a c o l l a b o r a t i o n o n G e m i n o s . J L B w o r k s o n b o t h m e d i e v a l A r a b i c mathe­ matics a n d ancient G r e e k mathematics. W i t h R . S . D . T h o m a s , he h a d published a translation of and commentary o n Euclid's Phenomena

Preface

·

xvii

(Berggren a n d T h o m a s 1 9 9 6 ) , w h i c h w e l l e q u i p p e d h i m for further w o r k o n G r e e k p h e n o m e n a literature. A n d , w i t h A l e x a n d e r Jones, he h a d p u b ­ lished a n a n n o t a t e d t r a n s l a t i o n o f the theoretical chapters o f P t o l e m y ' s Geography (Berggren a n d Jones 2 0 0 0 ) . T h r o u g h a h a p p y a l i g n m e n t o f their stars, J L B a n d J E h a d free t i m e , at the same t i m e , to devote to getting G e m i n o s i n t o final f o r m . J L B u n d e r t o o k a complete r e v i e w a n d r e v i s i o n o f the t r a n s l a t i o n . T h e t w o translators c o n s u l t e d r e g u l a r l y o n issues raised by the G r e e k text a n d its t r a n s l a t i o n , s t r i v i n g n o t o n l y for accuracy a n d readable E n g l i s h but also for fidelity to Geminos's style a n d cadences. J L B also reviewed a n d corrected the draft c o m m e n ­ tary, a d d i n g to it his o w n insights. T h e t w o authors w r o t e the i n t r o d u c ­ t i o n a n d appendices together. A l t h o u g h they were able to d o m u c h o f their w o r k apart, c o m m u n i c a t i n g by telephone, e - m a i l , a n d fax, they retain f o n d m e m o r i e s o f w o r k i n g together at the Evans's d i n i n g r o o m table i n Seattle, at the Berggren's house i n C o q u i t l a m , B r i t i s h C o l u m b i a , a n d their m o u n t a i n retreat i n W h i s t l e r , as w e l l as i n the bar o f the S y l v i a H o t e l i n V a n c o u v e r . T h e final p u s h to c o m p l e t i o n o f the m a n u s c r i p t w a s c a r r i e d out at the H e l e n R i a b o f f W h i t e l e y Center, i n F r i d a y H a r b o r , Washington. T h e authors are grateful to friends a n d colleagues w h o helped i n m a n y different w a y s i n the course o f this project. O u r greatest s c h o l a r l y debt is to G e r m a i n e A u j a c , w h o s e G r e e k text p r o v i d e d the basis for o u r transla­ t i o n . A l e x a n d e r Jones, L i b a T a u b , a n d N o e l S w e r d l o w read considerable p o r t i o n s o f the m a n u s c r i p t a n d were generous w i t h c o m m e n t s a n d sug­ gestions, m a n y o f w h i c h resulted i n i m p r o v e m e n t s o r saved us f r o m er­ rors. M a r i n u s T a i s b a k helped w i t h several translations f r o m the L a t i n , a n d T a s o u l a Berggren p r o o f r e a d the G r e e k of the glossary a n d t y p e d the i n d e x . J E remains grateful to the late W i l l H u m p h r e y s for a d a y - l o n g dis­ cussion o f P r o k l o s ' s citations of G e m i n o s i n the Commentary on the First Book of Euclid's Elements. D a r y n L e h o u x generously lent us his o w n translation of the G e m i n o s parapëgma i n advance of the p u b l i c a t i o n of his b o o k o n parapëgmata. O t h e r scholars t o o k the trouble to respond to questions, a m o n g w h o m we particularly thank L a w r e n c e Bliquez, D a v i d L u p h e r , a n d A . M a r k S m i t h . W e alone, o f course, are responsible for any errors or shortcomings i n the final product. Ernst Künzl a n d R u d o l p h Schmidt helped o b t a i n photographs for use as illustrations, a n d R o s s M u l h a u s e n aided w i t h p h o t o g r a p h i c w o r k a n d image processing. T h e U n i v e r s i t y of Puget S o u n d p r o v i d e d sabbatical leave that enabled Professor Evans to concentrate o n the project, as S i ­ m o n Fraser U n i v e r s i t y d i d for Professor Berggren. B o t h institutions also p r o v i d e d financial support for the payment o f fees for the r e p r o d u c t i o n of some o f the images appearing i n the b o o k . T h e H e l e n R i a b o f f W h i t e ley Center, at F r i d a y H a r b o r o n San J u a n Island, generously w e l c o m e d us

xviii

·

Preface

for a stay w h i l e w e were c o m p l e t i n g the final version o f the b o o k . It has also been a privilege a n d a pleasure to w o r k w i t h the capable staff o f P r i n c e t o n U n i v e r s i t y Press, i n c l u d i n g Ingrid G n e r l i c h , editor; J i l l H a r r i s , p r o d u c t i o n editor; a n d B i l l Carver, c o p y editor. W e are grateful to the f o l ­ l o w i n g institutions a n d their helpful staffs for s u p p l y i n g p h o t o g r a p h s a n d for p e r m i s s i o n to use them i n this b o o k : Musée d u L o u v r e , M u s e o A r c h e ologico Nazionale di N a p o l i , M u s e o Archeologico Nazionale di Aquileia, B i b l i o t h e c a N a z i o n a l e M a r c i a n a (Venice), B i b l i o t h e c a Apostólica V a t i ­ cana, Rômisch-Germanisches Z e n t r a l m u s e u m M a i n z , A n t i k e n s a m m l u n g (Berlin), U n i v e r s i t y of W a s h i n g t o n L i b r a r i e s , T h e B r i t i s h M u s e u m , T r i n i t y College D u b l i n , a n d the Musée N a t i o n a l d ' H i s t o i r e et d ' A r t L u x e m b o u r g . It remains o n l y to say that we o w e o u r deepest t h a n k s to S h a r o n E v a n s a n d T a s o u l a Berggren for their understanding a n d s u p p o r t d u r i n g the years it t o o k us to complete this w o r k . J.E. J.L.B. January 2006

Introduction

INTRODUCTION

G e m i n o s , a G r e e k scientific w r i t e r o f w i d e - r a n g i n g interests, has been assigned dates r a n g i n g f r o m the first century B.C. to the first century A . D . , w i t h , w e believe, the first century B.C. the m o r e likely. W e k n o w n o t h i n g of the circumstances o f his life. O f three w o r k s he is believed to have w r i t t e n , o n l y one, the Introduction to the Phenomena, has c o m e d o w n to us. (This w o r k is also frequently referred to as the Isagoge, f r o m the first w o r d o f its G r e e k title, Eisagôgêeis ta phainomena.) The translation o f his Introduction to the Phenomena here presented is the first complete E n g l i s h v e r s i o n ever p u b l i s h e d . F o r the m o d e r n reader, G e m i n o s provides a v i v i d i m p r e s s i o n o f a n educated Greek's v i e w o f the cosmos a n d o f a s t r o n o m y a r o u n d the beginn i n g o f o u r era. M o r e o v e r , he is frequently a graceful a n d c h a r m i n g writer, constantly aware o f his audience, a n d his b o o k remains quite readable today. Indeed, it is one of a very s m a l l n u m b e r o f w o r k s o f ancient a s t r o n o m y that c a n be read right t h r o u g h w i t h a p p r e c i a t i o n a n d u n d e r s t a n d i n g by a nonspecialist. Because G e m i n o s covers m o s t o f the central topics o f ancient G r e e k astronomy, his text provides a n excellent general survey o f those parts o f that a s t r o n o m y not dependent o n sophisticated m a t h e m a t i c a l models. A n E n g l i s h t r a n s l a t i o n o f the Introduction to the Phenomena s h o u l d thus be useful not o n l y to historians o f a s t r o n o m y but also to historians o f science m o r e generally, to those i n terested i n classical c i v i l i z a t i o n , a n d to astronomers w h o w o u l d like to k n o w m o r e a b o u t the h i s t o r y o f their discipline. W e have furnished o u r t r a n s l a t i o n w i t h a commentary, p r i n t e d at the foot o f the page a n d signaled i n the text by superscript numerals. T h e purpose of the c o m m e n t a r y is not to s u m m a r i z e a l l that is k n o w n o n the topics at h a n d , but to o p e n up G e m i n o s ' s text, to m a k e it m o r e c o m p r e hensible, a n d to reveal its connections w i t h other ancient sources— p h i l o s o p h i c a l a n d literary, as w e l l as scientific. It s h o u l d serve, as w e l l , to direct readers to the specialized s c h o l a r l y literature. T e x t u a l notes, signaled i n Geminos's text by superscript r o m a n letters, are g r o u p e d together i n a p p e n d i x 1.

2

·

Section 1

1. S I G N I F I C A N C E O F G E M I N O S ' S INTRODUCTION THE

TO

PHENOMENA

G e m i n o s ' s Introduction to the Phenomena, a competent a n d engaging i n t r o d u c t i o n to astronomy, w a s p r o b a b l y w r i t t e n i n c o n j u n c t i o n w i t h teaching. G e m i n o s discusses a l l o f the i m p o r t a n t branches o f G r e e k as­ t r o n o m y , except planetary theory. T h i s he promises to take up "else­ w h e r e . " Perhaps he d i d discuss planetary theory i n another w o r k , but i f so, it has not s u r v i v e d . T o p i c s covered i n G e m i n o s ' s Introduction include the z o d i a c , solar theory, the constellations, the theory o f the celestial sphere, the v a r i a t i o n i n the length o f the day, l u n i s o l a r cycles, phases o f the M o o n , eclipses, h e l i a c a l risings a n d settings o f the fixed stars, terres­ t r i a l zones, a n d an i n t r o d u c t i o n to B a b y l o n i a n l u n a r theory. Because the w o r k w a s w r i t t e n for beginners, it does not often get i n t o technical detail—except i n the discussion o f l u n i s o l a r cycles, where G e m i n o s does indulge i n a bit o f arithmetic. G e m i n o s ' s b o o k is i m p o r t a n t to the task o f filling gaps i n the h i s t o r y o f G r e e k a s t r o n o m y i n several w a y s . In general terms, G e m i n o s provides a n o v e r v i e w o f most o f a s t r o n o m y i n the p e r i o d between H i p p a r c h o s (second century B.C.) a n d P t o l e m y (second century A . D . ) , a n d thereby provides a g o o d deal o f insight i n t o w h a t was current a n d c o m m o n k n o w l e d g e i n G e m i n o s ' s o w n day. O n e o f the m o r e c h a r m i n g aspects o f his w o r k , frequently i n evidence, is his desire to set straight c o m m o n m i s c o n c e p t i o n s a b o u t a s t r o n o m i c a l matters. In this way, he offers us v a l u a b l e i n f o r m a t i o n a b o u t the beliefs o f his o w n audience. M o r e specifically, G e m i n o s provides detailed discussions o f several topics not very w e l l treated by other ancient sources. (1) H i s discussion o f B a b y l o n i a n l u n a r theory is a n i m p o r t a n t piece of the story o f the a d a p t a t i o n o f B a b y l o n i a n methods by G r e e k astronomers. (2) H i s dis­ c u s s i o n o f the 8- a n d 19-year l u n i s o l a r cycles is the most detailed by any extant G r e e k source. (3) H i s discussion o f H i p p a r c h o s ' s rendering o f the constellations provides i n f o r m a t i o n not f o u n d i n other sources. (4) H i s refutation o f the t h e n - c o m m o n v i e w that changes i n the weather are caused by the heliacal risings a n d settings o f the stars is the most patient a n d detailed such argument that has c o m e d o w n to us. In the extant m a n u s c r i p t s , G e m i n o s ' s b o o k concludes w i t h a parapëgma (star calendar) that permits one to k n o w the time o f year by o b s e r v a t i o n o f the stars. M a n y scholars believe that this c o m p i l a t i o n is older t h a n G e m i n o s by a century or m o r e . W h e t h e r by G e m i n o s o r not, this parapëgma is one of o u r most i m p o r t a n t sources for the early h i s t o r y of the genre. T h e G e m i n o s parapëgma w a s based substantially u p o n three earlier parapëgmata—those by E u k t ë m ô n (fifth century B.C.), E U d o x o s (early f o u r t h century B.C.), a n d K a l l i p p o s (late f o u r t h century B.C.). Because the G e m i n o s parapëgma s c r u p u l o u s l y cites its sources, it

Geminos's Other Works

·

3

permits us to trace the stages i n the e v o l u t i o n o f the parapëgma between the time o f E u k t ë m ô n a n d the time o f K a l l i p p o s . O u r b o o k includes a t r a n s l a t i o n o f the G e m i n o s parapëgma, as w e l l as a s y n o p t i c table o f its contents (appendix 2), w h i c h s h o u l d be useful i n the study o f this i m p o r ­ tant h i s t o r i c a l d o c u m e n t . A l t h o u g h ancient a n d m e d i e v a l G r e e k readers w o u l d have recognized G e m i n o s ' s b o o k as b e l o n g i n g to a class o f " p h e n o m e n a " literature (see sections 3 a n d 4 b e l o w ) , w e c a n n o t be sure that Introduction to the Phe­ nomena is the title that G e m i n o s himself gave it. T h i s is a c o m m o n diffi­ culty w i t h ancient scientific texts, the c o n v e n t i o n a l titles o f w h i c h are not always a u t h o r i a l . T h e G r e e k manuscripts o f G e m i n o s ' s text d o p r o v i d e g o o d evidence for the c o m m o n l y accepted title, a l t h o u g h there are sev­ eral variants. Indeed, the three best a n d oldest G r e e k m a n u s c r i p t s pres­ ent a bit o f a puzzle: one gives as its title Geminos's Introduction to the Phenomena-, another gives Geminos's Introduction to the Things on High (meteora); a n d still another gives neither title n o r author's name, since the copyist never filled i n this i n f o r m a t i o n . Some later G r e e k m a n ­ uscripts s i m p l y have The Phenomena" of Geminos} A s w e shall see be­ l o w (sec. 14), the L a t i n a n d H e b r e w translations made i n the twelfth a n d thirteenth centuries (from a n A r a b i c intermediary) also s h o w that there was considerable c o n f u s i o n about the title a n d a u t h o r o f the text. F o r the sake o f simplicity, w e shall a l w a y s refer to G e m i n o s ' s b o o k by the t i ­ tle c o m m o n l y used today, a n d best supported by the G r e e k manuscripts, Introduction to the Phenomena. ((

2. G E M I N O S ' S O T H E R W O R K S G e m i n o s w a s the a u t h o r o f t w o other w o r k s that have not c o m e d o w n to us. O n e was a m a t h e m a t i c a l w o r k o f considerable length that dis­ cussed, a m o n g other things, the p h i l o s o p h i c a l foundations o f geometry. Fortunately, a large n u m b e r o f passages f r o m this w o r k (whether i n q u o ­ t a t i o n o r i n paraphrase) are preserved by P r o k l o s i n his Commentary on the First Book of Euclid's Elements. T h e exact title o f G e m i n o s ' s b o o k is uncertain, but i n one passage P r o k l o s r e m a r k s , "so m u c h have I selected f r o m the Philokalia o f G e m i n o s . " (Philokalia means " L o v e o f the B e a u ­ tiful.") In one passage o f considerable interest, G e m i n o s discussed the 2

3

For the Greek titles, see the first textual note (appendix 1). Proklos (c. A.D. 41CM185) was a prolific Neoplatonist philosopher, best known for his Platonic Theology and his commentaries on Plato. His extant scientific works include a Commentary on the First Book of Euclid's Elements and a Sketch of Astronomical Hy­ potheses. Friedlein 1873, 177; M o r r o w 1970, 139. The title of Geminos's mathematical work has been disputed. See the introduction to fragment 1 for a discussion of this issue. 1

2

3

4

·

Section 3

branches o f m a t h e m a t i c a l science a n d their relationships to one another. T h i s is the m o s t detailed such discussion that has c o m e d o w n to us f r o m the G r e e k s . M o r e o v e r , it is clear that G e m i n o s was discussing, not merely abstract d i v i s i o n s o f mathematics, but actual genres o f mathem a t i c a l w r i t i n g . Because several o f G e m i n o s ' s branches o f mathematics p e r t a i n to a s t r o n o m y (e.g., sphairopoiïa, d i o p t r i c s , a n d g n o m o n i c s ) , his discussion sheds light o n the r e l a t i o n s h i p o f a s t r o n o m y to other mathem a t i c a l endeavors. Because o f its interest for the h i s t o r y o f astronomy, w e have i n c l u d e d a t r a n s l a t i o n o f this passage f r o m G e m i n o s ' s Philokalia as fragment 1. G e m i n o s w a s also the a u t h o r of a m e t e o r o l o g i c a l w o r k , w h i c h w a s perhaps a c o m m e n t a r y o n , o r a n a b r i d g e m e n t of, a n o w lost Meteorology o f P o s e i d ô n i o s . A fragment o f some length is preserved by S i m p l i k i o s i n his Commentary on Aristotle's Physics. A p p a r e n t l y , by S i m p l i k i o s ' s t i m e , G e m i n o s ' s m e t e o r o l o g i c a l b o o k h a d been lost, for S i m p l i k i o s makes it clear that he is q u o t i n g G e m i n o s , n o t f r o m G e m i nos's o w n w o r k , but f r o m some w o r k by A l e x a n d e r o f A p h r o d i s i a s . In the course o f his c i t a t i o n , S i m p l i k i o s says that A l e x a n d e r d r e w these r e m a r k s f r o m G e m i n o s ' s "Concise Exposition of the Meteorology of Poseidônios." T h e fragment f r o m G e m i n o s preserved by S i m p l i k i o s is o f c o n s i d e r a b l e interest, for it is devoted to the l i m i t s o f a s t r o n o m i c a l k n o w l e d g e . In this passage, G e m i n o s discusses the r e l a t i o n s h i p o f ast r o n o m y to physics (or n a t u r a l p h i l o s o p h y ) , a r g u i n g that a s t r o n o m y is, o f itself, unable to decide between c o m p e t i n g hypotheses a n d m u s t rely o n physics for guidance a b o u t first p r i n c i p l e s . W e include a transl a t i o n o f this passage f r o m G e m i n o s ' s lost m e t e o r o l o g i c a l w o r k as fragment 2 . 4

5

6

7

3 . O N " T H E P H E N O M E N A " IN G R E E K A S T R O N O M Y

G e m i n o s ' s Introduction to the Phenomena h a d its roots i n a w e l l established genre. In order to e x p l a i n w h a t the writers a n d readers o f this genre considered to be relevant, we must say a little about w h a t G r e e k Poseidônios (c. 135 to c. 51 B.C.) was a Stoic philosopher who wrote also on history, geography, and astronomy. N o complete works survive, but a large number of fragments have been collected. See Edelstein and K i d d 1989; K i d d 1999. Geminos's possible debt to Poseidônios w i l l be discussed below. Simplikios, a Neoplatonist of the sixth century A . D . , was the author of commentaries on Aristotle's Physics and On the Heavens and was one of the philosophers who left Athens after the emperor Justinian closed the pagan schools of philosophy in 529. Alexander of Aphrodisias, who flourished around A . D . 200, was the author of commentaries on Aristotle, many of which survive. Diels 1882, 291. See fragment 2, below, for the complete passage. 4

5

6

7

"The Phenomena" in Greek Astronomy

·

5

a s t r o n o m i c a l writers m e a n by the phenomena. T h e w o r d " p h e n o m e n a " is a p a r t i c i p l e o f the passive verb phainomai, w h i c h carries the meanings of " t o c o m e to light, c o m e to sight, be seen, appear." T h e last t w o are definitive for the a s t r o n o m i c a l sense o f the w o r d , w h i c h is "things that are seen/appear i n the heavens." A late source, S i m p l i k i o s , quotes Sosigenes as h a v i n g a t t r i b u t e d to P l a t o the statement that the task o f a s t r o n o m y w a s to s h o w h o w , by a c o m b i n a t i o n o f u n i f o r m c i r c u l a r m o t i o n s , one c o u l d "save (i.e., a c c o u n t for) the p h e n o m e n a . " T h e a s c r i p t i o n to P l a t o is c o n t r o v e r s i a l (see sec. 10 b e l o w ) , but i n a n y case the w o r d Phenomena appears as the title o f a w o r k by a n associate o f P l a t o , E u d o x o s o f K n i d o s (early f o u r t h c e n t u r y B.C.). E u d o x o s ' s w o r k has n o t s u r v i v e d , but its essence is preserved i n a p o e m o f the same n a m e by A r a t o s (early t h i r d c e n t u r y B.C.). T h e poetic Phenomena o f A r a t o s w a s the subject o f a c o m m e n t a r y by the great as­ t r o n o m e r H i p p a r c h o s o f R h o d e s (second century B.C.), w h o w a s able to c o m p a r e it w i t h the text o f E u d o x o s a n d demonstrate that A r a t o s h a d indeed r e l i e d u p o n E u d o x o s . It appears f r o m these sources that E u ­ d o x o s ' s w o r k w a s d e v o t e d to a detailed d e s c r i p t i o n o f the p l a c e m e n t o f the fixed stars a n d the c o n s t e l l a t i o n s , relative to some s t a n d a r d refer­ ence circles o n the celestial sphere. T h e f o l l o w i n g passages give a sense of the character o f E u d o x o s ' s b o o k , a n d also a n idea o f w h a t sort o f " p h e n o m e n a " it w a s o c c u p i e d w i t h . W e quote d i r e c t l y f r o m H i p p a r chos's Commentary, a n d i n each case H i p p a r c h o s has m a d e it clear that he is h i m s e l f d i r e c t l y r e p o r t i n g o n E u d o x o s ' s text: T h e r e is a certain star that remains a l w a y s i n the same spot; this star is the pole o f the universe.

8

Between the Bears is the t a i l o f the D r a g o n , the end-star o f w h i c h is above the head o f the G r e a t Bear. 9

A r a t o s , f o l l o w i n g E u d o x o s , says that it [the D r a g o n ' s head] moves o n the a l w a y s - v i s i b l e circle, using these w o r d s : "Its head moves where the l i m i t s o f r i s i n g a n d setting are c o n f o u n d e d . " 10

Because A r a t o s includes i n his p o e m a d i s c u s s i o n o f the p r i n c i p a l circles of the celestial sphere (ecliptic, equator, t r o p i c s , arctic c i r c l e , as w e l l as the M i l k y W a y ) , w e m a y surmise that the same m a t e r i a l w a s treated, i n m o r e d e t a i l , by E u d o x o s . S o , by the early f o u r t h century, the basic the­ ory o f the celestial sphere h a d been established, a n d a detailed descripHipparchos, Commentary on the Phenomena of Eudoxos and Aratos i 4.1. Hippar­ chos denounces this as erroneous, pointing out that the place of the celestial north pole was at that time not occupied by a star. Hipparchos, Commentary i 2.3. Hipparchos, Commentary, i 4.7. Quotation from Aratos: Phenomena 6 1 - 6 2 . 8

9

10

6

·

Section 3

t i o n o f the constellations g i v e n . S u c h were the p h e n o m e n a o f E u ­ doxos. T h e oldest extant w o r k n a m e d The Phenomena is that o f E u c l i d (c. 3 0 0 B . C . ) . U n l i k e the w o r k o f E u d o x o s , E u c l i d ' s b o o k has n o place for u r a n o g r a p h y . Rather, a short (and p o s s i b l y spurious) preface introduces the n o r t h celestial p o l e a n d the p r i n c i p a l circles o n the celestial sphere ( i n c l u d i n g the p a r a l l e l circles, the ecliptic, the h o r i z o n , a n d the M i l k y W a y ) . T h e a u t h o r also introduces the arctic a n d antarctic circles relative to a g i v e n l o c a l i t y a n d the consequent d i v i s i o n o f stars i n t o those that never rise, those that rise a n d set, a n d those that never set. T h u s E u ­ d o x o s ' s descriptions o f the constellations have been e l i m i n a t e d i n favor of a geometrical e x p l o r a t i o n o f the sphere. 1 1

12

1 3

After this b e g i n n i n g , E u c l i d ' s treatise proceeds by a series o f p r o p o s i ­ tions w i t h proofs a n d a c c o m p a n y i n g d i a g r a m s , i n the style o f his m o r e famous Elements. These begin w i t h p r o p o s i t i o n 1 o n the central p o s i t i o n of the E a r t h i n the c o s m o s , a n d then progress t h r o u g h three p r o p o s i t i o n s o n the risings a n d settings o f stars. P r o p o s i t i o n s 8 - 1 3 deal w i t h the ris­ ings a n d settings of arcs o f the ecliptic, p a r t i c u l a r l y the z o d i a c a l signs, a n d the w o r k concludes w i t h five p r o p o s i t i o n s o n h o w l o n g it takes e q u a l arcs o f the ecliptic to cross the visible a n d invisible hemispheres. T h e very format o f the w o r k illustrates w h a t h a d become a c o m m o n ­ place a m o n g G r e e k t h i n k e r s , n a m e l y that celestial p h e n o m e n a c a n be ex­ p l a i n e d rationally. O t h e r extant early G r e e k texts for w h i c h the celestial p h e n o m e n a f o r m the subject matter i n c l u d e t w o w o r k s o f E u c l i d ' s c o n t e m p o r a r y , A u t o l y k o s o f Pitanë, b o t h o f t h e m w r i t t e n i n the t h e o r e m - p r o o f style one finds i n E u c l i d ' s b o o k . In On the Moving Sphere, A u t o l y k o s treats some of the p h e n o m e n a arising f r o m the u n i f o r m r o t a t i o n o f a sphere a r o u n d its axis relative to a h o r i z o n that separates the visible f r o m the invisible p o r t i o n s o f the sphere. It is s t r i k i n g that i n On the Moving Sphere, the descriptions o f a l l circles other t h a n the h o r i z o n are as abstract a n d geo­ m e t r i c a l as possible, a n d there is n o e x p l i c i t m e n t i o n of the a s t r o n o m i c a l a p p l i c a t i o n s o f the theorems. A s a n e x a m p l e w e quote p r o p o s i t i o n 8: G r e a t circles tangent to the same [parallel circles] to w h i c h the h o r i z o n is tangent w i l l , as the sphere rotates, fit exactly o n t o the h o r i z o n . T h e ab­ stract character o f m a n y o f these p r o p o s i t i o n s illustrates h o w far the G r e e k g e o m e t r i z a t i o n o f a s t r o n o m y h a d been c a r r i e d by the time o f E u ­ c l i d a n d A u t o l y k o s . M a n y o f the p r o p o s i t i o n s are h a r d to p r o v e , but are easy to illustrate o n a celestial globe. Aristotle (On the Heavens i i 13), who was Eudoxos's younger contemporary, also uses the w o r d "phenomena" in its astronomical sense. For an English translation and commentary, see Berggren and Thomas, 1996. Here, as in Eudoxos's Phenomena, also claimed to be occupied by a star. 11

12

13

"The Phenomena" in Greek Astronomy

·

7

A u t o l y k o s ' s other b o o k , On Risings and Settings, is devoted to helia­ cal risings a n d settings—the a n n u a l cycle o f appearances a n d disappear­ ances o f the fixed stars. T h i s h a d been a part o f G r e e k p o p u l a r astron­ o m y f r o m the earliest days, as illustrated by H e s i o d ' s use o f the heliacal risings a n d settings o f the Pleiades, A r c t u r u s , a n d Sirius to tell the time o f year i n his p o e m , Works and Days (c. 6 5 0 B.C.). Clearly, the sidereal events i n the a n n u a l cycle were a part o f w h a t the G r e e k s considered " p h e n o m e n a . " A u t o l y k o s ' s g o a l i n On Risings and Settings is to p r o v i d e a m a t h e m a t i c a l f o u n d a t i o n , i n the f o r m o f theorems, for a field that h a d p r e v i o u s l y been i n the d o m a i n o f p o p u l a r lore. G e m i n o s devotes chapter x i i i o f his Introduction to the Phenomena to the same subject. Indeed, G e m i n o s ' s heading for chapter x v i i i is the same as the title o f A u t o l y k o s ' s b o o k . A s w e p o i n t out i n o u r c o m m e n t a r y o n that chapter, G e m i n o s f o l ­ l o w s A u t o l y k o s i n a l l significant details, but eliminates the proofs. The other major w r i t e r o n the p h e n o m e n a was T h e o d o s i o s o f B i t h y n i a (c. 100 B.C.), w h o s e On Habitations a n d On Days and Nights are the earliest extant w o r k s devoted to a discussion o f h o w the p h e n o m e n a change f r o m one l o c a l i t y to another: as an observer moves n o r t h o r south, the stars that are visible w i l l become different a n d the lengths o f the day a n d night m a y change. A n e x a m p l e o f a p r o p o s i t i o n f r o m the first of these is: F o r those l i v i n g under the n o r t h p o l e the same hemisphere o f the cosmos is a l w a y s visible a n d the same hemisphere o f the cosmos is a l ­ w a y s invisible, a n d none o f the stars either sets or rises for t h e m , but those i n the visible hemisphere are a l w a y s visible a n d those i n the i n ­ visible [hemisphere] are always i n v i s i b l e . 1 4

15

G e m i n o s ' s use o f T h e o d o s i o s is quite clear, for the G r e e k heading o f G e m i n o s ' s chapter x v i is the same as that o f Theodosios's On Habita­ tions, a n d the h e a d i n g o f chapter v i is o n l y t r i v i a l l y different (singular nouns instead o f plurals) f r o m that o f Theodosios's On Days and Nights. M a n y o f the f o u n d i n g w o r k s o n the p h e n o m e n a , such as those by E u ­ c l i d , A u t o l y k o s , a n d T h e o d o s i o s , s u r v i v e d because they were short enough a n d elementary enough for use i n teaching. T h e y became staples of the c u r r i c u l u m for mathematics a n d astronomy, a n d so survived t h r o u g h late A n t i q u i t y a n d i n t o the M i d d l e A g e s , i n b o t h the A r a b i c a n d Latin worlds. The m o t i o n s o f the S u n , M o o n , a n d planets a r o u n d the z o d i a c are also part of w h a t the G r e e k s considered " p h e n o m e n a . " Several features o f 16

Recall that for the Greeks the north pole was a point on the celestial sphere. Berggren and Eggert-Strand, forthcoming. But in our translation we have chosen the more descriptive rendering, " O n Geograph­ ical Regions," for the chapter title. 14

15

16

8

·

Section 4

p l a n e t a r y m o t i o n p o s e d challenges for e x p l a n a t i o n : the S u n appears to m o v e m o r e s l o w l y at some times o f year, a n d m o r e r a p i d l y at others. T h e planets are even m o r e p u z z l i n g , since they o c c a s i o n a l l y stop a n d reverse d i r e c t i o n i n w h a t is k n o w n as retrograde m o t i o n . M o s t scholars believe that the earliest G r e e k effort to e x p l a i n the c o m p l e x m o t i o n s o f the p l a n ­ ets w a s the b o o k On Speeds by E u d o x o s . It is lost, but w e have t w o rather lengthy discussions o f it, one by A r i s t o t l e , w h o w a s a c o n t e m p o ­ r a r y o f E u d o x o s , a n d one by S i m p l i k i o s , w h o l i v e d 9 0 0 years later, a n d w h o s e a c c o u n t must therefore be used w i t h c a u t i o n . P r o b a b l y b y the t i m e o f Apollônios o f Pergë (late t h i r d century B.C.) a n d c e r t a i n l y b y the t i m e o f H i p p a r c h o s , E u d o x o s ' s a p p r o a c h o f m o d e l i n g the p l a n e t a r y phe­ n o m e n a b y the gyrations o f nested, h o m o c e n t r i c spheres h a d g i v e n w a y to eccentric circles a n d epicycles l y i n g i n a plane. B u t this w a s d a u n t i n g m a t e r i a l to address i n a n elementary w o r k .

1 7

4. T H E G R E E K G E N R E O F A S T R O N O M I C A L SURVEYS

In the H e l l e n i s t i c p e r i o d , there emerged a d e m a n d for p o p u l a r surveys— w o r k s that w o u l d take students t h r o u g h the celestial p h e n o m e n a w i t h ­ o u t f o r c i n g t h e m t h r o u g h theorems a n d p r o o f s . T h e poetic Phenomena o f A r a t o s c a n be c o n s i d e r e d one o f the first s u c h p o p u l a r i z a t i o n s . T h e n e w p o p u l a r surveys eschewed the austere g e o m e t r i c a l d e m o n s t r a t i o n s o f E u c l i d , A u t o l y k o s , a n d T h e o d o s i o s tended s i m p l y to s u m m a r i z e m a t h ­ e m a t i c a l results i n p l a i n language. T h e y also tended to i n c l u d e a greater v a r i e t y o f subjects o f interest to the b r o a d p u b l i c — p h a s e s o f the M o o n , eclipses, a n d elements o f a s t r o n o m i c a l geography, s u c h as the t h e o r y o f terrestrial zones. O f course, a l l o f these topics h a d deep roots i n the his­ t o r y o f G r e e k science. W h a t w a s n e w w a s the attempt to p r o d u c e c o m ­ prehensive a s t r o n o m y t e x t b o o k s w r i t t e n at a n elementary level. T h e p o p u l a r surveys o f a s t r o n o m y c o u l d be read for their o w n sake, b u t some were clearly i n t e n d e d to f o r m part o f the c u r r i c u l u m o f studies expected o f a w e l l - b o r n student. T h e g e o g r a p h i c a l w r i t e r Strabo (c. 64 B.C. to c. A . D . 25) m e n t i o n s that students c a n learn i n the elementary m a t h e m a t i c s courses a l l the a s t r o n o m y they w i l l need for the study o f ge­ o g r a p h y . H e m e n t i o n s as a n e x a m p l e o f the s t a n d a r d a s t r o n o m i c a l c u r ­ r i c u l u m the theory o f the celestial sphere—tropics, equator, z o d i a c , arc­ tic circle, a n d h o r i z o n . T h e sort o f elementary a s t r o n o m y course that 1 8

O f all the elementary writers on astronomy, only Theón of Smyrna does a good job with planetary phenomena. Geminos (chapter i) gives only an explanation of the eccentriccircle theory of the Sun's motion, a vague reference to the sphairopoiia for each planet, and a brief mention of the basic planetary phenomena. Strabo, Geography i 1.21. 1 7

18

Greek Genre of Astronomical Surveys

·

9

Strabo h a d i n m i n d is w e l l represented by G e m i n o s ' s Introduction to the Phenomena. D i o g e n e s L a e r t i o s tells us that i n s t r u c t i o n i n basic astron­ o m y w a s part o f the c u r r i c u l u m o f Stoic teachers. A n d , o f course, as­ t r o n o m y h a d l o n g been part o f the q u a d r i v i u m o f m a t h e m a t i c a l studies i n the Platonist s c h o o l . W h e t h e r for the sake o f p o p u l a r reading, o r for liberal e d u c a t i o n , o r as part o f the p r e p a r a t i o n for m o r e a d v a n c e d stud­ ies, i n t r o d u c t i o n s to the a s t r o n o m i c a l p h e n o m e n a permeated G r e e k c u l ­ ture f r o m about 2 0 0 B.C. to the end o f A n t i q u i t y . 19

20

It is quite a p p r o p r i a t e , then, that G e m i n o s ' s w o r k is n a m e d Introduc­ tion to the Phenomena, for eisagôgë ( " i n t r o d u c t i o n " ) carries t w o mean­ ings. O n one h a n d , this is a regular w o r d for a n elementary treatise o n a subject; o n the other, it c a n denote a c o n d u i t , o r c h a n n e l , i n t o a harbor. T h u s an eisagôgë c o u l d serve either as a liberal arts survey of astronomy, complete i n itself, o r as the p r e p a r a t o r y course for higher studies i n the subject. G e m i n o s o c c a s i o n a l l y employs demonstrative m a t h e m a t i c a l argu­ ments (e.g., i n his treatment o f l u n i s o l a r cycles i n chapter v i i i ) , a n d he d i d not w r i t e his b o o k for those w h o were afraid o f numbers or geome­ try. H o w e v e r , his m o t t o seems to have been "mathematics i f necessary, but not necessarily m a t h e m a t i c s " — a n d i n any case he makes n o use o f f o r m a l m a t h e m a t i c a l proofs. N o r does G e m i n o s ' s w o r k smell o f the mathematics c l a s s r o o m . There is none o f the graded progression f r o m the easy to the c o m p l i c a t e d that one finds i n , for e x a m p l e , E u c l i d ' s Phe­ nomena. H a d G e m i n o s intended to w r i t e a t e x t b o o k of mathematics he w o u l d surely have put chapters iv (the axis a n d the poles) a n d ν (circles o n the sphere) at the b e g i n n i n g , a n d i n any case before chapter i (on the zodiac). A t h i r d feature o f his w o r k is its b l e n d i n g o f the topics o f the t w o earlier genres o f p h e n o m e n a literature (the descriptive u r a n o g r a p h y of E u d o x o s a n d the m a t h e m a t i c a l topics o f E u c l i d a n d his successors) w i t h topics outside o f these traditions, n a m e l y those he treats i n chapters v i i i - x i i , x v i i , a n d x v i i i . G e m i n o s even stretches the definition o f the phe­ n o m e n a to i n c l u d e the a s t r o l o g i c a l aspects o f the z o d i a c signs, i n chapter i . In summary, G e m i n o s , i n his account o f the celestial p h e n o m e n a , ex­ tended the t r a d i t i o n o f topics treated to include v i r t u a l l y a n y t h i n g hav­ i n g to d o w i t h the fixed stars, the S u n , a n d the M o o n . A n d he d i d so i n a w a y that w a s not s i m p l y systematic or m a t h e m a t i c a l , but discursive a n d , i n a b r o a d sense o f the w o r d , scientific. Geminos's Introduction to the Phenomena is but one o f several G r e e k elementary textbooks o f a s t r o n o m y that survive f r o m A n t i q u i t y . T h e t w o most nearly c o m p a r a b l e examples are T h e ó n o f Smyrna's Mathematical 1 9

2 0

Diogenes Laertios, Lives and Opinions vii 132. Plato, Republic vii 527d.

10

·

Section 4

Knowledge Useful for Reading Plato (second century A.D.) a n d K l e o m ë d ë s ' Meteoro ( p r o b a b l y early t h i r d to m i d - f o u r t h century A.D.). These three surveys have a fair a m o u n t o f o v e r l a p — f o r e x a m p l e , they a l l discuss the eccentric-circle theory o f the m o t i o n o f the Sun. B u t each o f the three also treats subjects not covered by the other t w o . F o r e x a m p l e , T h e ó n o f S m y r n a gives a n i n t r o d u c t i o n to the deferent-and-epicycle the­ ory o f planetary m o t i o n , a subject a v o i d e d by Kleomëdës a n d G e m i n o s . Kleomëdës, for his part, is o u r most detailed source for the famous mea­ surement o f the E a r t h by Eratosthenes. A n d G e m i n o s gives a detailed dis­ c u s s i o n o f l u n i s o l a r cycles, a subject a v o i d e d by T h e ó n a n d Kleomëdës. 21

11

These three t e x t b o o k s o f a s t r o n o m y also differ m a r k e d l y i n tone. W h i l e T h e ó n ' s b o o k is p e r v a d e d by P l a t o n i s m , K l e o m ë d ë s ' b o o k is steeped i n Stoic physics a n d concludes w i t h a savage attack o n the E p i ­ cureans. T h e ó n a n d Kleomëdës, then, give us nice examples o f h o w a n i n t r o d u c t i o n to a s t r o n o m y c o u l d be i n c o r p o r a t e d i n t o a general course i n p h i l o s o p h y — a n d w e have examples i n t w o flavors, P l a t o n i s t a n d Stoic. B y contrast, G e m i n o s ' s Introduction to the Phenomena is r e m a r k ­ able for its c o m p a r a t i v e freedom f r o m p h i l o s o p h y , for he is very m u c h a s t r a i g h t f o r w a r d astronomer. G e m i n o s does, however, d i s p l a y a certain literary bent, a n d is f o n d o f q u o t i n g poets, such as A r a t o s or H o m e r , i n i l l u s t r a t i o n o f a s t r o n o m i c a l p o i n t s . H i s Introduction to the Phenomena is also c o n s i d e r a b l y earlier t h a n the t e x t b o o k s o f T h e ó n a n d Kleomëdës, and sheds light o n the G r e e k s ' reactions to B a b y l o n i a n a s t r o n o m y a n d astrology, w h i c h , i n G e m i n o s ' s day, were i n the process o f being ab­ sorbed a n d adapted. A n earlier, t h o u g h shorter a n d m u c h less p o l i s h e d , survey o f astron­ o m y is the Celestial Teaching (Ouranios Didascalea) of Leptinës. See fig. L I . T h i s famous p a p y r u s , conserved i n the L o u v r e , is the oldest existing G r e e k a s t r o n o m i c a l d o c u m e n t w i t h i l l u s t r a t i o n s . It was c o m p o s e d i n the decades before 165 B.C. by a certain Leptinës as a n i n t r o d u c t i o n to as­ t r o n o m y for members o f the P t o l e m a i c c o u r t . (So it seems that, despite 23

For a French translation of Theón of Smyrna, see Dupuis 1892. For Kleomëdës, see Todd 1990 (text) and Bowen and Todd 2004 (translation). The original title of Kleomëdës' work is uncertain, and a number of different titles have been used by editors and translators. O n the title issue, see Goulet 1980, 35; Todd 1985; and Bowen and Todd 2002, l n l . The dating of Kleomëdës is also difficult. Kleomëdës says that Antares and Aldebaran are diametrically opposite in the zodiac, the first at Scorpio 15° and the second at Taurus 15°. Using this datum, Neugebauer (1975, 960) arrived at a date for Kleomëdës around A . D . 370. Bowen and Todd situate Kleomëdës around A . D . 200, be­ cause his w o r k reflects the Stoic polemics against the Peripatetics that began to fade after that period, and because works of Stoic pedagogy become rare after the second century. Earlier writers call this P. Parisinus 1, but it is now k n o w n in the Department of Egyptian Antiquities at the Louvre as Ν 2325. For the text, see Blass 1887. There is a French translation in Tannery 1893, 283-94. O n the history of this papyrus see Thompson 1988, 2 5 2 - 6 5 . 2 1

2 2

2 3

Greek Genre of Astronomical Surveys

·

11

Fig. 1.1. A portion of the Celestial Teaching of Leptinës on a papyrus, written shortly before 165 B.c. The left column treats the circles of the celestial sphere and the celestial poles. The right column explains that the stars are called fixed because the constellations always retain their forms and their relationships to one another. Département des Antiquités Egyptiennes, Inv. Ν. 2325, Musée du Louvre. Photo: Maurice and Pierre Chuzeville.

w h a t E u c l i d is supposed to have said a b o u t geometry, there was a r o y a l r o a d to astronomy.) M o d e r n writers sometimes refer to this tract as the " A r t o f E u d o x o s , " a n a m e that comes f r o m a n acrostic p o e m o n the verso o f the p a p y r u s , i n w h i c h the i n i t i a l letters o f the twelve lines o f

12

·

Section 5

verse spell out Eudoxou Techne. B u t the c o l o p h o n o n the recto clearly gives the title as the Ouranios Didascalea of Leptinës. In any case, the contents o f the treatise are certainly not by E u d o x o s . Rather, the tract is a brief a n d rather c h o p p y a c c o u n t o f s t a n d a r d a s t r o n o m i c a l matters. T h e text includes a short parapëgma, a n a c c o u n t o f the progress o f the S u n a n d M o o n a r o u n d the z o d i a c , descriptions o f the circles o n the ce­ lestial sphere, a discussion o f eclipses, a n d values for the lengths o f the four seasons a c c o r d i n g to v a r i o u s authorities. T h i s fare overlaps c o n s i d ­ erably w i t h the m a t e r i a l treated m o r e gracefully by G e m i n o s i n the next century. F i n a l l y , n u m e r o u s commentaries o n A r a t o s ' s p o e m Phenomena often served as i n t r o d u c t i o n s to astronomy. O n e o f the most complete is that of A c h i l l e u s (often called A c h i l l e s T a t i u s , p r o b a b l y t h i r d century A.D.), w h o s e Introduction to the "Phenomena" of Aratos f o r m e d a part o f his O n the A l l (Peri tou Pantos). * In o u r c o m m e n t a r y o n G e m i n o s , w e shall o c c a s i o n a l l y m a k e c o m p a r i s o n s to these other w o r k s , w h i c h c a n be t h o u g h t o f as c o n s t i t u t i n g a genre o f elementary a s t r o n o m y t e x t b o o k s . 1

5. G E M I N O S ' S SOURCES F O R H I S

INTRODUCTION

A p p e n d i x 4 lists the writers that G e m i n o s cites i n his Introduction to the Phenomena. H e enjoys q u o t i n g the poets H o m e r , H e s i o d , a n d A r a t o s i n i l l u s t r a t i o n of scientific p o i n t s . T h i s reflects not o n l y his o w n tastes but also his concession to the literary t r a i n i n g o f his students a n d readers. H e is not, however, one to ascribe t o o m u c h scientific k n o w l e d g e to H o m e r , a n d feels that critics such as Kratës have sometimes gone over­ b o a r d i n this regard. (The o c c a s i o n a l use o f poetry occurs i n other ele­ m e n t a r y surveys as w e l l , e.g., those o f Kleomëdës, T h e ó n o f S m y r n a , a n d Leptinës.) O f the a s t r o n o m i c a l w r i t e r s , G e m i n o s names E u k t ë m ô n , K a l l i p p o s , P h i l i p p o s , Eratosthenes, a n d H i p p a r c h o s , t h o u g h he m a y not have k n o w n the w o r k s of a l l these people firsthand. G e m i n o s was quite w e l l i n f o r m e d about l u n i s o l a r cycles, but w e c a n n o t tell f r o m his r e m a r k s o n those matters whose w o r k s he really h a d access to. H e seems to have used some w o r k o f H i p p a r c h o s o n the constellations that was different f r o m H i p p a r c h o s ' s Commentary on The Phenomena of Eudoxos and Aratos. For, i n chapter i i i , he mentions three decisions o f H i p p a r c h o s re­ g a r d i n g the constellations that have n o counterpart i n the Commentary. See Maass 1898, 25-85 for what remains of Achilleus's Commentary on Aratos. O n Achilleus, see Mansfield and Runia 1997, 2 9 9 - 3 0 5 . Hipparchos's extant Commentary on the Phenomena of Eudoxos and Aratos is not a part of this genre, since it is highly techni­ cal and numerical in its content. 2 4

Geminos's Sources

·

13

T h e clearest a n d m o s t significant o f these is the a t t r i b u t i o n o f the c o n stellation E q u u l e u s (Frotóme hippou) to H i p p a r c h o s . G e m i n o s ' s is the first m e n t i o n o f this c o n s t e l l a t i o n i n the G r e e k t r a d i t i o n . Perhaps it comes f r o m H i p p a r c h o s ' s star catalogue. In any case, P t o l e m y a d o p t e d this c o n s t e l l a t i o n name i n the Almagest. A m o n g writers o n such geog r a p h i c a l questions as m o u n t a i n heights, the extent o f O c e a n , a n d the arrangement a n d h a b i t a b i l i t y of the zones, G e m i n o s cites D i k a i a r c h o s , Pytheas, Kleanthês, a n d P o l y b i o s . G e m i n o s was clearly influenced by the Stoic Poseidônios i n his p h i l o s o p h i c a l musings a n d i n his w o r k o n meteorology. (See fragment 2.) In sec. 7 w e address the c o n t r o v e r s i a l question o f whether G e m i n o s , i n w r i t i n g the Introduction to the Phenomena, m i g h t have used a lost textb o o k of Stoic a s t r o n o m y a n d physics w r i t t e n by Poseidônios. H e r e , it suffices to p o i n t out that he does not m e n t i o n Poseidônios a single time i n the Introduction to the Phenomena. T h e m a t e r i a l o f G e m i n o s ' s Introduction consists largely o f notions that were the c o m m o n p r o p e r t y o f a l l astronomers. H i s c o n t r i b u t i o n was i n the selection a n d shaping o f mater i a l , i n his graceful prose, a n d i n the tasteful i n c o r p o r a t i o n o f literary exa m p l e s . H e w o u l d have needed n o help f r o m Poseidônios for this. 25

B u t G e m i n o s does leave some o f his most i m p o r t a n t sources u n n a m e d . F o r as w e have seen, a n d t h o u g h he does not cite t h e m by name, G e m i nos clearly k n o w s the m a t e r i a l i n E u c l i d ' s Phenomena, A u t o l y k o s ' s On the Moving Sphere a n d On Risings and Settings, a n d T h e o d o s i o s ' s On Habitations a n d On Days and Nights. W e shall see b e l o w that he p r o b a bly k n e w also Hypsiklës o f A l e x a n d r i a ' s Anaphorikos. G e m i n o s ' s merit as a teacher is to absorb a l l this rather d r y m a t h e m a t i c a l m a t e r i a l a n d t r a n s f o r m it i n t o graceful p r o s e — t h o u g h often at the expense o f the o r i g i n a l m a t h e m a t i c a l rigor. H i g h l y significant are G e m i n o s ' s citations of the " C h a l d e a n s , " by w h i c h he means B a b y l o n i a n astronomers. W e s h o u l d say a few w o r d s a b o u t this term. T h e C h a l d e a n s were a g r o u p of tribes w h o m o v e d i n t o southern M e s o p o t a m i a by about 1 0 0 0 B.C. T h e y assumed a g r o w i n g i m portance, a n d i n the eighth century succeeded i n p u t t i n g a k i n g o n the throne of B a b y l o n i a . W i t h i n a few decades, the C h a l d e a n kings lost c o n t r o l to the A s s y r i a n kings, w h o intervened repeatedly i n B a b y l o n i a n affairs. B u t under N a b o p o l a s s a r a n e w C h a l d e a n dynasty was established, w h i c h r u l e d B a b y l o n i a f r o m 6 2 5 B.C. u n t i l the Persian conquest i n 5 3 9 . A n c i e n t G r e e k writers often used the t e r m " C h a l d e a n s " (Chaldaioi) s i m p l y to m e a n B a b y l o n i a n s . B u t because B a b y l o n h a d a r e p u t a t i o n for arcane k n o w l e d g e , " C h a l d e a n " also came to m e a n an astronomer or 2 6

2 5

2 6

Compare Aujac 1975, lxxxviii, n l . O n the Chaldeans, see Oates (1986), 111-14.

14

·

Section 5

astrologer of Babylon. H e r e are a few e x a m p l e s that s p a n the range o f meanings f r o m " B a b y l o n i a n " to " a s t r o n o m e r o f B a b y l o n " to "as­ t r o l o g e r o r m a g u s " : In the Almagest, P t o l e m y refers to the " C h a l d e a n " (i.e., B a b y l o n i a n ) calendar. V i t r u v i u s says that Berossus came f r o m the " C h a l d e a n city o r n a t i o n " to spread the l e a r n i n g o f this people. T h e ó n of S m y r n a says that the C h a l d e a n s save the p h e n o m e n a by u s i n g a r i t h ­ metic procedures. F o r H e r o d o t o s , the C h a l d e a n s are priests o f B e l (i.e., M a r d u k ) . T h i s is quite reasonable, since a s t r o n o m y a n d a s t r o l o g y were c o n c e n t r a t e d i n the temples, a n d m a n y o f the practitioners were priestly scribes. In Daniel 2 . 2 - 4 , the C h a l d e a n s are interpreters o f dreams a n d are associated w i t h m a g i c i a n s a n d sorcerers. F o r Sextus E m p i r i c u s , C h a l d e a n s are a s t r o l o g e r s . 27

B y a b o u t 3 0 0 B.C. the B a b y l o n i a n s h a d d e v e l o p e d very successful the­ ories for the m o t i o n s o f the planets, S u n , a n d M o o n . These theories were based u p o n a r i t h m e t i c rules, rather t h a n o n the g e o m e t r i c a l m o d e l s that c h a r a c t e r i z e d the G r e e k a p p r o a c h . W h e n the G r e e k s began to deal q u a n ­ titatively w i t h planetary theory, they were able to base their g e o m e t r i c a l m o d e l s o n n u m e r i c a l parameters b o r r o w e d f r o m the B a b y l o n i a n s . T h i s process w a s w e l l under w a y i n the second century B.C. In the Almagest (second century A.D.), P t o l e m y begins w i t h planetary periods that he as­ cribes to H i p p a r c h o s (second century B . C ) . B u t i n fact these parameters were o f B a b y l o n i a n o r i g i n a n d t u r n u p o n c u n e i f o r m tablets. In his dis­ c u s s i o n o f the M o o n ' s m e a n m o t i o n s , P t o l e m y a g a i n starts w i t h H i p p a r ­ chos's values, but i n this case says e x p l i c i t l y that H i p p a r c h o s h a d m a d e use o f C h a l d e a n o b s e r v a t i o n s . H i p p a r c h o s ' s w o r k s o n l u n a r a n d p l a n e ­ tary t h e o r y have n o t c o m e d o w n to us, so w e d o n o t k n o w e x a c t l y h o w he came i n t o contact w i t h the B a b y l o n i a n parameters. 28

29

In the p e r i o d between H i p p a r c h o s a n d P t o l e m y , the G r e e k g e o m e t r i c a l p l a n e t a r y theories h a d n o t yet reached m a t u r i t y , a n d were n o t capable o f y i e l d i n g accurate n u m e r i c a l values for planet p o s i t i o n s . B u t the rise o f a s t r o l o g y ( w h i c h entered the G r e e k w o r l d f r o m B a b y l o n i a i n the s e c o n d o r first century B.C.) i m p o s e d a need for q u i c k , reliable m e t h o d s o f c a l ­ c u l a t i n g p l a n e t a r y p h e n o m e n a . G r e e k a s t r o n o m e r s a n d astrologers a d o p t e d the B a b y l o n i a n p l a n e t a r y theories w i t h enthusiasm. A s t r o n o m i ­ c a l p a p y r i f r o m E g y p t s h o w G r e e k s o f the first century A . D . u s i n g B a b y ­ l o n i a n p l a n e t a r y theories w i t h c o m p l e t e facility. Ptolemy's p u b l i c a t i o n o f his p l a n e t a r y theories a n d tables i n the Almagest a n d the Handy Tables Ptolemy, Almagest ix Mathematical Knowledge Sextus Empiricus, Against Ptolemy, Almagest ix Ptolemy, Almagest iv 2 7

2 8

2 9

7 and x i 7. Vitruvius, On Architecture ix 2.1. Theón of Smyrna, Useful for Reading Plato iii 30. Herodotos, Histories i 181-84. the Professors ν 2 - 3 . 3. For a discussion, see Neugebauer 1975, 150-52. 2. See Neugebauer 1975, 6 9 - 7 1 , 309-10.

Geminos's Country and Date

·

15

p r o d u c e d a major change i n the w a y p r a c t i c a l a s t r o n o m y w a s done. B u t c a l c u l a t i n g methods based o n B a b y l o n i a n procedures still existed side by side w i t h methods based o n Ptolemy's tables i n the f o u r t h century A . D . In chapter i i o f the Introduction to the Phenomena, G e m i n o s shows that he is f a m i l i a r w i t h some features o f C h a l d e a n astrology, t h o u g h he mentions o n l y a few doctrines i n passing, a n d does n o t seem intensely i n ­ terested i n the subject. In any case, n o t h i n g about the level o f his f a m i l ­ iarity w i t h C h a l d e a n astrology is s u r p r i s i n g for a w r i t e r o f his time. F a r m o r e detailed a n d m o r e h i s t o r i c a l l y significant is G e m i n o s ' s discussion of the B a b y l o n i a n l u n a r theory i n chapter x v i i i . H i s discussion there is i m p o r t a n t because his is the oldest extant classical text to display f a m i l ­ iarity w i t h the technical details o f a B a b y l o n i a n planetary theory based o n a n arithmetic progression. In particular, G e m i n o s explains a scheme for the m o t i o n o f the M o o n , a c c o r d i n g to w h i c h the d a i l y displacement increases by equal intervals f r o m day to day, u n t i l it reaches a m a x i m u m , then falls by equal increments f r o m one day to the next. T h e n u m e r i c a l parameters o f G e m i n o s ' s theory are i n exact agreement w i t h c u n e i f o r m sources. G e m i n o s ' s treatment o f the B a b y l o n i a n l u n a r theory is discussed b e l o w i n sec. 1 3 , below, where w e also address the question o f the f o r m that his source for the B a b y l o n i a n l u n a r theory m i g h t have taken. In chapter x i , G e m i n o s mentions that eclipses of the M o o n take place i n a n eclipse zone (ekleiptikon) that is 2 degrees w i d e . T h o u g h he does not m e n t i o n the C h a l d e a n s i n this passage, the 2-degree eclipse zone also comes f r o m B a b y l o n i a n astronomy. In t o t a l , G e m i n o s ' s r e m a r k s p r o v i d e i m p o r t a n t i n f o r m a t i o n about the a d o p t i o n a n d a d a p t a t i o n o f B a b y l o n ­ i a n k n o w l e d g e by the G r e e k s o f his time. B y contrast, G e m i n o s cites the " E g y p t i a n s " s i m p l y for the general structure o f the E g y p t i a n calendar a n d the circumstances o f a festival of Isis.

6. G E M I N O S ' S C O U N T R Y A N D D A T E

M o d e r n scholars sometimes refer to o u r astronomer as " G e m i n o s o f R h o d e s , " but there is n o ancient m e n t i o n o f his native l a n d o r city. T h e few ancient writers w h o cite h i m refer to h i m s i m p l y as G e m i n o s , or as " G e m i n o s the m a t h e m a t i c i a n . " T h e evidence for p l a c i n g h i m i n R h o d e s is suggestive, but not c o n c l u s i v e . In several passages i n the Introduction to the Phenomena, G e m i n o s uses R h o d e s as a n e x a m p l e i n m a k i n g some a s t r o n o m i c a l p o i n t — i n v o l v i n g the length o f the longest day, o r the p o r ­ t i o n of the s u m m e r t r o p i c cut off above the h o r i z o n , o r the m e r i d i a n alti3 0

3 0

For example, Sarton 1970, vol. 2, 305; Bowen and Todd 2004, 194n2.

16

·

Section 6

tude o f the star C a n o p u s , o r the date o f the m o r n i n g r i s i n g o f the D o g Star. B u t a l t h o u g h he does use R h o d e s m o s t frequently for such e x a m ­ ples, he also gives examples for A l e x a n d r i a , Greece, R o m e , a n d the P r o p o n t i s . D o e s his p r o c l i v i t y for u s i n g R h o d e s suggest a fondness for his native city, o r merely reflect Rhodes's usefulness i n a s t r o n o m i c a l e x a m ­ ples, o w i n g to its r o u g h l y central l o c a t i o n i n the G r e e k w o r l d ? G e m i n o s r e m a r k s (xiv 12) that celestial globes a n d a r m i l l a r y spheres were c o m ­ m o n l y constructed for this klima, or b a n d o f latitude. A n d it is n o t e w o r ­ t h y that i n the second century A . D . , Ptolemy, w h o l i v e d at A l e x a n d r i a , still f o u n d it n a t u r a l to construct examples for the p a r a l l e l t h r o u g h R h o d e s , " w h e r e the elevation o f the pole is 36 degrees a n d the longest d a y 14Vi h o u r s . " O r perhaps, as D i c k s suggests, G e m i n o s ' s use o f the klima o f R h o d e s reflects examples he f o u n d i n his sources, w h i c h m a y have i n c l u d e d the g e o g r a p h i c a l o r a s t r o n o m i c a l w o r k s o f H i p p a r c h o s . Blass makes a n interesting p o i n t a b o u t G e m i n o s ' s use o f t w o g e o g r a p h i ­ c a l e x a m p l e s . G e m i n o s ( x v i i 3) refers to M t . Kyllënë, a n d i m m e d i a t e l y specifies that it is "the highest m o u n t a i n i n the P e l o p o n n e s o s " ; but i n the very next sentence he refers to M t . A t a b y r i o n w i t h o u t m a k i n g any s i m i ­ lar specification that it is o n the i s l a n d o f R h o d e s . D o e s this suggest that he expected his readers to be f a m i l i a r w i t h R h o d i a n geography? 31

32

3 3

34

35

F i n a l l y , w e k n o w that G e m i n o s w r o t e some sort o f a b r i d g m e n t of, o r c o m m e n t a r y o n , the Meteorology o f Poseidônios, w h o s e native l a n d w a s R h o d e s . A n d a l i k e l y d a t i n g o f G e m i n o s ' s Introduction to the Phenom­ ena w o u l d m a k e G e m i n o s a y o u n g e r c o n t e m p o r a r y o f Poseidônios, a n d thus p o t e n t i a l l y his student. ( G e m i n o s ' s possible debts to Poseidônios w i l l be discussed below.) N e a r the end o f her o w n d i s c u s s i o n o f this is­ sue, A u j a c concludes, " L e t us a l l o w , then, since n o other better h y p o t h e ­ sis presents itself, that G e m i n o s w a s b o r n at R h o d e s a n d that he there re­ ceived his first i n s t r u c t i o n . " T h i s is not a n unreasonable p o s i t i o n to take, since n o c o n v i n c i n g evidence exists for p l a c i n g h i m e l s e w h e r e . 36

37

Introduction to the Phenomena i 10, 12; iii 15; ν 25; vi 8; xvii 40. Introduction to the Phenomena iii 15; ν 23; vi 8. Ptolemy, Almagest ii 2, trans. Toomer 1984, 76. Ptolemy's example continues, using the same latitude, in ii 3. Dicks 1972. Blass 1883, 5. The mss. actually call the mountain Satabyrion, which was corrected by Petau (followed by Aujac). See also Dicks 1960, 30n3. Aujac 1975, xv. However, Schmidt (1884b) argues that Geminos wrote the Introduction to the Phe­ nomena at Rome for a Roman audience. Against this, Tannery (1887, 37) points out that Geminos is not mentioned by Pliny, who would certainly have included him in his long list of authorities, if he had ever heard of him. Manitius (1898, 247) believes that Geminos wrote at Rhodes, but that the work we have is due to a much later excerptor, who lived in the klima of 1 5 , and most likely at Constantinople. 31

32

33

34

35

36

3 7

h

Geminos's Country and Date

·

17

B u t w e s i m p l y d o not k n o w . In any case, as T a n n e r y has p o i n t e d o u t , a l l the writers w h o cited G e m i n o s were associated w i t h A l e x a n d r i a or w i t h A t h e n s , w h i c h suggests that his w o r k s c i r c u l a t e d m a i n l y i n the G r e e k w o r l d o f the eastern M e d i t e r r a n e a n . W h e t h e r Geminos is a G r e e k name o r a H e l l e n i z a t i o n o f a L a t i n name (Geminus) has been the subject o f dispute. A s A u j a c r e m a r k s , " P e t a u made it a L a t i n name, M a n i t i u s a G r e e k name, T i t t e l a g a i n a L a t i n name ( ! ) " A c r u c i a l p o i n t i n the argument is the length o f the central v o w e l — a l o n g v o w e l f a v o r i n g the G r e e k . W h a t e v e r the o r i g i n o f his name, G e m i n o s was t h o r o u g h l y G r e e k i n e d u c a t i o n , intellectual interests, a n d m a n n e r o f expression. B u t w h e n d i d G e m i n o s write? There are t w o w a y s to n a r r o w the pos­ sibilities. A p p e n d i x 4 lists the writers that G e m i n o s m e n t i o n s . T h e latest datable writers cited i n the Introduction to the Phenomena are H i p p a r ­ chos, P o l y b i o s , a n d Kratës o f M a l l o s , w h o a l l flourished i n the m i d d l e o f the second century B.C. Conversely, G e m i n o s w a s q u o t e d by A l e x a n d e r of A p h r o d i s i a s , the A r i s t o t e l i a n commentator, w h o flourished at the end o f the second century A . D . T h u s w e m a y place G e m i n o s between 150 B.C. a n d A . D . 2 0 0 . It appears possible to date G e m i n o s m o r e closely by his r e m a r k (viii 2 0 - 2 2 ) c o n c e r n i n g the w a n d e r i n g year o f the E g y p t i a n s : 38

39

. . . most o f the Greeks suppose the w i n t e r solstice a c c o r d i n g to E u ­ d o x o s to be at the same time as the feasts o f Isis [reckoned] a c c o r d i n g to the E g y p t i a n s , w h i c h is c o m p l e t e l y false. F o r the feasts o f Isis miss the w i n t e r solstice by a n entire m o n t h . . . . 120 years ago the feasts of Isis happened to be celebrated at the w i n t e r solstice itself. B u t i n 4 years a shift o f one day arose; this o f course d i d n o t i n v o l v e a percep­ tible difference w i t h respect to the seasons o f the year. . . . B u t n o w , w h e n the difference is a m o n t h i n 120 years, those w h o take the w i n ­ ter solstice a c c o r d i n g to E u d o x o s to be d u r i n g the feasts o f Isis [reck­ oned] a c c o r d i n g to the E g y p t i a n s are n o t l a c k i n g a n excess o f i g n o ­ rance. T h e feasts o f Isis (ta Isia) were celebrated at a fixed date i n the E g y p t i a n year. B u t as the E g y p t i a n year consists o f 3 6 5 days ( w i t h n o leap day), the feast days shift w i t h respect to the solstice by 1 d a y every 4 years. Because the E g y p t i a n year is t o o short, the feast days g r a d u a l l y fall ear­ lier a n d earlier i n the n a t u r a l , o r solar, year. If w e k n e w the E g y p t i a n calendar date o n w h i c h this Isis festival w a s observed, it w o u l d be easy Tannery 1887, 37. Aujac 1975, xiv, n2. See Manitius 1898, 251; and Tittel 1910, col. 1027. Heath (1921, v o l . 2, 222) provides a summary of the debate up to his time. Dicks (1972) makes the name Latin, but stresses that Geminos's "works and manner are patently Greek." 3 8

3 9

18

·

Section 6

to calculate the year i n w h i c h the festival c o i n c i d e d w i t h the w i n t e r s o l ­ stice. W e w o u l d then place G e m i n o s 1 2 0 years after that year. M o s t writers o n the subject have t r i e d to date G e m i n o s by the use o f a r e m a r k by P l u t a r c h (late first to early second century A.D.): . . . they say that the disappearance of O s i r i s o c c u r r e d i n the m o n t h of A t h y r . . . . T h e n , a m o n g the g l o o m y rites w h i c h the priests p e r f o r m , they s h r o u d the g i l d e d image of a c o w w i t h a b l a c k l i n e n vestment, a n d d i s p l a y her as a sign o f m o u r n i n g for the goddess, i n a s m u c h as they regard b o t h the c o w a n d the E a r t h as the image o f Isis; a n d this is kept up for four days consecutively, b e g i n n i n g w i t h the seventeenth o f the m o n t h . 4 0

D e n i s Petau, i n his Uranologion o f 1 6 3 0 , used Plutarch's r e m a r k to date G e m i n o s ' s c o m p o s i t i o n , w i t h the f o l l o w i n g result: 4 1

year 4 5 3 7 o f the J u l i a n p e r i o d , f o u r t h year o f O l y m p i a d 1 7 5 , year 6 7 7 after the f o u n d i n g o f R o m e , or, as w e w o u l d say, 7 7 B.C. Petau was f o l l o w e d by m o s t later writers o n the subject, w i t h o n l y m i n o r adjustments. T h u s , most writers w h o have accepted this evidence put G e m i n o s ' s c o m p o s i t i o n o f the Introduction to the Phenomena i n the 60s o r 70s B . C . B u t as w e shall see, the m a r g i n o f error s h o u l d be t a k e n quite a bit wider. T h e reasoning is straightforward. Let us w o r k w i t h 19 A t h y r , the 3 r d day o f the 4-day festival. A t h y r is the 3 r d m o n t h o f the E g y p t i a n calendar, so 19 A t h y r is the 7 9 t h day o f the E g y p t i a n year. In Table 1.1, the first c o l u m n lists years o f the J u l i a n calendar. In the second c o l u m n , w e have w r i t t e n the date of 1 T h o t h , the 1st day of the E g y p t i a n year that began i n the course o f the given J u l i a n calendar year. T h u s , i n - 2 0 0 , a n e w E g y p t i a n year began o n 12 O c t o b e r . T o o b t a i n c o l u m n 3, w e a d d 78 days to the dates i n c o l u m n 2 . In this way, w e m o v e f r o m the 1st day o f the E g y p t i a n year (1 T h o t h ) to the 7 9 t h day (19 A t h y r ) . T h u s , i n the J u l i a n year - 2 0 0 , the 19th of A t h y r fell o n 2 9 December. T h e 4 t h c o l u m n 42

43

44

45

Plutarch, Isis and Osiris 39, 3 6 6 D - E , E C . Babbitt, trans. Petavius 1 6 3 0 , 4 1 0 - 1 1 . For summaries of multiple dating attempts by various writers, see Manitius 1898, 238, and Jones 1999a, 256n2. For an introduction to the Egyptian calendar, see Evans 1998. For Julian calendar equivalents of I T h o t h over the centuries, see Bickerman 1980, 110-11. We use "astronomical reckoning." The years A . D . are written as positive numbers. The B.c. years are shifted by one, in order to introduce a year zero, and written as negative numbers. Thus, +1 = A . D . 1,0 = 1 B.C., - 1 = 2 B.C., etc. 4 0

4 1

4 2

4 3

4 4

4 5

Geminos's Country and Date

·

19

TABLE 1.1

Comparing 19 Athyr and Winter Solstice Year

1 Thoth

19 Athyr

-250 -200 -150 -100 -50 0 +50

25 Oct 12 Oct 30 Sep 17 Sep 5 Sep 23 A u g 11 A u g

11 Jan 29 Dec 17 Dec 4 Dec 22 N o v 9 Nov 28 Oct

Winter solstice 25 24 24 23 23 22 22

Dec Dec Dec Dec Dec Dec Dec

gives the date of the w i n t e r solstice for each of the given J u l i a n calendar years. C o m p a r i n g the 3 r d a n d 4 t h c o l u m n s , w e see that the w i n t e r sol­ stice fell o n 19 A t h y r sometime between - 2 0 0 a n d - 1 5 0 . Interpolation gives - 1 7 9 . G e m i n o s w r o t e 120 years later, or a r o u n d the year - 5 9 . A n error o f 3 days i n the date o f the solstice c o u l d shift the date by ± 1 2 y e a r s . A g a i n , G e m i n o s speaks i n r o u g h fashion o f a w h o l e m o n t h , as the difference by w h i c h the Isis festival missed the solstice i n his o w n day. H e m i g h t have s p o k e n i n this same w a y i f the actual difference were, say, as s m a l l as 28 days or as great as 3 2 , w h i c h introduces an­ other ± 8 years o f uncertainty. Finally, the festival itself stretched over a p e r i o d o f 4 days, w h i c h gives us 4 m o r e years o f uncertainty after 60 B.C. a n d 8 m o r e years before. P u t t i n g a l l this together, w e find the p e r i o d 8 8 - 3 6 B.C. as the m o s t likely for the c o m p o s i t i o n o f the Introduction to the Phenomena, or, to speak i n r o u n d numbers, 9 0 - 3 5 B.C. 46

47

In 1 9 7 5 , O t t o N e u g e b a u e r p r o p o s e d a date for G e m i n o s about a cen­ tury later, a r o u n d A . D . 5 0 . A l t h o u g h Neugebauer's d a t i n g w a s influen­ tial for a w h i l e , w e shall see that it c a n n o longer be sustained. T h e argu­ ment that f o l l o w s w i l l be s o m e w h a t intricate. B u t at the end w e shall not a b a n d o n the d a t i n g of 9 0 - 3 5 B . C . that w e have just e x p l a i n e d . T h u s , readers w i t h little enthusiasm for details o f ancient c h r o n o l o g y s h o u l d feel n o guilt i n s k i p p i n g ahead to the next section. T h e key question that N e u g e b a u e r posed is whether Petau's argument, based o n Plutarch's r e m a r k , i n v o l v e d a c o n f u s i o n between the E g y p t i a n 4 8

The dates of the winter solstices have been calculated from the solar tables in N e w comb 1898. The reason why the date of the solstice slowly shifts is the incorrect length of the Julian calendar year (by approximately 3 days every 400 years), which was corrected by the Gregorian reform of 1582. Three days is not too large an error. Geminos refers to the winter solstice "according to Eudoxos." In the Geminos parapëgma, the winter solstice "according to Eudoxos" is separated by 3 days from the winter solstice "according to Euktëmôn." Neugebauer 1975, 5 7 9 - 8 1 . 4 6

4 7

4 8

20

·

Section 6

a n d the A l e x a n d r i a n calendars. After E g y p t became a p r o v i n c e o f the R o m a n empire, A u g u s t u s reformed the E g y p t i a n calendar by i n t r o d u c ­ i n g a leap day once every four years, the first such day being inserted at the e n d o f the E g y p t i a n year 2 3 / 2 2 B.C. In the reformed calendar, n o w u s u a l l y c a l l e d " A l e x a n d r i a n " to d i s t i n g u i s h it f r o m the o r i g i n a l E g y p t i a n calendar, three years o f 3 6 5 days were f o l l o w e d by a year o f 3 6 6 days. T h e reformed calendar thus was very s i m i l a r to the J u l i a n calendar, w h i c h h a d been used at R o m e since 4 5 B.C. O f course, the A l e x a n d r i a n calendar c o n t i n u e d to use the o l d E g y p t i a n m o n t h s o f 3 0 days each, as w e l l as the o r i g i n a l E g y p t i a n m o n t h names. F o r dates near 2 3 B . C . , a given day has nearly the same date i n b o t h the E g y p t i a n a n d the A l e x a n ­ d r i a n calendars. B u t gradually, at the rate o f 1 day i n 4 years, the calen­ dars diverge. M o r e o v e r , the t w o calendars c o n t i n u e d to be used side by side. F o r e x a m p l e , Ptolemy, i n the Almagest, used the o l d calendar for a s t r o n o m i c a l c a l c u l a t i o n , because o f its s i m p l e r structure, nearly t w o centuries after it h a d been a b a n d o n e d for c i v i l use. In his parapëgma, however, P t o l e m y a d o p t e d the A l e x a n d r i a n calendar, because the h e l i a ­ c a l risings a n d settings o f a g i v e n star have m o r e nearly fixed dates i n this calendar. W h e n an ancient writer, w r i t i n g after Augustus's r e f o r m , says "the 1 7 t h o f A t h y r , " it is not i m m e d i a t e l y clear whether he is ex­ pressing the date i n terms o f the E g y p t i a n calendar or the A l e x a n d r i a n calendar. O n e must e x a m i n e the context carefully. N e u g e b a u e r was t r o u b l e d by a second reference i n P l u t a r c h to w h a t w a s apparently the same Isis festival: . . . then O s i r i s got into [the chest] a n d lay d o w n , a n d those w h o were i n the p l o t r a n to it a n d s l a m m e d d o w n the l i d , w h i c h they fastened by nails f r o m the outside a n d also by u s i n g m o l t e n lead. T h e y say also that the date o n w h i c h this deed w a s done was the seventeenth o f A t h y r , w h e n the S u n passes t h r o u g h S c o r p i o . . . . 4 9

W e have again the date A t h y r 17, but n o w w i t h the added i n f o r m a t i o n that the S u n passes t h r o u g h S c o r p i o d u r i n g the m o n t h o f A t h y r . A s N e u g e b a u e r p o i n t e d out, this was true i n the A l e x a n d r i a n , but not i n the E g y p t i a n calendar, for Plutarch's time. T h e A l e x a n d r i a n m o n t h of A t h y r runs f r o m 28 O c t o b e r to 2 6 N o v e m b e r (Julian), w h i c h corresponds rather closely to the sign o f S c o r p i o . In Plutarch's time, say A . D . 1 1 8 , the E g y p t i a n a n d A l e x a n d r i a n calendars were out of phase by 35 days: 1 A t h y r (Egyptian) then fell o n September 2 3 (Julian), c o r r e s p o n d i n g to the Sun's entry i n t o L i b r a , not S c o r p i o . N e u g e b a u e r c o n c l u d e d that P l u t a r c h w a s using the A l e x a n d r i a n , a n d not the E g y p t i a n , calendar. M o r e o v e r , he surmised that P l u t a r c h (or his source) t o o k the o r i g i n a l 4 9

Plutarch, Isis and Osiris 13 3 5 6 C , E C . Babbitt, trans.

Geminos's Country and Date

·

21

date o f the Isis festival, as expressed i n the E g y p t i a n calendar, a n d c o n ­ verted it to a n A l e x a n d r i a n equivalent. T h e A l e x a n d r i a n calendar w a s , after a l l , the one i n official use, a n d the one m o r e l i k e l y to be u n d e r s t o o d by Plutarch's readers i n the w i d e r R o m a n w o r l d . N e u g e b a u e r f o u n d the E g y p t i a n date for w h a t he t o o k to be the same festival i n a h i e r o g l y p h i c text i n the East O s i r i s C h a p e l o n the r o o f o f the Temple of H a t h o r i n D e n d e r a . T h e text describes the rituals o f a n O s i r i s festival that lasted f r o m 12 to 30 C h o i a k . T h e text is not later t h a n 3 0 B . c . a n d thus predates the r e f o r m o f the calendar. M o r e o v e r , as N e u g e b a u e r also p o i n t e d out, the p a p y r u s H i b e h 2 7 (c. 3 0 0 B.C.) m e n ­ tions a n O s i r i s festival o n 2 6 C h o i a k . N o w i n Plutarch's time (A.D. 118), the date 2 6 C h o i a k (Egyptian) = 21 A t h y r ( A l e x a n d r i a n ) , w h i c h appeared to c o n f i r m Plutarch's use o f the A l e x a n d r i a n calendar w h e n he placed the rites o n 1 7 - 2 0 A t h y r . N e u g e b a u e r then c o m p u t e d the year w h e n the w i n t e r solstice fell o n 15 C h o i a k (Egyptian). (This date is w i t h i n the span o f rituals m e n t i o n e d by the text i n the East O s i r i s Chapel.) T h e answer is the year - 7 0 ; G e m i n o s w r o t e 1 2 0 years later, or a r o u n d A . D . 5 0 , a c c o r d i n g to Neugebauer. T h e n e w d a t i n g by N e u g e ­ bauer, p u s h i n g G e m i n o s f o r w a r d i n t o the first century A . D . , was g r a d u ­ ally a d o p t e d by historians o f ancient astronomy. 50

5 1

A l e x a n d e r Jones r e e x a m i n e d the question i n 1 9 9 9 . A s Jones points out, Neugebauer deserved credit for being the first to use p a p y r o l o g i c a l evidence for the date o f the Isis festival. T h e advantage of such evidence is that it comes f r o m a time w h e n the Isis festival was a l i v i n g c u s t o m , that it comes directly f r o m E g y p t w i t h o u t h a v i n g passed t h r o u g h the hands of other writers, a n d that some of it comes f r o m a date before the r e f o r m of the E g y p t i a n calendar, thus r e m o v i n g any possibility o f confusion be­ tween the calendars. B u t there was m u c h m o r e such evidence (in b o t h the G r e e k a n d E g y p t i a n languages) available t h a n Neugebauer h a d realized. Jones adduces a g o o d deal o f evidence s h o w i n g that N e u g e b a u e r h a d w r o n g l y t a k e n the O s i r i s festival o f 12 to 3 0 C h o i a k (Egyptian) to be the same festival as the Isia that P l u t a r c h mentions. Jones also points out that G e m i n o s refers to the festival s i m p l y as ta Isia, w i t h o u t any further specification. T h i s implies that the festival was so w e l l k n o w n that G e m i ­ nos h a d n o fear that it w o u l d be confused by his readers w i t h other festi­ vals associated w i t h Isis o r O s i r i s . N o w , as Jones points out, there are at least nineteen references i n G r e e k p a p y r i to a festival called the Isia (also spelled Iseia or Isieia). O n l y a few o f these p r o v i d e c a l e n d r i c a l i n f o r m a ­ t i o n . B u t e n o u g h d o that it is possible to c o n f i r m Plutarch's dates o f 52

5 0

5 1

5 2

For a bibliography pertaining to this text, see Porter and M o s s 1927, vol. 6, 97. Grenfell and H u n t 1906, 144, 148. Jones 1999a.

22

·

Section 6

1 7 - 2 0 A t h y r , a n d to be sure that these dates indeed a p p l y to the o l d (Egyptian) calendar. F o r e x a m p l e , several p a p y r i f r o m before the calen­ dar r e f o r m are private letters o r records, w i t h dates i n A t h y r , c o n c e r n e d w i t h o r d e r i n g o r issuing supplies (logs a n d l a m p oil) for the Isia. O n e p a ­ pyrus gives the dates of the Isia i n terms o f the M a c e d o n i a n calendar. These dates c a n be converted to the E g y p t i a n calendar ( w i t h an uncer­ tainty o f 1 day) a n d indeed c o r r e s p o n d to 1 7 - 2 0 A t h y r . Slightly altering the c h r o n o l o g i c a l assumptions a n d b r o a d e n i n g the error bars, Jones c o n ­ cludes that it is very p r o b a b l e that G e m i n o s w r o t e his Introduction to the Phenomena "between 9 0 a n d 2 5 B.C., a n d definitely not d u r i n g the first century o f o u r e r a . " There is an i r o n y i n the fact, c o n f i r m e d by the p a p y r i , that after the calendar r e f o r m , the Isia c o n t i n u e d to be celebrated o n days c a l l e d 1 7 - 2 0 A t h y r , but in the new calendar. T h u s , Plutarch's dates t u r n out to refer to the r e f o r m e d calendar after a l l ! (But they s h o u l d not be converted back i n t o the o l d calendar to o b t a i n the dates that G e m i n o s w o u l d have been f a m i l i a r w i t h . ) 53

O n e m i n o r p r o b l e m w i t h d a t i n g G e m i n o s to the first century B.C. i n ­ volves his m e n t i o n of H e r o of A l e x a n d r i a i n fragment 1. T h e d a t i n g of H e r o has been c o n t r o v e r s i a l , w i t h suggested dates f r o m the m i d d l e o f the second century B.C. to the m i d d l e o f the t h i r d century A . D . In Dioptra 3 5 , however, H e r o mentions a l u n a r eclipse observed s i m u l t a n e o u s l y i n A l e x a n d r i a a n d R o m e . A l t h o u g h H e r o does not m e n t i o n the year o f the eclipse, he is detailed about its other circumstances: 10 days before the v e r n a l e q u i n o x , 5 t h seasonal h o u r o f the night at A l e x a n d r i a . N e u g e ­ b a u e r has s h o w n that these circumstances were satisfied by o n l y one l u ­ nar eclipse between about - 2 0 0 a n d -f300, n a m e l y that o f M a r c h 1 3 , A . D . 6 2 . If H e r o used an eclipse of recent m e m o r y , w e must place h i m i n the second h a l f of the first century A . D . T h u s , i f the dating o f G e m i n o s to the first century B.C. is correct, w e must suppose that P r o k l o s o r a later c o p y i s t interpolated the name o f H e r o i n fragment 1. F i n a l l y , w e note that G e m i n o s writes about B a b y l o n i a n a s t r o n o m y a n d astrology as if they were still n e w to his G r e e k readers. T h i s w e l l suits a d a t i n g to the first century B.C., w h e n this m a t e r i a l was still being a b s o r b e d a n d adapted by the G r e e k s . 5 4

55

56

Jones 1999a, 266. For a summary of the older estimates, see Heath 1921, vol. 2, 298-307. Neugebauer 1938; with results summarized in Neugebauer 1975, 846. Some scholars have attempted to identify the author of the Introduction to the Phe­ nomena with other men named Geminos. Aujac (1975, xxii) suggests a certain Cnaeios Pompeios Geminos, active around A . D . 15. Reinhardt (1921, 178-83) prefers to identify the author of the Introduction with an earlier Geminos; Tannery (1887, 37) with a later one. Aujac (1975, x x - x x i i ) is one of the few who discount the usual dating argument based on the feasts of Isis, and seeks to explain this paragraph (viii 20-22) in a completely differ­ ent way. 5 3

5 4

5 5

5 6

Geminos and the Stoics

·

23

7. G E M I N O S A N D T H E STOICS G e m i n o s is u s u a l l y considered to have been a Stoic, a n d is often said to have been a disciple o f Poseidônios (c. 1 3 0 - 5 0 B . C . ) . K e y evidence for this involves t w o circumstances. First, S i m p l i k i o s tells us that G e m i n o s w r o t e a n abridgement of, o r perhaps a c o m m e n t a r y o n , a Meteorology by Poseidônios. (See fragment 2.) M o r e o v e r , S i m p l i k i o s says that G e m i ­ nos's discussion o f the r e l a t i o n s h i p between a s t r o n o m y a n d physics w a s based o n Poseidônios's. A n d , second, f o l l o w i n g Petau's analysis, m o s t scholars assigned a date o f a b o u t 7 0 B.C. to the c o m p o s i t i o n o f G e m i ­ nos's Introduction to the Phenomena. A c c o r d i n g to this d a t i n g , G e m i n o s w a s a younger c o n t e m p o r a r y o f Poseidônios, a n d c o u l d therefore be v i ­ sualized sitting at the feet o f the master o f Stoic p h i l o s o p h y i n R h o d e s . The fact that G e m i n o s w r o t e a Meteorology based o n Poseidônios's w o r k does seem to i m p l y , at the very least, a n interest i n Stoic physics. A n d G e m i n o s ' s dates may o v e r l a p Poseidônios's w e l l e n o u g h to p e r m i t a student-teacher r e l a t i o n s h i p . B u t because o f the m a r g i n o f error i n the date for the Introduction to the Phenomena, this is not certain. 57

58

Some have gone so far as to m a k e G e m i n o s ' s Introduction to the Phe­ nomena itself a v i r t u a l paraphrase o f Poseidônios's lost Peri Meteôrôn. T h i s v i e w w a s m a i n t a i n e d by B l a s s , w h o p o i n t e d to a n u m b e r o f pas­ sages i n w h i c h G e m i n o s ' s language closely resembles that o f c o r r e s p o n ­ d i n g passages o f Kleomëdës' Meteôra, a w o r k that o b v i o u s l y is depend­ ent o n Poseidônios a n d cites h i m m a n y times. Blass's a p p r o a c h is a n e x a m p l e o f the Quellenforschung ("source research") that characterized m u c h late-nineteenth-century G e r m a n p h i l o l o g y . T h e idea w a s to m i n e the s u r v i v i n g w o r k s o f ancient scientists o r p h i l o s o p h e r s , i n a n attempt to reconstruct the lost w o r k s o f their ancient precursors. Blass's thesis re­ g a r d i n g the Introduction to the Phenomena w a s refuted by Tannery, w h o carefully demonstrated its inconsistencies w i t h the e v i d e n c e . Blass's theory, i n its strong f o r m , has little, i f any, s u p p o r t a m o n g c o n ­ t e m p o r a r y scholars. B u t the parallels between G e m i n o s a n d Kleomëdës constitute the best evidence that b o t h made use o f the same earlier w o r k , 59

60

Writers who identify Geminos as a student of Poseidônios include Sarton (1959, 305) and Kouremonos (1994, 437). We also have the testimony of Diogenes Laertios (Lives and Opinions vii 132) that the relationship between mathematics (i.e., astronomy) and physics was, indeed, a topic discussed by Stoic writers. Blass 1883. Blass even proposed that the original title of Geminos's book was Gemi­ nos's exegesis of the phenomena from the Meteorológica of Poseidônios. Since Geminos modifies many of Poseidônios's doctrines, Geminos is therefore, according to Blass, a fraud, who could not have been a student of Poseidônios, and would not, in any case, have dared to publish such a work in Poseidônios's lifetime. Tannery 1887, 29-36. 57

58

59

60

24

·

Section 7

for w h i c h a l i k e l y candidate is the Peri Meteôrôn o f Poseidônios. T h i s is the t h i r d major argument for G e m i n o s ' s dependence o n Poseidônios. A g a i n s t this, as Tannery p o i n t e d out, there are other w a y s o f e x p l a i n i n g the p a r a l l e l passages. B o t h G e m i n o s a n d Kleomëdës d r e w u p o n m a t e r i a l that w a s the c o m m o n p r o p e r t y o f astronomers f r o m l o n g before Poseidônios's time. It is possible, t o o , that Kleomëdës made use o f other elementary surveys of astronomy, i n c l u d i n g G e m i n o s ' s , i n w r i t i n g his own. 6 1

A t the other end o f the o p i n i o n scale, R e i n h a r d t studied the uses o f " s y m p a t h y " i n G e m i n o s a n d i n the fragments of Poseidônios, a n d c o n ­ c l u d e d that G e m i n o s c o u l d not p o s s i b l y have been a student o f P o ­ s e i d ô n i o s . A n d Neugebauer, w h o was dismissive o f Poseidônios's c o m ­ petence i n astronomy, r e m a r k e d that G e m i n o s " h a d n o t h i n g to learn f r o m P o s i d o n i u s . " Indeed, the v i e w that Kleomëdës gives us o f P o ­ seidônios's c o n t r i b u t i o n to a s t r o n o m y is not very impressive. F o r e x a m ­ ple, Poseidônios's methods o f d e t e r m i n i n g the sizes a n d distances o f the S u n a n d M o o n , w h i c h N e u g e b a u e r described as " n a i v e , " represent a m e t h o d o l o g i c a l step b a c k w a r d f r o m A r i s t a r c h o s a n d H i p p a r c h o s . T a n ­ nery, t o o , judged G e m i n o s to be a m u c h better w r i t e r t h a n Kleomëdës, a n d argued that their b o o k s have n o m o r e i n c o m m o n t h a n one w o u l d expect to see i n t w o elementary t e x t b o o k s of o u r o w n day. T a n n e r y p o i n t e d also to the maladresse o f Kleomëdës' b o o k i i , chapter 1, a l m o s t certainly based o n Poseidônios, i n w h i c h the p h i l o s o p h e r tediously piles u p arguments against the E p i c u r e a n s ' doctrine o f the i m m e d i a t e v a l i d i t y of sense-evidence, a n d i n p a r t i c u l a r their v i e w that the S u n is the size it appears to be, n a m e l y 1 foot i n diameter. O n e c o u l d h a r d l y i m a g i n e G e m i n o s w a s t i n g his time o n such a m a t t e r . T a n n e r y concludes, "If, i n his Introduction to the Phenomena, G e m i n o s never cites Poseidônios, one must n o t at a l l c o n c l u d e that he is seeking to disguise a p l a g i a r i s m , but rather that he d i d not estimate the c o s m o g r a p h i c a l w o r k s of the p h i l o s o p h e r o f A p a m e a [Poseidônios] h i g h l y enough to rely o n their a u ­ thority." 62

63

64

65

Teachers o f Stoic p h i l o s o p h y c o m m o n l y offered a complete c u r r i c u l u m of logic, ethics, a n d physics. Diogenes Laertios gives an extensive s u m M a n y of the parallels between Geminos and Kleomëdës are pointed out in the notes to Todd's [1990] edition of Kleomëdës. Todd also identifies Kleomëdës' many parallels with other elementary writers on astronomy. Reinhardt 1921, 178-183; recapitulated in Reinhardt 1926, 5 1 - 5 3 . Reinhardt held that the Introduction to the Phenomena belongs to the Stoa of the period before Poseidônios, and that its author was not the same Geminos who wrote sophisticated math­ ematical works (the Philokalia). Neugebauer 1975, 579. For a more positive view of Kleomëdës' work, see Bowen and Todd 2004. Tannery 1887, 35. 6 1

6 2

6 3

6 4

6 5

Geminos and the Stoics

·

25

m a r y of this c u r r i c u l u m . A l t h o u g h , as he tells us himself, Diogenes shortens his discussion by m e r g i n g a n u m b e r o f Stoic writers, he does at times distinguish one Stoic f r o m another, i n instances where there were substantive disagreements over doctrine. It is also clear that Diogenes still h a d access to a n u m b e r of Stoic treatises a n d textbooks that are n o w lost. A s t r o n o m y was treated under the p h y s i c a l part o f the c u r r i c u l u m , a n d Diogenes gives a short s u m m a r y o f the teachings o f the Stoics i n astron­ o m y . T h e a s t r o n o m i c a l "teachings" are elementary facts that were the c o m m o n property o f a l l astronomers since the fourth century B.C. D i o ­ genes' list includes: a description o f the parallel circles o f the sphere, a n d of the oblique z o d i a c , the fact that the Sun is spherical a n d is larger than the E a r t h , that the M o o n gets its light f r o m the S u n , the causes of lunar a n d solar eclipses, that the M o o n ' s p a t h is oblique to the z o d i a c , that the E a r t h is i n the center o f the cosmos, a n d that it has five zones. T h e as­ t r o n o m y is c o m m o n p l a c e , a n d there is n o t h i n g p a r t i c u l a r l y Stoic about it. 66

67

T h e Stoic flavor is lent to the a s t r o n o m i c a l discussion by the i n c o r p o ­ r a t i o n of Stoic p h y s i c a l d o c t r i n e s . A m o n g these Diogenes mentions the f o l l o w i n g . There are t w o principles i n the w o r l d , the active a n d the pas­ sive. T h e passive p r i n c i p l e is substance w i t h o u t quality, that is, matter, w h i l e the active p r i n c i p l e is the reason inherent i n substance, that is, G o d . G o d is one a n d the same w i t h reason, fate, a n d Z e u s . T h e w o r l d is ordered by reason a n d p r o v i d e n c e , a n d harbors n o e m p t y space, but is held together by s y m p a t h y a n d tension. T h e m a t e r i a l w o r l d is finite, but b e y o n d it lies the infinite v o i d . T h e S u n is pure fire, a doctrine that D i o ­ genes attributes specifically to Poseidônios, " i n the seventh b o o k o f his Peri Meteôrôn." T h e b l e n d o f Stoic physics a n d elementary a s t r o n o m y that Diogenes describes corresponds w e l l w i t h Kleomëdës' Meteôra, but not so w e l l w i t h G e m i n o s ' s Introduction to the Phenomena, i n w h i c h the Stoic physics is c o m p l e t e l y l a c k i n g . 68

A s A l a n B o w e n has aptly said, S t o i c i s m w a s , i n G e m i n o s ' s day, " l i k e a c o l o r . " W e w o u l d not deny a Stoic c o l o r a t i o n to G e m i n o s ' s thought, as is apparent i n fragment 2 , a n d perhaps i n G e m i n o s ' s matter-of-fact accep­ tance o f " s y m p a t h i e s " between people b o r n i n certain z o d i a c a l r e l a t i o n ­ ships to one another (Introduction to the Phenomena i i ) . B u t it must be said that, i f G e m i n o s were t r u l y a Stoic, he w o r e his S t o i c i s m lightly. G e m i n o s ' s Introduction is r e m a r k a b l y free o f p h i l o s o p h i c a l i n t e r p o l a ­ tions, u n l i k e those o f Kleomëdës a n d T h e ô n o f S m y r n a , w h i c h s h o w us Diogenes Laertios, Lives and Opinions vii 38-160. Diogenes Laertios, Lives and Opinions. The summary of physical doctrine begins at vii 132, and the specifically astronomical and cosmographical parts are at vii 144-46 and 6 6

6 7

155-56. O n Stoic physics and cosmology, see Sambusky 1959, Todd 1976, Todd 1982, and Todd 1989. 6 8

26

·

Section 7

w h a t p h i l o s o p h i c a l l y oriented surveys o f a s t r o n o m y l o o k l i k e . Theón's text is m a r k e d by frequent, specific references to the w o r k s o f P l a t o . Kleomëdës begins his b o o k w i t h a n a c c o u n t o f Stoic physics a n d c o s m o l ­ ogy: the pneuma, the i m p o s s i b i l i t y o f a v o i d place inside the c o s m o s , the necessity for a n infinite v o i d place outside the c o s m o s , etc. M o r e o v e r , Kleomëdës frequently cites Poseidônios, w h i l e G e m i n o s never m e n t i o n s Poseidônios at a l l . G e m i n o s does cite three Stoic w r i t e r s , Kleanthës o f A s s o s , Kratës o f M a l l o s , a n d B o ë t h o s — b u t i n the first t w o cases, s i m p l y to take issue w i t h t h e m . A t x v i 2 1 , he mentions Kleanthës' v i e w that O c e a n spreads a l l over the t r o p i c a l zone, o n l y to refute it as "quite m i s t a k e n " a n d " a l i e n to b o t h m a t h e m a t i c a l a n d p h y s i c a l t h o u g h t . " In several passages (vi 1 0 , 16; x v i 2 7 ) , G e m i n o s refers to Kratës' efforts to attribute s o u n d a s t r o n o m i c a l a n d g e o g r a p h i c a l k n o w l e d g e to H o m e r . G e m i n o s admires H o m e r a n d enjoys m a k i n g use o f H o m e r i c verses w i t h a s t r o n o m i c a l ref­ erences, but he faults Kratës for r e a d i n g t o o m u c h i n t o H o m e r : " K r a t ë s , s p e a k i n g i n marvels, takes things said by H o m e r for his o w n purposes a n d i n archaic fashion, a n d transfers t h e m to the spherical system that accords w i t h reality." G e m i n o s (xvii 48) endorses Boëthos's use o f natu­ r a l signs i n forecasting the weather, rather t h a n a t t r i b u t i n g the weather to the influence of the stars. In this, says G e m i n o s , Boëthos is i n a c c o r d w i t h Aratos, Eudoxos, and Aristotle. M o r e o v e r , there are a n u m b e r o f t e c h n i c a l reasons that m a k e G e m i n o s a p o o r candidate for paraphraser o f Poseidônios i n particular, o r d e d i ­ cated Stoic i n general: (1) G e m i n o s (chapter x v ) adopts a scheme for the terrestrial zones different f r o m that o f Poseidônios. (2) G e m i n o s (vi 38) adopts a pattern for the v a r i a t i o n i n the length o f the day different f r o m that o f Kleomëdës. (See note 162 to the I n t r o d u c t i o n , p . 74.) (3) G e m i ­ nos ( x v i 6) f o l l o w s Eratosthenes rather t h a n Poseidônios o n the size o f the E a r t h . (4) G e m i n o s denies ( x v i i 2) that the exhalations f r o m the E a r t h reach as far as the stars, w h i c h is c o n t r a r y to a v i e w p o p u l a r a m o n g the Stoics, i n c l u d i n g Poseidônios, w h o c l a i m e d that the stars were n o u r ­ ished by these exhalations. (5) G e m i n o s devotes an entire chapter (xvii) to refuting the c o m m o n belief that changes i n the weather are caused by the h e l i a c a l risings a n d settings o f the stars. A g a i n , this puts h i m at odds w i t h the Stoics, w h o were sympathetic to the doctrine o f astral influ­ ences. (6) G e m i n o s (xvii 15) states that the stars have n o s y m p a t h y w i t h things o c c u r r i n g o n the E a r t h — a decidedly n o n - S t o i c view. (7) In the same passage, G e m i n o s is s i m p l y indifferent to whether the stars are m a d e o f fire o r aithër, w h i c h shows little regard for a question that P o ­ seidônios thought i m p o r t a n t . T h o u g h it is perfectly possible that G e m i n o s b o r r o w e d f r o m a n ele­ m e n t a r y survey o f a s t r o n o m y by Poseidônios o r someone else, p a r t i c u l a r

Astronomical Instruments and Models

·

27

chapters o f the Introduction to the Phenomena seem to depend rather o n earlier technical writers, such as A u t o l y k o s for chapter x i i i . A n d it is clear that G e m i n o s has made some use o f a w o r k by H i p p a r c h o s o n the fixed stars i n chapter i i i . It is also l i k e l y that G e m i n o s has put i n plenty of his o w n . T h e l o n g chapter (viii) o n l u n i s o l a r cycles is p r o b a b l y his o w n ; n o other extant G r e e k text treats the subject i n so m u c h detail, a n d there is n o evidence that Poseidônios w r o t e o n this subject. Similarly, the final chapter (xviii), o n the exeligmos, is w i t h o u t p a r a l l e l i n early G r e e k a s t r o n o m y ; w e d o not find another such discussion u n t i l Ptolemy's Almagest. Finally, G e m i n o s ' s i n c o r p o r a t i o n o f elements o f B a b y l o n i a n ast r o n o m y has n o p a r a l l e l i n other G r e e k w o r k s o f the p e r i o d , a n d for that reason is p a r t i c u l a r l y precious.

8. G E M I N O S O N A S T R O N O M I C A L I N S T R U M E N T S A N D M O D E L S

In the Introduction to the Phenomena, G e m i n o s mentions several k i n d s of a s t r o n o m i c a l instruments a n d models a n d assumes that his reader is familiar w i t h them: the s u n d i a l , the dioptra, a n d the celestial sphere, o f w h i c h there are t w o k i n d s , s o l i d a n d ringed. Today, w e refer to these last t w o as the celestial globe a n d the a r m i l l a r y sphere. G e m i n o s uses a l l these pieces o f apparatus m o r e as tools o f i n s t r u c t i o n t h a n as instruments o f o b s e r v a t i o n . H i s use o f instruments is thus p e d a g o g i c a l — exactly w h a t w e w o u l d expect i n a n i n t r o d u c t o r y t e x t b o o k .

Celestial

Globe and Armillary

Sphere

G e m i n o s mentions b o t h the " s o l i d sphere," sphaira sterea ( x v i 12), a n d the " r i n g e d sphere," sphaira krikôtê ( x v i 10, 12). B y his day, celestial globes were c o m m o n e n o u g h that a w r i t e r o n a s t r o n o m y or geography c o u l d assume his readers to be f a m i l i a r w i t h them a n d c o u l d therefore appeal to these instruments i n i l l u s t r a t i o n o r argument. Strabo devotes a chapter o f his Geography to the question o f w h a t sort o f a s t r o n o m i c a l k n o w l e d g e students o f geography w i l l require. H e assures his readers that they need not be experts i n astronomy, but w a r n s that they s h o u l d not be so simple o r lazy as never to have seen a globe a n d the circles i n scribed u p o n it, o r to have e x a m i n e d the positions o f the tropics, equator, a n d z o d i a c . Celestial globes were not o n l y useful tools o f instruct i o n , but also potent symbols that c o u l d carry p h i l o s o p h i c a l , religious, a n d (increasingly i n the R o m a n period) p o l i t i c a l meanings. F o r this reason they often figure o n coins a n d m u r a l s , as w e l l as i n s c u l p t u r e . 69

70

6 9

Strabo, Geography i 1.21.

28

·

Section 8

Readers w h o h a d not h a d the o p p o r t u n i t y to m a n i p u l a t e a celestial globe w o u l d at least have seen globes i n artistic representations. T h r e e complete celestial globes are extant f r o m A n t i q u i t y . T h e oldest a n d best k n o w n is the large m a r b l e sphere (fig. 1.2) s u p p o r t e d by a statue of A t l a s , n o w i n the M u s e o A r c h e o l o g i c o N a z i o n a l e i n N a p l e s . T h i s statue, transferred to its present l o c a t i o n f r o m the Farnese Palace i n R o m e , is c o m m o n l y c a l l e d the Farnese A t l a s . T h e Farnese globe, a b o u t 6 5 c m i n diameter, is a p r o d u c t o f the early R o m a n empire, but w a s p r o b a b l y m o d e l e d o n a H e l l e n i s t i c o r i g i n a l . T h e p o s i t i o n s o f the constel­ l a t i o n s place the o r i g i n a l globe m o s t l i k e l y i n the second century B.C. B u t since the artist i n c l u d e d the c o n s t e l l a t i o n o f the " T h r o n e o f Caesar," the Farnese globe m u s t be p l a c e d i n the reign o f A u g u s t u s or later. T h e F a r ­ nese globe is a magnificent d i s p l a y piece, but h a r d l y a suitable t o o l for classroom instruction. 71

A s m a l l b r o n z e R o m a n globe (fig. 1.3) i n M a i n z is p r o b a b l y m o r e t y p i ­ cal o f the globes that a s t r o n o m y students m i g h t have encountered i n G e m i n o s ' s day. F i g . 1.3 s h o w s the globe itself, a n d fig. 1.4 s h o w s a m o d ­ ern r e p l i c a , o n w h i c h details are m o r e easily seen. E r n s t K i i n z l has dated the g l o b e , o n the basis o f the e n g r a v i n g technique, to A . D . 1 5 0 - 2 2 0 . T h i s g l o b e , 11 c m i n diameter, is figured w i t h constellations, the p r i n c i p a l ce­ lestial circles, a n d the M i l k y W a y , as w e l l as i n d i v i d u a l s t a r s . A globe o f this size c o u l d easily have been h e l d i n the h a n d for study o f the constel­ l a t i o n s , as w e l l as the celestial circles that Strabo deemed so i m p o r t a n t for a s t r o n o m y i n s t r u c t i o n . T h i s sphere has a s m a l l square hole i n its t o p a n d a larger r o u n d hole i n its b o t t o m , w h i c h suggest that the globe fit a spike that served as a stand. O r perhaps the globe fit over the t i p o f a g n o m o n , as a decorative feature for a large s u n d i a l . 72

A t h i r d ancient celestial globe surfaced recently i n the Paris antiquities m a r k e t , o n l y to disappear a g a i n after its sale to a private collector. It is a s m a l l (6 c m i n diameter) a n d h a n d s o m e silver object, but its engraver O n the symbolism of the sphere, see Arnaud 1984. O n the date of the copy, see K i i n z l 2000, 535, and K i i n z l 2005, 63-66. O n the Throne of Caesar, see Pliny, Natural History ii 178, and K i i n z l 2000, 535. For a precession dating of the supposed Hellenistic original, see Schaefer 2005. The general forms of the constellations agree with the tradition established by Eudoxos and Aratos, but the artist made a number of modifications that point to the influence of later sources as well. Schae­ fer argues that the contours of some constellations were adjusted to accord with correc­ tions that Hipparchos introduced in his Commentary on the Phenomena of Eudoxos and Aratos. Schaefer's association of the Farnese globe with Hipparchos has been refuted by Duke 2006. See also Valerio 1987. See Kiinzl 2000 for a wonderfully detailed study of this object. One oddity of this globe is that its maker must have wished to show the 48 constellations made canonical by Ptolemy, but he missed two, and simply stuck in two round assemblages of stars in the southern hemi­ sphere to make up the difference. One of these can be seen in the lower right of fig. 1.3. 7 0

71

72

Astronomical Instruments and Models

·

29

Fig. 1.2. The Farnese globe. Roman copy of the first century A . D . or late first century B.c., after a Greek original of the second century B.c. The statue of Atlas was heavily restored in the Renaissance, and the arms and head are not original. Museo Archeologico Nazionale, Naples. The zodiac slants upward from left to right. The three broadly spaced parallel circles are, from top to bottom, the tropic of Cancer, the celestial equator, and the tropic of Capricorn.

30

·

Section 8

Fig. 1.3. A small, bronze Roman celestial globe from about A . D . 150-220. This view is centered on Ophiuchus. Rómisch-Germanisches Zentralmuseum, M a i n z ( R G Z M O . 41339). Photo by Volker Iserhardt.

m a d e a n u m b e r o f errors i n the constellations. It was most l i k e l y f r o m A s i a M i n o r , a n d was offered for sale w i t h t w o other m e t a l objects that c o u l d be dated confidently to the second century A . D . N o n e o f the extant globes are o f the k i n d most suitable for t e a c h i n g — i.e., none is free to t u r n o n a n axis that c a n m a k e an adjustable angle w i t h a fixed h o r i z o n . T w o ancient w r i t e r s , however, give directions for 7 3

Cuvigny 2004 offers a detailed description of this globe. The same author gives a shorter account, along with excellent color photographs, in Kugel 2002. 7 3

Astronomical Instruments and Models

·

31

Fig. 1.4. A galvanized plastic copy of the M a i n z globe. O n the lower half of the sphere may be seen the zodiac constellations Capricorn and Aquarius. The irregular chain of small dots crossing the upper part of the sphere is the M i l k y Way. Rómisch-Germanisches Zentralmuseum, M a i n z . Photo by Ernst K i i n z l . b u i l d i n g a globe, a n d b o t h m a k e these features p l a i n . L e o n t i o s the me­ c h a n i c , for e x a m p l e , says that the h o r i z o n r i n g is o f the same size as the base o f the g l o b e .

74

rings quite c l e a r l y .

75

A n d P t o l e m y describes the m e r i d i a n a n d h o r i z o n

Leontios, Construction of the Sphere of Aratos, in Halma 1821 or Maass 1898. Ptolemy, Almagest viii 3. Ptolemy's design is more complicated, since his globe is ad­ justable for precession. 74

75

32

·

Section 8

F i g . 1.5. A m o s a i c o f a n a r m i l l a r y sphere i n the C a s a d i L e d a , at S o l u n t o , near P a l e r m o . P h o t o courtesy o f R u d o l f S c h m i d t .

S i m i l a r to the celestial globe, but easier to construct, is the a r m i l l a r y sphere. In the a r m i l l a r y sphere, the sky is represented not as a s o l i d sphere, but by a n e t w o r k o f rings. (The L a t i n w o r d armilla signifies a n a r m b a n d o r bracelet.) These rings e m b o d y the most i m p o r t a n t circles o f the sphere: ecliptic, equator, t r o p i c s , arctic a n d antarctic circles, a n d the colures. N o n e of these delicate constructions has s u r v i v e d f r o m A n t i q ­ uity, but a n u m b e r of illustrations o f a r m i l l a r y spheres are preserved i n ancient art. A ceiling p a i n t i n g i n Stabiae (near P o m p e i i ) displays w h a t is almost certainly meant to be a n a r m i l l a r y sphere, but o n l y a p o r t i o n of the p a i n t i n g is extant a n d , moreover, the o r i g i n a l design seems to have been imperfect i n its d e p i c t i o n of the sphere. M o r e impressive is the m o s a i c i n S o l u n t o (near P a l e r m o ) , s h o w n i n F i g . 1.5, w h i c h depicts very p l a i n l y a l l the features o f a n a r m i l l a r y sphere, i n c l u d i n g w h a t is most 76

For pictures of the Stabiae sphere, see A r n a u d 1984, 73; or Carmado and Ferrrara 1989, 67-68. 7 6

A s t r o n o m i c a l Instruments a n d M o d e l s

·

33

F i g . 1.6. A R e n a i s s a n c e i l l u s t r a t i o n o f a n a r m i l l a r y sphere. F r o m Cosmographia . . . Petri Apiani & Gemmae Frisii ( A n t w e r p : 1 5 8 4 ) . Photo: Special Collections, University of Washington Libraries, Negative UW18183.

l i k e l y intended to be the h o r i z o n r i n g .

7 7

In fig. 1.6 we see a Renaissance

i l l u s t r a t i o n of an a r m i l l a r y sphere that preserves a l l the essential features of the ancient p r o t o t y p e that G e m i n o s w o u l d have k n o w n . In his teaching, G e m i n o s w o u l d have used a celestial globe to illustrate the circles o f the sphere (chapter v), the risings a n d settings o f the z o d i a c signs (vii), a n d the v a r i a t i o n of the p h e n o m e n a w i t h l o c a l i t y (xvi). A t x v i 12 he m e n t i o n s a l i m i t a t i o n of the globes that were c o m m o n l y 7 7

V o n Boeselager 1983, pp. 56-60, and Tafel X V .

available.

34

·

Section 8

A c c o r d i n g to G e m i n o s , the c o m m e r c i a l l y available globes were i n s c r i b e d w i t h arctic a n d antarctic circles for the single klima c o r r e s p o n d i n g to lat­ itude 3 6 ° . O f course, i f the globe were fitted w i t h a m e r i d i a n r i n g that w a s free to t u r n inside its h o l d i n g slots i n the h o r i z o n , one c o u l d adjust the globe for any latitude. B u t an arctic circle d r a w n o r engraved o n the surface o f the globe c o u l d c o r r e s p o n d o n l y to a single latitude. Sundial F o r " s u n d i a l " G e m i n o s uses three w o r d s m o r e o r less s y n o n y m o u s l y : hôroskopeion (ii 35 a n d x v i 13), hôrologion (ii 3 8 , 4 5 ; v i 3 3 , 4 6 ; v i i i 2 3 ; a n d x v i 18), a n d skiothëron (vi 32). G e m i n o s i n v o k e s the s u n d i a l , not for genuine o b s e r v a t i o n , but as an appeal to experience w i t h w h i c h his reader is familiar, for m o s t o f the arguments he makes f r o m the s u n d i a l w o u l d require the observer to m a k e observations separated by several m o n t h s i n time or by several h u n d r e d stades i n latitude. Clearly, G e m i ­ nos never intended his reader to m a k e these observations. Rather, he ap­ peals to sense evidence that is in principle easily o b t a i n e d , a n d the reader is supposed to grant h i m the argument. G r e e k dialers were very i n v e n t i v e , a n d m a n y different k i n d s are pre­ served a m o n g the 3 0 0 or so ancient dials extant t o d a y . F r o m the theo­ retical p o i n t o f view, the simplest is the s p h e r i c a l d i a l or skaphë. The s h a d o w - r e c e i v i n g surface is the i n t e r i o r o f a hemisphere, a n d the tip o f the g n o m o n lies at the center o f the s p h e r i c a l surface. Since the d i a l is m e r e l y a n inverted image o f the celestial sphere, the theory g o v e r n i n g the placement of the h o u r lines, as w e l l as the equator a n d t r o p i c s , is very s i m p l e . M o r e o v e r , because the S u n c a n n o t be f o u n d at just any place o n the celestial sphere, but must r e m a i n between the t r o p i c s , a n entire hemisphere o f stone is n o t r e q u i r e d . T h e l o w e r part o f the s o u t h face o f the d i a l , c o r r e s p o n d i n g to the part o f the sky above the t r o p i c o f Cancer, c a n be cut away. F i g . 1.7 s h o w s a n i d e a l i z e d v i e w o f this k i n d of d i a l . E l e v e n h o u r curves indicate seasonal hours. T h e p e r i o d f r o m sunset to sunrise consists a l w a y s o f 12 h o u r s , a l l e q u a l to one another. S i m i ­ larly, the night is d i v i d e d i n t o 12 e q u a l h o u r s . In the s u m m e r the day h o u r is l o n g a n d the night h o u r is short, whereas i n w i n t e r the opposite is true. T h e equinoctial hour that w e use t o d a y (one t w e n t y - f o u r t h o f the w h o l e d i u r n a l period) is a seasonal h o u r evaluated o n the day o f e q u i n o x . A l t h o u g h G r e e k astronomers d i d use the e q u i n o c t i a l h o u r 78

For a catalogue of 256 stone dials, see Gibbs 1976. For the portable dials and stone dials published since Gibbs's survey, see Arnauldi and Schaldach 1997; Catamo et al. 2000; Field and Wright 1984; Locher 1989; Pattenden 1981; Price 1969; Rohr 1980; Savoie and Lehoucq 2001; Schaldach 2004; and Evans and Marée (forthcoming). 7 8

Astronomical Instruments and Models

·

35

Fig. 1.7. The principle of the spherical sundial with cutaway south face. Eleven hour curves mark the seasonal hours. Three parallel circles represent (from top to bottom) the tropic of Capricorn, the celestial equator, and the tropic of Cancer. A n iron or bronze gnomon casts a shadow. w h e n they needed a u n i f o r m u n i t o f time for precise c a l c u l a t i o n , the seasonal h o u r w a s the o n l y one used i n everyday life. ( A l l s u r v i v i n g G r e e k a n d R o m a n sundials are i n fact m a r k e d i n seasonal hours.) T h e d i a l i n fig. 1.7 is also furnished w i t h three day curves, i n d i c a t i n g the track o f the s h a d o w ' s tip o n (from t o p to bottom) w i n t e r solstice, e q u i ­ n o x , a n d s u m m e r solstice. T o judge by the numbers preserved, the most c o m m o n d i a l was o f the conical type. In a c o n i c a l d i a l , the shadow-receiving surface is the inner surface o f a cone. Typically, the c o n i c a l surface was cut i n t o a r o u g h l y rectangular slab o f stone, as w i t h the d i a l s h o w n i n fig. 1.8. T h e stonew o r k i n g i n v o l v e d i n m a k i n g a c o n i c a l d i a l was easier t h a n that r e q u i r e d for a spherical d i a l . B u t by c o m p e n s a t i o n , the theory was m o r e c o m p l i ­ cated: it w a s necessary to project the celestial sphere o n t o a c o n i c a l sur­ face. T h e eleven m o r e o r less vertically oriented curves are the h o u r lines. O n the d i a l s h o w n i n fig. 1.8, the h o u r lines are labeled w i t h G r e e k n u ­ merals, w h i c h is a very rare feature. Large numbers o f plane dials have also been preserved. A l t h o u g h plane dials are simple f r o m a stonecutter's perspective, they are m u c h m o r e c o m p l i c a t e d mathematically, for the celestial sphere must be p r o ­ jected o n t o a plane surface. In the d i a l of fig. 1.9 w e see the t y p i c a l f o r m of a h o r i z o n t a l plane s u n d i a l . T h e upper curve is the s h a d o w track for 79

7 9

This is sundial no. 3086 in Gibbs 1976.

36

·

Section 8

Fig. 1.8. A conical sundial found near Alexandria at the base of Cleopatra's needle. 40 cm high χ 43 cm wide. The Greek numerals labeling the hour lines are an unusual feature and are probably Byzantine additions to the original Ptolemaic dial. British Museum no. 1936 3-9.1. Photo by permission of the Trustees of the British Museum. s u m m e r solstice; the l o n g straight line, the s h a d o w track for e q u i n o x ; a n d the l o w e r curve, that for w i n t e r solstice. T h e b r o k e n stub o f a n i r o n g n o m o n remains embedded i n the lead that fills the hole above the s u m ­ mer solstitial curve. T h e eleven m o r e o r less vertical lines represent the h o u r s . T h i s d i a l is decorated w i t h the names of eight w i n d s , i n L a t i n , i n the circle a r o u n d the outside o f the d i a l . A n u n u s u a l feature of this d i a l is the signature of the d i a l m a k e r , a certain M . A n t i s t i u s E u p o r u s . 80

8 0

N o . 4002 in Gibbs 1976.

A s t r o n o m i c a l Instruments a n d M o d e l s

·

37

F i g . 1.9. A h o r i z o n t a l - p l a n e s u n d i a l f o u n d i n A q u i l e i a i n the n o r t h o f Italy. T h e outer d i a m e t e r o f the c i r c l e o f w i n d s is 6 6 c m . P h o t o : M u s e o A r c h e o l o g i c o Nazionale, Aquileia.

Geminos's readers were certainly familiar w i t h dials of these basic types. A t i i 3 5 , G e m i n o s appeals to the evidence o f sundials to prove that the sign of C a n c e r is i n syzygy w i t h the sign of G e m i n i , saying that w h e n the Sun is i n Cancer, the tip of the gnomon's s h a d o w follows the same curve as it does w h e n the Sun is i n G e m i n i . (See also i i 38 a n d 4 5 , a n d v i 46.) A t v i 3 2 - 3 3 , G e m i n o s uses the s u n d i a l to s h o w that the rate o f the Sun's progress i n d e c l i n a t i o n (i.e., its m o t i o n t o w a r d the n o r t h or south) varies i n the course o f the year. T h u s , a r o u n d the time o f the solstice, the tip o f the s h a d o w f o l l o w s the same curve, as far as sense is concerned, for about 4 0 days. B y contrast, at the time o f the e q u i n o x , the tip o f the s h a d o w makes a perceptible departure f r o m the e q u i n o c t i a l track i n the course o f a single day. G e m i n o s also m e n t i o n s that a r o u n d the solstices the lengths o f the days a n d nights scarcely change for 4 0 days. T h e length o f the day, h o w ­ ever, was n o t s o m e t h i n g that most o f G e m i n o s ' s readers c o u l d have mea­ sured directly. T h e most direct m e t h o d o f measurement, using a s u n d i a l calibrated i n hours, was not o f m u c h help, since the dials were a l l d i v i d e d

38

·

Section 8

i n t o seasonal hours: every day was 12 hours l o n g . A n d the water c l o c k s m e n t i o n e d by scientific writers were neither c o m m o n n o r very re­ liable keepers o f time. G e m i n o s ' s argument based o n the m o t i o n o f the sundial's s h a d o w therefore p r o v i d e d a bolstering element o f objectivity. T h i s w a s s o m e t h i n g y o u c o u l d see. Similarly, at v i i i 2 3 , i n c r i t i c i z i n g the c o m m o n belief that the E g y p t i a n s celebrate the feasts o f Isis a r o u n d the w i n t e r solstice, G e m i n o s r e m a r k s that this c a n be refuted either by n o t i n g the lengths o f the days o r by l o o k i n g at the track of the s h a d o w tip o n a s u n d i a l . In G e m i n o s ' s time, the feast o f Isis differed f r o m the solstice by a w h o l e m o n t h . 8 1

F i n a l l y , at x v i 1 3 , 18, G e m i n o s appeals to the s u n d i a l to illustrate the m e a n i n g o f a geographic klima. T h e klima remains the same everywhere o n the same p a r a l l e l of latitude, a n d thus the lines engraved o n the sun­ dials r e m a i n the same. If w e travel n o r t h o r s o u t h , the klima changes, but the change is imperceptible for displacements o f less t h a n 4 0 0 stades. W i t h larger displacements, the lines engraved o n the sundials must be different. Dioptra The dioptra (i 4, ν 1 1 , a n d x i i 4) w a s o r i g i n a l l y a simple sighting tube, o r a r o d e q u i p p e d w i t h t w o sights. T h e w o r d itself indicates s o m e t h i n g that one " l o o k s t h r o u g h . " F o r compactness o f expression, w e shall refer s i m ­ ply to the sighting tube, a l t h o u g h a r o d fitted w i t h t w o sights is the m o r e l i k e l y f o r m . In its simplest v e r s i o n , the dioptra c o u l d be a i m e d at a star a n d c l a m p e d i n p o s i t i o n . A teacher c o u l d use it to p o i n t out a p a r t i c u l a r star to students. If one then w a i t e d for 10 or 2 0 minutes, the star w o u l d have m o v e d out o f the sighting tube: i n this w a y the m o t i o n o f the celes­ t i a l sphere c o u l d be demonstrated. A pedagogical use of the dioptra is m e n t i o n e d by E u c l i d i n his Phe­ nomena. E u c l i d wants to demonstrate that the E a r t h is at the center o f the celestial sphere. Suppose, as i n fig. 1.10, that ABC is the circle o f "the h o r i z o n i n the c o s m o s , " i.e., the intersection o f the h o r i z o n plane w i t h the celestial sphere. T h e E a r t h is at D . L o o k t h r o u g h a dioptra at C a n c e r w h e n it is rising at C . If y o u t u r n a r o u n d a n d l o o k t h r o u g h the other end of the dioptra y o u w i l l see C a p r i c o r n setting at A , so A , D , a n d C are three p o i n t s o n a straight line. B u t C a p r i c o r n a n d C a n c e r are d i a m e t r i ­ c a l l y opposite one another i n the z o d i a c , so ADC is a diameter o f z o d i a c . 82

Several ancient dials do call attention to the variation in the length of the day throughout the year, by comparing the length of the winter solstitial day to the other days of the year: see Gibbs 1976, nos. 1044, 1068, 3046, and 4001. For a discussion of the lat­ ter, see Evans 1 9 9 8 , 1 3 0 - 3 1 . Euclid, Phenomena, prop. 1. Berggren and Thomas 1996, 5 2 - 5 3 . 81

82

Astronomical Instruments and Models

·

39

Fig. 1.10. Euclid's argument, using a dioptra, to show that the Earth is at the center of the cosmos. In the same way, a i m the dioptra at Β w h e n L e o is rising there. If y o u then l o o k t h r o u g h the other end o f the dioptra y o u w i l l see A q u a r i u s set­ ting at E. Therefore EDB is also a diameter o f the z o d i a c . T h e t w o d i a m ­ eters o f the z o d i a c circle intersect at D , w h i c h is therefore the center o f that circle a n d o f the celestial sphere. Since one c a n m a k e the same argu­ ment at any place o n the E a r t h , the E a r t h is a geometric p o i n t — t h e cen­ ter o f the c o s m o s . N o w this is clearly o n l y a p e d a g o g i c a l argument, a n d not a sequence o f genuine observations. T h e c o n v e n t i o n a l character o f the argument is apparent i n E u c l i d ' s use o f L e o a n d C a n c e r as i f they were points rather t h a n extended constellations. T h e appeal to a n instru­ ment nevertheless lends the argument a n air o f authority. T h e dioptra was eventually equipped w i t h a protractor, w h i c h made it suitable for measuring angles. T h e dioptra i n this f o r m was used i n sur­ veying a n d i n measuring the heights o f m o u n t a i n s . It p l a y e d a role similar to that o f the m o d e r n surveyor's transit or theodolite. T h e most detailed extant discussion o f these surveying instruments is H e r o o f A l e x a n d r i a ' s Dioptra (first century A . D . ) . N O ancient dioptra has survived. M o s t dioptras must have been m u c h simpler t h a n the elaborate instrument de­ scribed by H e r o , a n d conjecturally reconstructed i n fig. 1.11. Yet another sort o f dioptra, described by Ptolemy, was used for measur­ ing the angular sizes o f the Sun a n d the M o o n . (See fig. 1.12.) T h e ob­ server l o o k s t h o u g h a small hole i n a plaque. A m o v a b l e cylinder is then slid a l o n g a r o d u n t i l the cylinder just barely covers the Sun or M o o n . Ptolemy makes it clear that H i p p a r c h o s h a d used a similar instrument i n 8 3

8 4

See Lewis (2001), which includes an English translation of Hero's description of the instrument, pp. 2 5 9 - 6 2 . Ptolemy, Almagest ν 4. The most detailed description of this instrument is given by Proklos, Sketch of Astronomical Hypotheses iv 87-99. 8 3

8 4

40

·

Section 8

Fig. 1.11. A conjectural reconstruction of Hero's dioptra. From H . Schône 1903. The reconstruction is based on Hero's own description, but also reflects nineteenth-century German engineering!

the second century B.C. M o r e o v e r , f r o m a brief r e m a r k i n his Sand Reck­ oner, it seems that A r c h i m e d e s used essentially the same instrument i n the t h i r d century B . C . T h u s the w o r d dioptra covers a b r o a d class o f instruments, f r o m quite simple to relatively sophisticated. T o g a i n an idea o f w h a t sorts o f diop85

8 5

Dijksterhuis 1987, 364-65.

Astronomical Instruments and Models

·

41

Fig. 1.12. A dioptra for measuring the angular diameters of the Sun and M o o n . The figure shows only a short length of the instrument, which would have extended much farther to the right. Hipparchos used a similar instrument that was 4 cubits long. tras G e m i n o s h a d i n m i n d , let us see w h a t uses he makes o f t h e m . A t i 4, he says that it is possible to divide the z o d i a c i n t o twelve equal parts by means o f the dioptra. H e r e , he is m a k i n g a p o i n t a b o u t the c o n v e n t i o n a l character o f the z o d i a c signs, i.e., that they are artificial 3 0 ° segments o f the z o d i a c . A n instrument rather like H e r o ' s w o u l d suffice for this pur­ pose. T h e table (the large r o u n d disk c a r r y i n g the sights at the t o p o f fig. 1.11) must be adjusted to lie i n the plane o f the ecliptic. O n e c o u l d then, i n p r i n c i p l e , t u r n the sights to m a r k off 3 0 ° segments. B u t it must be stressed that G e m i n o s is a p p e a l i n g to a process that c o u l d be imagined, a n d that his readers w o u l d therefore grant. T h i s process w o u l d have n o real a s t r o n o m i c a l utility. A t ν 11 G e m i n o s says that it is possible to trace out the celestial cir­ cles ( i n c l u d i n g the t r o p i c a n d arctic circles) by means o f the dioptra. A t x i i 4 he says that " a l l the stars observed t h r o u g h the dioptras are seen to be m a k i n g a c i r c u l a r m o t i o n d u r i n g the w h o l e r o t a t i o n o f the diop­ tras." B o t h o f these appeals require a m o r e a s t r o n o m i c a l l y specialized i n s t r u m e n t t h a n the s u r v e y i n g i n s t r u m e n t described by H e r o . It must be possible to fix the sighting tube at a given angle to the a x i s o f the cos­ m o s . A n d it m u s t be possible to rotate the w h o l e i n s t r u m e n t a b o u t the axis. T h i s is quite a n interesting a p p e a l o n G e m i n o s ' s part, since, as far as w e k n o w , n o other ancient w r i t e r m e n t i o n s a dioptra o f precisely this sort. T h e general p r i n c i p l e o f this instrument, w h i c h w e m i g h t c a l l the e q u a t o r i a l dioptra, is suggested i n fig. 1.13. T h e table is adjusted to lie i n the plane o f the celestial equator. T h u s the table must face t o w a r d the n o r t h a n d must m a k e a n angle w i t h the h o r i z o n that is e q u a l to the

42

·

Section 8

Fig. 1.13. A conjectural reconstruction of Geminos's equatorial dioptra.

observer's co-latitude. W i t h this o r i e n t a t i o n achieved, the table lies at right angles to the axis o f the c o s m o s . O n t o p o f the table is a d i s k that is free to t u r n a b o u t a p i v o t P. T o this d i s k is attached the s u p p o r t i n g m e c h a n i s m for a sighting tube OE. T h e sighting tube has a p i v o t at G a n d a m o v a b l e c l a m p at E. T h u s Ε m a y be slid up a n d d o w n a r m EE a n d c l a m p e d for any desired p o l a r angle a . T h e observer l o o k s t h r o u g h the tube at O , aims at a star S that is n o r t h o f the celestial equator, a n d c l a m p s the tube i n t o p o s i t i o n . T h e n , i n the course o f the night, the o b ­ server c a n keep S i n sight s i m p l y by t u r n i n g the sighting a p p a r a t u s a b o u t p i v o t P. T h e p o l a r angle o r the d e c l i n a t i o n o f the star c o u l d be read f r o m a p p r o p r i a t e m a r k i n g s a l o n g EE. T o sight a star that is s o u t h o f the celestial equator, the observer s i m p l y l o o k s t h r o u g h the other e n d , £ , o f the sighting tube.

Geminos on Mathematical Genres

·

43

Pure mathematics (concerned with mental objects only) Arithmetic (study of odds, evens, primes, squares, etc.) Geometry Plane geometry Solid geometry Applied mathematics (concerned with perceptible things) Logistic (practical calculation, which is analogous to arithmetic) Geodesy (practical mensuration, which is analogous to geometry) Canonics (theory of musical scales, an offspring of arithmetic) Optics (an offspring of geometry) Optics proper (straight rays, shadows, etc.) Catoptrics (mirrors, etc.) Scenography (perspective) Mechanics Military engineering (engines of war) Wonder-working (pneumatics applied to automata) Equilibrium and centers of gravity (theory of the lever, etc.) Sphere-making (sphairopoiïa=models of the heavens) Astronomy Gnomonics (sundials) Meteoroscopy (armillary spheres) Dioptrics (the dioptra and related instruments) Fig. 1.14. Geminos's branches of the mathematical arts. 9. G E M I N O S O N M A T H E M A T I C A L G E N R E S

G e m i n o s offers testimony to the i m p o r t a n c e o f genres i n m a t h e m a t i c a l w r i t i n g , i n a passage o f his Philokalia that discusses the subdivisions of pure a n d a p p l i e d mathematics. A l t h o u g h G e m i n o s ' s b o o k has not come d o w n to us, his discussion o f the o r g a n i z a t i o n o f the m a t h e m a t i c a l sciences is preserved by P r o k l o s i n his Commentary on the First Book of Euclid's Elements. T h i s passage is translated b e l o w as fragment 1. G e m i nos's o r g a n i z a t i o n o f the m a t h e m a t i c a l sciences is illustrated i n fig. 1.14. M a t h e m a t i c s is d i v i d e d first o f a l l into w h a t we m i g h t call the pure a n d the a p p l i e d . B u t G e m i n o s ' s o w n d i s t i n c t i o n is P l a t o n i c — t h a t one b r a n c h is concerned " w i t h m e n t a l things o n l y , " i n abstraction f r o m m a terial forms, w h i l e the other deals w i t h "perceptible things." O f sciences concerned w i t h m e n t a l objects he gives t w o examples: arithmetic a n d geometry. It is i m p o r t a n t to note that, for the G r e e k s , arithmetic (arithmëtikë) was n o t the elementary c o m p u t a t i o n taught i n the schools.

44

·

Section 9

T h i s d i s t i n c t i o n w a s made at a n early date, for P l a t o distinguishes the art of c a l c u l a t i o n f r o m the c o n t e m p l a t i o n o f pure n u m b e r . W h a t the G r e e k s called arithmetic w e m i g h t c a l l n u m b e r theory. T h u s a r i t h m e t i c is c o n c e r n e d w i t h the classes a n d properties o f w h o l e numbers a n d their re­ lationships to one another (for e x a m p l e , o d d , even, p r i m e , a n d perfect n u m b e r s , as w e l l as figured n u m b e r s such as the triangular, square, a n d pentagonal). G r e e k arithmetic o r i g i n a t e d w i t h the P y t h a g o r e a n t r a d i ­ t i o n , but b o o k s v i i - i x o f E u c l i d ' s Elements constitute the oldest s u r v i v i n g treatment o f the subject. T h e first part o f T h e ó n o f Smyrna's Mathemati­ cal Knowledge Useful in Reading Plato, is a survey o f arithmetic, w i t h ­ o u t p r o o f s , intended for b e g i n n e r s . 86

87

O f the sciences concerned w i t h perceptible things, G e m i n o s gives s i x examples: logistic, geodesy, c a n o n i c s , optics, mechanics, a n d astronomy. L o g i s t i c (logistikë) is the art o f p r a c t i c a l c o m p u t a t i o n . A s a subject o f i n ­ s t r u c t i o n i n the schools, logistic was c o n c e r n e d w i t h a d d i t i o n , subtrac­ t i o n , m u l t i p l i c a t i o n , a n d d i v i s i o n , i n c l u d i n g the h a n d l i n g o f fractions. E x a m p l e s o f s c h o o l p r o b l e m s are p r o v i d e d by the a r i t h m e t i c a l epigrams o f the Greek Anthology. A n d H e a t h argued that, despite its title, the Arithmetica o f D i o p h a n t o s is a w o r k o f l o g i s t i c . G e o d e s y (geodesia) is the science of m e a s u r i n g surfaces a n d v o l u m e s , i n c l u d i n g surveying. E x t a n t w o r k s by H e r o o f A l e x a n d r i a p r o v i d e g o o d examples o f this genre. In Dioptra, H e r o describes the c o n s t r u c t i o n o f a surveying instrument that m a y be used for leveling a n d for m e a s u r i n g angles. T h e a p p l i c a t i o n of the instrument to v a r i o u s surveying p r o b l e m s is treated i n some detail. A n o t h e r o f H e r o ' s w o r k s , Stereometrica, is de­ v o t e d to m e n s u r a t i o n . H e r o begins w i t h t e x t b o o k p r o b l e m s , i n v o l v i n g spheres, cones, a n d p y r a m i d s , but progresses to m o r e p r a c t i c a l p r o b ­ lems, i n c l u d i n g the c a l c u l a t i o n o f the seating capacity o f a theater f r o m the lengths of its highest a n d lowest r o w s . A c c o r d i n g to G e m i n o s , c a l c u l a t i o n a n d geodesy are o n l y " a n a l o g o u s " to arithmetic a n d geometry, since they d o not m a k e p r o p o s i t i o n s a b o u t m e n t a l numbers or figures, but rather a b o u t perceptible ones. T h u s , geo­ desy does not deal w i t h ideal cylinders or cones, but rather w i t h heaps as cones, a n d w i t h pits as c y l i n d e r s . A n d it does not use m e n t a l straight lines, but rather lines made perceptible. These perceptible lines c a n be rather precise representations o f ideal lines (e.g., rays o f sunlight) o r rather crude ones (ropes). Similarly, the c a l c u l a t o r (logistikos) does not consider the properties o f n u m b e r s i n themselves, but rather as present i n perceptible things. 88

89

8 6

8 7

88

8 9

Plato, Republic vii, 525a-26c. See also Heath 1921, vol. 1, 65-117. Greek Anthology xiv, in Paton 1918, vol. 5. Heath 1921, vol. 1, 15-16.

Geminos on Mathematical Genres

·

45

In a s i m i l a r but s o m e w h a t different way, G e m i n o s considers canonics a n d optics to be " o f f s p r i n g " o f arithmetic a n d geometry, respectively. C a n o n i c s (kanonikê) is the m a t h e m a t i c a l theory o f the m u s i c a l scale. A c a n o n (kanôn) was o r i g i n a l l y a rule o r line, as used by carpenters a n d masons. It is also the usual t e r m for the geometer's straightedge. T h e n , by metaphor, it refers to any n o r m o r standard. T h e stretched string, o r m o n o c h o r d , w a s the kanôn, o r basis, o f the theory o f m u s i c a l intervals. T h u s canonics is the theory o f the d i v i s i o n o f the octave. C a n o n i c s is a n offspring o f arithmetic because it is concerned w i t h r a t i o — b u t r a t i o made perceptible, either to the eye i n the f o r m o f the lengths of strings, or to the ear i n the aspect o f h a r m o n y . T h e oldest s u r v i v i n g treatise is the Division of the Canon, w h i c h is attributed to E u c l i d , a l t h o u g h this attrib u t i o n has been c h a l l e n g e d . T h e second part o f T h e ó n o f Smyrna's Mathematical Knowledge Useful for Reading Plato, is a n i n t r o d u c t i o n to canonics. 90

O p t i c s is a n offspring o f geometry, because it is concerned w i t h lines made perceptible. U n d e r optics, G e m i n o s ranges three s u b d i v i s i o n s . O p tics proper is concerned w i t h effects that m a y be e x p l a i n e d i n terms o f the straight-line p r o p a g a t i o n o f the v i s u a l ray. T h e E u c l i d e a n Optics (third century B.C.) is the p r o t o t y p e o f this genre. A m o n g its p r o p o s i tions w e find: p a r a l l e l lines seen f r o m a distance appear to converge (prop. 6), a n d a circle v i e w e d o b l i q u e l y appears flattened (prop. 36). T h e Optics has c o m e d o w n to us i n t w o forms. H e i b e r g conjectured that one is the o r i g i n a l , E u c l i d e a n v e r s i o n a n d the other a revised v e r s i o n due to T h e ó n o f A l e x a n d r i a (fourth century A . D . ) . T h u s it appears that this b r a n c h o f a p p l i e d science r e m a i n e d a part o f the mathematics c u r r i c u l u m for seven centuries. 91

T h e second s u b d i v i s i o n of optics is catoptrics (katoptrike), the theory of m i r r o r s (katoptron = " m i r r o r " ) . T h e prototype is the pseudo-Euclidean Catoptrics, w h i c h treats thirty propositions i n v o l v i n g plane, convex, a n d concave m i r r o r s . O t h e r i m p o r t a n t writers o n catoptrics include H e r o of A l e x a n d r i a a n d P t o l e m y , w h o also investigated refraction. T h e final s u b d i v i s i o n o f optics is scenography o r "scene-painting." Skênographikë is perspective. It is of use i n the p r o d u c t i o n of theater sets of pleasing a n d c o n v i n c i n g p r o p o r t i o n . N o G r e e k treatise o n scenogra92

93

For a recent study, see Bowen, "Euclid's Sectio canonis and the History of Pythagoreanism," in Bowen 1991. For Ptolemy's treatment of the subject, see Barker 2000. Both versions are found in Heiberg's text and in Ver Eecke's translation. M o r e recently, scholars have parted with Heiberg over which version is the older: see Jones 1994. The Catoptrics often attributed to Hero survives only in a medieval Latin translation, in which the work is wrongly ascribed to Ptolemy. Jones 2001b provides an edition and English translation and discusses the evidence for and against the attribution to Hero. For an English translation of Ptolemy's optical treatise, see Smith 1996. 9 0

9 1

9 2

9 3

46

·

Section 9

p h y is extant, but b o t h A n a x o g o r a s a n d D ë m o k r i t o s are said to have w r i t t e n o n this subject. H e r o o f A l e x a n d r i a ' s Mechanics is a g o o d e x a m p l e o f a treatise o n general mechanics. T h e G r e e k text has not c o m e d o w n to us, but the w o r k survives i n a m e d i e v a l A r a b i c t r a n s l a t i o n . H e r o ' s treatise deals w i t h gear ratios, the p a r a l l e l o g r a m o f forces, centers o f gravity, as w e l l as w i t h the basic m a c h i n e s — w i n d l a s s , lever, pulley, wedge, a n d screw, b o t h alone a n d i n c o m b i n a t i o n . B o o k v i i i o f Pappos's Mathematical Collection is a substantial treatise o n G r e e k m e c h a n i c s . A n d V i t r u v i u s gives a brief o v e r v i e w o f mechanics, based largely o n G r e e k sources, i n c l u d i n g a p p l i c a t i o n s to h o i s t i n g machines a n d w a t e r w h e e l s . 94

95

96

97

G e m i n o s points to m i l i t a r y engineering as a s u b d i v i s i o n of mechanics. T h i s art m a y be illustrated by H e r o o f A l e x a n d r i a ' s treatise o n catapults, Belopoiïkê ( m i s s i l e - m a k i n g ) . V i t r u v i u s is another g o o d source o n m i l itary a p p l i c a t i o n s o f mechanics. G e m i n o s ' s second s u b d i v i s i o n o f mechanics is w o n d e r - w o r k i n g . Thaumatopoiïkë, the " w o n d e r - m a k i n g " (art or science), was the craft o f dev i s i n g a u t o m a t a a n d other gadgets, often operated by means o f fluid o r air pressure. E x a m p l e s : m e c h a n i c a l b l a c k b i r d s that sing by means o f w a t e r w o r k s ; a p n e u m a t i c a l l y operated toy temple w h o s e d o o r s o p e n w h e n a fire is lit o n the altar; a sacrificial vessel f r o m w h i c h l i q u i d flows o n l y w h e n m o n e y is i n t r o d u c e d . M a n y such contrivances are described by H e r o i n his Pneumatics, w h i c h m a y be considered t y p i c a l o f this genre. Few, i f any, o f these inventions were of p r a c t i c a l use. T h e y were s m a l l scale toys intended to amuse a n d amaze. A l t h o u g h the theoretical discuss i o n is often defective, the Pneumatics nevertheless demonstrates a n i m pressive mastery o f hydrostatics i n concrete a p p l i c a t i o n s . 98

9 9

T h e earliest s u r v i v i n g w o r k o f the science o f e q u i l i b r i u m a n d centers of g r a v i t y is that o f A r c h i m e d e s (third century B.C.). In his treatise On Vitruvius, On Architecture v i i , introduction, 11. Heath (1921, vol. 1, 178) lists among Dëmokritos's works one titled Ekpetasmata, a w o r d which, he notes (p. 181), Ptolemy glosses in his Geography as referring to the projection of the armillary sphere on the plane. This is, of course, just a special case of the representation of three-dimensional bodies on a two-dimensional surface that scene painting would require, so Dëmokritos's Ekpetasmata may be the work to which Vitruvius refers. The Arabic text of Pappos's Book viii, the only one that seems to have been translated into Arabic, contains material that dropped out in the course of transmission of the Greek text. Unhappily, no translation of the Arabic version has yet been published. Vitruvius, On Architecture χ 1-9. O n ancient mechanics, see Landels 1978 or, more briefly, Lloyd 1973, 91-112. This treatise was omitted from the Teubner edition of Hero's works, but is available in English translation in Marsden 1971. Vitruvius, On Architecture χ 10-16. 94

95

96

97

98

99

Geminos on Mathematical Genres

·

47

the Equilibrium of Flanes™ A r c h i m e d e s demonstrates the l o c a t i o n s o f the centers o f g r a v i t y o f v a r i o u s plane figures, such as triangles, p a r a l l e l o g r a m s , a n d a p a r a b o l i c sector, a n d attempts a p r o o f o f the l a w o f the lever. T w o other w o r k s o f A r c h i m e d e s o n this science are lost: On Levers a n d On Centers of Gravity. S p h e r e - m a k i n g (sphairopoiïa) is the science o f c o n s t r u c t i n g m o d e l s o f the heavens, i n c l u d i n g celestial globes, a r m i l l a r y spheres, a n d orreries. A r e p u t a t i o n for b r i l l i a n c e i n this craft attached to the name o f A r c h i m e d e s . P a p p o s o f A l e x a n d r i a says that A r c h i m e d e s c o m p o s e d a treatise o n this subject, w h i c h appears, however, to have been lost already by Pappos's time. W h e n the R o m a n s c a p t u r e d Syracuse, M a r c e l l u s t o o k t w o o f A r c h i m e d e s ' devices b a c k to R o m e . O n e o f these seems to have been a celestial globe, w h i c h M a r c e l l u s p l a c e d i n the temple o f V e s t a , where it r e m a i n e d l o n g e n o u g h to be seen by O v i d . F r o m the a c c o u n t by C i cero, the other i n s t r u m e n t w a s a n o r r e r y that represented the m o t i o n s o f the S u n , M o o n , a n d p l a n e t s . T h i s orrery o f A r c h i m e d e s must have been quite a m a r v e l , for C i c e r o expresses d i s a p p r o v a l o f some " w h o t h i n k m o r e h i g h l y o f the achievement o f A r c h i m e d e s i n m a k i n g a m o d e l o f the revol u t i o n s o f the firmament t h a n that o f nature i n creating t h e m , a l t h o u g h the perfection o f the o r i g i n a l shows a craftsmanship m a n y times as great as does the c o u n t e r f e i t . " 101

1 0 2

1 0 3

104

105

U n d e r a s t r o n o m y , G e m i n o s ranges three s u b d i v i s i o n s , w h i c h are a l l c o n c e r n e d w i t h instruments o f some k i n d . G n o m o n i c s (gnômonikë) is the science o f designing a n d c o n s t r u c t i n g sundials. T h e earliest treatises o n this science are lost. V i t r u v i u s gives a list o f d i a l types, together w i t h the names o f their inventors, at least some o f w h o m p r e s u m a b l y w r o t e short treatises o n their i n v e n t i o n s . V i t r u v i u s also sets out the 1 0 6

100 Available in translation in Heath 1912 and Dijksterhuis 1987. Berggren 1976 argues that the extant version of this treatise contains introductory material (including an unsuccessful proof of the law of the lever) that was inserted by a later editor who wished to turn a research treatise of Archimedes into a text suitable for instruction. See Heath 1912, x x x v i i . Pappos, Mathematical Collection viii, 3; Hultsch, p. 1026; Ver Eecke, p. 813. Plutarch {Marcellus xvii 3-4), promoting his o w n philosophical line, says that Archimedes did not deign to leave behind any treatise on the mechanic arts, judging them ignoble and vulgar. Apparently, he made an exception in the case of sphairopoiïa, perhaps because of the nobility the subjects it seeks to represent. O v i d , Fasti vi 2 7 7 - 8 0 . Cicero, De república i 14. The more elaborate of Archimedes' two constructions was probably a gearwork mechanism. There are extant two ancient gearwork mechanisms for reproducing celestial motions: see Price (1974) and Field and Wright (1984). Both of these are, however, from well after Archimedes' time. Cicero, On the Nature of the Gods i i 35. 106 Vitruvius, On Architecture i x 8.1. 1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

48

·

Section 9

figure o f the a n a l e m m a , w h i c h is a f o l d i n g d o w n of the celestial sphere o n t o the plane, a n d w h i c h permits the c o n s t r u c t i o n o f the h o u r lines o f a s u n d i a l by techniques o f projective geometry. T h e a n a l e m m a described by V i t r u v i u s is certainly not o f his o w n i n v e n t i o n , but l i k e l y goes back at least to the t h i r d century B.C., w h e n G r e e k sundials began to appear i n substantial numbers. T h e other i m p o r t a n t s u r v i v i n g ancient w o r k o n g n o m o n i c s is Ptolemy's On the Analemma. 107

T h e b r a n c h o f a s t r o n o m y c a l l e d meteoroscopy (meteôroskopikê) s h o u l d not be confused w i t h m e t e o r o l o g y (meteorología). T h e w o r d meteoroscopy means " l o o k i n g at things o n h i g h , " a n d it almost certainly refers to the art o f m a k i n g a n d using a specialized instrument of observat i o n rather s i m i l a r to an a r m i l l a r y sphere. N o t e that G e m i n o s ' s other t w o branches o f a s t r o n o m y are also devoted to instruments—sundials a n d dioptra. In the Geography, Ptolemy uses meteoroskopion for a n instrument for t a k i n g celestial observations, a n d says that he has given its d e s c r i p t i o n . P r o k l o s uses meteoroskopeion for w h a t is some sort o f i m p r o v e m e n t o n Ptolemy's a r m i l l a r y astrolabe, described i n Almagest ν 1. A n d P a p p o s provides m o r e details for the c o n s t r u c t i o n o f the meteoroscope t h a n P t o l e m y gave i n his d e s c r i p t i o n of the astrolabe i n s t r u ­ ment i n the Almagest. F r o m a l l this it seems clear that P t o l e m y h a d w r i t t e n a short specialized treatise o n the c o n s t r u c t i o n o f a meteorosko­ pion. T h a t G e m i n o s already considered meteoroscopy a genre o f w r i t i n g s h o w s that the spherical instruments go back at least to his time. 1 0 8

1 0 9

1X0

F i n a l l y , d i o p t r i c s (dioptrike) is the science o f using the dioptra, a sight­ i n g a n d m e a s u r i n g instrument, several forms of w h i c h are described above. T h a t G e m i n o s ' s branches o f mathematics represent, not mere abstract categories, but actual genres is clear f r o m the fact that w e possess, o r at least have k n o w l e d g e of, actual G r e e k m a t h e m a t i c a l w o r k s that fit every one o f t h e m . N a t u r a l l y , each genre o f m a t h e m a t i c a l w r i t i n g h a d its o w n customs a n d c o n v e n t i o n s . M a n y o f these genres h a d l o n g lives. F o r ex­ a m p l e , w e have w o r k s o n optics by E u c l i d a n d by Ptolemy, w h o were separated by nearly 5 0 0 years. B u t , a l t h o u g h a l l the categories listed by G e m i n o s are true genres of m a t h e m a t i c a l w r i t i n g , it is clear that he has not listed every genre. T h e r e is, for e x a m p l e , n o t h i n g to c o r r e s p o n d to m a t h e m a t i c a l planetary theory (although the Introduction to the Phe­ nomena makes it clear that he k n e w o f such theories), for G e m i n o s ' s branches o f a s t r o n o m y are a l l c o n c e r n e d w i t h instruments o f some sort. 111

Ptolemy, Opera i i , 187-223. See Neugebauer 1975, 839-56. ios Ptolemy, Geography i 3. Proklos, Sketch of Astronomical Hypotheses vi 2. For a full discussion, see Rome 1927. For the text, see Rome 1931, 1-16. See K n o r r 1986 and Mansfeld, 1998. 1 0 7

1 0 9

1 1 0

1 1 1

Reality and Representation

·

49

1 0 . R E A L I T Y A N D R E P R E S E N T A T I O N IN G R E E K A S T R O N O M Y Hypotheses

and

Phenomena

In chapter i o f the Introduction to the Phenomena, G e m i n o s makes some r e m a r k s a b o u t the role o f hypotheses i n a c c o u n t i n g for the p h e n o m e n a , for w h i c h w e w o u l d like to p r o v i d e a context. A t i 1 9 , he says that "the hypothesis that underlies the w h o l e o f a s t r o n o m y is that the S u n , the M o o n , a n d the five planets m o v e circularly a n d at constant speed i n the d i ­ r e c t i o n opposite to that o f the c o s m o s . " M o r e o v e r , he attributes to the Pythagoreans the p r i n c i p l e that u n i f o r m c i r c u l a r m o t i o n is p r o p e r to ce­ lestial things. A t i 2 1 , still s p e a k i n g o f the Pythagoreans, G e m i n o s says, "they put f o r w a r d the question: h o w w o u l d the p h e n o m e n a be ac­ c o u n t e d for b y means o f u n i f o r m a n d c i r c u l a r m o t i o n s ? " O n the other h a n d , S i m p l i k i o s ( s i x t h c e n t u r y A.D.) a t t r i b u t e d to P l a t o the celebrated h o m e w o r k assignment for the a s t r o n o m e r s to "save the p h e n o m e n a " i n terms o f u n i f o r m c i r c u l a r m o t i o n s . B u t there is n o t h ­ i n g i n Plato's w o r k to c o n f i r m that he e n u n c i a t e d a p r i n c i p l e o f u n i f o r m c i r c u l a r m o t i o n , a n d m u c h that is inconsistent w i t h it. Indeed, K n o r r ar­ gued that the a t t r i b u t i o n o f the p r i n c i p l e to P l a t o w a s a m i s t a k e m a d e by the late G r e e k c o m m e n t a t o r s . I n any case, w e find the first clear, s u r v i v i n g statement o f the p r i n c i p l e i n A r i s t o t l e . B y Geminos's time, the p r i n c i p l e that u n i f o r m c i r c u l a r m o t i o n is a p p r o p r i a t e to heavenly bodies h a d become a c o m m o n p l a c e o f p h i l o s o p h y . O t h e r a s t r o n o m i c a l w r i t e r s w h o endorse the p r i n c i p l e i n c l u d e T h e ó n o f S m y r n a , P t o l e m y , and Proklos. 1 1 2

113

1 1 4

1 1 5

O n e i m p o r t a n t reason for clearly stating "hypotheses," o r assump­ tions, is that a s t r o n o m y w a s c o m m o n l y regarded as a b r a n c h o f mathe­ m a t i c s , a n d p a r t i c u l a r l y o f geometry. T h e s t a n d a r d f o r m o f a g e o m e t r i c a l d e m o n s t r a t i o n i n v o l v e d the clear statement o f hypotheses. A n excellent a s t r o n o m i c a l e x a m p l e comes f r o m A r i s t a r c h o s o f Samos's treatise On the Sizes and Distances of the Sun and Moon, i n w h i c h a l l the necessary a s t r o n o m i c a l hypotheses are c l e a r l y stated before the d e m o n s t r a t i o n be­ gins. T h e debt o f a s t r o n o m y to m a t h e m a t i c a l c o n v e n t i o n s does n o t nec­ essarily m e a n that G r e e k astronomers t h o u g h t o f their hypotheses as Simplikios, Commentary on Aristotle's O n the Heavens, comments on ii 12. Heiberg 1894, 488.18-24 and 492.15-493.5. The two short passages from Simplikios have been quoted and discussed by many, including Duhem (1908, 3), Sambursky (1956, 59) and Vlastos (1975, 59-61 and 110-11). For an English translation of Simplikios's commentary, see Mueller 2005. Knorr 1989. Aristotle, On the Heavens i 2 and ii 3, 12. Theón of Smyrna, Mathematical Knowledge iii 22. Ptolemy, Almagest ix 2. Proklos, Sketch of Astronomical Hypotheses i 1.7-9. 112

113

114

115

50

·

Section 10

arbitrary, o r as free choices, but it does raise the question o f w h a t t r u t h value o r epistemic status they attached to t h e m . W h e n G e m i n o s asks h o w "the p h e n o m e n a w o u l d be accounted f o r " (apodotheiê ta phainomena), this expression carries the same sense as the m o r e famous p r e s c r i p t i o n " t o save the p h e n o m e n a " (sôzein ta phainomena). B u t G e m i n o s ' s v e r s i o n o f the saying m a y w e l l be the older. A l t h o u g h there is n o reason to take seriously S i m p l i k i o s ' s a t t r i b u t i o n o f this p r o g r a m to P l a t o , the expression "save the p h e n o m e n a " does c r o p u p i n late writers w i t h P l a t o n i s t affiliations, such as T h ë o n o f S m y r n a a n d P r o k l o s . B u t a c c o r d i n g to G o l d s t e i n , the oldest extant text i n w h i c h the expression "save the p h e n o m e n a " occurs is o n l y o f the first century A . D . , n a m e l y Plutarch's On the Face in the Orb of the Moon. (We s h o u l d p o i n t out that G e m i n o s t w i c e uses the expression "save the p h e n o m e n a i n " fragment 2 , but w e c a n n o t exclude the p o s s i b i l i t y that this is a n i n t e r p o l a t i o n by S i m p l i k i o s , w h o is q u o t i n g him.) W r i t i n g i n the f o u r t h century B.C., A r i s t o t l e used apodósein ("to give b a c k , " o r " t o render a n account o f " ) w h e n he p o n d e r e d h o w the spheres o f E u d o x o s c o u l d best be m o d i f i e d " t o account for the p h e n o m e n a . " 1 1 6

117

118

T h e p r o g r a m of saving, o r a c c o u n t i n g for, the phenomena has given rise to a large m o d e r n literature o f interpretation, w h i c h bears strongly o n the question o f whether Greek a s t r o n o m y s h o u l d be thought of as a realist o r an instrumentalist endeavor. B y scientific realism w e m e a n the v i e w that a successful scientific theory contains at least some elements o f the true nature o f the w o r l d . A realist believes that one g o a l of science is to discover w h a t the w o r l d really is, a n d that at least some features o f a g o o d theory c o r r e s p o n d to things that truly exist i n nature. Scientific realism c a n be contrasted w i t h instrumentalism. In an instrumentalist a p p r o a c h to science, the practitioner m a y renounce (perhaps as an impossibility) the g o a l of ever discovering the real nature of things, a n d m a y be content to have a theory that " w o r k s , " i.e., that successfully accounts for k n o w n p h e n o m ena a n d a l l o w s the p r e d i c t i o n o f p h e n o m e n a yet to be investigated. T h e instrumentalist w o u l d not c l a i m that the i n d i v i d u a l features of a successful theory necessarily c o r r e s p o n d to things really existing i n nature. A t the b e g i n n i n g of the twentieth century, Pierre D u h e m constructed a n instrumentalist interpretation o f G r e e k a s t r o n o m y that has been very influential. In D u h e m ' s account, the testimony of G e m i n o s (in fragment 2) a n d o f P r o k l o s p l a y e d key roles. D r e y e r and Sambursky, t o o , represented G e m i n o s as an instrumentalist, w h o was c o n c e r n e d 119

1 2 0

1 1 6

1 1 7

1 1 8

1 1 9

1 2 0

1 2 1

Goldstein 1997, 7. Plutarch, On the Face in the Orb of the Moon 923A. Aristole, Metaphysics 1073b38 and 1 0 7 4 a l . Duhem 1908. Dreyer 1906, 132. Sambursky 1962, 135-37.

121

Reality and Representation

·

51

o n l y w i t h saving the p h e n o m e n a , a n d w h o r e n o u n c e d any attempt to find the true arrangement o f the cosmos. M o r e recently, L l o y d has argued that D u h e m misinterpreted some k e y evidence, a n d that a l l the G r e e k astronomers about w h o m w e k n o w enough to m a k e a judgment (including G e m i n o s ) were i n fact r e a l i s t s . In any case, D u h e m ' s picture of G r e e k a s t r o n o m y as a seamlessly d e v e l o p i n g science, w i t h a single set of goals that lasted f r o m the time o f P l a t o to the close o f A n t i q u i t y , is i n need o f strong c o r r e c t i o n . F o r the p h e n o m e n a judged to be i n need o f saving changed w i t h time as G r e e k a s t r o n o m y m a t u r e d . C o n c e r n w i t h the b r o a d features o f planetary p h e n o m e n a l o n g preceded c o n c e r n w i t h quantitative details o f planetary p o s i t i o n s . 122

123

M o r e o v e r , G r e e k a s t r o n o m y often exhibits a split personality, because of its o p p o r t u n i s m , a n d m a n y scholars have overestimated its consistency a n d allegiance to p h i l o s o p h i c a l p r i n c i p l e s . G r e e k writers steeped i n the p h i l o s o p h i c a l t r a d i t i o n w r o t e accounts o f the planets based o n the c i r c u l a r m o t i o n s of epicycles a n d deferents. A g o o d e x a m p l e is Theón of Smyrna's Mathematical Knowledge Useful for Reading Plato. H o w e v e r , i n T h e ó n 's day (early second century A . D . ) , the G r e e k geometrical theories were not capable o f y i e l d i n g accurate planet p o s i t i o n s . G r e e k astrologers, w h o needed convenient a n d reasonably accurate methods o f c a l c u l a t i n g planet p o s i t i o n s , relied instead o n arithmetic methods borr o w e d f r o m the B a b y l o n i a n s , a n d these methods h a d nonuniform mot i o n a r o u n d the z o d i a c b u i l t into t h e m . It w a s o n l y w i t h Ptolemy's p u b l i c a t i o n o f his planetary tables i n the Almagest that the G r e e k geometrical theories became capable o f p r o d u c i n g accurate planet positions. T h e n e w feature that made Ptolemy's planetary models quantitatively accurate was his equant p o i n t , w h i c h effectively i n c o r p o r a t e d n o n u n i f o r m motion. G e m i n o s exhibits the h a p p y coexistence of p h i l o s o p h i c a l p r i n c i p l e a n d arithmetic convenience that was characteristic of G r e e k ast r o n o m y i n his day. T h u s , i n spite o f his endorsement i n chapter i o f the p r i n c i p l e of u n i f o r m c i r c u l a r m o t i o n , i n chapter x v i i i he discusses the B a b y l o n i a n l u n a r theory, w h i c h involves n o n u n i f o r m m o t i o n m o d e l e d by an arithmetic progression. 124

1 2 5

Sphairopoiïa

as World and

Representation

Nevertheless, i n the Introduction to the Phenomena, as w e l l as i n fragment 2 , G e m i n o s provides us w i t h strong evidence of the p o s i t i o n he Lloyd 1978, with additions in Lloyd 1991, 248-77. O n these issues, see also Mittelstrass 1962, Goldstein 1980, Aiton 1981, Smith 1981, Smith 1982, Lloyd 1987, 285-336, Goldstein 1997, Evans 1998, 216-19, Bowen 2001, and Bowen 2003. Jones 1999b. See Evans 1984. 122

123

124

125

52

·

Section 10

w o u l d have t a k e n i n a debate about r e a l i s m a n d i n s t r u m e n t a l i s m ( w h i c h are not, o f course, ancient terms) i n astronomy. T o begin w i t h , w e c a n learn a g o o d deal f r o m his discussions of sphairopoua, the art o f m o d e l i n g the c o s m o s . Sphairopoua is a w o r d that exhibits an interesting range o f meanings. A s w e have seen, G e m i n o s uses it i n fragment 2 to indicate (1) a branch of mechanics—the art of c o n s t r u c t i n g w o r k i n g models o f the heavens. M o r e o v e r , sphairopoua was a genre of mathematical writing that w e n t back at least to A r c h i m e d e s . A n d 6 0 0 years later, Pappos o f A l e x a n d r i a still treated it as a genre. B u t sphairopoua c a n also m e a n (2) a particular mechanical model. F o r e x a m p l e , T h ë o n of S m y r n a says that he a c t u a l l y made a sphairopoiïan o f the nested spindle w h o r l s described by P l a t o i n his mystic v i s i o n o f the planetary s p h e r e s . 1 2 6

127

In the Introduction to the Phenomena, G e m i n o s uses sphairopoua to m e a n (3) a spherical theory of the world, w h i c h c a n be said to be i n agreement w i t h nature. A t x v i 2 8 - 2 9 , G e m i n o s remarks that " H o m e r a n d the ancient poets" believed the E a r t h to be flat, e x t e n d i n g a l l the w a y to the sphere of the c o s m o s . Consequently, a c c o r d i n g to these poets, the Aithiopians near the rising as w e l l as those near the setting are b u r n e d by the S u n . G e m i n o s says, " T h i s n o t i o n is consistent w i t h their p r o p o s e d arrangement, but alien to the spherical c o n s t r u c t i o n (sphairopoua) i n acc o r d w i t h nature." M o s t interestingly, G e m i n o s also uses sphairopoua to m e a n (4) a spherical system that actually exists in nature. A t x i i 2 3 , G e m i n o s says that "there is a certain spherical c o n s t r u c t i o n proper for each [planet]" that accounts for the m o t i o n s . A l t h o u g h G e m i n o s provides n o details a b o u t the sphairopoua a p p r o p r i a t e to the planets, he is p r o b a b l y t h i n k i n g o f epicycles a n d eccentrics e m b e d d e d i n three-dimensional spheres. T h a t he really means this as a c o n s t r u c t i o n existing i n nature is c o n firmed b y some of his other uses o f the same expression. A t x i v 9, he i n vokes the spherical c o n s t r u c t i o n (sphairopoua) o f the cosmos to e x p l a i n w h y the patterns o f risings a n d settings are not the same for a l l stars (e.g., the classes of stars c a l l e d doubly visible a n d night-escapers). At xvi 19, G e m i n o s i n v o k e s the spherical c o n s t r u c t i o n (sphairopoua) to s h o w that there exists another temperate zone i n the southern hemisphere o f 128

Theón of Smyrna, Mathematical Knowledge iii 16. Plato, Republic χ 614-17. Geminos's demonstration of the importance of genre in mathematical writing and his remark about sphairopoua that actually exist in nature bear on the history of threedimensional representations of deferent-and-epicycle theory. Evans 2003 argues that deferent-and-epicycle theory was introduced in three-dimensional form at its very be­ ginning. 1 2 6

1 2 7

1 2 8

Reality and Representation

·

53

the E a r t h . In these t w o passages, there is clearly n o question o f a " m o d e l " to save the p h e n o m e n a : G e m i n o s is speaking o f the sphairopoua o f the w o r l d itself. T h u s sphairopoua c a n m e a n (1) a b r a n c h o f mechanics; (2) a p a r t i c u ­ lar m e c h a n i c a l m o d e l ; (3) a theory o f the w o r l d , w h i c h c a n be said to be after, or a c c o r d i n g to, nature; or (4) the spherical arrangement o r system of the w o r l d itself ( i n c l u d i n g the spherical constructions that p r o d u c e the m o t i o n s o f the planets). These four uses are closely related a n d reveal a g r a d a t i o n i n m e a n i n g . A n d this is a sign o f the essential r e a l i s m o f G r e e k astronomy. N o better d e m o n s t r a t i o n c o u l d be w i s h e d o f G e m i n o s ' s o w n p o s i t i o n as a realist. F o r G e m i n o s , a s t r o n o m i c a l models are assertions about the nature o f the w o r l d : at x v i 2 7 , he speaks o f the " s p h e r i c a l c o n ­ struction that accords w i t h t r u t h [or r e a l i t y ] . "

Geminos's

Realism

G e m i n o s ' s realist stance is cogently developed i n a passage o f his Concise Exposition of the Meteorology of Poseidônios, p r i n t e d b e l o w as frag­ ment 2 . In this passage, G e m i n o s provides a discussion o f the r e l a t i o n ­ ship between a s t r o n o m y a n d physics that has m u c h to teach us about the goals a n d l i m i t a t i o n s o f a s t r o n o m y as the Greeks perceived them. Start­ ing f r o m a r e m a r k by A r i s t o t l e , G e m i n o s contrasts the methods of the astronomer a n d the physicist. Physics is concerned w i t h the very natures of the stars, their essential qualities, as w e l l as their o r i g i n a n d destruc­ t i o n . A s t r o n o m y , by contrast, is concerned w i t h such things as the size a n d shape o f the E a r t h , S u n , a n d M o o n , eclipses a n d conjunctions, a n d , generally, the m o t i o n s o f the heavenly bodies. A s t r o n o m y therefore must rely u p o n arithmetic a n d geometry for its demonstrations, rather t h a n o n arguments f r o m the nature o f things. 129

A s G e m i n o s points out, the astronomer a n d the physicist m a y address the same question, but w i l l proceed by different methods, a n d he gives t w o g o o d examples o f t h i s — p r o v i n g that the S u n is large, o r that the E a r t h is spherical. T h a t the Sun is large was a c o m m o n p r o p o s i t i o n i n b o t h G r e e k n a t u r a l p h i l o s o p h y a n d G r e e k astronomy. A m o n g the preSocratic p h i l o s o p h e r s , A n a x i m a n d r o s , Empedoklës, a n d A n a x a g o r a s a l l asserted that the S u n is large ( A n a x i m a n d r o s a n d Empedoklës that the Sun is as large as the E a r t h , a n d A n a x a g o r a s that it is larger t h a n the Peloponnesus). A n a s t r o n o m i c a l (or geometrical) m e t h o d o f d e t e r m i n i n g the size o f the S u n was first given by A r i s t a r c h o s o f Samos (third century B.C.). A l l the later G r e e k efforts, e.g., by H i p p a r c h o s a n d Ptolemy, were modifications o f Aristarchos's procedure. A r i s t a r c h o s ' s calculations led 129

Aristotle, Physics 193b22.

54

·

Section 10

to the c o n c l u s i o n that the r a t i o o f the diameter o f the S u n to the diame­ ter o f the E a r t h is greater t h a n 19/3 but less t h a n 4 3 / 6 : a large S u n , t h o u g h m u c h t o o s m a l l by m o d e r n s t a n d a r d s . A r i s t o t l e gives several arguments for the sphericity o f the E a r t h that nicely illustrate the differing approaches o f the physicist a n d the as­ tronomer. A r i s t o t l e begins w i t h a p h y s i c a l o r p h i l o s o p h i c a l argument, w h i c h he deems to be first i n force a n d i m p o r t a n c e . T h e E a r t h is spheri­ c a l because o f the center-seeking nature o f earth as an element: particles o f earth, s t r i v i n g to reach the center o f the universe, a n d c o m i n g t o w a r d that center f r o m a l l directions, n a t u r a l l y p r o d u c e a spherical shape. A r i s ­ totle does not, however, d i s d a i n to a d d three arguments based o n senseevidence. T h e first is based o n the shape o f the Earth's s h a d o w as seen d u r i n g a l u n a r eclipse. T h e second involves the change i n the v i s i b i l i t y o f certain stars as one moves f r o m E g y p t o r C y p r u s to m o r e n o r t h e r l y lands. T h e t h i r d argument f r o m sense-evidence is based o n elephants, w h i c h m a y be f o u n d at b o t h extremes o f the k n o w n w o r l d — i n India a n d i n the r e g i o n about the Pillars o f H e r c u l e s . T h e t w o a s t r o n o m i c a l argu­ ments are s o u n d ones a n d are clearly o f a different sort t h a n the p h y s i c a l argument based o n Aristotle's o w n theory o f n a t u r a l m o t i o n s . 130

131

M o r e o v e r , a c c o r d i n g to G e m i n o s , it is not for the astronomer to k n o w the causes o f things, o r to decide the true nature of the w o r l d . Rather, the a s t r o n o m e r must take his first principles f r o m the physicist, "that the m o t i o n s o f the stars are s i m p l e , u n i f o r m , a n d o r d e r l y . " B u t w e c a n n o t k n o w f r o m a s t r o n o m i c a l o b s e r v a t i o n whether the E a r t h goes a r o u n d the S u n o r the S u n goes a r o u n d the E a r t h , for the p h e n o m e n a c a n be saved (or a c c o u n t e d for) under either a s s u m p t i o n . In answer to the q u e s t i o n o f w h y the S u n , M o o n , a n d planets appear to m o v e irregularly, the as­ t r o n o m e r c a n o n l y say that, i f w e assume their circles are eccentric, or that the stars go a r o u n d o n epicycle, the apparent irregularity w i l l be ac­ c o u n t e d for. H e r e G e m i n o s is referring to the fact that there were t w o versions o f the G r e e k s ' g e o m e t r i c a l solar theory. F i g . 1.15 shows the eccentric-circle solar theory that G e m i n o s presents i n chapter i o f his Introduction. In the figure, w e l o o k d o w n o n the plane o f the ecliptic f r o m above its n o r t h pole. T h e S u n S moves at u n i f o r m speed o n a circle that is centered at a p o i n t C s o m e w h a t eccentric to the E a r t h O . T h u s angle α increases u n i f o r m l y w i t h time. In the solar theory of H i p p a r c h o s , w h o was apparently the first to specify n u m e r i c a l values for the parameters of this theory, the eccentricity O C is 4 . 1 5 % o f the r a ­ dius o f the circle. A l t h o u g h the S u n moves u n i f o r m l y o n its circle, to a n

O n Aristarchos, see Heath 1913. Van Helden 1985 gives a good brief account of several attempts by Greek astronomers to determine the size and distance of the Sun. Aristotle, On the Heavens 2 9 7 a 8 - 9 8 a l 5 . 1 3 0

1 3 1

Reality and Representation

·

55

Fig. 1.15. A n eccentric-circle model for the motion of the Sun. The Sun S moves uniformly around C in the course of the year. But because C is eccentric to the Earth O, the motion appears nonuniform to us.

observer at O it appears to m o v e m o r e s l o w l y o n some parts o f the circle a n d m o r e q u i c k l y o n others, because o f the v a r y i n g distance. T h e second v e r s i o n o f the solar theory is s h o w n i n fig. 1.16. T h e S u n S moves c l o c k w i s e a r o u n d a s m a l l epicycle, w h i l e the center Κ o f the epicycle moves c o u n t e r c l o c k w i s e a r o u n d a deferent circle that is c o n c e n ­ tric w i t h the E a r t h O . B o t h m o t i o n s are u n i f o r m a n d are c o m p l e t e d i n exactly 1 year. T h u s w e a l w a y s have β = α. It happens that the two theories are mathematically equivalent. See fig. 1.17, w h i c h examines the epicycle-plus-concentric theory more closely. Since β = α, line segment KS remains p a r a l l e l to O C as Κ moves a r o u n d the deferent. T h u s the p a t h a c t u a l l y described by the S u n is the circle s h o w n i n dashed line. If the radius KS o f the epicycle is chosen to be e q u a l to the eccentricity O C i n the eccentric-circle m o d e l , the t w o theo­ ries are g e o m e t r i c a l l y i n d i s t i n g u i s h a b l e . Apollónios o f Pergë p r o b a b l y p r o v e d this equivalence, a r o u n d 2 0 0 B . C . Both Theón of Smyrna and P t o l e m y give their o w n p r o o f s . A l t h o u g h the t w o forms o f the solar theory are m a t h e m a t i c a l l y equiv­ alent, nevertheless, a controversy arose over w h i c h c o r r e s p o n d e d to real­ ity. A c c o r d i n g to T h e ó n o f S m y r n a , H i p p a r c h o s expressed a prefer­ ence for the epicycle theory, saying that it w a s p r o b a b l e that the 1 3 2

133

1 3 4

1 3 2

1 3 3

1 3 4

See Neugebauer 1975, 262-65. Theôn of Smyrna, Mathematical Knowledge iii 26. Ptolemy, Almagest iii 3. Theón of Smyrna, Mathematical Knowledge iii 34.

56

·

Section 10

Fig. 1.16. A concentric-deferent-and-epicycle model for the motion of the Sun. The Sun S moves on an epicycle, while the center Κ of the epicycle moves around a deferent circle centered on the Earth O . Both motions are completed in 1 year, and angles β and α are always equal.

Fig. 1.17. Equivalence of the concentric-plus-epicycle model to the eccentriccircle model.

Reality and Representation

·

57

heavenly bodies are p l a c e d u n i f o r m l y w i t h respect to the center o f the w o r l d . Ptolemy, however, was later to endorse the eccentric-circle f o r m of the theory, saying that it w a s simpler, since it i n v o l v e d one m o t i o n rather t h a n t w o . Since the t w o models are m a t h e m a t i c a l l y equivalent, the choices o f H i p p a r c h o s a n d P t o l e m y for one or the other were clearly based o n criteria other t h a n agreement w i t h the p h e n o m e n a — c r i t e r i a that w e m a y describe as p h y s i c a l o r p h i l o s o p h i c a l . G e m i n o s goes o n to say, "It w i l l be necessary to fully examine i n h o w m a n y w a y s it is possible for these p h e n o m e n a to be b r o u g h t about, so that the treatment o f the planets m a y fit the causal e x p l a n a t i o n (aitiolo­ gia) that is i n a c c o r d w i t h acceptable m e t h o d . " T h e p r e s c r i p t i o n to seek m u l t i p l e geometrical e x p l a n a t i o n s echoes a r e m a r k o f H i p p a r c h o s , w h o said that research i n t o the e x p l a n a t i o n o f the same p h e n o m e n a by hy­ potheses that are quite different is a task w o r t h y o f the attention of the mathematician. A c c o r d i n g to G e m i n o s , the reason for d e v e l o p i n g a l l possible geometrical explanations is to be sure that w e w i l l find one that accords w i t h accepted p h y s i c a l p r i n c i p l e s , by p r o v i d i n g a causal e x p l a ­ n a t i o n o f the p h e n o m e n a . 1 3 5

136

The t e r m aitiologia, " c a u s a l e x p l a n a t i o n " or " i n v e s t i g a t i o n i n t o causes," comes f r o m G r e e k n a t u r a l p h i l o s o p h y , a n d the E n g l i s h w o r d " e t i o l o g y " descends f r o m it. Diogenes L a e r t i o s describes A r i s t o t l e as "ready at g i v i n g the cause," or " i n q u i r i n g i n t o causes," but Diogenes does so i n the superlative degree of the adjective (aitiologikôtatos), thus m e a n i n g that A r i s t o t l e w a s the most ready o f a l l p h i l o s o p h e r s to take up causes. A l s o a c c o r d i n g to Diogenes, aitiologikos was one o f three parts of Stoic physics a n d w a s concerned, i n part, w i t h the e x p l a n a t i o n of things such as v i s i o n , a n d w i t h g i v i n g the causes o f images i n m i r r o r s , c l o u d s , thunder, r a i n b o w s , a n d so o n . Aitiologia a n d related w o r d s t u r n u p several times i n the fragments a n d testimonia o f P o s e i d ô n i o s . O n e o f the m o r e c h a r m i n g a n d i l l u m i n a t i n g occurrences o f the w o r d is i n a passage o f S t r a b o i n w h i c h the latter criticizes Poseidônios for try­ ing to introduce too much o f this seeking after causes i n t o geography: " F o r i n h i m there is m u c h i n q u i r y i n t o causes, that is, ' A r i s t o t l e i z i n g , ' a t h i n g that o u r S c h o o l [i.e., the Stoics] avoids because o f the concealment of the causes." F r o m a l l this it is clear that aitiologia signifies an i n q u i r y i n t o the causes o f n a t u r a l p h e n o m e n a i n the w a y pioneered by A r i s t o t l e a n d c o n t i n u e d by some o f the Stoics, n o t a b l y Poseidônios. G e m i n o s uses 137

1 3 8

139

140

135

136

137

138

139

140

Ptolemy, Almagest iii 4. Theón of Smyrna, Mathematical Knowledge iii 26.3. Diogenes Laertios, Lives and Opinions ν 32. Diogenes Laertios, Lives and Opinions vii 132. Edelstein and K i d d 1989, fragments 18, 176, and 223. Strabo, Geography ii 3.8.

58

·

Section 11

the w o r d here to insist that it is n o t e n o u g h s i m p l y to save the p h e n o m ­ ena i n terms o f some c o m b i n a t i o n o f c i r c u l a r m o t i o n s . O n e must also have a causal e x p l a n a t i o n , that is, a n e x p l a n a t i o n based o n the nature o f things. T h e ô n o f S m y r n a gives another g o o d example of the importance to be placed o n physical explanation. T h e ó n remarks that there is more than one a p p r o a c h to e x p l a i n i n g the phenomena. T h e Greeks, o f course, used geo­ metrical methods. T h e B a b y l o n i a n s , using arithmetic methods, succeeded i n c o n f i r m i n g the observed facts a n d i n predicting future phenomena. B u t according to Theôn, the B a b y l o n i a n methods are nevertheless imperfect, because they were not based o n a sufficient understanding of nature, a n d "one must also examine these matters p h y s i c a l l y . " G e m i n o s ' s p o s i t i o n o n e x p l a n a t i o n has frequently been misstated. B u t as w e have seen, he was a t h o u g h t f u l realist. H e saw clearly that there were l i m i t s to w h a t w a s a s t r o n o m i c a l l y demonstrable, but by n o means r e n o u n c e d the effort to determine the true nature o f the w o r l d . 141

1 1 . H E L I A C A L RISINGS A N D SETTINGS

Sections 1 1 , 12, a n d 13 are the most technical parts o f o u r I n t r o d u c t i o n . Sections 12 a n d 13 are also s o m e w h a t m a t h e m a t i c a l (although o n l y a r i t h m e t i c a n d algebra are used). T h e present section (sec. 11) is devoted to h e l i a c a l risings a n d settings or, as they are sometimes called, phases o f the fixed stars. G e m i n o s treats this subject i n chapters x i i i a n d x i v o f the Introduction to the Phenomena. A n d , o f course, a n u n d e r s t a n d i n g o f the theory of star phases w i l l help the reader appreciate the G e m i n o s parapëgma, as w e l l as the analysis i n a p p e n d i x 2 . Sec. 12 illustrates the uses o f arithmetic progressions i n ancient G r e e k astronomy. Before the development o f trigonometry, arithmetic methods a l l o w e d for p r a c t i c a l s o l u t i o n o f p r o b l e m s i n a s t r o n o m y that c o u l d not be solved i n any other way. A n d even after the development o f trigonometry, the arithmetic methods c o n t i n u e d to be used because they were flexible a n d easy to ap­ ply. In several passages o f chapters v i , v i i , a n d x v i i i , G e m i n o s i n v o k e s the properties o f arithmetic progressions. Sec. 13 is devoted to l u n a r a n d l u ­ n i s o l a r cycles, topics that G e m i n o s addresses i n his chapters v i i i a n d x v i i i . F i x e d star phases, arithmetic progressions, a n d l u n i s o l a r cycles were i m p o r t a n t parts of ancient G r e e k astronomy, but have l o n g since d r o p p e d out o f the a s t r o n o m i c a l c u r r i c u l u m . Because these subjects w i l l not be familiar to most readers, w e t h o u g h t it i m p o r t a n t to p r o v i d e rea­ s o n a b l y complete discussions. B u t a reader w h o wishes to s k i p these 141

Theôn of Smyrna, Mathematical

Knowledge iii 30.

Heliacal Risings and Settings

·

59

three sections o n a first reading, a n d m o v e ahead to sec. 14, m a y d o so w i t h a clear conscience, a n d return to t h e m w h e n they become necessary for understanding parts o f G e m i n o s ' s text.

Star Phases and Star Calendars

in Early Greek

Astronomy

T h e a n n u a l cycle o f appearances a n d disappearances o f the fixed stars w a s a central part o f early G r e e k a s t r o n o m y . Indeed, one o f the oldest s u r v i v i n g w o r k s o f G r e e k literature is concerned w i t h this lore. H e s i o d ' s Works and Days (seventh century B.C.) is a didactic p o e m i n 828 lines. A substantial part o f the p o e m is a sort o f farmer's a n d sailor's calendar, w h i c h prescribes the labors to be performed i n each season o f the year. T h i s section o f the Works and Days begins w i t h the famous lines: 142

W h e n the Pleiades, daughters o f A t l a s , are r i s i n g , begin the harvest; the p l o w i n g , w h e n they s e t . 143

T h e rising o f the Pleiades here means their morning, o r heliacal rising, w h i c h o c c u r r e d i n late spring, the usual time for reaping w h a t w e t o d a y call w i n t e r wheat. T h e setting of the Pleiades means their morning set­ ting. If the farmer saw the Pleiades set i n the west i n the early m o r n i n g , just before the S u n came up i n the east, he k n e w that the time h a d ar­ rived for p l o w i n g the g r o u n d a n d s o w i n g the wheat (late fall). H e s i o d ' s theme was very p o p u l a r i n A n t i q u i t y , a n d was c o n t i n u e d by poets, ency­ clopedists, a n d a g r i c u l t u r a l writers d o w n to late R o m a n times. Star phases were also an early major c o n c e r n o f the G r e e k scientific writers. T h e G r e e k parapëgma o f later years w a s a m o r e complete a n d systematic v e r s i o n o f H e s i o d ' s star c a l e n d a r . It listed i n order t h r o u g h the year the heliacal risings a n d settings o f i m p o r t a n t stars o r constella­ tions, usually w i t h associated weather predictions. W e k n o w that parapëgmata were c o m p i l e d by M e t ô n , Euktëmôn, a n d Dëmokritos—all figures of the late fifth century B.C. A n d a l l of the most i m p o r t a n t of the later G r e e k astronomers devoted attention to this subject, i n c l u d i n g E u ­ doxos, Kallippos, Hipparchos, and Ptolemy. 144

145

142

For a more detailed introduction to star phases, see Evans 1998, 190-98, or Schmidt

1952. Hesiod, Works and Days 383-84. Systematic parapëgmata were compiled earlier in Babylonia. M U L . A P I N ("Plow Star") is the title of a Babylonian astronomical and astrological compilation, the oldest copies of which date to the seventh century B . c . M U L . A P I N includes a chronological list of the dates of the heliacal risings of various constellations. See Hunger and Pingree 1989. For a study of all known Greek and Latin parapëgmata, see Lehoux 2000 and Lehoux (forthcoming). For a recent work on Euktëmôn 's parapëgma, see Hannah 2002. For a study of the parapëgma in the transition between orality and literacy, see Hannah 2001. 143

144

145

60

·

Section 11

Fig. 1.18. Fragment of a stone parapëgma from Miletus (the "first Miletus parapëgma"), late second century B.c. About 3 5 χ 44 cm. Antikensammlung, Staatliche Museen zu Berlin—Preussischer Kulturbesitz (SK 1606 B).

In G r e e k cities, stone parapëgmata were sometimes set up i n p u b l i c places. T h e parapëgma fragment s h o w n i n fig. 1.18, f o u n d at M i l e t u s , has been dated to the second century B . C . F o r each day of the year, a s m a l l hole was d r i l l e d i n the stone, a n d next to the hole was inscribed a notice of the heliacal r i s i n g or setting o f some star or constellation. Someone h a d the job of m o v i n g a w o o d e n peg f r o m one hole to the next each day. (Alternatively, it is possible that n u m b e r e d pegs for a w h o l e m o n t h were inserted a l l at once.) T h e holes e x p l a i n the G r e e k name for these star calendars: the verb parapëgnumi means "to fix beside." T h i s parapëgma is arranged a c c o r d i n g to z o d i a c signs, just as is the G e m i n o s parapëgma. N e a r the top o f the center c o l u m n is the notice " 3 0 , " i n d i ­ cating that the Sun is i n the sign o f A q u a r i u s for 30 days. T h e next few lines o f text read: 1 4 6

oThe Sun i n A q u a r i u s . o [Leo] begins setting i n the m o r n i n g a n d L y r a sets, o o Diels and Rhem 1904. For a recent, very careful republication of the inscriptions, see Lehoux 2005. 146

Heliacal Risings and Settings

·

61

o T h e B i r d begins setting i n the evening. o o o o o o o o o

° A n d r o m e d a begins to rise i n the m o r n i n g . H o l e s w i t h n o adjacent w r i t i n g represent days o n w h i c h there are n o ce­ lestial p h e n o m e n a to note. A n u n u s u a l feature o f this parapëgma is that the text is confined to star phases a n d changes o f seasons, for e x a m p l e the b e g i n n i n g of the season of the west w i n d i n late winter. It does not attempt to m a k e day-by-day weather predictions, w h i c h is one o f the m a i n purposes o f most parapëgmata. M o s t G r e e k parapëgmata were p r o b a b l y not inscriptions o n stone, but documents o n p a p y r u s or other portable m e d i a . T h e parapëgma shown i n fig. 1.19 is the oldest k n o w n d o c u m e n t o f this type. It was w r i t t e n i n G r e e k E g y p t a r o u n d 3 0 0 B.C. a n d provides a nice e x a m p l e o f the c u l ­ t u r a l flexibility o f the format. T h i s parapëgma is arranged a c c o r d i n g to m o n t h s of the E g y p t i a n calendar. T h e fragment s h o w n i n the figure begins: 147

148

< C h o i a k 1> . . . the night is 13 4/45 hours, the day 10 4 1 / 4 5 . 16, A r c t u r u s rises i n the evening. T h e night is 12 3 4 / 4 5 hours, the day 11 1 1 / 4 5 . 2 6 , the C r o w n rises i n the evening, a n d the n o r t h w i n d s b l o w , w h i c h b r i n g the birds. T h e night is 12 8/15 hours a n d the day 11 7/15. O s i r i s circumnavigates a n d the g o l d e n boat is b r o u g h t out. T y b i , the S u n enters A r i e s . 2 0 , s p r i n g e q u i n o x . T h e night is 12 h o u r s a n d the d a y 12 hours. Feast o f P h i t o r o i s . T h i s parapëgma includes notices o f i m p o r t a n t events i n the E g y p t i a n reli­ gious cycle. It also includes notices o f the length o f the day a c c o r d i n g to an o l d E g y p t i a n scheme, w h e r e b y the day length changes by the constant increment o f 1/45 h o u r f r o m one d a y to the next. O n e c o u l d , o f course, c o m p i l e a list of heliacal risings a n d settings sim­ ply by observations made at d a w n a n d dusk over the course of a year. There is n o need for any sort of theory. In this sense, the parapëgma may be considered prescientific. B u t understanding the a n n u a l cycle of star phases was also a n i m p o r t a n t early g o a l of G r e e k scientific astronomy. In­ deed, one of the oldest surviving w o r k s o f G r e e k m a t h e m a t i c a l astronThis is the so-called first Miletus parapëgma. Found on the same site in Miletus were fragments of a second parapëgma that does include weather predictions. See Diels and Rhem 1904. Lehoux 2005 makes an important redivision of the four fragments among the two parapëgmata. O n ancient meteorology and its connections with the parapëgma tradi­ tion, see Taub 2003. Translation adapted from Grenfell and H u n t 1906, 152. 1 4 7

1 4 8

62

·

Section 11

Fig. 1.19. A portion of a papyrus parapëgma, written about 300 B.C. By permission of The Board of Trinity College Dublin (P. Hibeh i 27. T C D Pap. F 100 r). The diagonal strokes mark fractions. o m y is devoted to this subject. T h i s is A u t o l y k o s of Pitane's On and Settings of a b o u t 3 2 0 B . C .

1 4 9

Risings

A u t o l y k o s defines the various k i n d s o f

heliacal risings a n d settings, then states a n d proves theorems c o n c e r n i n g The Greek text and a French translation of the works of Autolykos are available in Aujac 1979. There is an English translation (Bruin and Vondijis 1971) based on the Greek 149

Heliacal Risings and Settings

·

63

their sequence i n time, a n d the w a y the sequence depends u p o n the star's celestial latitude. N o i n d i v i d u a l star is m e n t i o n e d by name. A u t o l y k o s ' s g o a l is to p r o v i d e a theory for understanding the p h e n o m e n a , a n d his style is that o f E u c l i d . Nevertheless, A u t l o y k o s ' s treatment established once a n d for a l l the G r e e k s ' a p p r o a c h to star phases, as w e l l as the tech­ nical v o c a b u l a r y o f the field. G e m i n o s ' s treatment o f the subject is c o n ­ ventional a n d agrees, i n b o t h substance a n d terminology, w i t h A u t o l y k o s .

True Star Phases A u t o l y k o s distinguished between true a n d visible star phases. A n e x a m ­ ple of a true star phase is the true morning rising ( T M R ) , w h i c h occurs w h e n the star rises at the same m o m e n t as the S u n . A t such a time the star is, o f course, invisible, o w i n g to the brightness o f the sky. T h e visible morning rising ( V M R ) occurs some weeks later, after the S u n has m o v e d a w a y f r o m the star. T h e visible phases are the observable events of inter­ est to farmers, sailors, a n d poets. H o w e v e r , the true phases are m o r e eas­ ily a n a l y z e d a n d described. There are four true phases: TMR TMS TER TES

True True True True

m o r n i n g rising m o r n i n g setting evening rising evening setting

(Star (Star (Star (Star

rises at sunrise) sets at sunrise) rises at sunset) sets at sunset)

F o r any star, the T M R a n d the T E R o c c u r half a year apart. Similarly, the T M S a n d the T E S o c c u r half a year apart. These p r o p o s i t i o n s are easily p r o v e d . Refer to fig. 1.20. Let a star S be rising i n the east w h i l e the Sun is rising at A . T h e star is thus m a k i n g its T M R . T h e T E R w i l l o c c u r w h e n the star is r i s i n g at S a n d the S u n is setting at B. T h e ecliptic is b i ­ sected by the h o r i z o n ; thus there are six z o d i a c signs between A a n d B. If w e suppose that the S u n moves u n i f o r m l y o n the ecliptic, it w i l l take the Sun h a l f a year to go f r o m A to B. T h u s the T M R a n d the T E R o c c u r six signs (about six months) apart i n the year. T h e same sort o f p r o o f is eas­ ily made for the T M S a n d the T E S . The stars have their true phases i n different orders, a c c o r d i n g to whether they are s o u t h o f the ecliptic, o n the ecliptic, o r n o r t h o f the ecliptic. E c l i p t i c Stars: If a star is exactly o n the ecliptic, its T M R a n d T E S w i l l occur o n the same day. Refer to fig. 1.21. Let the star S be at ecliptic p o i n t A ; then S a n d A rise together i n the east, thus p r o d u c i n g the star's T M R . In the evening, S a n d A w i l l set together i n the west, thus p r o d u c text in Mogenet 1950. But users of this translation should be forewarned that it often con­ fuses the invisibility of a star phase with the invisibility of the star itself.

64

·

Section 11

South

North Fig. 1.20. True star phases. If the Sun rises at A at the same time as the star rises at S, the star makes its true morning rising. i n g the star's T E S . (We assume that the S u n stays at the same ecliptic p o i n t for the w h o l e day.) In the same w a y , one m a y s h o w that for ecliptic stars, the T E R a n d T M S o c c u r o n the same day. N o r t h e r n Stars: If a star is n o r t h o f the ecliptic, the T M R w i l l precede the T E S . L e t the n o r t h e r n star S be m a k i n g its T M R , r i s i n g s i m u l t a n e -

South

North Fig. 1.21. For a star on the ecliptic, the true morning rising and the true evening setting occur on the same day.

Heliacal Risings and Settings

·

65

South

West

East

S lonzon

North Fig. 1.22. The situation shown in fig. 1.20, after some hours have gone by and star S is setting in the west. A is below the horizon, so S will not have its true evening setting until the Sun has moved from A to C on the ecliptic.

ously w i t h ecliptic p o i n t A , as i n fig. 1.20. N o w , o f any t w o points o n the celestial sphere that rise simultaneously, the one that is farther n o r t h w i l l stay up longer a n d set l a t e r . (In this a n d a l l that f o l l o w s , w e assume the observer is i n the n o r t h e r n hemisphere.) S a n d A rise together. B u t A w i l l set first. T h u s , w h e n S sets, the s i t u a t i o n w i l l resemble fig. 1.22. S is o n the western h o r i z o n . A , located farther south o n the sphere, w i l l a l ­ ready have set a n d w i l l be b e l o w the h o r i z o n . T h e T E S o f star S occurs w h e n the Sun is at C . T h u s w e must w a i t a few days o r weeks for the Sun to advance eastward o n the ecliptic f r o m A to C . T h e T E S therefore fol­ l o w s the T M R . T h e proofs given here are m o r e concise t h a n A u t o l y k o s ' s proofs o f the same p r o p o s i t i o n s , but f o l l o w his m e t h o d . S o u t h e r n Stars: If a star is south o f the ecliptic, the T M R w i l l f o l l o w the T E S . T h e p r o o f m a y be made i n the same w a y . T h e orders o f the true phases for n o r t h e r n , southern, a n d ecliptic stars are s u m m a r i z e d i n table 1.2. 150

Visible Star Phases The visible phases o c c u r i n the early d a w n (shortly before sunrise), or i n the late evening (shortly after sunset). In each case, the star is exactly o n Autolykos of Pitanë, On the Moving Sphere, Prop. 9.

66

·

Section 11 TABLE 1.2

Order of True Star Phases in the Year Star South of Ecliptic TMR TMS TER TES

Star on Ecliptic

Star North of Ecliptic

T M R and TES T E R and T M S

TMR TES TER TMS

the h o r i z o n ; but the S u n is just far e n o u g h b e l o w the h o r i z o n to p e r m i t the star to be seen. T h e r e are four visible phases: VMR

Visible m o r n i n g rising

VMS

V i s i b l e m o r n i n g setting

VER

V i s i b l e evening r i s i n g

VES

V i s i b l e evening setting

(Before sunrise, star for the first time) (Before sunrise, star for the first time) (After sunset, star is the last time) (After sunset, star is the last time)

is seen r i s i n g is seen setting seen r i s i n g for seen setting for

T h e visible m o r n i n g phases c o m e later t h a n the true ones. B u t the visible evening phases precede the true ones. These p r o p o s i t i o n s f o l l o w s i m p l y f r o m the fact that the Sun's m o t i o n o n the ecliptic is f r o m west to east. L e t star S be r i s i n g w h i l e the S u n is r i s i n g at A , as i n fig. 1.23. T h i s is therefore the time o f the star's T M R . T h e star's r i s i n g w i l l be i n v i s i b l e . B u t some weeks later, w h e n the S u n has a d v a n c e d f r o m A to D o n the ecliptic, the star's r i s i n g w i l l be visible for the first t i m e . T h e n the S u n w i l l be far e n o u g h b e l o w the h o r i z o n at the r i s i n g o f S for the star to be seen. T h u s , w h e n the S u n is at D , star S w i l l m a k e its V M R . N o t e that the same argument m a y be a p p l i e d to the star setting at T . T h i s star makes its T M S w h e n the S u n rises at A . T h e setting w i l l be i n ­ visible, however. T h e first setting o f Τ to be visible w i l l o c c u r w h e n the Sun has a d v a n c e d to D . T h u s the visible m o r n i n g phases (whether risings or settings) f o l l o w the true ones. A l s o , the m o r n i n g phases are the first events to be visible. T h a t is, the V M R is the first visible rising o f the star i n the a n n u a l cycle; the V M S is the first visible setting. In the same w a y it m a y be p r o v e d that visible evening phases precede the true ones. A l s o , the evening phases are the last events to be visible: the V E S is the last visible setting o f the star i n the a n n u a l cycle. Similarly, the V E R is the last visible rising i n the course of the year.

Heliacal Risings and Settings

·

67

South

Β

West Τ

East

horizon North Fig. 1.23. The visible morning phases follow the true morning phases in time. Star S makes its true morning rising when the Sun is at A , and its visible morning rising when the sun is at D .

Autolykos's

15° Visibility

Rule

The length o f time that separates a star's visible rising or setting f r o m its c o r r e s p o n d i n g true one depends o n m a n y factors: the brightness o f the star, the steepness w i t h w h i c h the ecliptic meets the h o r i z o n w h e n the star is rising o r setting, the observer's latitude, etc. A u t o l y k o s dispensed w i t h a l l these c o m p l i c a t i o n s by means o f one s i m p l i f y i n g a s s u m p t i o n : any star's rising o r setting w i l l be visible i f the S u n is b e l o w the h o r i z o n by at least half a zodiac sign measured along the ecliptic. In fig. 1.24, let star S rise s i m u l t a n e o u s l y w i t h p o i n t X o f the ecliptic. T h e n w h e n the Sun is at X , star S w i l l have its true m o r n i n g rising. A c c o r d i n g to A u ­ t o l y k o s , S w i l l have its visible m o r n i n g rising w h e n the S u n reaches Y , w h i c h is half a z o d i a c a l sign ( 1 5 ° ) f r o m X . A c c o r d i n g to m o d e r n astronomers, a s t r o n o m i c a l t w i l i g h t extends f r o m the Sun's setting u n t i l it reaches a p o s i t i o n 1 8 ° v e r t i c a l l y b e l o w the h o r i z o n , w h e n the sky becomes d a r k enough to p e r m i t o b s e r v a t i o n o f even the faintest stars. T h e brighter stars, however, c a n be seen w h e n the Sun is o n l y 1 0 ° o r 1 2 ° b e l o w the h o r i z o n , a n d these are precisely the stars that p l a y the m o s t p r o m i n e n t roles i n the ancient literature o n star phases. A u t o l y k o s ' s use o f 1 5 ° measured o b l i q u e l y to the h o r i z o n is then a fairly g o o d a p p r o x i m a t i o n to 1 2 ° measured vertically. Since the S u n moves r o u g h l y I per day o n the ecliptic, 1 5 ° corre­ sponds r o u g h l y to 15 days. T h u s , the V M R o f a star f o l l o w s the T M R by r o u g h l y 15 days. Similarly, the V M S f o l l o w s the T M S by r o u g h l y 15 o

68

·

Section 11

/ ecliptic °

7

15

horizon

e

Y Fig. 1.24. Autolykos's 15° visibility rule.

days. B u t the V E R precedes the T E R by 15 days, a n d the V E S precedes the T E S by 15 days. G e m i n o s does not m e n t i o n A u t o l y k o s ' s 1 5 ° v i s i b i l ­ ity rule. W e m e n t i o n it here because it affords a n easy w a y to estimate the dates o f the visible phases after one has f o u n d the dates o f the true phases by (for example) c o n s u l t i n g a celestial globe. Order of the Visible

Phases

The visible phases c a n come i n three different orders i n the year, depend­ ing o n where the star is located o n the celestial sphere. If the star lies far e n o u g h n o r t h of the ecliptic, the 15-day c o r r e c t i o n w i l l not change the or­ der f r o m that of the true phases. Similarly, if the star is far e n o u g h south of the ecliptic, the visible phases w i l l come i n the same order as the true ones. B u t for stars near the ecliptic, some o f the true phases o c c u r so close together i n the year that a p p l i c a t i o n o f the 15-day visibility rule causes a reversal o f order. It turns out that a l l stars near enough to the ecliptic (whether a little n o r t h or south of the ecliptic) have their visible phases i n the same order: V M R , V E R , V M S , V E S . These properties are s u m m a ­ rized i n table 1.3. T h e orders of the phases given here presuppose that the observer is i n the Earth's n o r t h e r n hemisphere, a n d at a moderate latitude (i.e., not i n the arctic zone). T h e same star m a y have its visible phases i n different orders at different latitudes. Stars near enough to the ecliptic are always " d o c k - p a t h e d " ; but w h a t is "near enough to the e c l i p t i c " varies w i t h the latitude of observation. In his w o r k o n star phases, the Phaseis, P t o l e m y i n t r o d u c e d some convenient nomenclature for the three star classes. Stars sufficiently far south o f the ecliptic are called " n i g h t - p a t h e d " (nuktidiexodos) because o f their behavior d u r i n g the interval between the V M S a n d the V E R . The 1 5 1

151

Ptolemy, Phaseis 6. Opera, vol. 2, p. 9.

Heliacal Risings and Settings

·

69

TABLE 1.3

Order of Visible Star Phases in the Year Star Well South of Ecliptic ("night-pathed") VMR VMS VER VES

Star near Ecliptic (" dock-path ed")

Star Well North of Ecliptic ("doubly visible")

VMR VER VMS VES

VMR VES VER VMS

first visible setting has already taken place, so the star's setting is visible. But the last visible rising has not yet o c c u r r e d , so the star's rising is also visible. Therefore, d u r i n g this p e r i o d , the star's w h o l e p a t h f r o m h o r i z o n to h o r i z o n is visible. T h e G e m i n o s parapëgma treats the D o g Star (Sirius) as night-pathed. (See a p p e n d i x 2.) Stars sufficiently near the ecliptic are called " d o c k - p a t h e d " (kolobodiexodos) by Ptolemy, because of their b e h a v i o r between V E R a n d the VMS. T h e last visible rising has already o c c u r r e d , so the star's rising is n o longer visible. T h e first visible setting has not yet o c c u r r e d , so the star's setting is not yet visible. T h u s , the star crosses the sky i n the night, but neither the rising n o r the setting c a n be seen. T h e ends o f its p a t h across the sky are d o c k e d , or cut off. T h e Pleiades p r o v i d e a g o o d e x a m ­ ple of the d o c k - p a t h e d . 1 5 2

N i g h t - p a t h e d a n d dock-pathed stars a l l have a period of complete invisi­ bility (between the V E S a n d the V M R ) , w h e n they are too near the Sun and therefore cross the sky i n the daytime. T h e classic example is the Pleiades, of w h i c h H e s i o d says, " F o r forty days a n d nights they hide themselves." Finally, P t o l e m y calls stars that are sufficiently far n o r t h o f the ecliptic " d o u b l y v i s i b l e , " o r "seen o n b o t h sides" (amphiphanës), because o f their b e h a v i o r between the V M R a n d the V E S . ( G e m i n o s uses the same t e r m at x i v 11.) T h e star appears i n the western sky after dusk. T h e n it is seen setting; but it rises i n the east before the night is over. P t o l e m y also calls such a star "visible a l l year l o n g " (eniautophanës), because it has n o p e r i o d o f invisibility. T h e star is visible at some time of night a l l year l o n g . A r c t u r u s is the classic e x a m p l e . T h e standard terms for the phases (true m o r n i n g rising, visible evening setting, etc.) were i n t r o d u c e d by A u t o l y k o s a n d were u n i v e r s a l l y f o l ­ l o w e d . A u t o l y k o s discussed the properties o f the stars that w e have 153

1 5 4

155

1 5 2

1 5 3

1 5 4

1 5 5

Ptolemy, Phaseis 5. Hesiod, Works and Ptolemy, Phaseis 6. Ptolemy, Phaseis 6.

Opera, vol. 2, p. 8. Days 385. Opera, vol. 2, p. 9. Opera, vol. 2, p. 9.

70

·

S e c t i o n 11

called d o c k - p a t h e d , night-pathed, a n d d o u b l y visible, but d i d not assign these names to them. Since every star that has risings a n d settings m a y be assigned to one o f these three classes, it is convenient to have names for the groups. T h i s rigorous systemization a n d n a m i n g appears i n Ptolemy's Phaseis, a n d m a y w e l l have been Ptolemy's o w n c o n t r i b u t i o n . These terms were, o f course, used by earlier w r i t e r s , but less systematically. A s w e have seen, G e m i n o s speaks o f d o u b l y visible stars at x i v 1 1 . H e also uses nuktidiexodos at x i v 12, but i n a sense completely different f r o m P t o l e m y ' s — w h i c h seems consistent w i t h the l a c k of standard terms for the three star classes before Ptolemy's time. Some Details

of Technical

Vocabulary

"RISING" AND "SETTING"

T h e u s u a l n o u n for the d i u r n a l r i s i n g o f a star, o r o f the S u n o r M o o n , is anatolè, a n d the associated verb is anatellein, " t o rise." These are the terms A u t o l y k o s uses to describe a d i u r n a l r i s i n g i n On the Moving Sphere. T h e verb is a c o m p o u n d o f a simple verb, tellein, "to arise," a n d a p r e p o s i t i o n , ana, " u p . " B u t there is another c o m m o n c o m p o u n d , f o r m e d w i t h the p r e p o s i t i o n epi. In c o m p o u n d , this p r e p o s i t i o n c a n i n d i cate a c c o m p a n i m e n t or o p p o s i t i o n . T h e epitolè is therefore the r i s i n g that the star makes " w i t h " (or, perhaps, "against") the S u n . T h e verb epitellein w a s used already by H e s i o d to indicate the first visible r i s i n g o f the Pleiades a n d also the last visible r i s i n g o f A r c t u r u s (i.e., not mere d i u r n a l risings). O d d l y , there is no s i m i l a r d i s t i n c t i o n between the terms used for the t w o k i n d s o f settings. O n e verb, dunein, serves for b o t h the d i u r n a l sett i n g a n d the settings i n c o n n e c t i o n w i t h the S u n . T h e n o u n is dusis, a "setting." A u t o l y k o s f o l l o w s H e s i o d ' s usage: epitolè is reserved for a r i s i n g that takes place w h i l e the S u n is either r i s i n g or setting. B y the time o f A u t o l y k o s , however, the true risings have been distinguished f r o m the v i s i ble ones. So a true m o r n i n g r i s i n g is heôia alëthinë epitolè (the E n g l i s h is a literal t r a n s l a t i o n o f the Greek) a n d a visible evening setting is hesperia phainomenè dusis. A u t o l y k o s ' s system o f t e r m i n o l o g y puts a l l eight phases o n a n equal f o o t i n g : the name o f each o f the eight possible phases is f o r m e d by selecting one w o r d f r o m each o f the three pairs true-visible, m o r n i n g - e v e n i n g , rising-setting. A u t o l y k o s ' s system p r o v e d to be definitive, but there was o c c a s i o n a l sloppiness i n usage. G e m i n o s ( x i i i 3-4) criticizes people w h o , out o f i g n o r a n c e , use anotóle a n d epitolè interchangeably. Such slips are n o t u n k n o w n even a m o n g a s t r o n o m i c a l writers. F o r e x a m p l e , A u t o l y k o s often

Heliacal Risings and Settings

·

71

uses anatellein for a heliacal rising, a n d even G e m i n o s , w h o makes such a fuss over the d i s t i n c t i o n , sometimes appears to lapse i n the same w a y , as A u j a c has p o i n t e d o u t . B u t most o f these apparent lapses are n o t re­ ally lapses at a l l . W h e n A u t o l y k o s or G e m i n o s uses anatellein i n a place that seems to require epitellein, w e u s u a l l y find that the m e a n i n g is made clear by a modifier. " R i s e s i n the m o r n i n g " makes perfect sense, regard­ less o f the f o r m o f the verb that is used. O n c e again w e find that P t o l e m y is the most consistent a n d l o g i c a l o f all writers o n the subject. If a t e m p o r a l modifier is a d d e d , P t o l e m y regu­ larly uses anatellein, as the modifier completes the sense: " A r c t u r u s rises i n the m o r n i n g " (anatellei). B u t i f n o modifier is added, P t o l e m y a l w a y s writes epitellei, o r else the reader w o u l d have n o w a y o f k n o w i n g that a rising w i t h respect to the S u n is meant, a n d not a mere d i u r n a l rising: "the bright star i n the Eagle rises" (epitellei). M o r e o v e r , Ptolemy's choice between these t w o styles is n o t h a p h a z a r d ; rather, it conveys a subtle dis­ t i n c t i o n . T h e Eagle emerges f r o m a p e r i o d o f complete invisibility, so P t o l e m y uses epitellei. A r c t u r u s (a d o u b l y visible star) does n o t have a p e r i o d o f complete invisibility, so it o n l y "rises (anatellei) i n the m o r n ­ i n g . " In this b o o k , w e translate b o t h anatellei a n d epitellei s i m p l y as "rises." 156

Sometimes, G r e e k writers w i l l say that a star "rises" (epitellei) o r "sets" (dunei) w i t h o u t b o t h e r i n g to say whether it is a m o r n i n g o r a n evening event. In such cases, the general rule is that a visible m o r n i n g phase is intended. E x c e p t i o n s to this rule are not u n k n o w n , but usually represent mistakes. T h e m o r n i n g rising is the first rising to be visible. T h e m o r n i n g setting is the first setting to be visible. First things (the m o r n i n g phases) n a t u r a l l y are o f greater i m p o r t a n c e , psychologically, t h a n last things (the evening phases). A variety o f usage is apparent i n the G e m i n o s parapëgma. For exam­ ple, at days 1 5 , 16, a n d 2 6 o f Sagittarius, w e read: "the Eagle rises" (epitellei), "the Eagle rises at the same time as the S u n , " "the Eagle rises i n the m o r n i n g . " These expressions a l l indicate the visible m o r n i n g rising of the Eagle ( A q u i l a ) , a c c o r d i n g to the v a r i o u s authorities cited i n the parapëgma. "HIDING" AND "APPEARING"

Besides the general terms for risings a n d settings, there are several m o r e specialized ones that a p p l y o n l y to the visible phases. W h e n a star makes its V E S , it is sometimes said that the star "hides i t s e l f " (kruptetai). B u t this t e r m indicates m o r e t h a n a simple evening setting: it implies that the 1 5 6

Aujac 1 9 7 5 , 6 8 η .

72

·

S e c t i o n 11

star goes i n t o a p e r i o d of complete invisibility. (The S u n moves so near the star that the star becomes lost i n its rays for a p e r i o d of time.) So, i n the G e m i n o s parapëgma w e have notices of the h i d i n g o f the Pleiades (day 10 o f A r i e s ) , the H y a d e s (day 2 3 o f A r i e s ) , a n d the D o g (day 2 o f Taurus). Strictly, then, " h i d i n g " s h o u l d be used o n l y o f night-pathed a n d d o c k - p a t h e d stars. ( D o u b l y visible stars c a n never hide themselves.) In the Phaseis, P t o l e m y is scrupulous i n o b s e r v i n g this d i s t i n c t i o n , but the same c a n n o t be said o f a l l ancient writers. F o r e x a m p l e , i n the Phaseis, P t o l e m y writes for the klima o f 15 hours: G e m m a "sets i n the e v e n i n g . " ( D a y 10 o f C h o i a k ) F o m a l h a u t "hides itself." ( D a y 4 o f Tybi) These are b o t h visible evening settings, but the d i s t i n c t i o n is deliberate. In the klima of 15 h o u r s , F o m a l h a u t enters i n t o a p e r i o d o f invisibility, but G e m m a (a d o u b l y visible star) does not. Similarly, w h e n a star emerges f r o m its p e r i o d of i n v i s i b i l i t y (at its V M R ) , it m a y be said that it " a p p e a r s " (phainetai), " s h o w s itself," o r "becomes v i s i b l e " (ekphanës ginetai). F o r e x a m p l e , i n the G e m i n o s parapëgma, the D o g is said to become visible i n E g y p t (day 2 3 o f C a n ­ cer) a n d the H a r b i n g e r o f the V i n t a g e is said to appear (day 10 o f V i r g o ) . These terms c a n strictly be used o n l y o f night-pathed a n d d o c k e d - p a t h e d stars, t h o u g h not a l l ancient writers are careful i n this regard. MODERN TERMINOLOGY

In m o d e r n w r i t i n g o n star phases, one encounters the terms: heliacal r i s i n g acronychal rising heliacal setting c o s m i c a l setting

VMR VER VES VMS

E x c e p t for " a c r o n y c h a l r i s i n g , " none o f these terms is used by the ancient Greek astronomers. A n d even " a c r o n y c h a l " poses a p r o b l e m . Akronychos = akron (tip, extremity) + nyktos (of the night). T h i s adjective is used by Greek writers o n star phases a n d , indeed, belongs to the v o c a b u l a r y of everyday speech. Usually, it means " i n the evening." But Theón of S m y r n a points out that the m o r n i n g is also an extremity of the night, a n d there­ fore, logically enough, he applies the same w o r d to b o t h evening risings a n d m o r n i n g settings. T h u s , it is far better, a n d clearer, to stick to A u ­ tolykos's technical v o c a b u l a r y (visible evening rising, etc.), w h i c h is fol­ l o w e d by G e m i n o s a n d Ptolemy, a n d w h i c h can h a r d l y be i m p r o v e d u p o n . 157

1 5 7

Theón of Smyrna, Mathematical

Knowledge iii 14.

Applications of Arithmetic Progressions

·

73

12. ASTRONOMICAL APPLICATIONS OF ARITHMETIC

PROGRESSIONS

A n arithmetic progression is a sequence o f terms a, b, c, d, . . . , i n w h i c h each p a i r o f successive terms differs by the same q u a n t i t y δ. T h u s b = a + 5;c = b + 5;d = c + δ; a n d so o n . G e m i n o s shows that he is f a m i l ­ iar w i t h the use o f arithmetic progressions i n the s o l u t i o n o f t w o closely related p r o b l e m s o f spherical a s t r o n o m y : d e t e r m i n i n g the rising times o f the z o d i a c signs a n d finding the length of the day. In this section, w e give a brief o v e r v i e w o f arithmetic progressions i n ancient astronomy. T h e n w e shall t u r n to G e m i n o s ' s o w n use o f arithmetic progressions. B y the " r i s i n g t i m e " o f a z o d i a c sign, w e m e a n the amount o f time that the sign takes to cross the eastern h o r i z o n . A list of the rising times for each sign (or for each 10-degree segment o f the z o d i a c , or for each single degree, depending o n the p r e c i s i o n desired) is called a table of as­ censions. O n e c a n use a table o f ascensions to solve m a n y different k i n d s of p r o b l e m s i n p r a c t i c a l astronomy. A p r i m e e x a m p l e is d e t e r m i n i n g the length o f the day at any time o f year. A s G e m i n o s r e m a r k s (vii 12), dur­ i n g any d a y l i g h t p e r i o d , six signs o f the z o d i a c rise a n d six set. T h e six signs that rise i n the course o f a day are the six b e g i n n i n g w i t h the Sun's p o s i t i o n a n d c o u n t e d eastward. So i f the Sun is at s u m m e r solstice (be­ g i n n i n g o f C a n c e r ) , the day lasts for the time it takes for the six signs starting w i t h C a n c e r to rise. O n e m a y therefore calculate the length o f the day f r o m a table o f ascensions by a d d i n g up the rising times o f the appropriate six signs. A table o f ascensions is also useful for astrologers, for it permits one to determine the h o r o s c o p i c p o i n t (the p o i n t of the ecliptic that is o n the eastern h o r i z o n ) , given the date a n d t i m e . 158

T h e rising times o f z o d i a c signs were never directly measured i n an­ cient astronomy. Rather, these rising times were a l w a y s the result o f the­ oretical c a l c u l a t i o n . Before the development o f trigonometry, G r e e k as­ t r o n o m e r s were forced to rely u p o n a p p r o x i m a t e , arithmetic methods, w h i c h were based u p o n a few properties o f arithmetic progressions. T h e arithmetic progression also gives a simple w a y o f i n t e r p o l a t i n g between the m a x i m u m a n d m i n i m u m values o f a quantity, such as day length, that varies i n a n o n l i n e a r way. A s w e shall see, the arithmetic methods often p r o d u c e very satisfactory results. Sophisticated a p p l i c a t i o n o f arithmetic progressions is a h a l l m a r k o f B a b y l o n i a n mathematics. A r i t h m e t i c solutions of p r o b l e m s i n v o l v i n g day lengths are f o u n d o n B a b y l o n i a n tablets of the Seleucid p e r i o d , 1 5 9

This point is emphasized by Manilius, Astronómica iii 295-97. For a discussion of the Babylonian material on day lengths, see Neugebauer 1975, 366-71. 1 5 8

1 5 9

74

·

Section 12

but the m a t h e m a t i c a l methods themselves (outside o f the a s t r o n o m i c a l context) are m u c h older. T h e B a b y l o n i a n arithmetic methods entered G r e e k a s t r o n o m y by the early second century B.C. T h e oldest s u r v i v i n g G r e e k treatise concerned w i t h n u m e r i c a l values for the r i s i n g times o f the signs is the Anapborikos of Hypsiklës. Hypsiklës states a n d proves several p r o p o s i t i o n s i n v o l v i n g the sums of arithmetic progressions, then applies t h e m to the p r o b l e m o f d e t e r m i n i n g the r i s i n g times o f the i n d i ­ v i d u a l z o d i a c signs. 160

T h e arithmetic methods became obsolete almost i m m e d i a t e l y after their i n t r o d u c t i o n i n t o G r e e k astronomy, for the development o f t r i g o n o m e t r y s o o n permitted exact solutions o f the t r a d i t i o n a l p r o b l e m s o f r i s i n g times a n d day lengths. P t o l e m y provides a table o f ascensions i n Almagest i i 8, gives directions for using it to solve six different k i n d s o f p r o b l e m s , a n d shows h o w to calculate the r i s i n g times trigonometrically for any desired latitude. A l t h o u g h Ptolemy's is the oldest extant t r i g o n o ­ metric treatment of the p r o b l e m , it is l i k e l y that the t r i g o n o m e t r i c s o l u ­ t i o n goes back to H i p p a r c h o s . Nevertheless, the arithmetic methods l i v e d o n a l o n g w h i l e , especially a m o n g a s t r o l o g i c a l w r i t e r s , most of w h o m never mastered trigonometry, the " n e w m a t h " of the second cen­ t u r y B.C. G e m i n o s ' s r e m a r k (vi 38) that the lengths of the days have c o n ­ stant second differences o f course implies an arithmetic progression, a n d there is n o t h i n g i n the Introduction to the Phenomena to suggest that G e m i n o s w a s a user o f trigonometry. T h e arithmetic methods c o n t i n u e d to exist side-by-side w i t h t r i g o n o m e t r i c methods w e l l i n t o the second century A . D . M o s t G r e e k writers w h o use arithmetic methods d o so w i t h o u t apology, or any i n d i c a t i o n that the methods are second-best ap­ p r o x i m a t i o n s to m o r e exact t r i g o n o m e t r i c methods. T h e methods based o n arithmetic progressions therefore constituted an independent ap­ p r o a c h to p r a c t i c a l a s t r o n o m i c a l c o m p u t a t i o n . 161

Rising

Times and Day Lengths in System A

T w o different patterns of day lengths (and u n d e r l y i n g patterns o f rising times) have been called System A a n d System Β by Neugebauer. In System A , the rising times o f the signs f o r m a strict arithmetic p r o g r e s s i o n . M o s t o f the B a b y l o n i a n m a t e r i a l o n day lengths belongs to this system, a n d Hypsiklës follows it as w e l l . T h e ecliptic makes its shallowest angle 162

160 p j Falco, Krause, and Neugebauer 1966. For a discussion of Hypsiklës' Anaphorikos, see Neugebauer 1975, 715-18. Neugebauer 1975, 719-20. In System B , the rising times of the signs almost form an arithmetic progression, but the changes in rising times are twice as big as normal between Gemini and Cancer, and between Sagittarius and Capricorn. There is no preserved Babylonian procedure text for the day lengths in System B, but isolated day lengths associated with System Β do appear e

1 6 1

1 6 2

Applications of Arithmetic Progressions

·

75

w i t h the h o r i z o n w h e n the s p r i n g e q u i n o c t i a l p o i n t is r i s i n g , a n d its greatest angle w i t h the h o r i z o n w h e n the fall e q u i n o c t i a l p o i n t is r i s i n g ( G e m i n o s , v i i 1 5 - 1 6 , 1 9 - 2 0 ) . T h u s , it is n a t u r a l to assume that A r i e s takes the least time to rise, a n d that L i b r a takes the most (vii 2 3 - 2 5 ) . L e t Τ be the a m o u n t o f time r e q u i r e d for A r i e s to rise, a n d Τ + D be the a m o u n t o f time r e q u i r e d for T a u r u s to rise. T h e n , a c c o r d i n g to System A , the time r e q u i r e d for C a n c e r to rise is T + 2 D ; for L e o , T + 3 D ; a n d so o n , as i n the second c o l u m n o f T a b l e 1.4. F r o m the r i s i n g times o f the signs i n the second c o l u m n o f table 1.4, one m a y o b t a i n the length o f the day w h e n the S u n is at the b e g i n n i n g o f any sign (third c o l u m n ) . L e t the S u n be at w i n t e r solstice (beginning o f C a p r i c o r n ) . In the course o f the day o n w i n t e r solstice, the h a l f o f the z o ­ diac that rises comprises the s i x signs f r o m C a p r i c o r n t h r o u g h G e m i n i . If w e a d d up the r i s i n g times for these s i x signs, w e o b t a i n 6 T + 6 D , as i n ­ dicated i n the t h i r d c o l u m n o f table 1.4. T h e f o u r t h c o l u m n o f table 1.4 gives the changes i n the lengths o f the days f r o m one z o d i a c sign to the next. These changes are c o m p u t e d by t a k i n g the differences between successive entries i n c o l u m n 3. These changes f o r m arithmetic progressions. M o r e o v e r , the changes i n day length stand to one another as successive o d d integers: D , 3 D , 5 D , etc. T h i s means, i n m o d e r n terms, that System A is a p a r a b o l i c a p p r o x i m a ­ t i o n : the day lengths are fitted by a p a r a b o l a that passes t h r o u g h a m a x i ­ m u m at the s u m m e r solstice a n d t h r o u g h 1 2 at the e q u i n o x e s . (Super­ script indicates h o u r s ; indicates minutes.) h

11

m

F i n a l l y , the fifth c o l u m n gives the second differences i n the lengths o f the days. T h e second differences are constant (in k e e p i n g w i t h a p a r a ­ b o l i c a p p r o x i m a t i o n ) , but change sign at the e q u i n o x e s . T o a p p l y System A to a p a r t i c u l a r klima, w e need o n l y specify the length M o f the s u m m e r solstitial day. Setting the expression for the length o f the s u m m e r solstitial day e q u a l to M , w e have M = 6T+24D. occasionally. Day lengths associated with System Β also turn up in Greek and Roman writ­ ers. A n especially relevant example is provided by Kleomëdës (Meteôra i 4.18-29), who gives day lengths for the klima of 1 5 that are in perfect accord with System B. Kleomëdës does not indicate any awareness of the derivation of day lengths from the rising times of the signs, and may simply have used the pattern of changes in day lengths without knowledge of its theo­ retical basis. The use of arithmetic patterns without understanding of their origins or their connections with other patterns is common, especially among astrological writers. Manilius gives a list of rising times that follows System A (Astronómica iii 275-94) and a list of day lengths that follows System Β [Astronómica iii 443-82). These are, of course, inconsistent. Aujac (1975, 3 8 n l ) and Manitius (1898, 261-62) both cite Kleomëdës as providing illustra­ tive detail for Geminos's remarks at vi 29 and 38. (Similarly, see Bowen and Todd 2004, 51n3.) But, in fact, Kleomëdës and Geminos follow different systems. Kleomëdës follows Sys­ tem B, while, as we shall see, Geminos uses a modification of System A . h

76

·

Section 12

TABLE 1.4

Rising Times and Day Lengths in System A Sign

Rising Time of Sign

Length of Day at Beginning of Sign

Aries

Τ

6T+ 15D

Taurus

T+D

6T + 20D

Diff

2nd Diff

5D -2D 3D Gemini

6T + 23D

T + 2D

-2D D

Cancer

6T+24D

T+3D

-2D -D

Leo

6T + 23D

T+4D

-2D -3D

Virgo

6T+20D

T+5D

-2D -5D

Libra

T+5D

6T+15D

0 -5D

Scorpio

6T+ 10D

T + 4D

2D -3D

Sagittarius

6T+7D

T+3D

2D -D

Capricorn

T+2D

6T+6D

2D D

Aquarius

T+D

6T+7D

2D 3D

Pisces

Τ

6 T + 10D

2D 5D

A l s o , the e q u i n o c t i a l day must be 12 hours, so 1 2 = 6T + 1 5 D . h

T h e length m o f the w i n t e r solstitial day is m =

6T+6D.

S u b t r a c t i n g the t h i r d e q u a t i o n f r o m the first gives D = (M-m)/18. S u b s t i t u t i o n o f this expression for D i n t o the second e q u a t i o n gives T=2 -(5/36)(M-m). h

Applications of Arithmetic Progressions

·

77

TABLE 1.5

Rising Times for the Klima of 1 5 Sign Aries Taurus Gemini Cancer Leo Virgo Libra Scorpio Sagittarius Capricorn Aquarius Pisces

Rising Time (System A) l 1 1 2 2 2 2 2 2 1 1 1

h

10 30 50 10 30 50 50 30 10 50 30 10

h

Rising Time (Trigonometric) ihi m

m

0

1 1 2 2 2 2 2 2 1 1 1

25 55 24 34 33 33 34 24 55 25 10

Hypsiklës applies his m e t h o d to the klima of A l e x a n d r i a ( M = 1 4 ) . F o r o u r e x a m p l e , w e w i l l use M = 1 5 . T h u s w e o b t a i n h

h

D = 1/3 h o u r Τ =7/6 hour.

(for System A , klima o f 1 5 ) h

If these parameters are inserted i n the expressions i n the second c o l u m n o f table 1.4, w e o b t a i n n u m e r i c a l values for the rising times s h o w n i n the second c o l u m n o f table 1.5. T h u s , i n System A , for the klima o f 15 hours, the r i s i n g times o f the signs increase by constant differences o f 2 0 . F o r c o m p a r i s o n , the t h i r d c o l u m n o f table 1.5 gives the rising times for the same klima that result f r o m exact, t r i g o n o m e t r i c c o m p u t a t i o n : these values are t a k e n f r o m Ptolemy's table o f ascensions i n Almagest i i 8 . T o o b t a i n day lengths, w e insert the n u m e r i c a l values for Τ a n d D i n t o the expressions i n the t h i r d c o l u m n o f table 1.4. T h e results are given i n the second c o l u m n o f the table 1.6. So, i n System A , for the klima o f 1 5 , the day lengths at the beginnings o f successive signs have constant m

163

h

The numerical results in the third column of table 1.5 reflect Ptolemy's value for the obliquity of the ecliptic. Both Hypsiklës and Ptolemy express their rising times in terms of degrees of time-. 360° correspond to one rotation of the celestial sphere, i.e., 24 hours of sidereal time. We have converted Ptolemy's figures to hours by dividing the angles by 15. Strictly, therefore, the rising times in table 1.5 are expressed in terms of sidereal time, rather than mean solar time. To convert to mean solar time, one should subtract l for every 6 . This small adjustment is without significance for the present discussion. 1 6 3

m

h

78

·

Section 12 TABLE 1.6

Day Lengths for the Klima of 1 5

h

Day Length at Beginning of Sign (System A)

Sign Cancer Leo Virgo Libra Scorpio Sagittarius Capricorn

15 14 13 12 10 9 9

h

00 40 40 00 20 20 00

m

Day Length at Beginning of Sign (Trigonometric) 15 14 13 12 10 9 9

h

00 31 22 00 38 29 00

m

second differences o f 4 0 . F o r c o m p a r i s o n , the t h i r d c o l u m n o f Table 1.6 gives the day lengths o b t a i n e d f r o m t r i g o n o m e t r i c c o m p u t a t i o n . Babylonian tablets o f the Seleucid p e r i o d (roughly the last three cen­ turies B.C.) preserve a text g i v i n g the procedure for c o m p u t i n g day lengths, at 3 0 ° intervals a l o n g the z o d i a c , i n System A . T h e longest day is t a k e n as 1 4 2 4 . T h i s corresponds to the ratio Ml m = 3/2, w h i c h is c o m m o n i n B a b y l o n i a n astronomy. T h e day lengths s h o w constant sec­ o n d differences o f 3 2 f r o m sign to sign. T h e preserved tablets give the day lengths directly, a n d d o not derive t h e m f r o m the r i s i n g times o f the signs. B u t there is little d o u b t that these day lengths were u n d e r s t o o d by the B a b y l o n i a n scribes as the r i s i n g times o f semicircles o f the ecliptic. A s already m e n t i o n e d , Hypsiklës applies System A to the klima o f A l e x a n ­ d r i a ( M = 1 4 ) a n d e x p l i c i t l y states the c o n n e c t i o n between the day lengths a n d the rising times. It is clear that G e m i n o s , t o o , used a n d u n d e r s t o o d the arithmetic m e t h o d s , f r o m his r e m a r k (vi 38) that the day lengths have constant sec­ o n d differences. Further evidence appears i n G e m i n o s ' s assertion (vii 36) that, for any sign, the s u m o f the r i s i n g time a n d the setting time is e q u a l to 4 e q u i n o c t i a l hours. In reality, this is a p p r o x i m a t e l y , but not r i g o r ­ ously, true; but this c o n d i t i o n is exactly satisfied i n the arithmetic systems (both A a n d B ) . T h i s is easily seen i n the second c o l u m n o f table 1.4. A r i e s sets w h i l e the d i a m e t r i c a l l y opposite sign, L i b r a , rises. So the s u m o f the r i s i n g a n d setting times for A r i e s is T + ( T + 5 D ) = 2 T + 5 D . T h e m

164

165

h

m

m

h

That is, by using Ptolemy's table of ascensions in Almagest i i 8. The day lengths ac­ cording to Ptolemy can also be obtained from his rising times in table 1.5, by adding the rising times for the appropriate six successive signs. Neugebauer [1955], vol. 1, 187 ( A C T 200, section 2) and i 214 ( A C T 200b, section 2). These are discussed in Neugebauer 1975, 3 6 9 - 7 1 . 1 6 4

1 6 5

Applications of Arithmetic Progressions

·

79

same t o t a l applies to every sign. If the expressions for Τ a n d D , given above, are substituted, w e see that the s u m o f the rising a n d setting times for any sign, a n d for any klima, is exactly 4 e q u i n o c t i a l h o u r s . Day Lengths Day by Day So far, w e have discussed arithmetic methods for finding the length o f the day w h e n the Sun is at the beginning of a z o d i a c sign. W h a t about other times o f year? O n e c o u l d proceed i n several different fashions. T h e c r u d ­ est possible treatment of the length o f the day w o u l d suppose a constant daily difference (i.e., constant first differences). T h e length o f the day w o u l d then increase by equal increments for each day between the w i n t e r solstice a n d the summer solstice. T h i s crude m e t h o d was a p p l i e d i n E g y p t as early as the twelfth century B . C . T h a t it was still i n use a m o n g G r e e k writers i n the second century B.C. is demonstrated by the Celestial Teach­ ing o f L e p t i n ë s . Leptinës assumes that the length o f the day increases by 1/45 h o u r for each of 180 days between w i n t e r a n d summer solstice. T h e Sun spends 2 days at the summer solstice a n d 3 days at the w i n t e r solstice w i t h o u t any change i n the length of the day, thus filling out the 3 6 5 days of the year. A s w e have seen, the same scheme was used i n the papyrus parapëgma o f about 3 0 0 B.C. s h o w n i n fig. 1.19. 1 6 6

167

But suppose w e w i s h to o b t a i n day lengths w i t h i n the B a b y l o n i a n scheme, i n w h i c h the rising times o f the signs are m o d e l e d by a n a r i t h ­ metic progression. A s w e have seen, this leads to day lengths at the be­ g i n n i n g o f successive signs that s h o w constant second differences. T h e s w i t c h f r o m a theory o f constant differences i n the length o f the day to a theory o f constant second differences represents a m a r k e d advance, not o n l y i n m a t h e m a t i c a l s o p h i s t i c a t i o n , but also i n the accuracy w i t h w h i c h the theory models the p h e n o m e n a . So, it m i g h t indeed be beneficial to find a w a y o f treating every successive day w i t h i n the scheme o f H y p ­ siklës a n d the B a b y l o n i a n s . H o w s h o u l d w e proceed i n this case? M o s t simply, one c o u l d interpolate linearly between the day lengths for the beginnings o f t w o successive z o d i a c signs. T h i s rule is given i n one o f the B a b y l o n i a n procedure t e x t s . L i n e a r i n t e r p o l a t i o n i n the day lengths a m o u n t s to assuming that the i n d i v i d u a l degrees o f a given sign a l l take equal times to rise. T h i s implies a step-function for the rising times o f i n d i v i d u a l degrees. A m o r e consistent a p p r o a c h w o u l d extend the premise about rising times f r o m w h o l e signs to i n d i v i d u a l degrees. T h i s m e t h o d is explicitly used by Hypsiklës for System A . T h u s , Hypsiklës assumes that the rising 168

Neugebauer 1975, 706. Blass 1887; Tannery 1893, 283-94, on p. 284. 168 Neugebauer [1955], vol. 1, 187 ( A C T 200, section 2). 166

1 6 7

80

·

Section 12

times o f the i n d i v i d u a l degrees f o r m an arithmetic progression. Suppose that the first degree o f A r i e s (from A r i e s 0 ° to A r i e s I ) rises i n time τ. T h e next degree (Aries I to 2 ° ) rises i n τ + δ, the next i n τ + 2 δ , a n d so o n , to the last degree o f V i r g o , w h i c h takes τ + 1 7 9 δ to rise. In the r e m a i n i n g six signs, there is a descending arithmetic progression. T h e first degree o f L i b r a takes τ + 1 7 9 δ to rise, a n d the last degree o f Pisces takes τ . T h e day length c a n be o b t a i n e d for any desired day by s u m m i n g the r i s i n g times for the a p p r o p r i a t e 1 8 0 ° . Hypsiklës does not go so far as to calculate i n d i v i d u a l day lengths, but this represents o n l y a simple exten­ s i o n o f his m e t h o d . L e t A denote the Sun's l o n g i t u d e , measured eastward from the winter solstice. T h e n it is s t r a i g h t f o r w a r d , t h o u g h tedious, to s h o w that, a c c o r d i n g to Hypsiklës' premises, o

o

1 6 9

L e n g t h o f day = m + Λ δ

(Λ a w h o l e number, 0 < A < 90)

2

w h i c h displays the q u a d r a t i c g r o w t h i n the length o f the day as w e m o v e f r o m w i n t e r solstice (A = 0) to s p r i n g e q u i n o x (A = 90). P u t t i n g A = 0 a n d A = 1 successively, w e see that the change i n the length o f the day f r o m one day to the next at the w i n t e r solstice is δ. B u t p u t t i n g Λ = 89 a n d A = 90 successively, w e find that the change i n the length o f the day f r o m one day to the next at the s p r i n g e q u i n o x is 179 δ. T h a t is, i n the degree-by-degree v e r s i o n o f System A , the d a i l y change i n the length o f the day at e q u i n o x is 179 times larger t h a n the d a i l y change i n the length of the day at the solstice.

Geminos's

Day Lengths,

Day by

Day

N o w w e are prepared to interpret G e m i n o s ' s remarks about day lengths. G e m i n o s f o l l o w s Hypsiklës i n t h i n k i n g a b o u t day lengths o n a degreeA s we did above when working with whole signs, we can obtain conditions on τ and δ by requiring that the length of the day be correctly represented at the equinox and the two solstices. The sums over the appropriate semicircles of the ecliptic give: 1 6 9

12 =180τ+16,110δ M = 1 8 0 τ + 24,210 δ m = 1 8 0 x + 8,010 δ. Η

These equations determine the parameters δ and τ: ô=(M-m)/16,200. τ = l / 1 5 - (179/32,400)(M - m). h

W i t h these two parameters, the rising time may be found for any single degree of the eclip­ tic. The rising times of whole signs may be found by summation. These w i l l agree exactly with the rising times given in the calculation above for System A , in which we worked with whole signs.

Applications of Arithmetic Progressions

·

81

TABLE 1.7

Day Lengths according to Geminos's Rule Day η

Length of Day

0 (winter) solstice) 1

Diff.

2nd Diff.

m % m +χ

χ 2x

2

m + χ + 2x

χ 3x

3

m + χ + 2x + 3x

88

ra

+ % - - - + 88% 89%

89

m + x · · · + 88* + 89%

χ 90x

90 (equinox)

m + χ · · · + 88% + 89% + 90%

by-degree or day-by-day basis. B u t rather t h a n d e r i v i n g his day lengths f r o m first principles (rising times), G e m i n o s s i m p l y i n v o k e s the "fact" that the changes i n the length of the day f o r m a n arithmetic progression. In v i 3 8 , G e m i n o s says that the d a i l y increase i n the length o f the day a r o u n d the e q u i n o x is about 9 0 times the d a i l y increase a r o u n d the sol­ stices. T h a t is, the difference between the e q u i n o c t i a l day a n d the f o l l o w ­ i n g day is 9 0 times larger t h a n the difference between the solstitial day a n d its f o l l o w i n g day. M o r e o v e r , he says that the second differences are constant. H i s w a y o f t h i n k i n g seems to require the scheme for the day lengths illustrated i n table 1.7. N o t e that table 1.7 satisfies b o t h o f G e m i ­ n o s ' assertions: (1) the d a i l y change i n the length o f the day a r o u n d the e q u i n o x is 9 0 times the d a i l y change a r o u n d the s o l s t i c e a n d (2) the d a i l y changes f o r m an arithmetic progression, i.e., have constant differ­ ences o f x . 170

1 7 1

In contrast, a strict application of System A w o u l d lead, as we have seen, to 179 rather than 90. The general expression for the length of day η in Geminos's scheme is thus 1 7 0

1 7 1

Length of day = m + Vi n(n + l)x. This should be compared with the corresponding expression from System A : Length of day = m + A

2

b.

Geminos's rule w i l l reproduce the 1 2 day at equinox (η = Λ = 90) if h

x = (180/91) δ, in which case the two systems are virtually identical.

82

·

Section 13 TABLE 1.8

Day Lengths in Geminos's Scheme for the Klima of 1 5

Day

η

Summer solstice

0 30 60 90

Equinox

h

Day Length in Geminos's Scheme 15 14 13 12

h

00.0 39.6 39.6 00.0

m

T h i s scheme m a y be used to determine n u m e r i c a l values for the lengths o f a l l the days o f the year, once m is specified. T h e procedure w o u l d be very s i m i l a r to that already illustrated for System A . So, rather t h a n g o i n g t h r o u g h it again, w e merely present some results by w a y o f e x a m p l e . It turns out that the parameter χ = ( M - ra)/8190. F o r the klima o f 1 5 this becomes χ = 3 / 4 0 9 5 . Table 1.8 gives some n u m e r i c a l results for this klima. These figures m a y be c o m p a r e d w i t h those i n table 1.6. G e m i n o s ' s scheme is nearly, t h o u g h not exactly, equivalent to the degree-by-degree version o f System A . h

h

T o s u m u p , G e m i n o s a p p r o p r i a t e d a c o n c l u s i o n f r o m System A — t h a t the changes i n day length f o r m an arithmetic progression f r o m solstice to e q u i n o x — a n d used the arithmetic progression as a w a y o f i n t e r p o l a t i n g between the e q u i n o c t i a l a n d solstitial day lengths. B u t he d i d not bother to go b a c k to first p r i n c i p l e s , i.e., to derive his day lengths f r o m the ris­ ing times o f i n d i v i d u a l degrees o f the ecliptic. Rather, he s i m p l y p o s t u ­ lated that the d a i l y change i n day length s h o u l d be 90 times larger at the e q u i n o x t h a n at the solstice.

13. L U N A R AND LUNISOLAR CYCLES

In chapter v i i i G e m i n o s takes up l u n i s o l a r cycles, n o t a b l y the 8- a n d 19year cycles. Scholars have disagreed i n their assessments o f h o w m u c h o f G e m i n o s ' s treatment represents faithful h i s t o r i c a l r e p o r t i n g a n d h o w m u c h is merely his o w n pedagogy. Nevertheless, G e m i n o s ' s account re­ mains o u r most detailed discussion o f this topic i n G r e e k . In chapter x v i i i , G e m i n o s returns to l u n a r cycles i n his discussion o f the exeligmos, a cycle useful i n the p r e d i c t i o n o f eclipses.

The

Octaetëris

T h e octaetëris is an 8-year l u n i s o l a r cycle. In p r i n c i p l e , the octaetëris contains a w h o l e n u m b e r o f t r o p i c a l years a n d a w h o l e n u m b e r o f syn-

Lunar and Lunisolar Cycles

·

83

TABLE 1.9

First N e w M o o n s of the Years 1961-90 Year 1961 62 63 64 65 66 67 68 69 70 71 72 73 74 75

Date of First New Moon January January January January January January January January January January January January January January January

16 6 25 14 2 21 10 29 18 7 26 16 4 23 12

Year

Date of First New Moon

1976 77 78 79 80 81 82 83 84 85 86 87 88 89 90

January January January January January January January January January January January January January January January

1 19 9 28 17 6 25 14 3 21 10 29 19 7 26

o d i c m o n t h s . (The t r o p i c a l year is the time f r o m one s u m m e r solstice to the next, i.e., the time for the S u n to m a k e one t r i p a r o u n d the z o d i a c . The s y n o d i c m o n t h is the time f r o m one n e w M o o n to the next.) T h u s , after 8 years, b o t h the Sun a n d the M o o n return, very nearly, to their o r i g i n a l positions i n the z o d i a c . T h e discussion c a n be m a d e m o r e c o n ­ crete by the use o f table 1.9, w h i c h gives the date o f the first n e w M o o n for each year f r o m 1 9 6 1 to 1 9 9 0 . The t r o p i c a l year does not c o n t a i n a w h o l e n u m b e r o f synodic m o n t h s . T h u s , the date o f the n e w M o o n o f J a n u a r y does not repeat f r o m one year to the next. Rather, as G e m i n o s says (viii 28), 12 synodic months c o n t a i n o n l y 3 5 4 days, w h i l e the solar year contains 3 6 5 / 4 . T h u s , there is an lVA-day difference. So, between 1963 a n d 1 9 6 4 , the date o f the J a n u ­ ary n e w M o o n suffers an 11-day shift. Table 1.9 shows that the shift f r o m one year to the next m a y be 10, 1 1 , or 12 days. Because 12 synodic m o n t h s are IIV4 days too short by c o m p a r i s o n w i t h the solar year, the n e w M o o n o f J a n u a r y occurs progressively earlier i n the m o n t h (until it reaches the beginning o f the m o n t h a n d w e start over). STRUCTURE OF THE

OCTAETËRIS

In 8 years, the 1 1 / 4 - d a y a n n u a l difference a m o u n t s to 8 x 1 1 / 4 = 9 0 days, o r very nearly 3 s y n o d i c m o n t h s (viii 2 9 ) . T h u s , w e m i g h t expect 8 years to c o n t a i n nearly a w h o l e n u m b e r o f s y n o d i c m o n t h s . There w o u l d

84

·

Section 13

be 8 χ 12 plus the 3 e m b o l i s m i c (or " l e a p " ) m o n t h s , f o r a t o t a l o f 99 m o n t h s (viii 2 7 ) . T h e existence o f a n 8-year cycle is c o n f i r m e d b y e x a m i ­ n a t i o n o f table 1.9. If w e p i c k a n y 2 years that are 8 years apart, w e find that the J a n u a r y n e w M o o n falls o n n e a r l y the same date i n each. H e r e are the essential features o f G e m i n o s ' s octaetëris: T H E OCTAETËRIS ACCORDING TO GEMINOS (VIII 27-31)

8 years = 99 m o n t h s = 2 , 9 2 2 days, c o m p r i s i n g 5 years o f 12 m o n t h s (354 days each) a n d 3 years o f 13 m o n t h s (384 days each). T h e r e are 51 full a n d 4 8 h o l l o w m o n t h s : the o r d i n a r y m o n t h s are alternately full a n d h o l l o w , but a l l the e m b o l i s m i c m o n t h s are full. A d o p t e d length o f year: ?>65 A days. R e s u l t i n g length o f m o n t h : 2 9 + 1/2 + 1/66 days. X

H e r e a n d i n w h a t f o l l o w s , " a d o p t e d l e n g t h " identifies a parameter se­ lected for the c o n s t r u c t i o n o f the cycle. " R e s u l t i n g l e n g t h " is a value that falls o u t as a result o f c o m p u t a t i o n f r o m the premises o n w h i c h the cycle is based. T h e resulting length o f the m o n t h i n the octaetëris is n o t m e n ­ t i o n e d b y G e m i n o s , b u t he is a w a r e that this m o n t h is t o o short, a n d he calculates the necessary c o r r e c t i o n at v i i i 37-39. DEFECTS OF THE

OCTAETËRIS

T a b l e 1.9 s h o w s that after 8 years the date o f the J a n u a r y n e w M o o n does n o t repeat exactly, b u t shifts f o r w a r d b y 1 o r 2 days. T h u s , the first n e w M o o n o f 1 9 6 1 fell o n J a n u a r y 1 6 , a n d the first n e w M o o n o f 1969 fell o n J a n u a r y 1 8 . G e m i n o s discusses the defects o f the octaetëris at v i i i 37-38. T o b e g i n , he i n t r o d u c e s a m o r e accurate v a l u e f o r the l e n g t h o f the s y n o d i c m o n t h . A c c o r d i n g t o G e m i n o s , the s y n o d i c m o n t h is n o t 29 /i days, b u t 2 9 + 1/2 + 1/33 days ( - 2 9 . 5 3 days, w h i c h is a g o o d value). So 99 s y n o d i c m o n t h s are therefore (29 + 1/2 + 1/33) x 99 = 2,923y2 days. B u t 8 solar years c o m e t o o n l y (365 /4) χ 8 = 2 , 9 2 2 days. T h u s , 99 synodic m o n t h s are longer t h a n 8 solar years b y lVi days. Because the 99 synodic m o n t h s are longer b y IV2 days i n c o m p a r i s o n w i t h the 8 solar years, the date o f the J a n u a r y n e w M o o n comes this m u c h later at the beginning o f the next 8-year cycle, as m a y be c o n f i r m e d by e x a m i n a t i o n o f table 1.9. A s G e m i n o s r e m a r k s , the octaetëris treats the solar year accurately, but involves a n error i n r e c k o n i n g the m o n t h s . l

!

The 16-Year

Period

G e m i n o s therefore proposes (viii 39) a m o d i f i c a t i o n o f the octaetëris: IV2 days must be a d d e d every 8 years, o r 3 days every 16 years, i n o r d e r t o

Lunar and Lunisolar Cycles

·

85

get a m o n t h o f the right length. T h i s results i n a 16-year p e r i o d , w h i c h consists o f 2 octaetêrides plus 3 extra days: GEMINOS'S FIRST MODIFICATION OF THE OCTAETËRIS:

16-YEAR PERIOD 16 years =

198 m o n t h s =

5 , 8 4 7 days

= 2 x 99 m o n t h s = 2 χ 2 , 9 2 2 + 3 days, c o m p r i s i n g 2 octaetêrides

plus 3 days.

A d o p t e d length o f m o n t h : 2 9 + 1/2 + 1/33 days. R e s u l t i n g length o f year: 3 6 5 + 1/4 + 3/16 days. T h e 16-year cycle is constructed to get the m o n t h exactly right (i.e., i n agreement w i t h the a d o p t e d value o f 2 9 + 1/2 + 1/33). B u t i n the process the year has been m a d e t o o l o n g . G e m i n o s does not m e n t i o n the resulting length o f the year, but i m m e d i a t e l y shows h o w to correct it.

The 160-Year

Period

G e m i n o s proposes (viii 4 0 - 4 1 ) a second m o d i f i c a t i o n o f the octaetëris, w h i c h results i n a 160-year cycle. In the octaetëris, the year w a s represented correctly. T h e n 3 days were a d d e d to each 16-year p e r i o d to get the m o n t h s right. T h u s the 16-year p e r i o d makes the year t o o l o n g by 3 days i n 16 years. Therefore, i n 160 years, 3 0 days t o o m a n y w i l l have been added, a n d it w i l l be time to d r o p a n entire m o n t h o f 3 0 days, to restore h a r m o n y w i t h respect to the S u n : GEMINOS'S SECOND MODIFICATION OF THE OCTAETËRIS:

160-YEAR PERIOD 160 years = 1,979 m o n t h s =

5 8 , 4 4 0 days

= 10 x 198 - 1 m o n t h s = 10 χ 5 , 8 4 7 - 3 0 days, c o m p r i s i n g ten 16-year periods m i n u s 1 full m o n t h . A d o p t e d length o f year: 3 6 5 -I-1/4 days. R e s u l t i n g length o f m o n t h : 2 9 + 1/2 + 1 1 9 / 3 , 9 5 8 days. T h e 160-year p e r i o d is again based o n a year o f 3 6 5 V i days. T h e m o n t h ( w h i c h was correctly represented i n the 16-year period) has been t h r o w n off, but o n l y by a little. G e m i n o s does n o t m e n t i o n the resulting length of the m o n t h . Table 1.10 s h o w s the cycles o f 8, 16, a n d 160 years as progressive (and progressively m o r e c o m p l i c a t e d ) a p p r o x i m a t i o n s to G e m i n o s ' s a d o p t e d lengths for the year a n d the m o n t h . A t each stage, one o f these t w o is represented c o r r e c t l y a n d the other is off a bit. B u t by the t h i r d iteration (the 160-year p e r i o d ) , b o t h the year a n d the m o n t h are very close to the a d o p t e d values. A t this stage (viii 4 2 ) , G e m i n o s t h r o w s u p his hands a n d

86

·

Section 13

TABLE 1.10 Year and M o n t h Lengths in the Octaetëris

and Related Cycles

Length of Year (days) Geminos's adopted value Octaetëris

365.25

16-year period

365 + 1/4 + 3/16 = 365.6875 365.25

365.25

160-year period

Length of Month (days) 29 + = 29 + = 29 +

1/2 + 1/33 29.53030 1/2 + 1/66 29.51515 1/2 + 1/33

29 + 1/2 + 119/3958 = 29.53007

declares the octaetëris a t o t a l failure, then passes o n to a d i s c u s s i o n o f the 19-year cycle. G e m i n o s ' s account o f the development o f the 8-, 16-, a n d 160-year cy­ cles s h o u l d not be t a k e n as history. Rather, his discussion is s i m p l y g o o d pedagogy. H e sets out the basic parameters o f the octaetëris, discusses the defects o f this cycle, then shows h o w one m i g h t i m p r o v e it by succes­ sive refinements. These successive refinements d o not depend o n succes­ sively better observations. Rather, they are successively better a r i t h m e t i ­ cal a p p r o x i m a t i o n s to p r e v i o u s l y a d o p t e d lengths for the s y n o d i c m o n t h a n d the t r o p i c a l year. Origin

of the

Octaetëris

G e m i n o s does not attribute the i n v e n t i o n o f the octaetëris to any p a r t i c ­ u l a r writer. Diogenes L a e r t i o s says that E u d o x o s w r o t e o n the octaetëris. Censorinus (third century A . D . ) , says that the octaetëris w a s c o m m o n l y attributed to E u d o x o s , but that some writers ascribed it to K l e o s t r a t o s (late s i x t h century B.C.?). C e n s o r i n u s goes o n to say that others have presented their o w n versions o f the octaetëris, based u p o n different patterns for the e m b o l i s m i c m o n t h s , a n d mentions H a r p a l u s , Nauteles, and Menestratus. C e n s o r i n u s concludes his list o f a t t r i b u ­ tions by saying that the octaetëris of E u d o x o s is frequently attributed to D o s i t h e u s (late t h i r d century B.C.). A n d G e m i n o s (viii 24) k n e w o f a treatise o n the octaetëris by Eratosthenes. A l l w e c a n be sure o f is that 171

173

174

Diogenes Laertios, Lives and Opinions viii 87. For the fragments of Eudoxos's work on this subject, see Lasserre 1966, 75ff. Censorinus, On the Day of Birth 18.5. Nauteles and Menestratus are otherwise unknown. For a summary of the specula­ tions on the identity and dates of Harpalus, see Samuel 1972, 39. 172

173

174

Lunar and Lunisolar Cycles ·

87

the octaetëris w a s i n t r o d u c e d earlier t h a n 4 3 2 B.C., the date associated w i t h the beginning o f M e t ô n ' s 19-year cycle. O n these g r o u n d s , the o n l y viable candidate o f those m e n t i o n e d by C e n s o r i n u s is K l e o s t r a t o s . Pliny claims that after A n a x i m a n d r o s discovered the o b l i q u i t y o f the z o d i a c , it w a s K l e o s t r a t o s w h o d i v i d e d it i n t o signs. O f course, this makes little sense, for we k n o w that the equal-sign z o d i a c w a s b o r r o w e d by the Greeks f r o m the B a b y l o n i a n s . F r o m a l l this, it is clear that by Censorinus's time, a n d even by the time o f G e m i n o s , n o one h a d any sure k n o w l e d g e o f the early h i s t o r y o f the octaetëris. T h e o n l y safe course is to leave the question o f its o r i g i n u n a n s w e r e d . 1 7 5

176

177

The 19-Year

Period

T h e existence o f a 19-year p e r i o d is easily discerned i n table 1.9. In any 2 years separated by 19 years, the date o f the J a n u a r y n e w M o o n repeats almost exactly: NINETEEN-YEAR PERIOD ACCORDING TO GEMINOS (VIII 50-54)

19 years = 2 3 5 m o n t h s = 6,940 days, c o m p r i s i n g 12 years o f 12 m o n t h s a n d 7 years o f 13 m o n t h s . There are 125 full a n d 110 h o l l o w m o n t h s . R e s u l t i n g length of year: 365 5/19 days. R e s u l t i n g length o f m o n t h : 2 9 + 1/2 + 3/94 days. G e m i n o s does n o t m e n t i o n the resulting length of the m o n t h , but m e n ­ tions the length o f the year at v i i i 5 1 . G e m i n o s asserts (viii 55) that the arrangement o f full a n d h o l l o w m o n t h s s h o u l d be as u n i f o r m as possible. There are 6,940 days i n the 19-year p e r i o d a n d 110 h o l l o w m o n t h s . If a l l the m o n t h s were tem­ p o r a r i l y considered as full, it w o u l d therefore be necessary to remove a day after every r u n o f 63 days ( 6 , 9 4 0 / 1 1 0 = 63). T h a t is, every 6 4 t h day n u m b e r w o u l d be r e m o v e d . A c c o r d i n g to G e m i n o s , the 3 0 t h day o f the m o n t h is not a l w a y s the one scheduled for r e m o v a l . Rather, the h o l l o w m o n t h is p r o d u c e d by r e m o v i n g w h i c h e v e r day falls after the r u n n i n g 63-day count. S u c h a procedure w o u l d have e n o r m o u s l y c o m p l i c a t e d the c o n s t r u c t i o n o f a calendar. N e u g e b a u e r therefore doubts that this rule w a s ever f o l l o w e d . G e m i n o s , however, is u n a m b i g u o u s o n this p o i n t , 178

Pliny, Natural History ii 6.31. A summary of the speculations of modern writers on the origin of the octaetëris given in Samuel 1972, 39-40. See also Heath 1913, 287-92. Neugebauer 1975, 620-21. Neugebauer 1975, 617. 1 7 5

1 7 6

1 7 7

1 7 8

is

88

·

Section 13

a n d t w o efforts at a r e c o n s t r u c t i o n o f the M e t o n i c cycle have t a k e n h i m at his w o r d .

1 7 9

HISTORY OF THE 19-YEAR PERIOD G e m i n o s (viii 50) attributes the 19-year p e r i o d to "the astronomers around Euktëmôn, Philippos, and K a l l i p p o s . " Philippos and Kallippos b o t h l i v e d a b o u t a century t o o late to have been i n v o l v e d i n the o r i g i n a l c o n s t r u c t i o n o f the 19-year p e r i o d . Perhaps G e m i n o s means o n l y to say that they w r o t e o n this subject. T h e n a m e K a l l i p p o s is, o f course, associ­ ated w i t h the 76-year p e r i o d , m a d e u p o f four 19-year p e r i o d s . T h e c o n ­ s t r u c t i o n o f the 19-year cycle is a t t r i b u t e d b y m o s t m o d e r n w r i t e r s to M e t ó n , w h o m G e m i n o s does n o t m e n t i o n , o r to M e t ó n a n d Euktëmôn. M o d e r n w r i t e r s often refer to this as the " M e t o n i c c y c l e . " T h e ancient t e s t i m o n y is r e a s o n a b l y c o n v i n c i n g . D i o d ô r o s says that M e t ô n i n t r o d u c e d the 19-year p e r i o d at A t h e n s , b e g i n n i n g at the s u m ­ m e r solstice o f the year 4 3 2 B.C. P t o l e m y says that this s u m m e r s o l ­ stice w a s observed by "those a r o u n d M e t ô n a n d E u k t ë m ô n . " C e n s o r i ­ nus attributes the 19-year cycle to M e t ô n o f A t h e n s . 180

181

1 8 2

1 8 3

K n o w l e d g e o f the f u n d a m e n t a l r e l a t i o n (19 years = 2 3 5 s y n o d i c months) is attested i n B a b y l o n i a f r o m a b o u t 4 9 0 B . C . In contrast to the G r e e k s , w h o never a d o p t e d a fixed rule for the e m b o l i s m i c m o n t h s i n their c i v i l calendars, the B a b y l o n i a n s actually used the 19-year p e r i o d as the regulating p r i n c i p l e for their o w n calendar, certainly by 3 8 0 B . C . (in the reign o f A r t a x e r x e s II), a n d perhaps as early as 4 9 7 ( D a r i u s I). W e d o n o t k n o w whether the G r e e k astronomers b o r r o w e d the 19-year cycle f r o m the B a b y l o n i a n s o r a r r i v e d at it independently. B o r r o w i n g m a y be considered likely i n v i e w o f other demonstrated debts o f G r e e k astron­ o m y to B a b y l o n i a n practice. O n the other h a n d , the fundamental r e l a t i o n (235 m o n t h s = 1 9 years) is very s i m p l e , a n d w e k n o w f r o m the existence o f the older octaetëris that the G r e e k astronomers o f the fifth century were interested i n l u n i s o l a r cycles. T h e octaetëris, at least, appears to be indigenous to the G r e e k s , for there is n o evidence for a n 8-year cycle i n the preserved B a b y l o n i a n m a t e r i a l . 1 8 4

185

Fotheringham 1924; van der Waerden 1960. There is a large modern literature on the 19-year, or Metonic, cycle. For an introduc­ tion, see the following: Neugebauer 1975, 622-24; Samuel 1972, 42-49; Toomer, article " M e t o n " in Gillispie 1970-80; Bowen and Goldstein 1989. Diodôros of Sicily, Historical Library x i i 36.2. 182 Ptolemy, Almagest iii 1; Heiberg, i p. 205. Censorinus, On the Day of Birth 18.8. Neugebauer 1975, 354-57. Neugebauer 1975, 354. 179

180

181

183

184

185

Lunar and Lunisolar Cycles

·

89

TABLE 1.11

Year and M o n t h Lengths in the 19- and 76-Year Periods

Geminos's adopted value 19-year period 76-year period

The 76-Year

Length of Year (days)

Length of Month (days)

365.25 days

29 + = 29 + = 29 + =

365 + 5/19 = 365.2632 365.25

1/2 + 1/33 29.53030 1/2 + 3/94 29.53191 1/2 + 29/940 29.53085

Period

M o d e r n w r i t e r s often refer to this p e r i o d as the K a l l i p p i c C y c l e . In the 19-year cycle, the solar year turns o u t to be 3 6 5 5/19 days, w h i c h is longer t h a n the a d o p t e d length o f the year (365V4 days) by 1/76 day. Therefore, after 76 years it w i l l be necessary to o m i t 1 d a y i n order to restore h a r m o n y w i t h the S u n . T h i s c a n be done by f o r m i n g a p e r i o d f r o m f o u r successive 19-year cycles, l e a v i n g o u t one d a y f r o m one o f the 19-year cycles: THE 76-YEAR PERIOD ACCORDING TO GEMINOS (VIII 59-60)

76 years =

940 months =

2 7 , 7 5 9 days,

= 4 x 2 3 5 m o n t h s = 4 χ 6,940 - 1 days, c o m p r i s i n g four 19-year periods, less 1 day. A d o p t e d length o f year: 3 6 5 + 1/4 days. R e s u l t i n g length o f m o n t h : 2 9 + 1/2 + 2 9 / 9 4 0 days. Table 1.11 shows the cycles o f 19 a n d 76 years as progressive a p p r o x i ­ m a t i o n s to G e m i n o s ' s a d o p t e d lengths for the year a n d the m o n t h . Historical

Purpose

of the 19- and 76-Year

Cycles

In B a b y l o n i a , the 19-year cycle was used to regulate the c i v i l calendar, perhaps as early as the fifth century B.C. In contrast, there is n o evidence that the A t h e n i a n calendar was ever regulated by any fixed l u n i s o l a r cy­ cle. T h i s raises the question o f w h a t role these cycles p l a y e d i n G r e e k as­ t r o n o m y . A n accurate l u n i s o l a r cycle h a d a p p l i c a t i o n s i n three d o m a i n s : the r e g u l a t i o n o f a parapëgma, the design o f artificial a s t r o n o m i c a l c a l ­ endars, a n d the e v a l u a t i o n o f parameters i m p o r t a n t i n the l u n a r theory. O f these three a p p l i c a t i o n s , the r e g u l a t i o n o f a parapëgma p r o b a b l y pre­ sented the earliest stimulus for the investigation o f l u n i s o l a r cycles.

90

·

Section 13

The early Greek calendars were lunisolar, a n d the first day of each m o n t h was, i n principle, the n e w M o o n . F o r the Greeks, this meant the first appearance of the crescent M o o n i n the west after sunset. A particular calendar year c o u l d c o n t a i n either 12 or 13 months. T h u s , a given star phase i n the a n n u a l cycle (e.g., the m o r n i n g rising of the Pleiades) d i d not occur o n a fixed calendar date. M o r e o v e r , each major city used a different calendar, w i t h different m o n t h names a n d different starting points for the year. W o r s e yet, the civil authorities h a d substantial freedom to intercalate days a n d months at w i l l . T h u s the calendar was often significantly out of step w i t h the M o o n . A t h e n i a n writers therefore sometimes distinguished between "the n e w M o o n a c c o r d i n g to the goddess" (Selënë, the M o o n ) and "the n e w M o o n a c c o r d i n g to the a r c h o n " (the head magistrate o f the c i t y ) . W e might call these the actual a n d the calendrical n e w M o o n s . 186

T y i n g a particular calendar year to a parapëgma was therefore a delicate t a s k . T h i s is h o w a lunisolar cycle might be of use. T o o m e r has argued that w h e n M e t ô n introduced his 19-year cycle i n Athens, a r o u n d 4 3 2 B.C., he published it together w i t h a parapëgma that covered 19 years. T h e l u ­ nisolar cycle served to connect t w o completely different classes of p h e n o m ­ ena o n w h i c h Greek time-reckoning was based: the cycle of heliacal risings and settings (used for telling the time o f year) a n d the cycle o f M o o n phases (used for telling the time of month). In Metôn's cycle, suppose that, i n a certain year, the peg goes into the first hole of the parapëgma o n the day o f the first new M o o n after the summer solstice (the traditional begin­ n i n g o f the year at Athens). N i n e t e e n years later, the peg w i l l again go into the same hole o n the first n e w M o o n after summer solstice. F r o m the use o f a l u n i s o l a r cycle i n the r e g u l a t i o n o f a parapëgma to the c o n s t r u c t i o n o f a n independent a s t r o n o m i c a l calendar is but a s m a l l step. T h i s step m a y w e l l have been t a k e n by M e t ô n . A b o u t a century 187

1 8 8

189

1 9 0

186

Pritchett and Neugebauer 1947 is the most thorough treatment of the Athenian cal­

endar. A similar difficulty was experienced in Babylonian applications of luni solar cycles. Thus, in the parapëgma that forms a part of M U L . A P I N (seventh century B.C.), the dates are expressed in terms of Babylonian month names; but these must be interpreted as ficti­ tious months of an ideal or artificial year—not the months of any particular civil calendar year. See Hunger and Pingree 1989, 4 0 - 4 7 , 139-40. G . J . Toomer, article " M e t o n " in Gillispie 1970-80. It seems unlikely that Metón would have set up an inscription covering 19 years. But one should consider the example of the bronze calendar of Coligny (second century A . D . ) , which carried a 5-year cycle showing the correspondences between the R o m a n calendar and the Gallic or Celtic calendar, complete with holes for each of 1,835 days. See M c Clusky 1998, 54-57, or, for more detail, Duval and Pinault 1986. In Almagest iv 11 (Toomer 1984, 211-13), Ptolemy cites three lunar eclipses from the fourth century B.C., details of which he got from Hipparchos. These three eclipses, though observed at Babylon, are all dated in terms of Athenian month names and the years of Athenian archons. Toomer (1984, 12nl8) suggests that these dates should be understood as 1 8 7

188

189

190

Lunar and Lunisolar Cycles

·

91

later, it certainly w a s t a k e n b y K a l l i p p o s . In P t o l e m y ' s Almagest, a n u m ­ ber o f the older observations cited are dated i n terms o f the K a l l i p p i c cy­ cle. F o r e x a m p l e , P t o l e m y says that T i m o c h a r i s , at A l e x a n d r i a , observed a n o c c u l t a t i o n o f the Pleiades by the M o o n i n the 4 7 t h year o f the F i r s t K a l l i p p i c 76-year p e r i o d , o n the 8th o f Anthestëriôn, w h i c h is A t h y r 2 9 i n the E g y p t i a n calendar, t o w a r d the end o f the 3 r d h o u r [of the n i g h t ] . . . .

m

In K a l l i p p o s ' s calendar, the years were expressed i n terms o f their p o s i ­ t i o n i n the r u n n i n g c o u n t f r o m 1 to 76 i n the 76-year cycle. T h e m o n t h names were b o r r o w e d f r o m the A t h e n i a n calendar. B u t the calendar be­ i n g used w a s a n i d e a l , s t a n d a r d i z e d calendar, designed by astronomers, a n d never used b y any city for c i v i l purposes. P t o l e m y ' s d u a l d a t i n g o f these observations (i.e., his c o n v e r s i o n o f the K a l l i p p i c date to a n equiv­ alent i n the E g y p t i a n calendar) p r o v i d e s m o s t o f the concrete i n f o r m a ­ t i o n w e have a b o u t the f u n c t i o n i n g o f K a l l i p p o s ' s calendar. S c h o l a r l y at­ tempts to reconstruct K a l l i p p o s ' s a s t r o n o m i c a l calendar have been based o n the general o v e r v i e w p r o v i d e d by G e m i n o s (chapter viii) a n d the h a n d f u l o f equivalent dates p r o v i d e d b y P t o l e m y .

192

P t o l e m y gives full K a l l i p p i c d a t i n g (year o f K a l l i p p i c cycle plus A t h e n ­ i a n m o n t h name a n d day) for o n l y four observations o f the t h i r d century B.C. In a d d i t i o n , he gives some seventeen o b s e r v a t i o n reports, gleaned f r o m the w r i t i n g s o f H i p p a r c h o s , i n w h i c h the year is given i n terms o f its p o s i t i o n i n the K a l l i p p i c cycle, but n o A t h e n i a n m o n t h name is given. T h e last o f these is for a n observation i n 128 B.C. F o r this reason, it seemed that the K a l l i p p i c calendar was a b a n d o n e d by the end o f the second cen­ tury B.C. H o w e v e r , several n e w K a l l i p p i c dates o n p a p y r u s , recently p u b ­ lished, s h o w that the K a l l i p p i c calendar was still used (always i n an astro­ n o m i c a l context) as late as the m i d d l e o f the first century A . D .

1 9 3

T h e final a p p l i c a t i o n o f l u n i s o l a r cycles w a s i n the e v a l u a t i o n o f p a ­ rameters o f m e a n m o t i o n for the l u n a r theory. L u n i s o l a r cycles p l a y e d

given in terms of the Metonic calendar. Hipparchos, or some other Greek astronomer, was presumably responsible for converting the original dates, expressed in terms of the Baby­ lonian calendar, to equivalent dates in the Metonic calendar, which was an idealized ver­ sion of the Athenian calendar. Ptolemy, Almagest v i i 3 (Toomer 1984, 334). For a convenient list of all the dated observations in the Almagest, see Pedersen 1974, 4 0 8 - 2 2 . Samuel 1972, 4 2 - 4 9 , provides a good overview. Fotheringham 1924 and van der Waerden 1960 make ambitious attempts to reconstruct the Kallippic calendar. One's judg­ ment of their success depends greatly on one's faith in Geminos. Toomer (1984, 13) re­ gards Geminos's account as "fiction." We would call it "pedagogical," but agree with Toomer that one should not take it literally. Jones 2000. 1 9 1

1 9 2

1 9 3

92

·

Section 13

just such a role i n B a b y l o n i a n astronomy. In G r e e k astronomy, there p r o b a b l y w a s n o l u n a r theory w i t h predictive capacity u n t i l the time o f H i p p a r c h o s , a n d this a p p l i c a t i o n is therefore rather late. H i p p a r c h o s be­ g a n his o w n l u n a r theory w i t h the exeligmos, then passed o n to i m ­ p r o v e d parameters. B o t h the exeligmos a n d the i m p r o v e d parameters were d e r i v e d f r o m B a b y l o n i a n s o u r c e s . 194

The Exeligmos

(Geminos,

Chapter

XVIII)

T h e exeligmos is a p e r i o d o f 6 6 9 s y n o d i c m o n t h s that plays a role i n eclipse p r e d i c t i o n . T w o l u n a r eclipses separated by one exeligmos will have very s i m i l a r circumstances. In particular, the t w o eclipses w i l l o c c u r at the same node of the M o o n ' s o r b i t , w i l l have a b o u t the same m a g n i ­ tude a n d d u r a t i o n , a n d w i l l even o c c u r at r o u g h l y the same time o f day. T h u s , eclipses m a y be g r o u p e d i n t o cycles that repeat, at least to a fair a p p r o x i m a t i o n , after an exeligmos. T h e G r e e k w o r d exeligmos means "revolution." Toomer translates it as " t u r n o f the w h e e l . " H i l d e r i c u s , i n his L a t i n t r a n s l a t i o n o f G e m i n o s (1590), used evolutio, Ά " r o l l i n g . " T h e t e r m thus indicates a cycle, a restoration o f things to their o r i g i n a l state. F o r the discussion that f o l l o w s , it is necessary to d i s t i n g u i s h several kinds of month. 1 9 5

SIDEREAL MONTH T h e average time required for the M o o n to go f r o m one fixed star, a l l the w a y a r o u n d the z o d i a c , a n d back to the same star is c a l l e d the sidereal month. T h i s is the p e r i o d that must be used i n d y n a m i c a l c a l c u l a t i o n s i n ­ v o l v i n g N e w t o n ' s l a w o f m o t i o n . A l t h o u g h the sidereal m o n t h m a y seem the m o s t n a t u r a l to m a n y m o d e r n readers, it w a s rarely used by the G r e e k astronomers. It is not used, for e x a m p l e , i n Ptolemy's Almagest. G e m i n o s (i 30) says that the M o o n moves a r o u n d the z o d i a c i n 2 7 V3 days. H o w e v e r , Leptinës gives a r o u n d figure o f 2 7 d a y s . 196

197

SYNODIC MONTH T h e time f r o m n e w M o o n to the next n e w M o o n is c a l l e d the synodic month. T h i s is always w h a t G e m i n o s means w h e n he refers to the Our source for Hipparchos's treatment of the Moon's mean motions is Ptolemy, Al­ magest iv 2. Ptolemy, however, did not appreciate the full extent of Hipparchos's depend­ ence on the Babylonian tradition for his numerical parameters. See Toomer 1984, 1 7 6 n l 0 , and Neugebauer 1975, 309-12. Toomer 1984, 175n8. Pedersen 1974, 160. Leptinës, Celestial Teaching, in Tannery 1893, 286. 194

195

196

197

Lunar and Lunisolar Cycles ·

93

" m o n t h " o r the " m o n t h l y p e r i o d . " If the M o o n starts out i n c o n j u n c t i o n w i t h the S u n , it must travel m o r e t h a n 3 6 0 ° w i t h respect to the fixed stars i n order to catch up w i t h the S u n again, for i n the course o f a m o n t h , the S u n moves nearly 3 0 ° f o r w a r d o n the ecliptic. So, as G e m i ­ nos says (viii 2), i n the course o f a synodic m o n t h the M o o n moves t h r o u g h a p p r o x i m a t e l y 13 z o d i a c signs. T h u s , the s y n o d i c m o n t h is r o u g h l y 2 days longer t h a n the sidereal m o n t h . TROPICAL MONTH

T h e time required for the M o o n to go f r o m the s u m m e r solstitial p o i n t , all the w a y a r o u n d the z o d i a c , a n d back to the same solstitial p o i n t is called the tropical month. If the equinoxes a n d solstices were fixed w i t h respect to the stars, the t r o p i c a l a n d sidereal m o n t h s w o u l d be the same. But the equinoxes a n d solstices precess w e s t w a r d at the rate o f I i n 72 years ( I i n 100 years a c c o r d i n g to H i p p a r c h o s a n d P t o l e m y ) . Therefore, if the M o o n starts at the s u m m e r solstitial p o i n t , it has to travel a t i n y bit less t h a n 3 6 0 ° w i t h respect to the fixed stars i n order to m a k e it back to the solstitial p o i n t . T h u s , the t r o p i c a l m o n t h is a little shorter t h a n the sidereal m o n t h . Because m o d e r n astronomers, like their G r e e k predeces­ sors, use the s p r i n g e q u i n o c t i a l p o i n t as the zero o f longitude, a l l p r a c t i ­ cal calculations o f l u n a r positions must m a k e use o f the t r o p i c a l m o n t h . F o r e x a m p l e , Ptolemy's tables for c a l c u l a t i n g the M o o n ' s longitude f r o m deferent-and-epicycle theory are based o n the t r o p i c a l m o n t h . B u t G e m i ­ nos never mentions precession, a n d thus makes n o d i s t i n c t i o n between the sidereal a n d t r o p i c a l m o n t h s . o

o

ANOMALISTIC MONTH

T h e m o t i o n o f the M o o n o n its p a t h a r o u n d the z o d i a c is, as G e m i n o s says, n o n u n i f o r m . In the course o f one 2 4 - h o u r day, the M o o n m a y m o v e as m u c h as 1 5 ° o r as little as 1 1 ° , i n r o u g h , r o u n d numbers. T h e time between least m o t i o n a n d the next least m o t i o n is the anomalistic month. G e m i n o s explains this clearly at x v i i i 2 . W h e n he mentions a "re­ t u r n " o f the M o o n , he always means the a n o m a l i s t i c m o n t h . F r o m a m o d e r n p o i n t o f v i e w the a n o m a l i s t i c m o n t h m a y be e x p l a i n e d as fol­ l o w s . T h e M o o n moves most s l o w l y w h e n it is at apogee o n its K e p l e r i a n ellipse. If the elliptical orbit were fixed i n space, the a n o m a l i s t i c m o n t h w o u l d be the same as the sidereal m o n t h . B u t the g r a v i t a t i o n a l influence of the S u n perturbs the orbit a n d causes the ellipse to rotate s l o w l y i n its o w n plane. T h e line o f apsides (the line d r a w n t h r o u g h the apogee, the E a r t h , a n d the perigee), makes one complete r o t a t i o n i n a b o u t 9 years. It turns eastward, i.e., i n the same d i r e c t i o n as the M o o n moves o n its or­ bit. Consequently, w h e n the M o o n is at apogee, it has to travel slightly more t h a n 3 6 0 ° w i t h respect to the fixed stars i n order to reach apogee

94

·

Section 13

again. T h u s , the a n o m a l i s t i c m o n t h is a little longer t h a n the sidereal month. DRACONITIC MONTH

Eclipses c a n o c c u r o n l y w h e n the M o o n is i n the plane o f the ecliptic, o r nearly so. Since the M o o n ' s o r b i t is i n c l i n e d to the ecliptic by a b o u t 5 ° , the o r b i t crosses t h r o u g h the ecliptic plane at t w o points called the nodes of the o r b i t . T h i s is w h y there are not eclipses o f the M o o n at every full M o o n . A t m o s t full M o o n s , the M o o n is either above o r b e l o w the plane of the ecliptic a n d so does not fall i n t o the Earth's shadow. A l u n a r eclipse c a n o c c u r o n l y i f the M o o n happens to be full w h e n it is near one of the t w o nodes. T h e time r e q u i r e d for the M o o n to go f r o m one node of its o r b i t , a l l the w a y a r o u n d the E a r t h , a n d back to the same node is c a l l e d a draconitic (or nodical) month. If M o o n ' s o r b i t were fixed i n space, the d r a c o n i t i c m o n t h w o u l d be the same as the sidereal m o n t h . But a p e r t u r b a t i o n by the S u n causes the plane o f the o r b i t to precess s l o w l y a b o u t the pole o f the ecliptic, at the rate o f one r e v o l u t i o n i n a b o u t 18 years. T h e nodes m o v e w e s t w a r d , opposite to the M o o n ' s m o ­ t i o n i n its o r b i t . Consequently, the d r a c o n i t i c m o n t h is a bit shorter t h a n a sidereal m o n t h . T h o u g h G e m i n o s does not allude to the d r a c o n i t i c m o n t h , it is an i m p o r t a n t part o f the full expression o f the exeligmos eclipse cycle, b o t h i n P t o l e m y a n d i n B a b y l o n i a n astronomy. Some n u m e r i c a l values (for the e p o c h A . D . 1900) m a y help to illustrate the relationships a m o n g the five m o n t h s : 198

draconitic month tropical sidereal anomalistic synodic

Geminos

on the

27.212 27.321 27.321 27.554 29.530

2 2 0 days 582 661 551 589

Exeligmos

G e m i n o s takes up the exeligmos i n chapter x v i i i . T h e essential statement, f r o m w h i c h a l l the rest o f that chapter f o l l o w s , is at x v i i i 3. G e m i n o s says that the exeligmos contains a w h o l e n u m b e r o f synodic months a n d a w h o l e n u m b e r of anomalistic m o n t h s . H e also gives the n u m b e r o f days i n the cycle a n d the m o t i o n o f the M o o n i n longitude d u r i n g this p e r i o d : 6 6 9 synodic m o n t h s = 7 1 7 a n o m a l i s t i c m o n t h s = 1 9 , 7 5 6 days = 7 2 3 t r o p i c a l r e v o l u t i o n s plus 3 2 ° . 198

Smart 1977, 420.

Lunar and Lunisolar Cycles

·

95

T h e s y n o d i c m o n t h a n d the a n o m a l i s t i c m o n t h represent t w o p e r i o d i c p h e n o m e n a w i t h slightly different periods. In such a case, one c a n a l ­ w a y s find a longer time interval that contains very nearly a w h o l e n u m ­ ber o f b o t h periods. It is a r e m a r k a b l e circumstance that the exeligmos also contains very nearly a w h o l e n u m b e r of draconitic months: 669 synodic - 7 1 7 anomalistic ~ 726 draconitic months. It is c u r i o u s that G e m i n o s does not m e n t i o n the equality w i t h 7 2 6 d r a ­ conitic m o n t h s , since this is o f central i m p o r t a n c e for the use o f the ex­ eligmos i n eclipse p r e d i c t i o n . Ptolemy, i n his o w n discussion o f the ex­ eligmos, gives a l l the n u m e r i c a l i n f o r m a t i o n that G e m i n o s ( x v i i i 3) does, but also includes the equality to 7 2 6 d r a c o n i t i c m o n t h s . 199

Astronomical

Significance

of the

Exeligmos

Suppose the M o o n is i n eclipse. Since exeligmos contains a w h o l e n u m ­ ber o f synodic m o n t h s , after one exeligmos the M o o n w i l l again be full, o r nearly so. T h e exeligmos also contains a w h o l e n u m b e r o f d r a c o n i t i c m o n t h s , so, after one exeligmos, the M o o n w i l l again be near the same node o f its o r b i t . T h u s , we have a very g o o d chance of finding a n eclipse. O n e possible difficulty is the M o o n ' s variable speed i n its o r b i t . Since the a n o m a l i s t i c m o n t h is o f a different length t h a n either o f the other t w o , it is possible that the M o o n w i l l be s o m e w h a t ahead o f or b e h i n d the place it needs to be to p r o d u c e an eclipse. Because o f the v a r i a t i o n i n speed, the M o o n , i n some parts o f its orbit, c a n be m o r e t h a n 6 ° ahead o f or be­ h i n d the place it w o u l d o c c u p y i f it m o v e d u n i f o r m l y . B u t this diffi­ culty is largely r e m o v e d by the fact that the exeligmos contains a w h o l e n u m b e r o f a n o m a l i s t i c m o n t h s . T h u s the c o n d i t i o n s o f a l u n a r eclipse s h o u l d repeat almost exactly after one exeligmos. M o r e o v e r , since an ex­ eligmos also contains a w h o l e n u m b e r o f days, the eclipse w i l l even oc­ cur at a b o u t the same time o f day as before. B u t because the M o o n moves 3 2 ° i n m e a n longitude, over a n d above complete cycles, d u r i n g an exeligmos, the second eclipse w i l l o c c u r a p p r o x i m a t e l y one z o d i a c sign farther east t h a n d i d the first one. 200

T h e exeligmos is three times as l o n g as a p e r i o d that t o d a y is usually called the saros. A l t h o u g h the saros cycle is ancient, the t e r m " s a r o s " is not: E d m o n d H a l l e y seems to have been the first to a p p l y this t e r m to a n Ptolemy, Almagest iv 2. In modern celestial mechanics, the leading term of the M o o n ' s equation of center is (in radian measure) 2esin(27ti/T4), where t is the time elapsed since the mean M o o n passed perigee, T is the anomalistic month, and e is the eccentricity of the M o o n ' s ellipse. Since e is about 0.054, the maximum value of this expression is about 6.2°, and obtains when the mean M o o n is ±90° from perigee. 1 9 9

2 0 0

A

96

·

Section 13

eclipse cycle, i n 1 6 9 1 . In the Almagest, P t o l e m y refers to the saros as the " p e r i o d i c t i m e " (periodikos chronos) a n d gives it the f o l l o w i n g p r o p ­ erties: 2 0 1

202

223 synodic = 239 anomalistic = 242 draconitic months = 6,585V3 days = 2 4 1 r e v o l u t i o n s i n longitude plus 1 0 / 3 ° . 2

Origin

of the Saros and

Exeligmos

P t o l e m y ascribes the saros to u n n a m e d astronomers " m o r e a n c i e n t " t h a n H i p p a r c h o s . T h e n , as P t o l e m y says, i n order to o b t a i n a p e r i o d w i t h a w h o l e n u m b e r o f days, they (the ancients), m u l t i p l i e d the saros by 3, w h i c h yields the exeligmos. G e m i n o s is nearer the m a r k . F o r a l t h o u g h he does not e x p l i c i t l y attribute the exeligmos to the C h a l d e a n s ( B a b y l o n ­ i a n astronomers), he does ( x v i i i 9) m e n t i o n t h e m i n c o n n e c t i o n w i t h the value o f the m e a n d a i l y m o t i o n deduced f r o m the exeligmos. Indeed, the saros is w e l l attested i n B a b y l o n i a n a s t r o n o m y . 203

Geminos's

Application

of the Exeligmos

to the Lunar

Theory

T h e b u l k o f G e m i n o s ' s chapter x v i i i concerns the a p p l i c a t i o n o f the ex­ eligmos to the B a b y l o n i a n l u n a r t h e o r y . First ( x v i i i 6 - 9 ) , G e m i n o s de­ duces the m e a n d a i l y m o t i o n m i n l o n g i t u d e : 204

m = (723 χ 3 6 0 ° + 3 2 ° ) / 1 9 , 7 5 6 = 1 3 ; 1 0 , 3 5 7 d , to the nearest second o f angle. d

N e x t ( x v i i i 10), G e m i n o s computes the length T month:

A

T

o f the a n o m a l i s t i c

= 1 9 , 7 5 6 / 7 1 7 = 2 7 ; 3 3 , 2 0 days. d

A

M o r e accurate arithmetic gives a substantially different value i n the sec­ o n d sexagesimal place: 2 7 ; 3 3 , 1 3 days. T h e reason for G e m i n o s ' s u n ­ u s u a l r o u n d i n g is i m m e d i a t e l y ( x v i i i 11) m a d e clear: he wants a r o u n d n u m b e r for AT : X

A

VAT = 2 7 ; 3 3 , 2 0 / 4 = 6;53,20 days. d

A

Neugebauer 1975, 497n2. 202 Ptolemy, Almagest i ν 2. Neugebauer 1975, 497-505. O n the Babylonian lunar theory, see Neugebauer 1975, 474-540, and Neugebauer [1955], 4 1 - 8 5 . 2 0 1

2 0 3

2 0 4

Lunar and Lunisolar Cycles

·

97

If the accurate value T = 2 7 ; 3 3 , 1 3 were used, the quotient AT w o u l d not truncate at the second sexagesimal place. M o r e o v e r , G e m i n o s is f o l ­ l o w i n g the B a b y l o n i a n l u n a r theory of system B , i n w h i c h the value T = 2 7 ; 3 3 , 2 0 days is standard. In a time equal to AT , the M o o n is as­ sumed to go f r o m least m o t i o n to m e a n m o t i o n , o r f r o m m e a n m o t i o n to greatest m o t i o n . G e m i n o s also assumes that the M o o n changes its speed o n the z o d i a c i n a linear way. L e t s be the least angle t h r o u g h w h i c h the M o o n moves i n the course o f a day. T h e next day, the M o o n is assumed to m o v e s + χ; the next day, s + 2x; then s + 3x a n d so o n . T h u s the d a i l y m o v e ­ ments f o r m a n a r i t h m e t i c p r o g r e s s i o n w i t h constant differences x. T h e r e m a i n i n g p o r t i o n o f C h a p t e r x v i i i is devoted to finding x, the d a i l y change i n the M o o n ' s d a i l y m o t i o n . Indeed, χ is the n u m b e r that "one must find," subject to the c o n d i t i o n s that G e m i n o s states i n x v i i i 1 6 - 1 7 . d

X

A

A

l

A

A

9

Let us define a few quantities: s, ra, g, T ,

the the the the

x,

the change i n the M o o n ' s d a i l y m o t i o n f r o m one day to the next.

A

M o o n ' s smallest daily m o t i o n M o o n ' s mean daily m o t i o n M o o n ' s greatest d a i l y m o t i o n anomalistic month

N o w w e m a y state the c o n d i t i o n s o f the p r o b l e m posed by G e m i n o s : ra = 1 3 ; 1 0 , 3 5 ° TA = 2 7 ; 3 3 , 2 0

( x v i i i 9) ( x v i i i 10)

ll°

\

\

Diagram 4. Trine aspect. Marcianus gr. 323, folio 482.

south w i n d b l o w s w h i l e the M o o n is i n one of the three signs, the same c o n d i t i o n lasts for m a n y days. 11 T h e next triangle, the one beginning w i t h G e m i n i , is called western for a similar reason. T h e final triangle, the one beginning w i t h Cancer, is called eastern for the same reason. I I 1 2 T h e triangles, t o o , are used i n c o n n e c t i o n w i t h sympathies i n n a ­ tivities. F o r those b o r n i n trine seem to be i n s y m p a t h y w i t h one another; a n d the situations o f the stars [i.e., planets] i n the same triangles seem to act together to i m p r o v e o r to w o r s e n the nativities.

Aspects of the Zodiacal Signs

·

129

I I 1 3 Sympathies arise i n three w a y s : i n o p p o s i t i o n , trine, a n d quartile: i n any other separation there is n o sympathy. 14 A n d yet it w o u l d be l o g i c a l that there be sympathies f r o m the signs l y i n g closest together; for the o u t p o u r i n g o r e m a n a t i o n proceeding f r o m the peculiar p o w e r o f each star s h o u l d most o f a l l c o l o r a n d m i x w i t h the n e i g h b o r i n g signs. 15 N o w , just as triangles a n d squares are inscribed i n the circle, so t o o [are the] hexagon, octagon, a n d dodecagon. H o w e v e r , there is n o sympathy corresponding to the i n s c r i p t i o n o f these, but o n l y for the aforementioned cases, since a certain natural sympathy exists [only] i n such intervals. 8

QUARTILE

II 16 In quartile are A r i e s , Cancer, L i b r a , C a p r i c o r n ; T a u r u s , L e o , S c o r p i o , A q u a r i u s ; G e m i n i , V i r g o , Sagittarius, P i s c e s — w h i c h is a l l 3 [possible] squares. T h e side o f the square subtends 3 signs o r 9 0 degrees. 17 C a l l e d the first square is the one starting f r o m A r i e s , i n w h i c h the [beginnings o f the] seasons o c c u r — s p r i n g , summer, f a l l , a n d winter. C a l l e d the second square is the one starting f r o m T a u r u s , i n w h i c h the seasons have a m i d p o i n t — o f s p r i n g , of summer, o f f a l l , o f w i n t e r . C a l l e d the t h i r d square is the one starting f r o m G e m i n i , i n w h i c h the sea­ sons end their times. b

c

II 18 T h e squares, t o o , are used, as has been s a i d , for sympathies i n the nativities. M o r e o v e r , the arrangement of the squares is used by some for another purpose. 19 F o r they supposed that, w h e n one o f the signs of the selfsame square is setting, the next sign c u l m i n a t e s the h e m i ­ sphere above the E a r t h , as w h e n C a p r i c o r n is setting, A r i e s c u l ­ minates, C a n c e r rises, a n d L i b r a culminates beneath the E a r t h . T h e same logic applies to the r e m a i n i n g s q u a r e s . 9

d

10

Sympathies arise in three ways: Geminos recognizes only three aspects as significant for sympathies in horoscopic astrology: opposition, trine, and quartile. Manilius (Astro­ nómica i i 270-432) recognizes four: opposition, trine, quartile, and sextile. H e considers two others, but rejects them as insignificant (neighboring signs and signs that are five signs apart). Ptolemy (Tetrabiblos i 13) recognizes the same four aspects as Manilius does. In ex­ plaining why these are the only significant aspects, Ptolemy appeals to a theory of har­ monies based upon musical intervals. H e treated this subject in more detail in his Harmon­ ics. See Neugebauer 1975, 931-34 and Swerdlow 2004. culminates. This is the standard term in modern astronomy. A star "culminates" when it crosses the celestial meridian, i.e., reaches the highest point of its passage across the visi­ ble hemisphere. Geminos's verb is mesouranein, "to mid-heaven." the remaining squares. Thus, for any square of signs, if one of the signs is setting, the next sign of the four will be culminating, the next will be rising, and the last will be cross­ ing the meridian beneath the ground. As Geminos goes on to explain, this proposition is only very roughly correct. However, "some people" applied it as if it were strictly accurate. 8

9

10

130

*

·

¡r ?

Chapter II

x

.m

%

^

%

^

:

\

+ \

rs£

Λ

Diagram 5. Quartile aspect. Marcianus gr. 323, folio 483.

II 2 0 S u c h a p r o p o s i t i o n is o n the w h o l e i n h a r m o n y w i t h the phe­ n o m e n a w h e n a p p l i e d to the one square c o m p r i s i n g the equinoxes

and

solstices; but, i n strict l o g i c , it is i n d i s a c c o r d [ w i t h the p h e n o m e n a ] .

21

F o r w h e n the first degree o f C a p r i c o r n is setting, the first degree o f A r i e s culminates, the first degree o f C a n c e r rises, a n d the first degree o f L i b r a culminates b e l o w the E a r t h

1 l e

; for the circle t h r o u g h the m i d d l e s o f the

signs is d i v i d e d into 4 equal parts by the colure c i r c l e s ,

12

so that i n this

the first degree of Libra culminates below the Earth. This whole statement (ii 21) is corrupt in all the manuscripts, for the astronomy is wrong. See textual note e, p. 258. the colure circles. There are two colure circles. One (the solstitial colure, in modern astronomical language) passes through the celestial poles and through the solstitial points of the ecliptic. The other (the equinoctial colure) passes through the poles and through the 11

12

Aspects of the Zodiacal Signs

·

131

s i t u a t i o n the distances f r o m the c u l m i n a t i n g [point] to the r i s i n g a n d set­ t i n g [points] o f the z o d i a c are e q u a l : for each o f t h e m is 3 signs. II 2 2

B u t i n the r e m a i n i n g p o s i t i o n s o f this square, a n d o f the r e m a i n ­

i n g [squares], it does not h a p p e n t h a t the z o d i a c circle is d i v i d e d i n t o 4 e q u a l parts [by the h o r i z o n a n d the m e r i d i a n ] . Because o f t h i s , the

dis­

tances f r o m the c u l m i n a t i n g [point] to the r i s i n g a n d setting [points] are not a l w a y s e q u a l , the distances b e i n g t a k e n o n the z o d i a c c i r c l e .

23

In

the case of a p a r a l l e l c i r c l e , o f course, the distances f r o m the c u l m i n a t i n g [point] to the r i s i n g a n d setting [points] are a l w a y s e q u a l ; a n d thus for the S u n , w h i c h m o v e s each day o n p a r a l l e l circles, the course f r o m r i s i n g to c u l m i n a t i o n is e q u a l to the one f r o m c u l m i n a t i o n to setting.

24

But

if the distances are t a k e n o n the z o d i a c c i r c l e , the distance f r o m the

cul­

m i n a t i n g to the r i s i n g [point] is u n e q u a l to the distance f r o m the

culmi­

n a t i n g to the setting because o f the o b l i q u i t y o f the z o d i a c c i r c l e . T h e r e are times w h e n , o f the six signs that are a l w a y s a b o v e the E a r t h , three a n d a h a l f are cut off f r o m the c u l m i n a t i n g to the r i s i n g [ p o i n t ] , a n d a n d a h a l f [from the c u l m i n a t i n g ] to the s e t t i n g .

13

25

two

I n d e e d , because

equinoctial points. The two colures divide the ecliptic into four equal 90° segments, which correspond to the four seasons of the year. These circles are said to be kolouroi ("docktailed") because their tails are cut off by the antarctic circle. That is, a portion of each colure lies permanently below the horizon and is never seen. The colures have little theoretical im­ portance. They are, however, important structural members of an armillary sphere. two and a h a l f . . . to the setting. The horizon and the meridian generally divide the ecliptic into two equal long segments and two equal short segments, the disparity between the long and the short segments depending upon both the geographical latitude and the orientation of the ecliptic. A t temperate latitudes, the disparity is greatest approximately (though not exactly) when the solstitial points are on the horizon. If, therefore, we assume that the winter solstitial point is setting, we find that Geminos's example (the meridian di­ vides the portion of the ecliptic that is above the horizon into parts that are 2V2 and 3V2 signs long) applies at latitude 2 8 ° . The state of mathematics in Geminos's day would have permitted calculation of the arc be­ tween the setting and the culminating point of the ecliptic for a given latitude. But in an ele­ mentary work such as the Introduction to the Phenomena, it was sufficient to state the case in words. A teacher could demonstrate the proposition to students with an armillary sphere or a celestial globe, and Geminos probably used just such an instrument in composing his example. The problem of calculating the arc of the ecliptic between the setting and the culminat­ ing point may also be solved with a table of ascensions. Composing such a table in turn re­ quires the calculation of the time required for each sign of the zodiac to rise above the hori­ zon. The oldest complete mathematical treatment of this problem is that of Hypsiklës of Alexandria (second century B.c.), who showed how to find the rising times of the signs by means of approximate arithmetical (rather than exact trigonometrical) methods. (See sec. 12 of the Introduction.) The trigonometric version of the problem probably was solved later in the same century, and perhaps by Hipparchos, although the oldest extant trigono­ metric treatment is Ptolemy's in Almagest i i 7-8. Geminos knows only the arithmetical methods, which he uses for related problems elsewhere in the text (e.g., vi 38), and we have no evidence to suggest that he was a user of trigonometry. 13

132

·

Chapter II

of the differences o f the klimata,

it [the zodiac] is [sometimes] d i v i d e d by

the m e r i d i a n circle i n t o parts that are m o r e u n e q u a l ; a n d it c a n h a p p e n that, o f the 180 degrees that are a l w a y s above the h o r i z o n , 120 degrees are cut off f r o m the c u l m i n a t i n g to the r i s i n g [point], a n d 6 0 [from the c u l m i n a t i n g ] to the s e t t i n g ,

14

o r vice versa.

2 6 A s there is such v a r i a t i o n

i n the d i v i s i o n of the z o d i a c circle, the error is entirely p l a i n . F o r w h e n A q u a r i u s is setting, Taurus does not c u l m i n a t e , but rather w i l l be a n en­ tire sign f r o m the c u l m i n a t i n g [ p o i n t ] ,

15

a n d sometimes even m o r e ; a n d

S c o r p i o does not c u l m i n a t e beneath the E a r t h , but rather w i l l be a n entire sign f r o m the m e r i d i a n , a n d sometimes even m o r e . A n d thus, i n general, the theory o f the squares [mentioned i n i i 1 8 - 1 9 ] is quite m i s t a k e n .

SYZYGY

II 2 7 S a i d to be i n syzygy are signs that rise f r o m the same place a n d set i n the same place; these are [signs] c o n t a i n e d by the same [two] par­ a l l e l circles. 2 8 N o w the ancients r e c k o n e d the syzygies i n the f o l l o w ­ ing f a s h i o n . T h e y p o s t u l a t e d C a n c e r to have n o syzygy w i t h a n y other sign, but to rise farthest n o r t h a n d to set farthest n o r t h , a n d acquiesced, p l a u s i b l y e n o u g h , i n some s u c h [reasoning] as this: 2 9 Since the s u m ­ m e r solstices o c c u r i n C a n c e r , a n d the S u n is farthest n o r t h at the s u m ­ m e r solstices, they therefore s u p p o s e d C a n c e r to rise farthest n o r t h , a n d to set i n the same w a y . 3 0 T h e same l o g i c [applies] also to C a p r i c o r n ; for they s u p p o s e d this one to rise farthest s o u t h a n d to have a syzygy w i t h n o other sign. 3 1 F o r since the w i n t e r solstices o c c u r i n C a p r i ­ c o r n , a n d the S u n is farthest s o u t h at the w i n t e r solstices, they therefore s u p p o s e d that C a p r i c o r n rises farthest s o u t h , a n d that n o other sign rises f r o m the same place a n d sets as C a p r i c o r n . 3 2 T h e y set o u t the r e m a i n i n g syzygies t h u s : L e o w i t h G e m i n i , V i r g o f

16

60° [from the culminating] to the setting. When the winter solstitial point is on the western horizon, 60° of the ecliptic will be cut off between the setting and the culminating point at a latitude of approximately 4 6 ° . an entire sign from the culminating [point]. Let the beginning of Aquarius be on the western horizon. The next sign in the square is Taurus, three signs farther east. According to the mistaken doctrine of the squares, which Geminos refutes, the beginning of Taurus should be on the meridian. However, if the latitude is chosen properly, it can happen that the beginning of Taurus will be an entire sign from the meridian (and so the beginning of Aries will be culminating). This occurs at latitude 5 5 ° . They set out the remaining syzygies thus. Geminos appears to be referring to an older division of the zodiac. According to this scheme, the equinoctial and solstitial points were placed at the midpoints of their signs (fig. 2.1). In this older way of doing things, Gemini and Leo are in syzygy, as are Taurus and Virgo, but Cancer and Capricorn are not in syzygy with any sign. Hipparchos {Commentary i 1.15) says that such a division of the zo­ diac was adopted by Eudoxos: "Eudoxos made the division thus, so that these points [the 14

15

16

A s p e c t s o f the Z o d i a c a l Signs

·

133

solstitial and equinoctial points] are the middles of Cancer and Capricorn in the one case, and of Aries and the Claws in the other." However, Jean M a r t i n (1998, vol. 1, 124-31) ar­ gues that Hipparchos was mistaken in attributing this convention to Eudoxos. See Neuge­ bauer 1975, 598-600 for other instances of this convention in Babylonian and Greek sources. In Geminos's day (as in our own), convention placed the equinoctial and solstitial points at the beginnings of the signs (fig. 2.2). It is curious that Geminos (ii 3 3 - 4 3 , vi 44-49) denounces the older convention so vocif­ erously. H e has already explained the conventional character of the zodiac signs and has given an example of a convention that differs from that of the Greeks—the "Chaldean" convention, according to which the equinoctial and solstitial points are 8° within their re­ spective signs (i 9). It appears likely that Geminos was unaware of Eudoxos's use of the 15° norm. This was Manitius's view (1898, 255n6); but see also Neugebauer 1975, 583. In any case, Geminos was striving to correct a common error made by laymen of his own day, who were probably not followers of Eudoxos's outdated convention, but were merely care­ less in their use of the term "syzygy."

134

·

Chapter II

w i t h T a u r u s , L i b r a w i t h A r i e s , S c o r p i o w i t h Pisces, Sagittarius w i t h Aquarius. II 33 B u t it happens that such a n account is c o m p l e t e l y erroneous. F o r solstices d o n o t o c c u r i n the w h o l e o f C a n c e r ; rather, there is one certain p o i n t , perceivable t h r o u g h r e a s o n , ' ε at w h i c h the S u n makes its t u r n i n g ; for the solstices take place i n a moment's time. 34 T h e w h o l e twelfth-part o f C a n c e r is situated i n the same w a y as G e m i n i , a n d each o f t h e m is e q u a l l y far f r o m the s u m m e r solstitial p o i n t . 35 F o r this rea­ s o n , the lengths o f the days are e q u a l i n G e m i n i a n d Cancer, a n d o n the s u n d i a l s the curves described by the g n o m o n s [when the S u n is] i n 17

18

Fig. 2.1. Division of the zodiac into signs in a system that has been attributed to Eudoxos.

one certain point, perceivable through reason. One cannot see the theoretical summer solstitial point in the sky, but one can, as we would say, "imagine it." on the sundials. Greek sundials were often marked, not only with hour lines, but also with solstitial and equinoctial curves (Gibbs 1976, 86). See the figures in sec. 8 of the In­ troduction. O n summer solstice, the tip of the gnomon's shadow traces out the curve corre­ sponding to the summer tropic. If the Sun were in the middle of Gemini or in the middle of Cancer, it would in either case be half a sign from the summer solstitial point. The curve 1 7

1 8

Aspects of the Zodiacal Signs

·

135

C a n c e r a n d i n G e m i n i are e q u a l l y distant f r o m the s u m m e r t r o p i c [curve]; 3 6 F o r the t w o twelfth-parts are situated e q u a l l y w i t h re­ spect to the s u m m e r [solstitial] p o i n t . H e n c e they are also c o n t a i n e d by the same [parallel] circles, because o f w h i c h G e m i n i a n d C a n c e r rise f r o m the same place a n d , similarly, set i n the same place. II 3 7 T h e same reasoning [applies] also to C a p r i c o r n . F o r it is n o t this [entire sign] that is southernmost, but rather one certain p o i n t , perceiv­ able t h r o u g h reason, w h i c h is c o m m o n to the end o f Sagittarius a n d the b e g i n n i n g o f C a p r i c o r n ; o n w h i c h account it [ C a p r i c o r n ] is situated equally to a n d has the same distance f r o m the w i n t e r solstitial p o i n t as Sagittarius. 38 A n d thus the lengths o f the days a n d o f the nights are the same i n Sagittarius a n d C a p r i c o r n , a n d the tip o f the g n o m o n o n the

traced out by the shadow tip would then lie a little inside the arc for summer solstice. In his explanation of the syzygies, Geminos draws on properties of sundials that would have been familiar to his readers.

Fig. 2.2. Division of the zodiac into signs by the Greek astronomers of Geminos's time. This is the same as the modern convention.

136

·

Chapter II

sundials describes the same curves. 3 9 T h e t w o twelfth-parts o f Sagit­ tarius a n d C a p r i c o r n are also c o n t a i n e d by the same p a r a l l e l circles; a n d , because o f this, Sagittarius a n d C a p r i c o r n rise f r o m the same place a n d set i n the same place. T h u s Sagittarius a n d C a p r i c o r n are i n syzygy. II 4 0 In the same w a y , it f o l l o w s that the r e m a i n i n g syzygies are erro­ neous. Clearest [of all] is the error c o n c e r n i n g the syzygy o f A r i e s . F o r they declare A r i e s to be i n syzygy w i t h L i b r a , o n the g r o u n d s that these signs rise f r o m the same place a n d set i n the same place. 4 1 B u t A r i e s rises a n d sets i n the n o r t h , for it lies n o r t h o f the e q u a t o r i a l circle; w h i l e L i b r a rises a n d sets i n the s o u t h , for it lies s o u t h o f the e q u a t o r i a l circle. 4 2 H o w then c a n A r i e s be i n syzygy w i t h L i b r a ? F o r they rise f r o m dif­ ferent places a n d set i n the same w a y . T h u s it c a n n o t be that these signs are c o n t a i n e d by the same p a r a l l e l circles. 4 3 In the same w a y , the re­ m a i n i n g syzygies also d o n o t m a t c h u p . F o r they [the ancients] have n o t perceived that they have t a k e n circumstances relating to the first degrees for signs i n syzygy a n d have extended t h e m to the w h o l e signs: it w o u l d have been better by far to keep to circumstances that a p p l y to the w h o l e twelfth-parts i n the d e s c r i p t i o n a n d the p r o p o s i t i o n s . II 4 4 T h e r e are, then, i n t r u t h , 6 syzygies: G e m i n i w i t h Cancer, T a u r u s w i t h L e o , A r i e s w i t h V i r g o , Pisces w i t h L i b r a , A q u a r i u s w i t h S c o r p i o , C a p r i c o r n w i t h Sagittarius. A n d so these rise f r o m the same place a n d set i n the same place; they are c o n t a i n e d by the same p a r a l l e l circles; a n d they are situated e q u a l l y w i t h respect to the solstitial p o i n t s . 4 5 A n d thus i n these the lengths o f the days a n d o f the nights are e q u a l , a n d the tips o f the g n o m o n s o n the sundials describe the same curves.

III. On the Constellations

III 1 T h e c o n s t e l l a t i o n s

1

are d i v i d e d i n t o three g r o u p s . S o m e lie o n

the z o d i a c c i r c l e , some are s a i d to be n o r t h e r n , a n d some are c a l l e d southern.

THE ZODIACAL CONSTELLATIONS III 2 T h e ones l y i n g o n the z o d i a c circle are the

12 signs, w h o s e

names w e have a l r e a d y m e n t i o n e d . A n d i n the 12 signs are c e r t a i n stars d e e m e d w o r t h y o f p r o p e r names because they are i n d i c a t o r s

2

for t h e m .

3 T h u s the stars l y i n g o n the

b a c k o f T a u r u s a n d 6 i n n u m b e r are c a l l e d the P l e i a d e s ; w h i l e the stars 3

l y i n g o n the h e a d o f T a u r u s , 5 i n number, are c a l l e d the H y a d e s .

4 The

l e a d i n g star o f the feet o f G e m i n i is c a l l e d P r o p u s . T h o s e i n C a n c e r re­ 4

s e m b l i n g a n e b u l o u s mass are c a l l e d the M a n g e r ; a n d the t w o stars l y i n g near it are c a l l e d the Asses.

5 T h e b r i g h t star l y i n g o n the heart o f L e o ,

n a m e d for the place it o c c u p i e s , is c a l l e d H e a r t o f the L i o n ; but it is c a l l e d b y some B a s i l i s k o s , because those w h o are b o r n a r o u n d this place 5

seem to have a k i n g l y geniture.

6 T h e b r i g h t star l y i n g o n the t i p o f the

the Constellations. See tables 3.1-3.6 at the end of this chapter for the correspon­ dences between Geminos's star and constellation names and modern nomenclature. For the Babylonian constellations, see: Gôssmann 1950; Hunger and Pingree 1999, 271-77; Van der Waerden 1974. Allen 1899 is still useful as an entrée to Greek and R o m a n sources (but not reliable for Arabic star names—for these, see Kunitzsch and Smart 1986). For Eratos­ thenes' Catasterisms, see Charvet 1998. Also useful are Le Boeuffle 1989; Bakhouche 1996; Boll and Gundel 1924-37. indicators. Episëmasia could also be translated "sign," except that this w o r d has al­ ready been used for zôdion, a sign of the zodiac. Episëmasia is also applied to stars whose heliacal risings and settings serve as signs of the seasons and of the weather, as in the parapëgma that follows Geminos's Introduction to the Phenomena. See Lehoux 2000, 130-45. Pleiades. Aratos {Phenomena 262-63) names all seven, though he says only six are visible. Propus. It is so called because it is the "forward foot"—the leading (or western) foot of the leading Twin. Basiliskos = "little king." Aratos does not mention the star, and neither Eratosthenes nor Hipparchos uses this name, which must have been fairly new among the Greeks in Gemi­ nos's day. The Babylonians called Regulus L U G A L , "king" (Gôssmann 1950, 89; Hunger and Pingree 1999, 273; van der Waerden 1974, 73). Ptolemy usually calls it "the star on the heart of Leo," but in the star catalog of the Almagest he adds that it is "called Basiliskos." 1

2

3

4

5

138

·

Chapter III

left h a n d of V i r g o is c a l l e d the W h e a t E a r ; a n d the s m a l l star l y i n g

be-

6

side the r i g h t w i n g o f V i r g o is n a m e d H a r b i n g e r o f the V i n t a g e . T h e

4

stars l y i n g o n the tip o f the r i g h t h a n d o f A q u a r i u s are c a l l e d the Pitcher. 7 T h e stars l y i n g i n a r o w s t a r t i n g f r o m the t a i l parts o f Pisces are c a l l e d the

Cords.

5 stars i n

the

n o r t h e r n c o r d ; the b r i g h t star l y i n g at the end o f the c o r d is c a l l e d

the

7

T h e r e are

9 stars i n the

southern c o r d and

Knot.

T H E NORTHERN CONSTELLATIONS III 8 T h e n o r t h e r n [constellations] are those that lie n o r t h o f the c i r c l e o f the

signs. T h e y are

D r a c o b e t w e e n the

the

f o l l o w i n g : the

Bears, the

G r e a t Bear, the

Bear Keeper, the

L i t t l e [Bear],

C r o w n , the

Kneeling

M a n , O p h i u c h u s , the Serpent, the L y r e , the B i r d , the A r r o w , the the

D o l p h i n , the

F o r e p a r t o f a H o r s e a c c o r d i n g to

H o r s e , C e p h e u s , C a s s i o p e i a , A n d r o m e d a , Perseus, the Triangle, and

the

c o n s t e l l a t i o n later established by

Eagle,

Hipparchos,

the

Charioteer,

the

Kallimachos,

the

L o c k o f Berenikê. I l l 9 A n d , a g a i n , i n these [constellations], c e r t a i n stars have p r o p e r n a m e s because they are i m p o r t a n t i n d i c a t o r s for t h e m . T h e n o t a b l e star l y i n g b e t w e e n the legs o f the Bear K e e p e r is n a m e d A r c t u r u s .

10

The

b r i g h t star l y i n g near the L y r e , n a m e d i n the same w a y as the w h o l e c o n s t e l l a t i o n , is c a l l e d L y r a .

8

T h e m i d d l e one o f the three stars i n the E a g l e ,

Wheat Ear. Virgo holds a wheat ear in her left hand. For the Babylonians, this constellation was usually the "furrow" (Hunger and Pingree 1989, 138; Hunger and Pingree 1999; Koch-Westenholz 1995, 207). But on one famous Babylonian astrological tablet, the constellation was pictured as a woman holding a sheaf of wheat. See van der Waerden 1974, 81, Plate 11c. the Cords. This may also be translated as "fishing lines." The two fish of Pisces are tied together by cords that come from their tails. The two cords form a V-shape and are joined in a knot at α Psc. In Table 3.4 we list the identifications according to Ptolemy's description in the Almagest star catalog (Toomer 1984, 379-80). However, Ptolemy puts ten stars in the southern cord and four in the northern, which differs from Geminos's count. Geminos is not clear about whether the Knot (a Psc) is included in his count of five for the northern cord. The Babylonians also had the cords (or lines or ribbons), but instead of two fishes, they had a fish and a swallow (Hunger and Pingree 1999, 148; Hunger and Pingree 1989, 138). Lyra. Aratos, while mentioning the constellation Lyra in several places, never singles out its brightest star. Eratosthenes mentions the "white and bright star" but gives it no name. Hipparchos {Commentary on the Phenomena of Eudoxos and Aratos i 6.15) refers to the star as "the bright one in the Lyre." Ptolemy {Almagest vii 5), like Geminos, calls the star Lyra. The association of the constellation name with the star may therefore be late. The constellation is small and contains only one bright star—circumstances that undoubt­ edly facilitated this transformation. 6

7

8

O n the Constellations n a m e d i n the same w a y as the w h o l e gle]. the

9

a

·

139

c o n s t e l l a t i o n , is c a l l e d A ë t o s [Ea­

11 T h e stars l y i n g o n the t i p o f the left h a n d o f Perseus are c a l l e d

G o r g o n ' s H e a d ; the

s m a l l , c l o s e l y p a c k e d stars at the

1 0

r i g h t h a n d o f Perseus are set i n the S i c k l e .

11

tip of

the left s h o u l d e r o f the C h a r i o t e e r is c a l l e d the G o a t ; a n d the t w o stars l y i n g at the t i p o f his h a n d

B

the

12 T h e b r i g h t star l y i n g o n small

are c a l l e d the K i d s .

T H E SOUTHERN CONSTELLATIONS III 13 T h e s o u t h e r n [constellations] are those t h a t lie s o u t h o f the c i r ­ cle o f the signs. T h e y are the f o l l o w i n g : O r i o n a n d P r o c y o n , the D o g ,

the

H a r e , A r g o , H y d r a , Crater, the R a v e n , C e n t a u r u s , the W i l d A n i m a l t h a t C e n t a u r u s is h o l d i n g , a n d the T h y r s u s - l a n c e that C e n t a u r u s is h o l d i n g a c c o r d i n g to H i p p a r c h o s ,

c

the Censer, the S o u t h e r n F i s h , the Sea

Mon­

ster, the W a t e r c o m i n g f r o m A q u a r i u s , the R i v e r c o m i n g f r o m O r i o n , the S o u t h e r n C r o w n , c a l l e d by some the c o r d i n g to H i p p a r c h o s .

C a n o p y , [and]

the

Caduceus

ac­

d

III 14 A n d a g a i n , c e r t a i n stars i n these [constellations] deserve to have their The

own

names. The

b r i g h t star i n the

bright

star

m o u t h o f the

in

Procyon

called

Procyon.

D o g , w h i c h seems to cause

i n t e n s i f i c a t i o n o f the heat, is c a l l e d the D o g , whole constellation.

is

1 3

i n the

same w a y

15 T h e b r i g h t star l y i n g o n the t i p o f the

o f A r g o is n a m e d C a n o p u s .

1 4

In R h o d e s it is h a r d to see,

as

1 2

the the

Rudder

o r is seen

Aëtos [Eagle]. Ptolemy (Almagest vii 5), like Geminos, gives the star the same name as the constellation. Aratos and Hipparchos do not mention this star. Eratosthenes mentions it but gives it no name. the Gorgon's Head. This is the head of Medusa, carried in Perseus's left hand. The story is told by Eratosthenes (Charvet 1998, 107-108). the Sickle. Perseus used it to cut off Medusa's head. Procyon. It is called Procyon ("before the Dog") because it makes its heliacal rising in advance of the D o g Star. Aratos {Phenomena 450, 595, 690) uses Procyon only for the en­ tire constellation. It is a small constellation, however, and contains only one prominent star. Thus the transfer of the constellation's name to the star parallels the cases of the Lyre and the Eagle. Eratosthenes and Ptolemy call the star Procyon. The constellation Procyon was also considered a dog: Eratosthenes makes it the dog of O r i o n (Charvet, 1998, 189). 9

1 0

1 1

1 2

1 3

the D o g (Kuan). Homer calls it the "dog of O r i o n " (Iliad x x i i 29). But the name Sir-

ius (Seirios) is also ancient (Hesiod, Works and Days 417, 587, 609). Canopus (or Canobus). The bright star on the steering oar of Argo. Canopus was invis­ ible in Greece and, as Geminos says (iii 15), was barely visible from the latitude of Rhodes. The name Canopus is used by Hipparchos, but not by Aratos or Eratosthenes. Hipparchos (Commentary i 11.6) quotes Eudoxos as placing this star upon the antarctic circle, and makes it clear that Eudoxos referred to it simply as "the star visible in Egypt." Strabo (Ge­ ography i 1.6) mentions Canobus and the Lock of Berenikë as recently invented star names. Ptolemy's use of the name in the star catalog of the Almagest helped secure the tradition. 1 4

140

·

Chapter III

c o m p l e t e l y [only] f r o m h i g h places; but i n A l e x a n d r i a it is c o m p l e t e l y evident, for it appears a p p r o x i m a t e l y one f o u r t h part o f a sign above the horizon. 1 5

G E M I N O S ' S STARS A N D C O N S T E L L A T I O N S : A S U P P L E M E N T T O CHAPTER

III

M a n y readers w i l l w a n t to have a q u i c k w a y o f m a t c h i n g G e m i n o s ' s names for the constellations a n d stars (most o f w h i c h were standard for his day) w i t h their m o d e r n counterparts. T o a v o i d the need for a large n u m b e r o f footnotes to G e m i n o s ' s chapter i i i , w e have g r o u p e d this i n ­ f o r m a t i o n i n t o the six tables below. Tables 3 . 1 , 3.2, a n d 3.3 deal w i t h c o n s t e l l a t i o n names, a n d are f o l l o w e d by r e m a r k s o n a few constella­ tions w i t h respect to w h i c h the ancient G r e e k writers expressed i m p o r ­ tant differences. Tables 3.4, 3.5, a n d 3.6 deal w i t h the stars that G e m i ­ nos m e n t i o n s by name.

TABLE 3.1

Geminos's Constellations: The Zodiac Translation

Geminos's Greek

Aries Taurus Gemini Cancer Leo Virgo Libra Scorpio Sagittarius Capricorn Aquarius Pisces

Krios Tauros Didumoi Karkinos Leon Parthenos Zugos Skorpios Toxotés Aigokerôs Hydr ochóos Ichthues

one-fourth part of a sign above the horizon. That is, at Alexandria, when Canopus culminates, it is one forty-eighth of a circle (7V2°) above the horizon. In the first century B.C., Poseidônios used this datum, and the fact that the star was on the horizon when ob­ served from Rhodes, to calculate the circumference of the Earth. Poseidônios assumed that Rhodes and Alexandria are on the same meridian, and that the distance between them is 5,000 stades. Thus he obtained 5,000 χ 48 = 240,000 stades for the circumference of the Earth. See Kleomëdës, Meteor a i 7.7-48. 15

TABLE 3.2 Geminos's Northern Constellations Translation

Geminos's Greek*

Modem

Name

Ursa M a j o r Ursa M i n o r

Draco

Megalë Arktos Mikra Arktos Drakôn

Bear Keeper

Arktophulax

Boôtes

Crown

Stephanos

Corona Borealis

Kneeling M a n

Engonasin

Hercules

Ophiuchus

Ophiouchos

Ophiuchus

Serpent

Ophis

Serpens

Lyre

Lyra

Lyra

Great Bear Little Bear

Draco

Bird

Omis

Cygnus

Arrow

Oïstos

Sagitta

Eagle

Aëtos

Aquila

Dolphin

Delphis

Forepart of a Horse

Frotóme

Horse

Hippos

Pegasus

Cepheus Cassiopeia

Këpheus

Cepheus

Delphinus Hippou

Equuleus

Cassiopeia

Andromeda

Kassiepeia Andromeda

Perseus

Perseus

Charioteer

Hëniochos

Auriga

Triangle

Deltôton

Triangulum

Lock of Berenikê

Berenikës

Andromeda Perseus

Plokamos

C o m a Berenices

* Geminos's list of northern constellations is almost identical with the list given by Aratos, and probably standardized a century earlier by Eudoxos. The order is different, however, for Aratos mixes the zodiacal constellations among the dnorthern ones. Only the following points should be noted. Pleiades. Geminos, like most later writers, puts the Pleiades in Taurus. Aratos (Phenomena 254) appears to give the Pleiades a separate status and does not mention them in connection with Taurus, thus following Homer and Hesiod, who always mention the Pleiades in their o w n right. A m o n g the Greeks the Pleiades—so singular in appearance—were probably older than Taurus. Eratosthenes gives them a separate paragraph, but says plainly that they are on the back of Taurus, where they have been ever since. Bear Keeper. This is the only name that Geminos uses. But Aratos knew the constellation by two names, for he speaks of the "Bear Keeper, w h o m men also call Bootes" (Phenomena 91). H e is a Bear Keeper because he follows close behind the Great Bear. But the Bears were also visualized as wagons or the carriages of plows (hamaxai) (Homer, Iliad xviii 487; Aratos, Phenomena 27). In this case the Bear Keeper is better thought of as boôtês, a " p l o w m a n " (Aratos, Phenomena 92). Homer uses only Bootes (Odyssey ν 272), so this may be the older name. Kneeling Man. Geminos follows Aratos (Phenomena 66 and 669), who did not identify the constellation with any mythological figure. Eratosthenes identifies the Kneeling M a n with Herakles (Charvet 1998, 41). But most Greek astronomical writers (including H i p ­ parchos and Ptolemy) do not. Forepart of a Horse. A protomê could be the "forepart" of anything, the "face" of an animal, or a "bust" in statuary. Ptolemy (Almagest v i i 5) lists the "constellation of the forepart [or bust] of a horse," which includes four stars, all faint. This is our modern

Equuleus, the colt. Ptolemy and Geminos are virtually alone among ancient writers in mentioning it. Geminos tells us that the name was devised by Hipparchos, which seems possible, since it is not mentioned by Aratos or Eratosthenes. It appears, then, that Ptolemy followed Hipparchos in recognizing the new constellation. Hipparchos, however, does not mention the Frotóme in his Commentary on the Phenomena of Eudoxos and Aratos, nor does it appear in any of the several versions of the list of constellations attributed to Hipparchos. Thus Geminos's attribution of the new constellation to Hipparchos cannot be confirmed. Lock of Berenikë. This constellation is due to the Alexandrian astronomer Konón, and the story goes like this. Berenikë was the cousin and wife of Ptolemaios III Euergetës, the third of the Macedonian kings of Egypt (ruled 247-22 B.c.). When her husband departed for war in Syria she vowed to make an offering to the gods of a lock of her hair if he should return safely. When her husband did return safely to Alexandria, she cut off a lock and placed it in a temple, from which it mysteriously disappeared. Konón, the court astronomer, consoled her by designating a new constellation the Lock of Berenikë. The story has come down to us because it inspired Kallimachos of Kyrënë, who also worked at Alexandria at this time, to compose a poem on the subject, which survives in a fragmentary state. (A Latin version made two centuries later by Catullus is intact. ) The new constellation had an unsettled history. Although it appears in the Catasterisms of Eratosthenes (in the paragraph on Leo), it was not included by Ptolemy among his 48 constellations. Ptolemy does, however, mention the "lock" in his description of the unconstellated stars around Leo. 16

17

18

TABLE 3.3 Geminos's Southern Constellations Translation

Geminos's Greek *

Modem

Name

Orion

Orion

Orion

Procyon

Frokuón

Canis M i n o r

Dog Hare Argo

Kuón Lagôos Argô

Hydra Crater

Hydros Kratër

Canis M a j o r Lepus Carina, Puppis, and Vela Hydra Crater

Raven

Korax

Corvus

Centaurus Wild Animal

Kentauros Thêrion

Centaurus Lupus

Thyrsus-lance

Thursologchos

part of Centaurus

Censer

Thumiatêrion

Ara

Southern Fish

Notios Icbthus

Piscis Austrinus

Sea Monster

Këtos

Cetus

Water

Hydôr

part of Aquarius

River

Potamos

Eridanus

Southern C r o w n

Notios Stephanos

Corona Australiis

Caduceus

Kêrukeion

—

* Again, Geminos's list is fairly standard: Aratos mentions all of these constellations except for the Thyrsus-lance and the Caduceus. For these and a few others, the following comments may be helpful. For a brief description of these manuscripts, see Neugebauer 1975, 285. For the texts, see Maass 1898, 136-39; Boll 1901; Weinstock 1951, 189-90. Kallimachos, Aetia 110. Catullus, poem 66. 1 6

1 7

1 8

O n the Constellations

·

143

Wild Animal and Thyrsus-Lance. A thyrsus, used in Bacchic rites, is a branch wreathed in ivy and vine-leaves with a pine cone at the top. Aratos {Phenomena 440-42; see also 662-63) says that Centaurus stretches his right hand toward the altar and holds i n it the W i l d A n i m a l , but does not mention the Thyrsus. Eratosthenes says that Centaurus (identi­ fied with Chiron) holds a wild animal, but that "some people say it is a wine-skin (askos) . . . , from which he pours an offering to the gods upon the Altar. H e holds it in his right hand and the thyrsus i n his left." Although Aratos mentions the W i l d A n i m a l only in conjunction with the Centaur, he seems to recognize it as a separate constellation. This sep­ arate status is clearer in Eratosthenes. For although Eratosthenes mentions the W i l d A n i ­ mal at the close of his paragraph on the Centaur, he gives a separate total for the number of stars that it contains: there are twenty-four stars in the Centaur and eight in the W i l d A n i m a l . In contrast, the Thyrsus is treated merely as a part of Centaurus. It seems from Geminos's remark that Hipparchos made the Thyrsus, or Thyrsus-lance, into a separate constellation. If so, he was not followed by Ptolemy, who kept to the older tradition: In Almagest viii 1, the Thyrsus is part of Centaurus, but the W i l d A n i m a l has the status of a separate constellation, which is the way things have remained. Unfortu­ nately, Hipparchos's designation of the Thyrsus-lance as a new constellation cannot be confirmed. The list of constellation names attributed to Hipparchos includes "the W i l d A n i m a l that Centaurus holds i n his right hand," but makes no mention of the Thyrsus. In Hipparchos's Commentary (ii 5.14; i i i 5.6) the Thyrsus is plainly considered to be a part of Centaurus. Water. This is the water that is poured out by Aquarius, the Water-Pourer. Geminos treats the Water as a separate constellation, "the Water from the Water-Pourer." Separate or semi-separate status for the Water is attested in Aratos {Phenomena 395-99). Eratos­ thenes mentions "the Pouring of the Water" in the course of his discussion of Aquarius, but he gives a separate tally for the number of stars in the Water, which demonstrates its semi-separate status in his arrangement. Hipparchos (Commentary i i 6.3) plainly makes it a part of Aquarius. Ptolemy's star catalog (Almagest viii 1) fixed the tradition by placing the water among the stars of Aquarius. The Water is an asterism of considerable size: in Ptolemy's catalog it contains twenty of Aquarius's forty-two stars and stretches over 22° in latitude. River. Although Geminos merely describes this constellation as a "river," the identifica­ tion with Eridanus already occurs i n Aratos (Phenomena 360). Southern Crown. This constellation appears to have been established rather late, and its tradition was long unsettled. Aratos (Phenomena 401) refers to it merely as a few stars "turned in a circle" (dinôtoi kuklôi) beneath the feet of Sagittarius. Eratosthenes does not mention it, nor does Hipparchos in his Commentary. But it does figure i n the list of con­ stellation names attributed to Hipparchos, where it is called the " C r o w n beneath Sagittar­ i u s . " Geminos says that some called it a Canopy (Ouraniskos). For Ptolemy it is Stephanos Notios, the Southern C r o w n (Almagest viii 1). Caduceus. A kërukeion was a herald's wand, of the sort sometimes carried by Hermes, often with two serpents wound around it. It is not k n o w n for certain where this constella­ tion might have been located, nor can Geminos's attribution of it to Hipparchos be con­ firmed, for Hipparchos breathes not a w o r d of it in the Commentary. Some scholars read this passage as if kërukeion were an alternative name for the Southern C r o w n (e.g., Tan­ nery 1893, 271; Allen 1899, 172), but considering the absence of any linking particle, the final phrase seems rather to indicate a separate constellation. Boll (1899) made a good case for placing the kërukeion in Orion's left hand, citing texts in which O r i o n carries a sword and a kërukeion, rather than the usual staff (or club) and pelt (e.g., Vettius Valens, An­ thologies, i 2; Bara 1989, 54). 19

1 9

Maass 1898, 138; Weinstock 1951, 190.

144

·

Chapter III

TABLE 3.4 Geminos's Stars: Zodiacal Constellations Translation

Modern Identification

Geminos's Greek

Early

Pleiades

Pleiades

Pleiades

Hyades

Hyades

Propus Manger

Propous Phatnè

Asses Heart of L i o n , or Basiliskos Wheat Ear Harbinger of the Vintage Pitcher

Onoi Kardia Leontos, Basiliskos Stachus Protrugëtër

Hyades, α, θ, γ, δ, ε Tau η Gem The cluster M 4 4 Beehive γ and δ Cnc Regulus, α Leo

Mentions*

11. xviii 486. O d . ν 272. W D 383, 572. A 262 11 xviii 486 Η iii 2.10, 4.12 A 892 A 898 H i 10.11; ii 5.7

Spica, α Vir Vindemiatrix, ε Vir

A 97 A 138

Kalpis

γ, ζ, η and (?) π A q r

Cords

Linoi

Knot

Sundesmos

Northern: ο, π, η , ρ and (?) α Psc Southern: 41, 51, δ, ε, ζ, 80, 89, μ, ν and ξ Psc α Psc

H i i 6.5; iii 1.9, 3.11,4.8 A 243 H i i 6.1; iii 3.9

A 245

* 11. = Homer, Iliad. O d . = Homer, Odyssey. W D = Hesiod, Works and Days. A = Aratos, Phenomena. Η = Hipparchos, Commentary on the Phenomena of Eudoxos and Aratos.

TABLE 3.5 Geminos's Stars: Northern Constellations

Translation

Geminos's Greek

Arcturus Lyra Eagle Gorgon's Head Sickle

Arktouros Lyra Aetos Gorgonion Harpe

Goat Kids

Aix Eriphoi

Modern Identification Arcturus, α Boo Vega, α Lyr Altair, α A q l β, ω, ρ, π Per galactic clusters 869, 884 Capella, α A u r ζ and η A u r

Early

Mentions

W D 566, 610 See comment 8 to ch. 3 See comment 9 to ch. 3 H i i 3.27, 6.15; iii 5.19 H i i 5.15, 6.1; iii 1.1,4.8 A 157 A 158

O n the Constellations

·

145

TABLE 3.6

Geminos's Stars: Southern Constellations Translation Procyon Dog Canopus

Geminos's Greek Prokuôn Kuàn Kanôbos

Modern Identification Procyon, α C M i Sirius, α C M a Canopus, α Car

Early

Mentions

See comment 12 to ch. 3 II. x x i i 29 H i 11.7; iii 2.14

IV. O n the Axis and the Poles

I V 1 T h e c o s m o s b e i n g o f s p h e r i c a l f o r m , the d i a m e t e r o f the c o s m o s a r o u n d w h i c h the c o s m o s t u r n s is c a l l e d the a x i s . T h e e x t r e m i t i e s o f the 1

a x i s are c a l l e d poles o f the c o s m o s .

2

2 O f the p o l e s , one is c a l l e d

n o r t h e r n , the other s o u t h e r n . T h e n o r t h e r n [pole] is the one t h a t is a l ­ w a y s v i s i b l e for o u r r e g i o n ; the s o u t h e r n [pole] is the one t h a t is a l w a y s axis, axon, originally an "axle," as of a cart. Geminos seems to be saying that, since the cosmos is round, it is natural to speak of an axle. poles of the cosmos. The axis of the cosmos passes through the center of the Earth. If the axis is extended, as in fig. 4.1, it pierces the celestial sphere at the north and south ce­ lestial poles (the "poles of the cosmos"), which lie directly above the corresponding poles of the Earth. The horizon plane is tangent to the surface of the Earth. In fig. 4.1, for clarity, we show an Earth of finite size. But the Earth is a mere point with respect to the vast sphere of the cosmos. So, for strict astronomy, one should imagine shrinking the Earth to a point, so that the horizon plane rigorously bisects the sphere of the cosmos. In subsequent figures, we w i l l always show the horizon as passing through the center of the cosmos. 1

2

Fig. 4.1. The Earth surrounded by the sphere of the cosmos. The axis passes through the poles of the cosmos and is (for an observer in Greece) inclined to the horizon approximately as shown.

O n the A x i s and the Poles i n v i s i b l e for

our

horizon.

3 T h e r e are,

·

147

however, some places o n

the

E a r t h w h e r e the p o l e that is a l w a y s visible for us happens to be i n v i s i b l e for t h e m , w h i l e the one that is i n v i s i b l e for us is visible for t h e m .

4

And

a g a i n there is a c e r t a i n place o n the E a r t h w h e r e the t w o poles lie, equally, o n the

horizon.

3

on the horizon. Geminos describes three possible orientations of the celestial sphere with respect to the local horizon. In the northern hemisphere ("our region"), the north ce­ lestial pole is always above the horizon, as in fig. 4.2. The angle marked θ is the altitude of the pole. The angle φ between the equator and the local zenith is, by definition, the geo­ graphical latitude of the place of observation. It is a simple matter to show from the dia­ gram that θ = φ. That is, the altitude of the pole is equal to the latitude of the place of ob­ servation. So, in Athens (latitude 38° N ) , the north celestial pole is seen 38° above the horizon. It follows that the south celestial pole, 38° below the southern horizon, is always invisible in Athens. In the southern hemisphere of the Earth, the south celestial pole is above the southern horizon, as in fig. 4.3. The altitude θ of the south celestial pole is equal to the south lati­ tude of the place of observation. A t the Earth's equator (latitude zero), both celestial poles lie on the horizon, as in fig. 4.4. 3

zenith

Fig. 4.2. The latitude φ of the observer is equal to the altitude θ of the celestial pole. Ν and S mark the north and south points of the horizon. NP and SP are the north and south celestial poles.

148

·

Chapter IV

South

Fig. 4.3. For an observer south of the equator, the north celestial pole is below the horizon. But the altitude θ of the south celestial pole 5? is equal to the south latitude of the observer.

zenith

Fig. 4.4. For an observer at the equator, the two celestial poles SP and NP lie on the horizon.

V. The Circles on the Sphere

V 1 O f the circles o n the sphere, some are p a r a l l e l , some are o b l i q u e , a n d some [pass] t h r o u g h the poles.

T H E PARALLEL CIRCLES

T h e p a r a l l e l [circles] are those that have the same poles as the cosmos. There are 5 p a r a l l e l circles: arctic [circle], summer t r o p i c , equator, w i n ­ ter t r o p i c , a n d antarctic [circle]. V 2 T h e arctic c i r c l e is the largest o f the always-visible circles, [the circle] t o u c h i n g the h o r i z o n at one p o i n t a n d situated w h o l l y above the 1

2

the parallel circles. See fig. 5.1. arctic circle. For the Greeks, the arctic circle is a circle on the celestial sphere, with its center at the celestial pole, and its size chosen so that the circle grazes the horizon at the north point. (See fig. 5.1.) The stars within the arctic circle are circumpolar, i.e., they never rise or set but remain above the horizon 24 hours a day. The arctic circle is therefore the d i ­ viding line between the stars that have risings and settings and those that do not. The size 1

2

Fig. 5.1. The celestial sphere for an observer at about 30° latitude.

150

·

Chapter V

E a r t h . T h e stars l y i n g w i t h i n it neither rise n o r set, but are seen t h r o u g h the w h o l e night t u r n i n g a r o u n d the p o l e . 3 In o u r oikoumenê, this circle is traced out by the forefoot o f the G r e a t Bear. 3,

4

of the local arctic circle depends on the latitude of the observer. Figs. 5.2 and 5.3 show the arctic circle for latitudes 40° Ν and 2 0 ° N . Because the size of the arctic circle in the Greek sense varies with the location of the observer, we shall sometimes refer to this as the local arctic circle. The modern arctic circle is fixed in size: it is a small circle, centered on the pole, whose radius is approximately 2 4 ° .

North

Fig. 5.2. The arctic circle for an observer at 40° latitude.

Fig. 5.3. The arctic circle for an observer at 2 0 ° latitude.

In our oikoumenê. The oikoumené is the "inhabited Earth." This term is used by Greek writers in two senses. It may designate the Greeks' portion of the Earth, as opposed to barbarian lands. But it may also mean the whole k n o w n inhabited world, namely Asia, Europe, and Africa (Ptolemy, Geography i 7; Berggren and Jones 2000, 64 and 2 0 - 2 2 . See also Dilke 1985). Oikoumenê is a participle meaning "inhabited," but the tacit noun " E a r t h " may usually be supplied, as pointed out in LSJ. Geminos first uses the term at ν 26, 29, 30, 43, and 45, an example being ν 29: "the sizes of some of the . . . [five] parallel circles remain the same for the whole inhabited Earth." This w o u l d include the parts south of the equator if, as some writers allowed as being possible, they should happen to be inhabited. The first appearance of the longer phrase, têt kath' hëmas oikoumenëi ("the Earth inhabited by us"), is at ν 39 and is a fair example of the more restricted meaning of the Greek portion of the Earth, as opposed to barbarian lands, for in this passage Geminos ex­ cludes the parts inhabited by, say, Scythians or Egyptians. The same phrase appears in xv 4, where we read that the northern temperate zone is inhabited by those " i n the Earth in­ habited by us." (We take the following passage on dimensions to refer to the northern tem­ perate zone, not just that part of it " i n the Earth inhabited by us.") But the phrase is also found in x v i 2 and 3, where the meaning is clearly the northern inhabited Earth. The pas­ sage x v i 5 is the one place in the work where the adjective appears in the neuter (in the phrase "the inhabited part of the Earth"). F r o m the value 2:1 given there for the ratio of the length to the breadth of this part it seems that, in light of the same ratio in xv 4, it means the northern inhabited Earth. Hence, in general, "the inhabited Earth" (with or 3

The Circles on the Sphere

·

151

V 4 T h e s u m m e r t r o p i c circle is the m o s t n o r t h e r n o f the circles de­ scribed by the S u n d u r i n g the r o t a t i o n o f the c o s m o s . W h e n the S u n is o n this c i r c l e , it produces the s u m m e r solstice, o n w h i c h occurs the longest o f a l l the days o f the year, a n d the shortest night.

5 A f t e r the s u m m e r

solstice, however, the S u n is n o longer seen g o i n g t o w a r d the n o r t h , but it turns t o w a r d the other parts o f the c o s m o s , w h i c h is w h y [this circle] is called " t r o p i c . "

5

V 6 T h e e q u a t o r circle is the largest o f the 5 p a r a l l e l circles. It is b i ­ sected by the h o r i z o n so that a semicircle is situated above the E a r t h , a n d a semicircle b e l o w the h o r i z o n . W h e n the S u n is o n this c i r c l e , it p r o ­ duces the e q u i n o x e s , b o t h the v e r n a l a n d the a u t u m n a l . V 7 T h e w i n t e r t r o p i c circle is the s o u t h e r n m o s t o f the circles de­ scribed by the S u n d u r i n g the r o t a t i o n o f the c o s m o s . W h e n the S u n is o n this circle it produces the w i n t e r solstice, o n w h i c h occurs the longest o f a l l the nights o f the year, a n d the shortest day.

8 A f t e r the w i n t e r s o l ­

stice, however, the S u n is n o longer seen g o i n g t o w a r d the s o u t h , but it turns t o w a r d the other parts o f the c o s m o s , for w h i c h reason this [circle] t o o is c a l l e d " t r o p i c . " V 9 T h e antarctic circle is e q u a l [in size], a n d p a r a l l e l t o , the arctic circle, being tangent to the h o r i z o n at one p o i n t a n d situated w h o l l y be­ neath the E a r t h . T h e stars l y i n g w i t h i n it are a l w a y s i n v i s i b l e to us. 6

V 10 O f the 5 aforementioned circles the e q u a t o r is the largest, the tropics are next i n size, a n d — f o r o u r region—the arctic circles are the smallest.

7

11 O n e m u s t t h i n k o f these circles as w i t h o u t thickness, per­

ceivable [only] b y reason, a n d delineated b y the p o s i t i o n s o f the stars, by observations m a d e w i t h the dioptra*

a n d by o u r o w n p o w e r o f t h o u g h t .

without the explicit qualifier "the whole") means "the whole inhabited Earth." However, context (as in xv 4) may restrict that to the northern inhabited Earth. A n d , by "the Earth inhabited by us," Geminos means the part of the Earth we live in, either "we Greeks" or in the wider sense of "we northerners." the Great Bear. For an observer in ancient Greece, the forefoot of the Great Bear (Ursa Major) did not quite set, but grazed the horizon in the north. The forefoot of the Bear was therefore situated on the local arctic circle; and, in the course of the night, it traced out this circle in the sky. O u r arctic is from arktos, "bear." which is why [this circle] is called "tropic." tropikos is the adjectival form of tropë, a "turn," "turning." beneath the earth. The antarctic circle—lying tangent to the horizon and below it—is a circle that we can never see. (Refer to fig. 5.1.) Geminos uses beneath "the earth" (as here) or under "the horizon" (v 6) interchangeably. the arctic circles are the smallest. This is true for ordinary latitudes, e.g., in Greece. However, Geminos w i l l give examples (v 29-34) of situations in which the arctic circles are larger than the tropic circles. dioptra. Several different kinds of sighting instruments were called by this name. For a dioptra to sweep out the parallel circles, as Geminos says, it would have to be rotatable about the diurnal axis. For a plausible reconstruction, see sec. 8 of the Introduction, p. 42. 4

5

6

7

8

152

·

Chapter V

F o r the o n l y circle visible i n the c o s m o s is the M i l k y W a y ; the rest are perceivable by reason. V 12 O n l y five p a r a l l e l circles are i n s c r i b e d o n the sphere, but n o t because these are the o n l y p a r a l l e l [circles] i n the c o s m o s . F o r , as far as the senses are c o n c e r n e d , each d a y the S u n describes a circle p a r a l l e l to the e q u a t o r i n the course o f the r o t a t i o n o f the c o s m o s , so that b e t w e e n the t r o p i c circles 1 8 2 p a r a l l e l circles are traced o u t by the S u n : for there are just this m a n y days between the solstices. 13 I n ­ deed, a l l the stars are c a r r i e d o n p a r a l l e l circles i n the course o f each day. A l l these circles are i n s c r i b e d o n the sphere because they c o n t r i b u t e m u c h to other u n d e r t a k i n g s i n a s t r o n o m y ; 14 for it is n o t p o s s i b l e to place the stars w e l l o n the sphere w i t h o u t a l l the p a r a l l e l circles, n o r to discover accurately the lengths o f the nights a n d days w i t h o u t the a f o r e m e n t i o n e d circles. H o w e v e r , because they c o n t r i b u t e n o result t o a first i n t r o d u c t i o n to a s t r o n o m y , they are n o t i n s c r i b e d o n the sphere. 9

3

V 15 B u t the 5 p a r a l l e l circles, because they d o c o n t r i b u t e c e r t a i n def­ inite results i n the first i n t r o d u c t i o n to a s t r o n o m y , are i n s c r i b e d o n the sphere. 16 F o r the arctic circle b o u n d s the stars that are a l w a y s visible; the s u m m e r t r o p i c circle c o n t a i n s the solstice a n d is the l i m i t o f the Sun's m o t i o n t o w a r d the n o r t h ; the equator circle c o n t a i n s the e q u i n o x e s ; the w i n t e r t r o p i c circle is the t e r m i n u s o f the Sun's advance t o w a r d the s o u t h a n d c o n t a i n s the w i n t e r solstice; a n d the antarctic cir­ cle b o u n d s the stars that are n o t visible. 17 Since, then, they have i m ­ p o r t a n t a n d definite results for the i n t r o d u c t i o n to a s t r o n o m y , they are w i t h g o o d reason i n s c r i b e d o n the sphere.

DIVISION O F T H E PARALLELS BY T H E H O R I Z O N

V 18 O f the 5 a f o r e m e n t i o n e d p a r a l l e l circles, the arctic circle is situ­ ated entirely above the E a r t h . V 1 9 T h e s u m m e r t r o p i c circle is cut by the h o r i z o n i n t o t w o u n e q u a l parts: the larger part is situated above the E a r t h , the smaller part b e l o w the E a r t h . 2 0 B u t the s u m m e r t r o p i c circle is not cut by the h o r i z o n i n the same w a y for every l a n d a n d city: rather, o n account o f the v a r i a t i o n s as far as the senses are concerned. If the Sun were fixed at one place on the celestial sphere, it w o u l d trace out a circle as the sphere rotates. But in one day the Sun shifts its po­ sition on the zodiac by about I , which causes it to move a little north or south on the sphere. Thus, the Sun's actual motion through the sky in the course of a day is not really a circle but one loop of a sort of spherical spiral. For Geminos's present purpose, the Sun's change in position may be ignored, and so the Sun does sensibly trace out a circle in the course of a day. 9

o

The Circles on the Sphere

·

153

i n the klimata, the excess o f the [one] part [over the other] differs. 2 1 F o r those w h o live farther n o r t h t h a n w e d o , it happens that the s u m m e r is cut b y the h o r i z o n i n t o parts that are m o r e u n e q u a l ; a n d the l i m i t is a certain place where the w h o l e s u m m e r t r o p i c circle is above the Earth. 2 2 B u t for those w h o live farther s o u t h t h a n w e d o , the s u m mer t r o p i c circle is cut by the h o r i z o n i n t o parts m o r e a n d m o r e equal; a n d the l i m i t is a certain place, l y i n g to the south o f us, w h e r e the s u m mer t r o p i c circle is bisected by the h o r i z o n . 10

1 1

1 2

V 2 3 it rains w i t h a southerly r a i n . O n the 9 t h , a c c o r d i n g to E u d o x o s , the G o a t rises i n the m o r n i n g . st L

M

N

p

R

s

the K n o t of Pisces, α Psc, the star at the knot where the two fishing lines join. O n the 6th, according to Eudoxos, equinox. Eudoxos's equinox falls late because he took the four seasons to be all of roughly equal length. (See comment 26.) According to the Celestial Teaching of Leptinës, however, it seems that Eudoxos's spring equinox should fall only 4 days later than that of the Geminos parapêgma. This w o u l d put Eudoxos's equinox on the 5th day of Aries. invisible for 40 nights. This is the classical length of the Pleiades' period of invisibility, as already in Hesiod: "For forty days and nights they hide themselves" (Works and 3 2

3 3

3 4

Days 385).

240

·

Parapëgma

O n the 1 1 t h , a c c o r d i n g to E u d o x o s , S c o r p i o begins to set i n the m o r n ­ i n g ; a n d there is r a i n . O n the 1 3 t h , a c c o r d i n g to Euktëmôn, the Pleiades rise; b e g i n n i n g o f s u m m e r ; a n d it signifies. A c c o r d i n g to K a l l i p p o s , the head o f T a u r u s rises; it signifies. O n the 21st, a c c o r d i n g to E u d o x o s , the w h o l e o f S c o r p i o sets i n the morning. O n the 2 2 n d , a c c o r d i n g to E u d o x o s , the Pleiades r i s e ; a n d it signifies. O n the 2 5 t h , a c c o r d i n g to Euktëmôn, the G o a t sets i n the evening. O n the 3 0 t h , a c c o r d i n g to Euktëmôn, rises i n the evening. O n the 31st, a c c o r d i n g to Euktëmôn, the Eagle rises i n the evening. O n the 3 2 n d , a c c o r d i n g to Euktëmôn, A r c t u r u s sets i n the m o r n i n g ; it signifies. A c c o r d i n g to K a l l i p p o s , T a u r u s finishes r i s i n g . A c c o r d i n g to Euktëmôn, the H y a d e s rise i n the m o r n i n g ; it signifies. 35

T

u

v

w

x

Y

T h e S u n passes t h r o u g h G e m i n i i n 3 2 days. O n the 2 n d , a c c o r d i n g to K a l l i p p o s , G e m i n i begins to rise; southerly [winds]. O n the 5 t h , a c c o r d i n g to E u d o x o s , the H y a d e s rise i n the m o r n i n g . O n the 7 t h , a c c o r d i n g to E u d o x o s , the Eagle rises at n i g h t f a l l . O n the 1 0 t h , a c c o r d i n g to Dëmokritos, there is r a i n . O n the 1 3 t h , a c c o r d i n g to E u d o x o s , A r c t u r u s sets i n the m o r n i n g . O n the 1 8 t h , a c c o r d i n g to E u d o x o s , the D o l p h i n rises at n i g h t f a l l . O n the 2 4 t h , a c c o r d i n g to Euktëmôn, the shoulder o f O r i o n rises. A c ­ c o r d i n g to E u d o x o s , O r i o n begins to r i s e . O n the 2 9 t h , a c c o r d i n g to D ë m o k r i t o s , O r i o n begins to rise; it is l i k e l y to signify u p o n this. z

the Pleiades rise; beginning of summer. The morning rising of the Pleiades was a tradi­ tional sign of the beginning of summer. See comment 21. 3 5

Fragment 1, From Geminos's Pbilokalia: Geminos on the Classification of the Mathematical Sciences

INTRODUCTION

G e m i n o s was the a u t h o r o f a m a t h e m a t i c a l w o r k o f considerable length w h i c h discussed, a m o n g other things, the p h i l o s o p h i c a l foundations o f geometry. T h i s b o o k t o o k up a great m a n y m a t h e m a t i c a l subjects that h a d been the subject o f dispute, for e x a m p l e the classification of lines, the r e l a t i o n a m o n g a x i o m s a n d postulates, a n d the status o f E u c l i d ' s par­ allel postulate. A large n u m b e r of passages f r o m this w o r k are preserved by P r o k l o s (fifth century A . D . ) i n his Commentary on the First Book of Euclid's Elements. Indeed, P r o k l o s cites G e m i n o s some t w o d o z e n times. T h e exact title o f G e m i n o s ' s w o r k is u n c e r t a i n , but at the end o f a dis­ c u s s i o n o f the classification o f lines that d o not meet, m o t i v a t e d by E u ­ clid's definition o f p a r a l l e l lines, P r o k l o s concludes, " S o m u c h I have se­ lected f r o m G e m i n o s ' s Philokalia to elucidate the subject before u s . " Philokalia means " l o v e o f the b e a u t i f u l . " 1

W h e t h e r Philokalia was the title o f G e m i n o s ' s entire m a t h e m a t i c a l treatise, a n alternative title, o r perhaps the title o f one o f its b o o k s , has been a subject o f debate. P a p p o s (third century A . D . ) , i n his Mathemati­ cal Collection, cites a w o r k o f G e m i n o s " o n the classification o f mathe­ matics. " ' B u t E u t o k i o s (sixth century), at the b e g i n n i n g o f his c o m ­ mentary o n the Conies o f Apollônios, cites G e m i n o s f r o m "the s i x t h b o o k o f the doctrine o f mathematics." T a n n e r y points out that Pappos's title suits the l o n g extract (our fragment 1) that P r o k l o s gives i n part 1 o f his P r o l o g u e , but not most o f the rest o f P r o k l o s ' s citations o f G e m i n o s . B o t h T a n n e r y a n d H e a t h therefore prefer the title given by E u t o k i o s , The Doctrine (or Theory) of Mathematics. B u t the title m e n t i o n e d by P r o k a

2

15

3

Friedlein 1873, 177; M o r r o w 1970, 139. Hultsch 1876-78, v o l . 3, 1026; Ver Eecke 1933, vol. 2, 813. This comes in the course of Pappos's o w n discussion of the branches of mechanics (Ver Eecke 1933, v o l . 2, 809-14). O n the title issue, see Tannery 1887,18-19; Heath 1926, vol. 1, 39; Aujac 1975, x i - x i i . 1

2

3

244

·

Fragment 1

los, Philokalia, is clearly a real title, a n d it seems b r o a d e n o u g h for, a n d even especially a p p r o p r i a t e t o , the fragments a n d citations o f the w o r k that w e possess. M o r e o v e r , P r o k l o s seems to have k n o w n G e m i n o s ' s w o r k intimately. F o r brevity a n d definiteness, w e shall refer to G e m i n o s ' s lost m a t h e m a t i c a l b o o k as the Philokalia, w i t h o u t m e a n i n g to i m p l y that the q u e s t i o n o f the title c a n definitely be decided. In 1 8 6 4 , H u l t s c h i n c l u d e d i n his e d i t i o n o f the g e o m e t r i c a l w o r k s o f H e r o o f A l e x a n d r i a several pages o f a n o n y m o u s r e m a r k s ( f o u n d i n some B y z a n t i n e m a n u s c r i p t s o f H e r o ' s Definitions) that he t o o k to be fragments o f the same w o r k b y G e m i n o s that P r o k l o s cites so often (our Philokalia). These a n o n y m o u s fragments c a r r y such titles as " D e f i n i t i o n o f geometry," " W h a t is the g o a l o f g e o m e t r y ? " " O n l o g i s t i c , " " W h a t is the subject matter o f l o g i s t i c ? " ( H i s t o r i a n s o f G r e e k m a t h e m a t i c s some­ times c a l l these a n o n y m o u s fragments the Variae Collectiones.) Hultsch a t t r i b u t e d some o f t h e m to H e r o himself, some to E u c l i d , A n a t o l i o s o f A l e x a n d r i a , o r P r o k l o s , as w e l l as some to G e m i n o s . In 1 9 1 2 , H e i b e r g , at the e n d o f his e d i t i o n o f H e r o ' s Definitions, attributed a substantially larger a r r a y o f these fragments to G e m i n o s . F o r e x a m p l e , some mate­ r i a l that H u l t s c h h a d a t t r i b u t e d to H e r o ' s Catoptrics H e i b e r g n o w as­ c r i b e d to G e m i n o s . O n e o f the sections n e w l y a t t r i b u t e d to G e m i n o s c a r r i e d the title " W h a t is s c e n o g r a p h y ? " T h e oldest m a n u s c r i p t s o f H e r o ' s Definitions are o f the fourteenth c e n t u r y a n d d o n o t a l l c o n t a i n the c o m p l e t e selection o f the Variae Collectiones. A n d , o f course, there is n o w a y to k n o w just w h e n i n the m a n u s c r i p t h i s t o r y o f the Defini­ tions the text a c q u i r e d this a d d i t i o n a l m a t e r i a l , w h i c h some c o p y i s t m u s t have seen as relevant to u n d e r s t a n d i n g H e r o . T h e a t t r i b u t i o n to G e m i n o s remains c o n t r o v e r s i a l , a n d , as w e have seen, authorities dis­ agree a b o u t w h i c h fragments s h o u l d be a t t r i b u t e d to w h i c h a u t h o r s . B u t w h e t h e r by G e m i n o s o r by another w r i t e r interested i n the classifi­ c a t i o n o f the m a t h e m a t i c a l sciences, a n d perhaps w r i t i n g u n d e r G e m i ­ nos's influence, these fragments d o shed l i g h t o n G e m i n o s ' s r e m a r k s i n fragment l . 4

5

6

7

Hultsch 1864, 246-49. Heiberg 1912, 96-108, 165 (Greek text and German translation). In making the attri­ bution to Geminos, Heiberg was influenced by the arguments of M a r t i n 1844, 113. See Tannery 1887, 4 3 - 4 6 . Tannery gives several arguments against Hultsch's attribu­ tion of much of this material to Geminos, preferring to ascribe a good deal of it instead to Anatolios. O n the other hand, Tannery sees the influence of Geminos on a fragment on op­ tics that Hultsch had ascribed to the Catoptrics of H e r o . Some of the same anonymous passages are also found at the end of fourteenth-century manuscripts of a short work on optics by Damianos, who was perhaps of the fourth to sixth centuries A.D. Schone 1897, 22-31 (Greek text and German translation). O n D a m i ­ anos, see Todd 2003a. 4

5

6

7

From Geminos's Philokalia

·

245

F i n a l l y , s c h o l i a t o the first b o o k o f E u c l i d ' s Elements preserve c o n ­ siderable p o r t i o n s o f G e m i n o s ' s r e m a r k s o n E u c l i d ' s d e f i n i t i o n s , a n d frequently m e n t i o n G e m i n o s b y n a m e . T h e scholiast a p p a r e n t l y used the same w o r k o f G e m i n o s t h a t P r o k l o s r e l i e d u p o n , b u t a c c o r d i n g t o H e a t h the s c h o l i a d r a w n f r o m G e m i n o s "are v a l u a b l e i n t h a t they give G e m i n o s p u r e a n d s i m p l e , w h e r e a s P r o k l o s i n c l u d e s extracts f r o m o t h e r a u t h o r s . " T h e s i t u a t i o n is n o t so s i m p l e , h o w e v e r , since the s c h o l i a s t r e l i e d o n P r o k l o s h i m s e l f (rather t h a n o n G e m i n o s directly) for a c o n s i d e r a b l e n u m b e r o f passages. T h i s is clear, f o r e x a m p l e , i n the passage " S o m u c h I have selected f r o m G e m i n o s ' s Philokalia to elucidate the subject before u s , " w h i c h o c c u r s i n the s c h o l i a as a ver­ b a t i m d u p l i c a t e o f the r e m a r k i n P r o k l o s . I n a n y case, G e m i n o s ' s Philokalia seems t o have attracted a fair a m o u n t o f a t t e n t i o n a n d c o m ­ m e n t i n A n t i q u i t y . T h i s stands i n c o n t r a s t t o G e m i n o s ' s o n l y e x t a n t w o r k , the Introduction to the Phenomena, w h i c h is n o t c i t e d b y a n y ancient writer. 8

9

1 0

L e t us n o w s k e t c h the c o n t e x t i n w h i c h P r o k l o s quotes G e m i n o s o n the c l a s s i f i c a t i o n o f the m a t h e m a t i c a l sciences. In the course o f the P r o ­ logue to his Commentary on the First Book of Euclid's Elements, P r o k ­ los has just c o m p l e t e d a d i s c u s s i o n o f the P y t h a g o r e a n d i v i s i o n o f m a t h e m a t i c s . A c c o r d i n g to P r o k l o s , the P y t h a g o r e a n s d i v i d e d mathe­ m a t i c a l science i n t o f o u r parts. F i r s t o f a l l , o n e - h a l f o f m a t h e m a t i c s is c o n c e r n e d w i t h q u a n t i t y (poson = " h o w m a n y ? " ) a n d the other h a l f w i t h m a g n i t u d e (pëlikon = " h o w l a r g e ? " ) . A n d then each o f these d i v i ­ sions is itself t w o f o l d . T h u s , a q u a n t i t y c a n be c o n s i d e r e d either i n its o w n r e g a r d o r i n r e l a t i o n to some other q u a n t i t y ; a n d a m a g n i t u d e c a n be regarded either as s t a t i o n a r y o r as i n m o t i o n . A r i t h m e t i c studies q u a n t i t y as s u c h , w h i l e m u s i c studies the relations between quantities. G e o m e t r y is d e v o t e d to m a g n i t u d e at rest, a n d spherics (i.e., a s t r o n ­ o m y ) to m a g n i t u d e i n m o t i o n . A s w e l e a r n f r o m Plato's Timaeus, a r i t h m e t i c is p r i o r to m u s i c , since n u m b e r comes i n t o b e i n g before r a t i o . S i m i l a r l y , g e o m e t r y is p r i o r to spherics because rest precedes m o ­ t i o n . It is after this d i s c u s s i o n o f the P y t h a g o r e a n d o c t r i n e that P r o k l o s takes u p G e m i n o s ' s p o s i t i o n o n the d i v i s i o n o f m a t h e m a t i c s . T h e reader m a y f i n d it h e l p f u l to refer to f i g . 1.14, w h i c h presents G e m i ­ nos's scheme i n d i a g r a m f o r m . Sec. 9 o f the I n t r o d u c t i o n p r o v i d e s a detailed d i s c u s s i o n . 11

8

9

1 0

11

See Heiberg 1888, 81 line 4; 82 line 29; 107 line 20; 108 line 17; 134 line 12. Heath 1921, v o l . 2, 224. Heiberg 1888, 108, lines 16-18. Compare with Friedlein 1873, 177, lines 2 4 - 2 5 .

Plato, Timaeus 35a-41a.

246

·

Fragment 1

TRANSLATION OF FRAGMENT

l

1 2

T h i s , then, is the p o s i t i o n o f the Pythagoreans a n d their d i v i s i o n o f the f o u r sciences. 13

B u t others, such as G e m i n o s , t h i n k it p r o p e r to d i v i d e m a t h e m a t i c s acc o r d i n g to another scheme. T h e y m a k e one b r a n c h c o n c e r n e d w i t h m e n t a l things only, a n d one c o n c e r n e d w i t h perceptible things o r t o u c h i n g o n t h e m . T h e y c a l l m e n t a l , o f course, those objects o f c o n t e m p l a t i o n that the s o u l itself calls u p , separating itself f r o m m a t e r i a l f o r m s . I n the b r a n c h engaged w i t h m e n t a l things, they place a r i t h m e t i c a n d geomet r y as the t w o first a n d m o s t i m p o r t a n t parts. In the b r a n c h that operates w i t h perceptible things they place s i x parts: m e c h a n i c s , a s t r o n o m y , o p tics, geodesy, c a n o n i c s , a n d l o g i s t i c . 14

15

16

O n the other h a n d , they d o not t h i n k it right to call tactics one o f the parts o f mathematics, as others d o . Rather, they h o l d that it uses sometimes logistic, as i n the tallying up o f the companies, a n d sometimes geodesy, as i n the division a n d measurement o f camps. Still less do they consider history or medicine to be part o f mathematics, even i f those w h o write histories often use mathematical theorems i n w r i t i n g o n the positions o f the klimata or i n inferring the sizes, diameters, a n d perimeters o f cities, a n d the physicians often clarify their o w n discussions by such methods: for Hippokratës a n d all those w h o have said anything o n seasons a n d places m a k e clear the utility o f astronomy to medicine. In the same way, then, the tactician w i l l use the theorems of mathematics, although he is not a mathematician, if, w h e n he wishes to s h o w the smallest size he s h o u l d f o r m the c a m p i n a circle, or to s h o w the largest, i n a square, pentagon, or some other p o l y g o n . 0

d

17

18

The translation is based on the text of Friedlein 1873, 38-42. A l l significant departures from Friedlein's text are indicated in the textual notes. In making our o w n translation of this passage we have benefited by consulting those of Tannery (1888, 38-42), Ver Eecke (1948, 31-36), Aujac (1975, 114-117) and M o r r o w (1970, 31-35). This passage was also discussed by Heath (1921, vol. 1, 10-18). the four sciences. See Plato (Republic v i i 520a-32c) for a famous justification of the Pythagorean quadrivium of arithmetic, geometry, music, and astronomy for the education of the guardians of the state. arithmetic, arithmétikë (technè understood) is the "arithmetical (art)," the science of pure number, which is not to be confused with the elementary computation taught to school children. We would characterize arithmétikë as number theory, of the sort that originated with the Pythagoreans. See sec. 9 of the Introduction. canonics. kanoniké (technè understood ) is the mathematical theory of musical scales. logistic. Logistikë is the art of practical computation. Hippokratës. In Greek medical writing, the heliacal risings and settings of the fixed stars were sometimes used to indicate the time of year. For examples, see Hippokratës, Regimen III, 68; Hippokratës, Epidemics i , 13-14. pentagon or some other polygon. This paragraph recalls Plato, Republic v i i 522c-26d, in which Socrates' interlocutors propose military applications of arithmetic and 1 2

1 3

1 4

1 5

1 6

1 7

1 8

From Geminos's Philokalia

·

247

These, then, are the species o f m a t h e m a t i c s as a w h o l e . G e o m e t r y is i n t u r n d i v i d e d i n t o the theory o f the plane a n d stereometry.

19

T h e r e is n o

b r a n c h o f study specially devoted to points a n d lines, i n a s m u c h as n o figure w o u l d arise f r o m t h e m w i t h o u t planes o r s o l i d bodies: a n d , indeed, it is a l w a y s t h e t a s k o f g e o m e t r y ,

b o t h plane a n d s o l i d , to c o n ­

s t r u c t [figures] o r to c o m p a r e o r to d i v i d e those that have already been constructed.

20

Similarly, there is a d i v i s i o n o f arithmetic i n t o the study o f linear n u m ­ bers, plane n u m b e r s , a n d s o l i d n u m b e r s : for it examines the classes o f 21

n u m b e r i n themselves, as they p r o c e e d f r o m the unit, a n d the generation o f the plane n u m b e r s , b o t h s i m i l a r a n d d i s s i m i l a r ,

22

a n d the p r o g r e s s i o n

to the t h i r d d i m e n s i o n . G e o d e s y a n d logistic are analogous to these [i.e., to geometry arithmetic], since they d o n o t discourse a b o u t m e n t a l n u m b e r s or

and fig­

ures, but rather a b o u t perceptible ones. F o r the task o f geodesy is to measure not a c y l i n d e r o r a cone, but rather heaps as cones a n d pits as cylinders; a n d to measure n o t by means o f m e n t a l straight lines, but by means o f perceptible ones, sometimes m o r e precise ones, such as rays o f sunlight, a n d sometimes coarser ones, such as ropes o r a carpenter's rule. N o r , a g a i n , does the c a l c u l a t o r (logistikos)

consider the very properties

of n u m b e r s i n themselves, but rather as present i n perceptible

things,

w h i c h is w h y he takes the names for t h e m f r o m the things being c o u n t e d , c a l l i n g some a p p l e s

e

a n d others b o w l s . A n d he does n o t concede that

geometry before Socrates convinces them that true arithmetic and geometry lead the soul to higher things. stereometry. Stereometria is the "measurement of solids," which Geminos uses here to mean "solid geometry." But Hero of Alexandria wrote a work under the title Stereometrica that was concerned more with practical applications, i.e., the mensuration of solids. See sec. 9 of the Introduction. to divide those that have already been constructed. This is an evident reference to an ancient tradition of division of figures, which goes back to Babylon and is witnessed also by Euclid's On the Division of Figures. See Heath 1921, vol. 1, 4 2 5 - 3 0 . linear, plane, and solid numbers. A linear number is a number regarded as a single factor. Linear numbers are analogous to lengths and are the measures of lengths. A plane number, analogous to an area, is a number regarded as a product of two factors. Thus, 6 regarded as 2 χ 3 is a plane number. A solid number, analogous to a volume, is the product of three factors. Thus, 6 regarded as 1 x 2 χ 3 is a solid number. See Euclid, Elements vii, def. 16. See also Theôn of Smyrna, Mathematical Knowledge Useful for Reading Flato i 7; i 22. similar and dissimilar plane numbers. Two plane numbers are similar (homoioi) if the factors composing them stand in the same ratio. Thus 6 = 2 x 3 and 24 = 4x6 are similar plane numbers, because 3:2 :: 6:4. If 6 and 24 are regarded as the areas of rectangles, one measuring 2 by 3 and the other 4 by 6, the two rectangles are similar in the geometer's sense. See Euclid, Elements vii, def. 21. See also Theôn of Smyrna, Mathematical Knowl­ edge Useful for Reading Flato i 22. 1 9

2 0

2 1

2 2

248

·

Fragment 1

s o m e t h i n g is least, as does the a r i t h m e t i c i a n , since he assumes his least i n r e l a t i o n t o a p a r t i c u l a r class: one m a n , as u n i t , is his measure for a c r o w d . A g a i n , o p t i c s a n d c a n o n i c s are o f f s p r i n g o f g e o m e t r y a n d a r i t h m e t i c . T h e f o r m e r uses v i s u a l lines a n d the angles f o r m e d f r o m t h e m . It is d i v i d e d i n t o : a p a r t c a l l e d o p t i c s proper, w h i c h e x p l a i n s the cause o f false appearances due t o the distances o f the things o b s e r v e d , s u c h as

the

m e e t i n g o f p a r a l l e l s o r the p e r c e p t i o n o f squares as c i r c l e s ; a p a r t c a l l e d general c a t o p t r i c s , w h i c h is c o n c e r n e d w i t h a l l the varieties o f reflections a n d i n v o l v e s the study o f images; a n d the p a r t c a l l e d s c e n e - p a i n t i n g ,

23

w h i c h e x p l a i n s h o w one m a y represent appearances i n pictures w i t h o u t d i s p r o p o r t i o n o r d i s t o r t i o n w h e n the things d r a w n are at a distance o r r a i s e d t o a height. C a n o n i c s , i n t u r n , treats the a p p a r e n t r a t i o s o f m u s i c a l notes b y d i s c o v e r i n g the d i v i s i o n s o f the kanôn,

24

sense-perception a n d , as P l a t o s a y s ,

25

relying always o n

p u t t i n g the ears before the m i n d .

Besides these sciences there is the one c a l l e d m e c h a n i c s , w h i c h is a p a r t o f the s t u d y o f perceptible a n d m a t e r i a l objects. U n d e r this c o m e s

the

c o n s t r u c t i o n o f i n s t r u m e n t s useful i n w a r , s u c h as the engines for defense that A r c h i m e d e s

26

is s a i d t o have b u i l t d u r i n g the siege o f Syracuse, a n d

a l s o the science o f w o n d e r - w o r k i n g , m e a n s o f air, as b o t h K t ë s i b i o s

28

2 7

w h i c h w o r k s its c o n t r i v a n c e s b y

and H e r o

2 9

describe, o r b y m e a n s o f

scene-painting. Skénographikê, or scenography, is the art of applied perspective, useful for the realistic representation of buildings i n theatrical sets. kanón (see comment 15, above) is here used as a synonym for monochordos, the monochord, the ancient instrument with a single taut string and a movable bridge. Plato {Republic vii 531a-c) disapproves of those w h o rely upon experiment to determine the rules governing musical harmony, rather than inquiring into number itself to determine which numbers are inherently concordant and why. Geminos himself does not seem to object to putting the ears before the mind. Archimedes is said to have contrived amazing engines for the defense of the city when the Romans under Marcellus were besieging Syracuse (212 B.c.). According to Plutarch (Marcellus xvii) these were so effective that the appearance of any stick or rope over the city walls was sufficient to terrify the R o m a n soldiers. Nevertheless, the city fell and Archimedes was among those killed during its sack. wonder-working. Thaumatopoiïkë was the art of devising automata and other gadgets, often operated by means of fluid or air pressure. Ktësibios. Ktësibios of Alexandria (second or third century B.c.) was the inventor of a force-pump, a water organ (a musical instrument in which a constant air pressure is supplied by means of a water pump), and several elaborate water clocks. One of the most difficult problems solved by Ktësibios i n his design of water clocks was the regulation of the length of the seasonal hour, which varies in the course of the year. Ktësibios's book on his inventions is lost. The fullest surviving account of them is given by Vitruvius, On Architecture i x 7; χ 7 - 8 , with excellent commentaries in the Budé volumes of Vitruvius. Brief de­ scriptions of the pump and the water clock are given in Landels 1978, 75-76, 193-94. See Drachmann 1948 for more detail. 2 3

2 4

2 5

2 6

2 7

2 8

H e r o . H e r o of Alexandria (probably mid-first century A.D.). If a date i n the first cen­ tury B.c. for Geminos's work is correct, this mention of Hero is an interpolation by Prok­ los or by a later copyist. See sec. 6 of the Introduction, i n which Geminos's date is dis­ cussed. H e r o is perhaps best k n o w n for his invention of a toy steam engine, described i n 2 9

From Geminos's Philokalia

·

249

weights w h o s e d i s e q u i l i b r i u m is the cause o f m o t i o n a n d w h o s e e q u i l i b r i u m is the cause o f rest, as the Timaeus has e s t a b l i s h e d , or, finally, by means o f c o r d s a n d ropes m i m i c k i n g the tugs a n d m o v e m e n t s o f l i v i n g creatures. A l s o u n d e r mechanics c o m e the general science o f things i n e q u i l i b r i u m a n d the d e t e r m i n a t i o n o f w h a t are c a l l e d centers o f gravity, as w e l l as s p h e r e - m a k i n g i n i m i t a t i o n o f the celestial m o t i o n s , such as Archimedes p r a c t i c e d , a n d , generally, a l l that is concerned w i t h matter i n motion. 30

31

T h e r e remains a s t r o n o m y , w h i c h treats the c o s m i c m o t i o n s , the sizes a n d shapes o f the heavenly bodies, their i l l u m i n a t i o n s a n d their distances f r o m the E a r t h , a n d a l l such questions. It benefits greatly f r o m sense perc e p t i o n but also has a g o o d deal i n c o m m o n w i t h p h y s i c a l theory. Its parts are: g n o m o n i c s , w h i c h is engaged w i t h the measurement o f the h o u r s t h r o u g h the placement o f g n o m o n s ; m e t e o r o s c o p y , w h i c h discovers the different altitudes [of the p o l e ] a n d the distances o f the stars a n d teaches m a n y other c o m p l e x matters f r o m a s t r o n o m i c a l theory; a n d d i o p t r i c s , w h i c h examines the positions^ o f the S u n , M o o n , a n d the other stars by means o f just such instruments [i.e., d i o p t r a s ] . 32

f

S u c h are the w r i t i n g s o n the parts o f m a t h e m a t i c s that w e have received f r o m the ancients. his Pneumatics. H e wrote on many of the branches of applied science listed by Geminos: mechanics (including both military engineering and wonder-working), optics, dioptrics, and geodesy. For a summary of Hero's works, see Landels 1978, 199-208. as the Timaeus has established. According to Plato {Timaeus 57d-58a), the cause of motion is always to be found in nonuniformity or unevenness {anômalotës), by which he means a nonuniformity in the arrangement of bodies. If all was uniform, it would be i m possible for one body to move and for another to be moved. Rest is therefore associated with uniformity. These remarks come in the midst of a discourse on the nature of the elements and of their changes and transformations. Plato had nothing to say on mechanics proper. sphere-making. Sphairopoii'a is the science of constructing models of the heavens, i n cluding celestial globes, armillary spheres, and orreries. meteoroscopy. Meteôroskopikê is most likely concerned with the armillary sphere as a specialized instrument of observation. See sec. 9 of the Introduction. 3 0

3 1

3 2

Fragment 2, From Geminos's Concise Exposition of the Meteorology of Poseidónios: Geminos on the Relation of Astronomy to Physics

INTRODUCTION

In a m e t e o r o l o g i c a l w o r k that has n o t c o m e d o w n to us, G e m i n o s gave a d i s c u s s i o n o f the r e l a t i o n s h i p between a s t r o n o m y a n d physics that is i m p o r t a n t for o u r u n d e r s t a n d i n g o f the G r e e k s ' attitudes t o w a r d these sciences. H i s d i s c u s s i o n is p r e s e r v e d , i n a m o r e o r less d i r e c t q u o t a ­ t i o n , b y S i m p l i k i o s i n his Commentary on Aristotle's Physics. T h e sup­ p o s e d title o f G e m i n o s ' s lost w o r k is g i v e n by S i m p l i k i o s , as he quotes G e m i n o s " f r o m the Concise Exposition of the Meteorology of Poseidônios." ** 1

S i m p l i k i o s , w h o flourished a b o u t A . D . 5 4 0 , p r o d u c e d commentaries o n a n u m b e r o f Aristotle's w o r k s , i n c l u d i n g the Physics a n d On the Heavens. Because S i m p l i k i o s still h a d access to a large p h i l o s o p h i c a l lit­ erature o f c o m m e n t a r y a n d c r i t i c i s m , m u c h o f w h i c h is n o w lost, he is a v a l u a b l e source for the h i s t o r y o f interpretations o f a n d reactions to A r i s t o t l e . B u t S i m p l i k i o s h a d n o direct k n o w l e d g e o f G e m i n o s ' s lost me­ t e o r o l o g i c a l treatise. Rather, as he makes clear, he quotes this extract f r o m a n earlier commentator, A l e x a n d e r o f A p h r o d i s i a s , w h o h a d c o p i e d it out o f G e m i n o s . Alexander of Aphrodisias, a philosopher and commentator on Aristo­ tle, flourished a b o u t A . D . 2 0 0 . H i s w o r k s are the earliest complete c o m ­ mentaries o n A r i s t o t l e that have c o m e d o w n to us, a n d are therefore o f great interest for the history o f p h i l o s o p h y . A m o n g the w o r k s w e have f r o m A l e x a n d e r are a treatise On Fate, as w e l l as commentaries o n the Metaphysics, the Meteorology, a n d the l o g i c a l w o r k s o f A r i s t o t l e . B u t the passage i n w h i c h A l e x a n d e r q u o t e d this extract f r o m G e m i n o s is n o t to be f o u n d i n A l e x a n d e r ' s extant w o r k . Since S i m p l i k i o s w a s w r i t i n g a c o m m e n t a r y o n Aristotle's Physics, it is l i k e l y that he t o o k this passage f r o m A l e x a n d e r ' s o w n c o m m e n t a r y o n the Physics, w h i c h has not sur­ vived. 2

1

Concise Exposition of the Meteorology of Poseidónios. We have followed Jones

1999a, 255, for this rendering of the title. O n Alexander of Aphrodisias and his place in the Aristotelian tradition, see Todd 1976, 1-20. 2

From Geminos's Concise Exposition

·

251

A l e x a n d e r does allude, however, to G e m i n o s i n one o f his extant w o r k s , i n a brief passage i n his Commentary on Aristotle's Meteorology: T h o s e w h o f o l l o w G e m i n o s a n d A i l i o s , i n p r o o f o f the r a i n b o w being a reflection, use the [fact that] w h e n [one is] a p p r o a c h i n g it, it seems to a p p r o a c h , a n d w h e n g o i n g away, to go away, just as one sees [things] m a k i n g [their] appearance i n m i r r o r s . 3

A i l i o s cannot be identified w i t h certainty, but Aristotle h a d already at­ tempted to e x p l a i n the r a i n b o w as a reflection. Geminos's explanation of the r a i n b o w presumably came from the same meteorological w o r k as frag­ ment 2. A c c o r d i n g to Simplikios, Geminos's b o o k was a n epitome of P o ­ seidônios's Meteorology. But i n the o n l y fragments of Geminos's b o o k that w e possess (that is, fragment 2 a n d the remark o n the r a i n b o w ) , G e m i n o s follows Aristotle rather closely. T h i s is not too surprising, since Poseidônios must have taken Aristotle's Meteorology for his o w n point o f departure. 4

There is a t h i r d , very brief, ancient c i t a t i o n o f G e m i n o s ' s w o r k , made by Priscianus L y d u s (sixth century A . D . ) , w h o says i n his Explanation of Problems for King Chosroes that he has t a k e n some things " f r o m G e m i ­ nos's C o m m e n t a r y o n the Peri Meteôrôn o f Poseidônios. " ' It m a y seem o d d that G e m i n o s a n d Poseidônios s h o u l d have discussed the r e l a t i o n between physics a n d a s t r o n o m y i n w o r k s c a l l e d Meteorol­ ogy. F o r the G r e e k s , however, m e t e o r o l o g y is the " s t u d y o f things raised o n h i g h . " It has p o t e n t i a l l y as m u c h to d o w i t h a s t r o n o m y as w i t h things i n the upper air, such as r a i n b o w s . A n c i e n t authors preserve several dif­ ferent titles for Poseidônios's lost m e t e o r o l o g i c a l w o r k a n d , indeed, there may have been m o r e t h a n one o f t h e m . T h e preserved fragments suggest that w h i l e Poseidônios's w o r k i n c l u d e d a treatment o f atmospheric phe­ n o m e n a , such as the r a i n b o w , it p r o b a b l y opened w i t h a general discus­ s i o n o f the c o s m o s , a n d also discussed the fiery nature o f the S u n . T h u s , the b e g i n n i n g p o r t i o n s o f Poseidônios's Meteorology m a y have some­ w h a t resembled the c o r r e s p o n d i n g parts o f Kleomëdës' Meteóra. Frag­ ment 2 c o u l d have fit quite l o g i c a l l y i n t o such a w o r k . Let us t u r n n o w to the intellectual context of fragment 2 . Just before giv­ ing his q u o t a t i o n f r o m G e m i n o s , S i m p l i k i o s is i n the course o f discussing a b

5

6

7

For the Greek text, see Hayduck 1899, or Manitius 1898, 285 (which quotes Ideler's older text). Aristotle, Meteorology 373a32-74a3. The passage is quoted in Edelstein and K i d d 1989, 22, and K i d d 1999, 52, from Bywater 1886, Solutiones ad Chosroem, Prooemium, 42.8-11. Priscianus wrote in Greek, but this work survives only in a Latin translation of the ninth century. See K i d d 1999, p. 72, and fragments 14, 15, 16, 17. But this is only one of several possible titles for Kleomëdës' book. See note 22 in our Introduction. 3

4

5

6

7

252

·

Fragment 2

passage f r o m Aristotle's Physics, i n w h i c h Aristotle distinguishes physics f r o m mathematics o n the one h a n d a n d f r o m metaphysics o n the other. In the course o f this investigation, Aristotle raises the question of whether as­ t r o n o m y is a separate science o r a part o f physics. For, as Aristotle points out, one w h o seeks to k n o w w h a t the Sun a n d the M o o n are c a n h a r d l y a v o i d i n q u i r y into their essential natures. M o r e o v e r , he continues, physical thinkers have discussed the shape of the Sun a n d the M o o n a n d have asked w h e t h e r the E a r t h a n d the c o s m o s are spherical. In c o n t r a s t i n g physics w i t h mathematics, A r i s t o t l e r e m a r k s that the m a t h e m a t i c i a n deals w i t h lines a n d surfaces, n o t as the boundaries o f n a t u r a l bodies, but rather i n a b s t r a c t i o n . H e also m e n t i o n s optics, h a r m o n i c s , a n d a s t r o n o m y as sci­ ences that partake o f b o t h the p h y s i c a l a n d the m a t h e m a t i c a l , t h o u g h they are m o r e p h y s i c a l t h a n m a t h e m a t i c a l i n nature. G e o m e t r y a n d optics stand, as it were, i n opposite relationships to the m a t h e m a t i c a l a n d the p h y s i c a l . T h u s the geometer m a y deal w i t h p h y s i c a l lines, but o n l y i n their m a t h e m a t i c a l , not their t r u l y p h y s i c a l , nature. In the opposite w a y , optics deals w i t h m a t h e m a t i c a l lines, but as p h y s i c a l lines, n o t t r u l y m a t h e m a t i c a l ones. 8

Some elements o f A r i s t o t l e ' s d i s c u s s i o n find echoes i n G e m i n o s , n o t o n l y i n fragment 2, under d i s c u s s i o n here, but also i n G e m i n o s ' s classifi­ c a t i o n o f the m a t h e m a t i c a l sciences (fragment 1). B u t G e m i n o s states the r e l a t i o n s h i p o f a s t r o n o m y to physics i n a m u c h clearer a n d s i m p l e r w a y . Indeed, his r e m a r k s constitute the clearest statement o f this r e l a t i o n ­ ship w e find i n any o f the G r e e k a s t r o n o m i c a l writers. T h o u g h he m a y have begun w i t h A r i s t o t l e (or w i t h Poseidónios, w h o began w i t h A r i s t o ­ tle), it is clear that G e m i n o s h a d done some h a r d t h i n k i n g o n this sub­ ject himself. F o r a detailed d i s c u s s i o n o f this passage, see sec. 10 o f the Introduction.

TRANSLATION OF FRAGMENT

2

9

A l e x a n d e r , i n his diligent w a y , p r o v i d e s , f r o m the Concise Account of the Meteorology of Poseidónios, a passage by G e m i n o s that takes its starting p o i n t f r o m A r i s t o t l e . H e r e it is: It is [the task] o f p h y s i c a l theory to i n q u i r e i n t o the essence o f the heaven a n d the stars, their p o w e r a n d quality, their o r i g i n a n d destruction; 10

8

Aristotle, Physics 193b22.

The translation of fragment 2 is based on the text of Diels 1882, 2 9 1 - 9 2 . English translations of this passage have been published by Heath 1932, 123-25, and by Bowen and Todd 2004, 199-204, which we have consulted with profit. takes its starting point from Aristotle. See Aristotle's discussion beginning at Physics 193b22. 9

1 0

From Geminos's Concise Exposition

·

253

a n d , b y Z e u s , it c a n even m a k e d e m o n s t r a t i o n s c o n c e r n i n g their size, f o r m , a n d arrangement. A s t r o n o m y , o n the other h a n d , does n o t attempt to speak a b o u t a n y such t h i n g , but demonstrates the a r r a n g e m e n t o f the heaven, presenting the heaven as a n o r d e r l y w h o l e , a n d speaks a b o u t the shapes, sizes, a n d distances o f the E a r t h , S u n , a n d M o o n , a b o u t eclipses a n d c o n j u n c t i o n s o f the stars, a n d a b o u t the q u a l i t y a n d q u a n t i t y o f their m o t i o n s . Therefore, since it deals w i t h the i n v e s t i g a t i o n i n t o quantity, m a g n i t u d e , a n d q u a l i t y i n r e l a t i o n t o f o r m , it n a t u r a l l y needed a r i t h ­ metic a n d geometry for this. A n d c o n c e r n i n g these things, the o n l y ones of w h i c h it u n d e r t o o k to give a n account, a s t r o n o m y has the c a p a c i t y to reach c o n c l u s i o n s b y means o f a r i t h m e t i c a n d geometry. N o w i n m a n y cases b o t h the astronomer a n d the p h y s i c i s t

11

will pro­

pose to demonstrate the same p o i n t , such as that the S u n is large o r that the E a r t h is spherical, but they w i l l not proceed by the same paths. O n e [the physicist] w i l l p r o v e each p o i n t f r o m considerations o f essence or i n ­ herent power, or f r o m its being b e t t e r

12

to have things thus, o r f r o m o r i ­

g i n a n d change; but the other [the astronomer] w i l l p r o v e t h e m f r o m the properties o f figures o r magnitudes, o r f r o m the a m o u n t o f m o t i o n a n d the time appropriate to it. A g a i n , the physicist w i l l often reach the cause by l o o k i n g to creative force; but the astronomer, w h e n he makes d e m o n ­ strations f r o m extrinsic properties, is n o t competent to perceive the cause, as w h e n , for e x a m p l e , he makes the E a r t h a n d the stars s p h e r i c a l .

13

Some­

times he does n o t even desire to take u p the cause, as w h e n he discourses about an eclipse;

14

but at other times he invents b y w a y o f hypothesis

the astronomer and the physicist. "Astronomer" = astrólogos, "physicist" = physikos, which could also be translated "natural [philosopher]," since, as the following passages from Geminos will make clear, the approach to nature of the physikos was that of a philosopher in the Aristotelian tradition, not that of a modern physicist. from its being better. This is a kind of argument recommended by Aristotle. See Physics i i 198b4.-9. the stars are spherical. Aristotle, On the Heavens ( 2 9 1 b l l - 2 3 ) , proves this, both by appeal to physical principles and by astronomical methods. Physically, the stars must be spherical. Nature does nothing without purpose. Therefore the stars, which do not move of themselves, should have the figure least suited to motion. The figure least suited to mo­ tion is the sphere, since it has no instrument to serve that purpose. (This recalls Plato's re­ mark about the cosmos as a whole: the demiurge made the cosmos without hands and feet, for it had no need to grasp anything, nor any need for stepping. Timaeus 33d.) Astronomi­ cally, it is clear that the M o o n is spherical, from the evidence of its monthly cycle of phases, and, Aristotle concludes, if one of the heavenly bodies is spherical, the others must be spherical also. eclipse. Geminos's remark here may seem slightly strange, since knowledge of the cause of eclipses goes back at least to Anaxagoras (fifth century B.c.). A n y astronomer of Geminos's day was perfectly capable of setting forth the cause of an eclipse. But in the use of eclipses in some practical calculation, he might very well omit to discuss the cause. Eclipses of the M o o n might be used, for example, in a discussion of lunisolar cycles. O r 11

1 2

13

1 4

254

·

Fragment 2

a n d grants c e r t a i n devices, by the a s s u m p t i o n o f w h i c h the w i l l be s a v e d .

phenomena

15

F o r e x a m p l e , w h y d o the S u n , M o o n , a n d planets a p p e a r to m o v e i r ­ r e g u l a r l y ? [The a s t r o n o m e r w o u l d answer] t h a t i f w e assume t h a t t h e i r circles are eccentric, o r t h a t the stars go a r o u n d o n a n e p i c y c l e , t h e i r ap­ p a r e n t i r r e g u l a r i t y w i l l be saved. A n d it w i l l be necessary to fully e x a m ­ ine i n h o w m a n y w a y s it is p o s s i b l e for these p h e n o m e n a to be b r o u g h t about,

1 6

so t h a t the t r e a t m e n t o f the planets befits a c a u s a l e x p l a n a t i o n

1 7

a c c o r d i n g to the accepted w a y . A n d thus a c e r t a i n p e r s o n , Hërakleidës o f Pontos, ' 1 8

c

c o m i n g f o r w a r d , says t h a t even i f the E a r t h m o v e s i n a cer­

t a i n w a y a n d the S u n is i n a c e r t a i n w a y at rest, the a p p a r e n t i r r e g u l a r i t y w i t h r e g a r d to the S u n c a n be saved. F o r it is c e r t a i n l y n o t for the a s t r o n o m e r to k n o w w h a t is b y n a t u r e at rest a n d w h a t s o r t [of b o d i e s ] are g i v e n t o m o v e m e n t . R a t h e r , i n t r o ­ d u c i n g h y p o t h e s e s t h a t c e r t a i n [bodies] are at rest a n d o t h e r s are

mov­

i n g , he c o n s i d e r s f r o m w h i c h h y p o t h e s e s the p h e n o m e n a i n the h e a v e n w i l l [actually] follow. principles,

1 9

t h a t the

But

he

m u s t t a k e f r o m the

m o t i o n s o f the

stars are

p h y s i c i s t the

simple, uniform,

first and

again, the time of onset of a lunar eclipse, observed at two different localities, might be used to deduce the difference in geographical longitude of the two places of observation. In either of these applications there w o u l d be no need to discuss the cause of eclipses. the phenomena w i l l be saved. O n the slogan and program of "saving the phenom­ ena," see sec. 10 of the Introduction. examine i n h o w many ways it is possible for these phenomena to be brought about. This echoes a remark of Hipparchos. According to Theôn of Smyrna (Mathematical Knowledge iii 26.3), Hipparchos said that research into the explanation of the same phe­ nomena by hypotheses that are quite different is a task worthy of the attention of the mathematician. Hipparchos, Theôn, and Geminos were all alluding to the fact that dif­ ferent mathematical models are capable of accounting for the same phenomena. causal explanation, aitiologia. For the meaning of this term, see the discussion in sec. 10 of the Introduction. Hërakleidës of Pontos. Thus Diels and the mss. M o s t scholars agree, however, that this name does not belong here. Probably, the text originally said that a "certain person (fis), coming forward, says . . . ," and then the name Hërakleidës of Pontos was interpo­ lated by some later copyist. Aristarchos of Samos, not Hërakleidës, was responsible for the view that the Earth moves in a circle about a stationary Sun. We have clear testimony from Aëtios (Opinions of the Philosophers i i i 13.3), as well as from Simplikios's Com­ mentary on Aristotle's O n the Heavens (293b30 and 2 8 9 b l ; Heiberg 1894, 519, and 444-45), that Hërakleidës espoused the daily rotation of the Earth on its axis, but not the motion of the Earth around the Sun. For translations of the relevant passages, see Heath 1932, 9 3 - 9 4 . A partially heliocentric system has sometimes been ascribed to Hërakleidës. In this system, Venus and M e r c u r y orbit the Sun while the Sun and the other planets orbit the Earth. Although such a system is described by several ancient writers (e.g., Theôn of Smyrna, Mathematical Knowledge Useful for Reading Plato i i i 33), the ascription of this view to Hërakleidës is mistaken. For a discussion of all the evidence, see Eastwood 1992. 1 5

1 6

1 7

1 8

first principles. See Introduction to the Phenomena of circular motion as the first principle of astronomy. 1 9

i 18-21 for Geminos's discussion

From Geminos's Concise Exposition

·

255

o r d e r l y , f r o m w h i c h he w i l l d e m o n s t r a t e t h a t the d a n c e o f a l l [the stars] is c i r c u l a r , w i t h some t u r n i n g r o u n d o n p a r a l l e l c i r c l e s , s o m e o n oblique ones. In this manner, then, does G e m i n o s , or rather Poseidônios i n G e m i n o s , give the d i s t i n c t i o n between physics a n d astronomy, t a k i n g his starting point from Aristotle. 2 0

21

dance, choreian. Plato (Timaeus 40c3) spoke of the choric dances (choretas) of the planets, and Geminos's readers might well have made this association. But the word could also be used, by analogy, of any circling motion. parallel and oblique circles. On the parallel circles see ν 1-9. The fixed stars move from east to west on parallel circles, i.e., circles parallel to the equator. In the course of any one day or night, the Sun, Moon, and each of the planets may be considered to move on a parallel circle. But the Sun, Moon, and planets each have an additional and slower motion from west to east along the zodiac, a circle that is oblique to the equator. 20

21

APPENDIX

1

Textual Notes to Geminos's Introduction to the Phenomena

TITLE

Geminos's Introduction to the Phenomena, Γεμίνου είσαγωγή εις τα φαινόμενα, Β and Μ. Γεμίνου είσαγωγή είς τα φαινόμενα, V, but with εισαγω­ γή inserted above the line in the same hand. The title and author's name are want­ ing in Α. Γεμίνου είσαγωγή εις τα μετέωρα, C . Γεμίνου τα φαινόμενα, A m brosianus C 263 inf. (Milan). a

CHAPTER I

The circle of the signs. Geminos refers to the zodiac indifferently as ó των ζωδίων κύκλος or ó ζωδιακός κύκλος. The first will be translated as "the circle of the signs," and the second as "the zodiac circle," but no significance should be attached to the distinction. Leo, ó Λέων, following the mss. ή Παρθένος ("Virgo"), Manitius, followed by Aujac. Virgo does extend over a greater range in longitude (some 46°) than any other zodiacal constellation. But because Leo also takes up more than the canonical 30°, there is nothing wrong with the reading of the mss. Autumnal equinox. Here begins a lacuna of some thirty lines in the Greek text, which Manitius (followed by Aujac) has filled from the medieval Latin translation. The text supplied from the Latin is enclosed between the marks < >. The Latin text of this sentence appears to be corrupt: Et conversio estiva fit, quando sol pervenit ad propinquiora loca sont capitum in habitationibus nostris et elevatur ab orizonte nostro ultimiore suis elevationibus. . . . The Latin word sont is glossed a little further on by cenit, i.e., zenith (Aujac 1975, 3n2). We con­ jecture that this originated as an imperfect transcription of the Arabic samt, which means "direction," and is the normal word for "zenith" in astronomical texts. . Added by Manitius (followed by Aujac) in analogy to Kleomëdës (Meteôra i 2.35), supported by the medieval Latin version of Geminos. a

b

c

d

e

CHAPTER II

antiskian, άντίσκια, our emendation, άντισυζυγίαν, the mss., Manitius, Aujac. Manitius interprets antisyzygy as if it were an alternative name, used by a

258

·

Appendix 1

some, for syzygy. Aujac treats antisyzygy as a fifth distinct aspect, added by some. But to our knowledge, άντισυζυγία occurs in no other astronomical or astrological writer, while άντίσκια is a common term that is indeed used syn­ onymously with " i n syzygy." occur, συντελούνται, following Aujac and the mss. άρχονται, Manitius. of spring, of summer, of fall, of winter. Thus the mss. and Aujac. Manitius brackets these words for deletion. . The passage in brackets < > is missing in the Greek mss. but has been supplied by Manitius (followed by Aujac) from the medieval Latin translation. the first degree of Libra culminates below the Earth. This whole statement (ii 21) is corrupt in all the manuscripts. For the statement to be correct, the first de­ gree of Capricorn must be at the meridian, and not on the horizon. Thus Gemi­ nos must have written either: (1) "when the first degree of Aries is setting, the first degree of Cancer culminates, the first degree of Libra rises, and the first de­ gree of Capricorn culminates below the Earth"; or (2) "when the first degree of Libra is setting, the first degree of Capricorn culminates, the first degree of Aries rises, and the first degree of Cancer culminates below the Earth." As may be con­ firmed with an armillary sphere or a celestial globe, it is only when the equinoctial points are on the horizon that the ecliptic is divided into four equal parts by the horizon and the meridian. In general, the ecliptic is not divided into four equal parts by the horizon and the meridian, as Geminos himself points out (ii 22). contained, έμπεριλαμβανόμενα, accepting Manitius's emendation. Geminos uses έμπεριλαμβάνεται in just this way at i i 36, 39, and 44. παραλαμβανόμενα, Aujac and the mss. s perceivable through reason, λόγω θεωρητόν. b

c

d

e

f

CHAPTER III

whole, όλω, accepting Manitius's emendation after the Latin version and in analogy to the preceding statement about Lyra. N o t in the Greek mss. or Aujac. hand. Before this word, Manitius adds αριστερά, "left," without ms. authority. the W i l d Animal that Centaurus is holding, and the Thyrsus-lance that Cen­ taurus is holding according to Hipparchos. θηρίον, δ κρατεί ó Κένταυρος, καΐ θυρσόλογχος, δν κρατεί ó Κένταυρος καθ' 'Ίππαρχον, following Manitius and A , B, C . Aujac, following I, Ρ, Τ, Μ and the medieval Latin version, deletes καί Θυρσόλογχος, δν κρατεί ó Κένταυρος, thus making the passage read "the W i l d Animal that Centaurus is holding according to Hipparchos." Aujac reads the deleted phrase as a spurious addition that originated as a gloss drawn from Ptolemy's or Hipparchos's mention of the thyrsus as a part of Centaurus. The best argument for this view is that the phrase is missing in the Latin version of Geminos, which was based (through an Arabic intermediary) on a Greek manuscript older than any we now possess. Deleting the phrase, however, produces a reading that attributes the W i l d Animal to Hipparchos, even though the W i l d Animal is well attested before Hipparchos (see the discussion following Table 3.3). a

b

c

Textual Notes

·

259

the Caduceus according to Hipparchos, Κηρύκειον (Κηρύκιον, Manitius) καθ* 'Ίππαρχον, following Manitius and the mss. et Abrachis nominavit earn purpuream, Latin. Aujac deletes this phrase as too problematical. d

CHAPTER V

all these circles are inscribed . . . in astronomy, Συγκαταγράφονται δέ ούτοι πάντες είς την σφαίραν δια τό προς μεν αλλάς πραγματείας των εν τη αστρολογία πολλά συμβάλλεσθαι, following Aujac and the mss. Manitius changes συγκαταγράφονται ("are inscribed together") to ού καταγράφονται ("are not inscribed") on the authority of the Latin, non signantur. It is unlikely that 182 parallel circles were drawn on many ancient celestial globes. But M a n i tius's reading seems to require an adversative or concessive construction of some kind in place of δια τό . . . πολλά συμβάλλεσθαι. In either case, Geminos's meaning is clear: the astronomer may need many parallels to specify the declina­ tions of stars, or to work out the lengths of all the days of the year; but for the beginner the five principal parallels will suffice, and these are the only ones cus­ tomarily shown on globes and armillary spheres. a

For the horizon in Greece. A , B, and C have a lacuna here. The words in brackets follow Aujac's guess, supported by the Latin version, orizon autem regionis grecorum qui nominantur elenes dividit orbem tropici estivi. Manitius prefers "For the horizon at the Hellespont." The Hellespont was traditionally as­ sociated with the klima of 15 hours, as in Ptolemy {Almagest ii 6) and Kleomëdës (Meteôra ii 1.442), and the 15-hour figure is more accurate for the Hellespont than for Greece. But Aratos did not specify the latitude to which his poem applies, and Neugebauer (1975, 711) points out the error of assuming the Hellespont whenever the 15-hour klima is mentioned. (In vi 8, Geminos puts Rome in the 15-hour klima.) Although Geminos interpreted Aratos as having intended a spe­ cific klima, we have no way of knowing whether Geminos meant to identify this klima narrowly with the Hellespont, or more generally with Greece. our oikoumenë, την ήμετέραν οΐκουμένην, following Aujac and A , B, C . Manitius prints οϊκησιν ("dwelling place"), which occurs in a ms. of the Sphere of Proklos (L). Even with a broad definition of the oikoumenë, the text is sensi­ ble. Moreover, Geminos uses oikoumenë in a similar way at ν 3. one makes the division in declination in the following way, διαιρείται ούτως κατά πλάτος, thus Aujac, following A , B, C . Manitius prints διαιρείται ούτως, " is divided in the following way." the one perceptible [to our sight] and the other perceivable by reason, είς μεν ó αίσθητός, έτερος δέ ó λόγψ θεωρητός. inclination, έγκλιμα. s whole, δλη, following Aujac and the mss. Manitius marks this word for deletion. slants the tropic circles, λελόξωται των τροπικών κύκλων, following Manitius and A ; added by Manitius. λελόξωται τω τροπικω κύκλω, Aujac, Β, and C . b

c

d

e

f

h

260

·

Appendix 1

CHAPTER V I

the longest day turns out to be two months long, διμηνιαίαν των ήμερων την μεγίστην ήμέραν συμβαίνει γίνεσθαι, Aujac. We follow Manitius in delet­ ing των ημερών. . Inserted by Manitius. This is why. Manitius brackets this whole statement (vi 31) for deletion, branding it an inept interpolation. But the same cause for the intensification of the heat around summer solstice is invoked at xvii 28, and we see no need to doubt that the argument is Geminos's. it (the subject of the verb simply understood), following Manitius. ó ήλιος, Aujac after the Latin version. N o t in A or Β. αυτός, C . The cause of the inequality in the lengthening of the days is the obliquity of the zodiac circle. The mss. read: Αίτία δε έστι της άνισότητος καί της των ήμερων παραυξήσεως ("The cause of the inequality and of the lengthening of the days is . . ."). Following Aujac, we suppress καί. Manitius suppresses της άνισότητος καί, on the grounds that the cause of the inequality of the days has already been given at vi 24. Thus Manitius would read, "The cause of the lengthening of the days is the obliquity of the zodiac circle." A t vi 24, however, Geminos is explaining only why some days are longer than others. By contrast, in the present paragraph he is addressing the fact that the changes in the length of the day (from one day to the next) are of different sizes at different times of the year. and the nights. Manitius brackets these words for deletion. 8 moreover, the increases have nearly a constant difference, καί σχεδόν την αυτήν παραλλαγήν εχουσιν αί παραυξήσεις, more literally, "the increases have nearly the same difference." Manitius brackets these words as a gloss. But this hy­ pothesis of constant differences is connected intimately with the remark that fol­ lows, i.e., that the daily increases around the equinox are ninety times as large as the daily increases around the solstices. See sec. 12 of the Introduction for Gemi­ nos's use of arithmetic progressions. (Note the typographical error in π α ρ α λ ­ λαγήν in Aujac's text.) Pisces. After the sentence, Manitius adds , "they are south­ ern," in analogy to the remark at the end of vi 40. a

b

c

d

e

f

h

CHAPTER VII

For this reason . . . in the same place. Manitius deletes this sentence as a gloss. arc, περιφέρεια, following Manitius and Aujac. περιφορά, the mss. degrees, μοιρών, following Manitius. μερών, Aujac and the mss. rising. In the mss. this is followed by καί τήν δύσιν, "and the setting." Fol­ lowing Manitius, we delete these words as an imprecise interpolation, for the signs that rise in the greatest time actually set in the least time. What these words are meant to suggest is that signs that set when the zodiac is upright take the most time to set; i.e., that what goes for rising goes for setting, too. , , following Manitius. a

b

c

d

e

Textual Notes

·

261

the rising. Followed in the mss. by καί της δύσεως, "and the setting," which we delete, following Manitius (see textual note d above). s as one another, κατ άλληλα, following Aujac. κατ άλλα, the mss., deleted by Manitius. Capricorn . . . Gemini. The mss., Manitius, and Aujac all have Κριοΰ . . . Παρθένου, "Aries . . . Virgo." The text states that the zodiac is lowest on the horizon when the first degree of Capricorn is culminating. This is correct for the northern midlatitudes. The problem is that the text goes on to claim that the semicircle of the zodiac from Aries 0° to Libra 0° therefore rises in a short time. In fact, this semicircle takes exactly 12 hours to rise at any latitude. This error is so elementary and so easily detected on a globe or armillary sphere that it is im­ possible to believe that Geminos could have committed it. A correct claim would be that the semicircle centered around Aries 0° (rather than starting from Aries 0°) would take a short time to rise. These six signs (from Capricorn 0° to Gemini 30°) are just those signs that rise in the course of the night on summer solstice. The version of the text is probably due to an early, and unskillful, emendation. Cancer . . . Sagittarius. The mss., Manitius and Aujac all have Ζυγού . . . Ιχθύων, "Libra . . . Pisces." This error is analogous to that committed in the preceding sentence. See textual note h. J or when the first degree of Libra is culminating. Manitius suppresses this paren­ thetical remark as an interpolation. This could well be right, for Geminos discusses the situation when Libra (the Claws) is culminating at vi 22, immediately below. . Inserted by Manitius after the model of the preceding statements (vii 19-20). they asserted, άπεφήναντο, adopting Manitius's emendation in analogy to the preceding statements, ώστε, Aujac and the mss. Virgo rises. Manitius inserts after these words, without manuscript author­ ity, . This emendation, like many others that Manitius makes to later parts of this passage, is not required for either meaning or clarity. Through­ out the remainder of the chapter, Manitius makes many additions that we have ignored, and it seems unnecessary to mention them all here. Manitius's emenda­ tions do make the discussion perfectly balanced and complete, but whether Geminos really so belabored the point is another question. [signs] that are equally distant from the solstitial and equinoctial points rise and also set in equal time. The text is sloppy here: see the discussion in comment 13. Still, Manitius's emendation of this sentence does not result in correct astron­ omy, but only states the error in more emphatic terms: "From these things it is clear that [signs] that are equally distant from the solstitial and the equinoctial points rise and set in equal time." f

h

i

k

1

m

n

CHAPTER VIII 29 + 1/2 + 1/33. Written κθ U λγ' in the mss., as is normal. fall. Before this word, Manitius inserts , thus giving the passage the sense, "and the summer festival to be also a winter one, and fall [festival to be] also a spring one. " a

b

262

·

Appendix 1

spring [festival]. Here the Latin version adds: immo permutantur in omnibus temporibus 4 et eorum revolutiones sunt in omnibus diebus anni, in omnibus mille et quadringentis et sexaginta annis. This appears to be a gloss based on viii 24. most of the Greeks suppose the winter solstice according to Eudoxos to be at the same time as the feasts of Isis [reckoned] according to the Egyptians, Ύπολαμβάυονσι γαρ οί πλείστοι των Ελλήνων άμα τοις Ίσίοις κατ' Αίγυπτίους καί κατ' Εΰδοξον είναι χειμερινός τροπάς. See the discussion in comment 10. again. Accepting Manitius's transposition of πάλιν from near the beginning of the sentence to near the end. they sought. Adopting Manitius's punctuation of this passage and his dele­ tion of όθεν immediately before these words. ε The time of the period, οτης περιόδου χρόνος, following Aujac and the mss. Manitius deletes these words without comment. , , accepting the insertion by Manitius and the ear­ lier editors. N o t in Aujac. The lunar year is reckoned at 354 days. Manitius deletes this entire para­ graph (viii 34-35) as spurious, but its role in the development is clear: it provides a capsule summary of the lunar year used in the octaetêris immediately before Geminos takes up the defects of the octaetêris. ' 30 days. These words are followed in the mss. and Aujac by προς τάς του ηλίου ώρας, "with respect to the seasons of the Sun." Following Manitius, we delete this phrase as a gloss. For the monthly period has not been taken accurately. After these words, the mss. contain two additional sentences: c

d

e

f

h

1

k

For the monthly period, taken accurately, is 29;31,50,8,20 days. Because of this, it w i l l sometimes be necessary to intercalate 4, instead of 3, embolismic days i n the 16 years.

We delete this passage as an interpolation. Manitius deletes all of viii 4 3 - 4 5 . A u ­ jac deletes only the third sentence of viii 43. The value for the length of the synodic month, 29;31,50,8,20 days, appears in the Latin translation and in some of the Greek manuscripts—the others showing variants. This is, in fact, a more accurate value for the length of the month than the value usually used by Geminos, 29 + 1/2 + 1/33 days. The value 29;31, 50,8,20 days is given by Ptolemy (Almagest iv 2), who says that it was obtained by Hipparchos, "by calculations from observations made by the Chaldeans and in his time." This parameter is actually of Babylonian origin and plays a role in system Β of the Babylonian lunar theory (see Neugebauer 1975, 483). In view of his familiarity with other details of Babylonian astronomy, there is no historical reason why Geminos could not have known the value 29;31,50,8,20. Neuge­ bauer (1975, 585) accepts Geminos's mention of this parameter as genuine. The difficulty is that the Hipparchian-Babylonian value is mentioned but once, while Geminos uses on several occasions the value 29 + 1/2 + 1/33 days (= 29;31,49,5,27), which he introduced at the beginning of chapter viii. When Geminos returns to lunisolar cycles in chapter xviii, he uses the value 29 + 1/2 + 1/33 days again, and does not mention the better Hipparchian-

Textual Notes

·

263

Babylonian value. Moreover, in the mss. the Hipparchian-Babylonian value for the synodic month is given in base-60 (as by Ptolemy), while throughout chapter viii Geminos operates with unit fractions. A n d when Geminos does introduce base-60 notation in chapter xviii, he is careful to explain his notation to his reader. It would be odd if Geminos used base-60 fractions earlier in the book without explanation. Finally, the Hipparchian-Babylonian month length has no function in the ar­ gument. Geminos states (viii 42) that the octaetëris fails in the length of the month, as well as in the number of embolismic months. Then (at viii 43-45) he points out that because the month is longer than 29Vi days, the octaetëris^ ba­ sic pattern of alternating full and hollow months makes the month too short. The already-mentioned length of the month, 29 + 1/2 + 1/33 days, is sufficient for this argument, and there is no need to introduce the HipparchianBabylonian value. Then (at viii 46-47) Geminos, fulfilling his promise in viii 42, shows how the octaetëris fails to give the correct number of embolismic months. Here he introduces a new parameter, the lunar year of 354V3 days, which implies a slight modification of his adopted length of the month. Again, there is no role in the argument for the Hipparchian-Babylonian value. Thus it appears that the Hipparchian, base-60 parameter in chapter viii was interpolated as a comment drawn from the Almagest, or some source dependent on it. 5 The year according to them is therefore 365 /i9 days. This sentence is rejected by Manitius as an inept addition. But it follows immediately from the length of the nineteen year period, 6,940 days, which Geminos has stated in the preceding sentence. and the days total 7,050. These words are followed in the mss. by the sentence: "But it was necessary to count 110 hollow, for which reason there are 6,940 days according to the M o o n for the nineteen-year period." Following Manitius, we delete this sentence as an interpolation and, moreover, one that breaks the flow of the argument and anticipates the conclusion reached in viii 54. therefore reckon, οΰν άγουσι, following Manitius. συνάγουσι, Aujac and the mss. 1

m

n

CHAPTER I X

faces toward the east. After these words, Manitius adds, without manuscript authority, We have no reason to think that Geminos so belabored the issue. . Added by both Manitius and Aujac, following the earlier editors. at the latest, βραδύτατον, following Aujac. Deleted by Manitius. becomes, γίνεται, following Manitius's conjecture, ανατέλλει, Aujac and the mss. The entire monthly period. Manitius rejects this entire paragraph (ix 16) as an inept repetition of the opening lines of chapter viii. a

b

c

d

e

264

·

Appendix 1

CHAPTER X I

for all the eclipses of the M o o n occur i n this space. Manitius rejects these words as a gloss; but they are an essential clarification of the meaning of τό έκλειπτικόν. for the magnitudes of the eclipses are in due relation to the M o o n ' s motion in latitude, προς λόγον γαρ της κατά πλάτος κινήσεως της σελήνης τα μεγέθη των εκλείψεων σύμφωνα γίνεται. M o r e literally, " . . . are harmonious i n relation to. . . . " Manitius deletes σύμφωνα because it normally governs the dative case. In the mss. and Aujac, κινήσεσως is followed immediately by της ημερησίου: thus, "the M o o n ' s 'daily' motion i n latitude." But as Manitius points out, this is surely a later and irrelevant addition, for the argument has nothing to do with 1 day's worth of the M o o n ' s motion in latitude. a

b

CHAPTER X I I

For the sign following the Sun is always invisible because of the rays of the Sun, while the one preceding it is visible. Manitius, without explanation, brack­ ets this sentence for deletion, and the sentence could be read as an awkward gloss. It does make sense, however, if we regard it (like many others in this pas­ sage) as applying to observations made at sunrise. The situation is reversed for observations made at sunset: then part of the sign following the Sun is visible, while the one preceding it is invisible, for it sets before the Sun does. to have stood away a distance of two signs, δύο ζωδίων διάστημα άφεστηκός. Manitius, without explanation, brackets these words for deletion. as the night advances it departs toward the east from the carefully noted star; and west . Accepting Manitius's emendation: while moving toward the same parts as the cosmos. Adopting, with Manitius and the earlier editors, κινούμενοι τω κόσμω. κινουμένου του κόσμου, Aujac. the motion proper to it were from west to east. Adopting Manitius's text, ίδίας ύπαρχούσης αύτω της κινήσεως [μέν άπ' ανατολής επί δύσιν, της δέ] άπό δύσεως έπ' άνατολήν ("the motion proper to it were [from east to west and] from west to east.") As Manitius conjectures, the phrase "from east to west" is most easily explained as a marginal note that worked its way into the text. Aujac prints the mss. text and translates "at the same time from east to west and from west to east." but nothing, ουδέν δέ, accepting Manitius's emendation, ουδέ, Aujac and the mss. a

b

c

d

e

f

CHAPTER XIII for this reason. Manitius brackets this entire statement (xiii 10) for deletion, because it is not found in the Latin version. again, πάλιν. Deleted by Manitius. a

b

Textual Notes

·

265

for the last time, τό εσχατον (our emendation). The mss. read πρώτος, "for the first time." However, "having escaped for the first time from the rays of the Sun," is inappropriate. The visible evening rising is the last, not the first, of the star's risings to be visible during the annual cycle. (Autolykos, On Risings and Settings i 2, as well as the initial definitions, Aujac 1979, 69. This is also apparent from fig. 13.1.) Statement 13 (on the last rising to escape the rays of the Sun) can be set off against statement 9 (on the first rising to escape the rays of the Sun). The concluding portion of statement 13 is correct. O n successive evenings follow­ ing the V E R , once the sky becomes dark enough for stars to be seen, star S will be seen higher and higher above the eastern horizon. Note that, in this situation, "es­ caping the rays of the Sun" can only be applied to the rising of the star and not to the star itself: the star itself is visible during some part of the night both before and after the visible evening rising. whenever both the Sun and the star are on the horizon. Following Manitius, who inserts immediately after όταν, by analogy to xiii 17. We point also to xiii 6, and 11, where κατά άλήθειαν is used for a simi­ lar purpose. for the first time, τό πρώτον. The mss. have τό εσχατον, "for the last time." Manitius suggested (p. 274) but did not actually adopt this emendation, lest it excuse the author (Geminos or his excerptor) of repeated error. Aujac tries to rescue the text by translating τό εσχατον as "à la dernière limite," meaning that the star is seen setting at the last moment before sunrise. It seems better to concede that the text is corrupt. For the . From here to the end of xiii 27, the text is badly corrupted. Every proposition in xiii 21-23 and 27 is either wrong or problematical as stated. c

d

e

f

The Difficulties The text does not state whether the propositions in xiii 21-27 apply to true or to visible phases. One might suppose that visible phases are meant, since in the parapëgmata and in literary references to heliacal risings and settings, unqualified phases are, almost invariably, visible ones. But the propositions in xiii 29 also are unqualified; and these propositions are correct as stated only for the true phases. A second difficulty with assuming that visible phases are meant is that the text never states a quantitative visibility rule. Autolykos, for example, assumes that the visible risings and settings occur when the star is on the horizon and the Sun is half a zodiac sign below the horizon, measured along the ecliptic. Thus, the visible phases precede or follow the true ones by about 15 days. This simple rule, a crude but reasonable approximation, allows quantitative predictions about the dates of the visible phases. That Geminos never mentions such a rule might seem to imply that his propositions apply to true rather than to visible phases. It should be pointed out, however, that a good deal can be done without a quantitative visibility rule, simply by using Geminos's proposition xiii 19 and some notions of symmetry. Let us now examine what the propositions assert.

266

·

Appendix 1

• For stars on the zodiac circle: x i i i 2 1 . T h e time f r o m evening r i s i n g to m o r n i n g rising is 6 m o n t h s , as is the time f r o m evening setting to m o r n i n g setting. In short­ hand notation: E R - » M R = 6 months ES —> M S = 6 months This proposition is false for visible phases. It is correct for true phases; but it is cor­ rect for all stars, and thus there would be no reason to single out zodiacal stars. • For stars north of the zodiac: xiii 22. xiii 27.

E R -> M R > 6 months M S —» E R < 6 months

The first proposition is false for true phases (the time should be = 6 months). It is correct for visible phases, but is correct for all stars, not just northern ones. The second proposition is false for both true and visible phases. • For stars south of the zodiac: xiii 23. xiii 27.

E R -> M R < 6 months M S -> E R > 6 months

The first proposition is false for true phases (the time should be = 6 months). It is also false for the visible phases (the time should be > 6 months). The second proposition is false for true phases (the time should be « 6 months). For the visible phases, the proposition is indeterminate: it is correct for some southern stars but false for others, depending upon just how far south the particular star is. Emendations A l l difficulties with these propositions would be solved by the following sim­ ple emendations. • For stars on the zodiac circle: xiii 21.

M R —> M S = 6 months ES —> E R = 6 months

• For stars north of the zodiac: xiii 22. xiii 27.

M R —> M S > 6 months ES —> E R < 6 months.

• For stars south of the zodiac: xiii 23. xiii 27.

M R —> M S < 6 months ES —> E R > 6 months.

A l l of the emended propositions are correct for both true and visible phases. Moreover, their applicability to the visible phases does not depend on adopting any particular numerical value for the visibility rule. Indeed, the emended propo­ sitions may be proved for the visible phases simply by invoking xiii 19 and a sim­ ple symmetry requirement. We need only assume that the visible morning rising

Textual Notes

·

267

follows the true one by just as much time as the visible morning setting follows the true one. Similarly, we assume that the V E S precedes the TES by just as much time as the V E R precedes the T E R . Finally, the emended propositions are equiv­ alent to those in Autolykos, On Risings and Settings i 4 - 5 . s extend above the Earth. Immediately after these words the mss. have τοις (τό Β) αεί μάλλον προς άρκτον κειμένοις, which, following Manitius, we delete as an accidental duplication of the beginning of the sentence. because smaller segments [of the star's diurnal circles] extend above the Earth, δια τό έλάττονα τμήματα υπέρ γην φέρεσθαι, following Manitius and C . έλάττονα γαρ τα τμήματα ή δμοια φέρονται οι προς μεσημβρίαν αστέρες κείμενοι, Aujac, A , and Β. Absent in the Latin version. In C , immediately after φέρεσθαι is the following, rejected by Manitius: τοις αεί μάλλον προς μ ε σ η μ ­ βρίαν κειμένοις· από δέ έφας δύσεως μεχρίς εσπερίας έπιτολής μείζων ó χρόνος γίνεται δια τό μείζονα τμήματα υπό γην φέρεσθαι. Although the exact form of the text is uncertain here, its basic meaning is clear. The expression έλάττονα . . . ή όμοια ("smaller than similar") in A and Β is meant to compare one arc to an­ other in terms of angular extent, rather than absolute length. The same expres­ sion is used by Autolykos (On Risings and Settings i l l ) and by Euclid (Phenom­ ena, Berggren and Thomas 1996, 74n47.) Arcs of circles of different radii are similar if they subtend the same angle from their respective centers. One arc is smaller than similar, compared to another, if it subtends a smaller angle, regard­ less of the absolute sizes of the circles to which the arcs belong. morning setting. Thus the manuscripts and Manitius. Aujac emends to "morning rising." The proposition stated by the manuscripts is certainly false, but Aujac's emendation makes the proposition correct for the visible phases of all stars, not just northern ones, for which reason we have not adopted it. (Aujac's emendation would not be correct for the true phases.) See textual note f for a proposed solution. » morning setting. Thus A , B, and Manitius. Aujac emends to "morning ris­ ing," but this still does not make the astronomy correct. See textual note f for a suggested emendation of this passage. while for those toward the south the time from morning setting till evening rising is more than 6 months. Thus A and B. Missing in C and the Latin version. h

1

k

CHAPTER X I V

the fixed stars. After these words, Manitius inserts by analogy to xiv 3. But see textual note b. those lying in the south, οί επί (προς, Manitius) μεσημβρίαν κείμενοι. We have punctuated the sentence differently than Manitius or Aujac by placing a comma before this phrase rather than after it. some rise at the same time, and some later, άμα μεν ανατέλλει [καί δύνει, τα μέν πρότερον], τα δέ ύστερον, following Manitius. άμα μέν ανατέλλει, τα μέν πρότερον, τα δ' ύστερον, Aujac. These. This word is followed in the Greek mss. by πάντα. Missing in the Latin version and deleted by Manitius, whom we follow. a

b

c

d

268

·

Appendix 1

CHAPTER X V

which is approximately 100,000 stades in length and about half that in width, επί μεν τό μήκος ούσα ώς εγγιστα περί I μυριάδας σταδίων, επί δέ τό πλάτος ώς εγγιστα τό ήμισυ. Aujac rejects this phrase as an early gloss that worked its way into the text. This may well be right, but it must have happened early, since the phrase also occurs in the Latin version. See Aujac 1975, 49n3. See also comment 3 to chapter xv, p. 209. a

CHAPTER X V I

world maps, γεωγραφίας. in proportion, κατά λόγον. . Accepting Manitius's emendation . T h e intervals between 8

Diels and Rehm 1904, 105. ¿Manitius 1898,214, 226. M a n i t i u s 1898, 232. For the text of the second Miletus parapêgma, see Diels and Rhem 1904. A complete English translation w i l l soon be available in Lehoux (forthcoming). 5

7

8

The Geminos Parapëgma

·

285

TABLE A 2 . 2

Comparison of the Second Miletus Parapëgma Second Miletus Parapëgma (456D Left Column) ° & Egyptians: ES o Eudoxos & Egyptians: d south winds o o o Eudoxos & Egyptians: es E R

(456A Right Column) o Euktëmôn: E

o Euktëmôn: Eagle E R o Euktëmôn: Arcturus M S , signifies

and the Geminos

Parapëgma

Geminos Parapëgma 1 7 ^ Eudoxos: Scorpio ES 18 19 Eudoxos: N o r t h and south winds 20 21 22 Eudoxos: Hyades E R 23

2 5 ^ Euktëmôn: ES 26 27 28 29 30 Euktëmôn: E R 31 Euktëmôn: Eagle E R 32 Euktëmôn: Arcturus M S , signifies

events i n the M i l e t u s parapëgma are m o r e certain t h a n the absolute day count: because o f m i s s i n g day holes, w e cannot be sure just where these phases were meant to fall i n the z o d i a c sign. W e have m a t c h e d the begin­ nings o f the t w o extracts against the c o r r e s p o n d i n g entries i n the G e m i ­ nos parapëgma. T h e close correspondence for the E u d o x o s entries i n the first extract suggest that, at least over short stretches o f time, the G e m i n o s parapëgma and the second M i l e t u s parapëgma are i n fair agreement. T h i s suggests that b o t h reproduce reasonably faithfully the sources o n w h i c h they were based. Since each includes notices o m i t t e d by the other, neither c a n have been derived f r o m the other. T h e E u k t ë m ô n entries i n the second extract s h o w a n imperfect corre­ spondence, w i t h a slip o f 1 day o c c u r r i n g between the first entry a n d the Eagle entry 5 days later. T h i s c o u l d be a simple c o p y i n g error by one o f the writers. A n alternative e x p l a n a t i o n o f the discrepancy is that the c o m p i l e r o f the G e m i n o s parapëgma has stretched out Euktëmôn's phases i n order to cover K a l l i p p o s ' s l o n g s p r i n g season. B u t w e shall see b e l o w that this is unlikely.

286

·

Ptolemy's

Appendix 2 Parapêgma

Ptolemy's parapêgma, w h i c h forms a part o f his Pbaseis, i n t r o d u c e d a n u m b e r o f i n n o v a t i o n s . N o t a b l y , P t o l e m y c a r r i e d to its l o g i c a l c o n c l u s i o n the i m p r o v e m e n t i n p r e c i s i o n that h a d been begun by K a l l i p p o s : P t o l e m y does not give the dates o f the h e l i a c a l risings a n d settings o f constellations o r parts o f constellations, but o n l y o f i n d i v i d u a l stars. H e includes fifteen stars o f the first m a g n i t u d e a n d fifteen o f the second. In this w a y , he eliminates the uncertainty i n the first or last appearances o f extended constellations, such as O r i o n o r C y g n u s . 9

T h u s , P t o l e m y was unable to use the t r a d i t i o n a l dates o f star phases due to E u k t ë m ô n , E u d o x o s , K a l l i p p o s , etc. Rather, he began w i t h the hel i a c a l risings a n d settings for the klima o f A l e x a n d r i a . H e then c o m p u t e d the dates o n w h i c h the stars ought to m a k e their heliacal risings a n d settings i n other klimata. T h e " c a l c u l a t i o n s " m a y w e l l have been perf o r m e d w i t h the a i d o f a celestial globe. T h u s , a l t h o u g h P t o l e m y gives a complete set o f heliacal risings a n d settings for five different klimata (from 13VÍ to \5Vi hours, by V i - h o u r steps), he does not report any star phases for the older authorities. H e does, however, give a n a m p l e selection o f weather predictions attributed to specific authorities. T h u s , i n c o m p a r i n g the G e m i n o s parapêgma w i t h Ptolemy's parapêgma, w e c a n n o t use the star phases, but must restrict ourselves to weather predictions. Unfortunately, i n the case o f weather signs a n d " s i g n i f y i n g , " it is often h a r d to identify c o r r e s p o n d i n g lines i n the t w o parapêgmata. T h i s is especially so for the predictions o f E u d o x o s . It is clear that neither P t o l e m y n o r the c o m p i l e r o f the G e m i n o s parapêgma i n c l u d e d a l l the weather predictions that E u d o x o s h a d put i n his o r i g i n a l parapêgma. T h u s it is often h a r d to k n o w w h i c h E u d o x a n " s i g n i f y i n g " i n P t o l e m y corresponds to w h i c h E u d o x a n " s i g n i f y i n g " i n G e m i n o s . So, i n attempting to m a t c h Ptolemy's parapêgma against the G e m i n o s parapêgma, w e must restrict ourselves to singular or otherwise especially clear events. T h i s turns out to be especially easy to d o for the predictions of K a l l i p pos. Table A 2 . 3 shows that the weather signs o f K a l l i p p o s reported by P t o l e m y bear a simple a n d steady relationship to those reported i n the G e m i n o s parapêgma. T h e left c o l u m n ( " D a y " ) is the day number i n the year, c o u n t i n g f r o m the summer solstice. T h e G e m i n o s parapêgma begins w i t h the summer solstice, but Ptolemy's does not. B u t c o u n t i n g i n this w a y a l l o w s us to at least r o u g h l y m a t c h the corresponding days of the year. T h e second c o l u m n gives the date as identified by Ptolemy, i n terms o f the A l e x a n d r i a n calendar, together w i t h the predictions of K a l l i p p o s that For the Greek text of Ptolemy's Phaseis, see Heiberg 1898-1903, vol. 2. A n English translation of the parapêgma w i l l be available in Lehoux (forthcoming). 9

The Geminos Parapëgma

·

287

TABLE A2.3

Weather Signs of Kallippos in Ptolemy and in the Geminos Day 1 67 68 69 70

Parapëgma

Geminos Parapëgma

Ptolemy's Parapëgma 1 Epiphi Summer solstice

1 £p Summer solstice

2 Thoth 3 4 Kallippos: stormy, etesian winds cease 5

5\¡l Kallippos: etesian winds cease 6 7 8 24 25 26

Kallippos: rain

86 87 88

21 22 23 Kallippos: rain

131 132 133

6 Athyr 7 8 Kallippos: rain

150 151 152

25 26 27 Kallippos: rain

28 29 30

168 169 170

13 Choiak 14 15 Kallippos: south wind, signifies

1 6 * Kallippos: south wind 17 18

196 197 198

11 Tybi 12 13 Kallippos: south wind

15^)° Kallippos: south wind 16 17

242

27 Mecheir

243 244

28 29 Kallippos: swallows appear, windy

273 274 275

28 Phamenoth 29 30 Kallippos: rain or snowstorm

91TL Kallippos: rain 10 11 Kallippos: rain

2 K Kallippos: swallow appears, signifies 3 4

3 °y° Kallippos: rain or snowstorm 4 5

288

·

Appendix 2

P t o l e m y selected for this time o f the year. T h e t h i r d c o l u m n gives the cor­ responding i n f o r m a t i o n f r o m the G e m i n o s parapëgma: the date i n terms of the day w i t h i n the zodiac sign, together w i t h the predictions attributed to K a l l i p p o s . Table A 2 . 3 includes a l l the weather predictions o f K a l l i p p o s for w h i c h it is possible to identify the corresponding entries i n the t w o parapëgmata. A s is readily seen, the predictions o f K a l l i p p o s reported by P t o l e m y r u n a steady 2 days b e h i n d those reported by G e m i n o s . T h e re­ t u r n o f the s w a l l o w , the cessation of the etesian w i n d s , etc., p r o v i d e clear correspondences. M o r e o v e r , the number of K a l l i p p o s ' s weather signs re­ ported by Ptolemy a n d by G e m i n o s is not great. T h u s , even rainstorms of­ ten clearly line up w i t h the usual 2-day offset. P t o l e m y a n d G e m i n o s each report weather signs due to K a l l i p p o s that the other leaves u n m e n t i o n e d , w h i c h proves that P t o l e m y d r e w u p o n some source other than the G e m i ­ nos parapëgma. In v i e w of this fact, the correspondence between P t o l e m y a n d G e m i n o s for the case o f K a l l i p p o s is very striking. T h i s shows that the G e m i n o s parapëgma reliably preserves the sequence of events i n K a l l i p ­ pos's parapëgma, a n d it shows that P t o l e m y does, t o o . 10

The i m p o r t o f this 2-day offset is not o b v i o u s , but it seems that one o f the c o m p i l e r s (and perhaps both) has adjusted the date o f K a l l i p p o s ' s s u m m e r solstice. If the predictions o f E u k t ë m ô n i n the G e m i n o s parapëgma are c o m p a r e d w i t h those i n Ptolemy's parapëgma, a similar offset c a n be discerned—but i n this case it is o n l y o f a single day. T h a t is, the predictions attributed to E u k t ë m ô n (counted f r o m the s u m m e r sol­ stice) tend to r u n 1 day later i n P t o l e m y t h a n i n G e m i n o s . There is some scatter, a n d the correspondence is not as perfect as for K a l l i p p o s . W h e n the predictions of E u d o x o s i n the G e m i n o s parapëgma are c o m p a r e d w i t h those i n Ptolemy's parapëgma, there is n o offset. T h a t is, the predic­ tions (counted f r o m the s u m m e r solstice) tend to occur o n the very same day i n the t w o parapëgmata, a l t h o u g h there is some scatter. B u t again, the pattern is m u c h less clear t h a n is the case w i t h the predictions o f Kallippos.

STRUCTURE OF THE GEMINOS

PARAPËGMA

N o w w e are i n a p o s i t i o n to say s o m e t h i n g about h o w the c o m p i l e r o f the G e m i n o s parapëgma m o s t l i k e l y proceeded. It seems clear that the One additional near-correspondence with a 2-day offset occurs on days 257 and 259 of the year. Geminos has, according to Kallippos, "north wind stops" (lëgei bóreas, day 17 of Pisces). But Ptolemy has, according to Kallippos, "cold north wind blows" {bóreas psuchros pnei, day 14 of Phamenoth). This seems to be an error in one of the two texts, perhaps occasioned by epipnei bóreas psuchros in the preceding entry in Geminos (day 14 of Pisces). 10

The Geminos Parapêgma

·

289

c o m p i l e r relied u p o n the parapêgma o f K a l l i p p o s for the basic structure. T h i s is suggested by the four e x p l i c i t mentions o f the beginnings o f the seasons " a c c o r d i n g to K a l l i p p o s " (at d a y 1 o f Cancer, L i b r a , C a p r i c o r n , a n d A r i e s ) . N o other a u t h o r i t y is m e n t i o n e d for the beginnings o f a l l four seasons. M o r e o v e r , the lengths o f the seasons are i n fair (though n o t perfect) a c c o r d w i t h the season lengths attributed to K a l l i p p o s i n the Celestial Teaching o f Leptinës. In the G e m i n o s parapêgma, the lengths o f the seasons are (in days, a n d b e g i n n i n g w i t h summer): 9 2 , 8 9 , 8 9 , 9 5 . T h e season lengths attributed to K a l l i p p o s by Leptinës are: 9 2 , 8 9 , 9 0 , a n d 9 4 . F i n a l l y , the close agreement between the parapêgmata of G e m i n o s a n d P t o l e m y regarding the predictions o f K a l l i p p o s suggests that the G e m i n o s parapêgma preserves the sequence o f events i n K a l l i p pos's o r i g i n a l calendar, t h o u g h w e c a n n o t k n o w w h i c h (if either) o f these texts preserves the exact starting d a y w i t h respect to the s u m m e r solstice. 1 1

E x a c t l y h o w the events due to E u d o x o s a n d E u k t ë m ô n were added is not certain. T h e c o m p i l e r m i g h t have proceeded i n t w o different w a y s . (1) H e m i g h t have s i m p l y s lip p e d each p r e d i c t i o n o f E u d o x o s o r E u k t ë m ô n i n at its a p p r o p r i a t e d a y n u m b e r i n the course o f the year, i g n o r i n g the fact that E u d o x o s a n d E u k t ë m ô n h a d different lengths for the seasons t h a n d i d K a l l i p p o s . T h i s is the same as preserving the time intervals between successive events. O r (2) he m i g h t have preserved the place o f a n event w i t h i n a p a r t i c u l a r season, o r (perhaps) w i t h i n a p a r t i c u l a r z o d i a c sign, e.g., m a k i n g sure that a n event scheduled for the 2 5 t h day o f T au r u s remains o n the 2 5 t h day o f T a u r u s . W e m i g h t already suspect that the c o m p i l e r f o l l o w e d the first course for E u d o x o s , i n v i e w o f the notice o f E u d o x o s ' s w i n t e r solstice (day 4 o f C a p r i c o r n ) . B u t n o w there is g o o d reason for believing that the c o m p i l e r also f o l l o w e d this course for E u k t ë m ô n . It is p r o b a b l e that Ptolemy's a n d G e m i n o s ' s parapêgmata are the results o f independent c o l l a t i o n s , i n v i e w o f the fact that each i n cludes m a n y notices that the other o m i t s . T h e reasonably consistent 1day offset for the predictions o f E u k t ë m ô n , w h i c h is m a i n t a i n e d over the course o f the year, seems then to i m p l y that each c o m p i l e r s i m p l y w r o t e the predictions out i n order f r o m the first to the last day o f the year. In any case, it is clear f r o m the scatter i n the c o m p a r i s o n between G e m i n o s a n d P t o l e m y that the texts o f the parapêgmata of Euktëmôn and E u d o x o s h a d already become less reliable by their time t h a n the text o f Kallippos. 12

Tannery 1893, 294. This was the position that van der Waerden (following Rehm) took in his attempted reconstruction of the parapêgma of Euktëmôn (van der Waerden 1983, 104). 1 1

1 2

APPENDIX

3

Glossary of Technical Terms in Geminos's Introduction to the Phenomena

T h e glossary is arranged topically. T h u s , under Sun w i l l be f o u n d the terms for the t r o p i c a l year, the four seasons, the equinoxes a n d solstices, etc. T h e glossary n o t o n l y enables a reader to determine w h i c h G r e e k w o r d corresponds to a p a r t i c u l a r E n g l i s h w o r d used i n the t r a n s l a t i o n , but also provides capsule definitions for most technical terms. F o r each technical t e r m , a reference is p r o v i d e d to a passage i n w h i c h the t e r m is used i n a defining o r characteristic w a y . (If several consecutive terms listed here o c c u r i n the same passage i n the text, a reference to that passage is given o n l y after the last t e r m , a n d o n l y a chapter n u m b e r is given if a t e r m appears t h r o u g h o u t a chapter.) N o attempt has been made to list every occurrence o f every t e r m , o r every l e x i c a l f o r m o f a given w o r d . (For these, see the " I n d e x graecitatis" at the b a c k o f M a n i t i u s 1898. A u j a c 1 9 7 5 includes a n a l p h a b e t i c a l l y arranged Lexique des Termes Techniques, w h i c h includes p h i l o s o p h i c a l vocabulary.)

T H E C O S M O S A N D T H E O B J E C T S W I T H I N IT

T h e a s t r o n o m i c a l scene for G e m i n o s ' s b o o k is the cosmos, kosmos, w h i c h has the Earth, gê, like a point, sêmeion, i n its middle, mesé ( x v i , 29). T h e cosmos is b o u n d e d by the sphere of the fixed stars, hê ton aplanan asterôn sphaira (i 2 3 ) . Its one m o t i o n is a d a i l y revolution, peristrophë (v 4), i n w h i c h the f o l l o w i n g heavenly bodies participate: The

Sun

T h e Sun, Helios (i 7), m a r k s the b e g i n n i n g o f the day, hêmera (vi 1), by its rising, anatolë ( x i i i , also a p p l i e d to the d a i l y rising o f stars), a n d the b e g i n n i n g o f night, nux (vi), by its setting, dusis (vi 1). " D a y " (vi 1) c a n also refer to the w h o l e p e r i o d f r o m one sunrise to the next. A n d dusis is used for b o t h the d i u r n a l setting o f stars (xiii 2) a n d their h e l i a c a l setting (xiii 4). The year, eniautos (i 7), a n d the annual period, eniausios (i 7), b o t h refer to the t r o p i c a l year o f the Sun's circuit,

chronos pendróme

292

·

Appendix 3

(viii 5), a r o u n d the z o d i a c a l circle, zôidiakos kuklos (i 3). A n o t h e r s y n o n y m for this k i n d o f year is the year by the Sun o r the solar year, kath hëlion eniautos (viii 5), w h i c h G e m i n o s uses w h e n he wishes to d i s t i n g u i s h the t r o p i c a l year f r o m a l u n a r year o f 12 s y n o d i c m o n t h s . W h e n it is a question o f c o u n t i n g years, however, the w o r d used t h r o u g h o u t the w o r k is etos (vii 2 4 , 33), year. G e m i n o s defines the seasons, s p r i n g , ear, s u m m e r , theros; a u t u m n , phthinopôron; a n d w i n t e r , cheimóna (i 9), by the e q u i n o x e s , isëmeriai, a n d solstices, trop ai. T h u s , the s p r i n g o r v e r n a l e q u i n o x (i 9) m a r k s the b e g i n n i n g o f s p r i n g , the a u t u m n a l e q u i n o x (v 6) that o f a u t u m n , w h i l e the s u m m e r solstice (i 9; ν 4) a n d w i n t e r solstice (v 7) m a r k the begin­ nings o f those seasons. O n the days s u r r o u n d i n g a solstice there is a n a p p a r e n t t a r r y i n g , epímone (vi 3 0 ) , o f the S u n at the c o r r e s p o n d i n g tropic. 3

The solar circle, bëliakos kuklos (i 33), refers to the actual circle o n w h i c h the a n n u a l , proper displacement, parodos (v 16), o f the S u n takes place. It lies closer to the E a r t h t h a n , a n d is eccentric, ekkentros (i 34), to the z o d i a c . M u c h looser i n m e a n i n g is the course of the Sun, bëliakos aromos (i 3 4 , referring to the Sun's eccentric circle; v i i i 3 2 , o r referring s i m p l y to the progress o f the S u n a r o u n d the z o d i a c ) . T h e w o r d dromos is also used for the Sun's (ii 2 3 ) , o r a star's (xiv 1) course o r p a t h across the sky f r o m r i s i n g to setting.

The

Moon

G e m i n o s describes the phases, schëmatismoi (ix 11), o f the M o o n , selënë (xi), as f o l l o w s : crescent M o o n , mënoides (lit., " M o o n - s h a p e d " ) ; first or t h i r d quarter, dichotomos (lit., " c u t i n h a l f " ) ; g i b b o u s , amphikurtos (lit., " c u r v e d o n b o t h sides"); a n d full M o o n , panselënos (lit., " w h o l e M o o n " ) (ix 11). T h e M o o n defines t w o periods o f time: the s y n o d i c m o n t h (from n e w M o o n to n e w M o o n ) , men (viii 1), a n d the l u n a r year, kata selënën eni­ autos (viii 4 6 ) , of 12 l u n a r m o n t h s . S y n o n y m o u s w i t h men is mëniaios obrónos (viii 2), the m o n t h y p e r i o d . Some days o f the m o n t h are n a m e d for the phases o f the m o o n , so the 1st d a y of the m o n t h is the noumënia (viii 11), a n d the m i d d l e o f the m o n t h the dichomënia (viii 1). T h e 3 0 t h d a y o f the m o n t h is c a l l e d the triakas (viii 12), after the w o r d " t h i r t y . " O n the last day of the m o n t h the M o o n runs under, hupotrochazei (ix 10), the S u n . T h e c i v i l m o n t h , kata polin men, is the m o n t h as r e c k o n e d i n a city's calendar, a n d it m a y be full, plërës; o r h o l l o w , koilos, i.e., it m a y c o n t a i n either 3 0 o r 2 9 days, respectively. A full a n d a h o l l o w m o n t h together constitute a d o u b l e m o n t h , dimënon, o f 59 days (viii 3).

Glossary of Technical Terms

·

293

The Stars Star, aster (iii 4 ) , o r astron ( x v i i 7). A l l but five stars are fixed or nonwandering, aplanéis (vii 2), as distinguished f r o m the five planets (below). L i k e other ancient c i v i l i z a t i o n s , the G r e e k s g r o u p e d the stars i n t o constellations, katastërigmena zôidia (iii 1). ( M a n i t i u s prefers the spelling katèsterismena.) T h e t e r m is also used (i 4) specifically for the zodiacal constellations, as distinguished f r o m the z o d i a c a l signs. F o r the names o f i n d i v i d u a l stars a n d constellations, see the tables i n chapter i i i . T h e r e is a r i c h v o c a b u l a r y associated w i t h the h e l i a c a l risings a n d settings o f the stars or, as they are sometimes c a l l e d , phases, phaseis ( x v i i 2 6 , where pbasis is translated as "appearance"). T h e same w o r d is also a p p l i e d to the phases o f the M o o n (viii 11). T h e terms for fixed star phases are discussed i n sec. 11 o f the I n t r o d u c t i o n .

The

Planets

T h e wandering stars, planètes ásteres ( x i i 2 2 ; x v i i 38), are sometimes s i m p l y called the planets, planètes (i 19). E a c h o f the five planets k n o w n to the ancients has a p r o p e r name, but is also called the star o f a certain g o d . F o r e x a m p l e , Saturn is Phainón ("Shiner") but also "the star o f K r o n o s " (i 2 4 ) . F o r further details o n the planet names, see c o m m e n t 2 0 to chapter i . T h e d e s i g n a t i o n " w a n d e r e r " o r i g i n a t e d i n the fact that, a l t h o u g h the planets share i n the w e s t w a r d d i u r n a l m o t i o n o f the cosm o s , they also have a p r o p e r e a s t w a r d m o t i o n a l o n g the z o d i a c , w h i c h is s a i d to be opposite that of the cosmos, bupenantiôs toi kosmói (xii 6). M o r e o v e r , the planets pass by the f i x e d stars w h i l e m o v i n g eastw a r d , a n d o c c a s i o n a l l y m a k e a retrograde m o t i o n to the west, w i t h respect to the f i x e d stars. B e t w e e n these t w o m o t i o n s they w i l l s t a n d still w i t h respect to the f i x e d stars at w h a t is c a l l e d a station, stërigmos (xii 2 2 ) . D e s p i t e this apparent i r r e g u l a r i t y , each planet's m o t i o n is someh o w circular, egkuklios, a n d uniform (or at constant speed), homalos (i 19). T o e x p l a i n the apparent anomaly, anomalía (i 2 0 ) , G e m i n o s refers vaguely to a spherical construction o r spherical system, sphairopoua ( x i i 2 2 , 2 6 ) , for each planet, w h i c h , t h r o u g h a c o m b i n a t i o n o f spheres (or perhaps o n l y circles), produces the apparently irregular m o v e m e n t o f a planet. G e m i n o s uses sphairopoua i n a m o r e general w a y , for the spherical arrangement o f the cosmos, t h r o u g h w h i c h some stars rise a n d set each day w h i l e others d o not (xiv 9; w i t h related uses at x v i 19, 2 7 , 2 9 ) . Sphairopoua c a n also indicate a b r a n c h o f a p p l i e d mechanics, devoted to the c o n s t r u c t i o n o f models o f the heavens (fragment 2). F o r a full discussion o f this t e r m , see sec. 10 o f the I n t r o d u c t i o n .

294

·

Appendix 3

E a c h o f the planets lies l o w e r , tapeinoteros (i 2 3 ) , t h a n the sphere o f stars, i n the sense that the planets are closer to the E a r t h . S a t u r n is higher, meteôroteros (i 2 3 ) , t h a n Jupiter.

ECLIPSES

A n eclipse, ekleipsis, m a y be solar (x) o r l u n a r (xi). N o t every conjunc­ t i o n , sunodos (viii 1), o f the S u n a n d M o o n gives rise to a n eclipse o f the S u n , because the M o o n ' s p a t h varies above a n d b e l o w the z o d i a c a l cir­ cle. T h e c o v e r i n g , epiprosthesis, o f the S u n by the M o o n causes a solar eclipse (x 1) w h i l e the c o v e r i n g o f the M o o n by the E a r t h (xi 3) causes a l u n a r eclipse. D u r i n g a l u n a r eclipse, the M o o n falls i n t o the s h a d o w o f the E a r t h , empiptei eis to skiasma tes gës ( x i 2.) A l l the eclipses o f the M o o n fall i n t o a n eclipse zone, ekleiptikon ( x i 6, 7), a belt that extends 2 ° above a n d b e l o w the ecliptic.

MATHEMATICS

T h e i m p l i c i t definition o f a sphere, sphaira (i 2 3 ) , is a three-dimensional figure w h o s e b o u n d a r y points are a l l equidistant f r o m its center, kentron (i 32). A n y straight line passing t h r o u g h the center a n d j o i n i n g t w o points o n the surface o f the sphere is a diameter, diámetros (ii 1). Periphereia c a n m e a n either an arc (i 18) o r the w h o l e circumference (v 49) o f a circle, kuklos (i 1; i 11). G e m i n o s sometimes divides a circle o n the sphere i n t o 60 parts, merai, w h i c h are c a l l e d sixtieths, hexëkosta (v 4 6 ) . B u t he also sometimes divides the circle i n t o 3 6 0 degrees, moipai (i 8), each o f w h i c h is d i v i d e d i n t o sixtieths, hexëkosta (xviii 7). A m i n u t e , o r 6 0 t h o f a degree, is also c a l l e d proton lepton ("first d i v i s i o n " ) , a n d a sec­ o n d , deuteron lepton ("second d i v i s i o n " ) ( x v i i i 8). B u t units o f time, e.g., days, c a n also be d i v i d e d i n t o sixtieths i n the same w a y ( x v i i i 10). A n y great circle, megistos kuklos (v 70), divides the sphere i n t o t w o e q u a l hemispheres, hëmisphairia (ν 54). O t h e r geometrical terms are: to cut, temnein (i 33); section (of a sphere), ektmëma (xvi 5); perimeter, perimetron (xvi 9); side, pleura (ii 7); o b l o n g , paramëkës (xvi 4); r o u n d , stroggulos (xvi 4, 5); square, tetragônon (ii 1); triangle, trigónon (ii 1); a n d b o u n d a r y , peras (i 40). A m o n g the a r i t h m e t i c a l terms are: to exceed, pleonazein (viii 4 0 ) ; excess, huperochë (ν 20); difference, parallagë (vi 33); a quarter, tetartëmorion (i 18); p r o p o r t i o n , logos ( x v i 4) but also summetria (xvi 5); d o u b l e , diplasios ( x v i 3); to m u l t i p l y , polu I polla-plasiazein (viii 4); to m u l t i p l y b y eight, oktaplasiazein (viii 38); a p p r o x i m a t e , oloscherës

Glossary of Technical Terms (ii 2 0 ) ; to add, suntithenai (i 17); an increase, parauxêsis diminution, meiôsis ( x v i i i 5).

·

295

(vi 2 9 ) ; a

T H E C E L E S T I A L S P H E R E A N D ITS C I R C L E S

T h e sphere of the fixed stars, hë ton aplanan asterôn sphaira (i 2 3 ) , is centered o n the E a r t h a n d has a l l the fixed stars o n its surface. It rotates each d a y a r o u n d a diameter k n o w n as the axis, axon (iv 1), w h i c h meets the surface o f the sphere at t w o p o i n t s k n o w n as the poles of the cosmos, poloi tou kosmou (iv 1). T h e one visible to us is the n o r t h pole, boreios polos, a n d the one invisible to us is the south pole, notios polos (iv 2). T h e plane tangent to the surface o f the E a r t h at a given l o c a l i t y defines the h o r i z o n , horizon (v 5 4 - 6 3 ) , w h i c h separates the part o f the c o s m o s that is visible, phaneron (v 5 4 ) , f r o m the part that is i n v i s i b l e , aphanes (v 54), at that l o c a t i o n . Because o f the m i n u t e size o f the E a r t h relative to the c o s m o s , the h o r i z o n m a y be regarded as bisecting, dichotomôn (ν 54), the c o s m o s . P e r p e n d i c u l a r to the h o r i z o n , passing t h r o u g h its south p o i n t , mesëmbria (v 8), a n d z e n i t h , kata koruphën sëmeion (ν 6 4 ) , is the local m e r i d i a n circle, mesëmbrinos kuklos (ii 2 5 , 2 6 ) .

Parallel

Circles

P a r a l l e l circles, parallëloi kukloi (ν 1), are those circles o n the sphere that are p a r a l l e l to the paths traced by the fixed stars i n the course o f the d i u r n a l r o t a t i o n o f the c o s m o s . T h e greatest o f these is the equator circle, isëmerinos kuklos (v 6). A n i m p o r t a n t g r o u p o f these circles is the always-visible circles, aei theôroumenoi kukloi (v 2), described by the m o t i o n s o f the stars that never rise o r set. T h e largest o f these circles is the arctic circle, arktikos kuklos (v 2), w h i c h separates stars that are a l ­ w a y s above the h o r i z o n f r o m the stars that rise a n d set. E q u a l i n size a n d p a r a l l e l to the arctic circle, but l y i n g a l w a y s b e l o w the h o r i z o n , is the antarctic circle, antarktikos kuklos (v 9). O f the p a r a l l e l circles, t w o m a r k the n o r t h e r n a n d southern limits o f the S u n d u r i n g its a n n u a l course, n a m e l y the s u m m e r t r o p i c circle, therinos tropikos kuklos (v 4 ) , a n d the w i n t e r t r o p i c circle, cheimerinos tropikos kuklos (v 7). These p a r a l l e l circles are generally to be u n d e r s t o o d as circles o n the celestial sphere. T h e h o m o n y m o u s circles o n the E a r t h w i t h w h i c h m o s t m o d e r n readers are familiar m a y be regarded as central projections o f their celestial counterparts, or—as G e m i n o s puts it—they lie under them. T h u s he says that the equator circle on the E a r t h , en tëi get isëmerinos kuklos, lies under, hupo, the equator circle in the cosmos, en

296

·

Appendix 3

tôt kosmôi isëmerinos kuklos (xv 3), i.e., under the celestial equator. T h e w o r d dunamis (v 4 1 , 4 2 ) , p o w e r , i n the context o f p a r a l l e l circles refers to their properties.

Oblique

Circles

A m o n g the o b l i q u e , loxos (v 5 1 ) , circles is the z o d i a c . (The aforemen­ t i o n e d h o r i z o n is another). T h e circle o f the signs, kuklos ton zôidion (i 1), a n d the z o d i a c circle, zôidiakos kuklos (i 3), are s y n o n y m o u s expres­ sions. T h e i n d i v i d u a l signs are m o s t often slanted, plagios (vii 11), to the h o r i z o n w h e n they rise o r set. A sign is said to be f o l l o w i n g , epomenon (xii 6), another i f it rises after the other. T h u s , Taurus f o l l o w s A r i e s , G e m i n i f o l l o w s T a u r u s , etc. A n y one of the twelve z o d i a c a l signs m a y be referred to either as a sign, zôidiôn (lit. " s m a l l figure") (i 2), or as a twelfth part, dôdekatëmorion (i 1), o f the z o d i a c . T h e latter t e r m is used to m a k e a clear d i s t i n c t i o n be­ tween a sign o f the z o d i a c , w h i c h is a 3 0 ° - l o n g geometrical segment, tmëma (i 1), a n d a z o d i a c a l c o n s t e l l a t i o n , w h i c h m a y have the same name but is o f irregular size a n d shape. T h e z o d i a c is a b a n d o f 1 2 ° w i d t h , platos (ν 5 3 ) , a n d is represented by three p a r a l l e l circles (v 51). T h e upper a n d l o w e r circles define the w i d t h o f the b a n d , a n d the t h i r d (middle) circle is the circle t h r o u g h the m i d ­ dles o f the signs, dia ton mesôn ton zôidiôn kuklos (ii 2 1 ) . T h i s is o u r " e c l i p t i c , " the apparent p a t h o f the S u n . Its p o i n t s o f section, tome (vi 36), w i t h the equator are the e q u i n o c t i a l p o i n t s . T h e angle at w h i c h the ecliptic is slanted w i t h respect to the equator is called its o b l i q u i t y , loxotëta (ii 2 4 ) . A n o t h e r p a i r o f great circles are the c o l u r e circles, kolouroi kukloi (v 4 9 ) , one o f w h i c h passes t h r o u g h the poles a n d the solstitial p o i n t s , a n d the other o f w h i c h passes t h r o u g h the poles a n d the e q u i n o c t i a l p o i n t s . B o t h are, therefore, p e r p e n d i c u l a r , orthos (vi 2 3 ; v i i 10), to the equator. T h e M i l k y W a y , ho tou galaktos kuklos (v 68), is also an o b l i q u e circle.

INSTRUMENTS

F o r a discussion of a s t r o n o m i c a l instruments, see sec. 8 o f the I n t r o d u c ­ t i o n . G e m i n o s k n o w s t w o k i n d s o f celestial globes, s o l i d globes, sphairai stereai ( x v i 12), a n d a r m i l l a r y spheres, krikôtai sphairai ( x v i 12). T h e s u n d i a l , hôrologion (ii 38) / hôroskopeion (ii 35) / skiothëron (vi 3 2 ) , m a r k s the passage o f the hours. It w a s c o m m o n e n o u g h that G e m i n o s re­ peatedly refers to p h e n o m e n a regarding the s h a d o w cast by its g n o m o n , gnomon (ii 35), to illustrate points a b o u t solar m o t i o n a n d geography

Glossary of Technical Terms

·

297

(xvi 13). T h e dioptra (i 4) was a n instrument that c o u l d be used for sighting objects o r m e a s u r i n g angles. Several different k i n d s o f instrument were c a l l e d by the same name, some w i t h a p p l i c a t i o n s to astrono m y ( i n c l u d i n g its teaching) a n d some a p p l i e d to surveying.

Z O D I A C A L ASPECTS AND O T H E R ASTROLOGICAL TERMS

Said to be in opposition, kata diametron (ii 1), are t w o d i a m e t r i c a l l y o p posite z o d i a c a l signs. S u c h a pair, e.g., A r i e s a n d L i b r a , are separated by five signs. A p a i r o f signs is in trine, kata trigônon (11 7), w h e n they are separated by three signs, as are A r i e s a n d L e o . B y extension, the t e r m m a y be used o f t w o z o d i a c a l points that are 1 2 0 ° apart, just as the p r e v i ous t e r m (in o p p o s i t i o n ) m a y be used for z o d i a c a l points 1 8 0 ° apart. A p a i r o f signs is in quartile, kata tetragônon (ii 13), w h e n they are separated by t w o others, as are A r i e s a n d Cancer. A g a i n , the t e r m m a y be used o f z o d i a c a l p o i n t s 9 0 ° apart. Said to be in syzygy, kata suzugian (ii 2 7 ) , are t w o signs that rise (and therefore set) o n the same arcs o f the h o r i z o n . F o r that reason they are c o n t a i n e d by the same t w o p a r a l l e l circles. G e m i n o s refers to the significance, episêmasia, o f certain celestial events, such as the h e l i a c a l risings a n d settings o f the stars, as weather signs. Power, dunamis ( i i , x v i i ) , refers to the a b i l i t y o f stars to exert i n fluences o n terrestrial life w h e n their positions, epocbai (ii 6), fall i n cert a i n signs. O n e e x a m p l e o f such p o w e r is the a b i l i t y o f stars to create a sympathy, sumpatheia (ii 5, 12), i.e., a resonance o r c o m m o n destiny, between t w o i n d i v i d u a l s b o r n under signs i n a certain r e l a t i o n to one a n other.

GEOGRAPHY

A spherical, sphairoeidës (vi 2 1 ) , E a r t h was a central tenet o f G r e e k a s t r o n o m y f r o m E u d o x o s o n w a r d . Its n o r t h a n d s o u t h poles lie directly b e l o w their c o s m i c counterparts. T h e meridians passing t h r o u g h these poles a n d a given l o c a l i t y define l o c a l north, boreios (ii 8), a n d south, notios (ii 10). (These t w o adjectival forms are derived f r o m the names o f the respective w i n d s , bóreas (ii 8) a n d notos (ii 10)). T h e phrase " t o w a r d the bear," pros arkton (v 16, 21), a n d related forms also indicate the n o r t h , arktos being the G r e e k for "bear," i.e., the G r e a t Bear, U r s a M a j o r . F o u r circles o n the surface of the E a r t h , p a r a l l e l to the equator, d i v i d e the E a r t h i n t o belts o r zones, zànai (xv 1). T h e n o r t h e r n m o s t zone is frigid, katapsugmenê (xv 1), a n d u n i n h a b i t e d because o f its extreme

298

·

Appendix 3

c o l d . N e x t is the n o r t h e r n temperate, eukratos (xv 2), zone, i n w h i c h the G r e e k s a n d m o s t other nations k n o w n to t h e m l i v e d . T h e n , between the n o r t h e r n a n d southern tropics, one finds the t o r r i d , diakekaumenë (xv 3), zone, thought by some (but n o t by G e m i n o s ) to be u n i n h a b i t a b l e be­ cause o f its extreme heat. P r o c e e d i n g s o u t h w a r d , one then comes to the southern temperate zone a n d , finally, to the southern frigid zone. These t w o share the c l i m a t i c characteristics o f their n o r t h e r n counterparts. A w o r d used of places o n the E a r t h is oikëseis (i 12), regions, or, m o r e t r a d i t i o n a l l y , habitations, f r o m the verb oikeô ("I i n h a b i t " ) . T h a t the use of the w o r d actually implies n o t h i n g about the h a b i t a b i l i t y o f the place is s h o w n by ν 3 8 , where G e m i n o s speaks, as T h e o d o s i o s does i n his Peri oikêseôn, o f the oikêsis "beneath the p o l e , " i.e. the n o r t h (or south) pole! A n i m p o r t a n t , t h o u g h s o m e w h a t flexible, g e o g r a p h i c a l t e r m is oikoumenê ( x v i 3), the i n h a b i t e d ("regions" understood). Oikoumenê can m e a n "the w h o l e i n h a b i t e d E a r t h , " "the n o r t h e r n i n h a b i t e d E a r t h , " o r "the E a r t h i n h a b i t e d by us G r e e k s . " F o r a detailed discussion, see c o m ment 3 to chapter 5. G e m i n o s has four special terms for the i n h a b i t a n t s o f the E a r t h : synoikoi, those d w e l l i n g w i t h i n 9 0 ° longitude o n either side o f us i n " o u r " zone (i.e., the n o r t h e r n temperate zone); perioikoi, those i n the other hemisphere i n o u r zone; antoikoi, those i n the southern temperate zone i n o u r hemisphere; a n d antipodes ( x v i 1), those i n the other h e m i sphere i n the southern temperate zone. T o specify l o c a t i o n n o r t h o r s o u t h a l o n g a m e r i d i a n , G e m i n o s e m p l o y s one o f t w o n o t i o n s . T h e first is the elevation o f the pole, exarma tou polou (vi 2 4 ) , or, the i n c l i n a t i o n o f the cosmos, egklima tou kosmou (vi 2 4 ) , b o t h referring to the acute (or, at the poles, right) angle that the axis o f the sphere makes w i t h the h o r i z o n o f a g i v e n locality, a n angle n u m e r ically equal to the g e o g r a p h i c a l latitude. T h e second n o t i o n G e m i n o s uses is that o f the longest day o r night, megistê hëmera I nucbta (i 10/12) at a g i v e n locality. K n o w l e d g e o f the length o f the longest day is e q u i v a lent to k n o w l e d g e of the latitude. A t h i r d , less t e c h n i c a l , t e r m for elevat i o n , meteorismos, is also a p p l i e d to the pole (vi 25) a n d (in a slightly different form) to stars ( x i i 1). T h e elevation o f the pole gave rise to the d i v i s i o n of the E a r t h a c c o r d i n g to klimata (i 10), belts p a r a l l e l to the equator consisting of localities where the pole has r o u g h l y the same elevation. A c o m m o n measure o f distance between localities is the stade, stadion (v 56). G e m i n o s refers to O c e a n , Okeanos ( x v i 2 8 ) , w h i c h H o m e r describes as encircling a flat E a r t h , but he denounces the error of those w h o take H o m e r at his w o r d a n d d r a w disklike maps. ( N o r d i d any m a t h e m a t i c i a n , mathëmatikos ( x v i 2 3 ) , G e m i n o s c l a i m s , believe that O c e a n spread out over the t o r r i d zone between the tropics.) T h e E a r t h s h o u l d be

Glossary of Technical Terms

·

299

represented, he says, by a w o r l d m a p , geógrapbia ( x v i 4), d r a w n o n a n o b l o n g p a n e l , pinax paramëkës ( x v i 4), w h o s e length, mëkos, is d o u b l e its w i d t h , platos.

TIME AND CALENDRICAL MATTERS

H o u r s , h or ai, are o f t w o k i n d s . T h e e q u i n o c t i a l h o u r , isëmerinë hora (vi 6), is 1/24 o f a night a n d day. T h e other, n o w c a l l e d a " t e m p o r a l " or " s e a s o n a l " hour, G e m i n o s refers to by saying (vii 33) that " i n every night s i x signs set i n 12 h o u r s . " T h a t is, they are twelfth-parts o f a night o r day. F i n a l l y , the w o r d hora m a y also denote one o f the four seasons (ii 17). A parapêgma (xvii 19) is a calendar that gave the dates, t h r o u g h o u t the year, o f the h e l i a c a l risings a n d settings o f p r o m i n e n t stars or constellations a n d associated weather p r o g n o s t i c a t i o n s . G e m i n o s ' s b o o k closes w i t h such a calendar. G e m i n o s discusses a n u m b e r of periods o r cycles, periodoi (viii 2 6 ) , w h i c h were designed to c o m p r i s e a w h o l e n u m b e r o f solar years a n d a w h o l e n u m b e r o f l u n a r m o n t h s . T h e 8-year cycle, oktaetëris (viii 2 7 ) , consists o f five years o f 12 m o n t h s a n d three years o f 13 m o n t h s . T h u s it includes 3 intercalary, embolimoi (viii 2 6 ) , m o n t h s . B u t it s l o w l y gets out of step w i t h the M o o n . G e m i n o s describes a 16-year p e r i o d , ekkaidekaetëris (viii 39), that attempts to correct this defect. A better cycle is the longer, so-called M e t o n i c cycle w i t h its 19-year p e r i o d , enneakaidekaetëris (viii 4 8 - 5 8 ) , i n c o r p o r a t i n g twelve years o f 12 m o n t h s a n d seven years o f 13 m o n t h s . Longest o f a l l is the 76-year p e r i o d , bekkaiebdomëkontaetëris (viii 59), o f four M e t o n i c cycles. U s e d i n the c o m p u t a t i o n o f eclipses is a p e r i o d called the exeligmos (xviii) w i t h its w h o l e n u m b e r of days (19,756), s y n o d i c m o n t h s (669), a n d r e v o l u t i o n s i n a n o m a l y (717). These cycles are discussed i n sec. 13 o f the Introduction. T o return, apokathistasthai, is used o f the S u n for its r e t u r n i n g to a given p o i n t i n the z o d i a c after a year (i 7), o r o f a n E g y p t i a n h o l i d a y ret u r n i n g to the same date i n the calendar after 1,460 years (viii 2 4 ) , or o f the M o o n c o m i n g back to the part o f its m o n t h l y o r b i t where it has its greatest speed ( x v i i i 2). E p a g o m e n a l days, epagomenai bernerai (viii 18), are days a d d e d acc o r d i n g to some regular calendric scheme. In the E g y p t i a n calendar o f 3 6 5 days the last 5 days were e p a g o m e n a l days.

APPENDIX

4

Index of Persons Mentioned by Geminos

INTRODUCTION

TO THE PHENOMENA

(EXCLUDING THE

PARAPËGMA) A r a t o s (fl. 2 7 0 B . C . ) ν 2 4 ; v i i 7, 1 3 ; v i i i 1 3 ; x i v 8; x v i i 4 6 , 4 8 . A r i s t o t l e (340 B . c . ) xvii 49. B o ë t h o s (second century B . C . ) xvii 48. Chaldeans i i 5; x v i i i 9. D i k a i a r c h o s (320 B . c . ) x v i i 5. Egyptians v i i i 16, 2 0 , 2 2 , 2 3 , 2 5 . Eratosthenes (240 B . C . ) viii 24. E u k t ë m ô n (430 B . C . ) viii 50. H e s i o d (650 B . c . ) x v i i 14. H i p p a r c h o s (150 B . c . ) i i i 8, 1 3 . H o m e r (750 B . c . ) v i 1 0 , 16; x v i 2 7 , 2 8 ; x v i i 3 1 . K a l l i m a c h o s (250 B . c . ) i i i 8. K a l l i p p o s (330 B . c . ) viii 50, 59. Kleanthës (270 B . c . ) xvi 21. K r a t ë s [of M a l l o s ] (160 B . c . ) v i 1 0 , 16; x v i 2 2 , 2 3 , 2 7 . P h i l i p p o s (330 B . C . ) viii 50.

302

·

Appendix 4

P o l y b i o s (160 B . c . ) xvi 32. Pythagoreans i 19. Pytheas (290 B . C . ) v i 9.

PARAPÊGMA D ë m o k r i t o s (400 B . c . ) D o s i t h e u s (230 B . C . ) E u d o x u s (360 B . c . ) E u k t ë m ô n (fl. 4 3 0 B . c . ) K a l l i p p o s (fl. 3 3 0 B . C . ) M e t ô n (430 B . C . )

FRAGMENT

1, FROM

PROKLOS

In the indexes for the two fragments, an asterisk * indicates a name that is used by the excerpter (either Proklos or Simplikios) and not by Geminos. A r c h i m e d e s (fl. 2 5 0 B . c . ) H e r o (first century A . D . ) Hippokratës (420 B . c . ) Ktësibios (250 B . C . ) P l a t o (390 B . C . ) Pythagoreans*

FRAGMENT

2, FROM

SIMPLIKIOS

A l e x a n d e r [of A p h r o d i s i a s ] * ( A . D . 200) A r i s t o t l e * (340 B . c . ) Hërakleidës P o n t i k o s ( f l . 3 4 0 B . c . ) (This name was p r o b a b l y i n t e r p o lated by a copyist.) Poseidónios* (90 B . c . )

Bibliography

Achilleus (Achilles Tatius), Introduction to the "Phenomena" of Aratos. See Maass 1898, 25-85. Aitón, E J . 1981. "Celestial Spheres and Circles," History of Science 19, 75-114. Alexander of Aphrodisias, Commentary on Aristotle's Meteorology. See Hayduck 1899. Allen, Richard Hinkley 1899. Star Names and Their Meanings (New York: G . E . Stechert). Reprinted as Star Names: Their Lore and Meaning (New York: Dover, 1963). Apollónios of Pergë, Conies. See Heiberg 1893. Aratos, Phenomena. See M a i r and M a i r 1955; K i d d 1997; J. M a r t i n 1998. Archimedes, Works. See Heath 1912; Dijksterhuis 1987. Aristotle, Metaphysics. See Tredennick 1961-62. , Meteorology. See Lee 1952. , On the Heavens. See Guthrie 1939. , Physics. See Wicksteed and Cornford 1960-63. Aristotle, Pseudo-, On the Cosmos. See Forster and Furley 1978. Arnaud, P. 1984. "L'image du globe dans le monde romain: Science, iconographie, symbolique," Mélanges de l'École Française de Rome 96, 53-116. Arnauldi, M . , and K . Schaldach 1997. " A Roman cylinder dial: Witness to a forgotten tradition," Journal for the History of Astronomy 28, 107-17. Ash, H . B . 1954-1960. ed. and trans. Columella, On Agriculture, 3 vols. (Cambridge: Harvard University Press; London: W. Heinemann). Aujac, G . 1970. " L a sphéropée, ou la mécanique au service de la découverte du monde," Revue d'Histoire des Sciences 23, 93-107. 1975. ed. and trans. Geminos, Introduction aux phénomènes (Paris: Les Belles Lettres). 1979. ed. and trans. Autolycos de Pitane, La sphère en mouvement, Levers et couchers héliaques, Testimonia (Paris: Les Belles Lettres). Aujac, G . , et al. 1987a. "The Foundations of Theoretical Cartography in Archaic and Classical Greece," in Harley and Woodward 1987, pp. 130-47. 1987b. "The Growth of an Empirical Cartography in Hellenistic Greece," in Harley and Woodward 1987, pp. 148-60. 1987c. "Greek Cartography in the Early Roman W o r l d , " in Harley and Woodward 1987, pp. 161-76. Autolykos of Pitanë. On the Moving Sphere. Risings and Settings. See Aujac 1970; Mogenet 1950; Bruin and Vondjidis 1971. Babbitt, E C . 1936. ed. and trans. Plutarch, Moralia, vol. 5. (Cambridge: Harvard University Press). Baccani, D . 1989. "Appunti per oroscopi negli ostraca di Medinet M a d i , " Analecta Papyrologica 1, 67-77.

304

·

Bibliography

1995. "Appunti per oroscopi negli ostraca di Medinet M a d i (II)," Analecta Papyrologica 7, 63-72. Bakhouche, B. 1996. Les astres: Actes du colloque international de Montpellier 23-25 mars 1995, 2 vols. (Montpellier: Université Paul Valéry). Bara, J.-F. 1989. Vettius Valens dAntioche: Anthologies, Livre I (Leiden: E.J. Brill). Barker, Α. 2000. Scientific Method in Ptolemy's Harmonics (Cambridge: Cam­ bridge University Press). Barton, T. 1994. Ancient Astrology (London: Routledge). Berggren, J.L. 1976. "Spurious Theorems in Archimedes' Equilibrium of Planes: Book I," Archive for History of Exact Sciences 16, 87-103. Berggren, J.L., and H . Eggert-Strand, forthcoming. "Theodosios on Habitations: A Translation and Commentary." Berggren, J.L., and A . Jones 2000. Ptolemy's Geography: An Annotated Transla­ tion of the Theoretical Chapters (Princeton: Princeton University Press). Berggren, J.L., and R.S.D. Thomas 1996. Euclid's Phaenomena: A Translation and Study of a Hellenistic Treatise in Spherical Astronomy (New York: Gar­ land). Bickerman, E.J. 1980. Chronology of the Ancient World, 2 ed. (Ithaca: Cornell University Press). Blass, F. 1883. De Gemino et Posidonio (Kiel: C F . M o h r ) . 1887. Eudoxi ars astronómica (Kiel: Schmidt and Klaunig). Reprinted in Zeitschrift fur Papyrologie und Epigraphik 115 (1997) 3-25. Bôckh, A . 1863. Über die vierjàhrigen Sonnenkreise der Alten, vorzüglich den Eudoxischen (Berlin). Boll, F. 1899. "Das Κηρύκιον ais Sternbild," Hermes 34, 643-45. 1901. "Die Sternkatalog des Hipparch und des Ptolemaios," Bibliotheca Mathematica, Folge 3, Band 2, 185-95. Boll, E , and W. Gundel 1924. "Sternbilder, Sternglaube und Sternsymbolik bei Griechen und Rômern," in W . H . Roscher, éd., Ausfiihrliches Lexikon der griechischen und romischen Mythologie, vol. V I (Leipzig: Teubner), cols. 867-1071. Bouché-Leclercq, A . 1899. L'astrologie grecque (Paris: Leroux). Bowen, A . C . 2001. " L a scienza del cielo nel periodo ptolemaico," in S. Petruccioli, ed., Storia della scienza, I. La scienza greco-romana (Rome: Instituto della Enciclopedia Italiana), pp. 806-39. "Simplicius' Commentary on Aristotle, De cáelo 2.10-12: A n Annotated Translation (Part 1)," SCIAMVS 4 (2003) 23-58. Bowen, A . C , and B.R. Goldstein 1989. " M e t o n of Athens and Astronomy in the Late Fifth Century B . C . , " in E. Leichty et al., eds., A Scientific Humanist: Studies in Memory of Abraham Sachs (Philadelphia: Samuel N o a h Kramer Fund), pp. 39-81. 1996. "Geminus and the concept of mean motion in Greco-Latin astron­ omy," Archive for History of Exact Sciences 50, 157-85. Bowen, A . C , and R . B . Todd 2004. trans. Cleomedes' Lectures on Astronomy (Berkeley: University of California Press). n d

Bibliography

·

305

Brown, D . 2000. Mesopotamian Planetary Astronomy-Astrology (Groningen: Styx Publications). Bruin, R, and A . Vondjidis 1971. trans. The Books of Autolykos (Beirut: Ameri­ can University of Beirut). Bury, R . G . 1987. ed. and trans. Sextus Empricus, Against the Professors, 4 vols. (London: William Heinemann; Cambridge: Harvard University Press). Bywater, I. 1886. ed. Prisciani Lydi quae extant. Metaphrasis in Theophrastum et Solutionum ad Chosroem liber, in Supplementum Aristotelicum, 3 vols. (Berlin: Deutsche Akademie der Wissenschaften, 1885-1903), vol. 1, pt. 2. Cajori, Ε 1993. A History of Mathematical Notations, 2 vols, bound as one (New York: Dover. First pub. L a Salle, Illinois: Open Court, 1928, 1929). Carmado, D . , and A . Ferrrara 1989. Stabiae: Le Ville (Castellamare di Stabia). Catamo, M . , et al. 2000. "Fifteen Further Greco-Roman Sundials from the Mediterranean Area and Sudan, Journal for the History of Astronomy 31, 203-21. Catullus. See Goold 1983. Censorinus, On the Birth Day. See Rocca-Serra 1980 or Sallmann 1988. Charvet, P., and A . Zucker 1998. Le ciel: Mythes et histoire des constellations: Les Catastérismes d'Érathosthène (Paris: N i L Éditions). Cherniss, H . , and W C . Helmbold 1957. eds. and trans. Plutarch, Moralia, vol. 12 (Cambridge: Harvard University Press). Cicero, De re publica. See Keyes 1961. , On the Nature of the Gods. See Rackham 1961. Columella, On Agriculture. See Ash 1954-60. Cumont, F 1935. "Les noms des planètes et l'astrolatrie chez les Grecs," D Antiquité Classique 4, 5-43. Cunliffe, B. 2002. The Extraordinary Voyage of Pytheas the Greek: The Man Who Discovered Britain (London: Penguin). Cuvigny, H . 2004. "Une sphère céleste antique in argent ciselé," in P. Horak, ed. Gedenkschrift Ulrike Horak (Florence: Edizioni Gonnelli). De Falco, V., M . Krause, and O . Neugebauer 1966. Hypsiklës. Die Aufgangszeiten der Gestirne. Abhandlungen der Akademie der Wissenschaften in Gôttingen. Philologisch-Historische Klasse, Dritte Folge, Nr. 62 (Gôttingen: Vandenhoeck & Ruprecht). Dicks, D . R . 1960. The Geographical Fragments of Hipparchus (London: Athlone Press). 1972. "Geminus," in Gillispie 1970-80, vol. 5, pp. 344-47. Diels, H . 1882. Simplicii in Aristotelis physicorum libros quattuor priores commentaria (Berlin: G . Reimer). 1924. Antike Technik (Leipzig: Teubner). Diels, H , and A . Rehm 1904. "Parapegmenfragmente aus M i l e t , " Sitzungsberichte der koniglich Preussischen Akademie der Wissenschaften, Jahrgang, 92-111. Dijksterhuis, E.J. 1987. Archimedes. W i t h a new bibliographic essay by Wilbur R. Knorr (Princeton: Princeton University Press. First pub. 1956). Dilke, O . A . W . 1985. Greek and Roman Maps (Ithaca: Cornell University Press).

306

· Bibliography

Diodoros of Sicily. See Oldfather 1947-67. Diogenes Laertios, Lives and Opinions of Eminent Philsophers. See Hicks 1958-59. Dodge, B. 1970. ed. and trans. The Fihrist of al-Nadim, 2 vols. (New York and London: Columbia University Press). Drake, S. 1989. "Hipparchus—Geminus—Galileo," Studies in History and Philosophy of Science 20, 47-56. Dreyer, J . L . E . 1906. History of the Planetary Systems from Thaïes to Kepler (Cambridge: Cambridge University Press). Reprinted as A History of Astronomy from Thaïes to Kepler (New York: Dover, 1953). Duhem, P. 1908. ΣΩΖΕΙΝ ΤΑ ΦΑΙΝΟΜΕΝΑ. Essai sur la notion de théorie physique de Platon à Galilée (Paris: A . Hermann et Fils). 1969. To Save the Phenomena: An Essay on the Idea of Physical Theory from Plato to Galileo, trans, by Edmund Dolland and Chaninah Maschler (Chicago and London: University of Chicago Press). Duke, D . 2006. "Analysis of the Farnese Globe," Journal for the History of Astronomy 37, 87-100. Dupuis, J. 1892, Théon de Smyrne, Philosophe platonicien: Exposition des con­ naissances mathématiques utiles pour la lecture de Platon (Paris: Hachette. Reprinted, Bruxelles: Culture et Civilisation, 1966). Duval, P . M . , and G . Pinault 1986. "Les calendriers (Coligny, Villards d'Héria)," Recueil des Inscriptions Gauloises, vol. 3 (Paris: Editions du Centre national de la recherche scientifique). Eastwood, B. 1992. "Heraclides and Heliocentrism: Texts, Diagrams, and Inter­ pretations," Journal for the History of Astronomy 23, 233-60. Edelstein, L . , and I.G. K i d d 1989. Posidonius, Vol. I, The Fragments, 2 ed. (Cambridge: Cambridge University Press). Engels, D . 1985. "The Length of Eratosthenes' Stade," American Journal of Philology 106, 298-311. Eratosthenes, Catasterisms. See Olivieri 1897; Charvet and Zucker 1998. Euclid, Phenomena. See Berggren and Thomas 1996. Evans, J. 1984. " O n the Function and the Probable Origin of Ptolemy's Equant," American Journal of Physics 52, 1080-89. 1998. The History and Practice of Ancient Astronomy (New York: Oxford University Press). 2003. "The Origins of Ptolemy's Cosmos," in S. Colafrancesco and G . Giobbi, eds., Cosmology through Time: Ancient and Modern Cosmologies in the Mediterranean Area (Milano: Mimesis), pp. 123-32. Evans, J., and M . Marée (forthcoming). " A Minature Ivory Sundial and Equinox Indicator from Greek Egypt." Field, J.V., and M . T . Wright 1984. "Gears from the Byzantines: A Portable Sun­ dial with Calendrical Gearing," Annals of Science 42, 87-138. Firmicus Maternus, Mathesis. See M o n a t 1992-97. Fischer, I. 1975. "Another Look at Eratosthenes' and Posidonius' Determination of the Earth's Circumference," Quarterly Journal of the Royal Astronomical Society 16, 152-67. n d

Bibliography

·

307

Forster, E.S., and D.J. Furley 1978. ed. and trans. Aristotle, On Sophistical Refutations and on Coming-to-Be and Passing Away; On the Cosmos (Cambridge: Harvard University Press; London: William Heinemann). Fotheringham, J.K. 1924. "The Metonic and Callipic Cycles," Monthly Notices of the Royal Astronomical Society 84, 383-92. Frazer, J . G . 1931. ed. and trans. Ovid, Fasti (London: Heinemann; N e w York: G.P. Putnam's Sons). Friedlein, G . 1873. Prodi Diadochi primum Euclidis elementorum lihrum commentant (Leipzig: Teubner). Geminos, Introduction to the Phenomena. See Manitius 1898; Aujac 1975. Gibbs, S. 1976. Greek and Roman Sundials (New Haven: Yale University Press). Gillispie, C . C . 1970-80. ed. Dictionary of Scientific Biography (New York: Charles Scribner's Sons). Goldstein, B. 1980. "The Status of Models in Ancient and Medieval Astronomy," Centaurus 24, 132-47. 1997 "Saving the Phenomena: The Background to Ptolemy's Planetary Theory," Journal for the History of Astronomy 28, 1-12. Goold, G.P. 1977. Manilius. Astronómica (Cambridge: Harvard University Press; London: William Heinemann). 1983. Catullus (London: Duckworth). Gôssmann, F. 1950. Planetarium Babylonicum oder Die Sumerisch-Bahylonischen Stern-Namen. A . Deimal, éd., Sumerisches Lexikon, Teil IV, Band 2 (Rome: Verlag des Pàpstlichen Bibelinstituts). Gould, J.B. 1970. The Philosophy of Chrysippus (Albany: State University of N e w York Press). Grant, E. 1974. ed. A Source Book in Medieval Science (Cambridge: Harvard University Press). Grenfell, B.P., and A.S. Hunt 1906. The Hibeh Papyri, Part I (London: Egypt Exploration Fund). Gundel, H . G . 1992. Zodiakos. Tierkeisbilder im Altertum (Mainz am Rhein: Philip von Zabern). Guthrie, W . K . C . 1939. ed. and trans. Aristotle, On the Heavens (London: William Heinemann; Cambridge: Harvard University Press). Halma, [Nicolas] 1821. Les Phénomènes dAratus de Soles, et de Germanicus César, avec les scholies de Théon, les Catastérismes d'Eratostbène, et la Sphère de Leontius. Paris: M e r l i n . Hannah, R. 2001. " F r o m orality to literacy? The case of the parapëgma," in J. Watson, ed., Speaking Volumes: Oralitiy and Literacy in the Greek and Roman World (Leiden: Brill), pp. 139-59. 2002. "Euctemon's parapëgma," in C.J. Tuplin and T.E. Rihll, eds., Science and Mathematics in Ancient Greek Culture (Oxford: Oxford University Press), pp. 112-32. Harley, J.B., and David Woodward 1987. The History of Cartography, vol. 1, Cartography in Prehistoric, Ancient, and Medieval Europe and the Mediterranean (Chicago and London: University of Chicago Press).

308

·

Bibliography

Hayduck, M . 1899. Alexandri in Aristotelis Meteorologicorum libros commentant*. Commentaria in Aristotelem Graeca, iii, 2 (Berlin). Heath, T.L. 1912. The Works of Archimedes, revised ed. (Cambridge: Cambridge University Press). 1921. A History of Greek Mathematics, 2 vols. (Oxford: Clarendon Press). Heiberg, J . L . 1885. Euclidis Elementa, vol. 5, Continens Elementorum qui feruntur libros XIV-XV et Scholia in Elementa (Leipzig: Teubner). 1893. ed. Apollonii Pergaei que graece exstant cum commentants antiquis (Leipzig: Teubner. Reprinted, Stuttgart: Teubner, 1974). 1894. ed. Simplicii In Aristotelis de Cáelo commentaria. Commentaria in Aristotelem Graeca, vii (Berlin). 1898-1907. ed. Claudii Ptolemaei Opera quae exstant omnia. V o l . 1 (2 parts), Syntaxis mathematical V o l . 2, Opera astronómica minora (Leipzig: Teubner). 1912. Heronis Alexandrini Opera quae supersunt omnia, V o l . 4, Heronis Definitiones cum variis collectionibus. Heronis quae feruntur geométrica (Leipzig: Teubner). Herodotos, Histories. See Sélincourt 1972. Hicks, R . D . 1958-59. ed. and trans. Diogenes Laertius, hives of Eminent Philosophers, 2 vols. (Cambridge: Harvard University Press; London: W. Heinemann). Hildericus, E . 1590. Gemini probatissimi philosophy ac mathematici elementa astronomiae Graece, & Latine interprete Edone Hïlderico. (Altdorf, 1590; Leiden, 1603). Hipparchos, Commentary on the Phenomena of Aratos and Eudoxos. See M a n itius 1894. Hippokratês, Regimen III. See W.H.S. Jones 1923-1931, vol. 4. Hôlbl, G . 2001. A History of the Ptolemaic Empire, trans. T. Saavedra (London and N e w York: Routledge). Hultsch, Ε 1864. ed. Heronis Alexandrini Geometricorum et stereometricorum reliquiae (Berlin: Weidmann). 1875-78. ed. Pappi Alexandrini Collectionis quae supersunt, 3 vols. (Berlin. Reprinted, Amsterdam: Verlag Adolf M . Hakkert, 1965). Hunger, H . , and D . Pingree 1989. MUL.APIN. An Astronomical Compendium in Cuneiform. Archiv für Orientforschung, Beiheft 24. (Horn, Austria: Ferdi­ nand Berger 8c Sonne). 1999. Astral Sciences in Mesopotamia (Leiden: Brill). Hypsiklës, Anaphorikos. See De Falco, Krause, and Neugebauer 1966. Jones, A . 1983. "The Development and Transmission of 248-Day Schemes for Lunar M o t i o n in Ancient Astronomy," Archive for History of Exact Sciences 29, 1-36. 1994. "Peripatetic and Euclidean theories of the visual ray," Physis 31, n.s., fase. 1, pp. 47-76. 1999a. "Geminus and the Isia," Harvard Studies in Classical Philology 99, 255-67. 1999b. Astronomical Papyri from Oxyrhynchus. Memoirs of the Ameri­ can Philosophical Society 233.

Bibliography

·

309

2000. "Calendrica I: N e w Callippic Dates," Zeitschrift fur Papyrologie una Epigrapbik 129, 141-58. 2001a. " M o r e Astronomical Tables from Tebtunis," Zeitschrift für Papyrologie und Epigrapbik 136, 211-20. 2001b. "Pseudo-Ptolemy ''De Speculis," S C I A M V S 2, 145-86. Jones, H . L . 1959-1961. The Geography of Strabo (Cambridge: Harvard University Press; London: William Heinemann). Jones, W.H.S. 1918-35. ed. and trans. Pausanias' Description of Greece, 5 vols. (London: W. Heinemann; N e w York: G.P. Putnam's Sons). 1923-1931. Hippocrates, 8 vols. (Cambridge: Harvard University Press; London: W. Heinemann). Kallimachos, Aetia. See Trypanis 1978. Keyes, C . W 1961. ed. and trans. Cicero, De re publica, De legibus (Cambridge: Harvard University Press). K i d d , I.G. 1988. Posidonius, V o l . II, The Commentary, 2 parts (Cambridge: Cambridge University Press). 1997. Aratus. Phenomena. (Cambridge: Cambridge University Press). 1999. Posidonius, V o l . I l l , The Translation of the Fragments (Cambridge: Cambridge University Press). Kleomëdës, Meteor a. See Todd 1990; Bowen and Todd 2004. Knorr, W 1989. "Plato and Eudoxus on the Planetary Motions," Journal for the History of Astronomy 20, 313-29. Koch-Westenholz, U . 1995. Mesopotamian Astrology: An Introduction to Babylonian and Assyrian Celestial Divination (Copenhagen: Carsten Niebuhr Institute). Kouremenos, T. 1994. "Posidonius and Geminus on the foundations of mathematics," Hermes: Zeitschrift für Klassische Philologie 122, 437-50. Kugel, A . 2002. Sphères. Uart des mécaniques célestes (Paris: J. Kugel). Kunitzsch, P., and T. Smart 1986. Short Guide to Modern Star Names and Their Derivations (Wiesbaden: Otto Harrassowitz). Kiinzl, E . 2000. Ein romischer Himmelsglobus der mittlern Kaiserzeit (Mainz: Rómisch-Germanisches Zentralmuseum). 2005. Himmelsgloben und Sternkarten. Astronomie und Astrologie in Vorzeit und Altertum (Stuttgart: Theiss). Landels, J . G . 1978. Engineering in the Ancient World (Berkeley and Los Angeles: University of California Press). Lasserre, Ε 1966. ed. and trans. Die Fragmente des Eudoxos von Knidos (Berlin: Walter de Gruyter & Co). Le Boeuffle, A . 1989. Le ciel des Romans (Paris: De Boccard). Lee, H.D.P. 1952. ed. and trans. Aristotle, Meteorológica (Cambridge: Harvard University Press). Lehoux, D . R . 2000. Parapëgmata, or, Astrology, Weather, and Calendars in the Ancient World (Ph.D. dissertation, University of Toronto). 2005. "The Parapëgma Fragments from Miletus," Zeitschrift für Papy­ rologie und Epigrapbik 152, 125-40. (forthcoming). Astronomy, Weather, and Calendars in the Ancient World (Cambridge: Cambridge University Press).

310

· Bibliography

Leontios, Construction of the Sphere of Aratos. See Halma 1821; Maass 1898, 561-67. Leptinës, Celestial Teaching (Ouranios Didascalea). See Blass 1887; Tannery 1893,283-94. Lerner, M.-P. 1996. Le monde des sphères, 2 vols. (Paris: Les Belles Lettres). Lewis, M . J . Taunton (2001). Surveying Instruments of Greece and Rome (Cambridge and N e w York: Cambridge University Press). Liddell, H . G . , and R. Scott, revised by H . S . Jones 1996. A Greek-English Lexicon, 9 ed. with N e w Supplement (Oxford: Clarendon Press). Lloyd, G . E . R . 1973. Greek Science after Aristotle (New York: Norton). 1978. "Saving the Appearances," Classical Quarterly 28, 202-22. 1987. The Revolutions of Wisdom: Studies in the Claims and Practices of Ancient Greek Science (Berkeley: University of California Press). 1991. Methods and Problems in Greek Science (Cambridge and N e w York: Cambridge University Press). Lucretius, On the Nature of Things. See Rouse 1982. Locher, K . 1989. " A further Hellenistic conical sundial from the theatre of Dionysius in Athens," Journal for the History of Astronomy 20, 60-62. Maass, E. 1898. Commentariorum in Aratum reliquiae (Berlin: Weidmann; Reprinted, 1958). Macrobius, Commentary on the Dream of Scipio. See Stahl 1952. Mair, A . W . , and G . R . M a i r 1955. Callimachus: Hymns and Epigrams. Lycophron. Aratus. (London: William Heinemann; Cambridge: Harvard University Press). Manilius, Astronómica. See Goold 1977. Manitius, Carolus 1894. ed. and trans. Hipparchi in Arati et Eudoxi Phaenomena commentariorum libri tres (Leipzig: Teubner). 1898. ed. and trans. Gemini Elementa astronomiae (Leipzig: Teubner; reprinted Stuttgart: Teubner, 1974). 1909. ed. and trans. Prodi Diadochi Hypotyposis astronomicarum positionum (Leipzig: Teubner; reprinted Stuttgart: Teubner, 1974). Mansfield, J., and D.T. Runia 1997. Aètiana: The Method and Intellectual Context of a Doxographer, vol. 1, The Sources (Leiden and N e w York: Brill). Marsden, E.W. 1971. Greek and Roman Artillery, vol. 2, Technical Treatises (Oxford: Oxford University Press). M a r t i n , J. 1998. ed. and trans. Aratos, Phénomènes, 2 vols. (Paris: Les Belles Lettres). M a r t i n , T . - H . 1854. Recherches sur la vie et les ouvrages d'Héron dAlexandrie, disciple de Ctésibius, et sur tous les ouvrages mathématiques grecs, conservés ou perdus, publiés ou inédits, qui ont été attribués à un auteur nommé Héron. Académie des Inscriptions et Belles-Lettres. Mémoires présentés pars divers savants. Première série, tome 4 (Paris: Imprimerie Impériale). McClusky, S. 1998. Astronomies and Cultures in Early Medieval Europe (Cambridge: Cambridge University Press). Mette, H . J . 1952. ed. Pytheas von Massalia (Berlin: W. de Gruyter). Mittelstrass, J. 1962. Die Rettung der Phànomene (Berlin: Walter de Gruyter). t h

Bibliography

·

311

Mogenet, J. 1950. Autolycus de Pitane, Histoire du Texte. Publications Universi­ taires de Louvain, 3e série, fasc. 37. Monat, P. 1992-97. ed. and trans. Firmicus Maternus, Mathesis, 3 vols. (Paris: Les Belles Lettres). M o r r o w , G . R . 1970. trans. Proclus, Commentary on the First Book of Euclid's Elements (Princeton: Princeton University Press). Mueller, I. 2004. trans. Simplicius, On Aristotle's "On the Heavens 2.1-9" (Ithaca: Cornell University Press). 2005. trans. Simplicius, On Aristotle's "On the Heavens 2.10-14" (Ithaca: Cornell University Press). Murray, A . T . 1936-39. ed. and trans. Demosthenes, Private Orations, 6 vols. (Cambridge: Harvard University Press; London: W. Heinemann). Neugebauer, O . [1955]. Astronomical Cuneiform Texts, 3 vols. (London: Lund Humphries). 1975. A History of Ancient Mathematical Astronomy (New York: Springer: Berlin). Newcomb, S. 1898. "Tables of the M o t i o n of the Earth on Its Axis and Around the Sun," Astronomical Papers Prepared for the Use of the American Ephemeris and Nautical Almanac 6. Oates, J. 1986. Babylon, 2 ed. (London: Thames and Hudson). Oldfather, C . H . 1947-67. Diodorus of Sicily, 12 vols. (Cambridge: Harvard University Press; London: W. Heinemann). Olivieri, A . 1897. Mythographi Graeci 16, 3, fasc. 1 (Leipzig). O v i d , Fasti. See Frazer 1931. Pappos of Alexandria, Commentary on the Almagest. See Rome 1931. , Mathematical Collection. See Hultsch 1875-78; Ver Eeeke 1933. Pattenden, P. 1981. " A late sundial at Aphrodisias," Journal of Hellenic Studies 101, 101-12. Pausanias, Description of Greece. See W.H.S. Jones 1918-35. Pedersen, O . 1974. A Survey of the Almagest (Odense, Denmark: Odense Uni­ versity Press). Petavius, Dionysius [Denis Petau] 1630. Uranologion sive systema variorum authorum, qui de sphaera ac sideribus eorumque motibus Graece commentati sunt. Cura & studio Dionysii Petavii Aurelianensis e Societate Iesu (Paris, 1630). Pfeiffer, R. 1968. History of Classical Scholarship: From the Beginnings to the End of the Hellenistic Age (Oxford: Clarendon Press). Pindar, Odes. See Sandys 1915. Pingree, D . 1986. Vettii Valentis Antiocheni Anthologiarum libri novem (Leipzig: Teubner). Pliny, Natural History. See Rackham 1947-63. Plutarch, Isis and Osiris. See Babbitt 1936. , On the Face in the Orb of the Moon. See Cherniss and Helmbold 1957. Porter, B., and R . L . B . Moss 1927-. Topographical Bibliography of Ancient Egyptian Hieroglyphic Texts, Reliefs, and Paintings, 7 vols. (Oxford: Claren­ don Press). Poseidônios, Fragments. See Edelstein and K i d d 1989; K i d d 1988; K i d d 1999. n d

312

·

Bibliography

Price, D J . 1969. "Portable Sundials in Antiquity," Centaurus 14, 242-66. 1974. Gears from the Greeks. The Antikythera Mechanism—A Calendar Computer from ca. 80 B.C. Transactions of the American Philosophical Society, vol. 64, pt. 7. Pricianus Lydus, Explanation of Problems for King Chosroes. See Bywater 1886. Pritchett, W.K., and O . Neugebauer 1947. The Calendars of Athens (Cambridge: Harvard University Press). Proklos, Commentary on the First Book of Euclid's Elements. See Friedlein 1878; M o r r o w 1970. , Sketch of the Astronomical Hypotheses. See Manitius 1909. Pseudo-Aristotle, On the Cosmos. See Forster and Furley 1978. Ptolemy, Almagest. See Heiberg 1898-1907, vol. 1; Toomer 1984. , Geography. See Berggren and Jones 2000. , Optics. See Smith 1996. , Phaseis. See Heiberg 1898-1907, vol. 2. , Tetrabiblos. See Robbins 1940. Pytheas of Massalia, On the Ocean. See Mette 1952; Roseman 1994. Rackham, H . 1947-63. ed. and trans. Pliny, Natural History, 10 vols. (Cambridge: Harvard University Press; London: W. Heinemann). 1961. ed. and trans. Cicero, De natura deorum; Académica (Cambridge: Harvard University Press). Rawlins, Dennis 1982. "The Eratosthenes-Strabo Nile M a p , " Archive for History of Exact Sciences 26, 211-19. Reiner, Erica 1995. Astral Magic in Babylon. Transactions of the American Philosophical Society, vol. 85, pt. 4. Reinhardt, K . 1921. Poseidónios. (Munich: C H . Beck). 1926. Kosmos und Sympathie (Munich: C H . Beck). Robbins, E E . 1940. ed. and trans. Ptolemy, Tetrabiblos. (London: William Heinemann; Cambridge: Harvard University Press). Rocca-Serra, G . 1980. ed. and trans. Censorinus, Le jour natal (Paris: J. Vrin). Rochberg-Halton, F. 1984. " N e w Evidence for the History of Astrology," Journal of Near Eastern Studies 43, 115-40. 1988. "Elements of the Babylonian Contribution to Hellenistic Astrology," Journal of the American Oriental Society 108, 51-62. Rohr, R.-J. 1980. " A unique Greek sundial recently discovered in central A s i a , " Journal of the Royal Astronomical Society of Canada 74, 271-78. Rome, A . 1927. "L'astrolabe et le météoroscope d'après le commentaire de Pappus sur le 5 livre de Γ Almagest," Annales de la Société Scientifique de Brux­ elles (Série A) 47, 77-102. 1931. Commentaires de Pappus et de Théon dAlexandrie sur VAlmageste. Tome 1. Pappus d'Alexandrie, Commentaire sur les livres 5 et 6 de lAlmageste. Studi e Testi 54 (Rome: Biblioteca Apostólica Vaticana). Roseman, C H . 1994. Pytheas of Massalia: On the Ocean." Text, Translation and Commentary (Chicago: Ares). Rouse, W . H . D . 1982. ed. and trans. Lucretius, De rerum natura, 2 ed., revised by M . F . Smith (Cambridge: Harvard University Press; London: William Heine­ mann). e

((

n d

Bibliography

·

313

Russo, L . 2004. The Forgotten Revolution: How Science Was Born in 300 B.C. and Why It Had to Be Reborn, trans, by Silvio Levy (Berlin: Springer). Sallmann, K . 1988. Betrachtungen zum Tag der Geburt =De die natali I Censor­ inus (Weinheim: V C H ) . Sambursky, S. 1959. Physics of the Stoics (Princeton: Princeton University Press). 1962. The Physical World of Late Antiquity (Princeton: Princeton Uni­ versity Press). Samuel, A . E . 1972. Greek and Roman Chronology: Calendars and Years in Classical Antiquity. Handbuch der Altertumswissenschaft, ser. 1, pt. 7 ( M u ­ nich: Beck). Sandys, J. 1915. The Odes of Pindar (London: W. Heinemann; N e w York: Macmillan). Sarton, G . 1970. A History of Science, 2 vols. (New York: W W Norton; first pub. 1959). Savoie, D . , and R. Lehoucq 2001. "Étude gnomonique d'un cadran solaire dé­ couvert à Carthage," Revue dArchéométrie 25, 25-34. Schaefer, B.E. 2005. "The Epoch of the Constellations on the Farnese Atlas and Their Origin in Hipparchus's Lost Star Catalogue," Journal for the History of Astronomy 36, 167-96. Schaldach, K . 2004. "The Arachne of the Amphiareion and the Origin of Gnomonics in Greece," Journal for the History of Astronomy 35, 435-45. Schmidt, M . C . P . 1884a. "Wann schrieb Geminus," Philologus 42, 83-110. 1884b. " W o schrieb Geminus," Philologus 42, 110-18. Schmidt, O . 1952. "Some Critical Remarks about Autolycus' On Risings and Settings," Den 11. Skandinaviske Matematikerkongress i Trondheim 22-25 August 1949 (Oslo: 1952), pp. 27-32. Schône, H . 1903. ed. Heronis Alexandrini Opera quae supersunt omnia, vol. 3, Rationes dimetiendi et commentatio dioptrica. Vermessungslehre und Dioptra (Leipzig: Teubner). Schône, R. 1897. Damianos Schrift Über Optik mit Auszügen aus Geminos (Berlin: Reichsdruckerei). Sélincourt, A . de 1972. trans. Herodotus, The Histories (Harmondswoth: Pen­ guin Books). Sextus Empiricus, Against the Professors. See Bury 1987. Sezgin, F. 1974. Geschichte des Arabischen Schrifttums. Band V: Mathematik bis ca. 430 H. (Leiden: E.J. Brill). Simplikios, Commentary on Aristotle's O n the Heavens. See Heiberg 1894; Mueller 2004; Mueller 2005. , Commentary on Aristotle's Physics. See Diels 1882. Smart, W . M . 1977. Textbook on Spherical Astronomy, 6 ed., revised by R . M . Green (Cambridge: Cambridge University Press). Smith, A . M . 1981. "Saving the Appearances of the Appearances: The Founda­ tions of Classical Geometrical Optics," Archive for History of Exact Sciences 24, 73-99. 1982. "Ptolemy's Search for a L a w of Refraction: A Case-Study in the Classical Methodology of Saving the Appearances," Archive for History of Exact Sciences 26, 221-40. t h

314

· Bibliography

1996. Ptolemy's Theory of Visual Perception: An English Translation of the Optics with Introduction and Commentary. Transactions of the American Philosophical Society, vol. 86, pt. 2. 1999. Ptolemy and the Foundations of Ancient Mathematical Optics: A Source Based Guided Study. Transactions of the American Philosophical Soci­ ety, vol. 89, pt. 3. Soubiran, J. 1969. Vitruve, De l'architecture Livre IX (Paris: Les Belles Lettres). Stahl, W . H . 1952. trans. Macrobius, Commentary on the Dream of Scipio (New York: Columbia University Press). Strabo, Geography. See H . L . Jones 1959-61. Swerdlow, N . M . 2004. "Ptolemy's Harmonics and the 'Tones of the Universe' in the Canobic Inscription" in C . Burnett, J.P. Hogendijk, K . Plofker, M . Yano, eds., Studies in the History of the Exact Sciences in Honour of David Pingree (Leiden: Brill). Tannery, P. 1893. L'histoire de l'astronomie ancienne (Paris: Gauthier-Villars & Fils). 1897. La géométrie grecque (Paris: Gauthier-Villars & Fils). Taub, L . 2003. Ancient Meteorology (London and N e w York: Routledge). Theôn of Smyrna, Mathematical Knowledge Useful for Reading Plato. See Dupuis 1892. Thompson, D.J. 1988. Memphis under the Ptolemies (Princeton: Princeton Uni­ versity Press). Tittel, K . 1910. "Geminos I," in Paulys Realencyclopadie der classsischen Altertumswissenschaft. Neue Barbeitung. Halbband XIII (Stuttgart: A . Druckenmiiller), cols. 1026-50. Todd, R. 1976. Alexander of Aphrodisias on Stoic Physics (Leiden: E.J. Brill). 1982. "Cleomedes and the Stoic Concept of the V o i d , " Apeiron 16, 129-36. 1985. "The Title of Cleomedes' Treatise," Philologus 129, 250-61. 1990. Cleomedis Caelestia (ΜΕΤΕΩΡΑ) (Leipzig: Teubner). 1993. "The Manuscripts of the Pseudo-Proclan Spbaera" Revue d'His­ toire des Textes 23, 57-71. 2003a. "Damianus (Heliodorus Larissaeus)," in Virginia Brown, ed., Catalogus Translationum et Commentariorum, vol. 8 (Washington, D . C . : Catholic University of America Press), pp. 1-5. 2003b. Geminus and the Ps.-Proclan Sphaera" in Virginia Brown, ed., Catalogus Translationum et Commentariorum, vol. 8 (Washington, D . C . : Catholic University of America Press), pp. 7-48. Toomer, G.J. 1984. trans. Ptolemy's Almagest (London: Duckworth). Tredennick, H . 1961-62. ed. and trans. Aristotle, The Metaphysics (Cambridge: Harvard University Press; London: W. Heinemann). Trypanis, C . A . 1978. ed. and trans. Callimachus, Aetia, Iambi, lyric poems, Hecale, minor epic and elegiac poems, and other fragments (Cambridge: Har­ vard University Press; London: Heinemann). Valerio, V. 1987. "Historiographie and numerical notes on the Atlante Farnese and its celestial sphere," Der Globusfreund 35/37, 97-124. u

Bibliography

·

315

Van der Waerden, B.L. 1960. "Greek Astronomical Calendars and Their Rela­ tion to the Athenian Civil Calendar," Journal of Hellenic Studies 80, 68-180. 1974. Science Awakening II. The Birth of Astronomy (New York: O x ­ ford University Press). 1984. "Greek Astronomical Calendars I. The Parapëgma of Euctemon," Archive for History of Exact Sciences 29, 101-14. Ver Eekke, P. 1933. trans. Pappus d'Alexandrie, La collection mathématique (Paris: Desclée De Brouwer). Vettius Valens, Anthologies. See Pingree 1986; Bara 1989. Vitruvius, On Architecture, IX. See Soubiran 1969. Vlastos, G . 1975. Plato's Universe (Seattle: University of Washington Press). Von Boeselager, D . 1983. Antike Mosaiken in Sizilien (Rome: Giorgio Bretschneider). Wachsmuth, C . 1863. Ioannis Laurentii Lydi liber de ostentis et Calendaría Graeca omnia (Leipzig: Teubner). 2 ed., 1897. Weinstock, S. 1951. Catalogus Codicum Astrologorum Graecorum; vol. 9, Codices Britannicos; pt. 1, Codices Oxonienses (Brussels: In Aedibus Academiae). Wicksteed, P . H . , and E M . Cornford 1960-63. eds. and trans. Aristotle, The Physics, 2 vols. (Cambridge: Harvard University Press). n d

Index

Achilleus, 210n2; On the All, 12 acronychal rising, 72 Aghaniyus, 103 Ailios, 251 aithêr, 2 1 5 n l 6 , 220 Aithiopians, 52, 215 aitiologia, 57 Alexander of Aphrodisias, 4, 250-52 Alexandria, 36fig, 140, 221 Alphonsine Tables, 271 amphiskian zones, 2 0 8 n l analemma, 48 anatolë, 200n3 Anatolios, 244 Anaxagoras, 46, 53, 186n2, 2 5 3 n l 4 Anaximandros, 53, 87, 220n9 Anaximenës, 220n9 annual period, 114, 184 anomaly, lunar, 188 Antares, 235n22 Antiochos of Commagene, 119 antipodes, 155, 210, 213 antiskian signs, 125 antoikoi, 210 Aphrodite, 119n20, 120 apogee, lunar, 93 Apollo, 119 Apollónios of Pergë, 8, 55, 198n9 appearance, 200n2 apsides, line of, 93 Aquileia, 37fig Aratos, 26, 141, 169, 206, 225, 236n24, 277; on names of days, 177; Phenomena, 5, 8, 153, 225n21, 232n5; on rising times, 171; on status of Pleiades, 141 Archimedes, 46, 2 2 0 n l l , 248; his celestial models, 47; Sand Reckoner, 40; on sphere-making, 47, 249 archon, 90 arctic circle, 41, 149-52, 154-56, 160, 2 0 8 n l ; modern, 150n2; terrestrial, 163n6 Ares, 119n20, 120 Aristarchos of Samos, 53, 2 2 0 n l l ; On the Sizes and Distances of the Sun and Moon, 49, 103n215

Aristotle, 26, 54, 119, 2 0 8 n l , 210n4, 220n9, 226, 232n5; on accounting for the phenomena, 50; On the Heavens, 6 n l l ; Meteorology, 2 1 7 n l , 237n28, 238n29, 251; on relation between physics and mathematics, 252; on terrestrial exhalations, 2 1 7 n l Aristotle, Pseudo-, On the Cosmos, 119, 220n9, 238n29 arithmetic, 43-44, 2 4 5 - 4 7 arithmetic progressions, 73-82; for lunar motion, 96-99; uses of in Greek astronomy, 58 armillary sphere, 27, 3 2 - 3 3 , 1 5 4 n l 7 , 156, 1 6 4 n l l , 212n8; construction of, 212-13; projection of, 46n94. See also globe, celestial Artaxerxes II, 88 Art of Eudoxos. See Leptinës, Celestial Teaching ascensions, tables of, 73-74, 7 8 n l 6 4 , 131nl3 Asia, 211 aspects, zodiacal, 125-36; diagrams for, 106-7, 126, 128, 130, 133; Ptolemy on, 126n5, 127n6; and sympathies, 129 astrologers, 73 astrólogos, 127n7 astrology: Babylonian, 127; Greek, 50; personal, 127 astronomer: contrasted with astrologer, 127n7; in relation to physicist, 53-58, 251-55 astronomical instruments, Geminos's use of, 27-42 astronomy, 25, 44, 246, 252; branches of, according to Geminos, 4 7 - 4 8 , 249; fundamental hypothesis of, 117-18; Greek surveys of, 8-12; its need of mathematics, 253; its relation to physics, 53-58, 251-55; its tasks, 253. See also Babylonian astronomy Atabyrion, 218 Athens, 88, 147

318

·

Index

Athyr, 18, 2 0 , 9 1 Atlas, 29, 219 Augustus, 28 Aujac, G . , 101-2, 104, 106-7, 274; on antisyzygy, 258; on birthplace of Geminos, 16 Autolykos of Pitanë, 62, 67, 69, 72, 2 0 0 n l , 2 0 3 n l 0 , 205n2, 206n7, 265; Geminos's use of, 27; On the Moving Sphere, 6; On Risings and Settings, 7, 62, 267n.h automata, 46 axis, 146 Azulafe, 271 nl Babylonian astronomy, 73-75, 78-79, 94, 96, 99-100; conventions for equinoxes and solstices in, 115n8; Geminos's knowledge of, 13-15, 125, 192n4, 229; in Greek astronomy and astrology, 14, 100; Hipparchos's use of, 92; rising times and day lengths in, 7 4 n l 6 2 ; zodia­ cal aspects in, 127nn5 and 6 Balance (Libra), Babylonian origin of, 117nl2 barbarians, 150, 162 barleycorn, 192 Basiliskos, 137 Bear, lands beneath the, 2 0 8 n l Berggren, J . L . , 4 7 n l 0 0 Blass, E , 23 Boëthos, 26, 226 Boll, E , 143 Bowen, Α., 25 calendar: Alexandrian, 20, 286-87; astronomical, 89-91; Babylonian, 88; Egyptian, 17-22, 178-80; Gallic, 90; Kallippic, 91; lunisolar, 90; star, 59. See also parapêgma Cancer, tropic of, 29, 34, 35 canonics, 44, 246, 248; and arithmetic, 45 Canopus Decree, 178n9 Capricorn, tropic of, 29, 35 catoptrics, 248 cause, 219; its absence in astronomy, 253 celestial globe. See globe, celestial celestial sphere, 5ff, 146; circles on, 32, 149-60; orientation of, 147 Censorinus, 86; On the Birth Day, 180nl7

centers of gravity, 47 Chaldeans, 13, 125, 229. See also Babylonian astronomy chapter divisions in Geminos, 105 Chiron, 143 Choiak, 61, 72 circle: antarctic, 1 3 9 n l 4 , 149, 151-53, 156; arctic, 41, 149-52, 154-56, 160, 2 0 8 n l ; colure, 130-31, 157; eccentric, 52, 54-57; equator, 29fig, 35, 149fig, 151-53, 208, 216; great, 160; meridian, 156, 159; through the middles of the signs, 157; parallel, 149, 169; of the signs, 113ff, 157, 257n.a; summer tropic, 151-53; tropic, 4 1 , 151, 166, 2 0 8 , 2 1 6 ; zodiacal, 257n.a Cleopatra's Needle, 36 Coligny, bronze calendar of, 90 Columella, De re rustica, 238n29 conjunction, 175, 177, 188, 190 constellations, 137-45, 235-36, 240; convention for names of, 109; in Introduction to the Phenomena, 140-43; northern, 138-39, 141-42; southern, 139-40, 142-43; zodiacal, 113, 125n3, 137-38, 140, 169-74; Andromeda, 61, 138; Aquarius, 31fig, 39, 113, 237; Argo, 139; Aries, 113, 206n4, 234, 239; Arrow, 138, 279; Bear Keeper, 138, 141; Bears, 5; Bird (Cygnus), 61, 138, 234, 238, 240, 277, 279, 284; Caduceus, 139, 143; Cancer, 38-39, 231, 237; Capricorn, 31fig, 38, 113, 237; Cassiopeia, 138; Censer, 139; Centaurus, 139; Cepheus, 138; Charioteer, 138-39, 206, 234; Claws (of the Scorpion), 1 1 7 n l 2 , 234, 239; Cords, 138; Crater, 139; C r o w n (Corona Borealis), 61, 138, 232-34, 237-38, 277; D o g , 72, 139; Dolphin (Delphinus), 233, 236-37, 240, 277; Dragon (Draco), 5, 138; Eagle (Aquila), 71, 138, 2 3 2 - 3 3 , 236-37, 240, 277, 285; Forepart of a Horse (Equuleus), 13, 138, 141-42; Fishes (Pisces), 238-39; Gemini, 113, 236, 240; Great Bear, 5, 138, 150; Hare, 139; Horse (Pegasus), 138, 233, 238, 277, 279; H y d r a , 139; Kneeling M a n , 138, 141; Leo, 39, 60, 113-14, 232-33, 237-38; Libra, 113, 1 1 7 n l 2 , 136, 233-34, 239-40; Little Bear, 138; Lock of Berenikë, 138,

Index 1 3 9 n l 4 , 142; Lyra, 60, 138, 233, 235, 237, 239, 277; Manger, 137; Ophiuchus, 30fig, 138; O r i o n , 139, 232, 235-36, 239-40, 277; Perseus, 138-39; Pisces, 113, 206n4, 238-39; Pitcher, 138; Procyon, 139; Raven, 139; River, 139, 143; Sagittarius, 113, 236; Scorpio, 113-14, 234-36, 238, 240, 277-78, 285; Sea Monster, 139; Serpent, 138; Southern C r o w n (or Canopy), 139, 143; Southern Fish, 139; Taurus, 113, 206, 235, 239-40; Throne of Caesar, 28; Thyrsuslance, 139, 143; Triangle, 138; Virgo, 113, 2 3 2 n l , 233-34; Water, 139, 143; W i l d A n i m a l , 139, 143 cosmical setting, 72 cosmos, 42; circular motion of, 195; inclination of, 165 Croesus, 1 8 0 n l 7 Cumont, F., on divine names of planets, 119 Cuvigny, H . , 30n71 cycle. See lunisolar cycles Damianos, 244n7 Darius I, 88 day, two senses of, 161; reckoned by phases of M o o n , 177 day length, 73, 179, 213; for Alexandria, 77; day by day, 79-80; Egyptian scheme for, 61; erroneous beliefs about, 167; according to Geminos, 80-82; latitude and maximum value of, 162-65; seasonal variation of, 115-16, 165-67, 260nn.e and g; in System A , 74-9; in System B, 74nl62 declination, 37 deferent, 198n9 deferent-and-epicycle theory, threedimensional form of, 5 2 n l 2 8 degree, definition of, 114 Dëmokritos, 46, 46n94, 59, 1 7 9 n l 0 , 197n5, 237-40; his parapêgma, 59 Diels, H . , 279 differences, constant, 167n18 Dikaiarchos, 218 Diodoros, 88 Diogenes Laertios, 86; on Stoic curriculum, 9, 24 Dionysios (astronomer), 119 dioptra, 27, 38-43, 113, 151, 195; equatorial, 41-42; in Euclid's Phenomena, 38;

·

319

Geminos's uses of, 41; Hipparchos's use of, 39 dioptrics, 48, 249 disappearance, of a star, 200n4 Dositheus, 2 2 4 n l 9 , 232, 275 doubly visible stars, 69 Duhem, P., 51 dunameis (powers), 155n21 dusis, 200n5 Earth: as center of cosmos, 38-39, 220; equatorial region of, 165; shadow of, 191; size of, 211; sphericity of, 54, 208; types of its inhabitants, 210 eclipses, 82; diagrams of, 107; etymology of, 190; Hipparchos's observations of, 190n4; geometry of, 190n4; lunar, 15, 191-94; magnitudes of, 190, 213; solar, 189-90; their uses in geography, 254 eclipse zone (ekleiptikon), 15, 192-93 ecliptic, 192; its division by horizon and meridian, 129-32, 258n.e; obliquity of, 77, 156 ecliptic arcs, risings and settings of, 6 Egypt, 61, 1 3 9 n l 4 eisagôgë, dual meaning of, 9 ekpetasmata, in Ptolemy's Geography, 46n94 ekphanês, 233n7 embolismic month, 84 Empedoklës, 53, 220n9 engineering, 46 epagomenal days, 178 epicycle, 52, 5 2 n l 2 8 , 1 1 8 n l 7 , 198n9 epitolë, 200n6 equator, 29fig, 35, 149fig, 151-53, 208,216 equilibrium, science of, 249 equinox, 36-37, 151, 216, 238-39; autumnal, 115, 234; changes in day lengths near, 166; perpetual, 165; vernal, 61, 114-15 Eratosthenes, 141, 163, 209n3; On the Octaetêris, 86, 179 Euclid, 171n9, 244; Division of the Canon, 45; Division of Figures, 247n20; Elements, 44; Optics, 45; Phenomena, 6, 103n215, 205n3, 267; scholia on his Elements, 245

320

·

Index

Euclid, Pseudo-, Catoptrics, 45 Eudoxos, 5, 26, 59, 119, 141, 178, 179nl0, 221nl2, 224nl9, 232-39, 2 8 5 - 8 6 , 288-89; his division of the zodiac, 132; homocentric spheres of, 8; on the octaetëris, 86; his parapëgma, 2, 277-78; On Speeds, 8; winter solstice of, 236n26 Euktëmôn, 88, 1 7 8 n l 0 , 2 2 1 n l 2 , 2 2 4 n l 9 , 232, 2 8 4 - 8 5 , 289; his parapëgma, 2, 277, 288 Europe, 211 Eutokios, 243 Evans, J . , 5 2 n l 2 8 exeligmos, 82, 92-94; in Geminos, 94-99, 227-30 exhalations, 217 explanations, multiple, 57 Farnese Atlas, 27, 29 fire, 220 Firmicus Maternus, 125 first principles, 54 first sixtieth, 228 forerunners, 232 forms, material, 246 fractions, base-60, 109, 269 Galen, 104 Geminos, 15n30, 17, 82, 86, 1 3 3 n l 6 , 207n8, 243-44, 246, 252; on the classification of mathematics, 43-48, 243-49; Concise Exposition of the Meteorology of Poseidônios, 4, 53, 2 5 0 - 5 5 ; date of, 17-22, 248n29; day lengths i n , 74, 80; on instruments, 2 7 - 4 3 ; Introduction to the Phenomena, 1-3, 226n23, 245; his mathematical methods, 78, 174; his name, 17; his parapëgma, 2, 69, 71-72, 275-89; pedagogy in, 1 8 0 n l , 187, 190n47; Philokalia, 3, 43, 2 4 3 - 4 5 ; his sources, 12-15, 301-2; and Stoicism, 23-27 geodesy, 44, 246-47 geometry, 245-47, 252; concerned with mental objects, 43; plane, 247 Gerard of Cremona, 104 globe, celestial, 6, 2 7 - 3 1 , 164, 1 7 3 n l 3 , 213, 286; Farnese, 28-29; Geminos's use of, 33-34; of M a i n z , 28, 31fig; parallel circles on, 152, 259n.a; of Paris, 28;

limitations of, 33; for modeling precession, 270. See also armillary sphere gnomon, 134-36, 166, 168, 249 gnomonics, 47, 249 gravity, centers of, 249 Greek Anthology, 44 halcyon days, 238 Halley, Edmond, 95 harmonics, 252 harmony, 45 Harpalus, 86 Heath, T. L . , 44, 243 Heiberg, J. L . , 244 heliacal risings and settings. See phases of the fixed stars Helios, 119n20 Hellespont, klima of, 259 Heósphoros, 119 Herakleidës, 121, 254 Hëraklës, 119n20, 141 Hermes, 119n20, 120, 217n2 Hero of Alexandria, 4 1 , 44, 248; Belopoiïkë, 46; Catoptrics, 244; date of, 248n29; and date of Geminos, 22; Definitions, 244; Dioptra, 39; Pneumatics, 46; Stereometrica, 44 Herodotos, 180n17 Hesiod, 219, 233n8, 236n24, 239n34; on the Pleiades, 69; Works and Days, 7, 2 2 n l 4 , 239n34, 259 heteroskian zone, 2 0 8 n l hiding, of a star, 71 Hildericus, E . , 101, 105 Hipparchos, 8, 54, 59, 91-92, 96, 124, 131, 206n4, 2 2 1 n l 2 , 275; Commentary on the Phenomena of Eudoxos and Aratos, 5n8, 224n19; his discovery of procession, 113; on Equuleus, 142; his figure for lunar diameter, 192; Geminos's use of, 27, 116n11; lunar theory of, 188; star catalogue of, 13; trigonometry in, 74 Hippokratës, 246 history, 246 Homer, 52, 1 3 9 n l 3 , 141, 163-4, 215, 223; Geminos's use of, 26 horizon, 38, 146, 149, 205, 213; its absence from spheres, 159; two kinds of, 157-59; its variation with the klima, 158

Index horizon ring, 33 hours, 1 1 6 n l 0 ; equinoctial, 34; seasonal, 34-35, 248n28 Hultsch, E , 244 hydrostatics, 46 hypotheses: epistemic status of, 51; investi­ gation of, 254; role of, 49 Hypsiklës, 77-80, 131; Anaphorikos, 13, 74, 103n215 indicators, for the constellations, 137-39 instrumentalism, 51 intercalation, 90 interpolation, 73 Introduction to the Phenomena, 1; astro­ nomical writers quoted in, 12; contrasts with books of Kleomëdës and Theôn of Smyrna, 25; editions of, 101-2; geogra­ phers in, 13; manuscripts of, 102-5; mathematical arguments in, 9; poets quoted in, 2; significance of, 2; title of, 3, 103, 257; topics treated i n , 2, 105; translations of, 3, 103-4; unique fea­ tures of, 27 Isagoge, 1 Ishtar, 119 Isis, feasts of, 17, 38, 178-79 Jones, Α., 2 1 - 2 2 , 2 5 0 n l Kallaneus the Indian, 284 Kallippos, 2, 88, 9 1 , 1 1 9 , 1 8 3 - 8 4 , 231-39, 224n19, 285-86, 288; beginnings of sea­ sons i n , 289; correspondence between his parapêgma and those of Geminos and Ptolemy, 288; lengths of seasons of, 289; his parapêgma, 278-79 kanon, 248 Kimmerians, 164 kingfisher, 238n30 kite, 238 Kleanthës, 26, 214, 220n9 Kleomëdës, 10n22, 197n5, 2 0 9 n l , 210n2, 220n11; on day lengths, 7 5 n l 6 2 ; Meteôra, 23, 251 Kleostratos, 86-87 klima, 38, 116n9, 154, 256; of Alexandria, 286; its effect on solar eclipses, 190; of Rhodes, 116n9 Kratës, 26, 163, 164, 214-15 Kronos, 119

·

321

krupsis, 200n4 Ktësibios, 248 Künzl, Ε., 28, 31fig Kyllënë, 217-18 Lehoux, D . , 2 3 2 n l Leptinës: Celestial Teaching, 10-11, 79, 1 1 9 , 1 7 9 n l 0 , 1 8 0 n l 4 , 239n33, 289; on sidereal month, 92 Leukonotos, 238n29 lines, physical and mathematical, 252 Lloyd, G . E . R . , on realism in Greek science, 50 logistic, 44, 2 4 6 - 4 7 logistikos, 247 Lucretius, 197n5 lunar theory: Babylonian, 50, 97, 100, 188n3, 192, 228-30; Greek, 92, 100n210; historical roots of Geminos's, 99; parameters of, 89; templates for, 100 lunisolar cycles, 58; exeligmos, 92-99, 227-30; octaetêris, 82-87, 179-83; purposes of, 89-92; 16-year period, 84-85, 182; 19-year period, 87-9, 183-84; 76year period, 89, 184-85; 160-year period, 85, 182 machines, 46 magnitudes, moving and stationary, 245 Manitius, C . , 101-2, 104-5, 119, 261,284 Marcellus, 47 M a r d u k , 119 M a r s , tropical period of, 120n21 M a r t i n , J., 1 3 3 n l 6 mathematicians, 214 mathematics, 243, 246; applied, 43; Babylonian, 73-4; Geminos's branches of, 43-48, 245-49; and perceptible things, 44; pure, 43; Pythagorean division of, 245 mean motion, 96, 228; of the M o o n , 91 mechanics, 44, 46, 246, 248 mechanisms, geared, 4 7 n l 0 4 medicine, 246 Menestratus, 86 Mercury, 119 meteoroscopy, 48, 249 meteoroskopion, 48 Metón, 87-88, 2 2 4 n l 9 , 232 Middle Books, 103

322

·

Index

military camps, 246 military engineering, 248 M i l k y Way, 28, 31fig, 152, 159-60 month, 84, 92; anomalistic, 93-94, 96, 227-29; Babylonian, 228; civil, 176; double, 176; draconitic, 94; embolismic, 180-84; full, 181, 87, 182, 184; hollow, 87, 181-82, 184; lunar, 181-82; sidereal, 92, 94; synodic, 83, 92-94, 175-76, 227, 261; tropical, 93-94 monthly period (= synodic month), 175 M o o n , 39, 117, 128, 176, 198, 220; angular diameter of, 192; eastward motion of, 196; its eclipse, 177, 191-94; its four phases, 186-87; illumination of, 186-88; non-uniform motion of, 227. See also lunar theory; lunisolar cycles motion, irregular, 254; of planets, 198; uniform and circular, 49; in width, 197nn6 and 7 music, 108, 125n4, 245 names, convention for Greek, 108 Nasïr al-DIn al-Tusï, 103 nativities, 125 Nauteles, 86 Neugebauer, O . , 19, 24, 261 nodes, lunar, 94 numbers, 109, 247 numerals, Greek, 35-36 observation, 228 occultation, 91, 120n21 Ocean, 214, 2 1 5 n l 5 octaetëris, 82-87, 180-83 octave, division of, 45 Odysseus, 214 oikëseis, 155n20 oikoumenê, 210n3; dimensions of, 209; divisions of, 211; in On the Cosmos, 2 1 7 n l ; our, 155; whole, 150, 156 opposition, 125, 126n5, 127 optics, 4 4 - 4 5 , 246, 248, 252 oracles, 176 ortive amplitude, 169 Osiris, 18, 61 Oxyrhynchus, 100 Pappos, 190n4 parabola, 75 parallax, 193

parallel circles, 131, 136, 149-52; coincidences of, 154; their division by horizon, 152-54; five main, 152; order of, 154; relative sizes of, 154-56; separations of, 156 parapëgma, 2, 3, 12, 59, 89, 90, 1 7 8 n l 0 , 225, 231; authorities for Geminos's, 275-79; dependence of on the klima, 221; of Dëmokritos, 275; of Dositheus, 275; of Eudoxos, 275, 2 7 7 - 7 8 , 285; of Euktëmôn, 5 9 n l 4 5 , 272, 275, 277, 285; evolution of, 277-79; of Geminos, 58, 275-89; of Kallippos, 275, 2 7 7 - 7 8 , 286-88; of Leptinës, 276; of Metón, 275; from Miletus, 60fig, 226n23, 272, 277, 2 8 4 - 8 5 ; in M U L . A P I N , 5 9 n l 4 4 , 9 0 n l 8 7 ; in P. Hibeh i 27, 61, 62fig, 79; of Ptolemy, 286-88; unscientific nature of, 218, 221 Parmenidës, 186n2, 2 0 8 n l pedagogy, 86, 98 perception, 162 perigee, lunar, 93 perioikoi, 210 periskian zones, 2 0 8 n l Perseus, 139 Petau, D . , 101; on date of Geminos, 18 Phaëthôn, 120, 224 Phainón, 119 phases of the fixed stars, 7, 5 8 - 6 3 , 1 3 9 n l 2 , 219n6, 2 3 4 n l 4 ; difficulties in Geminos's text concerning, 203n13, 265-67; kinds of, 200-203; order of, 68-70, 203; propositions concerning, 203-4; synopsis of those in Geminos's parapëgma, 279-83; terminology for, 70-72; theory of, 58-70; true, 63-65, 224; visibility rule for, 67; visible, 65-67, 231 phases of the M o o n , 186-88; explanation of, 187; intervals between, 177 pbasis, 200n2 phenomena: accounting for, 118, 185; dependence of on locality, 213; Geminos's use of, 9; in Greek astronomy, 5-8 Philippos (or Philip of Opus), 88, 119n20, 183,276 Phitorois, Feast of, 61 Phôsphoros, 120, 224 physicist, 54 physics, 252, 254

Index Pindar, 218 planets, 117-18; Babylonian theories of, 100; motion of, 198-99; names for, 119n20; order of, 120n21; tropical peri­ ods of, 118, 120n21 Plato, 44, 49; on dance of planets, 255n20; Epinomis, 119, 248; on musical experi­ ment, 248n25; on saving the phenom­ ena, 5; Timaeus, 249 Pliny, Natural History, 87, 237nn27 and 28; on effects of D o g Star, 2 2 2 n l 4 Plutarch, 51, 1 7 9 n l 0 pole, 146; elevation or altitude of, 165, 249 Polybios, 2 0 9 n l , 216 Pontos, 221 Poseidónios: on circumference of Earth, 1 4 0 n l 5 , 2 0 8 n l , 251; as possible source for Geminos, 4, 13, 23-27; on zones, 208nl precession, 93 Priscianus Lydus, 251 Proklos, 3, 43, 243 Ptolemaios II Euergetës, 178n9 Ptolemies, 214 Ptolemy, 57, 59, 77, 94, 104, 127, 2 2 1 n l 2 , 227, 278, 284; on Aithiopians, 2 1 5 n l 3 ; Almagest, 103; On the Analemma, 48; his celestial globe, 31n75; on heliacal risings, 200n2; introduces equant point, 50; his nomenclature for star phases, 68; on ortive amplitude, 169; Phaseis, 72, 2 2 1 n l 2 , 2 2 4 n l 9 , 226n23; Planetary Hypotheses, 198n9; pseudographia, 104; Tetrabiblos, 125, 126n5, 127; on types of stars, 207n8 Pyroëis, 120, 224 Pythagoras, 2 0 8 n l Pythagoreans, 44, 49, 117, 246 Pytheas, 162 quadrivium, 9, 2 4 6 n l 3 quantity, 245 quartile aspect, 125, 129-32 Quellenforschung, 23 rainbow, explanation of, 251 ratio, 45 realism, 51 refraction, 45 regions, geographical, 210 Rehm, Α., 284

·

323

Reinhardt, K . , 24 Rhodes, 15-16, 221, 224; klima, horizon or latitude of, 16, 153, 156n25, 212n8 risings, diurnal, 200 risings and settings, heliacal. See phases of the fixed stars risings and settings, simultaneous, 205-6 rising times of zodiac signs, 73-77; 170-72, 261; a common error concern­ ing, 172-73; their connection with set­ ting times, 173-74 Rome, 171n7, 176, 221; klima of, 259 sacrifices, 176, 217; to gods, 177-78, 181 Sambursky, S., 125n4 saros, 95-96 Saturn, 119 saving the phenomena, 51, 254 scale, musical, 45 scenography, 45, 244, 248 Schaefer, B., 28n71 seasons, 114, 116n9, 1 1 6 n l l second differences, 75 Seleucid period, 73, 78 senses, 152 setting, diurnal, 200 setting, heliacal. See phases of the fixed stars sexagesimal system, 228 Sezgin, E , 103n213 signifying, 231, 233-36, 286 signs (signifiers), and weather, 277; vs. causes, 219 signs (of the zodiac): circle of, 113; equipotent, 1 2 5 n l ; their inclination to the horizon, 170-71; names of, 113; in opposition, 125-26; preceding and fol­ lowing, 113; in quartile, 129-32; rising position of on horizon, 169-70, 219; ris­ ing times of, 170-71; in syzygy, 132-36, 168, 1 7 2 n l 0 ; in trine, 127-29; use of Claws (= Libra), 117, 172-73 Simplikios, 4, 23, 250 Sirius, 69, 72, 139, 2 2 2 - 2 5 , 2 3 2 - 3 3 , 236, 239, 277-78; its brightness, 223; its lack of inherent power, 222ff skaphê, 34 smaller than similar, 267 solar circle, eccentricity of, 124 solar theories, 54-57

324

·

Index

Solon, 1 8 0 n l 7 solstice, 37, 1 7 9 n l 0 ; changes in day lengths near; 166; extreme temperatures at, 166; Greek vs. Chaldean convention practice for, 115; length of the days and nights at, 37; its location perceivable only by reason, 167; summer, 36, 114, 151, 187; winter, 36, 38, 115, 151, 178-79, 186, 2 3 6 , 2 8 9 Solunto, 32 soul, 246 sphairopoua (= spherical construction or arrangement, or sphere making), 47, 5 1 - 5 3 , 198-99, 206, 2 1 4 - 1 5 , 249 sphere, armillary. See armillary sphere sphere, celestial, 52, 119 sphere making. See sphairopoua Sphere of Proklos, 103 spherical construction. See sphairopoua spherics, 245 squares, 129 Stabiae, 32 stars: always visible type, 69, 206n6; be­ havior relative to horizon, 206; blue, 271; convention for names of, 109; cul­ mination of, 129n9; distances to, 118; dock-pathed, 69, 68, 206n5; doubly visi­ ble, 206; fixed, 169; Geminos's list of, 137-44; kolobodiexodos, 207n8; magni­ tudes of, 286; as marks of seasons, 277; motion of, 195, 205; night-escapers, 52, 207n8; night-pathed, 68; their phases, 5 8 - 9 , 63, 65, 70, 90, 231, 2 4 6 n l 7 , 277; sphericity of, 253; as weather signs, 220; Aëtos, 139, 236n25; Antares, 235; Arc­ turus, 61, 69, 71, 138, 210n8, 206, 233n7, 2 3 4 - 3 5 , 238, 240, 277-78, 285; Asses, 137; Canopus, 139; D o g (Sirius), 69, 72, 139, 222-25, 2 3 2 - 3 3 , 236, 239, 277-78; Fomalhaut, 72; Gemma, 72; Goat (Capella), 139, 234, 236, 239-40, 277, 285; Gorgon's Head, 139; H a r b i n ­ ger of Vintage (Vindemiatrix), 233-34, 238, 277; Heart of the L i o n (Regulus), 137; Hyades, 72, 137, 235, 2 3 9 - 0 , 277, 285; Kids, 139, 164, 234, 277; Knot, 138, 239; Lyra, 139; Pleiades, 59, 69, 72, 9 0 - 9 1 , 137, 141, 219, 2 3 4 - 3 5 , 239-40, 270, 277-78; Procyon, 139, 233n7; Propus, 137; Sickle, 139; Wheat Ear, 138, 234

stations of planets, 118, 198 stereometry, 247 Stilbón, 120 Stoicism, 10, 25n68, 57, 125n4, 2 1 7 n l , 220nn 9 and 11; curriculum of, 2 4 - 2 5 ; Geminos and, 2 3 - 2 7 Strabo: on astronomy instruction, 27-28; on inquiry into causes, 57; on zones, 208nl Sun, 39, 117, 120, 176, 198, 216, 220, 222; apogee of, 124; eclipses of, 189-90; fiery nature of, 251; motion of, 114, 124, 195; symbol for, 106 sundials, 25, 27, 34-38, 1 3 4 n l 8 , 136, 166, 168, 179, 213; conical, 35; Geminos's pedagogical use of, 37-38; Geminos's vocabulary for, 34; plane, 35; portable, 34n78; spherical, 34; and variation in day length, 38n81 swallows, 238 sympathy, 24, 25, 125-26, 129, 220 synoikoi, 210 Syracuse, siege of, 248 System A for rising times, 74 System Β of Babylonian lunar theory, 98-99, 229n9 System Β for rising times, 174n162 syzygy, 37, 125, 132-36 tactics, 246 Tannery, P., 23, 24, 244n6 Tebtunis, 100 Theodosios of Bithynia, 7 Theokritos, 238n30 Theón of Alexandria, his revision of Euclid's Optics, 45 Theón of Smyrna, Mathematical Knowl­ edge Useful for Reading Plato, 9-10, 44, 50, 1 1 6 n l l , 198n9; on partially helio­ centric planetary system, 121n21; on physical explanation, 58; on planetary phenomena, 8 n l 7 ; Platonism i n , 10; on terms for star phases, 72 Theophrastos, On Winds, 237n27 time: degrees of, 77; sidereal, 77 Timocharis, 91 Toledo, 104 Toomer, G . , 90-91 Tower of Winds, 237n28 translation, approach to, 107 triakas, 177

Index triangles, astrological, 127-28 trigonometry, 74, 174 trine aspect, 125, 127-29 tropic: origin of name for, 151; summer, 29, 149, 156; winter, 149, 156 twilight, astronomical, 67 Tybi, 61, 72 uranography, 6 van der Waerden, B., 2 8 9 n l 2 Variae Collectiones, 244 visibility rule, 265 Vitruvius, 46, 47-48, 196n3 water clocks, 248n28 weather: not caused by stars, 217-26; prediction of, 61, 218, 225, 278, 281, 286,288 winds, 36, 240; bird, 61, 238; circle of, 37fig; etesian, 232; as signs of seasons,

·

325

61; and signs of zodiac, 127n6; Zephyros, 237n28 wonder-working, 248 world maps, circular, 211 Xenophanës, 220n9 year: definition of, 114; Egyptian, 178; lunar, 176, 181; solar, 84, 176; tropical, 83 Zeus, 120,218 zigzag function, 99-100 zodiac: Greek and Chaldean divisions of, 115; twelfth part of, 113, 135; where it rises, 38, 169-70 zodiacal signs, aspects of, 125-36 zodiac circle, 122, 131, 166, 169 zones, 208, 2 0 8 n l ; dimensions of, 211-12; frigid, 164, 208; southern, 213; torrid, 214,216