Under One Sky: Astronomy and Mathematics in the Ancient Near East

Alter Orient und Altes Testament Veròffentlichungen zur Kultur und Geschichte des Alten Orients und des Alten Testaments...

Author: John M. Steele | Annette Imhausen

44 downloads 851 Views 12MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Alter Orient und Altes Testament Veròffentlichungen zur Kultur und Geschichte des Alten Orients und des Alten Testaments

Band 297

Herausgeber

Manfried Dietrich • Oswald Loretz

Beratergremium R. Albertz • J. Bretschneider • St. Maul K.A. Metzler • H. Neumann • U. Riiterswòrden W. Sallaberger • G. Selz • W. Zwickel

2002 Ugarit-Verlag Miinster

Under One Sky Astronomy and Mathematics in the Ancient Near East edited by

John M. Steele - Annette Imhausen

2002 Ugarit-Verlag Munster

U n d e r O n e S k y . A s t r o n o m y and M a t h e m a t i c s in the A n c i e n t N e a r East, edited by J o h n M . Steele - A n n e t t e I m h a u s e n Alter O r i e n t u n d A l t e s T e s t a m e n t B d . 2 9 7

© 2002 Ugarit-Verlag, Munster (www.ugarit-verlag.de) Alle Rechte vorbehalten All rights preserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photo-copying, recording, or otherwise, without the prior permission of the publisher.

Herstellung: Hanf Buch und Mediendruck GmbH, Darmstadt Printed in Germany ISBN 3-934628-26-5 Printed on acid-free paper

Table of Contents

Preface

vii

O n C o l u m n s H and J in Babylonian Lunar Theory of System B Asger Aaboe

1

Predictions of L u n a r P h e n o m e n a in Babylonian A s t r o n o m y Lis Brack-Bemsen

5

T r e a t m e n t s of A n n u a l P h e n o m e n a in Cuneiform Sources John P. Britton

21

History of the heleq Leo Depuydt

79

M e a s u r i n g Egyptian Statues Friedhelm Hoffmann

109

H o w to E d u c a t e a K a p o or Reflections on the A b s e n c e of a Culture of Mathematical P r o b l e m s in Ur III Jens Hoyrup

121

T h e A l g o r i t h m i c Structure o f the Egyptian Mathematical P r o b l e m T e x t s Annette Imhausen

147

Babylonian Lunar T h e o r y in R o m a n Egypt. T w o N e w Texts Alexander Jones

167

Early Babylonian O b s e r v a t i o n s of Saturn: Astronomical C o n s i d e r a t i o n s Teije de Jong

175

T h e Eye of H o r u s and the Planet V e n u s : A s t r o n o m i c a l and Mythological References RolfKrauss

193

T h e Historicity Question in M e s o p o t a m i a n Divination Daryn Lehoux

209

G n o s i s and Astrology. ' B o o k I V ' of the Pistis Sophia Alexandra von Lieven

223

Ration C o m p u t a t i o n s at Fara: Multiplication or Repeated Addition Duncan J. Melville

237

S q u a r e T a b l e t s in the Yale Babylonian Collection Karen R. Nemet-Nejat

253

A G o d d e s s Rising 10,000 Cubits into the Air...Or Only O n e Cubit, O n e Finger? Joachim F. Quack

283

Aristarchos and the ' B a b y l o n i a n ' Month Dennis Rawlins

295

C l o s i n g the Eye o f H o r u s JimRitter

297

M o r e than M e t r o l o g y : M a t h e m a t i c s Education in an Old Babylonian Scribal School Eleanor Robson

325

A Study o f Babylonian N o r m a l - S t a r A l m a n a c s and Observational T e x t s Norbert A. Roughton

367

Egyptian Festival Dating and the M o o n Anthony Spa linger

379

A S i m p l e Function for the Length of the Saros in Babylonian A s t r o n o m y JohnM. Steele

405

T h e Earliest Datable Observation of the Aurora Borealis F. Richard Stephenson and David M. Willis

421

T h e ' T r a n s i t Star C l o c k ' from the Book of Nut Sarah Symons

429

EnQma A n u Enlil T a b l e t s 1-13 Lorenzo Verderame

447

T h e Role o f A s t r o n o m i c a l T e c h n i q u e s in Ancient Egyptian C h r o n o l o g y : T h e Use o f Lunar M o n t h Lengths in Absolute Dating Ronald A. Wells

459

Signs from the Sky, Signs from the Earth: The D i v i n e r ' s Manual Revisited Clemency Williams

473

Indices

487

Preface T h i s v o l u m e has its origin in a series of discussions between the t w o editors that t o o k p l a c e over dinner in a variety of Berlin restaurants during a w o r k s h o p on the " M a t e r i a l Culture o f C a l c u l a t i o n " held at the M a x Planck Institute for the History of Science in 1999. T h e s e conversations quickly led us to realise that although a s t r o n o m y and m a t h e m a t i c s are, and have been since antiquity, related disciplines for o n e c a n n o t d o a s t r o n o m y without mathematics, and whilst the reverse is not true, a s t r o n o m y did form one of the primary motivations for the d e v e l o p m e n t of m a t h e m a t i c s - historians of ancient mathematics and ancient a s t r o n o m y have often, particularly in recent years, undertaken their research in isolation from o n e another. Similarly, historians o f the sciences of M e s o p o t a m i a and those w o r k i n g on science in Egypt h a v e only rarely interacted. W e felt it w o u l d be of benefit to all researchers w o r k i n g on the history of Ancient N e a r Eastern science if there was a forum to discuss the results o f recent investigations across the w h o l e of the exact sciences in M e s o p o t a m i a and Egypt. This w o u l d not only allow the results of these investigations to be presented but w o u l d also p r o v i d e an opportunity for c o m p a r i s o n s to be m a d e between the different a p p r o a c h e s used by researchers in the different fields, hopefully inspiring n e w directions of research. W h e n w e a p p r o a c h e d him, Christopher W a l k e r generously agreed to help organise a conference based upon this t h e m e , and the resulting m e e t i n g took place at the British M u s e u m on 2 5 - 2 7 June 2001 under the title " U n d e r O n e Sky: A s t r o n o m y and Mathematics in the Ancient N e a r East". T h i s collection contains papers drawn from a m o n g those presented at the conference. W e h a v e e n c o u r a g e d the authors to reflect upon the issues raised at the m e e t i n g and to revise and expand their contributions into full-length research p a p e r s . A s such this v o l u m e not only contains several significant research contributions but when taken as a w h o l e can also, w e h o p e , p r o v i d e a snapshot of the r a n g e of research currently being undertaken within the history of m a t h e m a t i c s and a s t r o n o m y in Egypt and M e s o p o t a m i a . It therefore illustrates the historiographical d e v e l o p m e n t o f these four areas o f research, and m a y point the w a y t o w a r d s future research m e t h o d o l o g i e s . Acknowledgements: T h e U n d e r O n e Sky conference and this resulting v o l u m e w o u l d not have been possible without the help and support of a n u m b e r of individuals and organisations. First and foremost, w e would like to thank the British M u s e u m for hosting the meeting, and in particular the D e p a r t m e n t of the Ancient N e a r East and its staff for their help in m a k i n g the conference run smoothly. T h e British A c a d e m y generously provided financial support for the meeting. D u n c a n Melville was present during our initial discussions and provided frequent e n c o u r a g e m e n t . Alice Slotsky, A l e x a n d e r Jones, H e r m a n n H u n g e r and J o h n Britton all h e l p e d by either chairing sessions or undertaking the unenviable task of reading p a p e r s by colleagues w h o were unable to attend in person. Finally, and m o s t importantly, w e w o u l d like to express our deepest thanks to C h r i s t o p h e r W a l k e r w h o not only u n d e r t o o k the practical organisation at the British M u s e u m but also suggested several speakers. Without his help and e n c o u r a g e m e n t the conference could never h a v e taken place. John M. Steele and A n n e t t e Imhausen July 2 0 0 2

On Columns H and J in Babylonian Lunar Theory of System B Asger Aaboe,

New

Haven

W h a t follows is in the nature of a footnote to O. N e u g e b a u e r ' s Astronomical Cuneiform Texts ( A C T ) as is so m u c h of the w o r k o n B a b y l o n i a n a s t r o n o m y d o n e after its publication. M o r e precisely, I shall correct t w o statements in it. S o m e scholars take offence w h e n p r o v e n w r o n g , but N e u g e b a u e r w a s n o t a m o n g t h e m . Indeed, h e t o o k it as a sign that h e h a d b e e n taken seriously if o n e p o i n t e d out a m i s t a k e in his w o r k to him, a n d anyway, as h e often said, the only sure w a y of avoiding all error is to d o nothing at all. B a b y l o n i a n lunar t h e o r y deals almost exclusively with the m o o n at syzygy, i.e. w h e n it is either at conjimction or o p p o s i t i o n ( n e w or full) and o n the d a y s j u s t before a n d after these events. T h e first tasks in b o t h Systems A a n d B are to calculate w h e r e a n d w h e n syzygy occurs. F o r the sake of simplicity, I shall disregard oppositions, a n d w e n o w b e g i n b y finding the c o m m o n longitude of sim a n d m o o n at a s e q u e n c e o f c o n s e c u t i v e conjunctions. H e r e it turns out that the underlying s c h e m e s d e p e n d o n solar a n o m a l y only a n d so h a v e one year as their p e r i o d s - the B a b y l o n i a n a s t r o n o m e r s did not distinguish b e t w e e n sidereal, tropical, and anomalistic years. T h a t w e m a y ignore limar a n o m a l y w h e n w e w i s h to find a p p r o x i m a t e l y w h e r e a conjunction takes p l a c e c a n b e established on b o t h theoretical a n d heuristic g r o u n d s , ' b u t its plausibility is plain: If w e at first calculate the longitudes o f a s e q u e n c e o f conjunctions a s s u m i n g constant lunar velocity - which, in fact, w e d o in b o t h S y s t e m A a n d B - these positions will b e v e r y close to the o n e s of the actual conjunctions, for w h a t e v e r c h a n g e a variable lunar velocity m a y c a u s e in the time intervals b e t w e e n conjunctions, it will not allow the sun to m o v e v e r y far (it is about 13 times slower than the m o o n ) . Indeed, in a situation like this, it is the s l o w e r b o d y that has the greater s a y in w h e r e c o i n c i d e n c e occurs. It is far otherwise w h e n w e are c o n c e r n e d with the time of conjunction. In b o t h systems the time interval from one conjunction to the next (the synodic m o n t h ) is calculated as: A t = 29** + G + J w h e r e G d e p e n d s solely o n limar anomaly, a n d J solely o n solar a n o m a l y , so G and J p r e s u p p o s e constant solar and lunar velocity, respectively ( G a n d J are m e a s u r e d in time-degrees, l** = 3 6 0 ° = 6,0°).

See, for example, BERNSEN ( 1 9 6 9 ) and AABOE and HENDERSON ( 1 9 7 5 ) .

2

A. Aaboe

In lunar S y s t e m A -

so called b e c a u s e C o l u m n B , the longitude c o l u m n , is

derived from a ' s t e p function', a device characteristic of p l a n e t a r y S y s t e m A

-

C o l u m n G is d e r i v e d in a c u n n i n g fashion from the famous, or n o t o r i o u s , C o l u m n that h a s b e e n , and still is the object o f m u c h study, b u t w e h a v e m a d e substantial p r o g r e s s since the p u b l i c a t i o n of A C T . Colimin J is n o t strictly a step function in the true S y s t e m A sense, b u t is h a s s o m e o f its characteristics a n d c a n b e d e r i v e d from the implicit AÀ-column. I d i d n o t get a perfect click in m y a t t e m p t at s u c h a derivation, b u t J o h n B r i t t o n did, taking a different tack, a n d I leave it to h i m to tell a b o u t it. T h e p o i n t here is that in this instance, as e l s e w h e r e , S y s t e m A s h o w s internal consistency. It is quite different

in S y s t e m B , and I a m here c o n c e r n e d to s h o w

the

incompatibility of the s c h e m e for finding longitudes with C o l u m n J. Let m e r e m i n d you that in S y s t e m B , C o l u m n A ( m o n t h l y p r o g r e s s in longitude of the s y z y g y in degrees) a n d C o l u m n F (daily p r o g r e s s of the m o o n ) are o r d i n a r y z i g - z a g functions with appropriate p e r i o d s , a n d so is C o l u m n G. Its p e r i o d is the a n o m a l i s t i c m o n t h , as it s h o u l d b e , and its m a x i m a and m i n i m a c o r r e s p o n d very nearly to the m i n i m a a n d m a x i m a , respectively o f C o l u m n F. T h i s , too, is as it s h o u l d b e , for if w e a s s u m e that the m o n t h l y p r o g r e s s in longitude of the conjunction is constant, it s h o u l d t h e n be fiA, the m e a n v a l u e o f C o l u m n A, a n d w e might at first e x p e c t 2 9 ^ , G = (HA±360!) F w h e r e F is the variable lunar velocity, e x p r e s s e d as a zig-zag function. H o w e v e r , the reciprocal of such a function, t h o u g h still p e r i o d i c of the s a m e p e r i o d , is n o t a nice function at all. It is, then, v e r y natural that the texts p r e s e n t C o l u m n G as a zig-zag function, o n e of the standard devices for representing a p e r i o d i c function. B y the way, it is G ' s m e a n value that implies the famous value for the length o f the m e a n synodic m o n t h , 29;31,50,8,20'', u s e d b y H i p p a r c h u s a n d P t o l e m y . S o m u c h for C o l u m n G. C o l u m n J is a m o r e c o m p l i c a t e d affair. First w e m u s t c o m p u t e C o l u m n H as a zig-zag function with p a r a m e t e r s MH=21° mH

=-21°

dH

=6;47,30°^'"

(the texts actually list o n l y the absolute values o f this function, so we m u s t s u p p l y the sign). T h e p e r i o d of H (with sign) is 12;22,8"', the c a n o n i c a l v a l u e o f the year from S y s t e m A, found n o w h e r e else in lunar S y s t e m B e x c e p t h e r e in C o l u m n s H and J. C o l u m n J is the s u m o f C o l u m n H, b u t reflected in its o w n extrema: Mj

=32;28,6°

mj

= - Mj

It is a nice m a t h e m a t i c a l p r o b l e m to assure that H w 0 always c o r r e s p o n d s t o an e x t r e m u m o f J, a n d that a n extremimi o f H always c o r r e s p o n d s to J « 0, a n d further

On Columns H and J in Babylonian Lunar Theory of System B

3

that the m e a n p e r i o d of J is one year. N e u g e b a u e r discusses this p r o b l e m in A C T and again, in great detail, in H A M A , a n d I n e e d not take it u p here. N e u g e b a u e r says about C o l u m n J: " B e c a u s e of this character of A , c o l u m n J cannot b e a linear zigzag fimction but m u s t b e a s e q u e n c e of s e c o n d order"^ and again: " T h e essential p r o g r e s s of S y s t e m B b e y o n d S y s t e m A lies in the mastering of the m a t h e m a t i c a l difficulties w h i c h are the c o n s e q u e n c e of the consistent u s e of a solar velocity that varies from m o n t h to m o n t h instead of the simpler m o d e l o f S y s t e m A."^ W e l l , s o m e t i m e s e v e n g o o d old H o m e r n o d d e d , as the p o e t H o r a c e said. W h a t N e u g e b a u e r says here is w r o n g , a n d to s h o w that is the p o i n t of m y talk. First, the effect o f a variable lunar velocity on the length of the m o n t h has b e e n t a k e n care of b y C o l u m n G, so in w h a t follows w e a s s u m e that lunar velocity is constant, fip- A s w e saw, the effect o f solar anomaly, a n d it alone, w a s to p l a c e the conjunctions u n e v e n l y in the ecliptic, a n d w e shall n o w see h o w this u n e v e i m e s s affects the time interval from one conjunction to the next, i.e., to d e r i v e w h a t C o l u m n J should b e . T h e m o n t h l y p r o g r e s s in longitude o f the conjunction is g i v e n in C o l u m n A, a zig-zag function with: MA

= 3 0 ; 1,59°

M A = 2 8 ; 10,39,40° dA

=0;18°^"

W h e n A is o n a n ascending b r a n c h , the m o o n h a s 0;18° farther to travel e a c h m o n t h to c a t c h up with the sun, a n d likewise w h e n A is o n a d e s c e n d i n g b r a n c h . C o l u m n J then o u g h t to b e a zig-zag function with 0;18'

)/m

as its difference a n d of amplitude

w h e r e AA is C o l u m n A ' s amplitude. If w e n o w u s e the u n a b b r e v i a t e d value fip = 1 3 ; 10,35°^'' w e should h a v e :

dj=

' 13;I0,35'

= 0 ; 1 , 2 1 , 5 7 , 5 3 ...'''"' = 8;11,47,...°''"

and A, =

= 0;8,26,55,.. 13;10,35°^'*

^

ACT, p. 78.

^

ACT, p. 41.

= 50;41,30,...

4

A. Aaboe

Since w e w a n t the m e a n value to b e 0 for the sake o f G, w e should t h e n h a v e for the zig-zag function's extrema: M j = - m j = 2 5 ; 2 0 , 4 5 , . . . °. In sum, if w e d r a w the c o n s e q u e n c e s o f the m o n t h l y solar p r o g r e s s in longitude b e i n g represented as a zig-zag function, we find, c o n t r a r y to N e u g e b a u e r ' s claim, that also C o l u m n J o u g h t to b e a zig-zag function, a n d with the a b o v e p a r a m e t e r s . So whatever the m o t i v a t i o n for the complicated C o l u m n s H a n d J m a y b e - delight in m a t h e m a t i c a l complexity'* and dissatisfaction with the p a r a m e t e r s d e r i v e d a b o v e are a m o n g the p o s s i b l e candidates - it is surely n o t to b e found in the structure of Column A.

Abbreviations ACT

= NEUGEBAUER (1955)

H A M A = NEUGEBAUER (1975)

References A A B O E , Asger. 2 0 0 1 . Episodes Springer.

from

the Early

History

of Astronomy.

N e w York:

— a n d H E N D E R S O N , Janice A . 1975. " T h e B a b y l o n i a n T h e o r y of L u n a r Latitude and Eclipses A c c o r d i n g to S y s t e m A " . Archives Internationales d'Histoire des Sciences 25, No. 9 7 : 1 8 1 - 2 2 2 . B E R N S E N , Lis. 1969. " O n the Construction of C o l u m n B in S y s t e m A o f the A s t r o n o m i c a l C u n e i f o r m T e x t s " . Centaurus 14: 2 3 - 2 8 . N E U G E B A U E R , O t t o . 1955. Astronomical Cuneiform Texts. 3 vols. L o n d o n : L u n d H u m p h r i e s (Reprint N e w Y o r k : Springer 1983). — 1 9 7 5 . A History Springer.

of Ancient

Mathematical

Astronomy.

3 vols. N e w

York:

An extreme example of a mathematical scheme whose complexity far exceeds what could possibly be of astronomical interest is offered by Peter Huber's restoration of ACT, nos. 654 and 655 (AABOE (2001), p. 56). Here dated, daily positions of Jupiter are given to minutes, seconds, and thirds of arc, according to a scheme of constant third differences. Jupiter's computed travel through a loop around opposition is wonderfully smooth, but the retrograde arc is a full degree too short, as modem computations show. If this defect was tolerable, there could surely not have been a practical need for such refined positions.

Predictions of Lunar Phenomena in Babylonian Astronomy Lis Brack-Bernsen,

Regensburg

In this p a p e r I shall mainly b e c o n c e r n e d with predictions of the length o f the B a b y l o n i a n lunar m o n t h . T h e r e a s o n for this choice is the fact that in the important text T U 11 eight different m e t h o d s for predicting "full" or " h o l l o w " m o n t h s are collected. T h i s m e a n s that w e h a v e in this text a substantial a m o i m t of material to investigate in addition to w h a t c a n b e found o n the topic in other texts. T h e tablet A O 6 4 5 5 (hereafter referred to as T U 11) is perfectly p r e s e r v e d and was p u b l i s h e d in 1922 in an excellent c o p y b y F . T h u r e a u - D a n g i n as N o . 11 in Tablettes d'Uruk. It contains a mixture of primitive a n d a d v a n c e d a s t r o n o m i c a l rules alongside s o m e astrological p a s s a g e s . T h e tablet T U 11 is a c o p y written t o w a r d s the e n d of the 3rd century B . C . , a n d contains quite a nimiber of errors. Until n o w only short sections h a v e b e e n translated and c o m m e n t e d o n . ' A c o m p l e t e edition b y L. B r a c k - B e m s e n a n d H . H u n g e r will s o o n a p p e a r in SCIAMVS 3 . T h e r e a d e r is referred to that edition for a translation a n d interpretation of the text, a n d for a detailed discussion of its significance to the history of B a b y l o n i a n a s t r o n o m y . In the p r e s e n t p a p e r I shall only give a n o v e r v i e w of the a s t r o n o m i c a l content of T U 11 a n d then p r e s e n t all the rules w e k n o w o f for predicting the length o f the B a b y l o n i a n m o n t h . M o s t of these rules are written o n T U 1 1 , a n d at first glance s o m e o f t h e m s e e m quite strange. A r e they j u s t inventions a n d speculations b y s o m e Seleucid scribe, or are they a collection of rules w h i c h h a d really b e e n u s e d b y earlier B a b y l o n i a n a s t r o n o m e r s ? T h e present p a p e r will try to p r o v i d e a n a n s w e r to this question. If the text j u s t reflects the speculations of o n e p e r s o n , then it only tells us h o w he thought about the p r o b l e m , and the kind o f w a y s he t h o u g h t it might b e solved. B u t if it is a genuine collection of m e t h o d s that w e r e u s e d t h e n it gives us very fruitful hints a n d ideas a b o u t c o n c e p t s a n d m e t h o d s u s e d in intermediate astronomy.^ F u r t h e r m o r e , it w o u l d p r o v i d e a solid basis for efforts to reconstruct the d e v e l o p m e n t o f B a b y l o n i a n lunar theory.^ Since a great part o f m y discussion is b a s e d o n tablets from the cuneiform collection o f the British M u s e u m , I a m h a p p y to p r e s e n t t h e m here. A t this stage I w o u l d like to express m y w a r m e s t thanks to Irving Finkel a n d Christopher W a l k e r for their search for parallel texts a n d for d r a w i n g m y '

NEUGEBAUER ( 1947)

and VAN DER WAERDEN ( 1949,1951 ).

^ In LBAT Sachs classified some tablets as containing Intermediate Astronomy. He defined the term as follows: "This term refers to stages later than MUL.APIN and earlier than ACT. The boundaries in both directions are not sharp". ^ Another possible link between the non-mathematical astronomical texts and the ACT methods, is provided by John Steele in a tablet published in this volume: "A Simple Function for the Length of the Saros in Babylonian Astronomy".

6

L. Brack-Bemsen

attention t o the texts they identified, and to H e r m a n n H u n g e r a n d C h r i s t o p h e r W a l k e r for m a k i n g their translations o f the texts available to m e . W i t h o u t these translations I w o u l d not h a v e b e e n able to w o r k o n this topic in the first p l a c e .

Some Useful Preliminaries T h e B a b y l o n i a n m o n t h b e g a n on the evening after n e w m o o n (conjunction) on w h i c h the thin crescent w a s visible for the first t i m e . T h i s e v e n t of c o u r s e also indicated the e n d o f the current (old) m o n t h .

F i g u r e 1 : T h e situation at the western h o r i z o n o n the e v e n i n g w h e n the n e w crescent is visible for the first time after conjunction, a n n o i m c i n g the b e g i i m i n g of m o n t h I. T h e d a s h e d line depicts the ecliptic, the p a t h a l o n g w h i c h sim a n d m o o n m o v e . T h e direction o f m o t i o n is indicated b y the arrow, a n d O, the sun, s h o w s w h e r e t h e conjimction t o o k p l a c e s o m e 1 1/2 d a y s earlier. T h e m o o n , m o v i n g faster than the sun, h a s o n this evening r e a c h e d a position so far fi-om the sun, that it will b e visible at sunset. O n the p r e c e d i n g e v e n i n g it m i g h t h a v e b e e n in Position • at sunset, still too near to the sun to b e seen. T h e thick line is the equator, it s h o w s the direction along w h i c h all luminaries set. T h e time NA^j fi-om sunset imtil m o o n s e t is m e a s u r e d b y the arc o f the equator, w h i c h sets simultaneously with arc (OC ). T h e B a b y l o n i a n m o n t h h a d 2 9 or 3 0 days: if the m o o n was a l r e a d y visible at the begiiming o f day 3 0 in a month, this day 3 0 w a s rejected, w h i c h m e a n t that the m o n t h only h a d 2 9 d a y s . T h a t m o n t h w a s called G U R (rejected) w h i c h is n o r m a l l y translated as " h o l l o w " . W h e n the m o o n was still not visible after sunset o n d a y 3 0 , this d a y w a s confirmed as the last o f the (long) m o n t h . A m o n t h o f 3 0 d a y s w a s called G I N (confirmed), n o r m a l l y translated as "fiiU". O n the evening w h e n the n e w crescent indicated the b e g i n n i n g o f the n e w month, the time b e t w e e n simset a n d the setting o f the crescent w a s m e a s u r e d . In

Predictions of Lunar Phenomena in Babylonian Astronomy

7

the Astrotiomical Diaries'* this time interval w a s r e c o r d e d together w i t h the length of the m o n t h w h i c h h a d j u s t p a s s e d as follows: If the crescent w a s s e e n (and h e n c e A^^^v m e a s u r e d ) on d a y 3 0 of the last m o n t h M - 1 , M o n t h M w o u l d b e g i n w i t h " 3 0 NAÌ^ w h e r e a s in the case of the m o o n b e i n g s e e n only the d a y after day 3 0 , the n e w m o n t h w o u l d start with a statement like: " M o n t h M, \ NA^ ..." H e n c e w e see that 3 0 a n d 1 w e r e also u s e d as a n indicator for the h o l l o w a n d full m o n t h . S o m e t i m e s , N e u g e b a u e r translates 3 0 and 1 as "post h o l l o w " a n d " p o s t full" respectively, b e c a u s e , normally, these n u m b e r s tell u s the length o f the past m o n t h . B a b y l o n i a n t e r m i n o l o g y is, h o w e v e r , n o t very consistent in that " 3 0 " in T U 11 R e v . 2 2 is u s e d for saying that the current m o n t h will h a v e only 29 days. All n u m b e r s o n the tablets are given in the B a b y l o n i a n s e x a g e s i m a l s y s t e m (a positional n u m b e r s y s t e m with 6 0 as its basis). For e x a m p l e 2,15 c a n b e r e a d as 2 - 6 0 + 1 5 (or as (2 6 0 + 15)-60", since the system did not always specify the absolute value o f a n u m b e r ) . A text I shall also refer to is M U L . A P I N , ^ a n a s t r o n o m i c a l c o m p e n d i u m c o m p i l e d aroimd the end of the s e c o n d or the b e g i n n i n g of the first m i l l e i m i u m B . C . It is found in several copies, the oldest dating fi-om a r o u n d 7 0 0 B . C . A m o n g s t other things it gives the length o f day and night as (a linear zigzag) function o f the m o n t h ; d a y a n d night are m e a s u r e d in mana. D a y p l u s night equals 6 mana, the longest day is 4 mana a n d the shortest 2 mana (values w h i c h are v e r y inaccurate for the latitude of B a b y l o n ) . T h e daily retardation of the m o o n is also given as a function o f the m o n t h : it is calculated as 1/15 of the nightlength, b u t since the r e t a r d a t i o n is m e a s u r e d in us, while the night is m e a s u r e d in mana, it is found as 4 x the night (4 X N mana = 1/15 x N , 0 0 us). It is evident, therefore, that the text m u s t h a v e p u t 1,00 us = 6 0 us equal to 1 mana. T h e L u n a r Six, w h i c h are m o r e c o m p l i c a t e d o b s e r v a b l e p h e n o m e n a , a n d the G o a l - Y e a r m e t h o d for their prediction are p r e s e n t e d in the A p p e n d i x at the e n d of this paper.

The Astronomical Content of TU 11 T U 11 is d i v i d e d b y horizontal rulings into 2 9 sections. Sections 9 - 2 2 h a v e astronomical content; the r e m a i n i n g sections are astrological. T h e a s t r o n o m i c a l sections h a v e brief rules for predicting the time of eclipses, lunar p h a s e s a n d the length of the limar m o n t h s . Sections 9 - 1 3 are c o n c e r n e d with the times of (lunar) eclipses.^ T h e B a b y l o n i a n s specified the m o m e n t of a d a y b y its distance in time to or from sunrise or sunset. T i m e differences w e r e m e a s u r e d in us w h i c h are the s a m e as o u r time d e g r e e s : the daily revolution (by 3 6 0 ° ) of the sky takes 24 hours, so that 1° = 1 wi « 4 m i n u t e s . F o u r e x a m p l e s d e m o n s t r a t e t h r o u g h calculations h o w the time of a future eclipse can b e d e t e r m i n e d b y m e a n s o f the Saros cycle o f 18 y e a r s . T h e b a s i s o f the calculation is the time T of a n eclipse, w h i c h t o o k p l a c e 1 Saros'' earlier t h a n the "

SACHS and HUNGER ( 1 9 8 8 ) .

^

HUNGER and PINGREE ( 1 9 8 9 ) .

^ Since TU 11 mainly treats the moon, we read these examples as calculating lunar (and not solar) eclipses. Furthermore, the preceding astrological section 8 deals with lunar eclipses. '

The Saros is a period of 2 2 3 synodic months = 6 5 8 5 1/3 day w 1 8 years: In a good

8

L. Brack-Bemsen

eclipse to b e predicted. T o this time T is a d d e d o n e third of the d a y p l u s o n e third of the night ending u p with T + 2 [ , 0 0 ] , w h i c h is t h e n r e d u c e d to give the time in us after sunset or sunrise o f the n e w eclipse. T h e text here apparently u s e s the k n o w l e d g e that eclipses will r e p e a t after o n e Saros and that the time o f full m o o n will b e shifted b y about 1/3 of ( d a y plus night) after 2 2 3 synodic m o n t h s . T h e r e are m a n y other texts d e v o t e d to eclipses, e.g. lists of possible dates for eclipses, a r r a n g e d in Saros cycles.^ This k n o w l e d g e is also u s e d in the " G o a l - Y e a r " m e t h o d for p r e d i c t i n g lunar p h a s e s , w h i c h is u s e d a n d briefly described in sections 14 a n d 1 6 . ' All the r e m a i n i n g sections ( 1 4 , 15, a n d 17 t h r o u g h 22) give rules for determining the length o f the Babylonian month.

Duration of the Babylonian Month Before w e consider the different B a b y l o n i a n m e t h o d s for predicting full or h o l l o w m o n t h s , it is n e c e s s a r y to present s o m e b a c k g r o u n d k n o w l e d g e o n h o w to d e t e r m i n e the length of the synodic m o n t h . This " e m p i r i c a l " b a c k g r o u n d k n o w l e d g e has b e e n found b y analyzing c o m p u t e r simulated lunar data. T h e first crescent aimounces the n e w month, a n d b y so doing it also d e t e r m i n e s the length o f the former m o n t h . B u t the first crescent also contains information on the length of the m o n t h that has j u s t started: T h e size of NA;^ m e a s u r e d (or calculated) at the b e g i n n i n g o f a m o n t h is c o n n e c t e d to the length o f that current m o n t h . T h i s is illustrated in Figure 2 below. H e r e the time b e t w e e n the setting o f the s u n and the first crescent is depicted for a series of consecutive B a b y l o n i a n months.'° T h e full m o n t h s are m a r k e d with a black dot. N o t e : all m i n i m a o f the c u r v e h a v e a dot, but n o n e o f the m a x i m a has one. H e n c e the figure gives us a first, albeit rather crude, rule: a small NA^ indicates a long (full) m o n t h , while a large NA^^ a n n o u n c e s a short (hollow) m o n t h . F o r intermediate values o f NAf^, there is a p p a r e n t l y n o clear information o n m o n t h s length: in this figure NAf^ at lunation 10 is larger than its value at lunation 13, b u t m o n t h 10 is full while m o n t h 13 is h o l l o w . W e therefore h a v e a simple rule: IfNAj^

is large, then the m o n t h will b e c o m e h o l l o w is small, then the m o n t h will b e c o m e full

A closer analysis o f NA/^ reveals a very useful insight: it is the m a g n i t u d e or size of consecutive NA;^ w h i c h decides the m o n t h length. W h e r e NA^ for a m o n t h ( M ) is smaller than its value for the next m o n t h ( M + l ) , m o n t h ( M ) will b e full; w h e r e it is

approximation it also equals 239 anomalistic months and 242 draconitic months. The term "Saros" is modem; the Babylonians simply called it "18 years". *

See STEELE (2000a, 2000b) and AABOE et al, ( 1991 ), pp. 35-62.

' For a detailed presentation of the Goal-Year method, see the Appendix, which also introduces the Lunar Six time intervals. '° For the construction of the figures, I have used Peter Huber's computer file, creslong.dat, which among others gives for each month the magnitude of NA,^ and the length of the months. 1 warmly thank him for providing and allowing me to use his computed lunar files.

Predictions of Lunar Phenomena in Babylonian Astronomy

10

20

30

40

lunation i Figure 2: F o r consecutive m o n t h s / = 0, 1, 2,..., 70, the time NAj^i) from sunset to the setting o f the n e w crescent is plotted as function of the Iimation n u m b e r i. A black circle at a lunation i indicates that m o n t h ( 0 will h a v e 3 0 days.

35

• loDg month

NAN(Ì)

•

long month

30 25

^20

10

60

70

80

90

100

no

120

130

lunation i Figure 3 : F o r consecutive m o n t h s i = 60, 6 1 , 62,..., 130, the time NA/^i) from sunset to the setting of the n e w crescent is plotted as a function of the lunation nimiber /. A b l a c k circle at a lunation m a r k s a long m o n t h , while a triangle tells that at the first d a y o f the next m o n t h , the n e w crescent will b e visible for a longer time before setting. T h e dots a n d triangles occur at the same lunations, except for / = 120 which h a s a dot but n o triangle, a n d for i = 132, w h i c h has a triangle but n o dot.

10

L. Brack-Bemsen

larger t h a n the next, m o n t h ( M ) will b e c o m e h o l l o w . T h i s is formulated in the following R u l e R, w h i c h w o r k s in 9 6 cases out of h u n d r e d : If NAj^ (M) < NAf^ (M+ 1), then m o n t h ( M ) is full Rule R : If NA^ (M) > NAj^ (M+1),

then m o n t h (A/) is h o l l o w

T h e rule w a s found t h r o u g h analyzing figures like Figure 3 . H e r e the size of a NAj^ is c o m p a r e d graphically to its value for the n e x t m o n t h . W i t h very few e x c e p t i o n s it is true that a m o n t h is long ( m a r k e d b y a dot) w h e n e v e r its NAj^ is smaller than the NAi^ of the next m o n t h ( e a c h month(A/) for w h i c h NA^^M) < NAj^M+X) is m a r k e d b y a triangle). In Figure 3 the dots a n d triangles occur almost always at the same lunations. W i t h this " e m p i r i c a l " k n o w l e d g e w e shall n o w return to the B a b y l o n i a n texts.

Rules for predicting month lengths found in cuneiform texts M o s t of the rules that h a v e b e e n u n c o v e r e d in texts b e g i n b y finding in s o m e w a y or other the m a g n i t u d e of NA^, a n d use it for predicting the m o n t h length. B u t t w o very e a s y and rather primitive rules also exist, and w e shall start with these rules. In section 15 o f T U 11 the altitude of the n e w crescent is u s e d to foretell the length of the n e w m o n t h w h i c h has j u s t started: is h i g h over the horizon, t h e n the m o n t h will b e c o m e h o l l o w If the first crescent is l o w a b o v e the horizon, t h e n the m o n t h will b e c o m e full In the Reports^^ a n o t h e r very primitive rule s e e m s to h a v e b e e n used: T h e m o n t h length w a s c o n n e c t e d to the d a y at which the m o o n set for the first time after sunrise, w h i c h m e a n s that the (full) " m o o n could b e seen with the sun". T h i s event takes p l a c e in the m i d d l e of a m o n t h , shortly after opposition: o n the d a y before, the m o o n (in its full p h a s e ) sets before sunrise, o r in the t e r m i n o l o g y o f the R e p o r t s : " T h e m o o n d o e s n o t wait for the sun, b u t sets". If the m o o n is seen [early in the month, t h e n the m o n t h will b e c o m e h o l l o w with the Sim

late in the month, t h e n the m o n t h will b e c o m e full

In the Reports a n d Letters to the Assyrian kings E s s a r h a d d o n a n d A s s u r b a n i p a l , only the day near m i d d l e m o n t h w a s r e c o r d e d at w h i c h m o o n a n d sun w e r e seen together. B u t only a little later, texts r e c o r d also the time m e a s u r e d b e t w e e n the risings and the settings of s u n a n d m o o n . (See, for e x a m p l e . D i a r y - 5 6 7 I: " O n the 14th o n e g o d w a s seen with the other: NA=4 uf). Intuitively w e u n d e r s t a n d that w h e n NA o c c u r s early (at d a y n u m b e r 12 or 13), indicating that o p p o s i t i o n of sun and m o o n also o c c u r r e d early, t h e n the B a b y l o n i a n m o n t h will also e n d early. A n d if full m o o n takes p l a c e late, t h e n that B a b y l o n i a n m o n t h will also tend to e n d late. B u t this rule is v e r y crude: a study of 2 2 3 m o n t h s s h o w s that in 4 8 o f these m o n t h s , NA was m e a s u r e d o n d a y 12 or 13, b u t o n l y 3 8 o f these m o n t h s w e r e hollow. If NA was m e a s u r e d o n d a y 15 or 16, then the m o n t h was

"

HUNGER ( 1 9 9 2 ) .

Predictions of Lunar Phenomena in Babylonian Astronomy

full in

11

53 out o f 7 3 cases. T h i s is therefore a rather p o o r rule! T r a c e s of this rule are

also found in S e c t i o n 15 of T U 1 1 . I shall r e p e a t these first a n d r o u g h e m p i r i c a l rules in a s c h e m a t i c way: early : m o n t h short NA o c c u r s I late : m o n t h long Primitive Rules :

h i g h to the sim : m o n t h short Crescent « [ l o w to the s u n : m o n t h long J large : m o n t h short NAf, I small : m o n t h long

More Advanced Rules T h e n e w crescent a i m o u n c e s the e n d o f a B a b y l o n i a n m o n t h . B u t a b o v e w e h a v e seen that the t i m e o f its visibility, the quantity A'.^^', is also a n indicator for the length of the n e w m o n t h . T h e B a b y l o n i a n a s t r o n o m e r s also n o t i c e d this. T h e k n o w n textual material b e a r s witnesses to s e v e n different m e t h o d s for d e t e r m i n i n g NA;^, a n d t h e r e b y p r e d i c t i n g full or h o l l o w m o n t h s . B e l o w is a s u r v e y o f the m e t h o d s u s i n g the quantity NAj^ for p r e d i c t i n g the m o n t h length: -

NAN is found b y m e a n s of the G o a l - Y e a r m e t h o d (using lunar six data

from

lunations 18 years earlier) a n d a little detail within this c a l c u l a t i o n will d e t e r m i n e the length of the n e w m o n t h : Is an a d d i t i o n n e e d e d or not. -

NAN'IS

found from KUR t h r o u g h extrapolation. T h e s a m e v a l u e for the daily

r e t a r d a t i o n o f the m o o n is u s e d at the eastern a n d w e s t e r n h o r i z o n , n a m e l y a fifteenth of the d a y length. T h e size o f NAf^ d e c i d e s the d a y o n w h i c h the crescent is e x p e c t e d to b e c o m e visible. -

NAf^is found from its v a l u e s r e c o r d e d 1 Saros earlier. T h e difference b e t w e e n the t w o values o f NA^, situated 1 S a r o s apart, is d e r i v e d from the s c h e m a t i c length of the night.

-

NAMÌS

found ( b y different m e t h o d s ) from its value o n e m o n t h earlier. In the

A t y p i c a l text K

in the first a p p r o x i m a t i o n NAf^i+l)

is found b y a d d i n g

s o m e v a l u e t t o NAi^i), w h e r e t = t(X) is a function o f the lunar l o n g i t u d e . In section 17 of T U 11 NAN{11)

is found from NAf^l) a n d the v a l u e s of

NA(yU)

a n d NAÇVUÏ), m e a s u r e d 5 1/2 m o n t h s before the m o n t h s I and II o f interest. T h e sign o f NAiWU)-NA(yiU) seems

to

find

NA^ÇV)

from

d e t e r m i n e s t h e m o n t h length. S e c t i o n 2 0 NAf^YV)

by

calculations

which

must

be

erroneous'^ and which I cannot understand. -

NAN(1)

of the n e w year s e e m s to b e inferred from NA^il) of s o m e o l d year in

c o m b i n a t i o n with the t w o values of KUR(Xll)

w h i c h o c c u r r e d a few days

NEUGEBAUER and SACHS ( 1 9 6 9 ) , pp. 9 6 - 1 0 8 .

The text tells to calculate some quantity and compare it to 1/2 NA^ in order to decide between full or hollow month. The crucial criteria for full or hollow months is however worthless, since for the calculations reproduced in the text, only one of the inequalities can be true - the other will never occur. Therefore the text must be erroneous.

L. Brack-Bemsen

12

before the b e g i n n i n g o f m o n t h I(old) and m o n t h I(new). T h e relative size of KUR{o\d) and ArL7?(new) determines the length of the m o n t h . T h e text d o e s not h o w e v e r specify w h i c h year is m e a n t b y the " o l d " year. T h e rules from T U 11 for finding NA^ are p r e s e n t e d b e l o w in schematic form: addition : m o n t h h o l l o w Section 1 4 :

Goal - Year Method

->

NAj^j subtraction : m o n t h full

[Formulated and valid in case of the old m o n t h b e i n g full, NAj^i-223)

- > NAf^Ì)\

' NAj^(II) < A^^;v(7;: m o n t h h o l l o w Section 1 7 :

NAj^(I)

->

NAf^(II) NAj^(II) >

NAj^(II)-.monûiMX

if larger than 12 : h o l l o w Section 1 8 :

18years] ^

NAf^(II) if smaller than 1 2 : filli

Section 1 9 : KUR{i)

extrapolated

day o f n e w crescent addition : m o n t h h o l l o w

Section 22 : NAj^ (new) foimd through subtraction : m o n t h full W e can n o w ask " w h o invented these rules"? W a s it a scribe from the Seleucid p e r i o d w h o k n e w the G o a l - Y e a r m e t h o d and tried to p l a y a r o u n d with similar older s c h e m e s a n d m e t h o d s , or d o w e h a v e here a genuine collection o f n e w e r and older m e t h o d s ? T h i s question is v e r y important since s o m e of the m e t h o d s s e e m to apply to astronomical s c h e m e s from M U L . A P I N , so w e m i g h t learn here h o w such s c h e m e s w e r e used. H e r m a i m H i m g e r g a v e the s a m e answer to this question as I d o , h o w e v e r b y a r g u m e n t s w h i c h I n e v e r thought about. T h e fact that something is written o n a clay tablet gives it a certain value and importance. A scribe w h o tried out n e w ideas w o u l d n e v e r u s e s u c h a valuable material. H e n c e , w h a t occurs in cimeiform o n a nicely formed clay tablet is important and a c c e p t e d k n o w l e d g e . T h e r e a s o n s w h y I a m c o n v i n c e d that T U 11 contains a collection o f m e t h o d s a n d rules w h i c h w e r e actually developed and u s e d b y different a s t r o n o m e r s over time are the following: 1) Quite a lot of parallel texts in the British M u s e m n h a v e b e e n found, s o m e o f w h i c h are considerably older. M o s t o f these texts c a m e fi-om B a b y l o n , while T U 11 originates from U r u k . A l s o T U 11 itself is a c o p y , so s o m e o n e t h o u g h t that w h a t it contained w a s w o r t h copying. A n d 2) w h e n w e analyse the m e t h o d s , w e see that they all (as far as w e h a v e b e e n able to u n d e r s t a n d t h e m ) reflect the s a m e basic ideas w h i c h s e e m to h a v e b e e n refined o v e r time. T h e p r o c e d u r e s are b a s e d o n connections b e t w e e n the Limar Six, o n different m e t h o d s for determining the daily retardation of the m o o n , a n d o n "similar situations".

Parallel Texts and Texts with the Same Methods T h e tablets B M 4 2 2 8 2 + 4 2 2 9 4 were tentatively dated b y Irving Finkel to the fifth century B . C . T h e y h a v e passages w h i c h c o r r e s p o n d to section 14, 16, a n d 2 2 o f T U 1 1 . A p a r t from this, the text gives the G o a l - Y e a r m e t h o d in a m u c h clearer and m o r e

Predictions of Lunar Phenomena in Babylonian Astronomy

13

detailed formulation t h a n w h a t w e h a v e found o n T U 1 1 . A t o n e p o i n t the text disagrees with section 2 2 ( o f T U 11): at the p l a c e w h e r e T U 11 m e n t i o n s s o m e " n e w year", the parallel text h a s " o l d year". B M 3 6 7 8 2 h a s p a s s a g e s parallel to section 17, 18, a n d 19, a n d B M 3 6 7 4 7 w h i c h j o i n s to B M 3 7 0 1 8 h a s parts of sections 19 a n d 20.'"* E x c e p t for s e c t i o n 15 a n d 2 1 , t o all the other sections o n T U 11 dealing with lunar six a n d m o n t h lengths parallel p a s s a g e s h a v e b e e n found o n other tablets.

Structure of the Methods T h e different m e t h o d s ( r e p r o d u c e d in c o n d e n s e d f o r m a b o v e ) h a v e m a n y c o m m o n features. I n all cases the size o f (the established) NA^ is a n indicator for the m o n t h length. A n d NA^ for the m o n t h in q u e s t i o n is foimd from the value oïNAf^ 1 S a r o s earlier, 1(?) year earlier, or 1 m o n t h earlier; or it is found from KUR m e a s u r e d a few days earlier: ( S e c t i o n 14)

NA^II-

223)

( S e c t i o n 17)

AC^^I) ^

( S e c t i o n 18)

NAflll),^

( S e c t i o n 19)

KUR{i)

(Section 22)

NAilold)

NA^ii)

NA

till)

years back ->

NA^^)

NA^ii+l) NAi^nçw)

( S e c t i o n 2 0 a n d 2 1 also s e e m to u s e a NAj^ to d e t e r m i n e s o m e later NA/^ h o w e v e r b y arithmetic m a n i p u l a t i o n s w h i c h w e c a n n o t u n d e r s t a n d . ) F o u r o f the m e t h o d s listed a b o v e c o n n e c t values o f NA/^ m e a s u r e d at special intervals utilizing w h a t w e w o u l d call the daily c h a n g e o f NA^j, the m o n t h l y c h a n g e of NAi^, the y e a r l y ( ? ) c h a n g e o f NA/^, a n d the sarosly c h a n g e o f NAf^. T h e v a l u e s o f these

changes

are

either

determined

empirically

or

foimd

by

theoretical

c o n s i d e r a t i o n s , or b y a c o m b i n a t i o n o f both. W e u s e the t e r m ANA^ for the daily c h a n g e o f NAf^, a n d AKUR

for the daily c h a n g e of KUR, a n d r e m i n d the reader, that

these quantities m e a s u r e the daily retardation o f the m o o n ( m e a s u r e d in t h e w e s t b y setting, or in the east b y rising, respectively). I w a n t t o stress the fact that a l r e a d y o n the

earliest

astronomical

texts

we

find

these

daily

retardations

modelled

arithmetically: EnUma Anu Enlil T a b l e t X I V ' ^ a n d M U L . A P I N h a v e tables in w h i c h ANAf^ is a p p r o x i m a t e d b y 1/15 x length o f the night. A s u m m a r y o f t h e different w a y s o f finding the c h a n g e s m NA^ or KUR is given b e l o w : T h e daily c h a n g e of KUR: AKUR = 1/15 x length o f daylight ( S e c t i o n 19) T h e daily c h a n g e o f A^^A^: AA^^JV = 1/15 X length of night ( M U L . A P I N ) T h e daily c h a n g e o f A^^A<0: ANA!^ = {SU+ NA){i-6)

(Section 16 a n d 14)

I would like to express my thanks to Clemency Williams for this information. EAE is a great canonical omen series. Tablet XIV with astronomical schemes has been published in AL-RAWI and GEORGE ( 1 9 9 1 ) .

14

L. Brack-Bemsen

T h e m o n t h l y c h a n g e o f NAf^: function o f lunar longitude (Atypical T e x t K ) . T h e m o n t h l y c h a n g e : NAf^W) - NAf^l) = (M4(VIII) - NA{Wl\)) T h e yearly(?) c h a n g e oïNA^. T h e sarosly c h a n g e : NAdi+H'i)

{KUR{o\à) - NAfli)

- KUR{nQ^)) = ^IX^^

( S e c t i o n 17)

(Section 2 2 )

+ NA){i-6)

( S e c t i o n 14)

T h e sarosly c h a n g e of NAf^^ 1/30 x night ( S e c t i o n 18) O f course, the a c c u r a c y of the predicted value o f NAf^ d e p e n d s o n h o w g o o d the applied m e t h o d a p p r o x i m a t e s the changes o f NAj^. T h e best a p p r o x i m a t i o n to ANAi^j is foimd in the G o a l - Y e a r m e t h o d , w h i c h u s e s $U + NA, the daily retardation of the m o o n m e a s u r e d at full m o o n 5 1/2 m o n t h s earlier. Test calculations h a v e s h o w n that àNAfiï) is optimally a p p r o x i m a t e d b y (ÊU + NA){i-6)}^ T o m e this fine m e t h o d s e e m s to b e the e n d result o f refinements a n d a c o m b i n a t i o n o f older practices. In a n y case sections 9 - 1 2 , 15, 17, a n d 18 h a v e elements reflecting s o m e empirical k n o w - h o w , w h i c h in a skilled manipulation s h o w s u p in the G o a l - Y e a r m e t h o d . Section 9 - 1 2 a n d 18 coimects lunar events w h i c h are situated 1 S a r o s apart in order to m a k e p r e d i c t i o n s . T h e G o a l - Y e a r m e t h o d a n d the m e t h o d in Section 17 c o n n e c t NA/^ with quantities w h i c h o c c u r r e d half a year earlier.'^ It is wise to d o so, b e c a u s e the conditions u n d e r w h i c h the n e w crescent is seen in m o n t h (0 are v e r y similar to the conditions of the full m o o n w h e n it sets for the first time after sunrise in m o n t h ( i - 6 ) . ^ ^ This c a n b e illustrated b y c o m p a r i n g Figure 1 w i t h F i g u r e 4 (in the A p p e n d i x ) . F i g u r e 1 s h o w s the situation at the w e s t e r n h o r i z o n at the v e r y b e g i i m i n g o f m o n t h I, w h i l e F i g u r e 4 shows the situation a r o u n d full m o o n 5 1/2 m o n t h s earlier, in the m i d d l e o f m o n t h VII. T h e antisun © i n d i c a t e s the p l a c e w h e r e the o p p o s i t i o n t o o k p l a c e . T h e inclination of the ecliptic is the s a m e at the t w o events, a n d so is the m o v e m e n t of the m o o n relative to the s u n a n d antisun, respectively: A r c ( # , C ) « A r c ( C sû, C NA)- T h e s a m e will, for e x a m p l e , h o l d for the situation o f NAN in m o n t h V I I c o m p a r e d to NA in m o n t h I. In these cases, the ecliptic w o u l d b e low, inclined b y s o m e 3 4 ° to the horizon. W e h a v e ignored h e r e the lunar latitude; b u t r e m a r k that it will b e about the same in the t w o situations, since 5 1/2 m e a n synodic m o n t h s « 6 m e a n draconitic m o n t h s . Section 15 ( w h i c h also g a v e the simple rule: n e w crescent high, m o n t h short; n e w crescent low, m o n t h long), has traces o f this k n o w l e d g e a b o u t "similar s i t u a t i o n s " in its last p a s s a g e . It says: From month I onwards, the first days [the moon is] high, the fourteenth days [the moon is] low; from month VII on, the first days [the moon is] low, the fourteenth days [the moon is] high. W h a t is e x p r e s s e d h e r e reflects s o m e k n o w l e d g e about the inclination of the p a t h o f m o o n a n d sun: T h a t it stands steep to the h o r i z o n w h e n the n e w crescent o f m o n t h I is o b s e r v e d , a n d also w h e n the full m o o n sets in the m i d d l e of m o n t h VII. A n d that it is flat to the h o r i z o n w h e n the setting fiill m o o n is o b s e r v e d in m o n t h I, but also w h e n the n e w

See BRACK-BERNSEN (1999), Figure 7. Section 17 tells us to calculate NAflY) - {NAiyWl) - NAiyW)) [= NAf^lV)]. The longitude and latitude of the moon as well as the lunar velocity will be about the same in the two situations. For further details see BRACK-BERNSEN (1999).

Predictions of Lunar Phenomena in Babylonian Astronomy

15

crescent b e c o m e s visible n e a r the w e s t e r n h o r i z o n a n n o u n c i n g the b e g i i m i n g of m o n t h VIL In M e s o p o t a m i a there w a s a strong tradition of p r e s e r v i n g old w i s d o m . A n c i e n t tablets w e r e c o p i e d o v e r a n d o v e r again. I a m sure that T U 11 is a collection o f g e n u i n e rules from b o t h older and m o r e r e c e n t times, a n d I h o p e that m y a r g u m e n t s will c o n v i n c e the r e a d e r too.

Appendix The "Lunar Six" T h e B a b y l o n i a n s specified a m o m e n t of a d a y b y its time difference to sunrise or sunset. T h e y gave special attention to the m o v e m e n t o f the m o o n in the d a y s a r o u n d o p p o s i t i o n or conjimction. T h e time of these events is n o t directly o b s e r v a b l e , so w h a t the B a b y l o n i a n s o b s e r v e d w e r e the differences in time b e t w e e n the rising a n d setting of the sun a n d m o o n in the days aroimd o p p o s i t i o n a n d conjunction. A . S a c h s called these time differences the " L u n a r S i x " . ' ' T h e four intervals relating to the fiill m o o n , w h i c h w e call the " L u n a r F o u r " , are the following: = time from m o o n s e t to sunrise, m e a s u r e d at last m o o n s e t before sunrise. NA = time from sunrise to m o o n s e t , m e a s u r e d at first m o o n s e t after sunrise. ME = time from m o o n r i s e to sunset, m e a s u r e d at last m o o n r i s e before sunset. GEf, = time from sunset to m o o n r i s e , m e a s u r e d at first m o o n r i s e after sunset. T h e setting m o o n is n o t visible before conjimction, a n d its rising is n o t visible after conjunction. T h e r e f o r e only t w o time intervals w e r e o b s e r v e d a r o u n d n e w m o o n . 1) O n the e v e n i n g w h e n the n e w crescent b e c a m e visible, indicating the first d a y of the m o n t h : NAf^ = the time b e t w e e n sunset a n d the setting of the m o o n , w h e n it h a s b e c o m e visible for the first t i m e after conjunction^^ (Figure 1). A n d 2) at the e n d of the m o n t h : KUR = the time from m o o n r i s e to sunrise, w h e n the rising m o o n is visible for the last time before conjunction.

"

SACHS(1948),p. 281.

In the texts with which we are working, this interval is called NA, but it occurs always together with an indication that it is the NA of the first day or the NA at the beginning of the month. We put this identification into the name, calling it NA (of the new crescent), or NA^j. We do this in order to be as precise as the Babylonian texts. There the term NA is also used for a time interval in the middle of the month, but always identified by calling it the NA of day 14 or the NA opposite the sun.

16

L. Brack-Bemsen

The "Goal-Year" Method^' T h e s e t i m e intervals b e t w e e n the rising a n d setting o f the s u n a n d m o o n a r e o b v i o u s a n d e a s y t o o b s e r v e . F r o m a theoretical p o i n t o f view, h o w e v e r , t h e y a r e very complicated

quantities.

They

depend

on the time

o f the c o n j u n c t i o n

o r the

opposition: w h e n it takes p l a c e in c o m p a r i s o n to simset o r sunrise. T h e y a l s o d e p e n d o n the p o s i t i o n o f the full o r n e w m o o n in the ecliptic, a n d o n t h e lunar v e l o c i t y a n d latitude.^^ It w a s therefore v e r y surprising a n d exciting t o find o u t that t h e B a b y l o n i a n s h a d d e v e l o p e d a n easy, elegant a n d v e r y exact e m p i r i c a l m e t h o d for the p r e d i c t i o n o f these t i m e intervals. T h e y h a d n o t e d that, i n c o m p a r i s o n to sunset o r sunrise, a syzygy w o u l d o c c u r 1/3 d a y later than the o n e that t o o k p l a c e a S a r o s earlier. A n d they h a d realized that (SO + NA) m e a s u r e d the daily d e l a y o f the m o o n at setting, a n d that (ME + GE^) m e a s u r e d its delay at rising. I n addition, they m u s t h a v e r e m a r k e d that t h e daily d e l a y o f the m o o n w o u l d r e p e a t after o n e S a r o s . A l l these c o n n e c t i o n s are implicitly u s e d in w h a t w e h a v e called the G o a l - Y e a r

Method.

B e l o w the m e t h o d is r e p r o d u c e d in the form o f m a t h e m a t i c a l e q u a t i o n s : (NA^), = (NA^),.223-i/2(SÛ

+ NA)i.22,

(1)

ÉÛi = SÙi.221, + l / 3 ( ^ f / + NA)i.223

(2)

NAi = NAi.22^ - l / 3 ( ^ ( / + / V ^ ) , 2 2 3

(3)

MEi = MEi.22, + V3ÌME {GEe)i = iGEe),223

+ GE,)i.22i

- ^ME

KURi = KURi.223 + \/3iME

(4)

+ GEe),223

(5)

+ G£6),.229

(6)

C o r r e c t i o n s : S o m e t i m e s the results found b y these calculations a r e p r e l i m i n a r y . If NAN<

10 us, t h e n wait a day: c o r r e c t e d (NAi^)i = p r e l i m i n a r y (NAf^)i + (§Û + /V^)/-229-

( T h e visibility limit o f NA^ is given in b o t h sections 14 a n d 16 a s {NA!^)i = (NAM)Ì.223

+ 2/3(SÛ

+ NA)i.229

\Ous.) ( 1 corrected)

E x c e p t for the few cases o f eclipses, the time o f o p p o s i t i o n o r c o n j u n c t i o n c a n n o t b e directly o b s e r v e d . T h e B a b y l o n i a n s u s e d t h e L u n a r S i x t o g e t information o n t h e relative position o f t h e s u n a n d m o o n a r o u n d full m o o n a n d n e w m o o n . T h e y u s e d the o b s e r v a b l e {SÛ + NA) às the daily retardation o f the setting m o o n . T h e following c o n s i d e r a t i o n s will illustrate that it is quite o b v i o u s t o d o s o : In t h e very few " i d e a l " cases, in w h i c h o p p o s i t i o n takes p l a c e at the m o m e n t o f sunrise, t h e full m o o n will set at the W e s t e r n h o r i z o n while t h e s u n will rise at t h e E a s t e r n h o r i z o n . A t t h e next m o r n i n g , the m o o n will set a b o u t a n h o u r after sunrise. A n d this time interval from sunrise t o m o o n s e t e v i d e n t l y m e a s u r e s t h e r e t a r d a t i o n o f the m o o n o n the d a y o f opposition.

In T U 11 these rules were expressed more or less explicitly (in section 1 4 and 1 6 ) - some of them were reconstructed; on the parallel tablet BM 4 2 2 8 2 + 4 2 2 9 4 the rules are formulated more clearly. For more details, see BRACK-BERNSEN and SCHMIDT ( 1 9 9 4 ) .

Predictions of Lunar Phenomena in Babylonian Astronomy

17

N o r m a l l y , the o p p o s i t i o n d o e s not take p l a c e at the m o m e n t of sunset (or sunrise), therefore n o r m a l l y the daily retardation o f the sun is split u p into t w o intervals: ÉÛ, the time from m o o n s e t to sunrise m e a s u r e d o n the last m o r n i n g before opposition, and NA, the time from simrise to m o o n s e t m e a s u r e d o n the next m o r n i n g , the first after opposition. Obviously, their s u m (SO + NA) m e a s u r e s , h o w m u c h later, in c o m p a r i s o n to sunrise, the m o o n sets o n the m o r n i n g o f NA than o n the m o r n i n g before.

Figure 4 : T h e situation at the W e s t e r n h o r i z o n at simrise in the d a y s a r o u n d a n o p p o s i t i o n taking p l a c e in (the m i d d l e of) m o n t h VIL T h e d a s h e d line depicts the ecliptic, the p a t h along w h i c h sun a n d m o o n m o v e s , the direction o f m o t i o n is indicated b y the arrow. T h e p o s i t i o n s of m o o n a n d "anti-sun" are s h o w n on t w o m o r n i n g s o n w h i c h and NA are m e a s u r e d . Figure 4 illustrates the n o r m a l case w h e n opposition d o e s not take p l a c e at sunrise. F o r the sake o f simplicity w e introduce the symbol O for the "anti-sun", w h i c h w e define as the point o n the ecliptic situated directly o p p o s i t e the sun. A t the exact m o m e n t w h e n the sun rises, © sets, and vice versa. It m a r k s the p o i n t of the ecliptic at w h i c h the o p p o s i t i o n takes place. T h e daily retardation o f the setting m o o n is m e a s u r e d b y the arc (SÛ + NA) of the equator. T h e retardation of the rising m o o n at the d a y of o p p o s i t i o n is d e t e r m i n e d in a similar way: as the s u m (ME + GE^) of ME, the time from last m o o n r i s e before o p p o s i t i o n to sunset, and of GEf„ the time from sunset to first m o o n r i s e after opposition.

18

L. Brack-Bemsen

T h e E m p i r i c a l F o u n d a t i o n o f the G o a l - Y e a r M e t h o d T h e G o a l - Y e a r m e t h o d uses (SÛ + NA){i-223) a s the daily retardation o f the setting m o o n around o p p o s i t i o n /. It uses (SÛ + NA){i- 229) for the daily retardation o f the n e w crescent in m o n t h (i), and it takes a third o f the daily retardation a s the change after 1 Saros o f SO, NA, a n d NAi^, respectively. This m e t h o d is b a s e d o n the following empirical recognition: T h e daily retardation at full m o o n will in a g o o d a p p r o x i m a t i o n repeat after 1 Saros:

(SÛ + NA)ii) «

+ NA){i- 223)

T h e daily retardation o f t h e n e w crescent is a p p r o x i m a t e l y equal t o t h e daily retardation of the setting m o o n in its full p h a s e , m e a s u r e d 5 1/2 m o n t h s earlier. ANAfii)

« {ÈÛ + NA){i-6)

« {ÈÙ +

NA){i-229).

T h e time o f lunar eclipses in c o m p a r i s o n t o sunset is shifted b y 1/3 o f a d a y after 1 Saros. O r generally for e a c h lunar month: 2 2 3 synodic m o n t h s = 1 Saros is 1/3 d a y longer than a whole n u m b e r o f solar days.

Acknowledgements A t this place I express m y thanks t o the D F G (Deutsche Forschungsgemeinschaft) for supporting this work. I w a r m l y appreciate a careful reading o f the m a n u s c r i p t b y Claire O ' R e i l l y and thank Irving Finkel, Christopher W a l k e r , a n d H e r m a i m H u n g e r for identifying a n d translating parallel texts, a n d Peter H u b e r for m a k i n g his c o m p u t e d limar files available. Finally I thank J o h n Steele and H e r m a n n H i m g e r for valuable corrections and suggestions t o this p a p e r .

Abbreviations L B A T = PINCHES, STRASSMAIER, and SACHS (1955)

References A A B O E , Asger; B R I T T O N , J o h n P.; H E N D E R S O N , Janice A.; N E U G E B A U E R , O t t o , and S A C H S , A b r a h a m J. 1 9 9 1 . Saros Cycle Dates and Related Babylonian Astronomical Texts (Transactions o f the A m e r i c a n Philosophical Society 81/6). Philadelphia: A m e r i c a n Philosophical Society. B R A C K - B E R N S E N , L i s . 1 9 9 7 . Zur Entstehung der babylonischen Mondtheorie, Beobachtung und theoretische Berechnung von Mondphasen (Boethius 40). Stuttgart: F r a n z Steiner Verlag. — 1999. " G o a l - Y e a r Tablets: L u n a r Data a n d Predictions". In: N o e l M . S W E R D L O W , Ancient Astronomy and Celestial Divination: 149-177. Cambridge, MA: M I T Press. — a n d S C H M I D T , O l a f 1994. " O n t h e foundations o f the B a b y l o n i a n c o l u m n * : A s t r o n o m i c a l significance o f partial s u m s o f the L u n a r F o u r " . Centaurus 3 7 : 183-209.

Predictions of Lunar Phenomena in Babylonian Astronomy

H U N G E R , H e r m a n n . 1 9 9 2 . Astrological Reports to Assyrian Assyria 8). Helsinki: Helsinki University P r e s s .

19

Kings (State A r c h i v e s o f

—

and PINGREE, D a v i d . 1989. MUL.APIN, an astronomical compendium in cuneiform ( A r c h i v fur Orientforschung Beiheft 2 4 ) . H o r n : F e r d i n a n d B e r g e m & Sohne. N E U G E B A U E R , O t t o . 1947. "Studies in ancient a s t r o n o m y VIII. T h e water c l o c k in B a b y l o n i a n a s t r o n o m y " . Isis 3 7 : 3 7 - 4 3 .

— a n d S A C H S , A b r a h a m . 1969. " S o m e Atypical A s t r o n o m i c a l C u n e i f o r m T e x t s 11". Journal of Cuneiform Studies 22: 9 2 - 1 1 3 . P I N C H E S , T h e o p h o l u s G.; STRASSMAIER, J o h a n n N . , a n d S A C H S , A b r a h a m J. 1 9 5 5 . Late Babylonian Astronomical and Related Texts. P r o v i d e n c e : B r o w n U n i v e r s i t y Press. A L - R A W I , F a r o u k N . H . and G E O R G E , A n d r e w R . 1 9 9 1 . " E n u m a A n u Enlil X I V a n d O t h e r Early A s t r o n o m i c a l T a b l e s " . Archiv fur Orientforschung 38/39: 5 2 - 7 3 . S A C H S , A b r a h a m J. 1 9 4 8 . " A classification of the B a b y l o n i a n a s t r o n o m i c a l tablets of the Seleucid p e r i o d " . Journal of Cuneiform Studies 2: 2 7 1 - 2 9 0 . —

a n d H U N G E R , H e r m a n n , 1988, Astronomical diaries and related text from Babylonia, V o l u m e I ( 1 9 8 8 ) , II ( 1 9 8 9 ) a n d III ( 1 9 9 6 ) . W i e n : Osterreichische A k a d e m i e d e r Wissenschaften. STEELE, J o h n M . 2 0 0 0 a . Observations and Predictions of Eclipse Times by Early Astronomers. D o r d r e c h t : K l u w e r A c a d e m i c Publishers. — 2 0 0 0 b . " E c l i p s e P r e d i c t i o n in M e s o p o t a m i a " , Archive for History of Exact Science 54: 4 2 1 - 4 5 4 . T H U R E A U - D A N G I N , François. 1922. Tablettes d'Uruk (Textes Cunéiformes 6). Paris: M u s é e du L o u v r e . V A N D E R W A E R D E N , Bartel L. 1949. " D a u e r der N a c h t i m d Zeit d e s M o n d u n t e r g a n g s in d e n Tafeln des N a b u - z u q u p - G I . N A . " . Zeitschrift fur Assyriologie 4 9 : 3 0 7 ff — 1 9 5 1 . " B a b y l o n i a n A s t r o n o m y III: T h e earliest a s t r o n o m i c a l C o m p u t a t i o n s " . Journal of Near Eastern Studies 10: 2 0 - 3 4 .

Treatments of Annual Phenomena in Cuneiform Sources John P. Britton,

Wilson

(Wyoming)

Introduction This p a p e r essays a tentative timeUne o f the various treatments o f a n n u a l p h e n o m e n a e v i d e n c e d in c u n e i f o r m sources. A central issue is the e m e r g e n c e o f an increasingly precise imderstanding of the relationship b e t w e e n m o n t h s , years a n d days, w h i c h a p p e a r s to h a v e b e g u n in the 7 * century a n d to h a v e c o n t i n u e d t h r o u g h the transmission of B a b y l o n i a n expertise to Hellenistic a s t r o n o m e r s . A n o t h e r issue is the r e c o g n i t i o n a n d theoretical depiction of the seasonal variation d u e to solar a n o m a l y , w h i c h curiously seems to have coincided with the a d o p t i o n o f a n earlier a n d m u c h c r u d e r s c h e m e for determining the dates of the cardinal p h e n o m e n a . Finally, there is the transmission of this k n o w l e d g e to G r e e k a s t r o n o m e r / s c h o l a r s , w h i c h a p p e a r s to h a v e intensified during the 2"*^ century w i t h information s e e m i n g l y p a s s i n g in b o t h directions. In large p a r t this exercise draws o n p r e v i o u s l y p u b l i s h e d material significant textual qualification. H o w e v e r , four texts - B M 4 5 7 2 8 , B M W22801+22805,

and

U107+124

-

merit

amplified

comments

because

without 36731, close

examination, a n d in the case of B M 3 6 7 3 1 additional fragments, h a v e yielded hitherto u n r e m a r k e d details a n d n e w understandings. In brief, these are as follows. B M 4 5 7 2 8 : T h e p e r i o d relations p r e s e n t e d for the planets, excluding Jupiter, a n d Sirius are r o u g h b u t reasonable a p p r o x i m a t i o n s of actual p h e n o m e n a , dating p o s s i b l y to the 7'*' century. Attention is d r a w n to the c o l o p h o n , hitherto u n r e m a r k e d , ascribing the tablet to a Labasi, possibly related to the 7**' century scholar m e n t i o n e d in L A B S 160. B M 3 6 7 3 1 : (1) T h e dates o f solstices a n d equinoxes in B M 3 6 7 3 1 are c a l e n d a r dates in the civil lunar calendar, not tithis (see b e l o w ) as N e u g e b a u e r a n d S a c h s a s s u m e d , a n d reflect a year length of 3 6 5 ; 16 days with attested seasons o f 9 1 ; 19 d a y s . (2) H o w e v e r , the dates for the setting, rising a n d opposition o f Sirius are e x p r e s s e d in tithis a n d reflect a schematic 27-year cycle. (3) T h e original ( u n b r o k e n ) tablet p r o b a b l y b e g a n with the accession year of N a b o p o l a s s a r (-625) or p o s s i b l y the p r e c e d i n g year (21 K a n d a l a n u ) a n d c o n t i n u e d t h r o u g h the 43'^*' a n d last year o f N e b u c h a d n e z z a r (-561). (4) B y the e n d o f the text the dates for s u m m e r a n d winter solstice are r o u g h l y a d a y early a n d late respectively. W 2 2 8 0 1 + 2 2 8 0 5 : (1) T h e u n b r o k e n tablet p r o b a b l y b e g a n with year 0 or 1 of N a b o p o l a s s a r (-625/-624) a n d could h a v e continued t h o u g h year 8 or 9 o f C y r u s

22

J.P. Britton

( - 5 3 0 / - 5 2 9 ) . (2) T h e dates o f the solstices ( s u m m e r a n d winter) are e x p r e s s e d in tithis a n d reflect a s c h e m a t i c 19-year cycle w h i c h b e g a n w i t h y e a r 1 of N a b o p o l a s s a r . (3) T h e schematic dates for the first 8 years of the cycle are 1 d a y earlier than the c o r r e s p o n d i n g dates in the later " U r u k S c h e m e " , while the r e m a i n i n g 11 dates are identical a n d c o n n e c t a b l e with t h o s e firom the U r u k S c h e m e . (4) T h e sunmier solstice dates for the last 19 years o f the text are r o u g h l y VA d a y early o n a v e r a g e , reflecting a n i m p r o v e m e n t in ( a v e r a g e ) a c c u r a c y o v e r the dates o n B M 36731. U 1 0 7 + 1 2 4 : T h e u n b r o k e n text p r o b a b l y b e g a n with the s u m m e r solstice o f S.E. 113 ( - 1 9 8 ) o n I V 7 a n d c o n t i n u e d in 6 c o l u m n s (3 e a c h o n o b v e r s e a n d r e v e r s e ) t h r o u g h S.E. 2 5 6 ( - 5 5 ) , t h e r e b y explaining the u n i q u e a p p e a r a n c e of an integer d a t e in S.E. 113 and hopefiiUy b u r y i n g N e u g e b a u e r ' s a s s u m p t i o n - l o n g since d i s p r o v e d - that U r u k S c h e m e d a t e s in years with an intercalary U l u l (VI2), m a y h a v e b e e n o n e d a y earlier before S.E. 113 than after. T h e texts t h e m s e l v e s as collated a n d / o r reconstructed are p r e s e n t e d in A p p e n d i c e s A-D. D a y s a n d Tithis T h r e e units, d e s c r i b e d indistinguishably with the t e r m " d a y " (ud/u4), a p p e a r in B a b y l o n i a n a s t r o n o m i c a l texts. O n e is the c o n v e n t i o n a l " d a y " or n y c t h e m e r o n , w h i c h in B a b y l o n r a n from sunset to sunset, a n d w h o s e a v e r a g e length c o r r e s p o n d s to oiu- civil day. Subunits o f this d a y are typically found e x p r e s s e d in t i m e d e g r e e s (us) equal to 1/360* o f a day. A s e c o n d unit, the " s c h e m a t i c solar d a y " , w a s d e r i v e d from the a n c i e n t schematic administrative calendar a n d equal to 1/360'*' of a year, a n d t h u s implicitly to the time r e q u i r e d for the m e a n sun to travel the angular distance 1 u s . T h i s " d a y " figures o n l y p e r i p h e r a l l y in this story. Finally, a daily unit equal to 1/3 0th of a m o n t h a n d called " t i t h i " (x) b y N e u g e b a u e r , w a s frequently u s e d in c o m p u t a t i o n s to c i r c u m v e n t the difficulties arising from r a n d o m l y varying m o n t h lengths o f 2 9 or 3 0 days. I n calculating dates, multiples of 30T w e r e treated as w h o l e m o n t h s , w h a t e v e r their actual length, while a n y r e m a i n d e r ( u p to 3 0 ) w a s c o n s i d e r e d the " d a y " of the m o n t h in question, dates falling o n d a y 3 0 o f a 2 9 d a y m o n t h b e i n g a s c r i b e d to d a y 1 of the following m o n t h . T h e t e r m " e p a c t " (e) refers to the excess o f o n e year o v e r 12 m o n t h s . Generally, a n d unless o t h e r w i s e indicated, it is e x p r e s s e d in tithis. Other Definitions, Abbreviations and Notations T e r m s a n d notations u s e d h e r e generally follow the c o n v e n t i o n s of A C T . M o n t h n a m e s are transcribed b y r o m a n n u m e r a l s as follows. I = II = 111= IV= V =

bar gU4 sig su izi

(Nisan) (Ayar) (Siman) (Du'uz) (Ab)

VI = VI2 = VII = VIII= IX =

kin (Ulul) kin-a/kin-2-kam du6 (Tasrit) apin(Arahsamna) g a n (Kislim)

X = XI ^ XII = XIl2=

ab (Tebet) ziz ( S a b a t ) se ( A d a r ) dir[-se]/a

Treatments of Annual Phenomena in Cuneiform Sources

23

Intercalary years are indicated b y * w h e n a s e c o n d A d a r (XII2), is a d d e d a n d b y ** w h e n a s e c o n d U l u l (VI2) is a d d e d . C a r d i n a l p h e n o m e n a are s o m e t i m e s d e s i g n a t e d as S S ( s u m m e r solstice), F E (fall e q u i n o x ) , W S (winter solstice), a n d V E (vernalspring e q u i n o x ) . Visibilities o f Sirius are d e n o t e d b y G r e e k letters following A C T : F « first a p p e a r a n c e (igi); Q » d i s a p p e a r a n c e (sù). Finally, it s h o u l d b e n o t e d that our

source

material

is

exceptionally

spotty

and

incomplete,

and

that

qualification, " a s far as w e k n o w " , should b e u n d e r s t o o d to a c c o m p a n y

the every

generalization.

Early Sources (to -750) The Schematic Calendar P r i o r to the 7 * c e n t u r y B . C . a s t r o n o m i c a l texts a l m o s t exclusively reflect a schemafic c a l e n d a r of twelve 3 0 - " d a y " " m o n t h s " , ' u s e d for administrative p u r p o s e s from the 4 * m i l l e n n i u m onwards,"^ a n d still the basis for the t r e a t m e n t of risings a n d settings in the A s s y r i a n treatise M U L . A P I N , c o m p o s e d after - 1 1 0 0 . I n a s t r o n o m i c a l contexts this s c h e m e is first attested in the O l d B a b y l o n i a n text B M 17175,^ w h e r e the cardinal p h e n o m e n a are equidistantly s p a c e d at the m i d p o i n t s ( " d a y " 15) of m o n t h s III ( S S ) , V I ( F E ) , I X ( W S ) and X I I (VE).'* T h e s a m e s c h e m e is found in T a b l e t 14 o f the o m e n series, Enuma Anu Enlil ( E A E ) , in the series I . N A M . G I S . H U R . A N . K I . A ( K 2 1 6 4 + ) , a n d in the " A s t r o l a b e " texts, all n o later t h a n M i d d l e B a b y l o n i a n ( M B ) a n d p r o b a b l y of O l d B a b y l o n i a n ( O B ) origin.^ T h i s w o u l d also s e e m to b e the source o f the c o n v e n t i o n that 3 6 0 units c o m p r i s e a circuit, the c o r r e s p o n d i n g t e m p o r a l unit (us) reflecting the fact that t h e time n e c e s s a r y to c o v e r the distance 1 beru ( r o u g h l y 10 k m = 30"^ in linear m e a s u r e ) w a s r o u g h l y 1/12* of a day. In a n y event, a l r e a d y in E A E 14, w e find the d a y d i v i d e d into 3 6 0 t i m e d e g r e e s (6,0"^ MUL.APIN W i t h o n e significant modification the schematic c a l e n d a r also d o m i n a t e s M U L . A P I N , a n A s s y r i a n treatise c o m p o s e d a r o u n d the b e g i n n i n g o f the m i l l e i m i u m B . C . , w h i c h is largely d e v o t e d to describing risings a n d settings of constellations in relation to it. T h e c h a n g e is a shift in the d a t e s of the cardinal

' The singular exception is the "Venus Tablet of Amizaduga", Tablet 63 of the series Enuma Anu Enlil (EAE), which gives dates in the civil calendar of appearances and disappearances of Venus. ^ ENGLUND (1988), pp. 122 ff. describes evidence of its use in administrative texts from archaic Uruk HI and Jemdet Nasr at the end of the 4"" millennium B.C. and in administrative calculations throughout the 3'^'' millennium. For certain interest rate calculations it continued in use until the mid-1970s (A.D.), when the ubiquity of electronic calculators made it unnecessary, making this a uniquely long-lived administrative practice. ^

HuNGERandPiNGREE(1989), p. 163.

This was probably understood to mean simply that the phenomenon in question should occur in the designated month, since the actual dates of cardinal phenomena in the civil calendar would range throughout the month, assuming appropriate intercalations. ^

HOROWiTz(1998), pp. 158-159.

24

J.P. Britton

p h e n o m e n a b y o n e m o n t h so that the year b e g i n s o n e m o n t h earlier in the seasonal or Julian calendar a n d the e q u i n o x e s a n d solstices fall o n e m o n t h later in the A s s y r i a n calendar. T h i s a p p e a r s to h a v e b e e n a c o n s e q u e n c e o f the r e f o r m o f the A s s y r i a n calendar during the reign o f Tiglath Pileser I ( - 1 1 1 3 to - 1 0 7 5 ) , in w h i c h the intercalation practices a n d m o n t h n a m e s o f the S t a n d a r d M e s o p o t a m i a n C a l e n d a r - essentially the Simierian m o n t h n a m e s from the ancient N i p p u r calendar glossed with Semitic n a m e s from a variety o f local calendars - w e r e adopted.^ W h e t h e r this c h a n g e reflected a difference in harvest m o n t h s b e t w e e n n o r t h a n d south (where the Standard C a l e n d a r originated) or s o m e other c o n s i d e r a t i o n is u n k n o w n . W h a t e v e r the reason, in M U L . A P I N a n d related Neo-Ass3n-ian texts such as B E 13918 a n d the " I v o r y P r i s m " , the solstices a n d e q u i n o x e s r e m a i n equally spaced, b u t fall ( o n e m o n t h later) o n " d a y " 15 o f m o n t h s I ( V E ) , I V ( S S ) , V I I ( F E ) a n d X ( W S ) in the schematic calendar. Scant awareness o f actual periodicities is e v i d e n c e d in these texts. In M U L . A P I N the helical rising ( F ) o f Sirius is p l a c e d o n I V 15, t h e date o f s u m m e r solstice, w h e r e a s it ought to h a v e followed S S b y a p p r o x i m a t e l y 2 0 days, suggesting that the empirical foundations of this treatise w e r e extremely m o d e s t . N e v e r t h e l e s s , the first e v i d e n c e w e e n c o u n t e r for a n actual, rather t h a n schematic year length is the statement in M U L . A P I N (ii. 12, amplified in ii. 16) that 1 year = 12 m o n t h s plus 10 "additional d a y s " (ud""'^ diri""'^)

(1.1)

or, equivalently as in ( i i . l 1 a n d 17) that a n intercalary m o n t h is to b e a d d e d e v e r y 3 years, so that 3 years = 3 7 m o n t h s ; € = 1 Ox.

(1.2)

Evidently these are actual rather than schematic m o n t h s , a n d a n a s s u m e d m o n t h length o f 2 9 V 2 d a y s a n d a 3 7 * m o n t h o f 30 days, implies that 3 years = 1092 days, whence 1 year = 3 6 4 d a y s .

(1.3)

K9794/A06578 Additional e v i d e n c e for a n assimied year length o f 3 6 4 d a y s is found in K 9 7 9 4 / A 0 6 5 7 8 , a N e o - A s s y r i a n text (probably later than M U L . A P I N b u t e a r h e r than the 7 * century)'' in w h i c h the intervals b e t w e e n successive ziqpu (culminating) stars e x p r e s s e d in " u s ina qaqqari" (earthly u s ) a d d u p to 3 6 4 over a c o m p l e t e circuit. H e r e the unit " u s ina qaqqari" evidently coimotes (the time r e q u i r e d for the rotation o f the celestial sphere through) that part o f a circle e q u a l to the average p r o g r e s s o f the sim in a n actual rather than schematic day, suggesting t h e b e g i n n i n g s of a shift o f e m p h a s i s a w a y from the schematic calendar t o w a r d s actual days. W h a t e v e r the precise circumstance, this a w k w a r d definition w a s evidently a b a n d o n e d as impractical, in favour o f the m o r e convenient 360"^ circle, distinguished as the " 1 2 beru circle" in a n o t h e r N e o - A s s y r i a n ziqpu star list, B M 3 8 3 6 9 + a n d similiarly m e n t i o n e d in S T T 340.^

^ '

COHEN (1993), pp. 299-301. See discussion in HUNGER and PINGREE ( 1999), p. 84. HOROWITZ (1998), p. 186.

Treatments of Annual Phenomena in Cuneiform Sources

25

Neo-Assyrian Period (-750 to -625) Intercalations-I T h e v a r i o u s rules d e s c r i b e d in M U L . A P I N (II, G a p A 8 - 1 5 ; ii 1 - 1 2 ) for d e t e r m i n i n g w h e n to a d d an extra m o n t h s h o w that it w a s already r e c o g n i z e d that intercalations e v e r y three years w e r e i n a d e q u a t e to m a i n t a i n a consistent c a l e n d a r after 3 0 years s u c h a c a l e n d a r w o u l d b e out o f w h a c k b y a w h o l e m o n t h - a n d equally that n o g e n e r a l s c h e m e for such intercalations w a s yet r e c o g n i z e d . B y the 3'^'' m i l l e i m i u m m o r e frequent intercalations are attested in administrative texts, n o t a b l y o f 2 intercalary m o n t h s in 5 years a n d 3 intercalary m o n t h s in 8 y e a r s , a l t h o u g h n o regular practice is e v i d e n c e d . T h u s a w a r e n e s s that o c c a s i o n a l l y a n extra m o n t h s h o u l d b e a d d e d e v e r y 2 rather t h a n 3 years w a s certainly o f ancient origin, a n d the intercalation rules d e s c r i b e d in M U L . A P I N amoimt to g u i d e s for d e t e r m i n i n g w h e n a 2-year intercalation w a s required, a b s e n t w h i c h a n intercalation e v e r y 3 years w o u l d b e assumed.^ T h e r e are relatively few c o m b i n a t i o n s of 2 - and 3-year intervals from w h i c h to fashion a short p e r i o d intercalation cycle. T h e m o s t o b v i o u s of these are s h o w n in the a c c o m p a n y i n g table together with their a p p r o x i m a t e errors o v e r r o u g h l y 3 0 years. Evidently, 2-years is far too frequent for regular intercalations, while 3 - a n d 5-year cycles p r o d u c e gross errors in o p p o s i t e directions a m o u n t i n g to r o u g h l y a m o n t h in 3 0 years. T h e 8- a n d 11-year cycles are better, yielding offsetting errors of r o u g h l y 5 or 6 d a y s in 32 or 33 years, a m o u n t s large e n o u g h to b e p e r c e p t i b l e from e v e n r o u g h o b s e r v a t i o n s of stellar risings a n d settings. C o m b i n i n g these p r o d u c e s the 19-year cycle, w h i c h has n o sensible error in 2 full cycles a n d r e q u i r e s 5 cycles ( 9 5 years) to r e a c h an error of 1 full day. H o w e v e r , from o b s e r v a t i o n s of risings a n d Intercalations

Total

at intervals o f 2-yrs 3-yrs

extra months

Total years

Total months

Error: o b s e r v e d Epact

s c h e m a t i c date years ÔT

[1

0

1

2

25

15;0

30

-116T]

0

1

1

3

37

10;0

30

+ 3 IT

1

1

2

5

62

12;0

30

-28T

1

2

3

8

99

11;15

32

+6T

1

3

4

11

136

10;54..

33

-5T

2

5

7

19

235

11;3,9..

38

OT

3

7

10

27

334

11;6,40

27

-IT

3

8

11

30

371

11;0

30

+2T

settings its a c c u r a c y w o u l d b e h a r d l y distinguishable from that of either the 2 7 - or 30-year cycles ( b e t w e e n w h i c h the 27-year cycle is slightly better), e x c e p t o v e r several cycles. In the N e o - A s s y r i a n p e r i o d (-746 to - 6 2 5 ) our k n o w l e d g e of actual intercalations is exfremely sketchy before ca. -685 and still i n c o m p l e t e thereafter. H o w e v e r ,

^ Attested intercalations after -750 reflect only three 4-year intervals between intercalations and one 5-year interval, al! between -587 and -536 when irregular intercalations were singularly common, two of which were followed by the only instances of intercalations at 1year intervals.

J.P. Britton

26

certain observations c a n b e m a d e . Figure 1 s h o w s attested a n d inferred intercalations during the N e o - A s s y r i a n period, arranged in c o l u m n s such that the first y e a r in e a c h c o l u m n b e g i n s earlier in the Julian year than a n y of the s u c c e e d i n g y e a r s in that colunm. Intercalary Ululs (VI2) are attested in years 1 a n d 3 o f N a b o n a s s a r (-746 a n d - 7 4 4 ) , e v i d e n c i n g a 2-year interval b e t w e e n intercalations at that date. O n ( m o n t h ) I ( d a y ) l o f 1 N a b o n a s s e r (-746 F e b 2 1 ) the s i m ' s longitude at simset was 3 2 6 ° , its earliest fi^om - 7 5 0 o n w a r d s . B y -685 (3 Seimacherib) there m u s t h a v e b e e n the equivalent o f eight 2-year intervals a n d fifteen 3-year intervals b e t w e e n intercalations ( r o u g h l y 1:2 as in the 8-year cycle), w h i c h shifted the m i n i m a l solar longitude at the b e g i i m i n g of the year (sunset o n I 1) b y a p p r o x i m a t e l y h a l f a sign to 3 4 2 ° . Finally, from - 6 8 5 to -628 ( 1 9 K a n d a l a n u ) the r e c o r d is m o r e c o m p l e t e and reflects three 19-year cycles in the course o f w h i c h the m i n i m a l initial solar longitude increased o n l y 1° to 3 4 3 ° . T h e frequent a p p e a r a n c e of 2-year intercalation intervals suggests a n i m p r o v e d estimate of the relationship b e t w e e n m o n t h s a n d years a n d a better y e a r length than 3 6 4 d a y year reflected in M U L . A P I N . Less clear, h o w e v e r , is w h a t this relationship w a s thought to b e , for a suspicion arises that the B a b y l o n i a n s intermittently allowed the earliest begiiming of the year to m o v e later in the seasonal c a l e n d a r t h r o u g h the intercalation p r o c e s s , to gradually eliminate the effect of the A s s y r i a n r e f o r m o f the Standard Calendar. T h u s w e c a n conclude with confidence o n l y that b y the 8 * century it w a s u n d e r s t o o d that the epact was greater than 10 days or tithis, a n d that the year w a s longer t h a n 3 6 4 days. It is also evident that n o single fixed cycle o f intercalations w a s used, although attested intercalations during this p e r i o d are always separated b y t w o or three years. B M 4 5 7 2 8 = S H . 1 3 5 (81-7-6) T h e first explicit m e n t i o n o f a better annual p e r i o d relation is found in B M 4 5 7 2 8 , a text p u b l i s h e d b y K u g l e r ' ° and described in A p p e n d i x A . T h e text w a s p o s s i b l y written in the 2"** half o f the 7 * century a n d relates c r u d e b u t r e a s o n a b l e sidereal p e r i o d s for the m o o n , planets excepting Jupiter, a n d Sirius in years a n d integer d a y s . T h e stated sidereal m o n t h is simply 2 7 days, while the p e r i o d for Sirius is 2 7 years, a c c o m p a n i e d b y the c o m m e n t " d a y for day y o u see it", implying that 2 7 years = 3 3 4 m o n t h s precisely. T h i s in turn implies a n u n d e r s t a n d i n g that the year w a s a p p r o x i m a t e l y 3 6 5 d a y s long, a n d is the earliest direct e v i d e n c e " w e e n c o u n t e r o f M e s o p o t a m i a n a w a r e n e s s o f this fact. W h i l e the dating o f this text is hardly solid, all its elements s e e m consistent with a date the late 7 * century. A s such it suggests that the simplest short t e r m p e r i o d relations for the m o o n , sun a n d planets w e r e r e c o g n i z e d b y the e n d o f the N e o -

'°

SSB, 45-47; copy Nr. 3, Tafel 2.

" The assumption of a Neo-Assyrian date for this text rests on two (insecure) bits of evidence. One is the preliminary character of the described period relations, and especially that for Mercury, which is otherwise unknown and replaced in later Goal Year texts by the more accurate period relation, 145 phenomena = 46 years . The second is the colophon, which ascribes the tablet to a Laba§i, son of a Babylonian temple official, who may also be mentioned in a letter (LABS 160) from Marduk-âapik-zeri, a Babylonian, to an unnamed Assyrian king as one "proficient in the Series (EAE) and exorcism" and among "twenty able scholars" recommended to the king.

Treatments of Annual Phenomena in Cuneiform Sources

"a jO

o

>n

27

a\ Vî m (S ^ PO IS
"2 00 5 --^
g

^

I

(N

00 so <S fS <s Vû Sû s ç

SA ^

•S

^ 3 o
os

Il

i II

D ^ ^

5 < <: P < < b VO fS s O " n r t r » - ) ( N ' — i O O v O O r - s O < n ' > * r < i « N i - H O O s O O vû s o \ 0 ^ s o \ o 'O «n T) «n T) «n "/i «n ir> CO \ 0 ^ v o s O S Ç v Ç * O S D S ( p s Ç v o s D S O M p s i p s 4 p s i p * Ô S Ç

^^1 I ^

•S f*^
1 ^ § ?^

^ Il Q .

o o s 00 r~- SO Os 00 00 00 00 s^ s ç SO SO SO

S o

S < o o r - s o « n ' ! } ' f n < N ^ o o s o o r - v o « n ' ! f f n < N ^ O O O O O O O O O O O s O s O s O s O s O s O s O s O s ^ r—1 o fO

<4-l Os

c

I I

es

«M y

m 00 r-

3 « o c r - s o ' / ^ ' t m f s

—

3 « « <3 OONOor^-soiOTtmcN-^o

S ^ § S

^

s

so m ri

^7

-H s o

z

•o ^ jD ^

3 Cd « — sO'^'+fOCN'—lOOs 1

1

1

1

1

1

I

S Cd ^ o Os 00 rrj- rf m m m ^ T T T T

r1

1

1

c

G fi > .2 <

fS Tt r-1 1

§

Il

w «3 c < ë S ô û, - s w u c

Z o o • Cd E

^ >^ Cd C/5

' S I - —<-

g

1

s o «A 'tt m

^7

g)

Il ™

»

s^ ^ z

Os

28

J.P. Britton

A s s y r i a n p e r i o d , reflecting a significant a d v a n c e o v e r the a s t r o n o m y of M U L . A P I N b u t n o meaningful m a t h e m a t i c s . T h i s c h a n g e s dramatically in the following p e r i o d .

Neo-Babylonian and early Achaemenid Sources (-625 to ca. -450) B M 36712 (80-6-17, 445+820+919) T h i s text, p u b l i s h e d b y S a c h s and N e u g e b a u e r ( 1 9 5 6 ) , c o m e s fi-om a u n i q u e archive e x c a v a t e d b y R a s s a m in 1879,'^ w h i c h has b e e n the p r e d o m i n a n t s o u r c e of atypical, p r e - A C T m a t h e m a t i c a l a s t r o n o m i c a l texts. T h e text describes t w o v a l u e s for the length o f the year: the first a n d evidently o l d e r value, is 1 year = 364'/2 d a y s , w h i c h the text describes as "falling short" (umattu), w h i c h it corrects b y a d d i n g 0;40 d a y s to m a k e 1 year = 3 6 5 ; 10 d a y s , an interval w h i c h the text explicitly identifies with successive risings of Sirius. T h e following section states that 1 year = 13;22 sidereal months, whence 1 year = 12;22 synodic m o n t h s = 3 6 5 ; 1 0 d a y s ; e = 1 1 ; 0 \

(2.1)

T h e older year length, 364'/2 days, is p e r h a p s associated with an a s s u m e d m o n t h length of 29/4 d a y s a n d the relationship, 59 years = 7 2 9 (=3^) m o n t h s a s c r i b e d in Greek

s o u r c e s to Pythagoras.'^ H o w e v e r ,

it m a y also m e r e l y reflect

a

close

a p p r o x i m a t i o n w h i c h is also a regular n u m b e r , since it is u s e d for that p u r p o s e in the calculations w h i c h follow. W h a t e v e r the case, it implies that 1 year = 1 2 ; 2 1 , 2 1 . . m o n t h s , w h e n c e e = 10;40,40..^, a n d also a m e a n solar m o t i o n o f 80° in 81 d a y s , as n o t e d in the text. T h e revised year length, 3 6 5 ; 10 days together with a n e p a c t o f 11;0^

is

equivalent to the relationship 3 0 years = 3 7 1 m o n t h s = 10,955 days,

(2.2)

1 month = 29;31,41..days,

(2.3)

w h i c h implies that

the first value w e e n c o u n t e r w h i c h clearly reflects a w a r e n e s s that the m e a n m o n t h length e x c e e d s 291/2 days, a n d w h i c h is in fact correct in its first fractional p l a c e . T h e r e m a i n d e r of the text is d e v o t e d to c o m p u t i n g s o m e of the c o n s e q u e n c e s o f these p a r a m e t e r s . T h e final section states that 3 0 years = 4 0 1 sidereal m o n t h s , i m p l y i n g that 1 sidereal m o n t h = 2 7 ; 1 9 , 9 . . d a y s ,

(2.4)

a l t h o u g h e l s e w h e r e the text uses the value 2 7 ; 2 5 days in calculation. A c c o r d i n g to N e u g e b a u e r a n d Sachs the text is written in a Late B a b y l o n i a n script, uses the old fashioned s y m b o l for " n i n e " , a n d contains several instances o f u n u s u a l and s e e m i n g l y archaic t e r m i n o l o g y a l o n g with at least o n e error suggesting that it m a y h a v e b e e n c o p i e d from an even o l d e r tablet. T h e authors n o t e that b o t h its

READE(1986), pp. xixff. Aetius ii. 32.2; Censorinus xviii, 8; xiv, 2; c.f Plato, Republic, 588A; OLMSTEAD (1948), p. 210,n.39.

Treatments of Annual Phenomena in Cuneiform Sources

29

p a r a m e t e r s a n d m e t h o d o l o g y are inferior to those characteristic o f A C T m a t e r i a l , b u t profess uncertainty w h e t h e r the text is " a n evolutionary s t e p " p r e c e d i n g the A C T stage or " a not quite professional p i e c e o f w o r k o f the Hellenistic p e r i o d " . I n fact there s e e m s little r e a s o n to d o u b t that the text reflects a n early effort to estimate the length of the year c o m b i n e d with a mathematical analysis w h i c h g o e s well b e y o n d the use o f simple cycles and w h o l e n u m b e r s found in B M 4 5 7 2 8 . A t the s a m e time it is b o t h less accurate a n d seemingly less confident than B M 3 6 7 3 1 , d i s c u s s e d b e l o w . All in all the text s e e m s m o s t likely to date from the early years o f the N e o B a b y l o n i a n period, m a k i n g it the earliest evidence of the application o f precise m a t h e m a t i c a l p r o c e d u r e s to periodic p h e n o m e n a . B M 36731 (80-6-17, 464+A+B) F r o m the s a m e archive as B M 3 6 7 1 2 , this text contains c o m p u t e d instants (dates a n d times) of solstices a n d e q u i n o x e s together w i t h schematic dates for the settings (sii) and risings (igi) of Sirius. P r e s e r v e d data e x t e n d from year 8 N a b o n a s s a r (-617) to 25 N e b u c h a d n e z z a r (-578), b u t it is likely that the c o m p l e t e tablet c o v e r e d the reigns of N a b o p o l a s s a r a n d N e b u c h a d n e z z a r (-625 to -561). T h e principal fragment of the text w a s p u b l i s h e d b y N e u g e b a u e r a n d Sachs ( 1 9 6 7 , T e x t A , 1 8 3 - 1 9 0 ) , w h o identified its contents a n d dates. Unfortimately, they m i s t o o k the dates o f the solstices a n d equinoxes to b e g i v e n in tithis, w h i c h led to a n e r r o n e o u s depiction of b o t h the underlying structure o f the text a n d the c o m p e t e n c e of its execution. S u b s e q u e n t discovery of t w o small fragments e x t e n d e d the attested range o f the text a n d shed light o n the structure o f the c o l u m n dealing w i t h the settings a n d risings o f Sirius. A reconstruction of the text a n d related discussion is p r e s e n t e d in A p p e n d i x B . A p a r t from the a p p a r e n t extent of the text, three n e w findings e m e r g e d . First, the dates o f the solstices a n d e q u i n o x e s t i u n out to b e d a y s in the actual civil c a l e n d a r (not tithis) w h i c h reflect the n e w p a r a m e t e r , 1 year = 3 6 5 ; 16 d a y s ,

(2.5)

w h i c h is d i v i d e d into equal seasons of 9 1 ; 1 9 days. S e c o n d , a n d in confrast, the dates for the settings a n d risings of Sirius are schematic dates e x p r e s s e d in tithis, w h i c h reflect the familiar p e r i o d relation, 2 7 years = 3 3 4 m o n t h s , implying that 1 synodic m o n t h = 2 9 ; 3 1 , 3 9 . . d a y s a n d 1 sidereal m o n t h = 2 7 ; 19,9..days, essentially the s a m e as in B M 3 6 7 1 2 . Finally b y the e n d of the text, p r e s u m a b l y n e a r the time o f its c o m p o s i t i o n , the s c h e m e ' s s u m m e r solstices are r o u g h l y a d a y t o o early, w h i c h error increases for earlier dates b y r o u g h l y 1 d a y every -40 years. '"^ A p a r t from initial dates for a cardinal p o i n t ( a l m o s t certainly a s u m m e r solstice) and o n e a p p e a r a n c e or d i s a p p e a r a n c e of Sirius, the s c h e m e w o u l d a p p e a r to rest o n an o b s e r v a t i o n that 3 0 years = 10958 days = 371 m o n t h s + 3 d a y s ,

(2.6)

Since a single date and time determines the entire scheme, it is tempting to search for likely anchor points, one candidate being the summer solstice of -557 at dawn (3° nim), not long after the end of the text and corresponding to the midpoint of the Babylonian day in question.

30

J.P. Britton

a n increase of 3 days from the relationships reflected in B M 3 6 7 1 2 . F o r sidereal years, e.g. risings o f Sirius, this is correct to t h e nearest d a y ( a c c u r a t e l y 1 0 9 5 7 . 7 d ) , while for tropical years it is nearly a d a y too h i g h (accurate, 1 0 9 5 7 . 3 d ) . E v i d e n t l y , a c o m p l e t e d a y c o u n t w a s available from at least the b e g i i m i n g o f the N e o - B a b y l o n i a n period. W22801+22805 R o u g h l y 3 0 years later, a r o u n d the e n d of the r e i g n of C y r u s ( - 5 2 9 ) , w e e n c o u n t e r the first explicit u s e of the 19-year cycle in a r e c e n t l y e x c a v a t e d text from U r u k , ' ^ s h o w n as r e c o n s t r u c t e d in A p p e n d i x C. T h e text consists o f t w o c o l u m n s s e p a r a t e d b y 4 8 years, e a c h r o w of w h i c h contains the r e g n a l y e a r a n d s c h e m a t i c d a t e s ( m o n t h a n d tithi) o f s i u n m e r a n d winter solstice. W h e r e relevant, intercalary m o n t h s a r e indicated b y the insertion o f "kin-dir"(Vl2) or "dir"(XIl2) i m m e d i a t e l y after t h e date of the p r e c e d i n g solstice. T h e text c o n t a i n e d data from at least 2 N a b o p o l a s s a r (-623) to 4 A m e l - M a r d u k (-555), b u t s e e m s likely to h a v e b e g u n with year 1 or 0 of N a b o p o l a s s a r (-624 or -625), a n d to h a v e c o n t i n u e d t h r o u g h y e a r s 9 or 8 o f Cyrus (-529 or - 5 3 0 ) . N o dates o f winter solstices are p r e s e r v e d , b u t b e g i n n i n g w i t h t h e s u m m e r solstice o n " d a y " 9 in 1 N a b o p o l a s s a r , s u b s e q u e n t dates reflect the reflect the a d d i t i o n ( m o d u l o 3 0 ) o f a consistent epact o f 11 tithis for 18 y e a r s followed b y a singular e p a c t o f 12 tithis in the 1 9 * year, after w h i c h , as H u n g e r ( 1 9 9 1 ) n o t e s , the s c h e m e r e p e a t s . T h e structmre of this s c h e m e parallels that o f the later " U r u k S c h e m e " w i t h the e x c e p t i o n that the early s c h e m e (henceforth " U l " ) a p p l i e s t o the d a y n u m b e r only; there is n o c o o r d i n a t i o n with intercalations, a n d t h e s c h e m e s i m p l y b e g i n s with the (calculated) solstitial date for 1 N a b o p o l a s s a r . In contrast the later U r u k S c h e m e (henceforth " U 2 " to distinguish it) reflects a settled a n d consistent p r o c e s s of intercalation, w h e r e i n the latest date for s u m m e r solstice in the local calendar (i.e. the year w h i c h b e g i n s earliest in t h e seasonal-Julian c a l e n d a r ) , is I V 7. E i t h e r as a result o r as p a r t o f the systematization o f intercalations, U 2 b e g i n s with this date ( I V 7) a n d p r o c e e d s in identical fashion to U l with a n e p a c t o f I T (12^ in year 19), with a n intercalary m o n t h infroduced w h e n e v e r the d a t e w o u l d e x c e e d I V 7. In contrast t o this o r d e r e d p r a c t i c e the following table s h o w s the r a n g e of S S dates for e a c h cycle of U l c o n t a i n e d in W 2 2 8 0 1 . T h e p r o g r e s s i o n o f earliest dates reflects a steady u p w a r d s shift, eventually stabilized at III 8 in the U r u k S c h e m e , b u t there is n o c o r r e s p o n d i n g c o n v e r g e n c e of latest dates. C o n s e q u e n t l y , d a t e s r a n g e o v e r m o r e than a m o n t h a n d a half, a n d a given " d a y " date c a n often fall in different m o n t h s in successive cycles. In U 2 a single " d a y " date a l w a y s falls in the s a m e m o n t h , I V 7 is a l w a y s the latest S S date, a n d all dates fall within 2 9 tithis o f e a c h other.

Copies of the two surviving fragments were published in VON WEIHER ( 1 9 9 3 ) , No. 1 6 9 , and HUNGER ( 1 9 9 1 ) gives a transliteration and brief commentary establishing the dates of the preserved entries and noting the evidence of a 19-year scheme.

Treatments of Annual Phenomena in Cuneiform Sources

U l Cycle

Median Date (Range) -615 (-624 t o - 6 0 6 ) - 5 9 6 (-605 to -587) - 5 7 7 (-586 to -568) -558 (-567 to - 5 4 9 ) -539 (-548 to -530)

Earliest S S

1 2 3 4 5 U2

- 3 6 0 (-369 to -350)ff

III 8

III III III III III

1 2 5 5 9

31

Latest S S IV IV IV IV IV

23 15 12 23 21

IV 7

T h e resulting s c h e m e s are c o m p a r e d in Figure 2 , a c c o m p a n i e d b y the Julian dates for successive cycles of U l , w h e r e underlinings m a r k dates p r e s e r v e d in the text. O n e c o n s e q u e n c e of the different starting points is that U l avoids " d a y " 3 0 , w h i c h in h o l l o w m o n t h s b e c o m e s d a y 1 of the following m o n t h . H o w e v e r , for the last 11 years of U l the " d a y " dates o f the t w o s c h e m e s are identical, while those for the first eight years o f U l are o n e d a y later than the c o r r e s p o n d i n g dates in U 2 . Interestingly, the dates in c o n m i o n are connectable in the sense that p r o c e e d i n g in 19-year increments from a n y o f such day date in one s c h e m e yields the s a m e d a y date in the other. T h i s is s o m e w h a t u n e x p e c t e d since U 2 reflects a settled intercalation p r o c e s s , w h o l l y absent in the p e r i o d c o v e r e d b y the present text. Evidently there w a s m o r e e n ^ i r i c a l continuity b e t w e e n the t w o s c h e m e s than the b r o a d difference in intercalation practice might suggest. I n any event for 11 o f the 19 years in e a c h cycle there w a s continuity of schematic simrmier solstice dates e x t e n d i n g firom the reign o f N a b o p o l a s s a r to the e n d o f cimeiform astronomy. T h e s u m m e r solstice dates in W 2 2 8 0 1 are o n a v e r a g e d a y t o o early for the last 19 years c o v e r e d b y the text - substantially better than the c o r r e s p o n d i n g error in B M 3 6 7 3 1 - a n d r o u g h l y '/2 a day t o o early over the entire text. B e c a u s e the yearlength implied b y the 19-year cycle is slightly too long, this error d i m i n i s h e s w i t h time, trending to z e r o a r o u n d - 4 7 0 a n d b e c o m i n g half a d a y late o n a v e r a g e b y - 3 6 0 . T h u s for all intents a n d p u r p o s e s the s c h e m e was accurate in the p e r i o d i m m e d i a t e l y following the contents of the text a n d r e m a i n e d so until r o u g h l y the m i d d l e of the 4 * century, w h e n - p e r h a p s not coincidentally - the earliest e v i d e n c e o f the U r u k S c h e m e a p p e a r s . In contrast U r u k S c h e m e dates are accurate to within h a l f a d a y o n average from - 5 0 0 to - 3 0 0 , are r o u g h l y a d a y late b y - 2 0 0 , a n d are b e t w e e n 2 a n d 3 days late b y P t o l e m y ' s time. T h e a p p e a r a n c e o f the 19-year cycle in schematic use b y the last quarter o f the 6 * century, suggests that its superiority over the 27-year cycle w a s r e c o g n i z e d during the r o u g h l y 3 0 years b e t w e e n the latest contents of B M 3 6 7 3 1 a n d W 2 2 8 0 1 . T h i s is consistent with r o u g h l y a century having p a s t since the b e g i n n i n g of N a b o p o l a s s a r ' s reign, e n o u g h time to m a k e clear the inferiority of the 2 7 - a n d 3 0 year cycles w h i c h w o u l d exhibit errors o f -4 a n d + 6 days respectively o v e r that interval. F r o m this p o i n t on only the 19-year cycle figures in theoretical and practical d e v e l o p m e n t s .

32

J.P. Britton

Q

r)vC>t-OOOsO>—I

o ON oo vo v-> 0\ On 00 00 00 00 00

T t

t

t

T t

T

(^«N-HOONOOr^^OW^-^rO

oooooooor-~r-t^r~r~r~-t^

'-HOOOOOOOOOOOOsOnOvOnOnOsOn

oor-vû«nTtmr-)-HOONOor-vo>n-<+r) m 'o «n «n
o

oor^^w^-'tr'ifS-^oovoor-vo'OTtmcs-HO

00

I

I

I l l so

Tt CO (N O ON m m m in ir> IT) IT) IT) m IT) IT)

NO'^^'îl-r'ieN'—lOONOOt-voio-^m oooooooooooooor^t^r-r-r^r-ri/imw^mmi/^v^w^ininir^ininin fO«N — O 0 N 0 0 r - ^ O » O ' * r < - i c M ' — o ON 00 r~ OOOOOnOnOnOnOnONOnOnOnw>««WW s o ^ M p s c v i ^ i / j ^ i / l i i / j i v ^ - i n i o i n i r ^ i /O jn^ 00 i n i 00 / ^ i00 o ••i-mfs^ooNoor-NOi/^-rtrofN

— o

^

??? —<«NroTtinNor~-oooNO^cNro"<*i/i^r-oo

3^

Treatments of Annual Phenomena in Cuneiform Sources

33

Intercalations-II B y early in the N e o - B a b y l o n i a n period, it w a s evidently u n d e r s t o o d that p e r i o d relations o f 19, 2 7 , or 3 0 years best d e s c r i b e d the relationship b e t w e e n m o n t h s a n d years, a n d b y late in the 6^ centiuy, it h a d b e c o m e clear that the 19-year cycle w a s superior to the others. Therefore, it is curious that attested intercalation p a t t e r n s of the p e r i o d reflect inferior cycles o f 5 a n d 8 years. Figure 3 s h o w s the intercalations of the period, arranged, as before, in coliunns begiiming with the year w i t h the lowest initial solar longitude (i.e. b e g i n n i n g earliest in the seasonal/Julian calendar) in that c o l u m n . T h e r e are t w o e x c e p t i o n s , 41 N e b u c h a d n e z z a r (-563) a n d 1 Cyrus (-537), b o t h involving VI2S ( U ) w h i c h s h o u l d h a v e b e e n in the p r e c e d i n g year and w h i c h are followed i m m e d i a t e l y (and exceptionally) b y intercalary XII2S in their n o r m a l p l a c e s . T h e first o f these is a n o m a l o u s ; the s e c o n d p e r h a p s d u e to s o m e disruption of the n o r m a l p r o c e s s s t e m m i n g from the a c c e s s i o n of Cyrus. F r o m colimin to c o l u m n the initial solar longitude in year 1 a d v a n c e s steadily from 3 3 9 ° in - 6 2 5 to 3 5 4 ° in - 5 8 3 , w h e r e it p a u s e s for several cycles o f 19 years before a d v a n c i n g to 3 5 7 ° in - 4 9 1 . In the s h a d e d coliunns w e find instances of successive t w o year intervals b e t w e e n intercalations (exceptionally as far as w e k n o w for the entire p e r i o d from - 7 5 0 o n ) , as well as similarly u n u s u a l intervals o f 4 years b e t w e e n intercalations. T h e effect of the former (and the excess of 2-year intervals in the m i d d l e s h a d e d c o l u m n ) w a s to accelerate the shift o f the b e g i n n i n g o f the year so that the e q u i n o x (give or take 2 days) always falls in m o n t h X I I as in the O B version of the schematic calendar; the effect o f the latter w a s to restore the pattern o f intercalation to a m o r e m e a s u r e d p a c e of a d v a n c e . W a s there a dispute b e t w e e n a d v o c a t e s of a n u n m e d i a t e return to the traditional B a b y l o n i a n c a l e n d a r a n d a faction favouring a m o r e gradual transition? W h a t e v e r the case, during this p e r i o d the b e g i n n i n g o f the B a b y l o n i a n year continued to m o v e p r o g r e s s i v e l y later in the seasonal calendar, I n t e r c a l a t i o n s - I l l : S t a n d a r d i z a t i o n of the 1 9 - y e a r cycle B e g i n n i n g in - 4 8 3 (2 X e r x e s ) the p a t t e r n o f intercalations c h a n g e s a p p r e c i a b l y , as m a y be s e e n from Figure 4 . F r o m this p o i n t o n w a r d s intercalations o c c u r in regular cycles of 19 years, a n d the u p w a r d shift in the earliest solar longitude o n I 1 from intermittent successive cycles o f 5 or 8 years ceases. R e m a r k a b l y , it also c e a s e s at the p o i n t w h e r e this longitude is 3 5 9 ° , o n l y 1° short o f the actual V E , thus very nearly restoring p r e c i s e l y the tradition of the O B schematic c a l e n d a r that I 1 o c c u r s after V E (or equivalently that V E occurs in m o n t h X I I ) . T h i s suggests a better k n o w l e d g e of the date o f the actual e q u i n o x t h a n is otherwise attested, since virtually all p r e s e r v e d reports o f e q u i n o x dates are later a n d schematically calculated from the dates of S S a s s u m i n g seasons of equal length. A s e c o n d c h a n g e is that intercalary Ululs (VI2) d i s a p p e a r e x c e p t in the first year o f each cycle, in contrast to prior years w h e n they usually m a r k e d the earliest y e a r in a s e q u e n c e , b u t also a p p e a r e d irregularly at other p o i n t s . D u r i n g the r e i g n of A r t a x e r x e s 1 they d i s a p p e a r altogether, p e r h a p s a c o n s e q u e n c e o f his p r e d e c e s s o r

34

D ^

•r

r

J.P. Britton

o

m

<

<

o

Û

c/o

00

»n O S O N O O O O O O O O O O O O

g *^ P o

13

<

<

^

<

<

o

^

1

^

<

1

1

1

1

1

1

1

^

^

^

><

<

1

1

1

1

1

1

1

1

OS I

L

I

1

I

<

<

<

<

1

^

1

I

1

1

<

1

<

1

>^

1

I

<

D

<

D in

(4-1

;D

ID

1

<

1

1

<

1

1

1

1

1

1

ONCTNOSOSONONOSONÌJVOSOOOOOOOOOOOO

1

13 1

<

to

<

1

1

1

1

<

1

1

1

<

1

1

1

<

I

<

<

<

is t:

«n

o s o o r - s o m - ^ m r s i . — i O O s o o r - s o i n ' ^ f < ^ ( N < — 1 >n s o s o ^ ^ s o s o s o s o s o s o m i n i n m i n i n i n i n m

<

<

<

<

<

<

CM

<

O

OS

1

1

D >< s o

^ " ^

1

1

<

1

1

1

1

£ <

1

1

1

<

1

1

<

1

1

I

1

<

•

1

<

O o o r ^ s o m ' ^ m c S ' - o o s o o r ^ ^ i n ' s t m f S — ' O o o o o o o o o o o o o o o o o o o r - - r - - r - > r - r ^ r ~ - t ^ t ^ r ~ - r 1

1

1

0 :2 S ^

1

1

<

1

'

<

<

<

<

<

t—1

i< t

1

O O O O O O O O 0 s 0 s 0 s O > 0 s C T \ O \ O \ O > 0 s 0 0

^

0^

<

^

1

<

o o s o o r - ^ s o m - ^ r ' ^ f s ^ o o s o o r ^ s o i n - ^ f o f s so i n ' ^ ^ ^ ' ^ ' ^ ' ^ ^ ' ^ ' ^ ^ f * ^ » ^ < ^ f ^ < ^ f ^ ' ^ ' ^ <^ 1 1 1 1 1 1 1 1 1

o 1

1

<

o

o

<

1

<

L

- r ^ t ^ r ^ r - t ^ r - s o s o s o s o s o 1

D <

L

<

«ri »n s o s o v o s o s o m i n i n i n » n » n > n i n » N M T T T I - T T - 5 T 1

3

<

00

00

^

C P <

2

1

D

o

O ^

<

i<

S

3

<

1

<

\D t: •<^

s O i n - ^ R O F N R - i O O s

â

<

^ O c j s o o t ' - s o i n ' ^ r ^ f S . — l O O s o o r ^ - s o i n - ^ m

^ OS

•g o

<

1

so

M

<

P

CO

JO

^

< s ^ o c 3 s o o r ^ s o » N T I - f O f S ' — l O O N O o r ^ s o m - ^

00

&

Q

^^

^ ^ ^ O O O O O O O O O O O s O s O s O s O s O s

•72 (N s o

m

S 7 ^ * ^ ° ° ^

"S

1

W

•72

35

Treatments of Annual Phenomena in Cuneiform Sources

<

1

so

1

< ^

<

<

<

<

<

s o m ' ^ m f s . — l O O s o o r - s o m - ^ m c S r - H O c ^ s o o 00 OS ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ m in ^

OS OS

1

1

1

<

1

1

1

1

'

CO

<

<

<

<

i n ' ^ r o f s — ' O o s o o r ^ - ^ m - ^ r ^ c s ^ o o s o o r * -

<^ 1

o

o

o

o

o

o

o

o

fc<

S Û V O V O S Û S O S O V O S Û I

1

1

1

1

1

1

1

-

OS

a> cs

in

^

OS

X CO

v O s O v n s o s o v o ^ v S

CS

X D ON

»n
m
•RI

•S

^ K>

CO ^->

m

•S

cS

C/3

1

1

1

<

CO

I

<

<

L

L

<

• ^ ^ f v j ^ o o s o o r - ^ s o i n r r r o r N ' — « o o \ o o r - - s o s o s o s o s o s o i n i n i n i n m i n i n i n i n i n ' ^ ^ ' ^ ' ! ! "

P < c o £ < < < < f < ^ < s . — ' O O v o o r ^ S O M T F F ^ - i r ^ R - i O o s o o r ^ s o i n o o o o o o o o r - r ^ r ^ t ^ r ^ r ^ r ^ r ^ t ^ t ^ s o s o s o ^ s o •

^ 1

•

^ 1

'

^ • 1

^ • 1

^ • 1

^ " 1

^ • 1

^ • 1

^

1

•

^

1

•

^

1

•

^

1

'

^

" 1

^

" 1

^

• 1

^

• 1

^

• 1

^

• ^ 1 I

<

fc^ II

40

1

<

^

00

Z

1

P

00

' — ' ( N m r f i n s o r ^ o o o s o ^ r s r ^ ^ - ' i - m ^ r ^ o o o N

*j § R - H F S R O ' ^ i n s o r - o o o s o . — i < s < ^ " ^ i n s o t ^ o o o s

36

J.P. Britton

having b e e n m u r d e r e d in one, but thereafter w h a t is b y n o w a n o r m a l pattern resumes. Finally, the dates o f s u m m e r solstice (SS) b e g i n n i n g with IV 7 in year 1 and p r o c e e d i n g with an epact of 11 tithis, stabilize in the pattern s h o w n in the rightmost c o l u m n o f Figure 4 , w h i c h is identical with the dates found later in the U r u k S c h e m e , a n d w h i c h differ in years 12 through 19 from the day dates in the U l . T h e difference arises from begiiming with day 7 (instead of d a y 9 in U l ) a n d p r o c e e d i n g as before in a n n u a l increments of 11 tithis ( m o d u l o 30), but with the additional difference of assigning d a y 7 to m o n t h IV as the latest date for SS a n d intercalating w h e n e v e r this date w o u l d b e e x c e e d e d . T h u s w a s the schematic 19-year cycle seen in W 2 2 8 0 1 transformed into an intercalation s c h e m e w h i c h r e m a i n e d essentially u n c h a n g e d until the d i s a p p e a r a n c e of cuneiform a s t r o n o m y m o r e t h a n five centuries later.

Late Achaemenid Sources (ca -450 to ca -330) Uniform Solar Motion A r o u n d the m i d d l e of the s e c o n d half of the 5 * cenhiry the infroduction o f a u n i f o r m z o d i a c c o m p r i s e d of twelve 3 0 ° signs p e r m i t t e d the calculation of longitudes a n d the d e v e l o p m e n t o f fimctions with longitude as an argument. T h e earliest o f these fimctions a p p e a r s in B M 3 6 5 9 9 a n d B M 3 6 7 3 7 ( A A B O E a n d S A C H S ( 1 9 6 9 ) , T e x t s B , C , D ) also k n o w n as T e x t S, w h i c h p r e s e r v e s calculated fimctions and observational r e m a r k s for 3 8 solar eclipse possibilities over one saros or 2 2 3 m o n t h s from - 4 7 4 to - 4 5 6 . L o n g i t u d e s of the syzygies are c o m p u t e d b y w h a t is essentially a m e a n m o t i o n s c h e m e b a s e d o n the 19-year cycle w h i c h assumes that A?ii2m =

1''" - 1 0 ; 3 0 ° (V = 12;21,37.."'), a n d

A?i223m = 1 8 ' ' V 1 0 ; 3 0 ° (V = 12,22,7,50..""), w h e n c e 19''"+ 0;0°

(3.1)

(1^=12,22,6,18..'").

W h a t is n e w h e r e is n o t the p e r i o d relation, b u t the formulation of a ftinction to describe c h a n g e s in the positions o f syzygies. It is also n o t e w o r t h y that the e s t i m a t e d excess over c o m p l e t e revolutions for 2 2 3 m o n t h s is v e r y nearly accurate, the correct (sidereal) a m o u n t b e i n g 10;33°. W e shall e n c o u n t e r this p a r a m e t e r again in System B . C o l u m n * and the S y s t e m A M o n t h L e n g t h A far m o r e radical a d v a n c e is found in C o l u m n 4>, w h i c h describes the variation in the length of 2 2 3 m o n t h s d u e to limar a n o m a l y a n d serves as the a r g u m e n t o f lunar a n o m a l y in the S y s t e m A lunar theory. R e m a r k a b l y , C o l u m n $ a l o n g with intimately related C o l u m n F, w h i c h describes the lunar velocity at syzygy, are b o t h attested in T e x t S together with a primitive fimction for eclipse m a g n i t u d e s w h i c h could n o t (easily) h a v e b e e n c o m p o s e d too m u c h later than the contents of the text.'^ C o l u m n ^ ' s next a p p e a r a n c e is in B M 3 6 8 2 2 , a n early lunar e p h e m e r i s for - 3 9 7 / 6 ,

See BRITTON (1999), n. 46, p. 252.

Treatments of Annual Phenomena in Cuneiform Sources

37

p u b l i s h e d b y A A B O E and S A C H S ( 1 9 6 9 ) as T e x t A . T h i s i m p o r t a n t text includes a variant of C o l u n m F , the first e x a m p l e of a S y s t e m A s c h e m e d e s c r i b i n g the effects of solar a n o m a l y , a n d seemingly primitive s c h e m e s for the length of daylight and the variable l e n g t h of the month. C o l u m n $ serves principally as a n a r g u m e n t in d e t e r m i n i n g the v a r i a t i o n d u e to lunar a n o m a l y in o n e , six, 12, 2 2 3 , a n d 2 3 5 m o n t h s , b u t b u r i e d in its structure is a substantially i m p r o v e d value for the m e a n synodic m o n t h . M A = 2 9 ; 3 1 , 5 0 , 6 days,

(3.2)

w h i c h is correct in its second fractional p l a c e (and n o t far off in its third). E v i d e n c e for this p a r a m e t e r is found in B M 3 6 3 1 1 , p u b l i s h e d b y A A B O E ( 1 9 6 8 ) as T e x t E, w h i c h led h i m to identify securely the physical significance of $ . Details o f the derivation m a y b e found in B R I T T O N ( 1 9 9 9 ) , b u t briefly T e x t E contains a c o l u m n of $ values distributed over a cycle of lunar a n o m a l y and representing the length o f 223 m o n t h s following an initial s y z y g y followed b y a c o l u m n called A of c o r r e s p o n d i n g values for the length of the 12 m o n t h s p r e c e d i n g that syzygy. B o t h c o l u m n s vary o n l y with lunar velocity, and b o t h require offsetting corrections to accurately reflect the effects o f solar a n o m a l y . H o w e v e r , the s u m of the two c o l u m n s gives the length of 2 3 5 m o n t h s or 19 years, o v e r w h i c h the effects of solar a n o m a l y vanish, since syzygy recurs at (very nearly) the s a m e longitude. T h e resulting function is perfectly s y m m e t r i c a l a b o u t its e x t r e m e s a n d h a s the median value, 2 3 5 m o n t h s = 254 sidereal m o n t h s = 6 9 3 9 days + 4 , 7 ; 2 1 , 1 7 , 4 6 , 4 0 ° ,

(3.3)

implying that 1 m e a n synodic m o n t h = 29;31,50,6,0...days,

(3.3.1)

1 sidereal m o n t h = 2 7 ; 1 9 , 1 7 , 4 6 . . . d a y s ,

(3.3.2)

j [ i ( v j = m e a n lunar velocity = 13;10,34,59,30..°/d = 1 3 ; 1 0 , 3 5 7 d , a n d fiiAK)

= m e a n synodic arc = 2 9 ; 6 , 1 9 , 2 2 , 5 8 . . 7 m

(3.3.3) (3.3.4)

T h e m e a n daily lunar velocity turns u p later in the S y s t e m B lunar theory, b u t it is the substantially accurate m o n t h length w h i c h is the m o s t interesting a d v a n c e . H o w it w a s a c h i e v e d is imcertain, b u t it is suggestive that 4 2 6 7 s u c h m o n t h s contain a l m o s t e x a c t l y 1 2 6 0 0 7 days or that 126007 days

4267months = 29;31,50,6,2..days/month.

(3.4)

T h i s interval, w h i c h corresponds to 17 2 5 1 - m o n t h p e r i o d s of limar a n o m a l y , is the only interval o f practical length in which there is a n e a r return (ô^sid =^ -6;21°) in solar p o s i t i o n so that the effects of b o t h solar a n d lunar a n o m a l y are v e r y n e a r l y eliminated.'^ Usefully, as noted b y P t o l e m y , it is also often b o u n d e d b y eclipses.

The next closer return in X requires 41 periods (10291 months=832 years) for which ô>.s,n^2;30°, but òri =91°, hence no eclipse pairs except at twice the interval. As AABOE (1955) p. 124 notes, intervals of 13 and 18 anomalistic periods result in closer returns in nodal elongation, but yield substantial differences in longitude and thus solar anomaly. Ptolemy's discussion emphasizes the frequent presence of eclipses at the ends of such intervals, obscuring the fact that it is also the smallest interval with a near return in solar as well as

J.P. Britton

38

since t h e ( m e a n ) c h a n g e in n o d a l e l o n g a t i o n o v e r this interval, ôr|4267m = 11;19°, is less than half the z o n e within w h i c h lunar eclipses c a n o c c u r ( c a . 2 4 ; 2 2 ° ) . If this w a s in fact the origin o f M A - a n d it is difficult t o think o f a m o r e plausible alternative'^ - the e v i d e n c e o f T e x t A t o g e t h e r with t h e fact that eclipse r e c o r d s a p p e a r t o b e g i n with t h e lunar eclipse o f - 7 4 6 F e b 6 i m p o s e s tight limits o n the p o s s i b l e time frame for t h e determination o f this p a r a m e t e r a n d the invention o f Column

O n e t h e o n e h a n d it could n o t h a v e b e e n earlier t h a n t h e eclipse o f - 4 0 1

F e b 2 ( 4 2 6 7 m o n t h s after t h e first r e c o r d e d eclipse). O n t h e other h a n d , if w e a s s u m e that T e x t A w a s c o m p o s e d n o t t o o long after its contents (-397), t h e n M A a n d C o l u m n «ï> w o u l d h a v e h a d t o b e established s o m e t i m e a r o u n d this date a n d after - 4 0 1 . B e t w e e n - 4 0 1 a n d - 3 9 5 there w e r e 3 pairs o f eclipses s e p a r a t e d b y 4 2 6 7 m o n t h s w h o s e b e g i i m i n g s w e r e visible at B a b y l o n , as s h o w n in F i g u r e 5. Unusually, e a c h pair w a s r o u g h l y t h e s a m e m a g n i t u d e , a l l o w i n g meaningfiil c o m p a r i s o n s o f b e g i n n i n g times, a n d all b e g a n in the m o r n i n g w a t c h - m o s t l y at least 3 h o u r s before sunrise. F o r the three pairs the a v e r a g e e x c e s s o v e r w h o l e days c o m p u t e d from eclipse b e g i n n i n g s w a s slightly m o r e than 1 h o u r ( 1 5 ° ) with a n a m p l i t u d e o f very n e a r l y 1 hoiu*. Julian Date GN2

Y

M

Babylonian Date

D

Reign

Yr Mo d

Syz

Beg End

Mag

(UT) (Bab Loc T)

E

A

SR

(° f. north) (BLT)

3142

-746 FEB

6 NBNSR

0

XII 14

1.44 2.35 6.01

0.93

139

264

6.71

3148

-746 AUG 2 NBNSR

1

VI 14

0.82 2.12 5.49*

-1.03

46

277

5.02

3189

-743 NOV 25 NBNSR

4

IX 14

1.54 3.04 6.38

0.97

109

235

6.79

7409

-401 FEB

2 ARTX2

2

XI 14

2.39 3.24 6.65

7415

-401 JUL 29 ARTX2

3

IV 14

7456

-398 NOV 21 ARTX2

6 VIII 14

0.74

134

245

6.73

3.79 7.09*

-0.95

55

291

5.00

2.49 3.81 7.21*

1.03

95

225

6.77

2.63

7409-3142

Excess over whole days (hrs)

0.95

7415-3148

Excess over who le days (hrs)

1.81

1.67

7456-3189

Excess over whole days (hrs)

0.95

0.77

1.24

1.11

Average

0.89

F i g u r e 5 : L u n a r eclipses separated b y 4 2 6 7 m o n t h s visible in B a b y l o n from -746 to -395

lunar anomaly. In BRITTON ( 1 9 9 9 ) , pp. 2 1 9 - 2 2 1 I suggest that this parameter derived from the theoretical structure of Column O and an estimate that the longest interval of 2 3 5 months was < 1 0 ° in excess of 6 9 4 0 days, while suggesting in note 2 7 that the process might have begun with ^(Vm) = 13;10,35°/d since either procedure entails small roundings to arrive at 235Mmax= 6939**+ 1245*, as we find from Text E. However, I now believe that the derivation proceeded from an assumption that MA=29;3 1,50,6'' - derived from 4 2 6 7 months as suggested - and that 235M™x was derived strictly from the sum of 2 3 5 M A (834;48,21..*) and the theoretical quantity, 2 3 5 M „ ^ - 235M,vg

(410;10,37,30<'), whence 235M™x= (6939**+) 1244;58,58..'' s

1245*, without appreciable rounding. 1 am indebted to Dennis Rawlins for conversations which led to this understanding.

Treatments of Annual Phenomena in Cuneiform Sources

39

A s it h a p p e n s r e c o r d s of the first t w o of these eclipses b e g i n the c o m p i l a t i o n p r e s e r v e d in L B A T 1413 ( B M 4 1 9 8 5 ) , recently p u b l i s h e d in H U N G E R ( 2 0 0 1 ) , T e x t 1. T h e text b e g i n s as follows:'^ L B A T 1413: edge ina a-mat ^EH u G A S A N - m lis-lim obv. 1 2

1,40 M U - S A G N A M - L U G A L - L A [o o o] § E 5 I T U 14 U4-ZAL G A R àd '^x\o o]

3

2,10MU 1 KIN[1]V G A R m a S I S A R [o o]

4

[ x ] ULÙ D[IR]

At the c o m m a n d of Beltija, m a y it go well

"^Bel

and

1,40 A c c e s s i o n year of [o o o] (Month) morning eclipsed [ Year 1 occurred;

XII, 5 m o n t h s , 14*, watch, it occurred, x 0 o]. ( m o n t h ) V I [1/4'''^ it it b e g a n in the n o r t h

[00]

"^GIN^

àd

§Û

KIN

[x], the south w i n d b l e w . It set eclipsed, ( m o n t h ) V I 2 .

T h e first eclipse is only reported as b e g i n n i n g in the m o r n i n g watch,^° w h i l e the same w o u l d b e a s s u m e d for the second eclipse (no time is given), since it r e p o r t e d l y set eclipsed. T h e s e reports lack sufficient precision to p e r m i t a n acciurate a s s e s s m e n t o f the excess o v e r w h o l e days b e y o n d observing that b o t h sets b e g a n in the s a m e watch, a n d that the m o o n set eclipsed in b o t h of the s e c o n d pair. T h u s the inference that 4 2 6 7 m o n t h s = 126007 days with n o sensible excess, w o u l d b e r e a s o n a b l e from these early r e c o r d s g i v e n their imprecision, but difficult to m a i n t a i n from m o r e precise r e c o r d s w h i c h w o u l d reflect an excess ranging from r o u g h l y 0 to 2 h o u r s a n d averaging v e r y n e a r l y 1 hour. H o w e v e r achieved, the p a r a m e t e r M A represents a n e w level o f b o t h theoretical sophistication a n d empirical a c c u r a c y which, c o m b i n e d with the 19-year cycle, implies that 1 year = 2 3 5 / 1 9 m o n t h s = 365;14,48,4..days,

(3.5)

a p a r a m e t e r a d o p t e d b y H i p p a r c h u s (-140) a n d P t o l e m y ( + 1 4 0 ) as the length o f the tropical year. B M 3 6 8 2 2 -First C u t at S o l a r A n o m a l y B M 3 6 8 2 2 also contains the earliest e v i d e n c e of a S y s t e m A s c h e m e depicting the variation d u e to solar a n o m a l y o n the m e a n synodic arc as a function o f longitude. T h e central feature o f such s c h e m e s is an a s s u m p t i o n (period relation) that the p h e n o m e n o n in question repeats after H events c o r r e s p o n d i n g to Z cycles o f variation. T h e z o d i a c is then divided into H intervals, distributed a m o n g s o m e

" My translation differs in minor details from Hunger's, principally in being slightly more literal. Other differences include adding the word "eclipsed {âdy\ at the end of line 2 and substituting "14", the date of the actual eclipse for the broken "15" in line 3. Morning watch began at 2.23*" (Babylonian Local Time) by modem calculation.

40

J.P. Britton

n u m b e r o f z o n e s o f constant interval length, Ij, w h e r e Z such intervals c o r r e s p o n d to the difference in longitude b e t w e e n two c o n s e c u t i v e p h e n o m e n a . H e r e the s c h e m e consists o f 2 z o n e s a n d a s s u m e s the rather p o o r p e r i o d relation, n = 7 4 3 ( 1 2 , 2 3 ) m o n t h s = 6 0 (1,0) = Z years; 1 year = 12;23 m o n t h s ,

(3.6)

together with the p a r a m e t e r s :

Interval, Ij N u m b e r , Vj

Zone I

Zone 2

0;28,20° 414

0;30° 329

ratio 17:18

(3.6.1) (3.6.2)

ttj = Ij-Vj

195;30°

164;30°

(3.6.3)

B e g i n n i n g at A,o

343;30°

179;0°

(3.6.4)

Wi = Z ì i

28;20''

30;0°

(3.6.5)

T h e effective a p o g e e of the s c h e m e ( m i d p o i n t o f the s l o w [zone 1] arc) is 8 1 ; 1 5 ° , w h i c h is (surprisingly) close to the 8 0 ° found in the c o m p l e t e d S y s t e m A . T h e vernal p o i n t m a y b e at 10° as in S y s t e m A , although 12° is n o t i m p o s s i b l e . W i t h a 10° vernal point, the s c h e m e yields the seasons s h o w n in F i g u r e 6. A p a r t from the year length, these are scarcely distinguishable from those o f S y s t e m A or H i p p a r c h u s / P t o l e m y , a n d only s u m m e r is a w h o l e d a y too long. T h e year length is a c o n s e q u e n c e o f the p o o r p e r i o d relation, w h i c h m a y h a v e n e v e r b e e n c o n s i d e r e d a serious p a r a m e t e r , b u t s i m p l y a convenient a p p r o x i m a t i o n for w o r k i n g o u t the principal features o f the s c h e m e eventually found in S y s t e m A.

BM Spring (VE~SS) Summer (SS~FE)

36822 93;48..'' 9 3 ; 9.."* 88;35..'' 90;

System A

Hipparchus/ Ptolemy

(-400) Modern

Fall ( F E ~ W S ) Winter (WS~VE) Year

365;41..'*,

94;29..' 92;43..'' 88;35..*' 89;26..'* 365;15,37..'*

94;30' 92;30'^

94;6..' 92;7..''

8 8 ; 7,30"^ 9 0 ; 7,30" 365;14,48'*

88;33..'* 90:26..'* 3 6 5 ; 14,34..'*

VE~FE SS~WS

186;58..'' 181;45..'*

187;13..'' 181;19..''

187;0'* 180;37,30''

186;14..'' 180;41..''

Apogee

81;15°(sid)

80;0°(sid)

70;25°(sid)^'

70;24°(sid)

Figure 6. L e n g t h s o f S e a s o n s C o m p a r e d System A S o far n o t h i n g w e h a v e e n c o u n t e r e d has m o v e d b e y o n d the p e r i o d relation o f the 19year cycle, w h i c h is strictly assvmied in b o t h C o l u m n a n d the m e a n m o t i o n s c h e m e found in T e x t S. T h i s c h a n g e s with the frill lunar S y s t e m A , w h i c h is first attested in B M 4 0 0 9 4 ( A a b o e , 1969) for the y e a r s - 3 1 8 t o - 3 1 5 , b u t w h i c h w a s

Computed from Hipparchus's tropical apogee of 65;30° (-140) by X.sid=X.trop + 9.9° 1.3825°(Y+500)/100. Ptolemy's retention of the same tropica! apogee would be equivalent to a sidereal longitude of 66;33°. The modem sidereal longitude is calculated in the same fashion for -400.

42

J.P. Britton

I = w / Z t o b e a terminating fraction. A s it h a p p e n s for I > 0;3°, there are o n l y two close a p p r o x i m a t i o n s , n

Z

Y(months)

h

1971 ( 3 2 , 5 1 )

160(2,40)

12;22,7,30

0;11,15°

- M =-0;11..°

AÀ.ssm

2783(46,23)

225(3,45)

12;22,8

0;8

-2-I = - 0 ; 1 5 °

(3.10)

O f these 11= 2 7 8 3 , Z = 2 2 5 leads to the closest a p p r o x i m a t i o n a n d also yields the c o n v e n i e n t interval, I2 = 0;8°. T h e alternative is m o r e accurate b u t a less g o o d a p p r o x i m a t i o n a n d leads to the less convenient interval I 2 = 0;11,15°.^^ T h i s leaves the ratio I i : l 2 as a free choice, subject to the constraint that if Ii:l2 = (^-^):^ (^ a n d k integers), q m u s t b e a regular n u m b e r . F o r k=l, Vi m o s t n e a r l y equals V2 if ^ = 16 ( I i : l 2 = 1 5 : 1 6 ) , and a m o n g the p r o x i m a t e alternatives this is the o n l y c h o i c e w h i c h results in integer 05s. T h a t I | a n d I2 are m u t u a l r e c i p r o c a l s , is an irrelevant, b u t aesthetically p l e a s i n g additional characteristic of the a d o p t e d scheme.^^ T h u s the S y s t e m A s c h e m e reflects a n optimally b a l a n c e d c h o i c e of p a r a m e t e r s from a m o n g the available alternatives, leaving it q u e s t i o n a b l e w h e t h e r it reflected a n y further empirical input - apart from its i m p r o v e d p e r i o d relation b e y o n d the variation a s s u m e d in B M 3 6 8 2 2 . T h e modification of the p e r i o d relation t o a c c o m m o d a t e constraints of the S y s t e m A m o d e l , infroduces small changes into the empirical e q u i v a l e n c i e s s h o w n in (3.9) a b o v e . A s a result for S y s t e m A w e h a v e A,À = 2 9 : 6 . 1 9 . 0 . 5 5 . . 7 m .

(3.10.1)

A223?^ = 10:28,40..° ( m o d . 3 6 0 ) ,

(3.10.2)

Msid

= 27;19,18,2..^and

^(vj=13;10,34,51,27..°/d,

(3.10.3) (3.10.4)

w h e r e u n d e r l i n i n g s highlight c h a n g e s from (3.9.x) a b o v e . T h e relationship, 1 year = 2 7 8 3 / 2 2 5 = 12;22,8 m o n t h s , b e c a m e the c a n o n i c a l definition o f the year in B a b y l o n i a n asfronomy, a p p e a r i n g in virtually all v e r s i o n s o f planetary theory, and imderlying - as we shall see - the s e e m i n g l y different p a r a m e t e r s found in the S y s t e m B lunar theory. T h e derivation o f this p a r a m e t e r m u s t h a v e p r e c e d e d the d e v e l o p m e n t of the latitude t h e o r y in S y s t e m A a n d also C o l u m n G, a n d s e e m s likely to h a v e o c c u r r e d early in the 4"* century, a n d p r o b a b l y not t o o l o n g after the invention of C o l i u n n $ a n d the derivation o f M A . A l t h o u g h variant year-lengths a p p e a r in S y s t e m B contexts, these a p p e a r to result from

For I2 = 0;11,15° only l,:l2 = 14:15 yields a two digit interval for I, (0;10,30°). U is noteworthy that for Z regular and <10,0 (l2>0;3°), these are the only two period relations which lead to a short fall, ÒX, such that 0;5°<6>.<0;30°. Thus the empirical observation need not have been precise, but simply sensible (>0;5°) and less than half a degree. Had the objective been simply to approximate the 19-year cycle, the period relation n=6679 (1,51,19), Z = 540 (9,0) yields a serviceable I2 = 0;3,20° with ÒX = - I I , so an empirical adjustment was evidently intended. For q=18 as in BM 36822, I, = (inconveniently) 0;7,33,20°, while for q=15, I,=0;7,28° butV2:v,= 1538:1245 (cf 1455:1328 in Sys A) and = 205;4°.

Treatments of Annual Phenomena in Cuneiform Sources

43

different arithmetical m a n i p u l a t i o n s , a n d there is n o e v i d e n c e o f a n y s u b s e q u e n t attempt to empirically i m p r o v e o n this p a r a m e t e r for the sidereal year. The Uruk Scheme Curiously, a b o u t t h e s a m e time as the S y s t e m A t h eo r y o f solar a n o m a l y w a s d e v e l o p e d , w h i c h realistically d e s c r i b e d the variable lengths o f the s e a s o n s , w e find e v i d e n c e o f the systematic a d o p t i o n o f a m u c h cruder s c h e m e for calculating the dates o f the c a r d m a l p h e n o m e n a a n d visibilities o f Sirius. T h i s s c h e m e , d u b b e d t h e " U r u k S c h e m e " b y N E U G E B A U E R ( 1 9 4 8 ) , is s h o w n in Figure 7. B a s e d o n the 19-year cycle, its b a c k b o n e is a set o f s u m m e r solstice dates calculated b y starting w i t h I V 7 as the latest p e r m i s s i b l e date a n d applying a n epact o f I T ( m o d u l o 3 0 ) as in U l a b o v e , with a n e p a c t o f 12^ in year 19 restoring the s c h e m e to its starting point.^'* S u b s e q u e n t cardinal p h e n o m e n a ( F E , W S , a n d V E ) are o b t a i n e d b y a d d i n g 3 m + 3 T , leaving 3 m + 2 T b e t w e e n V E a n d the following S S . Sirius visibility p h e n o m e n a are d e r i v e d from S S dates b y the relationships First A p p e a r a n c e

igi ( F ) = S S + 2 I T

Opposition

m e ( 0 ) = SS + 6 m 1 I T

Disappearance

sù ( Q ) = SS + 1 0 m 27T

leading to r(igi) - Q ( s ù ) = 2 m 5T (as in B M 3 6 7 3 1 ) e x c e p t in y e a r 1 o f the cycle w h e n the difference is 2 m 6T. T h e date for r(igi) is 3 T later than o n a v e r a g e in B M 3 6 7 3 1 w h e r e it o c c u r s 1 7 T - 1 9 T later than S S , w h i c h is consistent w i t h the effect o f p r e c e s s i o n over the intervening t w o centuries. F r o m a p e r s p e c t i v e o f a c c u r a c y a n d m a t h e m a t i c a l sophistication the U r u k S c h e m e w a s a distinct step b a c k w a r d s from the S y s t e m A solar m o d e l . N e v e r t h e l e s s , from r o u g h l y - 3 5 0 onwards,^^ virtually all r e c o r d e d solstices a n d e q u i n o x e s a n d nearly all r e c o r d e d Sirius p h e n o m e n a are consistent with a n d evidently c o m p u t e d from this s c h e m e . Henceforth, w e find n o evidence that either solstices o r e q u i n o x e s w e r e actually o b s e r v e d , a l t h o u g h Diaries regularly r e c o r d their c o m p u t e d d at es in the midst o f other o b s e r v a t i o n s .

NEUGEBAUER ( 1 9 4 8 ) and HAMA, pp. 3 5 7 - 3 6 3 believed that the scheme was based on a set of S S dates calculated as in U 1 0 7 + 1 2 4 with a precise epact of 11 ; 3 , 1 0 tithis, with fractions truncated to yield the schematic dates. Accordingly he surmised that S S in year 1 should fall on IV 6 instead of IV 7 for cycles beginning in - 2 1 7 ( 9 4 S.E.) and earlier. Evidence against this was published by AABOE and SACHS ( 1 9 6 6 ) , Texts G and H, 1 2 and recognized and confirmed by SLOTSKY ( 1 9 9 3 ) .

Variants from the scheme are found in the Diary for - 3 7 2 and earlier texts. S S dates agreeing with the scheme are preserved in Texts G and H for - 3 5 7 to - 3 3 1 (AABOE and SACHS ( 1 9 6 6 ) , p. 12). From - 3 3 0 onwards nearly all dates agree with the scheme.

44

J.P. Britton

(S ^

II ®

fS

^

( - - • O C O N O R - i f N r O ' T i n ' O

^ rs

^

fi-i «

(s)

— f
>^><X><XX><XX><X><><X><XX><X 2 'o

?5

.S

I ^

^

O

o

I OH

—

SO

«S

-H

rs

^ rn ^

2

J3

G

i ID

X«XXgX

gx o

^ x x x x x x x x x x x x x x x x x x 8

X

W I

o

s Ô

5

U

>

D

•Hi 60

Treatments of Annual Phenomena in Cuneiform Sources

45

Early Seleucid Astronomy - System B (ca -330 to ca-150) System B P e r h a p s as m u c h as a century after the c o m p l e t i o n of the S y s t e m A lunar t h e o r y , a n alternative t h e o r y k n o w n as S y s t e m B w a s d e v e l o p e d , e v i d e n c e for w h i c h first a p p e a r s in a n auxiliary text ( A T ' ) firom B a b y l o n for - 2 5 7 to - 2 4 4 . S u b s e q u e n t e v i d e n c e is p r o v i d e d b y texts firom U r u k , w h o s e earliest contents r a n g e from - 2 0 5 to - 1 6 3 . Thereafter S y s t e m B texts c o m e from B a b y l o n , w i t h initial contents r a n g i n g from-135 t o - - 1 3 . S y s t e m B is distinctive, n o t m e r e l y b e c a u s e it reflects different p a r a m e t e r s a n d m e t h o d o l o g i e s t h a n S y s t e m A , b u t b e c a u s e o f the u b i q u i t y o f s u c h differences. Indeed, e x c e p t for the c o m m o n ratio of 3:2 for longest to shortest daylight/night the t w o theories share n o t a single explicit p a r a m e t e r or p r o c e d u r e . Indirect e x c e p t i o n s to this rule include t w o disguised intrusions o f the S y s t e m A annual p e r i o d relation in c o l u m n s J a n d A4^, b u t otherwise S y s t e m B a v o i d s all e l e m e n t s o f S y s t e m A , as if they w e r e p a t e n t e d . L u n a r V e l o c i t y a n d Sidereal M o n t h - C o l u m n F In S y s t e m B the d a i l y lunar velocity is tabulated in C o l u m n F as a linear z i g z a g function with the anomalistic p e r i o d relation n = 2 5 1 ( 4 , 1 1 ) m o n t h s ( m ) = 2 6 9 (4,29) = Z a n o m a l i s t i c m o n t h s (m»)

(4.1)

and parameters M a x = 15;16,5°/d

d,F = 0;367d(4.1.1)

(4.1.1)

min=ll;4,5

II/(Z-n)= 4 , 1 1 / 1 8 = 13;56,40

(4.1.2)

AF =

n/Z = 0;55,59,6,28..m/ma

(4.1.3)

M^id = 2 7 ; 1 9 , 1 7 , 4 5 . . ' '

(4.1.4)

4 ; 11

/z(F)= 13;10,357d

T h e p e r i o d relation, w h i c h w a s certainly k n o w n to the author o f C o l u m n b u t d o e s n o t a p p e a r in S y s t e m A , is identical with that for C o l u m n G B a n d a n excellent a p p r o x i m a t i o n o f the anomalistic p e r i o d relation. T h e a m p l i t u d e , AF, is s i m p l y 0;1° X n, w h i c h results in the m i n i m a l increment, Ô=di4=0;2°/d, a n d is s e e m i n g l y b u t a c o n v e n i e n t refinement of an a p p r o x i m a t e a m p l i t u d e of 4°/d. T h e m e a n value, 1 3 ; 1 0 , 3 5 7 d , is as w e h a v e seen, e m b e d d e d in C o l u m n $ o f S y s t e m A , w h e r e it is d e r i v e d from M A a n d an a s s u m e d strict 19-year cycle, b u t o t h e r w i s e plays n o explicit role in S y s t e m A . It is therefore older t h a n the m o d i f i e d 19-year cycle reflected in the solar m o d e l s of b o t h systems. M e a n Synodic M o n t h - Column GB In S y s t e m A the variable m o n t h - l e n g t h d u e to lunar a n o m a l y is p r e s e n t e d in C o l u m n G , n o r m e d for a n a s s u m e d synodic arc of 3 0 ° / m . In S y s t e m B this variation is also d e s c r i b e d in C o l u m n G , b u t i n d e p e n d e n t l y of a n y variation d u e to solar a n o m a l y , so that / Ì ( G B ) is in fact the m e a n synodic m o n t h . G B is a linear z i g z a g function w h o s e units are time d e g r e e s in e x c e s s of 2 9 days (as in S y s t e m A ) b a s e d o n the s a m e p e r i o d relation as C o l u m n F , with the p a r a m e t e r s

46

J.P. Britton

n

=251(4,ll)m

Z = 269 (4,29)ma

(4.1)

M a x = 4,29;27,5°

d,GB = 22;307m

(4.2.1)

m i n = 1,52;34,35<'

P = 2A/di = 13;56,40

(4.2.2)

A G B = 2,36;52,30°

p = W{Z-U) = 0 ; 5 5 ,5 9 ,6 ,2 8 ..m/ m a

(4.2.3)

/X(G) = 3 , 1 1 ; 0 , 5 0 ° .

(4.2.4)

T h e amplitude and m o n t h l y increments, A G B a n d d i G s , are a constant multiple (37;30x)

o f their counterparts

in C o l u m n F , a p p a r e n t l y c h o s e n to m a k e

the

a m p l i t u d e of 2 2 3 m o n t h s precisely 15;0°,^*^ w h i c h leaves only the m e a n a n d one initial value to fully d e t e r m i n e the function.

T h e m e a n value c o r r e s p o n d s to the

famous p a r a m e t e r , MB = 29;31,50,8,20^

(4.3)

w h i c h w a s a d o p t e d b y H i p p a r c h u s a n d retained b y P t o l e m y in the Almagest, and w h i c h is accurate to its third fractional place. T h e empirical origins of this p a r a m e t e r w o u l d s e e m to lie in a n i m p r o v e d estimate of the length o f 17 anomalistic p e r i o d s as 17-251 = 4 2 6 7 m o n t h s = 126007** + 15°(1;0''),

(4.4)

an interval k n o w n to H i p p a r c h u s b y P t o l e m y ' s accoimt, b u t almost certainly of B a b y l o n i a n origin. F o r the p e r i o d - 7 5 0 to - 3 0 0 ( e p o c h - 5 2 5 ) the actual m e a n value o f the excess o v e r w h o l e days for this interval is 18;39° ± 1 5 ° , so the a p p r o x i m a t i o n (4.4) is as accurate as m i g h t b e r e a s o n a b l y e x p e c t e d from the errors characteristic o f eclipse reports.^^ T h e i m p r o v e m e n t over the a p p a r e n t S y s t e m A estimate p r e s u m a b l y resulted from the availability of a substantially greater n u m b e r o f e x a m p l e s t o the author o f S y s t e m B . T h u s while the author o f S y s t e m A w o u l d h a v e h a d o n l y a handful o f eclipse pairs separated b y this interval a n d visible in B a b y l o n before - 3 9 0 , b y - 3 1 4 (24 saroi after 0 N a b o n a s s a r ) there w o u l d h a v e b e e n 31 o f s u c h pairs and 50 such by -260. H o w e v e r , as c o m m e n t a t o r s from C o p e r n i c u s o n w a r d s h a v e n o t e d ( a n d P t o l e m y h i m s e l f a p p e a r s aware^^), division yields not M B b u t rather 29;31,50,8,9^**, raising the question of w h y the foiuth fractional p l a c e is ..,20, instead o f 0 or 10 as m i g h t b e expected. T h e a n s w e r lies m the specific nature o f the B a b y l o n i a n function, w h o s e units are time d e g r e e s rather than fractional d a y s , so that (4.4) c o r r e s p o n d s to 1 m o n t h = 29**+ 3,11 ;0,48,56..°. H o w e v e r , if one desires to avoid imits in the last

fractional

(4.5) s p a c e (saving one

c o l u n m ) , /t(G) m u s t e n d in a multiple o f 0;0,5°, resulting in a (natural) r o u n d i n g to / Ì ( G B ) = 3,11;0,50° = 0;31,50,8,20''

BRITTON ( 1 9 9 9 ) , p. 1 9 7 . STEELE ( 2 0 0 0 ) , pp. 5 7 - 6 8 .

Suggested by the term "approximately" (Aim, I V 2 , 1 7 6 ) .

(4.6)

Treatments of Annual Phenomena in Cuneiform Sources

47

as we find. This value, unlike multiples o f 0;0,15° (e.g. 3,11;0,45), also p e r m i t s values of G o n a s c e n d i n g a n d d e s c e n d i n g b r a n c h e s a n d at n e w a n d full m o o n s to h a v e distinctive e n d i n g s , a useful attribute for anyone w o r k i n g w i t h s u c h tables. Solar Model - Column A T h e S y s t e m B solar m o d e l is reflected in C o l u m n A, w h i c h describes the variable synodic arc,

as a linear z i g z a g function with the p a r a m e t e r s M a x = 3 0 ; 1,59, 0 7 m

d, = 0 ; 1 8 7 m

(4.5.1)

min

H = 2,46,59 ( 1 0 , 0 1 9 )

(4.5.2)

Z=

(810)

(4.5.3)

WZ = 12;22,8,53..m

(4.5.4)

AA=

= 28;10,39,40 1;51,19,20

^(A) = 29; 6,19,207m

13,30

Y = 3 6 0 7 / x ( A ) = 12;22,7,51..m = ( ~ M B ) 365;15,34..'* 2 2 3 - /x(A) = 10;29,51..° (mod. 3 6 0 ° )

(4.5.5) (4.5.6)

H e r e w e find t w o a p p a r e n t l y different p e r i o d relations: o n e strictly defining the a n o m a h s t i c year as WZ = 1 0 0 1 9 / 8 1 0 = 12;22,8,53..m; the other, defining the sidereal y e a r as 360°//i(A) = 12;22,7,51..m. Both, h o w e v e r , result from a p p r o x i m a t i o n s a n d r o u n d i n g s introduced to facilitate tabulation. T h e nature a n d order o f these a p p r o x i m a t i o n s c a n b e g a u g e d b y first considering the hypothetical p a r a m e t e r s o f C o l u m n A , h a d it e m u l a t e d the p e r i o d relation of S y s t e m A , while retaining the m o n t h l y i n c r e m e n t di = 0;18°/m, to wit: Max = 30; l,58,36°/m

di=0;18°/m

(4.6.1)

min

H = 46,23 (2783)

(4.6.2)

Z=

(4.6.3)

AA=

= 2 8 ; 10,39,24 1;51,19,12

/x(A) = 2 9 ; 6 , 1 9 , 0 ° / m

3,45

(225)

H/Z = 12;22,8m

(4.6.4)

Y = 360°//x = 12;22,8,0..m,

(4.6.5)

2 2 3 - /x(A) = 10;28,37..° ( m o d . 3 6 0 ° )

(4.6.6)

Instead fi(A) has b e e n increased b y 0;0,0,20°/m, s e e m i n g l y from a s s u m i n g that the m e a n increment in longitude over 2 2 3 m o n t h s , A223X = 18*^*^+ 10;30°, w h e n c e A\X = H(A) = 29;6,19,22..°/m, w h i c h r o u n d s to the attested 2 9 ; 6 , 1 9 , 2 0 ° / m if o n e w i s h e s to avoid units in the third fi-actional place. Similarly, the a m p l i t u d e c o r r e s p o n d i n g to the S y s t e m A p e r i o d relation also generates units in the third fractional p l a c e w h i c h are eliminated b y an increase of 0;0,0,8°/m to l ; 5 1 , 1 9 , 2 0 ° / m . It is the c o m b i n e d effect o f these two r o u n d i n g s that results in the a p p a r e n t anomalistic p e r i o d relation, w h i c h is clearly of n o physical significance. It a p p e a r s , therefore, that the fundamental a s s u m p t i o n u n d e r l y i n g the S y s t e m B annual p e r i o d relation is that A223>^= 1 8 ' + 1 0 ; 3 0 ° ,

(4.7)

48

J.P. Britton

or that 2 2 3 m o n t h s = 18+7/240 years = 2 4 1 + 7 / 2 4 0 sidereal m o n t h s ,

(4.7.1)

Y B = 365;15,33,49..^

(4.7.2)

Msid = 2 7 ; 1 9 , 1 8 , 3 . . ^

(4.7.3)

/ i ( v j = 1 3 ; 1 0 , 3 4 , 5 1 ( , 8 . . ) 7 d , and

(4.7.4)

/i(vs) = 0;59,8,9,48(,6..)7d.

(4.7.5)

whence

T h e last t w o p a r a m e t e r s are m e n t i o n e d explicitly in a p r o c e d u r e text ( B C M 18451982.2+) from the collection of Sir H e n r y W e l l c o m e in the B i r m i n g h a m City Museum,^^ w h i c h w o u l d a p p e a r to confirm the role of expression (4.7) in the derivation of S y s t e m B parameters.^° Ul07+124 T h i s text, first p u b l i s h e d in N E U G E B A U E R ( 1 9 4 7 ) and subsequently as A C T 199, contains c o m p u t e d dates of consecutive simimer solstices a s s u m i n g a n epact derived from the 19-year cycle of 11;3,10 tithis. A l t h o u g h N e u g e b a u e r a s s u m e d that the text contained t w o c o l u m n s of dates on e a c h side, it is m o r e likely that it c o n t a i n e d three colimms o f 2 4 (or p o s s i b l y 2 5 ) years to a side a n d b e g a n with the simfimer solstice o n IV 7 in year 1,53** S.E.(-198).^' If so, its latest entry w o u l d h a v e b e e n for 4,16 S.E. (-55), the fiill text b e i n g s h o w n in A p p e n d i x D . T h e text is c o m p u t a t i o n a l l y trivial, but interesting nevertheless for the fact that it reflects r e n e w e d attention to the 19-year cycle long after a better estimate o f the sidereal year length h a d b e e n achieved. It is also a singlularly precise m a t h e m a t i c a l treatment of the dates of a cardinal p h e n o m e n o n . Since the resulting dates with fractions truncated n o w h e r e differ from U r u k S c h e m e dates, the calculation w o u l d h a v e h a d n o practical c o n s e q u e n c e . Nevertheless, for the first time w e find solstice dates clearly identified with the 19-year cycle, in constrast to the sidereal y e a r o f Systems A and B . Since it w o u l d h a v e b e e n computationally simpler to u s e a n epact o f 11 ;4T, w e are left to w o n d e r what was the intended p u r p o s e o f the text. W a s it simply a n exercise in pointless precision, or w a s there recognition o f a (possible) difference b e t w e e n the sidereal and tiopical y e a r s ?

Briefly described with photograph in HOROWITZ (2000). ^'^ M(Vm) can be simply a truncation of the parameter derived directly from the adjusted 19year cycle, but /x(Vs) only follows precisely from expression (4.7). Writing dates of ACT texts - preserved only in texts from Uruk - range from (ca) S.E. [106] (-205) to S.E. 130 (-181). Thus a composition date around S.E. 113 (-198) would fall within the range of attested or inferred writing dates. See ACT, p. 6-7 for details.

Treatments of Annual Phenomena in Cuneiform Sources

49

Late Seleucid-Arsacid Sources System B ' - Abbreviated Column A F o u r late texts,^"^ all from B a b y l o n , three o f w h i c h can b e securely d a t e d to the 1'' century (ca. - 9 0 , - 7 6 , a n d -75) reflect a b b r e v i a t e d p a r a m e t e r s for C o l u m n

A

(henceforth S y s t e m B ' ) , to wit: Max

=30; 2,07m

d, = 0 ; 1 8 7 m

(5.1.1)

min

= 2 8 ; 10,40

H = 5,34 (334)

(5.1.2) (5.1.3)

AAB=

1;51,20

Z=

27

/X(AB)=

29;6,207m

I I / Z = 12;22,13,20

(5.1.4)

1 year = 3 6 0 7 / i ( A ' ) = 12;22,7,34..m = 3 6 5 ; 1 5 , 2 5 . . d a y s ,

(5.1.5)

w h e n c e from / Ì ( A B ) a n d ^ ( G B ) Msid=

27;19,18,0..'' a n d /x(v„,) = 1 3 ; 1 0 , 3 4 , 5 2 . . 7 d

(5.1.6)

T h e p e r i o d relation is e x t r e m e l y c r u d e , reflecting the 2 7 - y e a r cycle, last s e e n in B M 3 6 7 3 1 (ca. - 5 6 0 ) . I n d e e d the s c h e m e w o u l d b e entirely iminteresting, b u t for the fact that its m e a n v a l u e yields a value for the sidereal year w h i c h is essentially identical with P t o l e m y ' s a n d v e r y n e a r l y accurate as well. T h e year-length Y = 365'/4+'/,44 = 365;15,25**

(5.2)

is also m e n t i o n e d in a problematical text b y Vettius V a l e n s (ca. +160),^^ w h o attributes it to " B a b y l o n i a n s " as distinguished from " C h a l d e a n s " , a n d m a y also h a v e b e e n the i n t e n d e d sidereal year length attributed to H i p p a r c h u s b y Galen.^'* In addition, the implicit m e a n daily lunar velocity, |x(Vn,)=13;10,34,527d, serves as the m e a n v a l u e of the " S t a n d a r d L u n a r S c h e m e " in G r e e k sources for daily lunar motion,^^ a l t h o u g h n o w h e r e d o e s it a p p e a r explicitly in B a b y l o n i a n sources. S u m m a r y of Principal Parameters from Systems A, B and B ' T h e s e e m i n g profusion o f p a r a m e t e r s from Systems A a n d B c a n b e simplified a n d s u m m a r i z e d as follows:

Sidereal Year Synodic Month Sidereal Month Mean lunar velocity Mean synodic arc

(1) Syst A (*) 365;14,48..'* 29:31.50,6'' 27;19,17,46.."* 13;10,357d 29;6,22,57..°/m

(2) System A 365:15,37.. 29;31,50,6 27;19,18,2.. 13;10,34,51.. 29;6,19,0..

(3) System B 365;15,33.. 29:31,50.8.20 27;19,18,3.. 13;10,34,51.. 29;6,19,20

(4) System B' 365:15.25.." 29;31,50,8,20'* 27:19,18,0..'' 13;10,34,52..°/d 29;6,207m

ACT 122a (BM 35753), ACT 123 (BM 45694), ACT 123aa (BM 34627), and ACT 123a (BM 45849). See ACT, pp. 146-152 for details. "

HAMA,p. 6 0 1 , n . l . HAMA, p. 293.

JONES (1997). It is perhaps noteworthy that the mean daily lunar motion derived from the relationship, 669 months = 723+''/45 sidereal months = 19756 days is Vm= 13;10,34,51,55..7d.

50

J.P. Britton

I n t e r m s of empirical input these reflect really only three e m p i r i c a l steps: (1) derivation o f a n i m p r o v e d value for the m e a n synodic m o n t h

from

an

early

(ca. -400) estimate o f the length of 4 2 6 7 m o n t h s in cormection w i t h C o l u m n * ; (2) correction of the 19-year cycle to reflect a shortfall from c o m p l e t e return after 2 3 5 m o n t h s o f '/4°; a n d (3) d e v e l o p m e n t of a substantially accurate v a l u e for the m e a n s y n o d i c m o n t h i n c o r p o r a t e d in S y s t e m B from an i m p r o v e d estimate of the length of 4 2 6 7 m o n t h s resulting from the availability of m o r e a n d p o s s i b l y b e t t e r data. All other differences a p p e a r to result simply from n u m e r i c a l m a n i p u l a t i o n , r o i m d i n g s a n d different a p p r o x i m a t i o n s , w h i c h in the case o f a b b r e v i a t e d S y s t e m B , led to an essentially accurate v a l u e for the sidereal year. B M 55555+ = A C T 210 So far I h a v e omitted m e n t i o n o f the p r o c e d u r e text A C T 2 1 0 , a l t h o u g h c h r o n o l o g i c a l l y it p r o b a b l y b e l o n g s b e t w e e n U 1 0 7 a n d the first c e n t u r y texts containing S y s t e m B ' . T h e text, from B a b y l o n b u t reflecting u s a g e of the s p a c e sign m o r e characteristic of texts from Uruk,^^ contains a n u n u s u a l c o l l e c t i o n o f p r o c e d u r e s a n d p a r a m e t e r s from b o t h S y s t e m A a n d B lunar t h e o r i e s , s h o w i n g that it w a s c o m p i l e d after the creation o f b o t h theories. Section 3 contains a series of statements of periodicities a n d intervals, m a i n l y reflecting S y s t e m B a n d o t h e r w i s e unattested p a r a m e t e r s , a n d r e a d s as follows: 5. 6. 7. 8.

.. .concerning the revolutions o f the m o o n [returning] t o its p l a c e .... [29;31,]50,.8,20 d a y s , the m e a s u r e of the m o n t h o f the m o o n [ 5 , 5 4 ; 2 ] 2 , 1 , 4 0 d a y s , (the m e a s u r e ) of 12 m o n t h s of the m o o n [5,27;5] 1,20 d a y s , the m e a s u r e o f 12 m o n t h s o f the m o o n returning to its place

9. 10.

[ 1 , 4 ] 9 , 4 4 ; 3 1 , 2 0 d a y s , (the m e a s u r e ) of 18 years of the m o o n r e t u r n i n g to its place [ 1 , 4 ] 9 , 4 5 ; 1 9 , 2 0 d a y s , (the m e a s u r e ) o f 18 years of the m o o n

11. 12.

[ 1 , 4 ] 9 , 3 4 ; 2 5 , 2 7 , 1 8 days, (the m e a s u r e ) of 18 years of the sun returning to [its p l a c e ] in 18 revolutions

A s N e u g e b a u e r n o t e s , lines 6 a n d 7 state the lengths o f 1 a n d 12 S y s t e m B synodic m o n t h s ( M B ) precisely. Line 8 impUes that 1 sidereal m o n t h = 2 7 ; 1 9 , 1 6 , 4 0 d a y s , an otherwise unattested a n d relatively c r u d e p a r a m e t e r . P r o b a b l y it is corrupt, reflecting either an error in the last fractional p l a c e - r e p l a c i n g 2 0 with 3 0 leads to the m o r e r e a s o n a b l e value 2 7 ; 19,17,30 - or s o m e effect of successive roimdings.^^ Line 9 gives the length of 241 sidereal m o n t h s ( 2 2 3 m o n t h s p l u s 18 rotations) as 6 5 8 4 ; 3 1 , 2 0 d a y s , s o m e w h a t abstractly since the m o o n d o e s n o t return to its p l a c e in 2 2 3 m o n t h s . P r e c i s e c o m p u t a t i o n with S y s t e m B ' p a r a m e t e r s leads t o rrisid [= M B - 3 6 0 / ( 3 6 0 + ^ ( A B ))] = 27;19,18,0,53..'*, w h e n c e 241msid = 6584;31,2L,**. W h i l e this c o u l d r e a s o n a b l y b e r o u n d e d to 65 84 ;31,20**, it is n o t e w o r t h y that a s s u m i n g

Texts from Uruk write employ a space sign (transcribed ".") when a sexagesimal "digit" containing no units is followed by one contining no tens, e.g. write 10,. 1 to distinguish 601 from 11. Texts from Babylon generally omit the space sign, the exceptions being principally texts from the first century, including those reflecting abbreviated Column A' (System B'). e.g. 12x27:19,17'* = 5,27:51,24''..s 5,27:51,20'*.

Treatments of Annual Phenomena in Cuneiform Sources

51

Y B = 365;15,25'* yields nisid [ = Y B X M B / ( Y B - + M B ) ] = 2 7 ; 1 9 , 1 8 , 0 , 3 4 . . ^ w h e n c e 241nisid= 6584;31,20..*' without r o u n d i n g , suggesting that Y B = 365;15,25'* w a s a r e c o g n i z e d rather than simply implicit p a r a m e t e r . T h e c o r r e s p o n d i n g m e a n daily lunar m o t i o n o f 1 3 ; 1 0 , 3 4 , 5 2 , 3 6 . . 7 d , is essentially identical w i t h P t o l e m y ' s sidereal parameter of 13;10,34,52,35..7d. Line 10 brings us b a c k to S y s t e m B with the length o f 2 2 3 S y s t e m B m o n t h s , curtailed after 2 fractional p l a c e s (the next fractional p l a c e is ..20,58..). T h e n lines 11 a n d 12 state with a n u n u s u a l p r e c i s i o n of 3 fractional p l a c e s that 18 years = 6574;25,27,18 days, whence 1 year = 3 6 5 ; 14,44,51 days.

(5.3)

T h i s is a n e x t r e m e l y curious result, first b e c a u s e it is far r e m o v e d from a n y of the values associated with Systems A, B , B ' or their f o r e n m n e r s , a n d s e c o n d b e c a u s e it is a c o n s p i c u o u s l y p o o r representation o f the sidereal year, if that is in fact intended. A s usual, the critical sign in line 12 is b r o k e n : N e u g e b a u e r restores " [ a n a ki-s]ù", to its p l a c e (longitude), b u t there is scarcely space for these signs, and " [ a n a g u ] b - to the solstice - w o u l d fit the p r e s e r v e d traces a n d space as well if n o t better. E v e n m o r e curious, h o w e v e r , is the fact - d i s c o v e r e d b y D e n n i s Rawlins^^ - that 2 9 7 s u c h years e q u a l 108,478 integral d a y s almost exactly.^^ This c o r r e s p o n d e n c e is o t h e r w i s e unattested in cimeiform sources, but, as R a w l i n s notes, it is precisely the interval in w h o l e d a y s b e t w e e n the sunmier solstice o b s e r v e d b y "the school o f M e t o n a n d E u k t e m o n " on -431 J u n e 27 a n d that o b s e r v e d b y H i p p a r c h u s o n - 1 3 4 J u n e 2 6 as related in Aim. I l l 1, 1 3 8 - 1 3 9 . P t o l e m y d o e s n o t r e p o r t the time o f H i p p a r c h u s ' s solstice observation, w h i c h should h a v e o c c u r r e d at n o o n , 108,478% d a y s after M e t o n ' s , for c o n s i s t e n c y with his year length and solar m o d e l , b u t P t o l e m y ' s a c c o u n t suggests s o m e inconsistency in the data, a n d R a w l i n s h a s a r g u e d that H i p p a r c h u s m a y h a v e o b s e r v e d the solstice a quarter d a y earlier t h a n c o n s i s t e n c y requires.'*" A n additional intriguing fact is that the SS dates a c c o r d i n g to the U r u k S c h e m e for the t w o solstices b e g a n on the dates cited b y P t o l e m y ( a l t h o u g h in the afternoon).'*' W h a t e v e r the precise circumstances, it is difficult to a c c o u n t for this p a r a m e t e r without a s s u m i n g that k n o w l e d g e of these observations s o m e h o w m a d e its w a y to B a b y l o n , implying a t w o - w a y transmission o f information b e t w e e n B a b y l o n i a n a n d Hellenistic astronomers, after at least -134. W h a t w e can say w i t h s o m e confidence r e g a r d i n g the Hellenistic p a r a m e t e r s is that the 19-year cycle c o m b i n e d with either the S y s t e m A or S y s t e m B m o n t h length gives the value o f the tropical year a d o p t e d b y H i p p a r c h u s a n d a c c e p t e d b y P t o l e m y ,

RAWLINS (1991), pp. 4 9 - 5 1 . The multiplication yields 108,478;0,0,27 days. RAWLINS (1991), pp. 5Iff. '" According to the Uruk Scheme, SS in -431 occurred on III 11 which began on June 27, while SS in -134 occurred on III 24 which began on June 26. In U107 SS for -134 (SE 2,57) was computed to occur on III 24;22,40 equivalent to 4; 16am June 27, nearly a day later than Hipparchus's ostensible observation. However Ptolemy's solstice "observation" on +140 June 25: 2am falls on III 22 in the Babylonian calendar, precisely in accordance with the Uruk Scheme. In -134 the summer solstice actually occurred at roughly 7am June 26 (mean civil time Rhodes), so whether Hipparchus determined the solstice to have occurred at dawn or noon, he was accurate to within a day.

52

J.P. Britton

and that this c o r r e s p o n d e n c e was already implicit in U 1 0 7 + 1 2 4 , h a l f a century before H i p p a r c h u s . F u r t h e r m o r e , the sidereal year of 3 6 5 ; 15,25 d a y s , implicit in S y s t e m B ' is consistent with the p r e c e s s i o n a n d sidereal year a d o p t e d b y P t o l e m y a n d v e r y p r o b a b l y k n o w n to H i p p a r c h u s , w h o s e lower limit for p r e c e s s i o n is essentially the difference b e t w e e n this and the 19-year cycle year length. If this p a r a m e t e r was s i m p l y the result of various r o u n d i n g s , t h e n its a c c u r a c y is fortuitous, a n d from P t o l e m y ' s a c c o u n t it a p p e a r s that H i p p a r c h u s was u n a b l e to decide w h e t h e r this or the m o r e traditional B a b y l o n i a n year length w a s m o r e accurate.'*^ T h i s argues against H i p p a r c h u s b e i n g the author of this p a r a m e t e r , a n d in favour of Vettius V a l e n s ' attribution, although H i p p a r c h u s m a y h a v e found r e a s o n to favour it, and P t o l e m y certainly a d o p t e d it.'*^ All o f w h i c h leaves, the a n o m a l o u s year length from A C T 2 1 0 seemingly p o s t - H i p p a r c h a n , but still u n e x p l a i n e d . W a s it an i n c o m p e t e n t estimate o f the sidereal year or a refined estimate o f the tropical year, reflecting - p e r h a p s m o r e accurately than P t o l e m y implies - the actual interval b e t w e e n the solstice o b s e r v e d b y H i p p a r c h u s in - 1 3 4 and M e t o n ' s ostensible o b s e r v a t i o n in - 4 3 1 ? O n b a l a n c e , the issue a n d especially the latest texts from B a b y l o n d e s e r v e m o r e detailed scrutiny.

Summary Before the 7 * century the schematic calendar d o m i n a t e s the a s t i o n o m i c a l literature a n d the o n l y e v i d e n c e of an actual p e r i o d relation is the exceptionally c r u d e 3 year cycle found in M U L . A P I N , w h i c h implies a year length o f r o u g h l y 3 6 4 d a y s . B y late in the N e o - A s s y r i a n p e r i o d the first e v i d e n c e of sensible, if primitive, p e r i o d relations for the sun, m o o n a n d planets a p p e a r s , b u t it is n o t until the N e o B a b y l o n i a n p e r i o d a n d p r o b a b l y the 6 * century, that arithmetical analysis is applied to better observational data to obtain m o r e accurate a p p r o x i m a t i o n s o f the lengths o f year and m o n t h s - n a m e l y 3 6 5 ; 16 days = 1 year a n d 2 9 ; 3 1 , 4 0 ± = 1 m o n t h . B y late in the 6 * century the superiority o f the 19-year cycle over the 2 7 a n d 3 0 - y e a r cycles h a d b e e n recognized, a n d b y early in the 5 century incorporated into a fixed s c h e m e o f intercalations w h i c h persisted to the end of the Seleucid Era. T h e next a d v a n c e c o m e s a r o u n d the begiiming o f the 4 * century, following the infroduction o f a u n i f o r m z o d i a c , w h e n a substantially m o r e accurate v a l u e for the m e a n synodic m o n t h , 2 9 ; 3 1 , 5 0 , 6 d a y s was d e r i v e d (I believe) from eclipse r e c o r d s spanning 4 2 6 7 m o n t h s , an interval c o r r e s p o n d i n g to 17 anomalistic p e r i o d s a n d very nearly 3 4 5 c o m p l e t e years, thus largely eliminating the effects o f b o t h lunar a n d solar anomaly. T h i s value, c o m b i n e d with the 19-year cycle in the d e v e l o p m e n t of Column led precisely to the lunar m e a n daily m o t i o n o f 1 3 ; 1 0 , 3 5 7 d , later found in C o l u m n F of S y s t e m B , a n d implied a year length of 3 6 5 ; 14,48..days, a p a r a m e t e r

""^ According to Ptolemy Hipparchus investigated the period relation, 4267 months = 345 years -T/°. From System A parameters the shortfall is -7;45..°; from System B, -7;20..°; from System B', -6;30°. Thus at this point in his researches he had not embraced the System B' parameter. Ptolemy's precise value for the sidereal year, 365; 15,24,31,32,27,7 days (HAMA, p. 902 but note typo of 22 for 32) is simply the arithmetic result of applying his rounded value of precession, 1° per (Egyptian) century, to his tropical year length derived from the 19-year cycle.

Treatments of Annual Phenomena in Cuneiform Sources

53

s u b s e q u e n t l y a d o p t e d b y H i p p a r c h u s and retained b y P t o l e m y for the length of the tropical year. C o n c u r r e n t l y w e find the first evidence of a theoretical treatment o f solar anomaly, implying recognition of the variable length o f the seasons. A n i m p r o v e d estimate of the sidereal year followed from an adjustment of the 19-year cycle to reflect a shortfall from c o m p l e t e return in 2 3 5 m o n t h s o f a d e v e l o p m e n t w h i c h s e e m s to have o c c u r r e d early in the 4'*' century, after the establishment o f C o l u m n 4» b u t before the c o m p l e t i o n o f the S y s t e m A latitude theory a n d C o l u m n G, b o t h of which d e p e n d on it. T h i s implied a y e a r length of 3 6 5 ; 1 5 , 3 6 . . days a n d led to the famous aimual relationship, 1 year = 12;22,8 m o n t h s , w h i c h a p p e a r s t h r o u g h o u t B a b y l o n i a n planetary theory a n d underlies e v e n S y s t e m B , w h e r e it a p p e a r s covertly in several places. A b o u t the s a m e time as the c o m p l e t i o n of the S y s t e m A lunar t h e o r y - p e r h a p s around or slightly before the m i d d l e o f the 4 * century - w e find e v i d e n c e o f a rather regressive s c h e m e for c o m p u t i n g the dates o f the cardinal p h e n o m e n a a n d Sirius visibilities, a s s u m i n g seasons of equal length and the 19-year cycle. D e s p i t e an a w a r e n e s s of the inequality of the seasons, evident in the S y s t e m A solar s c h e m e , the " U r u k S c h e m e " r e m a i n e d the basis for all r e p o r t e d solstice a n d e q u i n o x dates until the e n d of c u n e i f o r m astronomy. S o m e t i m e after the c o m p l e t i o n o f the S y s t e m A lunar theory, but before the middle of the 3'^'' century, a n alternative S y s t e m B lunar theory w a s d e v e l o p e d e m p l o y i n g a l m o s t universally different m e t h o d s a n d p a r a m e t e r s . A significant empirical a d v a n c e w a s an essentially accurate value for the m e a n synodic m o n t h , 2 9 ; 3 1 , 5 0 , 8 , 2 0 d a y s - a p a r a m e t e r k n o w n to H i p p a r c h u s a n d retained b y P t o l e m y . It s e e m s likely that this w a s derived from a better estimate of the length of 4 2 6 7 m o n t h s t h a n in S y s t e m A m a d e possible b y a larger n u m b e r o f qualifying eclipse pairs. S y s t e m B ' s year length, 3 6 5 ; 1 5 , 3 3 . . days also differs from S y s t e m A ' s , but the difference is entirely d u e to different a p p r o x i m a t i o n s in its derivation, a n d n o t to a n y n e w empirical input. In U 1 0 7 + 1 2 4 , written p r o b a b l y in - 1 9 8 , precise m a t h e m a t i c a l m e t h o d s are applied for the first time to the dates of s u m m e r solstice, taking the date ( I V 7) for the first year of a n U r u k S c h e m e cycle as a starting p o i n t a n d p r o c e e d i n g with a n epact b a s e d o n the 19-year cycle rather than the sidereal epact of 1 1;4T long u s e d in S y s t e m A , in p l a n e t a r y theories, a n d indirectly in S y s t e m B . N o e v i d e n c e o f associated observational activity is apparent, b u t it is interesting to find a cardinal p h e n o m e n o n treated with the formal precision of other astronomical events w i t h a periodicty b a s e d on the 19-year cycle and distinct from the sidereal year. Finally, S y s t e m B ', the form o f S y s t e m B u s e d in s o m e late e p h e m e r i d e s from B a b y l o n after material from U r u k ceases, uses a b b r e v i a t e d p a r a m e t e r s for the solar m o t i o n w h i c h implied a n essentially accurate value for the length o f the sidereal year, 3 6 5 ; 1 5 , 2 5 d a y s , w h i c h ultimately b e c a m e the basis for P t o l e m y ' s sidereal y e a r a n d value for precession. O n the other h a n d the a n o m a l o u s year length of 3 6 5 , 1 4 , 4 4 , 5 1 days, reflected in A C T 2 1 0 , a n d seemingly related to the 2 9 7 year interval b e t w e e n solstices o b s e r v e d b y M e t o n and H i p p a r c h u s , suggests that the transmission of astronomical k n o w l e d g e was not solely from B a b y l o n i a n to Hellenistic a s t r o n o m e r s in the late and 1'' centuries, and that at least s o m e astronomical information flowed b a c k to B a b y l o n .

54

J.P. Britton

Years, Months and Days Figure 8 s u m m a r i z e s the relationships b e t w e e n years m o n t h s a n d days e n c o i m t e r e d in this review. A s reflected in the rightmost c o l i m m s there are in all only four distinct values for the m e a n synodic m o n t h 29'/2 days 2 9 ; 3 1 , 4 0 days 29;31,50,6 days 2 9 ; 3 1 , 5 0 , 8 , 2 0 days

(<-600) (ca-600) (ca-400) (ca.-300)

b u t eight distinct year lengths 3 6 4 days 3 6 4 ; 3 0 days 365;10days 3 6 5 ; 16 days 3 6 5 ; 14,48.. d a y s

(<-750) (?<-600) (ca.-600) (ca.-560) (ca. - 4 0 0 sidereal)

3 6 5 ; 1 5 , 3 5 . . ± 0;0,2.. [ 3 6 5 ; 1 4 , 4 8 . . days 3 6 5 ; 15,25 3 6 5 ; 1 4 , 4 4 , 5 1 days

(ca. - 3 8 0 to ca. -280) (ca. - 2 0 0 tropical)] ( c a . - 1 5 0 ? sidereal) (ca.-130? tropical?)

T h e two series e n d w i t h essentially accurate v a l u e s for the m o n t h a n d sidereal year, while after - 2 0 0 y e a r lengths either derived fi-om or closely related to the 19-year cycle a p p e a r associated with cardinal p h e n o m e n a . Finally, w e see that all three p a r a m e t e r s a d o p t e d b y P t o l e m y - the m e a n synodic m o n t h , the tropical y e a r a n d the sidereal year w e r e o f B a b y l o n i a n origin, trivially modified b y arithmetical a p p r o x i m a t i o n s b u t unaffected b y Hellenistic empirical refinement.

55

Treatments of Annual Phenomena in Cuneiform Sources

VO so

« 3

^1 ON
00 VO

tt^
cn

I 1^

n VO VO VO m f'i

«n >r-i >r> VO VO VO m

"1VO VD

VI

TF

>n

oo (N

>o v-i >r>
Ti

VO VO

o

IO fS

«n

o o 00| fS^ (N^ "^J oo" oe" 00 o" o" o

o

VO VO VO VO

o" o" o o~ o'

«n^ «n^ v-i^
rrv

I

'TV

ro cn

m

ON Ov <^ 0^

eN CM

. d 0 _ -«^t

II

o

O

en

cT o~

I

p; ^ ^ rIl

VO cn^

—; o u-T
os o

2

O

00^

0\

VO

o

I

eN

««
ON ON VO VO
rt >n «n cô ro co en
^

ON"

On" en O enn" 00 1-^

m

O

>n o en 00 CM •n-

m en fN e s

VO l

O

o

"C

en

^

On

£2j CM —<

ON

00

^ 00 ON O

II

0::

00

o

II

o

. 1 ^

-a 2

^

00 00

I §V"r r'i ON

00 ON en en

-r ^ + +

1^

A

w

"1 "1

•<3

S" S* —

tu

0 , 0 - 0

ffiffia: es

3 o.

I

00

o o o o o o o 'a- en en
00 CM «n V O en

CN

1—1

O} O> s

1-^

en t>- t-VO VO en en

2 CQ QQ CQ

es CQ

c:

f-H

56

J.P. Britton

Abbreviations A C T = NEUGEBAUER (1955) A D = SACHS and H U N G E R ( 1 9 8 8 - 1 9 9 5 ) Aim.

=T00MER(1984)

H A M A = NEUGEBAUER (1975) LABS = PARPOLA (1993) L A S = PARPOLA (1970, 1983) L B A T = PINCHES, STRASSMAIER, and SACHS (1955)

SbTU = v o N WEIHER (1993) S S B I = KUGLER(1907)

References A A B O E , A s g e r . 1 9 5 5 . " O n the B a b y l o n i a n Origin o f S o m e H i p p a r c h i a n P a r a m e t e r s " . Centaurus 4 : 1 2 2 - 5 . — 1968. Some Lunar Auxiliary Period.

(Kongelige

Tables and Related

Danske

Texts from

Videnskabemes

the Late

Selskab,

Babylonian

Matematisk-fysiske

Meddelelser 36:12). Copenhagen: Munksgaard. — 1969. A Computed B.M.

40094.

List of New Moons for 319 B.C. to 316 B.C. from

( K o n g e l i g e D a n s k e V i d e n s k a b e m e s Selskab,

Babylon:

Matematisk-fysiske

Meddelelser 37:3). Copenhagen: Munksgaard. —

; BRITTON, John

P.; H E N D E R S O N , Janice A.; N E U G E B A U E R , Otto, and S A C H S ,

A b r a h a m J. 1 9 9 1 . Saros

Cycle

Dates

and Related

Babylonian

Astronomical

Texts ( T r a n s a c t i o n s o f the A m e r i c a n P h i l o s o p h i c a l Society 8 1 / 6 ) . P h i l a d e l p h i a : A m e r i c a n P h i l o s o p h i c a l Society. — a n d S A C H S , A b r a h a m J. 1966. " S o m e D a t e l e s s C o m p u t e d Lists o f L o n g i t u d e s o f Characteristic P l a n e t a r y P h e n o m e n a from t h e L a t e - B a b y l o n i a n P e r i o d " . of Cuneiform

Studies

Journal

20: 1-33.

— and S A C H S , A b r a h a m J. 1969. " T w o L u n a r T e x t s o f the A c h a e m e n i d P e r i o d B a b y l o n " . Centaurus 14: 1 - 2 2 .

from

B R I T T O N , J o h n P . 1 9 9 9 . " L u n a r A n o m a l y in B a b y l o n i a n A s f r o n o m y " . In: N o e l M . SWERDLOW

(ed.), Ancient

Astronomy

and

Celestial

Divination:

187-254.

Cambridge, M A and London, England: M I T Press. C O H E N , M a r k E. 1 9 9 3 . The Cultic

Calendars

of the Ancient

Near East.

Bethesda,

MD: C D L Press. ENGLUND,

Robert

K.

M e s o p o t a m i a " , Journal

1988.

"Adminisfrative

of the Economic

Timekeeping

and Social

History

in

Ancient

of the Orient

31 :

121-85. GALTER,

Hannes

Mesopotamiens,

D . ( e d ) . 1 9 9 3 . Die Rolle

der Astronomie

in den

Kulturen

Beifrâge z u m 3 . G r a z e r M o r g e n l a n d i s c h e n S y m p o s i o n ( 2 3 . - 2 7 .

S e p t e m b e r , 1991). G r a z : R M - V e r l a g . H O R O W I T Z , W a y n e . 1 9 9 8 . Mesopotamian

Cosmic

Geography.

W i n o n a L a k e , fN:

Eisenbraims. — 2 0 0 0 . " A s t r o n o m i c a l C u n e i f o r m T e x t s in the B i r m i n g h a m City M u s e u m " . In: A n d r e w R. G E O R G E a n d Irving FINKEL, e d s . , Wisdom.Gods Studies

in Assyriology

Eisenbrauns.

in Honour

of WG. Lambert:

and

Literature:

3 0 9 - 3 1 4 . Winona Lake, IN:

Treatments of Annual Phenomena in Cuneiform Sources

57

H U B E R , Peter J. 1982. Astronomical Dating of Babylon I and Ur III ( O c c a s i o n a l P a p e r s o n the N e a r E a s t 1.4). M a l i b u , C A : U n d e n a . HUNGER, Hermann. 1 9 9 1 . "Schematische Berechnungen der Soimenwenden". Baghdader Mitteilungen 22:513-19. — 1 9 9 2 . Astrological Reports to Assyrian Kings (State A r c h i v e s o f Assyria 8 ) . Helsinki: Helsinki University Press. — 2 0 0 1 . Astronomical Diaries and Related Texts from Babylonia, Vol. V: Lunar and Planetary Texts. V i e n n a : V e r l a g d e r Ôsterreichischen A k a d e m i e d e r Wissenschaften. — a n d PINGREE, D a v i d . 1989. MUL.APIN: An Astronomical Compendium in Cuneiform (Archiv fiir Orientforschung Beiheft 24). H o r n : F e r d i n a n d B e r g e m & Sohne. — a n d PINGREE, D a v i d . 1999. Astral Sciences in Mesopotamia (Handbuch der Orientalistik, Abteilimg 1, V o l u m e 4 4 ) . Leiden, B o s t o n , K o l n : Brill. JONES, Alexander. 1983. "The Development and Transmission of 248-Day Schemes for L u n a r M o t i o n in A n c i e n t A s t r o n o m y " . Archive for History of Exact Sciences 2 9 : 1 - 3 6 . — 1997. "Studies in t h e A s t r o n o m y o f the R o m a n P e r i o d S c h e m e " . Centaurus 3 9 : 1 - 3 6 .

I. T h e S t a n d a r d L u n a r

K U G L E R , F r a n z X . 1907. Sternkunde und Sterndienst in Babel, V o l . I. Miinster: Aschendorff. N E U G E B A U E R , Otto. 1947. " A T a b l e o f Solstices from U m k " . Journal of Cuneiform Studies 1: 1 4 3 - 1 4 8 . — 1 9 4 8 . "Solstices a n d E q u i n o x e s in B a b y l o n i a n A s t r o n o m y during t h e Seleucid P e r i o d " . Journal of Cuneiform Studies 2: 2 0 9 - 2 2 . — 1955. Astronomical Cuneiform Texts. L o n d o n : L u n d H u m p h r i e s ( R e p r i n t e d N e w York, H e i d e l b e r g , Berlin: Springer-Verlag 1983). — 1 9 7 5 . A History of Ancient Berlin: Springer-Verlag.

Mathematical

Astronomy.

N e w York, Heidelberg,

— a n d S A C H S , A b r a h a m J. 1956. " A P r o c e d u r e T e x t C o n c e m i n g Solar a n d L u n a r M o t i o n : B . M . 3 6 7 1 2 " . Journal of Cuneiform Studies 10: 1 3 1 - 1 3 6 . — a n d S A C H S , A b r a h a m J. 1967. " S o m e Atypical A s t r o n o m i c a l C u n e i f o r m T e x t s I " , Journal of Cuneiform Studies 2 1 : 1 8 3 - 2 1 8 . — a n d S A C H S , A b r a h a m J. 1969. " S o m e Atypical A s t r o n o m i c a l C u n e i f o r m T e x t s II", Journal of Cuneiform Studies 22, 9 2 - 1 1 3 . O L M S T E A D , A l b e r t T . 1948. History of the Persian Empire. C h i c a g o a n d L o n d o n : University o f C h i c a g o Press. P A R P O L A , S i m o . 1970. Letters from Assyrian Scholars to the Kings Esarhaddon and Assurbanipal, I: Texts. N e u k i r c h e n - V l u y n : N e u k i r c h e n e r V e r l a g . — 1 9 8 3 . Letters from Assyrian Scholars to the Kings Esarhaddon and Assurbanipal, II: Commentary and Appendices. N e u k i r c h e n - V l u y n : N e u k i r c h e n e r V e r l a g . — 1 9 9 3 . Letters from Assyrian and Babylonian Scholars (State A r c h i v e s o f Assyria 10). Helsinki: Helsinki University Press. P I N C H E S , T h e o p h o l u s G . ; S T R A S S M A I E R , J o h a n n N . , a n d S A C H S , A b r a h a m J. 1 9 5 5 .

Late Babylonian Press.

Astronomical

and Related

Texts. P r o v i d e n c e : B r o w n U n i v e r s i t y

58

J.P. Britton

R A W L I N S , D e n n i s . 1 9 9 1 . " H i p p a r c h o s ' s Ultimate Solar Orbit a n d the B a b y l o n i a n T r o p i c a l Y e a r " , DIO (The International Journal of Scientific History) 1.1: 49-66. R E A D E , Julian E . 1 9 8 6 . " R a s s a m ' s B a b y l o n i a n Collection: the E x c a v a t i o n s a n d the A r c h i v e s " . Introduction to Erie LEICHTY, Catalogue of the Babylonian Tablets in the British Museum, Volume VI: Tablets from Sippar I. L o n d o n : British M u s e u m Publications. R O C H B E R G , Francesca. 2 0 0 0 . " S c r i b e s and Scholars: T h e tupsar E n u m a A n u Enlil". In J o a c h i m M A R Z A H N a n d H a n s N E U M A N N (eds.), Assyriologica et Semitica: Festschrift fiir Joachim Oelsner (Alter Orient i m d Altes T e s t a m e n t 2 5 2 ) : 3 5 9 - 3 7 5 . Miinster: Ugarit Verlag. S A C H S , A b r a h a m J. 1 9 5 2 . "Sirius D a t e s in B a b y l o n i a n A s t r o n o m i c a l T e x t s o f the Seleucid P e r i o d " . Journal

of Cuneiform

Studies

6: 1 0 5 - 1 1 4 .

— a n d H U N G E R , H e r m a i m . 1 9 8 8 . Astronomical Diaries and Related Texts Babylon, Volume I(-651 to -261). V i e n n a : Osterreichische A k a d e m i e Wissenschaften. — a n d H U N G E R , H e r m a n n . 1989. Astronomical Diaries and Related Texts Babylon, Volume II (-260 to -164). V i e n n a : Osterreichische A k a d e m i e Wissenschaften.

from der from der

— a n d H U N G E R , H e r m a i m . 1 9 9 5 . Astronomical Diaries and Related Texts from Babylon, Volume III (-163 to end). V i e n n a : Osterreichische A k a d e m i e der Wissenschaften. S L O T S K Y , Alice. 1 9 9 3 . " T h e U r u k Solstice S c h e m e Revisited". In: H a n n e s D . G A L T E R ( e d ) , Die Rolle der Astronomie in den Kulturen Mesopotamiens, Beitràge z u m 3. Grazer Morgenlandischen

Symposion (23.-27.

September,

1991): 3 5 9 - 3 6 6 . G r a z : R M - V e r l a g . STEELE, J o h n M . 2 0 0 0 . Observations and Predictions of Eclipse Times by Early Astronomers. Dordrecht, Boston, London: Kluwer. T O O M E R , G e r a l d J. 1 9 8 4 . Ptolemy's ALMAGEST, translated a n d annotated. N e w Y o r k , Berlin, H e i d e l b e r g , T o k y o : Springer-Verlag. — 1 9 8 0 . " H i p p a r c h u s ' Empirical Basis for H i s Limar M e a n Motions". Centaurus 2 4 : 9 7 - 1 0 9 . V O N W E I H E R , Egbert. 1 9 9 3 . Spatbabylonische Texte aus UI8. Teil IV ( A u s g r a b u n g e n in U r u k - W a r k a , E n d b e r i c h t e 12). M a i n z : P h i l i p p v o n Z a b e m .

Treatments of Annual Phenomena in Cuneiform Sources

59

Appendix A: BM 45728 = SH.135 (81-7-6) P r e v i o u s l y p u b l i s h e d : K u g l e r , S S B I, 4 5 - 7 ; c o p y Nr. 3 , Tafel 2. R e v i s e d collation: B r i t t o n 6-28-01 Translation

Text 1. 2.

' /GUR-ÓR

^[

... a]-na àr-ki-ka/

'

] YOU GO BACK BEHIND YOU.

ir-BU-LÌ

^

]

]

]x' hu-sù mu-Sà ina MU-^[A

3.

4. IGI.DUG.A sà SIN] 27 u^-MU"** IT-TI I-TA-ÂR

'

] ITS [

THEY SET ] ITS YEAR WITH YOUR YEAR

^OBSERVATION of THE MOON] (IN) 27 ADDITIONAL DAYS IT RETIMIS (TO ITS PLACE).

5. IGI.DUG.A sà '^]dili-bat 8 MU™' a-na àr-ki-ka ' OBSERVATION of] VENUS. 8 YEARS YOU GO BACK 6. ' IgvT-àr ' ....]4 U4-mu"^ tu-maf-fa igi-mar BEHIND YOU '....4 DAY(S) YOU SUBTRACT (AND) YOU SEE (IT).

IGI.DUG.A sà ]'*gU4-UD 6 mu-ka a-na àr-ki-ka ^ OBSERVATION of] MERCURY. YOUR 6 YEARS YOU GO

7.

8. ' /GUR-ÓR '

] it-ti-sù ta-fip-pit

... ]I0 IT-TI IGI.DUG.A ta-tip-pii IGI.LÀ

9.

IGI.DUG.JA sà ^$al-bat-a-nu 47 MU""'

10.

BACK BEHIND YOU.

'

...] YOU ADD TO IT

'...10 YOU ADD TO THE OBSERVATION (AND YOU) SEE (IT). '"OBSERVATION OF MARS. 47 YEARS "YOU GO BACK

11. a-na âr-ki]-ka GUR-ÓR 12 UD it-ti-i

BEHIND. 12 DAY(S) IN ADDITION...'^ 12 DAY(S) YOU ADD

12. ...]12 UD it-ti IGI.DUG.A ta-tip-pig-ma IGI.LÀ

TO THE OBSERVATION (AND YOU) SEE (IT).

13.

IG]I.DUG.AIA "SAG.US 5 9

MU""

14. [a-na àr-k]i-GUT-àr UI-MU a-na U^-mu IGI.LÀ

"OBSERVATION of SATURN, 5 9 YEARS '*YOU GO BACK BEHIND YOU. DAY FOR DAY YOU SEE (IT).

15. ^IGI.du8.A^ sà """KAK.SI.SÂ 27 MU™*

" ..OBSERVATION of THE ARROW. 27 YEARS '*YOU GO

16. [a-na àr-k]i-ka gur-àr UD anà UD IGI.LÀ

BACK BEHIND YOU. DAY FOR DAY YOU SEE (IT).

17.^DUB sà^ "'la-ba-Si DUMU Sà '"'EN.MAN-xV / '^DUMU SÀ^ KA-NIK Sà KÀ 18. [ 19. [

] 31 MU TA LA I-TAB-BAL E""' "*^GAN^ X x/TAR^ x 28 x x

" [TABLET of] LABASI, SON OF BEL-§AR-(?), DESCENDENT OF THE SEALER OF DOORS. '*....]31YEAR(S) SINCE IT' WAS NOT CARRIED AWAY; BABYLON MONTH I X ' "

x' x' 28 x x

Critical A p p a r a t u s : 8,9,12: pig written B U . 1 3 : 9 written with 3 d i a g o n a l w e d g e s . 15: old form o f " m u l " . 17: n o " n i " following " m a n " as in K u g l e r ' s c o p y (Nr. 3 , Tafel 2 ) ; " s a " for " n u m u n " following 2"** " d u m u " . 18: " 3 0 " c o u l d b e "itu", b u t n o t 4 0 o r 5 0 . 19: K u g l e r ' s c o p y s h o w s " 1 2 " , but if so, a c r o o k e d o n e w h i c h looks m o r e like "tar"; line h a s b e e n erased.

60

J.P. Britton

B M 4 5 7 2 8 o b v e r s e and reverse. (Copyright T h e British M u s e u m )

61

Treatments of Annual Phenomena in Cuneiform Sources

Periods: Planet Venus

Period

Modern

Text

[ 5 syn. p h e n . =] [19 syn. phen. =]

8 years -

4 days

8y-

2.4°

6 years + 10 d a y s

6 y + 10°

Mars

[22 syn. phen. =] 4 7 years + 12 d a y s [15 syn. p h e n = 32 years + 12 days

4 7 y - 9°

Saturn

[57 syn. phen. =] 59 years to the d a y

Sirius

[334 m o n t h s =

Mercury

v p a r s tn ]1 29 7 years to the

day

32y+ir] 59y+

r

2 7 y + 1;16^

Equivalent from 27 y = 3 3 4 m Venus

[ 5 syn. p h e n . =]

9 9 m - 5.IT

99m-4d

Mercury

[19 syn. p h e n . =]

7 5 m - 13.3T

75m-13d

Mars

[22 syn. phen. =] 5 8 1 m + 24T

581m + Id

[ 15 syn. p h e n =

396m +5d]

Saturn

3 9 6 m + 7.5T

[57 syn. p h e n . = ] 7 3 0 m - 4 . 5 T

7 3 0 m - 6d

Notes: T h e p e r i o d for V e n u s a p p e a r s to a s s u m e 8 years = 9 9 m o n t h s , or alternatively states the shortfall from 9 9 m o n t h s . T h e rest o f the increments in " d a y s " clearly relate to z o d i a c a l returns rather than w h o l e m o n t h s . Evidently the t w o G o a l Y e a r p e r i o d s for M a r s have b e e n conflated so that either " + 1 2 d a y s " is a n error for " - 9 ( ± 1 ) d a y s " or " 4 7 y e a r s " is an error for " 3 2 y e a r s " . Either way, the text s e e m s e v i d e n c e of early k n o w l e d g e of b o t h these p e r i o d s . T h e a b s e n c e of Jupiter is p u z z l i n g . O n e w o u l d expect 12y + 5d, if n o t the G o a l Y e a r p e r i o d s o f 7 1 y - 6d or 8 3 y - I d . A Labasi is m e n t i o n e d in the Assyrian R e p o r t 4 5 5 ( H U N G E R ( 1 9 9 2 ) , p . 2 5 6 ) as father o f Bel-ahhe-eriba, author o f a report o n an occultation o f the P l e i a d e s ; a n d also as an e p o n y m for - 6 5 6 . F r o m his a n c e s t o r ' s position, h o w e v e r , the a u t h o r o f the text w a s p r o b a b l y firom B a b y l o n . In L A B S 160 ( L A B S , p p . 122-124) M a r d u k - s a p i k zeri, a B a b y l o n i a n writing to a n i m n a m e d Assyrian king m e n t i o n s a ""la-a-ba-si as one of 2 0 fellow students of a s t r o n o m y w h o h a v e r e t i u n e d from E l a m (or from Assyria as refugees) w h o is "proficient in the Series a n d e x o r c i s m " a n d potentially "useful to the k i n g " . T h e n a m e Labasi also o c c u r s frequently in later texts from U r u k a n d also in C T 4 4 49,'*'' a 2"*^ century text from B a b y l o n , w h e r e it is written NU.UR.'*^ H o w e v e r , from the father's profession a n d the t a b l e t ' s p r o v e n a n c e , it s e e m s u n l i k e l y that the a u t h o r is from U r u k , while the contents are m o r e consistent with an earlier rather t h a n later date. In particular, the writing of S A G . U S for Saturn a p p e a r s in the D i a r y for - 6 5 1 , a p p e a r s together w i t h G E N N A in the D i a r y for - 5 6 7 , a n d has b e e n r e p l a c e d b y G E N N A in 5'*' century a n d subsequent Diaries."*^

See RocHBERG (2000) for a recent discussion of this text. ''^ Reference and detail courtesy of J. Steele. 46

Detail again courtesy of J. Steele.

62

J.P. Britton

Appendix B: BM 36731+ (= 80-6-17, 464+A+B) This h a n d s o m e l y written text contains c o m p u t e d instants (dates a n d times) o f solstices a n d e q u i n o x e s together with schematic dates for the settings (sii) a n d risings (igi) of Sirius. P r e s e r v e d data extend from year 8 N a b o p o l a s s a r (-617) to 2 5 N e b u c h a d n e z z a r (-579), b u t it is very likely that the c o m p l e t e tablet c o v e r e d the reigns of N a b o p o l a s s a r a n d N e b u c h a d n e z z a r (-625 to - 5 6 1 ) . T h e text is u n i q u e in several respects. First it is the earliest e p h e m e r i s w e p o s s e s s - imderstanding the t e r m in its general sense as a series o f c o m p u t e d successive asfronomical p h e n o m e n a - although it covers m u c h the s a m e p e r i o d as W 2 2 8 0 1 discussed b e l o w . It is also the only k n o w n text w h i c h calculates the t i m e s o f day a n d actual dates of the cardinal p h e n o m e n a , rather than simply their s c h e m a t i c dates, expressed in tithis. Finally, it p r o v i d e s u n i q u e e v i d e n c e of early efforts to establish the lengths of the year a n d m o n t h , a n d thus also of the c h r o n o l o g y o f this undertaking. T h e p r o v e n a n c e of the text is all b u t certainly B a b y l o n , a n d m o s t p r o b a b l y from a h o u s e abutting the iimer wall near the south w e s t c o m e r of the m o u n d c o v e r i n g the Esagila, w h e r e R a s s a m e x c a v a t e d j u s t prior to the s h i p m e n t of tablets, i n c l u d i n g o u r text, w h i c h w e r e a c c e s s i o n e d at the British M u s e u m o n June 17, 1880."*' T h e principal fragment w a s originally p u b l i s h e d b y N e u g e b a u e r a n d Sachs,'*^ w h o identified its contents a n d dates. Unfortunately, they m i s t o o k the dates o f the solstices a n d e q u i n o x e s to b e given in tithis (as in fact are the dates o f the Sirius p h e n o m e n a ) , w h i c h led to a n erroneous depiction o f b o t h the u n d e r l y i n g s t m c t u r e of the text a n d the conq)etence of its execution. Subsequently, t w o additional small fragments found a m o n g the imaccessioned fragments of the British M u s e u m ' s 80-6-17 archive e x t e n d e d the attested r a n g e of the text to at least - 5 7 8 a n d s h e d light on the s t m c t u r e o f the c o l u n m dealing with the settings a n d risings o f Sirius a n d other details of its contents. Text T h e original tablet c o n t a i n e d at least five a n d p r o b a b l y six c o l u m n s , o f w h i c h elements o f only the last four are preserved. O n the leftmost e d g e , n o w missing, w a s p r o b a b l y a c o l u n m [(i)] of successive regnal years. F o l l o w i n g w e r e four c o l u n m s giving for e a c h y e a r the dates ( m o n t h a n d day) a n d times (time d e g r e e s relative to sunset or sunrise) o f [(ii-SS)] s u m m e r solstice (also missing), (iii-FE) fall e q u i n o x , ( i v - W S ) winter solstice, a n d ( v - V E ) vernal e q u i n o x . Finally, the last c o l u n m o n the right (vi-Sirius) contains the dates in tithis o f the setting (su) a n d rising (igi) of Sirius in e a c h year. T h e c o l u m n s are delimited b y vertical scorings, a n d e v e r y third line is m l e d off horizontally. T h e m a i n fragment m e a s u r e s r o u g h l y 9 c m (w) x 8 c m (h) w i t h very nearly 3 r o w s / c m of height a n d c o l u m n - w i d t h s of 2 . 5 c m , the last c o l u n m ending 2 c m from the right e d g e . T h u s if the original tablet c o n t a i n e d 33 lines to a side, a n estimate consistent with the curvature o f the m a i n fragment, it w o u l d h a v e m e a s u r e d a p p r o x i m a t e l y 1 3 c m (w) x 1 1 c m (h).

READE (1986), pp. xixff. NEUGEBAUER and SACHS (1967), Text A, pp. 183-190.

Treatments of Annual Phenomena in Cuneiform Sources

Figure B s h o w s the reconstructed text. P r e s e r v e d text is highlighted and partially p r e s e r v e d signs are underlined. A n apparent scribal error (GU4(II) for B A R ( I ) ) is s h o w n as read, b u t in italics. T i m e s of d a y are time degrees (us), transcribed as ° and c o r r e s p o n d i n g to 1/360th o f a to the nearest sunrise o r simset. T h e s e designations are abbreviated in transcription as follows. g < gin:

after sunset (until midnight)

63

in b o l d type, at rev. vi, 11 e x p r e s s e d in day, referred

< [° of night] h a v i n g g o n e (gin «

z < zalag: until sunrise (from midnight)

< [until] brightening (zalag «

n < nim:

after sunrise (until n o o n )

< [after sun]rise ( n i m « saqu)

s < su:

until simset (from n o o n )

< [until sun]set (su » rabu)

alaku)

namaru)

T h e unit us (°) is explicitly expressed at every instance o f 3°. T h e principal fragment, B M 3 6 7 3 1 , includes the right e d g e and p r e s e r v e s contents from obverse, cols, iii (FE) to vi (Sirius), for years 8 N a b o p o l a s s a r (-617/6) to 8 N e b u c h a d n e z z a r (-596/5) and from reverse, col. vi (Sirius) for years 11-17 N e b u c h a d n e z z a r (-593 to-587/6), with t w o lines missing at the b o t t o m o f the obverse and n o e d g e s h o w i n g at the top of the reverse. F r a g m e n t A contains the u p p e r e d g e o f the reverse, fraces of time designations from col, iii ( F E ) a n d four years o f c o l u m n iv ( W S ) for 11-14 N e b u c h a d n e z z a r . F r a g m e n t B p r e s e r v e s p a r t of col. vi-Sirius a n d fraces of col. v - V E for years 21-24 of N e b u c h a d n e z z a r (-583/2 to - 5 8 0 / 7 9 ) , and loosely j o i n s the m a i n fragment at a point c o r r e s p o n d i n g to N a b o p o l a s s a r years 16-21 o n the obverse. T h e t a b l e t ' s thickness and curvature suggest that n o t m o r e than 9 lines are missing at the top of the o b v e r s e , while the ruling e v e r y 3 lines suggests either 6 or 9 lines are missing. Since the latter w o u l d p l a c e the V E for the accession year of N a b o p o l a s s a r (-625/4) in line 1, while allowing the text to extend through the last year (43) of N e b u c h a d n e z z a r ' s reign (-561/0), it s e e m s the m o r e likely alternative. Critical A p p a r a t u s Obv.

Rev.

iii, 18: 2 5 is confirmed iv, 2 0 : 2 0 . Sachs r e a d 2 [ 1 ] , but the 2 0 covers the full space leaving n o r o o m for a T . iv, 3 0 : H m o r e likely than 12 vi, 16: GU4(II) p r o b a b l e , n o t S I G (III). iv, 1: 2 5 , 2 6 o r 2 4 also possible. vi, 14: GU4 (II) confirmed, error for B A R (I).

D a y s or T i t h i s ? N e u g b a u e r a n d S a c h s interpreted the " d a y s " in the dates for solstices a n d e q u i n o x e s as tithis and a s s u m e d that the difference in time b e t w e e n successive lines for a given p h e n o m e n o n w a s intended to b e AT = 1 year = 12*" 1 T 1,36° = 12*" 11 ; 1 6 \ Since the p r e s e r v e d day n u m b e r s s o m e t i m e s increase b y 10 from r o w to r o w , this m e a n t that the author of the text c o m m i t t e d n u m e r o u s errors in adding 1 1 , despite getting the c o n s i d e r a b l y m o r e c o m p l i c a t e d times correct and the a b s e n c e of s u c h errors in the Sirius dates. T h i s interpretation also m a d e the text a u n i q u e e x a m p l e o f using time

J.P. Britton

64

51 '3) 'a 'Sb 'oil 'oil •& '& 'OB sù '5b 'Si — (N

— CM

> > >

> >

>

ON O

—

— (S

> > > > > >

S

-,s 'Si '3i SIS S s S S S

t— 00

ON

^

2

r- 90 Oj

Si 'w 5b

& ^'3'

ou o£ oc

> > >

MM

m

VÌ

ou N ON

>M 61)

r- r-

DU N

e

ON

r*-

g s

il ilia

O SI G

III

HP

•5 ;3 S o — (N

m »o — (S

iiiis

MU

«1^ M ci m — in

^

r-

SS ^

< 60 N.:H >« « C M

+

M ML

ML

«0 N

c

^ ç; S ^ Î2 (N — X X X X X X

o N

> a

1/Ì

MI 60

m

5: N a — in

^ > ?

PQ

X X X

E3 <^ ^ E > > 5:

U

5 " r-I r-' ? >= £3 > > > 5 > >M

60 N

m — ON

—

C

>

r- r- ON

c | WL — m

> 5: ?

? 5 N

0\

c

«/1

m

— es

— <s >

N

— in

N

S

>

^1

6.^

w> 60::j4) — ON

>

> = >

\0 *n "«t m
> > a > >

> > a

>

=

>

>=>

Q> 00 r;^ ^ »/i ON ON ^ OS m in »/i

Treatments of Annual Phenomena in Cuneiform Sources

65

5b 'Sb 'Eb 5b'«b 'Sb fib 5b'5b•5b'Eb'5t 5b'a'5t 5b'5b'Ei> 'Sb'a'5b

ss

2 a

mm

2:

g

IQ VO

=

>

i l i i l "

^1 »s

>

(S — CN

— <s > >

SMS

15-Si SIS 2: ?S ^

s

-

2

I I

s

ON T^ M

s

R><x

00 g

s

MM

K ><

>5

X X 3

^ — fS

X X X

X X X

+

1 ^

— <s — X X

>5

X X

>5

R " s (N — (N — X X X X X X >5 X X

>

N «

00 c

UL

> 00 N B

tri fS — <s

> ? > > ?>

? >?

> 5

> >>

—
2

5: > > ^ ?

^

u

3 OD s

^

2

— (N > 3

— Q 00 00 00 00 00

VO

f- 00 0\ o —

ON

a

— U-I

>< R

?I 2

=

^

00

6b N

01) N C

2

5

S S SS S S S S S^ 00 OS S S S -«T UL VO T^ Tt LA) VO s

M

> >>

>

= > = ;=

ON

00 r-

T)- m IN — o

(Nm-5tuivot^ooov lN
66

J.P. Britton

degrees to denote subdivisions of tithis, w h i c h e l s e w h e r e are r e p r e s e n t e d b y sexagesimal fractions. Finally, it implied that the seasons should b e 3"" T 4 , 5 4 ° long, w h e r e a s the p r e s e r v e d e x a m p l e s reflected increments of only 1,54° o v e r w h o l e days/tithis, resulting in a cumulative error of T p e r year, w h i c h w a s a s c r i b e d to b e i n g " i n c o m p e t e n t l y c o m p e n s a t e d for b y incorrect d a y nimibers". T h e s e difficulties all disappear w h e n it is r e c o g n i z e d that in fact, the dates in the text reflect actual days in the limar calendar. T h i s results in t w o attested s e a s o n s o f 9 1 ' ' 1,54° = 9 1 ; 1 9 d a y s a n d a consistent interval b e t w e e n cardinal p h e n o m e n a of the same sort o f AT = 1 year = 365*' 1,36° = 3 6 5 ; 1 6 d a y s . T h i s is demonsfrated in Figure B 2 , w h i c h gives: first, the p r e s e r v e d dates from the text; next, the c o r r e s p o n d i n g calculated Julian dates for the b e g i i m i n g of that B a b y l o n i a n day, followed b y the h o u r (H**) c o r r e s p o n d i n g to the d e s i g n a t e d time, but adjusted for winter solstice so as to b e stated relative to 18:00 ( 6 : 0 0 P M , as at the e q u i n o x e s ) , followed b y the Julian D a y n u m b e r c o r r e s p o n d i n g to the b e g i i m i n g of the B a b y l o n i a n day; a n d finally the time intervals c o r r e s p o n d i n g to a year a n d the season (where p r e s e r v e d ) . T h e calculated dates are from a p r o g r a m b y Peter H u b e r ( C R E S D A T ) using visibility criteria for the beginnings of the m o n t h d e s c r i b e d in H U B E R ( 1 9 8 2 ) . It is r e m a r k a b l e that a m o n g the p r e s e r v e d dates there are only t w o instances w h e r e the calculated date is at variance with the B a b y l o n i a n date, o b v . iv, 13 a n d v, 2 9 for W S -614 and V E - 5 9 8 , w h i c h are m a r k e d b y u n d e r s c o r i n g in Figure B 2 , with the discordant data highlighted in bold. B o t h turn out to h a v e b e e n instances of threshold visibility c i r c u m s t a n c e s at the b e g i n n i n g of the m o n t h . F o r W S in - 6 1 4 ( X 3), the difference b e t w e e n the calculated negative altitude of the s u n at m o o n s e t a n d the threshold value for visibility (dh) o n the d a y prior to calculated d a y 1 w a s negligible (0.0°), a n d the actual m o n t h evidently b e g a n o n e d a y earlier, m a k i n g the p r e v i o u s m o n t h 2 9 instead of 3 0 days. F o r V E - 5 9 8 the opposite w a s true; d h w a s only + 0 . 2 ° o n calculated d a y 1, a l ^ the actual m o n t h evidently b e g a n the following day, m a k i n g the p r e v i o u s m o n t h 3 0 instead o f 2 9 d a y s . T h u s there is n o d o u b t that the text reflects d a y s rather than tithis in the dates of the solstices a n d e q u i n o x e s . It also s h o w s that H u b e r ' s visibility criteria for the b e g i i m i n g of the m o n t h closely m o d e l whatever w a s the actual practice.

67

TREATMENTS OF ANNUAL PHENOMENA IN CUNEIFORM SOURCES

VI2 VI

D

---TIME---

15 25

D

D*

H**

JD*

- 6 1 0 SEP

26

15.4

1478524

0

33 S

- 6 0 9 SEP

26

21,8

1478889

CALCULATED JULIAN DATES

- - -TIME - - -

IX IX

21

IX

4

IX

14

1

IX IX IX IX

25 7 19 29

21 1 9 27 1 33

M

D

1

1

1 1 1

1

1

0 1 0 0 0 0 0 0 1 0 1 0 1

INTERVAL (DAYS) JD*

- 6 1 7 DEC

27

2.2

1476059

- 6 1 6 DEC

26

8.6

1476424

- 6 1 5 DEC

26

15

1476789

3 6 5 16

- 6 1 4 DEC

27

21.4

1477155

- 6 1 3 DEC

27

3.8

1477520

366 16 364 16

- 6 1 2 DEC

26

10.2

1477885

3 6 5 16

-611 DEC

26

16.6

1478250

3 6 5 16

27

- 5 9 6 DEC

26

3°G 39 g

- 5 9 5 DEC

27

- 5 9 4 DEC

Z

WS-FE

3 6 5 16 DH(-L)=0,0,27 > 2 6

- 6 1 0 DEC

27

-1

1478616

3 6 5 16

91 19

- 6 0 9 DEC

27

5.4

1478981

3 6 5 16

9 1 19

- 6 0 8 DEC

26

11,8

1479346

3 6 5 16

- 6 0 7 DEC

26

18,2

1479711

3 6 5 16

- 6 0 6 DEC

27

0.6

1480077

3 6 5 16

- 6 0 5 DEC

27

7

1480442

3 6 5 16 3 6 5 16

- 6 0 4 DEC

26

13,4

1480807

- 6 0 3 DEC

26

19,8

1481172

3 6 5 16

- 6 0 2 DEC

27

2.2

1481538

3 6 5 16

-601 DEC

27

8.6

1481903

3 6 5 16

- 6 0 0 DEC

26

15

1482268

3 6 5 16

- 5 9 9 DEC

26

21,4

1482633

3 6 5 16

- 5 9 8 DEC

27

3.8

1482999

3 6 5 16

10,2

1483364

3 6 5 16

16,6

1483729

365 16

-1

1484095

3 6 5 16

27

5,4

1484460

3 6 5 16

- 5 9 3 DEC

27

11.8

1484825

3 6 5 16

$ g

- 5 9 2 DEC

26

18,2

1485190

3 6 5 16

-591 DEC

27

0.6

1485556

3 6 5 16

Z

- 5 9 0 DEC

27

7

1485921

3 6 5 16

CALCULATED JUUAN DATES

15 9 27 57 39 45 51 33 3 21 15 9 27

DT =Y

H**

- 5 9 7 DEC

FE-SS

3 6 5 16

D*

51 N

- - -TIME

DT=Y

M

JY

g z 27 n 21 S 15 g 45 Z 51 n 3°g 39 g 21 Z 9S 27 g 33 Z 3 °n 45 S 51g 9Z 27 n 21 « 15 g 45 Z 51 9

TEXT - V E

I 1 I 12 XII 24 I 5 XII2 15 XII 26 I 8 XII 18 XII 29 I 10 xn 22 3 I XII 14

M

51

1 11 21 3 14 25 6 18 29 10 20 2 12 23 4 16 27 8 19 1 11

X X IX X X IX X IX IX X IX X X IX X IX IX X IX X

JY

0

TEXT - W S M

INTERVAL (DAYS)

CALCULATED JULIAN DATES

TEXT - F E M

JY

M

INTERVAL (DAYS)

— n - 6 0 8 MAR

D* 27

13

1479072

S

- 6 0 7 MAR

27

19.4

1479437

3 6 5 16

g

- 6 0 6 MAR

28

1,8

1479803

3 6 5 16

91 19

Z

- 6 0 5 MAR

28

8.2

1480168

3 6 5 16

91 19

n

- 6 0 4 MAR

27

14,6

1480533

3 6 5 16

9 1 19

S

- 6 0 3 MAR

27

21

1480898

3 6 5 16

9 1 19

g

- 6 0 2 MAR

28

3,4

1481264

3 6 5 16

9 1 19

Z

-601 MAR

28

9,8

1481629

3 6 5 16

91 19

n

- 6 0 0 MAR

27

16,2

1481994

3 6 5 16

9 1 19

S

- 5 9 9 MAR

27

22.6

1482359

3 6 5 16

9 1 19

g

- 5 9 8 MAR

28

5

1482725

3 6 5 16

9 1 19

Z

- 5 9 7 MAR

27 11.4

1483089

90 19

n

- 5 9 6 MAR

27

1483455

364 16 366 16

H**

17,8

JD*

D T = Y VE-WS 9 1 19 91 19

DH=0.2, 2 7 > 2 8

Figure B 2 . C a l c u l a t e d Julian dates for p r e s e r v e d B a b y l o n i a n dates in B M 3 6 7 3 1 . D * a n d J D * = Julian dates a n d d a y n u m b e r s of the b e g i n n i n g of the indicated B a b y l o n i a n day. H * * = h o u r s from 18:00 h o u r s ( 6 : 0 0 P M ) m e a n time at the longitude of B a b y l o n (-44.5).

J.P. Britton

68

Times SS (a) Txt: rei s-set/rise 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

45 z 51 n 69 § 27 g 21 z 75 n 45 § 51 g 3 °n 99 n 21 .s 69 z 27 n 93 §

FE (b,c) from (a) s-set Txt; rei 19; 12 s-set/rise 45 z 3 99 51 n 33 s 195 291 63 g 27 21 z 75 n 123 219 9S 87 g 315 51 3 "n 147 81 § 243 15 g 69 z 339 27 n 75 171 57 § 39 g 267

(c) (b) (a) from from Txt: rei s-set 19;12 s-set/rise 135 117 51 n 231 213 3 "g 327 309 99 g 45 63 21 z 159 141 69 § 255 237 27 g 351 333 93 z 87 69 3 °n 183 165 45 § 279 261 51 g 15 357 69 z 93 27 n 111 207 189 21 § 303 285 75 g 21 39 45 z

WS (b) (c) (a) from from Txt: rei s-set 19; 12 s-set/rise 267 231 3°g 81 z 3 327 99 15 n 63 195 159 69 § 291 255 27 g 27 351 57 z 123 87 39 n 45 s 219 183 315 279 51 g 51 15 33 z 147 111 63 n 243 207 21 s 339 303 75 g 75 39 9z 171 135 87 n

VE (b) (c) from from s-set 19;12 3 345 99 81 195 177 291 273 27 9 123 105 219 201 315 297 51 33 147 129 243 225 339 321 75 57 171 153 267 249

F i g u r e B 3 . Instants (us) o f solstices and e q u i n o x e s relative to (a) sunset a n d sunrise (text); (b) sunset; a n d (c) sunset at simimer solstice ( 1 9 ; 12 p m ) T h e instants o f solstices a n d equinoxes are stated relative to the nearest sunset or sunrise, in k e e p i n g w i t h observational practice reflected in eclipse reports a n d predictions as early as - 6 8 5 a n d c o n t i n u e d t h r o u g h o u t the Seleucid Era. In F i g u r e B 3 these instants are shown: (a) as reported in the text; (b) as related to theoretical sunset for that p h e n o m e n o n (18;00'' ± 1;12''); a n d (c) as related to simset at s u m m e r solstice ( 1 9 ; 1 2 ). M e a s u r e d relative to simset, these a d v a n c e yearly b y 0;16 d a y = 1,36°, a n d thus r e p e a t after 15 years. B e t w e e n seasons times a d v a n c e b y 0;19'^ = 1,54° relative to the s a m e instant (e.g. sunset at SS in Figure B 3 ) a n d b y 1,54° ± 1 8 ° relative to sunset for e a c h p h e n o m e n o n . Since 1 9 x 1 , 5 4 ° = 6° ( m o d . 3 6 0 ° ) , the resulting times, m e a s u r e d from any given simset are all distinct, a n d together c o m p r i s e a series at intervals o f 6° b e g i n n i n g 3° after sunset. T h i s n o r m , equal to half a n interval b e t w e e n collective instants, evidently reflects a desire to a v o i d the a m b i g u i t y o f times w h i c h coincide with simrise or sunset, and clearly is of n o observational significance. Still, it reflects a d e g r e e of care in the construction of the scheme. It is p e r h a p s n o t e w o r t h y that the s c h e m e for times p r e s e n t e d in F i g u r e B 3 , c o r r e s p o n d s also to that begiiming with line 1 in the text. Accuracy T h e a c c u r a c y of t h e s c h e m e c a n b e g a u g e d from the following c o m p a r i s o n of schematic instants at the b e g i n n i n g a n d end o f the text with calculations from m o d e m theory. O n a v e r a g e the s c h e m e is a p p r o x i m a t e l y accurate for dates shortly after it ends, with solstices being r o u g h l y a d a y t o o early (SS) or late ( W S ) a n d e q u i n o x e s r o u g h l y 2 days early (FE) or late ( V E ) . F o r earlier dates the errors

Treatments of Annual Phenomena in Cuneiform Sources

69

increase algebraically b y roughly 1 d a y every 4 0 years. Since a single date and time determines the entire s c h e m e , it is t e m p t i n g to search for likely a n c h o r points, candidates for w h i c h include the s u m m e r solstices o f - 5 6 6 at sunset (3° gin) and 5 5 7 at d a w n (3° n i m ) , the latter b e i n g presimiably near the time the text was c o m p o s e d . It is doubtful that observations o f equinoxes p l a y e d a n y role in the formulation o f this s c h e m e , if indeed a n y such were m a d e at this early date.

Year -625 -561

SS +2.5 +0.9

E r r o r (Cale - Text) in days for: WS Avg FE +0.3 +1.4 +3.1 -1.2 -0.2 +1.6

SE -0.4 -2.0

Avg +1.4 -0.2

Sirius Dates T h e dates for the d i s a p p e a r a n c e Q ( s ù ) and r e a p p e a r a n c e r(igi) of Sirius follow a different s c h e m e , m a r k e d b y the absence of fractional " d a y s " a n d a m o r e regular p r o g r e s s i o n of armual increments o f 11 " d a y s " . B o t h characteristics suggest that the units are tithis rather than calendar days, w h i c h p r o v e s to b e the case, as s h o w n b y the constant interval o f 2'"5^ from Q ( s ù ) to r(igi). In the original fragment there is a single p r e s e r v e d increment of 12^ b e t w e e n years -603 and - 6 0 2 ( o b v . 2 4 - 2 5 ) , following 9 dates separated b y increments of 1 T . T h e r e m u s t also b e a n increment o f 12^ in the t w o missing lines b e t w e e n obverse and reverse, but w e cannot b e sure o f m o r e than that the p r e c e d i n g g r o u p o f dates separated b y I T contains at least 7 a n d not m o r e than 9 dates. This suffices to rule out regular cycles o f 8, 19 and 3 0 years as the basis of the s c h e m e , b u t leaves the precise structure o b s c u r e . F r a g m e n t B , h o w e v e r , s h o w s that the s c h e m e repeats after 27 years, and that a third interval of 12^ m u s t fall in the missing t w o lines for -585 and -584 (rev. 9-10). If o n e calculates with a n epact of I T , after 8 years the resulting date will fall 2 ' short o f the original date; after 11 years, V over; after 19 years V short; a n d after 2 7 years, 3 ' short. T h u s a s c h e m e b a s e d on a 2 7 year cycle m u s t contain three 1 2 ' increments disfributed in s o m e fashion. T h e options are: every 9 years; 8-9-10 years; or 8-8-11 years. A s reconstructed in Figure B the 1 2 ' intervals are uniformly disfributed every 9 years, w h i c h in fact seems m o s t likely. Other possibilities include: a d d i n g a tithi to the dates s h o w n for -594 (obv. 33) a n d / o r - 5 8 5 (rev. 9), which together or singly w o u l d result in groups separated b y 8-9-10 years; or adding tithis to each of the dates for - 5 9 4 , - 5 8 6 and - 5 8 5 , w h i c h w o u l d yield g r o u p s separated b y 8-8-11 years. N o n e of these alternatives s e e m likely, let alone compelling, and in light o f the t e x t ' s scorings every 3 lines, I believe the 1 2 ' intervals were p r o b a b l y evenly disfributed after groups o f 9 dates, and specifically after " d a y s " 2, 12 and 2 2 for Q(sii) and 7, 17 and 2 7 for r(igi). T h i s s c h e m e is r e p r o d u c e d in Figure B 4 , w h e r e intervals of 1 2 ' separate the last date in e a c h c o l u m n from the first date in the next. Attested and inferable dates are s h o w n in bold, while dates w h i c h could b e increased b y 1' are underlined. In contrast to the U r u k S c h e m e , this s c h e m e h a d n o coimection with the p r o c e s s o f intercalation in the p e r i o d c o v e r e d b y the text, h e n c e the a b s e n c e of fixed m o n t h s associated with e a c h date. Q occurs as early as m o n t h I and as late as m o n t h III, although usually in m o n t h II, while F occurs in m o n t h s III, I V and V , although mainly in m o n t h IV. T h e last is m o r e or less consistent with M U L . A P I N (ii, 4 2 - 4 3 ) ,

70

1

J.P. Britton

yr

r

/2

yr

r

n

/2

r

1

4

9

10

14

19

19

24

2

15

20

11

25

30

20

5

10

3

26

1

12

6

11

21

16

21

29

4

7

12

13

17

22

22

27

2

5

18

23

14

28

3

23

8

13

6

29

4

15

9

14

24

19

24

7

10

15

16

20

25

25

30

5

8

*2i

*26

17

1

6

26

11

16

9

2

7

18

12

17

27

22

27

* i n r e a s e r e q u i r e s t h a t 2 2 / 2 7 b e also i n c r e a s e d F i g u r e B 4 . Sirius Visibility S c h e m e - B M 3 6 7 3 1 . w h e r e F-Sirius is p l a c e d at I V 15 in the schematic calendar, which, h o w e v e r , is also and a n o m a l o u s l y , the stated date of the s u m m e r solstice. H e r e F-Sirius follows SS b y r o u g h l y 17 d a y s n e a r the b e g i n n i n g of the text w h i c h increases t o a p p r o x i m a t e l y 20 d a y s b y its end. I n the U r u k S c h e m e this interval is fixed at 2 1 ' .

Treatments of Annual Phenomena in Cuneiform Sources

71

Appendix C: W22801+22805 T h e t w o fragments c o m p r i s i n g w h a t r e m a i n s o f this text w e r e u n e a r t h e d at U r u k b y the G e r m a n A r c h a e o l o g i c a l Expedition. C o p i e s b y E. v o n W e i h e r a r e p u b l i s h e d in S b T U I V ( 1 9 9 3 , N o . 169), a n d HUNGER ( 1 9 9 1 ) g i v e s a transcription a n d b r i e f c o m m e n t a r y . T h e text, s h o w n in F i g u r e C, consists of t w o c o l u m n s , s e p a r a t e d b y d o u b l e scoring, w i t h e a c h r o w also ruled o f f W i t h i n a coltmin e a c h line c o n t a i n e d a statement of: (a) the date (regnal year, m o n t h a n d tithi, a c c o m p a n i e d b y the k i n g ' s n a m e in year 1) o f s u m m e r solstice, d e n o t e d b y " g u b " (~ izzuzu, stand), followed b y the notation " k i n - d i r " for years containing an intercalary m o n t h VI2; t h e n (b) the date ( m o n t h [and tithi]) of winter solstice (also " g u b " ) ; followed b y " d i r " for years with a n intercalary m o n t h XII2. T h e larger fragment ( 2 2 8 0 1 ) p r e s e r v e s dates from 3 0 N e b u c h a d n e z z a r t h r o u g h 4 A m e l - M a r d u k , while o n the smaller fragment the intercalary VI2 in year 18 establishes that its dates m u s t refer to the r e i g n of N a b o p o l a s s a r as H u n g e r o b s e r v e s . A t the left-hand e d g e of the larger fragment, following traces o f the " g u b " for winter solstice are t w o " d i r " s d e n o t i n g XII2 intercalations, s e p a r a t e d b y 4 y e a r s a n d followed b y 7 years without such intercalations. T h e s e suffice to establish that the entry in c o l u m n [i] c o r r e s p o n d i n g to y e a r 34 N e b u c h a d n e z z a r in c o l u n m [ii] m u s t h a v e b e e n for 7 N a b o p o l a s s a r , a n d thus that the t w o fragments w e r e p o s i t i o n e d as s h o w n in F i g u r e C. F u r t h e r confirmation o f this configuration is p r o v i d e d b y year 18 N a b o p o l a s s a r (col [i]) w h e r e the winter solstice m o n t h ( X ) is p u s h e d slightly out o f alignment b y the "kin-dir", w h i c h mis-alignment c o n t i n u e s a n d is reflected in the c o r r e s p o n d i n g " g u b " p r e c e d i n g year 2 of Neriglissar in c o l u m n [ii]. T h e tablet thus c o n t a i n e d data for the years from at least 3 N a b o p o l a s s a r t h r o u g h 4 A m e l - M a r d u k , with 2 8 years separating 1 N e b u c h a d n e z z a r in c o l u m n [i] {opposite 4 A m e l - M a r d u k in col [ii]} a n d 3 0 N e b u c h a d n e z z a r , the first p r e s e r v e d year in col. [ii]. Since n o e d g e s are p r e s e r v e d , s o m e of these m i s s i n g years m u s t h a v e b e e n included in c o l u m n [ii], w h i c h w o u l d also extend the earliest dates in c o l u n m [i]. O n l y one side o f the text is p r e s e r v e d o n b o t h fragments, so in t h e o r y the text c o u l d h a v e b e g u n as early as - 6 5 0 , with the p r e s e r v e d fragments c o m p r i s i n g parts o f the reverse. M o r e p r o b a b l y , h o w e v e r , the fragments p r e s e r v e parts o f the o b v e r s e in w h i c h case, the text c a n only h a v e h a d 24 lines o n b o t h o b v e r s e a n d r e v e r s e a n d m u s t h a v e b e g u n with years 0 or 1 N a b o p o l a s s a r and ended, if c o m p l e t e , in y e a r s 8 or 9 Cyrus, as s h o w n . Text See Figure C. Critical A p p a r a t u s N o d a y dates are p r e s e r v e d for W S ; dates s h o w n a s s u m e W S = SS + 6"* 6\ " 9 " is written in the old, 9 - w e d g e form throughout. i, obv. 18 ( N b p l s 18): A B , error for G A N , is offset half a sign b y the p r e c e d i n g K I N - D I R in W 2 2 8 0 5 ; so is G U B in W 2 2 8 0 1 . ii, obv. 10 ( N b k d r 3 7 ) : G A N (per H i m g e r ) is correct, b u t c o p y l o o k s m o r e like A B . ii, o b v . 12 ( N b k d r 39): S I G , error for § U . ii, o b v . 2 0 ( A m l d k 2): 2 8 , error for 2 9

72

73

Treatments of Annual Phenomena in Cuneiform Sources

J.P. Britton

I j t t t t t t t t t t t t t t t t t t t t t t t t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 K>2x><xx><xx><x><

X x>< ><><>< ><>^>S

XXXx><><X><><X><XX><XX><X><><X><X

11

Ï

IA- ^ ^ X! ^ BUJ AI> ^ M> 3^ 3 «N — ^^

F -

f3:;:;:i 1-* *S

. . . fs

t

«M

3 3 3 | i | | i | | i | i i ; i ; | ; i | i | i

a1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1

a1 1i

I

> s S K 5 5 « « > > 3 > S S > H 5 > > > 3 S > S 3 cN

3

P

3::i3::BSI9:;;qt::à;;ià;;:ii;i

P P 3

g l i a

3

3

3

3

VO r-

3

e 6 S S6

1 1 1 I 1 1 1 I 1 1 i 1 i 1 1 i 1 i i i i i

i

I

O 00

^

.O^

J 3 J 3 X> : : J Ì : i J S Ì ; Ì ; i Ì : : Ì i i ; i Ì ; : ; ; Ì ; ; Ì j Ì J : Ì J ^

Xl x : X)

.1,

.c

I

ttttttttttttttttttllttl|t X xxxx><xxxx><xx

I

><x><><x'^'xx><x><><

î t ttttttttlttt ttfIHiiiittl ^

3

3

?q

3

3

3

3

3

3

3

3

3

3

3

3

3 3

S ^ :ÌS^ : tf ^ii:: 3

3

3

.b

I t t t t t t t t t t t t t t t t t t t t t t t t

3

i 1 1 1 i 1 1 i 1 1 i 1 i i i i 1 i i i i i

2;sO

VO^OVOVOVÇSÇVÇVOVÇVOVOVO

vûvovoîS^BisCSvÇCsSSS

i

74

J.P. Britton

19-year Cycle T h a t the " d a y " dates are in fact tithis is s h o w n b y the constant e p a c t of 11 " d a y s " , except b e t w e e n 27 a n d 9 w h e r e the difference is 12. A s H u n g e r notes, this m e a n s that the s c h e m e r e p e a t s after 19 years, although with s o m e dates w h i c h differ from those found in the later U r u k S c h e m e . H e r e the s c h e m e b e g i n s w i t h a solstice o n " d a y " 9, w h i c h also c o r r e s p o n d s to year 1 o f N a b o p o l a s s a r , w h e r e a s the U r u k S c h e m e b e g i n s w i t h the latest solstice date, (IV) 7, following w h i c h b o t h p r o c e e d in steps of 1 r ( m o d . 3 0 ) , for 18 years, followed b y a n interval o f 1 2 ' at the e n d o f the cycle. T h e date differences in the t w o s c h e m e s are s h o w n b e l o w , the d a y n u m b e r s of the first 8 years o f the p r e s e n t s c h e m e (henceforth here " U l " ) b e i n g 1' later (higher) than those for the last 8 years of the U r u k S c h e m e (henceforth s o m e t i m e s " U 2 " ) . U n l i k e the U r u k S c h e m e U l avoids " d a y " 30, w h i c h in h o l l o w m o n t h s b e c o m e s day 1 of the following m o n t h . Solstice " d a y " dates: W 2 2 8 0 5 + 0 1 ( U 1 ) a n d U r u k S c h e m e ( U 2 ) Ul 9 - 2 0 - 1- 1 2 - 2 3 - 4 - 1 5 - 2 6 - 7- 18- 2 9 - 10- 2 1 - 2 - 13- 2 4 - 5- 16- 2 7 / U2 8- 1 9 - 3 0 - 1 1 - 2 2 - 3 - 1 4 - 2 5 / 7- 18- 2 9 - 10- 2 1 - 2 - 13- 2 4 - 5- 16- 2 7 Accuracy In B M 3 6 7 3 1 the instants o f the solstices were stated in relation to actual d a y s , so that c o m p a r i s o n o f a single date c o m b i n e d with the error in the a s s u m e d year-length defines the error in the s c h e m e at any given date. In W 2 2 8 0 5 the dates are stated in (whole) tithis, so the errors for individual years are subject to a n uncertainty o f ± 1 day, and a different c o m p a r a t i v e p r o c e d u r e is required. F o l l o w i n g are the disfributions of errors (cale - text) o v e r successive 19-year cycles, o b t a i n e d b y c o m p a r i n g calculated B a b y l o n i a n dates for S S (using H u b e r ' s C r e s d a t p r o g r a m again to o b t a i n the Julian dates for the b e g i n n i n g s of the B a b y l o n i a n m o n t h s ) with the dates o f U - 1 .

Cycle 1 2 3 4 5

# of Errors (Cale - Text) = Median Date (Range) +2d +ld Od -615 (-624 t o - 6 0 6 ) 1 12 6 - 5 9 6 (-605 t o - 5 8 7 ) 12 7 - 5 7 7 (-586 t o - 5 6 8 ) 10 9 - 5 5 8 (-567 t o - 5 4 9 ) 9 10 - 5 3 9 (-548 t o - 5 3 0 ) 6 11

-Id

2

Avg 0.74 0.63 0.53 0.47 0.21

Error days days days days days

T h e s c h e m e is r o u g h l y half a d a y early o n average for the p e r i o d c o v e r e d b y the text ( m e d i a n date: - 5 7 7 ) , a b o u t half the c o r r e s p o n d i n g error for S S in B M 3 6 7 3 1 . B e c a u s e the year-length implied b y the 19-year cycle is slightly t o o long, this error diminishes w i t h t i m e , frending to z e r o b e t w e e n cycles 8 a n d 9 ( m e d i a n d a t e s : - 4 8 2 and - 4 6 3 ) a n d b e c o m i n g half a d a y late o n average b y cycle 14 ( m e d i a n date: - 3 6 8 ) . T h u s for all intents a n d p u r p o s e s the s c h e m e was essentially accurate in the p e r i o d i m m e d i a t e l y following the contents o f the text a n d r e m a i n e d so until r o u g h l y the m i d d l e o f the 4 * c e n t u r y B . C . , w h e n - p e r h a p s not coincidentally - the earliest e v i d e n c e o f the U r u k S c h e m e a p p e a r s . U r u k S c h e m e cycles b e g i n 8 years later a n d increase the average errors over a cycle for U l b y 0.42 d a y s . T h e resulting a v e r a g e errors are within h a l f a d a y for cycles with begiiming dates b e t w e e n - 5 0 2 a n d - 3 0 3

Treatments of Annual Phenomena in Cuneiform Sources

75

( m e d i a n dates: - 4 9 3 to - 2 8 4 ) , trending t h r o u g h z e r o in - 3 8 8 ( m e d i a n date). B y c a - 1 9 0 U 2 ' s s c h e m a t i c dates are 1 d a y late o n a v e r a g e a n d b y P t o l e m y ' s time b e t w e e n 2 a n d 3 d a y s late. In Short W h e n c o m p l e t e , W 2 2 8 0 5 + 0 1 from U r u k p r o b a b l y gave schematic dates of s u m m e r a n d winter solstices for 9 6 years from 1 N a b o p o l a s s a r (-624) t h r o u g h 9 C y r u s (5 2 9 ) , a l t h o u g h p o s s i b l y starting a n d ending o n e year earlier. T h e dates for s u m m e r solstice are d e r i v e d from a s c h e m e similar to the later U r u k S c h e m e , a l t h o u g h differing in its starting p o i n t a n d c o n s e q u e n t location o f the 1 2 ' epact, w h i c h is the earliest attested a p p l i c a t i o n o f the 19-year cycle. T h i s cycle b e g a n with year 1 N a b o p o l a s s a r , a n d s h o w s n o c o n n e c t i o n with the p r o c e s s of intercalation o v e r the interval c o v e r e d b y the text. T h e s c h e m e shares 11 dates w i t h the U r u k S c h e m e , a n d is c o n n e c t a b l e to it for t h o s e dates. D a t e s for the s c h e m e ' s first 8 years are 1 d a y later (higher) t h a n t h o s e in the U r u k S c h e m e , b u t it coimects with the latter for the r e m a i n i n g 11 m u t u a l l y shared dates. O n a v e r a g e the schematic s u m m e r solstice dates are h a l f a d a y early for the p e r i o d c o v e r e d b y the text, b u t less t h a n % d a y early for the last cycle c o v e r e d b y the text. T h u s they are n o t i c e a b l y m o r e a c c u r a t e t h a n the c o r r e s p o n d i n g SS dates n e a r the e n d o f B M 3 6 7 3 1 , w h i c h are r o u g h l y 1 d a y early. It w o u l d s e e m t h e n that b e t w e e n c a . - 5 6 1 , the last year of B M 3 6 3 7 1 , a n d - 5 2 9 , the last year o f W 2 2 8 0 5 + 0 1 ( b o t h b e i n g the earliest years the respective texts c o u l d h a v e b e e n written), the 27-year cycle w a s a b a n d o n e d in favoiu- of the m o r e accurate 19-year cycle, a c c o m p a n i e d b y a sensible i m p r o v e m e n t in the a c c u r a c y o f c o m p u t e d S S dates from r o u g h l y 1 d a y early to r o u g h l y accurate o n a v e r a g e .

76

J.P. Britton

o O03 O o o <s 00 •-H

RÎ

Oo Oo >O

CO

00 ON

O

«s

O

m

>r) VO O

r- 00 ON

o OO M > o ON

O

O

O

I—1

G O •G

S3 o oo O •t o «N
W

O

77

Treatments of Annual Phenomena in Cuneiform Sources

'2

1—< FN RO

to
O 00 ON <—1 M

WI

>N Oo o o O

o o o o o «N

FI

r~ o 00 ON FN

IN 1—1 F S

O

FN

FS

ED

FN

FN

O o FN O o 00 • N «N

o RO o to o OO

FN

IN R<^

00 M

00 ON

S -S

•
;3

O

ON

e

«N FN

FN —< FN

O

s o

03 ^ "S

rr

W R-J

«N

00 ON o

R.)

> n ITI

NO

in

3 > : S 3 > 3 3 S 3 > 3 3 > 3 S 3 3 3 K 3 > S S >

«o

r*

<S(:

I

r~00 ON

FN FN
FN

W

M

<s

FN

00 FN

CL F^

R-1

FO

FI

C/3

UH

NO

CI CNJ

>^

o CN)

O

"H a H-H 1

S 1—1

Q

+

NO

EU

O Mo O o O

I

ET

il il

o M

00

00 ON

NO 1—(

1 S

N

««s

O O o MO O >N » W( «fi

M

h>

ill

» FFS J-L M

M

«N

-H NO ON
OO

NO

op o o o

o oooo

FN

FN R<-i

«R>

—" FN

|> s s > s s « l l l l l l l p ^ ^ ^ r - - O O O N O - H < N R O ' * < O

s > s « > T)-

^IIO'-r*: ?5 ?î ^ ^

M

NO

F'-i

«

r- 0M0 MON n-O

ofo o o o - H JFN

o oooo

TT "N <-H
"N

o

' S T R ^ O R ^ ^ O N T ' Ì N O O N ^ ^ ^ ^ < S F N F N

FNMOOFNITIOC'—

0 0 0 N O ^ F S < ^ " * ' ' I ' O FN ^ FN FS

r^oooNO—"FNM-^IO r~|oo ON FN

3 3 S S > 3 3 ^ ! 3

- H C N - ^ F N

-^r^

—'

-HFN

o

O N O - ^ F N F I ' ^ - I R I N O t^"ôô"joN F N F ^ M R O R ^ R ^ F ^ F ^ R ^ R<^IR^ ^

!-HFN

2 2 12 !Û ^

ci

Oo Oo O CI

«N NO ON

«N

m

ON

00 ON O

EO C3 :3 Oo o o O O O O r, o o o O O FN «N O FMI •n o NO ON FN 00 FN FN FN en TT
NO

r~ 00 ON O
FS FN

FN

00 ON O FN FN

o o FN R<-i

NO ON FN M FN FN FN PO NO FN

00 ON S^ rRT TT TT

-^—I

—

- « N R ^ T F .

ON

r- 00 ON

1—1

{o5^!oio5n8^?î

o oooo

00 ON FN

3 oIC N

CO

o o oRO

O O FN CO

O M

O

O

M

NO ON FN O FN FN
O

L-H FN FN

M (N

•-I

.R)NO(-00ONp^FNRNT^U-»^o

S

NO

FN

M

w 1 C/3 -O

-

S

s'^

-s

O

I ^

(50 -fi S^

2» ;r:

< .a .S (N<^ w

00 ON M

M

I -CL

T—1

«N NO R~IN M m M

F N < N F N F N < N F N F N F N F N < N F N F N < N F N F N F N F N

oj oo ii =1 1

FN

> '—'

I W vi

oM

O O FN R<^

â

i l

> a H a K >

3 3 3 3 3 3

•S

M NO

< N ( N < N < N « N < N < N < N < N < N < N < N < N ( N F N < N < N < N < N < N < N < N R N | F S

Q

^

l î

^

en m

2 5 5 - H F S R N R T I / I N O R ^ O O O N O — FNR<^T}-«/INOR~-OOONO-^CN|ROTT ' ^ " " - ^ - " - ^ ' - ' - ^ " ^ — FNFNFNFNFN

FN

M

NO

o
D
M OJ

O O

78

J.P. Britton

Appendix D: U107+124 (=ACT 199) O u r p r e s e n t u n d e r s t a n d i n g o f the treatment of solstices a n d e q u i n o x e s in late Babylonian

astronomy

fragmentary

text from U r u k . T h e text contains the c o m p u t e d instants (year, m o n t h ,

tithi and

fraction)

began

with N E U G E B A U E R ' S

(1947)

publication

of

this

of c o n s e c u t i v e s u m m e r solstices a s s u m i n g a n e p a c t o f 11;3,10

tithis. N o p h o t o g r a p h h a s b e e n published, b u t N E U G E B A U E R ( ( 1 9 4 7 ) , p p . 1 4 3 - 1 4 6 a n d A C T , plate 136) d e s c r i b e s the c o m b i n e d

fragment

as a b o u t 5 c m b y 4 c m with

partial entties from 13 lines of o n e c o l u m n w i t h fraces of " 2 " or " 3 " at t h e b e g i i m i n g of a s u b s e q u e n t colimrai o n the o b v e r s e , a n d p a r t s of 4 lines o n the r e v e r s e . T h e tablet turns a r o u n d its right e d g e , so c o l i m m s o n the r e v e r s e p r o c e e d from right to left. Since the

fraces

o n the reverse underUe the first p r e s e r v e d c o l u m n o n the

o b v e r s e , the original text m u s t h a v e h a d at least four c o l u m n s .

NEUGEBAUER

( ( 1 9 4 7 ) , p . 146) a s s u m e s that this w a s in fact the extent of the text a n d c o n c l u d e d that it c o m p r i s e d four c o l u m n s o f a p p r o x i m a t e l y 2 3 lines, b e g i i m i n g a r o u n d year 2 , 1 8 S.E. (-173) a n d e n d i n g a r o u n d 3,49 S.E. (-82), give o r take a few years. H o w e v e r , it s e e m s m o r e likely that the original text b e g a n with y e a r

1,53**

S.E.(-198) and c o n t a i n e d three c o l u m n s of 2 4 lines o n e a c h side. If s o , its latest entry w o u l d h a v e b e e n for 4 , 1 6 S.E., c o r r e s p o n d i n g to the year - 5 5 , a n d the size o f the tablet w o u l d h a v e m e a s u r e d r o u g h l y 9 c m (h) b y 1 2 c m ( w ) or a p p r o x i m a t e l y 314 x AVA inches. A r e c o n s t r u c t i o n o f the text o n this a s s u m p t i o n is s h o w n i n F i g u r e D . T h e p r i n c i p a l a r g u m e n t for a 6-column text is that 1,53** S.E. is t h e initial year of a n U r u k S c h e m e cycle, for w h i c h s u m m e r solstice falls o n I V 7. T h e t e x t ' s epact, 1 1 ; 3 , 1 0 \ is e v i d e n t l y d e r i v e d from the 19-year cycle (€ = 1 1 ; 3 , 9 , 2 8 . . ' ) , w h e r e the slight r o u n d i n g results in a n i n c r e m e n t of 0 ; 0 , 1 0 ' in

19 years. T h i s i n c r e m e n t is

c o n s p i c u o u s l y reflected in the text's c o m p u t e d date for 2 , 3 1 * * S.E. ( I V 7 ; 0 , 2 0 ' ) , w h i c h implies the following s e q u e n c e of v a l u e s for initial years in the U r u k S c h e m e . Year S.E 1,34 ( - 2 1 7 ) * * 1,53 ( - 1 9 8 ) * * 2,12 ( - 1 7 9 ) * * 2,31 ( - 1 6 0 ) * *

Mo IV

Day(T) 6;59,50

IV IV

7;0

IV

7; 0,10 7; 0,20

N e u g e b a u e r b e l i e v e d that this s c h e m e with t r u n c a t e d fractions f o r m e d t h e b a c k b o n e o f the U r u k S c h e m e , a n d c o n c l u d e d that for ** y e a r s p r i o r to 1,53 (-198) the date o f s u m m e r solstice s h o u l d fall o n I V 6 a n d all d e p e n d e n t p h e n o m e n a o n e d a y earlier than otherwise. T h i s w a s confradicted b y e v i d e n c e p u b l i s h e d b y A A B O E a n d S A C H S ( ( 1 9 6 6 ) , T e x t s G + H ) s h o w i n g c o m p u t e d S S dates o f I V 7 for - 3 5 0 a n d - 3 3 1 , and conclusively d i s p r o v e d b y SLOTSKY ( 1 9 9 3 ) . T h u s I V 7 - the S S d a t e in the initial year of a n U r u k S c h e m e cycle for the duration o f its u s a g e - w o u l d h a v e b e e n a natural b e g i n n i n g for the c o m p u t a t i o n reflected in this text. F u r t h e r m o r e , it results in precisely the spatial relationships of the p r e s e r v e d

fragment.

T h i s w o u l d p l a c e the

p r o b a b l e c o m p o s i t i o n o f the text within the 19 y e a r s from 1,53 to 2,11 S.E. (-198 to - 1 8 0 ) , an interval w h i c h s p a n s all k n o w n c o m p o s i t i o n dates o f o f A C T texts

from

Uruk. O n e fiirther detail is n o t e w o r t h y . I n year 2,25 (col. II) the instant o f s u m m e r solstice is g i v e n as 3 0 ; 4 1 , 2 0 \ T h i s implies that the fractional tithis relate t o a sunset e p o c h a n d thus that S S in 1,53 (-198) w a s a s s u m e d to o c c u r exactly at simset.

History of the heleq Leo Depuydt,

Providence

T h i s is a survey of the history of a singular time-unit, the H e b r e w c a l e n d a r ' s \?^J\ heleq " p a r t . " T h e heleq h a s n o w b e e n in use for a b o u t a milleimium, p e r h a p s a little longer. Its history r e a c h e s b a c k to the eighth century B.C.E. T h e heleq is VA s e c o n d s long. N o natural event lasts exactly 3'/3 s e c o n d s . Rather, 3'/3 s e c o n d s are

a

subdivision of a t i m e - p e r i o d that d o e s relate t o a natural event. 3'/3 s e c o n d s fit exactly 7 6 5 4 3 3 times into 2 551 443'/4 s e c o n d s or into 2 9 days 12 h o u r s 4 4 m i n u t e s 3'/3 s e c o n d s . T h i s latter t i m e - p e r i o d is a value for the a v e r a g e s y n o d i c limar m o n t h u s e d in B a b y l o n i a n a n d G r e e k a s t r o n o m y . T h e following historical outline sfraddles several subjects, e a c h w i t h its o w n experts. A . W . C r o s b y ' o p i n e d that writing history w i t h o u t m i s t a k e s is like l e a d i n g a life without sin: w o r t h the effort, b u t i m p o s s i b l e . O n e of the subjects at h a n d is the M o o n ' s celestial m e c h a n i c s . T h e M o o n plays an i m p o r t a n t role in the s t u d y o f c a l e n d a r s a n d c h r o n o l o g y . M o s t o f the w o r l d ' s c a l e n d a r s h a v e a l w a y s b e e n or are lunar. T h e M o o n is also the m o s t studied o f celestial b o d i e s , o w i n g to its vicinity to the Earth. Full c o m m a n d of lunar theory, including familiarity w i t h w o r k s s u c h as P i e r r e - S i m o n L a p l a c e ' s Mécanique

céleste

( 1 7 9 9 - 1 8 2 5 , 5 volumes), can hardly be

e x p e c t e d from historians. T h e p r e s e n t s u r v e y tries to b r i n g together w h a t b e l o n g s t o g e t h e r b u t is n o w scattered. O p p o r t u n i t i e s for m o r e in-depth w o r k m a y therefore p e r h a p s b e m o r e readily discerned. A s with a n y philological-historical inquiry, the p r i m a r y focus is o n the extant sources in the original l a n g u a g e s a n d o n the history o f d i s c o v e r y in p a s t research. B i b l i o g r a p h i c a l references p o i n t to m o r e detailed a c c o u n t s . T h e title I a i m o u n c e d for " U n d e r O n e S k y " w a s " T h e R o l e of A n c i e n t A s t r o n o m y in A n c i e n t Chronology." But I ended up speaking about a narrower topic, "The Historical V e r a c i t y of P t o l e m y ' s R o y a l C a n o n as B a s i s of First M i l l e n n i u m B.C.E. C h r o n o l o g y : T o w a r d s A F o r m a l P r o o f " H o w e v e r , this topic w a s p l a n n e d for p o s s i b l e p u b l i c a t i o n e l s e w h e r e . I therefore t h a n k the a c t s ' editors for a l l o w i n g m e to p r e s e n t h e r e instead the results o f a similar investigation, fitting b o t h the a n n o u n c e d title's t h e m e a n d the c o n f e r e n c e ' s cross-cultural spirit.^

'

As quoted by RICHARDS (1999), p. xxi.

^ The topic was also discussed in a communication to the joint meeting of the Midwest branches of AOS, SBL, and ASOR, held at the Hebrew Union College, Jewish Institute of Religion in Cincinnati, Ohio on 14-16 February 1999.

80

L. Depuydt

1. Direct and Indirect Time-units In his Sternkunde und Sterndienst in Babel, K u g l e r calls a s t r o n o m y the " m o t h e r of chronology."^ T h e close relation b e t w e e n a s t r o n o m y a n d c h r o n o l o g y manifests itself in that the time-units b y w h i c h w e live are astronoinical in origin. O n closer inspection, different time-units relate differently to the m o v e m e n t s of the h e a v e n s . T h e relation c a n b e direct or indirect. A direct time-unit is the day. D u e to the E a r t h ' s rotation, the p l a c e s w h e r e w e live alternately turn t o w a r d a n d a w a y from the Sim. T h e alternation o f light a n d d a r k is a n event of nature i n d e p e n d e n t of our will. O t h e r direct time-units are the year, w h i c h relates to the cycle o f the seasons, a n d the lunar m o n t h , w h i c h relates to the p h a s e s o f the M o o n . A n e x a m p l e o f a n indirect time-unit is the hour. N o recurrent e v e n t o f n a t u r e lasts exactly a n hour. B u t as subdivision o f a p e r i o d w h o s e length d o e s relate to a natural event, the h o u r is related mdirectly to astronomy. O t h e r indirect time-units are the w e e k , the hour, the m i n u t e , a n d the second. Further distinctions naturally i m p o s e t h e m s e l v e s . F o r e x a m p l e , the d a y is the o n l y direct time-unit relating to a single i m p r e s s i o n m a d e o n the senses b y asfronomical events. T h e event is the alternation o f light a n d dark. Incidentally, n o frill alternation o f light a n d d a r k lasts exactly 2 4 h o u r s or 1440 m i n u t e s . 2 4 h o u r s is an average. T h e relation o f year a n d lunar m o n t h t o asfronomy is m o r e c o m p l e x . T h e c o r r e s p o n d i n g asfronomical events last on a v e r a g e a b o u t 3 6 5 . 2 4 2 2 d a y s a n d a b o u t 2 9 . 5 3 0 5 9 d a y s . B u t m o d e m years last 365 or 3 6 6 days a n d lunar m o n t h s a l m o s t always 2 9 or 3 0 days. Y e a r a n d lunar m o n t h therefore differ from the d a y in t w o respects. First, their lengths t h e m s e l v e s m a y n o t b e a v e r a g e s . B u t t h e alternations of 3 6 5 a n d 3 6 6 a n d o f 2 9 a n d 3 0 over time p r o d u c e fairly accurate a v e r a g e s . Second, they are related, n o t to o n e repetitive event of n a t u r e , b u t t o t w o . T h e s e c o n d in b o t h cases is the alternation o f light and dark. Y e a r a n d lunar m o n t h b o t h last a frill n u m b e r of days. T h e d a y is the d o m i n a n t time-unit o f chronology. First, it is the o n l y calendrical time-unit w h o s e length relates to only one i m p r e s s i o n m a d e o n the senses b y astronomical events. S e c o n d , any time-units larger than the d a y ( w e e k , m o n t h , year, century) consist of a w h o l e n u m b e r of days. T h e d a y ultimately o w e s its d o m i n a n c e to the fact that n o other repetitive event of nature impresses itself as i n e s c a p a b l y on our senses as the alternation of light a n d dark. O u r lives are d e e p l y affected b y it. O n e example is the sleep cycle. T h e m o d e m m o n t h s of 2 8 , 2 9 , 3 0 , and 31 days are a case apart. T h e y reflect the length o f the asfronomical lunar m o n t h . B u t they h a v e also b e c o m e i n d e p e n d e n t from the lunar cycle. A s time-units, they are therefore pseudo-direct. Time-units smaller than the d a y are indirect. N o p e r i o d i c event o f nature lasts exactly a n hour, a m i n u t e , or a second. T h e s e three units share three p r o p e r t i e s . First, they are arbifrary. O t h e r divisions m i g h t h a v e w o r k e d equally well. S e c o n d , they m e a s u r e the p r o g r e s s of time. T h a t is o b v i o u s from h o w e v e r y o n e glances at wristwatches. Third, they are subdivisions o f the day. T h e focus o f this p a p e r .

KUGLER ( 1 9 0 7 - 1 9 2 4 ) , vol. 2 . 1 , p. 3 .

History of the heleq

81

however, is o n o n e specific time-unit smaller than the d a y that exhibits n o n e of these three p r o p e r t i e s , n a m e l y the H e b r e w

heleq.

2. The heleq as a Time-unit T h e p^J] heleq (plural Wp^n tflâqïm) " p a r t " is a pivotal imit o f the H e b r e w " f i x e d " lunar calendar. T h i s c a l e n d a r is called fixed for the following reason. It follows the M o o n . B u t there is n o n e e d to o b s e r v e the M o o n to b e g i n a lunar m o n t h . T h e begiimings o f lunar m o n t h s are fixed b e f o r e h a n d by calculation. T h e fixed c a l e n d a r c a m e into use p r o b a b l y in the late first m i l l e n n i u m C.E., e v e n if s o m e traditions date its institution earlier. T h e heleq equals o f a n hour. It is exactly VA s e c o n d s long. A subdivision o f the pVn heleq " p a r t " is the y^T rega ^ " m o m e n t . " T h e r e are exactly 76 D^yn r^gâ fm " m o m e n t s " in a heleq, or 82 Ò80 (76 X 1080) in an hour. O n e " m o m e n t " is exactly ^ s e c o n d s long. T h e rega is d e r i v e d from the heleq with certainty inside the H e b r e w calendar. It is therefore of s e c o n d a r y i m p o r t a n c e . Like all time-units smaller than the day, as well as m a n y larger o n e s , the heleq relates indirectly to a s t r o n o m y . N o recurrent a s t r o n o m i c a l event lasts 3'/3 s e c o n d s . T h r e e properties set apart the heleq from other parts o f the d a y such as hour, m i n u t e , a n d second. First, its length is not arbitrary. It is d e t e r m i n e d b y m a t h e m a t i c a l considerations. S e c o n d , the heleq is n o t u s e d for m e a s u r i n g the p a s s a g e o f t i m e in daily life. Third, a n integer n u m b e r of Iflâqïm m a y fit inside the day. B u t in fact, the heleq is a subdivision o f a unit larger t h a n the day, the limar m o n t h . T h e heleq o c c u p i e s a u n i q u e p o s i t i o n at the intersection of a s t r o n o m y a n d c h r o n o l o g y . It is the only non-arbitrary subdivision of the day. T h e tithi of the H i n d u c a l e n d a r is similar. A t 0.98435 days, it is also smaller than the day. A n d it also derives its length from the lunar m o n t h , b e i n g one thirtieth of the m e a n synodic m o n t h long. B u t it is not a subdivision of the day. T h e n u m b e r s 1080 a n d 76 m a y s e e m r a n d o m at first. B u t they are p r e c i s e l y as large a n d as small as they n e e d to b e . T h e i r p u r p o s e is not to k e e p t i m e in daily life, as w h e n w e w a t c h the s e c o n d s ticking a w a y o n a m o d e m clock. T h e i r p u r p o s e is to k e e p J e w i s h m o n t h s a n d h o l y days in line with the lunar cycle without observing the Moon, a n d that for m a n y t h o u s a n d s o f years to c o m e — " t o the e n d o f t i m e " in M a i m o n i d e s ' words."* A s in a n y historical essay, b e it a b o u t a n individual, a n institution, a nation, or a concept, m u c h will b e said that is well k n o w n b y s o m e or better k n o w n b y others. W h a t is m o r e , in o r d e r to m a k e this p a p e r self-sufficient, m u c h that is o b v i o u s h a s b e e n stated explicitly. T h e a i m is to m a k e the topic accessible to t h o s e with little or n o prior k n o w l e d g e of it. W i t h subjects like c h r o n o l o g y , omitting a step in the a r g u m e n t m i g h t c a u s e confusion.

Sanctification of the New Moon (see

GANDZ ( 1 9 5 6 ) ) , V I . 7

and 8.

82

L. Depuydt

3. The Synodic Lunar Month T h e p h a s e s of the M o o t i are the night-sky's m o s t striking display. N o t surprisingly, m o s t ancient p e o p l e s h a d lunar calendars. T h e H e b r e w s w e r e n o e x c e p t i o n . T h e lengths o f calendrical lunar m o n t h s are ultimately d e t e r m i n e d b y the time that elapses b e t w e e n syzygies of the same kind. A syzygy ( G r e e k for " c o n j o i i m i e n f ) is w h e n Sim, Earth, a n d M o o n are aligned or in the s a m e p l a n e . T h e r e are t w o k i n d s of syzygies. A t n e w m o o n or conjunction. M o o n is b e t w e e n Sun a n d Earth. W h e n the three b o d i e s are in addition r o u g h l y on the s a m e line, the M o o n o b s c u r e s the Sun. T h i s is a solar eclipse. A t full m o o n or opposition. Earth is b e t w e e n Sim a n d M o o n . W h e n the three b o d i e s are in addition roughly o n the same line, the E a r t h casts its s h a d o w o n the M o o n . T h i s is a lunar eclipse. Conjimction c a n fall at a n y time o f the day. T h e time in w h i c h the M o o n returns to conjimction is the synodic m o n t h . T h e r e are other a s t r o n o m i c a l lunar m o n t h s : tropical, anomalistic, sidereal, dracontic. T h e y are not i m m e d i a t e l y relevant to calendrics, n o r to the present argument. " A s t r o n o m i c a l lunar m o n t h " therefore henceforth m e a n s " s y n o d i c lunar m o n t h . " Synodic m o n t h s all vary in length. T h e m a i n r e a s o n is that the M o o n accelerates a n d decelerates o n its elliptic orbit a r o u n d the Earth. I n the p e r i o d from 1900 to 2 1 0 0 C.E., the shortest lunar m o n t h lasts about 2 9 . 2 7 days, or a little a b o v e 2 9 days a n d óVi h o u r s . T h e longest is about 2 9 . 8 3 d a y s long, or a little u n d e r 2 9 d a y s a n d 2 0 h o u r s . T h e difference b e t w e e n the longest a n d shortest synodic m o n t h s is therefore a b o u t 13'/2 h o u r s . T h e s e n u m b e r s are taken from M E E U S ( 1 9 8 8 ) , w h o also lists the four shortest a n d t w o longest m o n t h s in the said p e r i o d = days; ^ = hours; = minutes). Shortest synodic lunar m o n t h s in 1 9 0 0 - 2 1 0 0 C E . 25 J u n e - 2 4 July 1903 6 J u n e - 5 July 2 0 3 5

2 9 " 6*' 35"" 29^ 6** 39""

16 J u n e - 1 5 J u l y 2 0 5 3 27 J u n e - 2 7 July 2071

29'* 6'' 3 5 ' " 29*' 6'' 36""

L o n g e s t synodic lunar m o n t h s in 1 9 0 0 - 2 1 0 0 C E . 14 D e c e m b e r 1 9 5 5 - 1 3 January 1956 2 4 D e c e m b e r 1 9 7 3 - 2 3 January 1974

29^ 19*" 5 4 " 2 9 " 19'' 55""

M e e u s estimates that his formulas guarantee a c c u r a c y to " w i t h i n 1 or 2 m i n u t e s . " T h e shortest m o n t h s fall in the s u m m e r , the longest in the winter. T h a t is b e c a u s e , as M e e u s explains, in the simmier, the Earth is farthest a w a y from the S u n o n its elliptic orbit a r o u n d the Sun. T h e E a r t h then also m o v e s m o r e slowly. A c c o r d i n g l y , looking from the Earth, the S u n m o v e s m o r e slowly against the b a c k g r o u n d o f the stars. T h i s a p p a r e n t p a t h o f the S u n in the stars is the ecliptic, the b a n d of the sky close to w h i c h is traditionally d i v i d e d into 12 parts, the signs o f the z o d i a c . T h e S u n m o v e s about o n e d e g r e e in that p a t h every day from w e s t to east, in a direction o p p o s i t e to its daily revolution firom east to west. B e c a u s e the S u n m o v e s m o r e slowly in the s u m m e r , the M o o n catches u p faster with the S u n to return to the p o s i t i o n o f conjunction. T h a t m a k e s lunar m o n t h s shorter. Short m o n t h s c a n o c c u r at a n y time o f

History of the heleq

83

the year d u e to o t h e r factors. B u t short o n e s o c c u r r i n g in the s u m m e r h a v e a d d i t i o n a l shortening factors. T h e shortest m o n t h s therefore fall in the simimer.

4. Observed Calendrical Lunar Months T h e vast majority o f the w o r l d ' s calendars h a v e always b e e n or are lunar. H o w e v e r , cycles o f lunar p h a s e s d o n o t last a full n u m b e r o f d a y s . Y e t , c a l e n d r i c a l units larger t h a n the d a y m u s t consist of a w h o l e n u m b e r o f d a y s . T h e alternation of light a n d d a r k leaves t o o p o w e r f u l an i m p r e s s i o n o n the senses t o b e ignored. " D a y ( l i g h t ) p l u s night are to m o n t h s , years, a n d e r a s , " writes A l b i r u n i ( 9 7 3 - a f l e r 1 0 5 0 C E . ) at t h e e n d of the Preface t o his Chronology,

" w h a t the n u m b e r 1 is to n u m b e r s : that o f

w h i c h t h e y are c o m p o s e d a n d that into w h i c h t h e y are resolved."^ O n e solution is to find a striking lunar e v e n t that h a p p e n s o n c e e v e r y m o n t h . T h a t event c a n t h e n serve as the m a r k e r o f the b e g i n n i n g of calendrical lunar m o n t h s . O n e s u c h e v e n t is the limar c r e s c e n t ' s first visibility. T h e first crescent c a n first b e seen in the e v e n i n g a b o u t a n h o u r after sunset a b o v e the w e s t e r n h o r i z o n a b o u t o n e to t w o days after conjunction. T h e M o o n s o o n after sets, following the Sun. T h e n e x t daylight p e r i o d c o u l d t h e n b e called daylight o f D a y 1. H o w e v e r , b y o n e of ancient h i s t o r y ' s m o s t o v e r r a t e d a s s u m p t i o n s , lunar m o n t h s typically caimot b e g i n before the first crescent is sighted. Positive proof, h o w e v e r , exists only for B a b y l o n i a n a s t r o n o m i c a l texts a n d the M u s l i m calendar. T h e s e t w o c a l e n d a r s are strictly s u p e r v i s e d b y councils o f clerics o r scholars. B y contrast, e v i d e n c e from all o v e r the ancient w o r l d suggests that the first c r e s c e n t s e r v e d h a r d l y a n y w h e r e else as m a r k e r of the b e g i n n i n g of the m o n t h . It is clear from t h e s o u r c e s that p e o p l e d i d take n o t i c e of the r e - a p p e a r a n c e of the M o o n , p e r h a p s as a r o u g h m a r k e r that the m o n t h s h a d n o t drifted far from the lunar cycles. B u t there is n o p o s i t i v e e v i d e n c e a n y w h e r e , other t h a n in B a b y l o n i a n asfronomy a n d the M u s l i m calendar, that the daylight p e r i o d following the sighting of the first c r e s c e n t in the e v e n i n g w a s systematically called D a y 1 T h e a v e r a g e s y n o d i c m o n t h lasts b e t w e e n 29 a n d 3 0 d a y s . C a l e n d r i c a l lunar m o n t h s will therefore a l m o s t always h a v e 2 9 or 3 0 full d a y s . T h e a v e r a g e is j u s t a little longer t h a n 2 9 . 5 d a y s . T h e r e will therefore b e slightly m o r e m o n t h s o f 3 0 d a y s t h a n of 2 9 d a y s , at a ratio o f a b o u t 53 to 4 7 . H o w s o o n after conjunction or n e w m o o n can the first crescent b e s e e n ? N e w m o o n c a n fall at a n y t i m e o f the day. T h e first crescent m a k e s its first a p p e a r a n c e in the early e v e n i n g . M u c h h a s b e e n written a b o u t the p r o b l e m o f

first

crescent

visibility. K e p l e r thought it c o u l d n o t b e solved, a n d to this day, it has n o t b e e n . T h e r e are t o o m a n y variables. B u t m u c h m o r e is k n o w n n o w t h a n in K e p l e r ' s t i m e .

^

SACHAU (1878), p. o; SACHAU (1879), p. 4 .

^ For a more detailed discussion, see DEPUYDT (2002), where the problem is also addressed how the ancients did establish the beginnings of lunar months. This problem was also discussed in a paper entitled "The Day in Antiquity," presented at the Two Hundred and Twelfth Annual Meeting of the American Oriental Society, held in Houston, Texas, on 22-25 March 2002. (There are indications that, even in Babylonian astronomy, first crescent visibility was not the marker. On this, more hopefully elsewhere.)

84

L. Depuydt

F o r e x a m p l e , the r e c o r d for soonest n a k e d - e y e sighting o f the crescent w a s recently still a b o v e 15 h o u r s after n e w m o o n / Increased

astronomical

understanding

o f first

crescent

visibility

does

not

necessarily p r o d u c e i n c r e a s e d historical u n d e r s t a n d i n g . T h e a s t r o n o m e r first a n d foremost studies w h e n the first crescent can b e seen. T h e historian is p r i m a r i l y c o n c e r n e d with w h e n the first crescent was seen at s o m e p a s t p o i n t in t i m e . It is n o t b e c a u s e a s t r o n o m e r s d e t e r m i n e that the first crescent could

h a v e b e e n s e e n at a

certain m o m e n t in time that it historically was. V e r y little is k n o w n a b o u t h o w the crescent w a s o b s e r v e d in antiquity. T h e r e are n o detailed reports o n w h i c h efforts o n e precisely u n d e r t o o k or h o w the d e g r e e o f a c c u r a c y o f sightings w a s p e r c e i v e d .

5. Computed Calendrical Lunar Months O b s e r v i n g the M o o n for the p i u p o s e o f establishing the lengths o f the m o n t h s is a c u m b e r s o m e p r o c e d u r e . T h e r e is also s o m e t h i n g unsatisfying a b o u t n o t k n o w i n g w h a t the calendar will l o o k like in the i m m e d i a t e future. T h e p r o b l e m is acute for a nation inhabiting different parts o f the world. A n o b v i o u s solution is to a b a n d o n lunar m o n t h s altogether. T h e ancient Egyptians

first

d i d so in the early third

m i l l e n n i u m B.C.E., m o r e than t w o t h o u s a n d years before a n y o n e else.^ B u t w h a t if o n e wishes t o retain lunar m o n t h s , s a y in deference to tradition, a n d yet b e liberated from the complexities involved in o b s e r v i n g the M o o n ? T h e solution is to fix the alternation o f 2 9 - d a y m o n t h s a n d 3 0 - d a y m o n t h s b e f o r e h a n d

by

c o m p u t a t i o n . T h e r e are t w o w a y s o f achieving this. O n e c a n u s e the actual length o f the sjmodic m o n t h or a n av er ag e length. Synodic m o n t h s all differ in length, r a n g i n g from a b o u t 2 9 . 2 7 d a y s to a b o u t 2 9 . 8 3 days (see section 3 a b o v e ) . T o u s e t h e m , o n e m u s t b e able to c o m p u t e true n e w m o o n s . F o r e x a m p l e , in 6 1 9 C.E., true n e w m o o n s w e r e introduced in the C h i n e s e calendar.^ D a y 1 o f the calendrical limar m o n t h is the day, counting firom midnight, o n w h i c h n e w m o o n falls in Beijing. T h e other o p t i o n is to use a m e a n o r av er a g e length for the s y n o d i c lunar m o n t h . U s i n g this a v e r a g e , o n e fixes points in time that are e a c h r e m o v e d from the next b y o n e m e a n synodic m o n t h . T h e s e p o i n t s never c o i n c i d e with true n e w m o o n . B u t if the value for the m e a n synodic m o n t h is fairly accurate, they a l w a y s r e m a i n close. T h e p o i n t s in time are a k i n d o f artificial conjimction or n e w m o o n . T h e y oscillate in relation t o true n e w m o o n , b u t n e v e r stray far fi-om it. T h e differences d o n o t a c c u m u l a t e over the m o n t h s . T h e y average out. T h e H e b r e w fixed c a l e n d a r is b a s e d o n this principle.

6. 29** 12" 44™ 373* and iVs' in the Hebrew Fixed Calendar T h e H e b r e w c a l e n d a r is the only fixed lunar calendar in w e s t e r n civilization. It is u s e d m a i n l y for religious p u r p o s e s , to date J e w i s h holidays i n d e p e n d e n t l y o f the

^

Among recent contributions to the topic are DOGGETT and SCHAEFER ( 1 9 9 4 ) , ILYAS

( 1 9 9 4 ) , PEPIN ( 1 9 9 6 ) , SCHAEFER ( 1 9 9 6 ) , and SCHAEFER, AHMAD, and DOGGETT ( 1 9 9 3 ) . These

papers should lead to most else that has been written before. ^

Cf

'

DERSHOWITZ and REINGOLD ( 1 9 9 7 ) , p. 1 6 9 , note 1, with bibliography at p. 1 8 6 .

DEPirfDT(2001).

History of the heleq

85

M o o n while yet following the M o o n . Its basis is the m e a n synodic m o n t h o f 2 9 " 12^ 44"" 3'/3^ T h i s m e a n length is u s e d to fix e a c h molad. Molad

is H e b r e w for " p l a c e o f

birth (of the M o o n ) . " E a c h molad is r e m o v e d firom the p r e v i o u s a n d the n e x t b y 2 9 " \2^ AAT VA'. T h e first molad

has b e e n fixed at 20 s e c o n d s after 1 1 : 1 1 P M , S u n d a y

night, 6 S e p t e m b e r , 3 7 6 1 B.C.E. T h e c a l e n d a r d a y b e g i n s imiformly at 6 : 0 0 P M . T h u s , the first molad falls 5 h o u r s a n d 2 0 4 p a r t s (11 m i n u t e s 3 s e c o n d s , or 2 0 4 x 3'/3 s e c o n d s ) after the b e g i n n i n g o f the first d a y o f the first year o f the H e b r e w era. 29" \2^ 44"" 3'/3' naturally p r o d u c e s

of an h o u r , that is, VA s e c o n d s , as a

time-unit, as follows. T h e p o r t i o n that d o e s n o t fill a n hour, 4 4 m i n u t e s 3'/3 s e c o n d s , c a n b e e x p r e s s e d b y the fraction

(of an h o r n ) . N e i t h e r 7 9 3 n o r 1080 are p r i m e

n u m b e r s . 7 9 3 h a s 13 a n d 61 as factors. B u t 7 9 3 and 1080 d o not share a n y factors. ^

c a n n o t therefore be r e d u c e d to lower t e r m s . A g a i n ,

is therefore e x a c t l y as

large and as small as it n e e d s to b e . T h e rega

" m o m e n t , " w h i c h is ^

of the heleq, is d e r i v e d from the heleq

inside

the H e b r e w calendar. A t the origin lies the n e e d to o b s e r v e , n o t only the p h a s e s of the M o o n , b u t also the cycle o f the seasons o r the solar y e a r . ' " T h e s e a s o n a l year i m p r e s s e s itself sufficiently o n the senses to m a k e incorporation into c a l e n d a r s desirable. M o s t c a l e n d a r s d o incorporate it. T h e true length of the solar year is a b o u t 3 6 5 . 2 4 2 2 days. In the H e b r e w calendar, the length o f the solar year is d e r i v e d , n o t from the true solar year, b u t from the lunar m o n t h of 2 9 " 12'' 44*" V/z. T h e d e r i v a t i o n is b a s e d o n a fact k n o w n as early as the fifth century B.C.E. in B a b y l o n a n d A t h e n s : 2 3 5 lunar m o n t h s are a b o u t as long as 19 solar years. 19 solar years, o r a b o u t 6 9 3 9 . 6 0 1 8 d a y s ( 3 6 5 . 2 4 2 2 x 19), are a b o u t 2 h o u r s 5 m i n u t e s shorter t h a n 2 3 5 lunar m o n t h s , or a b o u t 6 9 3 9 . 6 8 8 6 5 days ( 2 9 . 5 3 0 5 9 x 2 3 5 ) : 6 9 3 9 . 6 8 8 6 5 - 6 9 3 9 . 6 0 1 8 = 0 . 0 8 6 8 5 d a y s ; 0 . 0 8 6 8 5 x 1440 m i n u t e s - a b o u t 125.064 m i n u t e s - a b o u t 2 h o u r s 5 minutes. T h e rega"

is o b t a i n e d in three steps. First, o n the p r e m i s e that 2 3 5 lunar m o n t h s

equal exactly 19 solar years, the length of the lunar m o n t h is multiplied b y 2 3 5 to obtain the length of 19 solar years: 29 days 12 h o u r s 7 9 3 " p a r t s " x 2 3 5 = 6 9 3 9 days 16 h o u r s 5 9 5 " p a r t s . " S e c o n d , 6 9 3 9 " 16'' 5 9 5 " is divided b y 19 to o b t a i n the l e n g t h of o n e solar year. T h e result is 3 6 5 " 5'' 997'' + ] | - o f a heleq.

T h e solar y e a r of the

H e b r e w c a l e n d a r is therefore six to s e v e n m i n u t e s l o n g e r than the actual solar year. A t this point, a division o f the heleq into 19 p a r t s is still a d e q u a t e . H o w e v e r , a fiirther division b y 4 is n e e d e d to obtain the length of the season, t a k e n at o n e fourth o f the year. T h e result is 9 1 " 7'' 5 1 9 " + ^ r e d u c e d to l o w e r t e r m s . T h e fraction ^

o f a heleq.

T h e fi-action ^

cannot be

therefore e m e r g e s as a n e c e s s a r y time-unit.

76 is the smallest p o s s i b l e d e n o m i n a t o r after a n u m b e r has b e e n d i v i d e d first b y 19 and t h e n b y 4 ( 1 9 X 4 = 7 6 ) . T h e H e b r e w fixed calendar b o r r o w s only a single time-value from asfronomy, n a m e l y 2 9 " 12** 44*" V/^. T h i s value p r o d u c e s all the others, including the year. T h e year is a little l o n g e r than the solar year. T h e c a l e n d a r ' s spring e q u i n o x therefore slowly m o v e s t o w a r d s s u m m e r . T h e c a l e n d a r therefore stays close to the M o o n , b u t is slowly m o v i n g a w a y from the Sun. T h e c a l e n d a r c a n b e k e p t in line, n o t o n l y with

'° SWERDLOW ( ( 1 9 8 0 ) , pp. 3 0 7 - 3 0 9 ) has recently discussed the length of the Hebrew year in the context of ancient astronomy.

86

L. Depuydt

the M o o n , b u t also with the Sun, as h a s long b e e n realized, ' ' b y o m i t t i n g o n e lunar m o n t h of 2 9 " 12^ 44*" 3'/a' w h e n the difference b e t w e e n c a l e n d a r solar y e a r a n d real solar year h a s r e a c h e d a fiill lunar m o n t h of 2 9 " 12^ 44*" SVJ', that is, e v e r y six to seven thousand years. T h e details of the fixed c a l e n d a r ' s structure e x c e e d the s c o p e o f this p a p e r . T h e y are d e s c r i b e d in h a n d b o o k s o f general c h r o n o l o g y a n d J e w i s h c h r o n o l o g y (see b e l o w ) . A m o n g r e c e n t descriptions are R e s n i k o f f s a n d A k a v y a ' s . ' ^ T h e p o s i t i o n o f D a y 1 of the m o n t h in relation to the molad rules that h a v e different

is d e t e r m i n e d b y

rationales, such as p r e v e n t i n g certain h o l i d a y s

fi-om

coinciding with the S a b b a t h . Typically, h o w e v e r . D a y 1 a p p e a r s t o b e the d a y after the d a y of the molad.

It is a fact that the fu-st crescent is n o t imtypically s e e n in the

evening o f the d a y that follows the d a y of conjunction. T h i s m e a n s that the

first

crescent is not u n t y p i c a l l y s e e n in the e v e n i n g after daylight o f D a y 1 (not the evening before

daylight o f D a y 1!). It h a s b e e n a s s u m e d that D a y 1 w a s d e t e r m i n e d

with this effect in m i n d , p o s s i b l y as a reflection o f a p r a c t i c e o b s e r v e d in the earlier, observational H e b r e w lunar calendar. T h e surviving sources a l l o w neither d e n y i n g nor confirming this a s s u m p t i o n . T h e years o f the H e b r e w fixed c a l e n d a r are lunisolari limar b e c a u s e the m o n t h s are lunar; solar b e c a u s e the specific alternation o f 1 2 - m o n t h a n d 1 3 - m o n t h years k e e p s u p with the cycle o f the seasons.

7. The True Length of the Mean Synodic Month T h e focus h a s n o w naturally shifted to the time-unit to w h i c h the heleq

o w e s its

existence, n a m e l y 2 9 " 12*" 44"" 3 V 3 ' . T h e a c c u r a c y o f this v a l u e is a d d r e s s e d first. N o k n o w l e d g e o f p h y s i c s or a s t r o n o m y is n e e d e d to realize that the v a l u e is v e r y accurate, d e v i a t i n g b y less t h a n a s e c o n d from the true v a l u e . O n l y after m a n y t h o u s a n d s o f years d o e s this m i n u s c u l e deviation a d d u p to a full day. I n w o r k s o n c h r o n o l o g y , the a v e r a g e m o s t often cited is 2 9 . 5 3 0 5 9 days. B e c a u s e the third n u m b e r after the p e r i o d is a z e r o , 2 9 . 5 3 is often a d e q u a t e for the p u r p o s e s o f history. 2 9 " 12^ 44*" 3'/3' is a little m o r e t h a n 2 9 . 5 3 0 5 9 d a y s , a b o u t 2 9 . 5 3 0 5 9 4 . In o r d e r to p r o v i d e the r e a d e r w i t h a fully c o m p l e t e u p d a t e o n the t o p i c at h a n d , the following q u e s t i o n is w o r t h asking: H o w long exactly is the m e a n s y n o d i c m o n t h ? T h e q u e s t i o n is treated in the following e x c u r s i o n . It is o t h e r w i s e p o s s i b l e , h o w e v e r , t o p r o c e e d to section 8 at this point. E x c u r s i o n : R e c e n t W o r k o n t h e P r e c i s e L e n g t h o f the S y n o d i c L u n a r M o n t h No one knows how long the mean synodic month is. No formula is perfect. But formulas have only been improving over time. Owing to technological advances such as laser and satellites, estimates for modem times are extremely accurate. It is now certain that the mean synodic month has been lengthening slightly in one respect and shortening slightly in another respect. What follows in this excursion is derived from Stephenson's recently published detailed investigation of what can be derived from ancient and medieval observations of eclipses about

"

ScHWARZ(1872),p. 121. RESNIKOFF (1943) and AKAVYA( 1953).

History of the/je/e^

87

long-term variations in the length of the day or the Earth's rate of rotation.'^ The matter is of some complexity. But in the interest of completeness and self-sufTiciency, it will be necessary to elaborate somewhat. Such elaboration is all the more necessary because it is possible to read in recent accounts on chronology that the synodic month is lengthening and in others that it is shortening. It is in fact both. Different numbers following different models have been postulated for both the lengthening and the shortening. It is not known how close these numbers are to the true values. Britton found "substantial differences among determinations by different investigators."''* But the long-term tendency by which the Earth's rotation is gradually slowing to the point where there will be just one day or one rotation per lunar month is generally accepted. First, in one respect, the synodic month is lengthening. The change is due to the fact that the Moon is slightly accelerating in its orbit around the Sun. This is the Moon's "secular acceleration." By the laws of mechanics, a satellite (for example, the Moon) that gains angular momentum both recedes from its primary, in this case the Earth, and loses angular velocity. In other words, because the Moon's speed increases, it moves into an orbit a little farther away from the Earth. As a result, the Moon needs more time to pass through an angle, such as the specific angle larger than a full circle that brings the Moon back to new moon. The Moon's secular acceleration therefore causes the mean synodic month to lengthen slightly over the centuries. This variation is cyclical. The mean synodic month will at some point again shorten. The length of the cycle is in the order of 100 000 years. Edmund Halley first proposed the secular acceleration in 1695, but without comment. Richard Dunthome first proved its existence in 1749. The following chart illustrates the change in the length of the mean synodic month. 4000 B.C.E. 3000 B.C.E. 2000 B.C.E. 1000 B.C.E. Ic.E. 1000 C E . 2000 C E .

29.53057457 29.53057714 29.53057962 29.53058204 29.53058439 29.53058665 29.53058885

29" 12^ 4 4 " 1.64^ 29" 12" 44"" 1.86' 29" 12''44"" 2.08' 29" 12*'44"^ 2.29' 29" 12''44*" 2.49' 29" 12''44"" 2.69' 29" 12''44"^ 2.88'

These are the numbers computed by Richards using the following formula: 29.5305888531 + 0.000000216217--3.64 X lO^'V.'^ T is the time measured in centuries from 2000 CE.'^ The formula is found in the Explanatory Supplement to the Astronomical Almanac of 1992." It is based on lunar theory ELP 2000 of

STEPHENSON (1997), pp. 6-17 and 23-26. BRITTON (1992), p. ix. '^

RICHARDS (1998), p. 389.

For example, the first number is obtained as follows. T is - 6 0 centuries. The formula becomes: 29.5305888531 + (0.00000021621 x - 6 0 ) - (3.64 : 10 000 000 000 x 3600). And hence: 29.5305888531 - 0 . 0 0 0 0 1 2 9 7 2 6 - 0 . 0 0 0 0 0 1 3 1 0 4 = 29.5305745701. To measure the length of the mean synodic month for any day in history, the distance in days needs to be known first. It is obtained by subtracting 2 451 545.0 (that is, the j.d. or Julian day number for midnight as a point in time between 31 December 1999 and 1 January 2000 C E . ) from the j.d. number of the day in question. The result will be negative for days before 1 January 2000 C E . The distance in days then needs to be converted into a distance in centuries by dividing the result by the number of days in a century, 36 525. The deviation resulting from the fact that 2 451 545.0 is Gregorian 1 January 2000 C E . whereas any j.d.

88

L. Depuydt

Chapront-Touzé and Chapront of the Bureau de Longitudes in Paris. Version ELP 2000-82'* is more precise for modem times. Version ELP 2000-85'^ has been tested over a longer period of time and is more suitable for historical work. Second, in another respect, the mean synodic month is shortening. The cause is a slowdown in the Earth's rotation. One rotation equals one alternation of light and dark, that is, one solar day. The day is therefore lengthening as the Earth's rotation is slowing. No periodic motion in the universe seems truly uniform. The same could have been expected a priori about the Earth's rotation. But serious speculation about variation in the Earth's rate of rotation did not begin before the mid-eighteenth century. Euler once expressed the hunch that the rate of rotation was increasing. By the early twentieth century, it had become certain that it is decreasing. The day is the basic time-unit of chronology, for obvious reasons (see section 1 ). But it was certain by the early twentieth century that the day itself is not constant. The need arose for a universal time system independent from the variation in the Earth's rotation, a standard by which the variation itself might then be measured. Decade-long efforts led to the adoption, at the Intemational Conference on Weights and Measures in 1967, of the definition of the second as "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium atom 133." Two concepts relating to the length of day (LOD) are ALOD and Ar. Over time, the tiny daily differences in LOD accumulate. The LOD at the beginning of 1900 C E . can serve as the standard. Its length is fixed at 86 400 seconds SI (Système intemational). The cumulative change (A) of the LOD is measured for any given time in history in relation to the LOD of 1900 C E . For example, in - 5 0 0 or 501 B.CE., by Stephenson's model,^° ALOD amounted to - 4 2 milliseconds. This means that the day was then about 0.042 seconds shorter than at the beginning of 1900 C E . Over time, there is also a change in the difference between the time it takes the Earth to rotate a given number of times and the time it would have taken the Earth to rotate the same number of times at the rate of rotation of 1900 C E . This cumulative difference (A) is Ar. For example, in - 5 0 0 , by the same model, Ar amounts to 16 800 seconds. That is the additional amount of time the Earth would need to make the same number of rotations that it has made back to the year - 5 0 0 , but at the rate of rotation of 1900 C E . ^T'\s therefore called the Earth's rotational clock error. The deviation is in relation to the beginning of 1900 C E . The effect of the lengthening of the day may be illustrated as follows. Noon, when the Sun is at its highest point, returns every day because of the Earth's rotation. Noon is also a point in time that can easily be measured, even with instruments that are more primitive than modem clocks. In - 5 0 0 , A f w a s 16 800 seconds or 4 to 5 hours. If one counts back from 1900 C E . using the LOD of 1900 C E . , noon would fall 4 to 5 hours earlier than it in fact did. How does the lengthening of the day make the mean synodic month longer over time? Mean or average synodic months are presumably obtained by selecting pairs of lunar eclipses that have similar properties and by then dividing the long time-spans between two similar eclipses by the number of lunar months that separate them. The result is an average lunar month. There are between 1236 and 1237 (365.2422 x 100 : 29.53059) lunar months in a century. If the time measured between two eclipses is in error by an hour and the eclipses are 3600 lunar months or close to 300 years apart, the error of 3600 seconds results in a one-

number before 1582 C E . is Julian is negligible, as is the fact that a century lasts closer to 36 524.22 days. SEIDELMANN(1992), p. 576. CHAPRONT-TOUZÉ and CHAPRONT (1983). CHAPRONT-TOUZÉ and CHAPRONT (1988). ^°

STEPHENSON(1997),p. 515.

History of the Ae/eç

89

second deviation from the true average. It appears, then, that it is relatively easy to obtain an average of the synodic month that is correct to within a second. The difficulty and effort is in keeping careful records of eclipses and their properties over long periods of time. The Babylonians did just that from about the eighth century B.CE. onwards. Days are alternations of light and dark caused by the Earth's rotation. If, in a given period, the Earth's rotation has slowed dovm slightly, then slightly fewer rotations or days will separate two distantly removed eclipses. The average number of rotations per lunar month will also be slightly smaller. The mean synodic month thus appears shorter when counted in days. Two factors have been described that affect the length of the mean synodic month. The Moon's secular acceleration makes the mean synodic month lengthen in universal time and therefore also in day-length. The slowdown of the Earth's rotation makes the mean month shorten in day-length. What matters to the chronologist is day-length. Which of the two factors is larger in the historical period: the lengthening or the shortening? It is not easy for historians to interpret the numbers offered by astronomers, not only because a more advanced knowledge of physics is required. The numbers are based on theoretical models that are not definitive. Lunar theory is still evolving. As usual, the quest is for that last small refinement of a number that is already fairly precise. It will be useful to begin by leaving numbers aside and describing where the variations will lead in the long run. By all current theories and observations, the Earth's rotation will after billions of years slow down to the point where it always presents its same face to the Moon. The Earth is subject to forces similar to those that once caused the Moon to turn its same face permanently to the Earth. To the chronologist, the end-scenario means that there will then be exactly as many days as there are lunar months. The Earth and the Moon will be locked onto one another face to face. From one conjunction to the next, the Earth's rotation will follow the Moon's revolution. Consequently, earthlings will experience just one alternation of light and dark, that is, one day, in the time that the Moon circles the Earth to return to its position in front of the Sun, that is, to conjunction. The following tendency seems certain. Over the millennia, the mean synodic month will gradually decrease in terms of number of alternations of light and dark, or days, and finally reach an average of exactly one day per synodic month. The Moon's secular acceleration counteracts this gradual decrease to some extent by making the mean synodic month longer in absolute time, and therefore also in day-length. This variation is cyclical. A deceleration phase will follow an acceleration phase. All of human history belongs to an acceleration phase. As regards numbers, Stephenson recently calculated the average increase of the mean synodic month at about 38 milliseconds (0.038 seconds) per century and the average decrease in the Earth's rotation at about 1.7 milliseconds (0.0017 seconds) per century.^' This increase and this decrease both accumulate. 38 milliseconds is about 20 times larger than 1.7 milliseconds. But the former accumulates only monthly, every 29 to 30 days; the latter, daily. The mean synodic month decreases in number of solar days in the long run. Moreover, the numbers seem to indicate that it has also been decreasing in the past 5000 years, the historical period. The net effect for clocks calibrated to noon, a point in time easy to determine, is that the mean synodic month is shortening.

8. 29** 12" 44"* 3%' in Babylonian Astronomy 2 9 " 12** 4 4 " 3'/3* first occurs in B a b y l o n i a n astronomical texts. A distinction is n e c e s s a r y b e t w e e n the v a l u e ' s ancient discovery a n d its r e d i s c o v e r y in the extant sources. T h e m o d e m discovery b e g a n with the d e c i p h e r m e n t of cimeiform writing m STEPHENSON (1997), pp. 36 and 513.

90

L. Depuydt

the m i d 1800s. B a b y l o n i a n a s t r o n o m y w a s d e c i p h e r e d in outline n e a r t h e e n d o f the s a m e century. A s for t h e m e a n synodic m o n t h , four dates are relevant: 1 8 8 1 , 1 8 8 9 , 1890, a n d 1900. In 1 8 8 1 , E p p i n g p r o v i d e d a first outline o f B a b y l o n i a n a s t r o n o m y o n the basis o f texts c o p i e d from clay tablets at the British M u s e u m b y Strassmaier, w h o h a d s u s p e c t e d their a s t r o n o m i c a l purport.^^ In 1 8 8 9 , E p p i n g first a n a l y z e d the n e w m o o n table that w o u l d yield the B a b y l o n i a n value o f the m e a n s y n o d i c month.^^ B u t the value 2 9 " 12*' 4 4 " 3'/3' did n o t yet surface. In 1890, then, E p p i n g , in c o l l a b o r a t i o n w i t h A u g u s t Lorenz,^'' first identified the v a l u e 2 9 " 12*' 4 4 " 3'/3' in the c o l i m m n o w l a b e l e d G o f the lunar theory n o w labeled B , to distinguish it

from

another t h e o r y labeled A.^^ T h e m e a n synodic m o n t h ' s length is n o t explicitly g i v e n in the B a b y l o n i a n text. It n e e d s to b e inferred from a c o l u n m o f n u m b e r s that f o r m a so-called z i g z a g function in w h i c h the n i u n b e r s increase a n d d e c r e a s e b y a certain interval b e t w e e n a m a x i m u m a n d a m i n i m u m . Interval, m a x i m u m , a n d m i n i m u m also n e e d to b e inferred. W h a t follows is a simplified e x a m p l e o f a z i g z a g function: 2 1 , 3 3 , 4 5 , 5 7 , 5 5 , 4 3 , 3 1 , 19, 1 3 , 2 5 , 3 7 , 4 9 , 6 1 , 5 1 , 3 9 , 2 7 , 15, 17, 2 9 , 4 1 , 5 3 , 5 9 , 4 7 , 3 5 , 2 3 , 1 1 , 2 1 (cycle o f 2 6 n u m b e r s starts o v e r ) . F o u r values c a n b e inferred:

interval,

m a x i m u m , m i n i m u m , a n d a v e r a g e . T h e interval b y w h i c h the n u m b e r s c h a n g e is o b v i o u s l y 12. T h e m a x i m u m o c c u r s b e t w e e n 5 7 a n d 5 5 a n d b e t w e e n 61 a n d 5 1 . It is o b t a i n e d b y dividing 5 7 + 55 + 12 = 61 + 51 + 12 = 124 in h a l f T h e r e s u h is 6 2 . T h e m i n i m u m o c c u r s b e t w e e n 19 a n d 13 a n d b e t w e e n 15 a n d 17. It is o b t a i n e d b y dividing 1 9 + 1 3 - 1 2 = 15 + 1 7 - 1 2 = 2 0 in h a l f T h e r e s u h is 10. T h e a v e r a g e is o b t a i n e d b y dividing 6 2 + 10 = 7 2 in h a l f T h e result is 3 6 . A d d i n g u p the 2 6 n u m b e r s listed a b o v e a n d dividing the s u m b y 2 6 also yields the a v e r a g e 3 6 . In the s a m e w a y , interval, m a x i m u m , m i n i m u m , a n d the a v e r a g e o f 2 9 " 12** 4 4 " 31/3* c a n b e d e r i v e d from the following e l e v e n n u m b e r s found in c o l u n m G o f limar theory B.^^ 1

3

59

52

30

2

4

22

3 4

4

14

22 1

40

3

51

31

40

5

3

29

1

40

6

3

6

31

40

7

2

44

1

40

8

2

21

31

40

9

1

59

40 30

10

2

8

1 37

11

2

31

7

30

30

EPPING and STRASSMAIER (1881 ).

"

EPPING (1889); KUGLER (1900), p. 11.

EPPING (1890), p. 225 describes Lorenz's contribution as a "grossere Vertiefung.' "

EPPING ( 1890); KUGLER ( 1900), p. 11.

KUGLER(1900),p. 13.

History of the heleq

The

numbers

sexagesimally

are sexagesimal. or

60

decimally,

For example, 2 00

91

1 + 59 d e c i m a l l y

sexagesimally

is

equals

1 00

120 decimally,

and

sexagesimal 5 9 5 9 is d e c i m a l 3 5 5 9 ; the result o f a d d i n g 1 to 5 9 5 9 is s e x a g e s i m a l 1 0 0 0 0 , or d e c i m a l 3 6 0 0 . T h e following four n u m b e r s c a n b e inferred from the c o l u m n a b o v e . First, the interval is 2 2 3 0 0 0 . Seco n d , the m i n i m u m o c c u r s b e t w e e n n u m b e r s 9 a n d 10 a n d a m o u n t s t o 1 52 3 4 3 5 . Third, the m a x i m u m o c c u r s b e t w e e n n u m b e r s 2 a n d 3 a n d a m o u n t s to 4 2 9 2 7 5. Fourth, the a v e r a g e o f m i n i m u m a n d m a x i m u m is 3 11 0 5 0 . T h e cycle h a s 2 5 1 n u m b e r s . C o l u m n G gives the a m o u n t o f time that t h e s y n o d i c m o n t h is d e e m e d to e x c e e d 2 9 full days ( o n the a s s u m p t i o n that the S u n ' s m o t i o n is n o t variable, b u t this detail plays n o role in the p r e s e n t line o f a r g u m e n t ) . 2 9 days plus 3 11 0 5 0 is therefore the length o f the m e a n s y n o d i c m o n t h . T h e basic time-unit o f B a b y l o n i a n a s t r o n o m y is the us. T h e us equals four m i n u t e s o r so. Its origin is n o t certain. P e r h a p s , it is in origin the a p p r o x i m a t e time in w h i c h the Sim m o v e s in relation t o the b a c k g r o u n d o f the stars e v e r y d a y . It is also r o u g h l y the time in w h i c h the stars at night m o v e o n e d e g r e e . 3 11 0 5 0 c a n b e interpreted as 3,11;0,50 us.^^ T h e s e m i c o l o n separates the units from the p a r t s . T h e d e c i m a l equivalent is 3 x 6 0 + 1 1 + - ^ + 191

us or

X 4 m i n u t e s = 12 h o u r s 4 4 minutes 3'/3 seconds. K u g l e r first n o t e d that P t o l e m y ( s e c o n d century CE.) attributes this value to

Hipparchus

(second

century

B.C.E.)

in his Almagest?^

However,

Ptolemy's

statement, as it survives, is slightly i n c o n g r u o u s . P t o l e m y also s e e m s t o ascribe i m p r o v e m e n t o f the B a b y l o n i a n value to H i p p a r c h u s . N e u g e b a u e r d i s c o v e r e d this s a m e value in a z i g z a g function inscribed o n a G r e e k p a p y r u s from E g y p t .

This was

the first direct e v i d e n c e to e m e r g e for a k i n d o f contact b e t w e e n B a b y l o n a n d E g y p t w h o s e existence h a d until then o n l y b e e n suspected; in the m e a n time, o t h e r e v i d e n c e of the s a m e k i n d h a s surfaced.^" T h e B a b y l o n i a n synodic m o n t h is discussed b y N e u g e b a u e r . ^ ' It is n o t k n o w n h o w precisely 2 9 " 12" 4 4 " 3'/3' w a s obtained. B u t a n outline o f the general m e t h o d c a n b e inferred Almagest}^

w i t h h i g h probability

from

Book 4 Chapter 2 of Ptolemy's

T h i s m e t h o d e n c o m p a s s e s five m a i n tenets. First, b e c a u s e all synodic

m o n t h s differ in length, a time-period covering m a n y synodic m o n t h s is d i v i d e d b y the n u m b e r o f lunar m o n t h s occurring in it in order t o obtain a n a v e r a g e . S e c o n d , conjunctions are u s e d as m a r k e r s o f the begiimings a n d e n d s o f m o n t h s . S y n o d i c m o n t h s are after all b y definition returns to the s a m e syzygy, b e it c o n j u n c t i o n or opposition. Third, eclipses are u s e d as m a r k e r s b e c a u s e the m o m e n t o f conjunction is otherwise n o t visible from the Earth, Fourth, solar eclipses, w h i c h o c c u r at n e w m o o n , are a v o i d e d . T h e i r timing is t o o m u c h affected b y the o b s e r v e r ' s specific location o n Earth, that is, parallax. T h i s leaves lunar eclipses. Fifth, p a i r s o f lunar eclipses that share the s a m e properties are selected from accurate r e c o r d s s p a n n i n g m a n y centuries. T h e B a b y l o n i a n s b e g a n k e e p i n g s u c h r e c o r d s in a b o u t the eighth JONES ( 1 9 9 7 ) , p. 1 7 1 . KUGLER ( 1 9 0 0 ) , p. 2 4 . Cf T O O M E R ( 1 9 8 4 ) , p. 1 7 6 . NEUGEBAUER ( 1 9 8 8 ) ; see now also JONES ( 1 9 9 7 ) .

JONES ( 1 9 9 7 ) , p. 1 6 9 with note 1 0 . NEUGEBAUER ( 1 9 7 5 ) , pp. 4 8 2 - 4 9 6 ; for lunar theory B , see pp. 4 8 3 - 4 8 4 . ToOMER(1984),pp. 1 7 4 - 1 7 9 .

92

L. Depuydt

century B.C.E. C h o o s i n g dissimilar eclipses m i g h t affect the a v e r a g e . F o r e x a m p l e , the M o o n exhibits a cycle o f acceleration a n d d e c e l e r a t i o n in a p e r i o d o f a b o u t 2 7 " jgm tjje anomalistic m o n t h . A n average t a k e n from a t i m e - p e r i o d b e t w e e n t w o eclipses that c o n t a m s a "fasf ' p o r t i o n of a n anomalistic m o n t h w o u l d differ from an a v e r a g e taken from a time-period b e t w e e n t w o eclipses that contains a " s l o w " p o r t i o n of a n anomalistic m o n t h . T h e n e e d is for a t i m e - p e r i o d b e t w e e n t w o eclipses that also h o l d s close to a frill n u m b e r of anomalistic m o n t h s . Z i g z a g fimctions o n l y a p p r o x i m a t e reality. T h e real n u m e r i c a l p r o g r e s s i o n is trigonomefrical. T r i g o n o m e t r y , the b r a n c h o f m a t h e m a t i c s that relates angle a n d distance, w a s formulated in the s e c o n d c e n t u r y B.C.E.^^ It w a s n e v e r part o f B a b y l o n i a n m a t h e m a t i c s . P t o l e m y ' s n u m b e r s for limar m o t i o n are in e s s e n c e real due to frigonometry. G a l i l e o ' s telescopes, N e w t o n ' s laws, infinitesunal analysis, laser technology, a n d artificial satellites only refined them.

9. 29** 12" 44" Astronomy

373*

and ca, 29'' 12" 44™ 3.262245' in Greek

A b o u t one or t w o centuries after it first a p p e a r s in c u n e i f o r m texts, 2 9 " 12" 4 4 " 3'/3" s h o w s u p again as K 0 v T| K ( K 0 = 2 9 , A,a = 3 1 , V = 5 0 , T) = 8, K = 2 0 ) , that is, 2 9 ; 3 1 , 5 0 , 8 , 2 0 , in G r e e k asfronomy. In C h a p t e r 8 of his Introduction to Astronomy, G e m i n u s (first century B.C.E.) m e n t i o n s the value in p a s s i n g as m o r e accurate t h a n 29j+jj days, b u t without a source reference.^'* T h e similarities b e t w e e n B a b y l o n i a n asfronomy a n d G r e e k asfronomy h a v e always b e e n interpreted as d e p e n d e n c y o f the latter o n the former. K u g l e r ' s thesis o f B a b y l o n i a n priority has n e v e r b e e n disputed.^^ N o reports a b o u t p e r s o n a l contacts b e t w e e n B a b y l o n i a n a n d G r e e k asfronomers h a v e survived. T h e relation b e t w e e n H i p p a r c h u s and B a b y l o n has b e e n e x a m i n e d b y Toomer.^^ A s in c u n e i f o r m texts, the p o r t i o n of a d a y b e y o n d 2 9 d a y s is r e p r e s e n t e d sexagesimally. B u t in the cuneiform notation 3 11 0 50, the n u m b e r 11 refers to a unit, n a m e l y the us (4 minutes). In the G r e e k n o t a t i o n 31 5 0 8 2 0 , there are n o units. All the numbers are parts o f the day or parts thereof, that is, -^ +^ +^ + ^days. T w o centuries after G e m i n u s , P t o l e m y o f A l e x a n d r i a ( s e c o n d century CE.) cites the same value 2 9 ; 3 1 , 5 0 , 8 , 2 0 in his Almagest, a s f r o n o m y ' s bible for m o r e t h a n a millennium, at B o o k 4 C h a p t e r 2.^^ It has often b e e n o b s e r v e d that the p a r a g r a p h in w h i c h P t o l e m y m e n t i o n s the value is slightly i n c o n g r u o u s . P t o l e m y begins b y noting that H i p p a r c h u s , that other great asfronomer o f antiquity, h a d i m p r o v e d o n the B a b y l o n i a n m e a n v a l u e s o f lunar motion. A c c o r d i n g to P t o l e m y , this i m p r o v e m e n t includes the o b s e r v a t i o n that there

33

Cf DEPUYDT ( 1 9 9 8 ) .

34 35

36 37

KUGLER ( 1 9 0 0 ) , p. 5 ; D U K S T E R H U I S ( 1 9 5 7 ) , p. 3 6 . KUGLER ( 1 9 0 0 ) , pp. 5 0 - 5 3 . TOOMER(1988). TooMER ( 1 9 8 4 ) , p. 7 6 , top; cf

164.

NEUGEBAUER ( 1 9 7 5 ) , p. 6 9 ; PEDERSEN ( 1 9 7 4 ) , pp.

161-

History of the heleq

93

are 4 2 6 7 s y n o d i c m o n t h s in 126 0 0 7 d a y s p l u s 1 hour. P t o l e m y t h e n states that H i p p a r c h u s d i v i d e d 126 0 0 7 d a y s p l u s 1 h o u r b y 4 2 6 7 to find 2 9 ; 3 1 , 5 0 , 8 , 2 0 as the a p p r o x i m a t e result. T h e r e are t w o p a t e n t p r o b l e m s with this a c c o u n t as w e h a v e it. First, P t o l e m y s e e m s t o p u t the cart before the h o r s e . W e k n o w n o w that 2 9 ; 3 1 , 5 0 , 8 , 2 0 w a s a B a b y l o n i a n v a l u e . If the B a b y l o n i a n v a l u e w a s historically a n t e c e d e n t , it w o u l d h a v e to b e the i m p r o v e m e n t ' s target, n o t its result. S e c o n d , t h e result o f d i v i d i n g 126 0 0 7 days p l u s 1 h o u r b y 4 2 6 7 is n o t 2 9 ; 3 1 , 5 0 , 8 , 2 0 . It is a little less. C o p e r n i c u s p o i n t s out the s e c o n d p r o b l e m in his De coelestium,

revolutionibus

orbium

at I V . 4 , a n d g i v e s the result m o r e accurately as 29;31,50,8,9,20.^* T h a t

is: 2 9 " 12" 4 4 " 3 . 2 6 2 2 2 . . O n e also finds this v a l u e earlier in Isaac I s r a e l i ' s Sefer C.E.).^^

Also

before

Copernicus,

Savasorda

associated

Yesod Olam

(ca.

1310

29;31,50,8,9,20

with

H i p p a r c h u s in a treatise o n the H e b r e w calendar d a t e d to 1122/23 CE."*" E v e n earlier, in his Al-Qânûnu C E . ) g i v e s the v a l u e as

l-Mas'^ûdï, at V I I . 2 , A l b i n m i ( 9 7 3 - a f t e r

^ L ^

^

1050

v I ^ ' * ' o r 2 9 ; 3 1 , 5 0 , 8 , 9 , 2 0 , 1 3 , that is: a b o u t

2 9 " 12" 4 4 " 3 . 2 6 2 2 4 6 3 ' . ^ T h e exact r e s u h of dividing 126 0 0 7 d a y s a n d 1 h o u r b y 4 2 6 7 m o n t h s is a d d i n g the fraction ^ fraction

is

o f a s e c o n d , o r j u s t a b o v e 0 . 2 6 2 2 4 5 1 3 7 ' , to 2 9 " 12" 4 4 " 3 ^ T h i s irreducible.

The

corresponding

Babylonian

value

is

2 9 ; 3 1 , 5 0 , 8 , 9 , 2 0 , 1 2 , 2 2 , 2 5 with 2 5 b e i n g a p p r o x i m a t e . In trying t o establish w h a t c a u s e d the conflision in Almagest p r o p o s e s that H i p p a r c h u s did n o t divide instead multiplied

IV.2, Aaboe

126 0 0 7 d a y s p l u s 1 h o u r b y 4 2 6 7 , b u t

the B a b y l o n i a n v a l u e 2 9 ; 3 1 , 5 0 , 8 , 2 0 b y 4 2 6 7 , w i t h 126 0 0 7 " l " 5 "

3'/3' as the result.''^ H i p p a r c h u s then r o u n d e d this n u m b e r off to the closest h o u r . B y this theory, 126 0 0 7 " l " w a s n o t b a s e d o n o b s e r v a t i o n . It w a s n o t a p o i n t of departiu-e,

but

the

result

of

a

multiplication.

That

would

make

Albiruni's

29;31,50,8,9,20,13 and Copernicus' and Savasorda's 29;31,50,8,9,20 ghost values n e v e r in the m i n d s of ancient a s t r o n o m e r s , j u s t results o f divisions p e r f o r m e d in a n effort to interpret Almagest

IV.2.

A n alternative t h e o r y is to give H i p p a r c h u s m o r e credit. H i p p a r c h u s m a y h a v e inferred

from

comparing

eclipse

observations

that

the

Babylonian

value

2 9 ; 3 1 , 5 0 , 8 , 2 0 w a s a little t o o long. T h e gist of P t o l e m y ' s a c c o u n t is after all that H i p p a r c h u s i m p r o v e d o n the B a b y l o n i a n s " b y calculations from o b s e r v a t i o n s m a d e b y the C h a l d e a n s a n d in his t i m e . " It is e a s y t o establish b y simple m u l t i p l i c a t i o n that 4 2 6 7 m o n t h s o f 2 9 ; 3 1 , 5 0 , 8 , 2 0 d a y s are j u s t a b o v e 126 0 0 7 " l". In c o m p a r i n g eclipses ( r e c o r d e d b y the B a b y l o n i a n s a n d in H i p p a r c h u s ' o w n t i m e , as P t o l e m y says), H i p p a r c h u s m a y h a v e foimd that 4 2 6 7 m o n t h s t e n d to b e shorter, s a y slightly

COPERNICUS (1543), f 101V.

As noted by BANETH (1898-1903), p. 79, note 1, refen-ing to the Berlin edition of 1848.1 have not seen this work. Baneth gives 29;31,50,8,9,20.2, that is, 29;31,50,8,9,20,12 (29" 12*" 44"" 3.2622444...'). FiLiPOWSKi(1851),p. 37. TOOMER(1980), p. 108, note 6. ALBIRUNI, Al-Qânûnu (Hyderabad edition), vol. 2 (1955), p. 730, lines 10-11. AABOE (1955).

94

L. Depuydt

less than 126 0 0 7 " l". T o o m e r tries to identify pairs of shnilar lunar eclipses separated b y 4 2 6 7 limar m o n t h s that m a y have inspired H i p p a r c h u s / ' * T h e pairs that T o o m e r d e e m s w o r t h y o f consideration are i n d e e d s e p a r a t e d b y less t h a n 126 0 0 7 " l". H i p p a r c h u s w o u l d h a v e n e e d e d eclipse r e c o r d s accurate to within h a l f an h o u r or so. S u c h a c c u r a c y m a y h a v e b e e n p o s s i b l e . A c c o r d i n g to B a b y l o n i a n theory, of t w o similar eclipses separated b y 3 4 5 E g y p t i a n years of 3 6 5 d a y s p l u s 82 days, the s e c o n d eclipse o u g h t to fall at least a n h o u r later in the d a y t h a n the first. G i v e n r e a s o n a b l y accurate records, H i p p a r c h u s c o u l d easily h a v e n o t i c e d that the s e c o n d eclipse fell a little earlier than B a b y l o n i a n values said it w o u l d . H i p p a r c h u s w o u l d then h a v e faced the p r o b l e m of p r o d u c i n g a n e w n u m b e r for the m e a n sjmodic m o n t h . 29;31,50,8,9,20,... is not in the sources b u t c o u l d b e inferred as b e i n g that number. P t o l e m y , then, reports t w o items of information: the t i m e - p e r i o d 126 0 0 7 " l " a n d the fact that H i p p a r c h u s h a d i m p r o v e d o n the B a b y l o n i a n s . H e d o e s n o t m e n t i o n 2 9 ; 3 1 , 5 0 , 8 , 9 , 2 0 , 1 2 , 1 8 b u t seems to consider 2 9 ; 3 1 , 5 0 , 8 , 2 0 sufficiently accurate for use. A s for 29;31,50,8,9,20,..., P t o l e m y and H i p p a r c h u s w e r e a w a r e o f the false impression o f exactitude such precise n u m b e r s give. T h e y w e r e a w a r e that additional observations over time always bring refinement to that last little quantity that separates a n already precise n u m b e r from the elusive true value. T h e y w e r e a w a r e that their m o d e l s w e r e i n c o n ^ l e t e b u t that t u n e w o u l d b r i n g a n answer. A similar case illustrating the same awareness is discussed in the following excursion. E x c u r s i o n : T h e L e n g t h of the S o l a r Y e a r Another case in which one sees Ptolemy and Hipparchus struggle with the suspicion that their numbers show potential for getting better while for the time being remaining inaccurate is the analysis of that other chronologically significant large time-unit, the mean solar year, in Almagest IIl.l.'*' Ptolemy and Hipparchus were puzzled by the seemingly solid report that, in 4 3 2 B.C.E., Meton had observed the summer solstice in the morning. They knew that, 1 5 2 years later, in 2 8 0 B.CE., the summer solstice had fallen in Alexandria on 2 9 Pharmouthi (that is, 2 7 June). Ptolemy does not give the time of day. But it is possible that he knew that the solstice had occurred in the morning. The sole means available for determining the time of the solstice of 4 3 2 B.CE., 1 5 2 years earlier, was the length of the solar year as Hipparchus and Ptolemy knew it. Hipparchus knew that the mean solar year was a little shorter than the Callippic year of 3 6 5 . 2 5 days. He estimated the difference at about a day in 3 0 0 years. That is about half a day in 1 5 0 years. His solar year was therefore about 3 6 5 . 2 4 6 6 6 day long. Counting back 1 5 2 Callippic years, or 5 5 5 1 8 full days ( 1 5 2 X 3 6 5 . 2 5 ) , from the morning of 2 7 June 2 8 0 B.CE., one obtains the morning of 21 Phamenoth (11 June) in 4 3 2 B.CE. Ptolemy and Hipparchus assumed that the solar year was about half a day shorter than the Callippic year over 1 5 0 years. The summer solstice of 4 3 2 B.CE. therefore needed to be moved half a day closer to, or to about 5 5 5 1 7 . 5 days before, the solstice of 2 8 0 B.CE., that is, roughly to the evening of 21 Phamenoth {11 June) in 4 3 2 B.CE. Ptolemy and Hipparchus dated the solstice to 2 1 Phamenoth ( 2 7 June). But they were left wondering about morning as the time of day transmitted by reliable sources. Morning time

^

TOOMER ( 1 9 8 0 ) ; cf TOOMER ( 1 9 8 4 ) , p. 1 7 6 , note 1 0 . Cf DEPUYDT ( 1 9 9 6 ) .

History of the heleq

95

did not match there computations. That may be the origin of their criticism in Almagest III.l that the solstice of 432 B.CE. was observed "rather crudely.'"*^ Only after Ptolemy's time was the solar year's length determined to be about 365.2422 days long. The difference with the Callippic year therefore amounts to a full day or so in 152 years, not half a day. The solstice of 432 B.CE. therefore belongs an additional half a day or so closer to, or roughly 55 517 days before, the solstice of 280 B.CE., that is, in the morning of 22 Phamenoth (28 June) 432 B.CE. The report that Meton observed the solstice in the morning agrees with modem theory. There are therefore no obstacles to assuming that Meton observed the solstice on the correct day in 432 B.CE., namely 28 June. Ptolemy and Hipparchus date the solstice to 27 June. But they had no other choice with the tools available to them. Furthermore, the lunar date of the same solstice makes two assumptions nearly impossible. The first is that Meton observed the solstice on 27 June, on a day when it did not happen. The second is that the Athenian lunar month began with first crescent visibility. The lunar date is Day 13 of the lunar month Skirophorion. Conjunction fell around 10:00AM on 16 June. The first crescent could not have been visible before the evening of 17 June. If, by the first assumption, 27 June is 13 Skirophorion, then lunar Day 1 fell on 15 June, the day before conjunction. That is abnormal by any kind of lunar calendar. If, by the second assumption, the Athenian lunar month began with first crescent visibility, then daylight of Day 1 fell at the earliest on 18 June, and daylight of Day 13 therefore at the earliest on 30 June. If first crescent sighting was the practice, then Meton would have observed the solstice at the earliest two days after it actually happened, namely 28 June. That too seems improbable for what is described in the sources as an achievement. But it can be stated with higher probability which method was not used for determining the beginning of the month than exactly which method was. In sum, the association of 21 Phamenoth (27 June) and 13 Skirophorion in the sources appears to be false. We know that 13 Skirophorion was a transmitted date. And we know that 21 Phamenoth was a computed date. The different ways in which the two dates were obtained would explain their being falsely associated in later times. Nothing is more widespread than the assumption that the Athenians began their lunar months with first crescent visibility. But this assumption has never been positively proven. There are no sources that explicitly support it. Among the students of chronology who rejected the assumption was Epping. Epping noted, though without further comment, that the Greeks, in contrast with the Babylonians, determined the number of days of lunar months "nach dem mittleren Neumond.'"*' Logically speaking, at the outset, the question is open as to when the Athenians began their lunar months. The facts presented above can therefore be used logically in support of the notion that Greek lunar months did not begin with first crescent visibility. As to when they did begin, indications are that it was typically the day of conjunction. That is certainly the case with Greek astronomical lunar months, those of the Callippic cycles. If Day 1 of Skirophorion in 432 B . C E . was the day of conjunction, 16 June, then 13 Skirophorion fell on the day of the solstice, 28 June. Two items can then easily be reconciled. First is the ancient report that Meton observed the solstice in the morning of 13 Skirophorion. Second is the modem computation that the solstice did fall in the morning of 28 June in 432 B.CE.

While the specifics remain unclear because Almagest III.l is not fully transparent, the gist of the above argument is as follows. Hipparchus' measuring-rod accepted by Ptolemy, namely 365'/i days minus 1/300 of a day, was too long. If one extends such a measuring-rod back into the past in order to date equinoxes or solstices, it will be extended too far. The dates of the equinoxes or solstices obtained by means of it will therefore fall too far into the past, or TooMER(1984),p. 137. EPPING (1889), p. 179.

96

L. Depuydt

earlier in time than they should. The converse of this scenario is as follows. If one extends the same measuring-rod into the future, the dates of equinoxes and solstices obtained by means of it will fall too far into the future, or later in time than they should. Ptolemy adds equinox and solstice dates from his own time to those available to him from past records. It has long been known that these added dates are too late in time by about a day. That is exactly the margin of error that one would expect from using the said measuring-rod that is too long. In fact, Ptolemy states explicitly that 365% days - 1/300" is the measure by which the dates of his own time relate to those of the past. "Although the conclusion that Ptolemy's equinox-observations can scarcely have been more than the results of computations [and not of actual observations by Ptolemy himself) is unsatisfying," writes Britton, "I can find no other explanation of the errors in his reported times and their agreement with Hipparchus' observations and year-length [of 365Vi days - 1/300'*].'"** Therefore, "[l]acking reliable early observations of tropical phenomena to settle the question, Ptolemy may well have chosen to sacrifice the accuracy of his equinox for theoretical clarity."^'

10. Adoption of 29** 12" 44" 3%' into the Hebrew Fixed Calendar T h e history of the H e b r e w fixed calendar is a topic of large p r o p o r t i o n s . T h e principal

surveys

remain

the

chapters

on

Jewish

chronology

in the

general

h a n d b o o k s o n c h r o n o l o g y b y Ideler a n d G i n z e l a n d the t w o h a n d b o o k s o f J e w i s h c h r o n o l o g y b y S c h w a r z a n d Mahler.^° A m o n g earlier w o r k o n the H e b r e w c a l e n d a r , S c h w a r z m e n t i o n s in his preface three m e d i e v a l s o u r c e s : S a v a s o r d a ' s Sefer %bur

ha-

(early twelfth c e n t u r y C.E.), the calendrical w o r k s o f M a i m o n i d e s ( 1 1 3 5 - 1 2 0 4

C.E.), and Isaac IsraeU's Sefer might add Albinmi's

freatise

Yesod

'Olam (early fourteenth c e n t i u y C.E.). O n e

o n c h r o n o l o g y ( 1 0 0 0 CE.) (see s e c t i o n 11 b e l o w ) ,

p u b l i s h e d b y S a c h a u after S c h w a r z ' s h a n d b o o k appeared.^' A m o n g s c h o l a r s o f the m o d e m age, S c h w a r z a c k n o w l e d g e s debts to D e R o s s i , S l o n i m s k i , P i n i l e s , R e g g i o , and Steinschneider. Earlier b i b l i o g r a p h y o n J e w i s h c h r o n o l o g y is also listed b y Ginzel." B u t the focus o f this investigation is n a r r o w e r . It is o n a c o r e p r i n c i p l e o f the H e b r e w fixed calendar. T h i s p r i n c i p l e is a chain of p r e c i s e p o i n t s in t i m e . E a c h o f these points is s e p a r a t e d from the next b y exactly 2 9 " 12" 4 4 " VA'. S u c h a p o i n t in time is called a molad molad

" b i r t h ( o f the M o o n ) , " that is, (average) n e w m o o n . T h e first

w a s fixed at 2 0 s e c o n d s after 1 1 : 1 1 P M , S u n d a y night, 6 S e p t e m b e r , 3 7 6 1

B.C.E. T o that fu-st molad, s u b s e q u e n t molads,

o n e k e e p s a d d i n g 2 9 " 12" 4 4 " 3'/3' t o o b t a i n all the

in perpetuity. T h e a c c u r a c y o f the v a l u e 2 9 " 12" 4 4 " 3'/3'

guarantees that the molads

will stay close t o the alignments o f Sun, M o o n , a n d E a r t h ,

that is, n e w m o o n s , for m a n y t h o u s a n d s o f years. T h e k n o w l e d g e o f 2 9 " 12" 4 4 " 3'/3* is n o t w h a t is o f interest h e r e . It is p o s s i b l e to b e a w a r e o f the v a l u e w i t h o u t a p p l y i n g it t o a calendar. W h a t m a t t e r s is the c o n s c i o u s c h o i c e a n d d e t e r m i n a t i o n o f fixed p o i n t s in time that are all 2 9 " 12" 4 4 "

"*

BRnTON(1992),p. 36. BRITTON ( 1 9 9 2 ) , p. 3 7 .

'°

IDELER ( 1 8 2 5 - 1 8 2 6 ) ,

vol.

1, pp.

4 7 7 - 5 8 3 ; GINZEL ( 1 9 0 6 - 1 9 1 4 ) , vol.

SCHWARZ ( 1 8 7 2 ) ; and MAHLER ( 1 9 1 6 ) . SACHAU ( 1 8 7 8 , 1 8 7 9 ) . "

GINZEL ( 1 9 0 6 - 1 9 1 4 ) , vol. 2 , pp.

115-119.

2 , pp.

1-119;

History of the heleq

97

3'/3* r e m o v e d from o n e another. T h a t d e c i s i o n is so specific that it o u g h t to h a v e o c c u r r e d at a specific time a n d in a specific p l a c e . B u t the extant sources are such that w e will p r o b a b l y n e v e r b e able to establish with certainty w h e n that d e c i s i o n w a s m a d e a n d p u t into effect. T h e t i m e - p e r i o d within w h i c h the d e c i s i o n must h a v e fallen c a n b e d e l i n e a t e d as follows. O b v i o u s l y , the value 2 9 " 12" 4 4 " 3'/3' n e e d e d to exist first. E s t a b l i s h i n g the value w a s n o t p o s s i b l e without detailed r e c o r d s o f limar o b s e r v a t i o n s spaiming several centuries. O n the o n e h a n d , the B a b y l o n i a n s b e g a n k e e p i n g s u c h r e c o r d s from a b o u t the eighth century B.C.E. o n w a r d s . B y a b o u t the third to s e c o n d centuries B.C.E., the value 2 9 " 12" 4 4 " S'/a' is attested in cuneiform sources. O n the other h a n d , A l b i n m i ' s treatise o n c h r o n o l o g y o f 1 0 0 0 C E . describes the H e b r e w fixed c a l e n d a r as w e still k n o w it today. T h e calendrical application o f the v a l u e 2 9 " 12" 4 4 " SVs' m u s t therefore fall b e t w e e n the third century B . C E . a n d the e n d o f the tenth c e n t u r y CE.

T h r e e m a i n chaimels o f transmission naturally c o m e to m i n d . T h e y differ in script a n d l a n g u a g e . First, there w a s a J e w i s h p r e s e n c e in B a b y l o n w h e r e the value w a s inscribed in A k k a d i a n o n cimeiform tablets. A c c o r d i n g to N e u g e b a u e r , A l b i r u n i " r e m a r k s that the J e w s t h e m s e l v e s say that they r e c e i v e d the v a l u e for the synodic m o n t h firom the B a b y l o n i a n s . " ^ ' B u t the p a s s a g e in question^'* c o n c e r n s the intercalation o f a 1 3 * m o n t h , n o t the m e a n synodic m o n t h . S e c o n d , there w a s a J e w i s h p o p u l a t i o n in A l e x a n d r i a w h e r e P t o l e m y w r o t e his Almagest. T h i r d , in the late eighth a n d t h e ninth centuries C . E . , the study o f a s t r o n o m y flourished in the A r a b i c - s p e a k i n g world. A r a b i c a s t r o n o m y largely follows P t o l e m a i c astironomy. T h e r e are s o m e i m p r o v e m e n t s , b u t also retirogressions. It is difficuh to identify a n a r g u m e n t that w o u l d settle the matter o n c e a n d for all. B u t c o n s e n s u s h a s in r e c e n t d e c a d e s m o v e d t o w a r d t h e third possibility, that the value w a s b o r r o w e d in J u d e o A r a b i c circles firom A r a b i c translations o f G r e e k w o r k s o n a s t r o n o m y . T h e first A r a b i c translations o f the Almagest date t o the eighth a n d the early n i n t h centuries C E . "[I]t w a s this surge o f G r e e k - A r a b i c a s t r o n o m y , " O b e r m a i m writes,^^ "that precipitated t h e e m e r g e n c e o f a calendaric system b a s e d o n m e a n a s t r o n o m i c a l values." T w o alternative dates are as follows. First, a c c o r d i n g t o t h e T a h n u d i c treatise Rosh ha-Shanah, at 2 5 a , pati-iarch G a m a l i e l {ca. 100 C E . ) r e c e i v e d t h e v a l u e " 2 9 " 12" 73P" ( p = " p a r t s " or iflâqïm) fi-om the school o f h i s grandfather. ^ or 720" equals %". T h e r e f o r e , " ^ 3 " 7 3 " " is the s a m e as 7 9 3 " . B u t tìie formulation " ¥ 3 " 7 3 " , " instead o f 793", suggests that "73P" h a s b e e n inserted later into the text, as m a n y h a v e argued. In favor o f this v i e w is the fact that the value 2 9 " 12^3" is attested elsewhere.^^ S e c o n d , in a tiradition traced t o H a i G a o n , p a t i i a r c h Hillel II instituted the fixed calendar, including the value 2 9 " 12" 4 4 " 3V3', in t h e fourth c e n t i u y C E . S c h w a r z defends this date in his h a n d b o o k , against Slotiimski's tenth c e n t u r y C E . ^ ^

NEUGEBAUER ( 1 9 8 9 ) , p. 3 9 9 . See SACHAU ( 1 8 7 9 ) , p. 6 5 . In GANDZ ( 1 9 5 6 ) , p. xlviii.

SCHWARZ ( 1 8 7 2 ) , p. 2 0 with note 1. SCHWARZ ( 1 8 7 2 ) , p. 4 5 , top.

98

L. Depuydt

11. Albiruni's 29" 12" 44"" 2.289222.. T h e earUest k n o w n m e n t i o n o f the value 29** 12" 4 4 " VA' in coimection with t h e H e b r e w calendar a p p e a r s t o b e in a short A r a b i c treatise that is attributed t o t h e n o t e d m a t h e m a t i c i a n a n d astronomer A l - K h w a r i z m i (ninth c e n t i u y C E . ) a n d for w h i c h the date 824/25 C E . has b e e n proposed.^* T h e earliest extant detailed description o f the H e b r e w calendar r e m a i n s a chapter in A l b i r u n i ' s treatise o n chronology. T h e treatise is written in A r a b i c and b e a r s t h e title Al-'âtâru l-bâqiya "ani l-qurûni l-kâliya " M o n u m e n t s R e m a i n i n g from P a s t G e n e r a t i o n s . " A b u R a i h a n M u h a m m a d ibn A h m a d A l - B e r u n i (973-after 1050 C E . ) was a native o f C h o r a s m i a , located south o f the C a s p i a n Sea. S a c h a u edited a n d franslated his freatise o n c h r o n o l o g y / ' which w a s c o m p l e t e d in 1000 CE.^'^ T h e relevant p a s s a g e is a s follows.^' A n e m e n d a t i o n m a d e b y S a c h a u stands in italics b e t w e e n square brackets.^^ T h e precise w o r d i n g o f the restored p o r t i o n is n o t certain. B u t its contents is. First w e n e e d t o k n o w the conjimction o f S u n and M o o n o n D a y 1 o f (the m o n t h ) Tishri, according t o their v i e w ( t h e o n e o f the J e w i s h calendar), n o t a c c o r d m g t o t h e v i e w o f t h e asfronomers (jL-ajVl ')• T h e r e are differences b e t w e e n the t w o v i e w s , a m o n g t h e m that, for t h e lunar m o n t h from conjunction t o conjunction, t h e y h a v e 29 days, 12 hours, a n d 793 parts (Liia.), that i s , 44 m i n u t e s , 3 seconds, a n d 20 thfrds [of an hour, whereas the astronomers have 29 days, 12 hours, 44 minutes, 2 seconds, 17 thirds, 21 fourths,] a n d 12 fifths. T h e difference b e t w e e n t h e m (the t w o values) is 1 second, 2 thirds, 38 fourths, a n d 48 fifths o f a n hour. T h e following c o m m e n t s are in order. First, the value associated b y Albiruni w i t h the H e b r e w fixed calendar is in fact, a s w e k n o w n o w , a B a b y l o n i a n v a l u e that is also u s e d i n G r e e k asfronomy. Second, A l b i r u n i ' s representation o f this value a s 29" 12;44,3,20" o r 29 days a n d 12 + -*^ + ^ h o u r s is a step closer t o the m o d e m notation 29" 12" 4 4 " 3'/3" than "Ptolemy's'^9;31,50,8,20 days. T h e B a b y l o n i a n notation w o u l d b e 29 days 3,11;0,50 us (see section 8 a b o v e ) . Third, d u e t o a c o p y i s t ' s error, t h e value that Albfruni attributes t o t h e asfronomers o f his time w a s omitted in the manuscripts accessible t o Sachau. B u t the value can easily b e r e c o n s t m c t e d , a s S a c h a u s h o w e d , b e c a u s e the difference b e t w e e n it and the B a b y l o n i a n value has b e e n p r e s e r v e d . " T h e difference is 0;0,1,2,38,48 o f an hour. F r o m the subfraction 29 days 12 hours

'*

44

3

20

KENNEDY (1964).

Edition in SACHAU (1878), translation in SACHAU ( 1879). ^

SACHAU (1878), pp. x x i v - x x v . SACHAU(1878), p. u o , line u - p . > i t , line ^ SACHAU(1879), p. 143.

"

SACHAU ( 1878), pp. LIX-LX.

"

SACHAU (1878), pp. LIX-LX.

History of the heleq

minus

99

1

2

38

48

3

19

59

60

1

2

38

48

2

17

21

12.

or also 2 9 d a y s 12 h o u r s

44

minus o n e obtains the v a l u e 2 9 d a y s 12 h o u r s

44

T h a t is, 2 9 days p l u s

+

A n o t h e r n o t a t i o n is 2 9 " 12" 4 4 " 2 . 2 8 9 2 2 2 . .

h o u r s , o r 2 9 " 12;44,2,17,21,12". P t o l e m y w o u l d h a v e written this t i m e -

period as 29;31,50,5,43,23". T h e m a n u s c r i p t s h a v e " 4 4 m i n u t e s , 3 s e c o n d s , 2 0 thirds,

12 fifths." T h e

discontinuity from " 2 0 t h i r d s " to " 1 2 fifths" is o b v i o u s . In the restoration, " 2 0 thfrds [of a n hour, w h e r e a s the asfronomers h a v e 2 9 d a y s , 12 hoiu-s, 4 4 m i n u t e s , 2 s e c o n d s , 17 thirds, 2 1 (in A r a b i c : 1 a n d 2 0 ) fourths,] a n d 12 fifths," o n e easily r e c o g n i z e s , with S a c h a u , a t e x t b o o k case o f h o m o i o t e l e u t o n : the e y e s k i p p e d from t h e first t o the second " 2 0 . " T h e nimiber 2 9 " 1 2 ; 4 4 , 2 , 1 7 , 2 1 , 1 2 " d e s e r v e s t o b e t a k e n seriously in light o f its p r e c i s i o n a n d A l b i n m i ' s i m p e c c a b l e reputation. A l b i r u n i is n o t specifically k n o w n as a speculative or iimovative thinker. B u t h e fully m a s t e r e d the b e s t k n o w l e d g e o f his time in technical fields s u c h as m a t h e m a t i c s a n d asfronomy. H e m a y well h a v e b e e n the m o s t l e a r n e d m a n o f his generation. S a c h a u r e c o g n i z e s in h i s writings a critical sense m o r e characteristic o f m o d e m scholarship t h a n o f the e l e v e n t h c e n t u r y C.E.^ A m o n g A l b i n m i ' s vast o p u s is a freatise in p r a i s e o f A l b a t t a n i (ninth c e n t u r y C.E.), s o m e t i m e s called t h e A r a b i c P t o l e m y . A systematic search m i g h t r e v e a l other r e p o r t s of the s a m e v a l u e . Albattani h i m s e l f u s e s P t o l e m y ' s value,*^^ If A l b i m n i ' s 2 9 " 1 2 ; 4 4 , 2 , 1 7 , 2 1 , 1 2 " or 2 9 ; 3 1 , 5 0 , 5 , 4 3 , 2 3 " is to b e t a k e n seriously, then so is the B a b y l o n i a n value o f 2 9 " 12;44,3,20" or 2 9 ; 3 1 , 5 0 , 8 , 2 0 " . H i p p a r c h u s studied the B a b y l o n i a n value

and improved

it o n l y m i n i m a l l y . H i s v a l u e

is

2 9 ; 3 1 , 5 0 , 8 , 9 , 2 0 , . . . or a b o u t 2 9 " 12" 4 4 " 3 . 2 6 2 2 4 5 ' (see section 9 a b o v e ) . T h a t is only a b o u t 0.071 s e c o n d s o r 71 m i l l i s e c o n d s less t h a n the B a b y l o n i a n v a l u e . A s r e g a r d s the m e a n s y n o d i c m o n t h , there is a sfriking difference o f o n e s e c o n d or so b e t w e e n the b e s t v a l u e s o f B a b y l o n i a n a n d G r e e k asfronomy a n d the b e s t value of A r a b i c asfronomy a m i l l e n n i u m o r so later. S u c h a difference is significant b e c a u s e the v a l u e s are p r e s u m a b l y a v e r a g e s . H i p p a r c h u s c o m p a r e d similar eclipses s e p a r a t e d b y 3 4 5 years or 4 2 6 7 lunar m o n t h s . I n this period, a difference o f o n e s e c o n d p e r m o n t h a c c u m u l a t e s to 1 h o u r 11 m i n u t e s 7 s e c o n d s ( 4 2 6 7 s e c o n d s ) . B e t w e e n the earliest B a b y l o n i a n eclipse o b s e r v a t i o n s a n d those r e c o r d e d b y A r a b i c asfronomers, there a r e at least four such p e r i o d s o f 3 4 5 years. I n that length o f time, a difference o f o n e s e c o n d p e r m o n t h a m o u n t s to a l m o s t five h o u r s .

^

SACHAU(1878),p. X.

"

BANETH (1898-1903), p. 81.

100

L. Depuydt

A c c o r d i n g to the values c o m p u t e d for different years b y lunar t h e o r y E L P 2 0 0 0 85 (see section 7 ) , A l b i r u n i ' s value of about 2 9 " 12" 4 4 " 2 . 2 9 ' is a little l o w in relation to the m e a n synodic m o n t h of his time as measm-ed in universal time. H i p p a r c h u s ' value 2 9 " 12" 4 4 " 3 . 2 6 ' is high for his time. B u t these c o m p a r i s o n s are not meaningful b e c a u s e A l b i r u n i ' s a n d H i p p a r c h u s ' values are p r e s m n a b l y a v e r a g e s d r a w n from time-periods spaiming centuries, during w h i c h the values t h e m s e l v e s changed. T h e question r e m a i n s : W h e n c e the difference of one s e c o n d b e t w e e n B a b y l o n i a n - G r e e k a s t r o n o m y a n d A r a b i c a s t r o n o m y ? A decrease in the length o f the synodic m o n t h at first sight gives the impression that the lunar m o n t h b e c a m e shorter over the centuries. It is a fact, however, that, o w i n g to the M o o n ' s secular acceleration, lunar m o n t h s lengthened in absolute time t h r o u g h o u t hirnian history (section 7). H o w c a n lunar m o n t h s so d e m o n s t r a b l y lengthen a n d yet b y the b e s t information of ancient a n d m e d i e v a l a s t r o n o m y so ostensibly s h o r t e n ? T h e search is for a factor involving differences that accumulate to t h o u s a n d s o f s e c o n d s o v e r the centuries. T h e slowing of the E a r t h ' s rotation is s u c h a factor. N o factor other t h a n the E a r t h ' s rotational c l o c k error (AT) k n o w n at this time c o u l d a c c o u n t for a shortening in the m e a n synodic m o n t h from the b e s t o f B a b y l o n - G r e e k a s t r o n o m y to the b e s t of A r a b i c a s t r o n o m y . It is therefore tentatively p r o p o s e d here to k e e p this factor in m i n d as o n e p o s s i b l e contributing factor to the o n e - s e c o n d shortening. T h e effect of the slowing of the E a r t h ' s rotation is that n o o n - t i m e falls later in relation to a g i v e n eclipse than it w o u l d h a v e if the rotation h a d not slowed. O r the eclipses fall earlier in relation to c l o c k s that m e a s u r e rotations. T h e times b e t w e e n distant eclipses therefore a p p e a r shorter. Therefore, so w o u l d a v e r a g e m o n t h lengths. I refrain from discussing n u m b e r s in m o r e detail, such as those p r o p o s e d in S t e p h e n s o n ' s model.^^

12. Theories about the Origin of the heleq T h e r e is m o r e t h a n o n e theory o n the origin of the heleq. In c o n s i d e r i n g theories, the focus is n o t on 31/3 s e c o n d s , which is j u s t a m o d e m notation, but on the n u m b e r 1080, the d e n o m i n a t o r o f a fraction denoting part o f a n hour. T h e t h e o r y outlined a b o v e m a y b e s u m m a r i z e d briefly. It is a fact that the d e n o m i n a t o r 1080 follows b y m a t h e m a t i c a l inevitability from the time-period 2 9 " 12" 4 4 " 3'/3*. A l l it takes is to realize that the

fraction

cannot b e further r e d u c e d . T h e value 2 9 " 12" 4 4 " 3'/3'

was k n o w n from G e m i n u s a n d P t o l e m y before its B a b y l o n i a n origin w a s discovered. T h e lack o f B a b y l o n i a n e v i d e n c e was therefore n e v e r a n obstacle to c o n c e i v i n g the theory. In fact, the t h e o r y is p r o p o u n d e d as early as the twelfth c e n t i u y C E . b y S a v a s o r d a , j u s t a c o u p l e o f h u n d r e d years after the H e b r e w fixed c a l e n d a r w a s instituted. T h e theory did receive a boost, h o w e v e r , w h e n B a b y l o n i a n asfronomy w a s d e c i p h e r e d a n d the ultimate origin a n d prior existence o f 2 9 " 12" 4 4 " 3'/3' was revealed. N o other e v i d e n c e has ever e m e r g e d for a unit involving the

fraction

a n d antedating the H e b r e w fixed calendar. O t h e r w i s e , o n e m i g h t b e t e m p t e d to a s s u m e that 2 9 " 12" 4 4 " 3'/3' h a d b e e n determined, or p e r h a p s e v e n r o u n d e d off, with -fô^ in m i n d . T h a t leaves t w o possibilities. O n e is that 2 9 " 12" 4 4 " 3'/3'

STEPHENSON ( 1 9 9 7 ) .

and-r^

Historyof the Ae/eq-

101

c a m e into existence i n d e p e n d e n t l y . It s e e m s unlikely, h o w e v e r , that t w o irregular time-units g e n e r a t e d i n d e p e n d e n t l y w o u l d m a t c h perfectly b y s h e e r c o i n c i d e n c e . T h a t leaves j u s t o n e possibility: that the relation b e t w e e n 2 9 " 12" 4 4 " VA' a n d

is

causal, w i t h t h e former h a v i n g p r o d u c e d the latter. P e r h a p s t h e m o s t striking characteristic o f p a s t p r o n o i m c e m e n t s o n t h e origin o f the heleq

is t h e l o w profile o f t h e t h e o r y j u s t d e s c r i b e d in writings o n t h e H e b r e w

calendar. T h i s l o w profile h a s foin characteristics. First, t h e t h e o r y is either m a r g i n a l or a b s e n t in s t a n d a r d m a n u a l s o f chronology.^^ S e c o n d , other t h e o r i e s h a v e b e e n v i g o r o u s l y defended, e v e n after the d i s c o v e r y o f 2 9 " 12" 4 4 " VA' i n c u n e i f o r m tablets. T h i r d , t h e t h e o r y preferred h e r e h a s b e e n subject n o t o n l y t o tacit o m i s s i o n b u t also t o explicit resistance. F o u r t h , slight inconsistencies i n t h e p r e s e n t a t i o n o f the a b o v e t h e o r y b y those that a c c e p t it m i g h t affect its credibility. T h e following selective a c c o u n t illustiates these foiu- characteristics. It follows a c h r o n o l o g i c a l order. T h e r e a r e references t o contributions b y S a v a s o r d a , M a i m o n i d e s , Israeli, Slonimski, H i l d e s h e i m e r , S c h w a r z , C o h n , M a h l e r , N e u g e b a u e r , a n d O b e r m a i m . A m o n g early a u t h o r s w h o s e w o r k o n t h e H e b r e w c a l e n d a r is n o w lost, O b e r m a i m m e n t i o n s S a a d i a G a o n (d. 9 4 2 ) , H a s s a n H a d d a y y a n o f C o r d o v a (fl. 9 7 2 ) , a n d Isaak b . B a r u c h i b n A l b a l i a o f C o r d o v a (d. 1094).^* T h e oldest extant ti-eatise o n t h e H e b r e w fixed c a l e n d a r is A l b i n m i ' s ( s e e section 11 a b o v e ) . A l b i m n i says n o t h i n g a b o u t t h e origin o f the heleq,

however.

T h e s e c o n d oldest tieatise, a n d the oldest extant in H e b r e w , is S a v a s o r d a ' s Sefer ha-7bbur

" B o o k o f Intercalation" o f 1122/23 C E . A b r a h a m b a r H i y y a h a - N a s i , o r

A b r a h a m J u d a e u s , m o s t often referred t o b y his h o n o r a r y title S a v a s o r d a (a LatinS p a n i s h form o f A r a b i c sahib

as-surta

" l o r d o f t h e royal suite"), w a s a J u d e o -

S p a n i s h scholar from B a r c e l o n a . T h e t e r m Hbbur " i n t e r c a l a t i o n " in the w o r k ' s title is from

"iTy " c a u s e t o c o n c e i v e . " M o n t h s h a v e 2 9 o r 3 0 days a n d y e a r s 12 o r 13

m o n t h s . T h e s t u d y o f the c a l e n d a r is all a b o u t k n o w i n g w h e n t o insert a 30*" d a y o r a 1 3 * m o n t h , that is, m a k i n g a m o n t h o r a year m e t a p h o r i c a l l y " p r e g n a n t . " A s O b e r m a i m n o t e s , ^ ' S a v a s o r d a explains the division o f the hoiu: into 1080 p a r t s in relation t o r e s o l v i n g the s e x a g e s i m a l

fractions

of 29;31,50,8,20.'^ Likewise,

D e r s h o w i t z a n d R e i n g o l d n o t e that S a v a s o r d a " s u g g e s t e d that t h e r e a s o n for t h e c h o i c e o f 1 0 8 0 p a r t s p e r h o u r is that it is the smallest n u m b e r that a l l o w s this particular v a l u e o f the length o f the m o n t h t o b e e x p r e s s e d w i t h a n integral n u m b e r of parts (in other w o r d s , 7 9 3 / 1 0 8 0 is irreducible)."^' S a v a s o r d a ' s text a s w e h a v e it associates the heleq with P t o l e m y a n d H i p p a r c h u s a n d w i t h 2 9 ; 3 1 , 5 0 , 8 , 9 , 2 0 , b u t n o t with 2 9 ; 3 1 , 5 0 , 8 , 2 0 as t h e a r g u m e n t requires.^^ B y confrast, in t h e Almagest, Ptolemy

gives

29;31,50,8,20,

but

his

argument

requfres

at I I I . l ,

something

like

2 9 ; 3 1 , 5 0 , 8 , 9 , 2 0 ( s e e section 9 ) . S a v a s o r d a ' s text s e e m i n g l y exhibits t h e o p p o s i t e

"

Such

as iDELER's (1825-1826),

SCHWARZ'S (1872),

MAHLER'S (1916).

InGANDZ(1956),p. xlv,note35. In GANDZ ( 1956), pp. xlviii, 89-90. FiLiPOWSKi(1851),p. 37. DERSHOWITZ and REINGOLD ( 1997), p. 87, note 4. "

Cf SCHWARZ ( 1872), p. 48, note 2.

GINZEL'S (1906-1914), and

102

L. Depuydt

i n c o n g r u e n c e of P t o l e m y ' s text. P e r h a p s , S a v a s o r d a ' s i n c o n g r u e n c e s o m e h o w originated in P t o l e m y ' s . S a v a s o r d a also cites the T a l m u d i c tradition that patriarch G a m a l i e l (ca. 100 C E . ) received the value 29'' 12^ 7 3 " from the school o f his grandfather.^' G a m i H e l ' s forebears m a y h a v e lived before H i p p a r c h u s ( s e c o n d century B.C.E.). S a v a s o r d a therefore suggests that G r e e k asfronomers a d o p t e d information a b o u t asfronomy from J e w i s h asfronomers: i r n m o N'nn noDnn nn*? " t h e y learned that w i s d o m from our rabbis."^" T h e r e w a s a J e w i s h c o m m u n i t y in Seleucid B a b y l o n w h e n G r e e k asfronomers, p e r h a p s H i p p a r c h u s , p r e s u m a b l y visited the city and learned that w h i c h eventually p a s s e d from B a b y l o n i a n asfronomy into G r e e k asfronomy.^' A s f r o n o m y was v a l u e d b y J e w i s h scholars w h o p r o b a b l y entertained close ties w i t h J e w i s h c o m m i m i t i e s elsewhere in the G r e e k - s p e a k i n g w o r l d . T h e n e e d is for sources confirming S a v a s o r d a ' s account, A r o i m d 1180 C E , , in C h a p t e r 6 Section 2 o f Treatise 8 (freatise entitied Qiddush ha-Hodesh "Sanctification o f the N e w M o o n " ) o f B o o k 3 o f his Code o r Mishneh Torah, M a i m o n i d e s ( 1 1 3 5 - 1 2 0 4 C E . ) suggests that 1080 w a s c h o s e n b e c a u s e of its m a n y factors, 2, 3 , 4 , 5, 6, 8, 9, a n d 10. Ginzel a n d Ideler s u p p o r t this t h e o r y in their h a n d b o o k s . Ideler d e e m s it " n o d o u b f correct.^^ G i n z e l refers to a variation o n it p r o p o s e d b y H i l d e s h e i m e r focusing o n 3 5 4 " 8*' 8 7 6 " as the length o f a 1 2 - m o n t h lunar year.^^ N e u g e b a u e r objects that 3 6 0 might t h e n as well h a v e b e e n c h o s e n instead of 1080 b e c a u s e it c a n b e divided b y the s a m e numbers.^* O b e r m a i m interprets M a i m o n i d e s ' t h e o r y as a terse version o f the theory preferred h e r e . ^ ' In deriving 1080 as a smallest c o m m o n d e n o m i n a t o r from 2 9 ; 3 1 , 5 0 , 8 , 2 0 , o n e u s e s the n u m b e r s 2, 3 , 4, 5, 6, 8, 9, a n d 10. T h e t h e o r y preferred here also appears in Isaac Israeli's Sefer Yesod 'Olam (ca. 1310). Likewise, in the early nineteenth century, Ch, S. Slonimski a s s u m e s that "the length of the s y n o d i c m o n t h was found b y c h a n g i n g the s e x a g e s i m a l fractions g i v e n b y P t o l e m y into c o m m o n fractions."*" C o h n is the m o s t r e c e n t p r o p o n e n t o f the s a m e tiieory, " T h e H i p p a r c h a n length o f tiie synodic m o n t h o f 2 9 d a y s 4 4 m i n u t e s 3,33 s e c o n d s , " h e writes, "lies at the b a s i s o f the J e w i s h calendar. B u t to avoid the d o u b l e division into m i n u t e s and s e c o n d s , the J e w i s h chronologists preferred to equate 3'/3 s e c o n d s with h o u r s a n d therefore a c c e p t e d 2 9 d a y s 12 hoiu-s and 7 9 3 Iflâqïm as the value of the m e a n synodic month,"*' A s h o r t c o m i n g in C o h n ' s version of the theory is that minutes a n d s e c o n d s p l a y a role. M i n u t e s a n d s e c o n d s did not yet exist w h e n the H e b r e w fixed c a l e n d a r w a s instituted,

FiLiPOWSKi ( 1 8 5 1 ) , p. 3 7 , lines 3 7 - 3 8 . FiLiPOWSKi ( 1 8 5 1 ) , p. 4 1 ; cf OBERMANN in GANDZ ( 1 9 5 6 ) , p. 9 0 , top. Cf TOOMER ( 1 9 8 8 ) . IDELER ( 1 8 2 5 - 1 8 2 6 ) , vol. l , p . 5 3 8 . GINZEL ( 1 9 0 6 - 1 9 1 4 ) , vol. 2 , p. 8 3 , note 3 ; H I L D E S H E I M E R ( 1 8 8 Q - 1 8 8 1 ) . NEUGEBAUER ( 1 9 4 9 ) , p. 3 2 5 ( a l s o in GANDZ ( 1 9 5 6 ) , p. 1 1 7 ) . I n G A N D Z ( 1 9 5 6 ) , p. 8 9 . *°

ScHWARZ(1872),p.40.

*'

COHN ( 1 9 1 4 ) .

History of the heleq

103

M a h l e r criticizes C o h n ' s v i e w a n d therefore also t h e theory preferred here ("Dies, m i t V e r l a u b , ist d o c h keine Losimg d e r F r a g e ! " ) / ^ A t t h e s a m e time, he p r o p o s e s a different t h e o r y for t h e origin o f the heleq. H e a c k n o w l e d g e s that " t h e idea that t h e horn- w a s d i v i d e d into 1080 parts in o r d e r t o express t h e ( H i p p a r c h a n ) length o f the m e a n sjoiodic m o n t h i n fractions o f h o u r s is not n e w " ; b u t the following statement is n o t entirely clear t o m e : Precisely t h e fact that J e w i s h scholars u s e d t h e H i p p a r c h a n value for the m e a n synodic m o n t h a n d d i d n o t express it in m i n u t e s o r s e c o n d s b u t in Iflâqïm p r o v e s m o s t clearly a n d convincingly that they a l r e a d y u s e d t h e division o f h o u r s into Iflâqïm w h e n they a d o p t e d t h e H i p p a r c h a n value. T h e y c o n s i d e r e d it n e c e s s a r y to c o n v e r t t h e H i p p a r c h a n value, w h i c h they k n e w in minutes a n d s e c o n d s , into their o w n Iflâqïm P T h e m a i n p r o b l e m with M a h l e r ' s v i e w has a l r e a d y b e e n n o t e d a b o v e . 1080 a n d the m e a n synodic m o n t h are such a perfect m a t c h n u m b e r - w i s e that it is difficult t o assiune that they w e r e c o n c e i v e d independently. O n e m u s t h a v e b e e n d e r i v e d from the other. M a h l e r also notes s o m e w h a t cryptically, a n d n o t helpfully, "I b e l i e v e that d e e p e r p h e n o m e n a intimately c o n n e c t e d w i t h the e s s e n c e o f cultural e l e m e n t s a r e t o b e t a k e n m t o consideration."*" H e then p r o c e e d s t o his o w n e x p l a n a t i o n o f the origin o f t h e heleq.^^ H e notes that there a r e 2 5 9 2 0 ( 2 4 x 1080) iflâqïm in a d a y a n d coimects this n u m b e r s o m e h o w with the n u m b e r o f years, 2 6 0 0 0 o r s o , i n w h i c h a full cycle o f t h e p r e c e s s i o n o f e q u i n o x e s n m s its course a c c o r d i n g t o t h e B a b y l o n i a n s . B u t a s N e u g e b a u e r observes,*^ the B a b y l o n i a n s d i d n o t k n o w t h e p r e c e s s i o n o f the equinoxes.*' T h i s survey o f theories o n t h e origin o f the heleq c o n c l u d e s w i t h the m o s t recent one, that o f N e u g e b a u e r , w h o writes a s follows: A c t u a l l y w e h a v e h e r e a imit o f time c o r r e s p o n d i n g t o the " b a r l e y c o r n " o f the O l d B a b y l o n i a n p e r i o d . O n e b a r l e y c o m (called se i n S u m e r i a n ) w a s originally Vigo o f a shekel. I n late B a b y l o n i a n a s t r o n o m i c a l texts it is also a m e a s u r e for Ve o f o n e fmger. Since 1 finger c o r r e s p o n d s t o V,2 o f 1°, w e h a v e 1° = 7 2 b a r l e y c o m s a n d 15° = 1080 b a r l e y c o m s . B u t 15° o f the e q u a t o r c o r r e s p o n d t o the 2 4 t h p a r t o f the daily rotation, or t o o n e hoiu". T h u s , t h e " p a r t s " o f the J e w i s h t i m e r e c k o n i n g c o r r e s p o n d exactly t o the smallest imit, the b a r l e y c o m , i n B a b y l o n i a n a s t r o n o m y o f the last centuries B . c . T h e fundamental relation 1 finger

MAHLER ( 1 9 1 6 ) , pp. 2 0 - 2 3 . "

MAHLER(1916),p. 2 1 .

*'

M A H L E R ( 1 9 1 6 ) , pp. 2 2 - 2 3 .

*'

As first shown by KUGLER ( 1 9 0 7 - 1 9 2 4 ) , vol. 2 . 1 , pp. 2 4 - 3 2 .

MAHLER(1916),p. 2 1 .

In GANDZ ( 1 9 5 6 ) , p. 1 1 7 .

104

L. Depuydt

= 6 b a r l e y c o r n s is also well k n o w n in A r a b i c , Syriac, a n d Sanskrit a s t r o n o m y , thus leaving n o d o u b t of an u n b r o k e n tradition.** N e u g e b a u e r later reaffirmed "the imquestionable B a b y l o n i a n d e r i v a t i o n fi-om the ' B a r l e y c o m ' . " * ' T h e w e a k n e s s e s of this theory are as follows. First, there is n o e v i d e n c e for a division in 1080 parts in any units o f a n y kind before the H e b r e w fixed calendar. S e c o n d , there is the afore-mentioned p r o b l e m with the i n d e p e n d e n t c o n c e p t i o n o f 1080 a n d the value o f the m e a n synodic m o n t h . Third, as a unit, the " b a r l e y c o m " always m e a s u r e s v o l u m e , never time.

References A A B O E , Asger. 1 9 5 5 . " O n tìie B a b y l o n i a n O r i g i n of S o m e H i p p a r c h i a n P a r a m e t e r s " . Centaurus 4: 1 2 2 - 1 2 5 . A K A V Y A , a.a. 1 9 5 3 . Ha-luah ve-simmuso be-kronologia (The Calendar and Its Use in Chronology). Jemsalem: The Magnes Press. A L B I R U N I . 1 9 5 4 - 1 9 5 6 . Al-Qânûnu ( H y d e r a b a d edition): Al-Qânûnu l-Mas'^ûdï (Canon Masudicus. An Encyclopedia of Astronomical Sciences). 3 vols. H y d e r a b a d - D n . : O s m a n i a Oriental Publications B u r e a u . (In A r a b i c ) B A N E T H , E d u a r d . 1 8 9 8 - 1 9 0 3 . Maimuni's Neumondsberechnung ( B e r i c h t e iiber die L e h r a n s t a h fiir die Wissenschaft des J u d e n t h u m s in B e r l i n 16 ( 1 8 9 8 ) , 17 ( 1 8 9 9 ) , 2 0 ( 1 9 0 2 ) , a n d 21 ( 1 9 0 3 ) ) . Berlin: Itzkowski. B R I T T O N , J o h n P . 1992. Models and Precision: The Quality of Ptolemy's Observations and Parameters ( S o u r c e s a n d Studies in the H i s t o r y a n d P h i l o s o p h y of Classical Science). N e w Y o r k a n d L o n d o n : G a r l a n d P u b l i s h i n g Inc. C H A P R O N T - T O U Z É , M i c h è l e and C H A P R O N T , Jean. 1 9 8 3 . " T h e L u n a r E p h e m e r i s E L P 2 0 0 0 " . Astronomy and Astrophysics 124: 5 0 - 6 2 . — 1988. " E L P 2 0 0 0 - 8 5 : A Semi-analytical L u n a r E p h e m e r i s A d e q u a t e for Historical T i m e s " . Astronomy and Astrophysics 190: 3 4 2 - 3 5 2 . C O H N , Berthold. 1914. " D i e Stundenteile i m jtidischen K a l e n d e r " . Zeitschrift

der

Deutschen Morgenlandischen Gesellschaft 6 8 : 3 7 5 - 3 7 6 . C O P E R N I C U S , N i c o l a u s . 1 5 4 3 . De Revolutionibus orbium coelestium. Facsimile R e p r i n t o f the first edition of 1543 (fi-om the c o p y at the U n i v e r s i t y L i b r a r y of Leipzig). N e w Y o r k and L o n d o n : J o h n s o n R e p r i n t C o r p o r a t i o n 1 9 6 5 . D E P U Y D T , L e o . 1996. " T h e Egyptian a n d A t h e n i a n D a t e s of M e t o n ' s O b s e r v a t i o n of the Solstice". Ancient Society 2 7 : 2 7 - 4 5 . [ T h e following n u m b e r s are w r o n g , without the a r g u m e n t b e i n g affected: p . 3 7 , line 2 0 , for " 2 7 , 7 5 8 ' / ^ " r e a d "27,758y4"; p . 4 3 , 1. 2 0 , for " 1 3 S k i r o p h o r i o n or 3 J u l y " r e a d " 1 3 S k i r o p h o r i o n or 15 July"; p . 4 5 , 1 . 6, for "for 2 7 , 7 5 9 " read " 5 5 , 5 1 8 . " ] — 1 9 9 8 . " G n o m o n s at M e r o ë and Early T r i g o n o m e t r y " . Journal of Egyptian Archaeology

S4: 1 7 1 - 1 8 0 .

NEUGEBAUER ( 1 9 5 6 ) , p. 1 1 7 . See also NEUGEBAUER ( 1 9 4 9 ) and ( 1 9 8 9 ) , p. 3 9 9 . 89

NEUGEBAUER ( 1 9 8 9 ) , p. 3 9 9 .

History of the heleq

105

— 2 0 0 1 . " W h a t Is Certain about the O r i g m o f the E g y p t i a n Civil C a l e n d a r ? " . In: H e d v i g G Y O R Y (ed.), Â-^ï^4.^^<=, "le lotus qui sort de terre". Mélanges offerts à Edith Varga: 8 1 - 9 4 . B u d a p e s t : M u s e u m o f Fine Arts. — 2 0 0 2 . " T h e D a t e o f D e a t h o f Jesus of N a z a r e t h " . Journal of the American Oriental Society (forthcoming). DERSHOWITZ, N a c h u m and REINGOLD, E d w a r d M . 1997. Calendrical Calculations. C a m b r i d g e : C a m b r i d g e University Press. DIJKSTERHUIS, E d u a r d J. 1957. Gemini Elementorum Astronomiae Capita I, III-VI, VIII-XVÎ (Textus m i n o r e s 2 2 ) . Leiden: E.J. Brill. D O G G E T T , L e r o y E. and SCHAEFER, B r a d l e y E. 1994. " L u n a r C r e s c e n t Visibility". Icarus 107: 3 8 8 - 4 0 3 . EPPING, Joseph. 1889. Astronomisches aus Babylon (Erganzimgshefte z u den S t i m m e n aus M a r i a - L a a c h 4 4 ) . Freiburg: H e r d e r ' s c h e V e r l a g s b u c h h a n d l i m g . — 1890. " D i e b a b y l o n i s c h e R e c h n i m g des N e u m o n d e s " . Stimmen 39: 2 2 5 - 2 4 0 .

aus

Maria-Laach

—

a n d STRASSMAIER, J o h a i m N . 1 8 8 1 . "Zur Entzifferung d e r a s t r o n o m i s c h e n Tafeln der C h a l d a e r " . Stimmen aus Maria-Laach 21: 277-292. FILIPOWSKI, H e r s c h e l l (Zvi b e n Y e h e z q e l ) . 1 8 5 1 . Abraham bar Chyiah the Prince Who Flourished in Spain in the IIth century on the Mathematical and Technical Chronology of the Hebrews, Nazarites, Mahommetans, etc. L o n d o n : L o n g m a n , B r o w n , Green, a n d L o n g m a n s . (In H e b r e w ) G A N D Z , S o l o m o n ; O B E R M A N N , Julian, and N E U G E B A U E R , Otto. 1956. The Code of Maimonides, Book Three, Treatise Eight: Sanctification of the New Moon (Yale Judaica Series 11). N e w H a v e n : Y a l e University Press. GINZEL, F r i e d r i c h K a r l . 1 9 0 6 - 1 9 1 4 . Handbuch der mathematischen und technischen Chronologie: Das Zeitrechnungswesen der Volker. 3 vols. ( 1 9 0 6 , 1 9 1 1 , 1914). Leipzig: J.C. Hiiu-ichs'sche B u c h h a n d l u n g . HILDESHEIMER, J. 1 8 8 0 - 1 8 8 1 . Die astronomischen Kapitel in Maimonidis Abhandlung iiber die Neumondsheiligung (Jahres-Bericht d e s R a b b i n e r Seminars z u B e r l i n p r o 5 6 4 1 ) . Berlin. I h a v e n o t h a d access to this w o r k . IDELER, L u d w i g . 1 8 2 5 - 1 8 2 6 . Handbuch der mathematischen und technischen

Chronologie, aus den Quellen bearbeitet. Berlin: A u g u s t R u c k e r . ILYAS, M o h a m m a d . 1994. " L u n a r Crescent Visibility Criterion a n d Islamic C a l e n d a r " . Quarterly Journal of the Royal Astronomical Society 3 5 : 4 2 5 - 4 6 1 . JONES, Alexander. 1997. " A G r e e k Papyrus Containing B a b y l o n i a n Limar T h e o r y " . Zeitschrift fur Papyrologie und Epigraphik 119: 1 6 7 - 1 7 2 . K E N N E D Y , E d w a r d S. 1964. " A l - K h w a r i z m i o n the Jewish C a l e n d a r " . Scripta Mathematica 11: 5 5 - 5 9 . K U G L E R , F r a n z X . 1900. Die babylonische Mondrechnung: Zwei Système der Chaldaer iiber den Lauf des Mondes und der Sonne. F r e i b u r g i m B r e i s g a u : Herder'sche Verlagsbuchhandlimg. — 1 9 0 7 - 1 9 2 4 . Sternkunde und Sterndienst in Babel 2 vols. (1 ( 1 9 0 7 ) , 2.1 ( 1 9 0 9 ) , 2.2.1 ( 1 9 1 2 ) , 2.2.2 ( 1 9 2 4 ) ) with supplements 1 ( 1 9 1 3 ) and 2 ( 1 9 1 4 ) . M u n s t e r in Westfalen: Aschendorffsche Verlagsbuchhandlimg. A third s u p p l e m e n t , b y J o h a n n S C H A U M B E R G E R , a p p e a r e d in 1 9 3 5 .

106

L. Depuydt

LEICHTY, Erie; E L U S , M a r i a de J., a n d G E R A R D I , P a m e l a (eds.). A Scientific Humanist: Studies in Memory of Abraham Sachs ( O c c a s i o n a l P u b l i c a t i o n s o f the S a m u e l N o a h K r a m e r F u n d 9). Philadelphia: n o publisher. M A H L E R , E d u a r d . 1916. Handbuch der jtidischen Chronologie ( G r u n d r i s s der Gesamtwissenschaft des Judentiuns). Leipzig: G u s t a v F o c k . M E E U S , Jean. 1988. " L e s durées extrêmes de la limaison". L'astronomie 102: 2 8 8 289. N E U G E B A U E R , Otto. 1949. " T h e A s t r o n o m y o f M a i m o n i d e s a n d Its S o u r c e s " . Hebrew Union College Annual 2 2 : 3 2 2 - 3 6 3 . R e p r i n t e d in NEUGEBAUER ( 1 9 8 3 ) , pp. 3 8 2 - 4 2 3 . — 1 9 7 5 . A History of Ancient Mathematical Astronomy (Studies in the H i s t o r y of M a t h e m a t i c s a n d Physical Sciences 1). 3 vols. N e w Y o r k a n d Berlin: S p r i n g e r Verlag. — 1 9 8 3 . Astronomy Verlag.

and History:

Selected

Essays.

N e w Y o r k a n d Berlin: Springer-

— 1 9 8 8 . " A B a b y l o n i a n L u n a r E p h e m e r i s from R o m a n E g y p f . In: LEICHTY ( 1 9 8 8 ) , pp. 3 0 1 - 3 0 4 . — 1 9 8 9 . " F r o m A s s y r i o l o g y to R e n a i s s a n c e A r t " . Proceedings of the American Philosophical Society 133: 3 9 1 ^ 0 3 . P E D E R S E N , O l a f 1974. A Survey of the Almagest (Acta Historica Scienfrarum Natiuraliiun et M e d i c i n a l i u m 30). O d e n s e : O d e n s e University P r e s s . PEPIN, M . B a r l o w . 1996. " I n Q u e s t o f the Y o u n g e s t M o o n " . Sky and Telescope D e c e m b e r 1996: 1 0 4 - 1 0 6 . RESNIKOFF, Louis A . 1 9 4 3 . " J e w i s h C a l e n d a r C a l c u l a t i o n s " . Scripta Mathematica 9: 191-195 and 2 7 4 - 2 7 7 . R I C H A R D S , E d w a r d G. 1999. Mapping Time: The Calendar and Its History. Oxford: Oxford U n i v e r s i t y P r e s s . Reprint o f 1998 edition w i t h corrections. S A C H A U , E d u a r d . 1 8 7 8 . Chronologie orientalischer Volker von Alberuni. Leipzig: F.A. B r o c k h a u s . ( A r a b i c text) — 1 8 7 9 . The Chronology of Ancient

Nations.

L o n d o n : P u b h s h e d for the Oriental

T r a n s l a t i o n F i m d o f G r e a t Britain & Ireland b y W . H . A l l e n a n d c o . E n g l i s h franslation of the A r a b i c text in S A C H A U ( 1 8 7 8 ) . SCHAEFER, B r a d l e y E. 1996. " L u n a r Crescent Visibility". Quarterly Royal Astronomical

Society

Journal

of the

37: 759-768.

— ; A H M A D , I m a d A., a n d D O G G E T T , L e r o y E . 1 9 9 3 . " R e c o r d s for Y o u n g M o o n Sightings". Quarterly Journal of the Royal Astronomical Society 3 4 : 5 3 - 5 6 . S C H W A R Z , A d o l f 1872. Der Jiidische Kalender historisch und astronomisch untersucht. Breslau: S c h l e t t e r ' s c h e B u c h h a n d l u n g . S E I D E L M A N N , P . K e n n e t h (ed.). 1992. Explanatory Supplement to the Astronomical Almanac. Mill Valley: University Science B o o k s . S T E P H E N S O N , F . R i c h a r d . 1997. Historical Eclipses and Earth's Rotation. C a m b r i d g e : C a m b r i d g e University P r e s s . S W E R D L O W , N o e l M . 1980. " H i p p a r c h u s ' s D e t e r m i n a t i o n o f the L e n g t h o f the T r o p i c a l Y e a r a n d the R a t e o f Precession". Archive for History of Exact Sciences 21:291-309.

History of the/je/e^

107

I

T O O M E R , G e r a l d J. 1 9 8 0 . " H i p p a r c h u s ' E m p i r i c a l Basis for H i s L u n a r

Mean

M o t i o n s : A Historical F o o t a o t e to O l a f P e d e r s e n , A Survey of the Almagest,

161-

164". Centaurus

— 1 9 8 4 . Ptolemy's

24: 9 7 - 1 0 9 .

Almagest.

Translated and Annotated by — . N e w York and

Berlin: Springer-Verlag. —1988. 362.

"Hipparchus and Babylonian Astronomy". In LEICHTY ( 1 9 8 8 ) , pp. 3 5 3 -

Measuring Egyptian Statues* Friedhelm

Hoffmann,

Wiirzburg

In a c c o r d a n c e with o n e of the objectives o f the U O S m e e t i n g (i.e. to e x p l o r e the c o n n e c t i o n o f matheinatics with other issues o f ancient cultures), I w o u l d like to deal with the question of h o w the Egyptians treated height.' A l t h o u g h at first g l a n c e this question a p p e a r s v e r y simple it p r o v e s instead to b e not at all trivial b u t has m a n y facets. A t the centre o f m y p a p e r will b e the heights o f statues as r e c o r d e d in the crypts of the t e m p l e o f H a t h o r at D e n d e r a in Egypt. T h i s t e m p l e w a s built in the s e c o n d half o f the first c e n t i u y B C a n d the first half o f the first century A D a n d is v e r y well p r e s e r v e d . M a n y h i d d e n c h a m b e r s , so-called crypts, are built into the walls and foundations o f the temple. In the subterranean crypts there are h u n d r e d s o f depictions of divine statues ( c f Figure 1).^

Figure 1 N o r m a l l y , t w o types of text a c c o m p a n y these representations. Firstly, as usually found in t e m p l e scenes, there are inscriptions indicating the n a m e s o f the k i n g a n d the g o d s w h o are represented and w h a t they are saying. T h e s e texts are a r r a n g e d in short c o l u m n s or r o w s a b o v e the figures (Figure 1, c o l u m n s 1-7, 9 - 1 0 ,

12-13,

1 5 - 1 6 etc.). S e c o n d l y - a n d this is w h a t interests u s here - there are scattered little bits of text floating

aroimd these representations, often near the h e a d o f a g o d o r b e h i n d it,

I would like to thank M. Meyer and an anonymous person from the audience for corredi ng my Engl i sh. ' I have examined this and other problems in more detail in my still unpublished Habilitationsschrift Wort und Bild. Texte und Untersuchungen zur agyptischen Statuenbeschreibung. Wiirzburg University, 2001. 2

Taken from CHASSINAT ( 1947), pi. C C C X X I X .

F. Hoffmann

!10

s o m e t i m e s also m i d e m e a t h it (Figure 1 coluinns 8, 1 1 , 14, 17, 2 0 ) . T h i s s e c o n d type of inscription n a m e s typically the material a n d height of the statue a n d might therefore b e called a technical description. F o r e x a m p l e , in Figure 1 w e h a v e for each god: "gold; height: 1 c u b i f . ' I w o u l d like to concentrate on the different heights only. Before g o i n g into details, I think it is useful to m e n t i o n the E g y p t i a n m e a s u r e s of length u s e d in these texts. T h e basic unit is the divine cubit (equalling a p p r o x i m a t e l y 5 2 . 5 c m ) w h i c h is divided into 7 p a l m s ; 1 p a l m is divided into 4 digits. T h u s 1 cubit is e q u a l to 2 8 digits. + 1 cubits

+ 2 cubits

+ 3 cubits

+ 4 cubits

Od.

265

10

6

3

1 d.

4

1

2d.

4

+ 0 cubits Op.

Ip-

1 2

3d. Od.

2

Id.

1

4

1

2d. 3d.

2 5

Od.

1

Id. 2d.

1

2

3d. 3 p.

Od.

18

5 4

1 d.

4 p.

9

1

2d.

4

3d.

3

Od.

12

5

1 d. 2d.

5

2

3d. 5 p.

Od.

6

1

1 d. 2d.

2

3d. 6 p.

Od.

2

1 d. 2d. 3d. Figure 2 (p. - p a l m s , d. = digits) Collecting together all the indications of the heights of statues in the D e n d e r a t e m p l e w e obtain the result represented in Figure 2. T h i s table displays the frequency of all heights ranging from z e r o to 4 cubits 6 p a l m s 3 digits. T h e r e are for e x a m p l e 2 6 5

Cf CHASSINAT (1947), pi.

CCCXXXIV.

Measuring Egyptian Statues

ii:

o c c u r r e n c e s of statues that are exactly 1 cubit high, only four that are 1 cubit a n d 1 digit high, t w o that are 1 cubit 1 p a l m , a n d five that are 1 cubit 3 p a l m s 2 digits in height, etc. It is o b v i o u s that s o m e heights o c c u r m o r e often than others - i n d e e d m a n y d o not s h o w u p at all. T h e m o s t c o m m o n l y o c c i u r i n g height is 1 cubit. In v i e w o f the fact that this is the basic unit of the E g y p t i a n s y s t e m of m e a s u r i n g lengths, this is not u n e x p e c t e d . F o r the s a m e r e a s o n the integer multiples 2, 3 a n d 4 cubits are quite frequently given as well. + 0 cubits Op.

+ 1 cubits

Od.

28

265

Id.

29

4

2d.

30

4

3d. Ip.

Od.

32

2

Id.

33

1

36

5

37

1

38

2

39

1

40

5

41

4

42

5

+ 2 cubits 5 6 10 57

1

59

2

60

4

66

9

+ 3 cubits 84

88

6

+ 4 cubits 112

3

114

1

1

2d. 3d. 2 p.

7

2

Od. 1 d. 2d.

10

1

3d. 3P-

Od.

12 18

Id.

4 p.

2d.

14

4

3d.

15

3

Od.

16 12

1 d. 2d.

18

5

20

6

102

2

3d. Od.

48

1

1 d. 2d.

22 2

3d. 6 p.

Od.

24 2

1 d. 2d. 3d. Figure 3 ( p . = p a l m s , d. = digits)** B u t is it not sfrange to also find m a n y other heights - s o m e o f t h e m o c c u r r i n g rather frequently - s u c h as, for e x a m p l e , 1 cubit 2 p a l m s or 1 cubit 3 p a l m s 2 digits, a n d others? O n the other h a n d , m a n y n u m b e r s d o not occiu* at all, a l t h o u g h at first sight they s e e m as p r o b a b l e as those attested. W h y ? I think it is b e c a u s e the figures that d o

The individual entries represent 1 ) the height in digits, 2) the frequency of this height.

112

F.Hoffmann

occur represent a c o h e r e n t system.^ In order to m a k e this m o r e transparent, I h a v e c o n v e r t e d all measiu-es to the smallest unit, the digit, a n d I h a v e listed t h e m in a modified table (Figure 3 ) . N o w it is easier to see the relationships. W e h a v e for e x a m p l e 8 8 , 6 6 , 3 3 a n d 22 digits, apparently forming a series o f multiples o f 1 1 . T h e r e is also a series with 7: 7, 14, 2 8 , 4 2 , 56, 84 a n d 112; another with 10: 10, 2 0 , 3 0 , 4 0 , 6 0 , into w h i c h s h o u l d b e included also 15, the h a l f of 3 0 . Y e t another series starts with 12: 12, 2 4 , 3 6 , a n d 18, half its value, t h e n 4 8 a n d 6 0 ; another with 16: 16, 3 2 , 4 8 ; a n d there is a 38-series: 38, 114 a n d 5 7 , its h a l f W e are only left with 6 niunbers that caimot b e linked to any series: 2 9 , 3 7 , 3 9 , 4 1 , 5 9 a n d 102 digits. T h e fact that the vast majority of n u m b e r s c a n b e g r o u p e d into o n l y a few series o f n u m b e r s s h o w s , in m y opinion, that Egyptian statue heights c o n f o r m to a system. T h e depictions p r o v e that the several series d o n o t d e p e n d o n the d e p i c t e d deity, their kind o f representation (e.g., standing or sitting) or their material. Therefore, the relationship b e t w e e n the series c a n b e found o n l y in the n u m b e r s t h e m s e l v e s . A n d it is n o t difficult to find their connection. B y m e a n s o f multiplication b y a different multiple o f one seventh, o n e c a n r e a c h each series directly starting from the 7-series or the n u m b e r 2 8 - i.e., 1 cubit - respectively: 2 8 (28 digits = 1 divine cubit) 28 • 7

= 1 2

28 • 7

= 1 6

28 • 7

= 2 0

28 • 1 ^

=

3 8 ( ^ c u b i t s = 10 digits)

28-3-

=

88

7 N o doubt, d e p a r t m g from 1 cubit b y u s i n g sevenths suggested itself to the E g y p t i a n s b e c a u s e the (divine) cubit was d i v i d e d into 7 p a l m s . T h e few n u m b e r s that d o n o t fit into this overall s y s t e m c a n b e e x p l a i n e d v e r y easily b y looking at the relevant pictures. T o take j u s t o n e e x a m p l e : 4 1 digits is the height o f the statue r e p r e s e n t e d in Figure 4.^ T h i s statue is a c o m b i n a t i o n of a falcon b o d y a n d a h u m a n h e a d . M o s t o f the figures w i t h irregular heights are representations in w h i c h h u m a n a n d a n i m a l parts are c o m b i n e d as in this e x a m p l e . A n o t h e r type o f irregular r e p r e s e n t a t i o n s h o w s a divmity in a n a w k w a r d b o d y position like this semi-raised Osfris (Figure 5 ) . ' It is not ' The distribution is definitely not simply by chance (p < 0.001). The One-Sample Kolmogorov-Smimov Test was used to compare the observed cumulative distribution function of the data with a uniform distribution. I would like to thank A. Spahn of the Computing Centre of Wiirzburg University for his kind help. ^ Taken from CHASSINAT (1935), pi. CLXXVIIl; for the wording of the accompanying text see belovsf, note 18. '

After CAUVILLE (1997), pi. 107 (by kind permission of the Institut français d'archéologie

Measuring Egyptian Statues

113

h e r e the c o m b i n a t i o n o f different elements b u t rather the extraordinary p o s i t i o n that a p p e a r s to b e r e s p o n s i b l e for the u n u s u a l height.

Figure 4

Figiu-e 5

But, as I said, the v e r y large majority - in fact m u c h m o r e than 9 5 % - fall into one o f the series o f heights. O n e v e r y i m p o r t a n t p o i n t n e e d s to b e stressed h e r e . T h e heights g i v e n in the D e n d e r a t e m p l e are n o t the heights o f real statues that h a d b e e n m e a s u r e d , b u t rather theological m o d e l s a n d h o l y p l a n s . T h i s is s h o w n b y a note referrmg to a textual variant,* w h i c h is o n l y plausible if the texts w e r e h a n d e d d o w n i n d e p e n d e n t l y o f a n y existing statues (otherwise one c o u l d h a v e c h e c k e d doubtful specifications against the statues), therefore p r o v i n g that these texts are defmitely not the result o f a n y kind of inventory t a k e n from existing objects, at least not at the time w h e n the t e m p l e w a s built. O f c o u r s e , statues could b e created a c c o r d i n g to the texts. B u t it is i m p o r t a n t to realize that the texts are p r i m a r y a n d the statues only secondary. T h i s m e a n s that the m e a s u r e s are exactly w h a t they are m e a n t to b e a n d n o t the o u t c o m e o f accidental or incorrect w o r k b y the sculptor. In o r d e r to a p p r o a c h the question o f the intention o f these heights, t w o points m u s t b e clarified. First, is the E g y p t i a n w o r d that I h a v e b e e n franslating tacitly as " h e i g h t " really the e x p r e s s i o n for the p e r p e n d i c u l a r d i m e n s i o n o f the statues? Second, w h i c h parts of a statue did the Egyptians include into its h e i g h t ? L a c k o f s p a c e p r e v e n t s m e from demonsfrating in detail that the E g y p t i a n w o r d qi, w h i c h is u s e d m the D e n d e r a texts a n d also e l s e w h e r e to indicate the m o s t important d i m e n s i o n o f statues, n o r m a l l y refers to thefr height. T h e m a t t e r is s o m e w h a t m o r e c o m p l i c a t e d than the dictionaries m a k e u s s u p p o s e . F o r as I h a v e s h o w n in a n o t h e r p a p e r , ' the use o f the Egyptian w o r d s for the d i m e n s i o n s d e p e n d s o n the line o f sight ( c f Figiu"e 6). qi m e a n s w h a t is p e r p e n d i c u l a r in relation to the line o f sight.

orientale). *

CHASSINAT ( 1934), p. 194, lines 3 - 4 : " H a r s o m t u s g o l d - variant: iron - ; ..."

'

Presented on July 3rd, 1999 during the Standige Àgyptologenkonferenz at Trier.

F. Hoffmann

114

wsh

line of sight Figure 6 T h e m a i n r e a s o n s for a s s u m i n g such a relative s y s t e m a r e t h e following: T h e r e is firstly a n inscription o f the famous architect a n d sage A m e n o p h i s , s o n o f H a p u . H e tells o f a colossal royal statue w h o s e three d i m e n s i o n s wsh, Sw a n d qi a r e g i v e n a s : ' ° "large in respect t o the wsh ( " b r e a d t h " ) , q? ("high") t o its p i l l a r , . . . its iw ("length") 4 0 cubits". It is absolutely i m p o s s i b l e that these 4 0 cubits o f the iw " l e n g t h " a r e a n y t h i n g b u t the actual height. E v e n this - m o r e than 2 0 m - is gigantic. B u t w e c a n safely e x c l u d e that it refers to t h e distance from t o e t o heel o r from t h e front e d g e o f the b a s e t o its rear e d g e . B u t if Sw is u s e d h e r e for t h e height, qi, n o r m a l l y franslated a s " h e i g h t " , m u s t refer to this e x t e n s i o n from toe t o heel. H o w c o u l d this h a p p e n ? W e l l , t h e text states clearly that this is " t o its p i l l a r " (r jwn-f).

This used to b e

franslated

incorrectly as " m o r e t h a n its pillar". B u t for E g y p t i a n statues it is n o r m a l that the h e a d o f t h e d e p i c t e d is higher than the b a c k pillar. S o , this s h o u l d n o t b e w o r t h m e n t i o n i n g in a n a r c h i t e c t ' s inscription. It h a d n o t b e e n t a k e n into a c c o i m t i n franslations

so far that t h e E g y p t i a n p r e p o s i t i o n r c a n also refer t o t h e m e e t i n g o f

d i m e n s i o n s in a right a n g l e : " qi stands in a right angle t o t h e b a c k pillar a n d the w h o l e o f t h e shift i n t h e w o r d s for t h e d i m e n s i o n s " h e i g h f a n d " l e n g t h " c a n b e e x p l a m e d o n t h e a s s u m p t i o n that t h e statue is still lying in t h e q u a r r y w h e n it is d e s c r i b e d ( c f F i g u r e 7 ) . ' ^ In fact, t h e fransport t o a n d t h e e r e c t i o n m a t e m p l e a r e n a r r a t e d later o n in t h e text. T h e s e c o n d mstructive e v i d e n c e occurs i n t h e Embalming

Ritual for the Apis

Bull. F o r e m b a l m i n g this a n i m a l , m a n y large vessels a r e u s e d a n d e n u m e r a t e d in t h e

'°

H E L C K ( 1 9 5 8 ) , p . 1 8 2 2 , line 1 9 - p . 1 8 2 3 , l i n e 2 .

''

Cf PEET ( 1 9 2 3 ) , pp. 8 5 - 8 6 (no. 4 5 ) .

I have combined one figure from ÉPRON ( 1 9 3 9 ) , pl. LUI with another one from WILD ( 1 9 5 3 ) , pi. CXXIII.

115

Measuring Egyptian Statues

Figure 7 text that also gives their d i m e n s i o n s . T h e diameters o f their m o u t h s are called qi " h e i g h t " . ' ' I n the light o f w h a t I said before this c a n easily b e i m d e r s t o o d ( c f Figure 8 ) . ' " L o o k i n g to the vessels from a b o v e is not at all imusual for the E g y p t i a n s as these scenes from the t o m b of T i p r o v e . T h i s position, h o w e v e r , m e a n s that the m o u t h of a vessel is p e r p e n d i c u l a r to the line o f sight. Therefore the d i a m e t e r is called qi "height".

^^^\ii&\urmfi

^^^^ ^"^

Figiu-e 8 Let u s n o w return to the statues in the D e n d e r a temple that are n o t lying. Since they are o n the confrary in the n o r m a l position o f statues a n d since they are n o r m a l l y seen from the side, from the front or from b e h i n d , b u t not from a b o v e or from b e l o w , qi i n d e e d refers to the height o f these statues ( c f Figiu-e 9 ) .

V o s ( 1 9 9 3 ) , p p . 169ff. Taken from ÉPRON (1939), pl. LXVIl and modified.

F. Hoffmann

116

Figure 9 Let u s n o w t i u n to the s e c o n d question: W h a t is m c l u d e d into t h e height o f a statue b y the E g y p t i a n s ? M o d e m art historians are a c c u s t o m e d to give the total height o f E g y p t i a n statues. B u t is this also w h a t the E g y p t i a n s m e a n t ? If n o t , t h e m o d e m m e t h o d o f measiuring c a n n o t b e u s e d to grasp the intention o f the sculptors. E v e n a simple question w h e t h e r statues are o f equal height - like those in F i g u r e lO'^ m i g h t n o t b e a n s w e r e d correctly although a right a n s w e r w o u l d b e essential for seeing coimections b e t w e e n statues o f a n e n s e m b l e . This s h o w s h o w important it is to establish h o w t h e heights w e r e u n d e r s t o o d b y the E g y p t i a n s t h e m s e l v e s . T h e e x a m p l e o f Figiu-e 10 gives already a first clue. T h e inscriptions referring to the h m n a n figures m e n t i o n 10 digits for each o f them. C o m p a r i n g this with the picture, o n e gets the i m p r e s s i o n that raised limbs a n d w e a p o n s are n o t i n c l u d e d into the height.

1

I n i

^

Figure 10 T h e s a m e s e e m s to b e t m e o f c r o w n s . T h e last b u t o n e shield h a s a h e a d w i t h a h i g h feather crown. B u t h e r e again the text gives the s a m e height for all shields. S o the c r o w n is o b v i o u s l y n o t c o n s i d e r e d as part o f the height.

Taken from NAVILLE ( 1 8 8 8 ) , pi. 6.

Measuring Egyptian Statues

117

Eveti ears w e r e n o t included into w h a t the Egyptians c o n s i d e r e d to b e the height. In addition to the shields c o m p a r e also the carefully w o r k e d set of c a n o p i c j a r s depicted in SEIPEL ( 1 9 8 9 ) , p . 193. It includes as usual o n e lid with a j a c k a l h e a d with tall ears. W h i l e all others are 21 c m in overall height, this o n e is 28.5 c m . Since I d o u b t that there c o u l d h a v e b e e n any r e a s o n for not m a k i n g all the j a r s o f the set the s a m e height, the o n l y r e a s o n a b l e but seemingly p a r a d o x i c a l c o n c l u s i o n is that all four j a r s are of the s a m e height - at least for the E g y p t i a n s . T h i s m e a n s that the tips o f the ears w e r e n o t c o n s i d e r e d as the u p p e r b o i m d a r y line of the height. R a t h e r , the b o u n d a r y line m u s t b e the t o p of the skull, w h i c h is i m e x p e c t e d in that this line d o e s not p l a y any role in the Egyptian c a n o n o f p r o p o r t i o n . ' ^ In m y opinion, h o w e v e r , w e h a v e n o other c h o i c e but to accept the t o p o f the skull as the relevant p o i n t o f m e a s u r e m e n t . In addition to the evidence g i v e n a b o v e , the d i m e n s i o n s o f a figm^e a n d its n a o s found in the crypts of D e n d e r a c a n b e a d d u c e d ( c f F i g u r e 4 ) . A c c o r d i n g to the i n s c r i p t i o n s ' ' the statue is 1 cubit 3 p a l m s a n d 1 digit in height,'* its n a o s 1 cubit 3 p a l m s 2 d i g i t s . " S o the statue is only 1 digit shorter t h a n the n a o s into w h i c h it c a n b e placed. G i v e n the p r o p o r t i o n s o f the statue, this implies, firstly, that it w a s to b e p u t into the n a o s without its h e a d d r e s s . S e c o n d l y , 1 digit or less than 2 c m are less than the distance from the r o o t o f the n o s e , w h i c h m a r k s the u p p e r m o s t line o f the late c a n o n of proportion,^" to the t o p of the h e a d . T h e statue m e a s u r e s 1 cubit 3 p a l m s a n d 1 digit, that is 76.9 c m . O f these r o u g h l y (or m o r e than 7 c m ) constitute the distance from the root o f the n o s e to the c r o w n , w h i c h is o b v i o u s l y m u c h m o r e than 1 digit. Therefore it is evident that the height of the statue g i v e n in the text m u s t b e the height u p to the t o p of the skull. N o w w e h a v e to l o o k for the bottom b o r d e r l i n e in the E g y p t i a n c o n c e p t of the height of statues. T h e answer is not difficult to find. O n the one h a n d the E g y p t i a n construction grid starts from the groimd line, w h i c h p r o v e s its o b v i o u s i m p o r t a n c e not only in reality b u t also in E g y p t i a n art. O n the other h a n d w e d o find in the crypts o f D e n d e r a figiu-es standing o n a statue b a s e to w h i c h a separate height is g i v e n as in the case of the statue of F i g u r e 1 1 . ^ ' A c c o r d i n g to the text, the statue p r o p e r m e a s u r e s 1 cubit 3 p a l m s 1 digit, the b a s e 2 cubits 1 digit. I h a v e n o doubt, therefore, that the Egyptians r e c k o n e d the height o f their statues from the b a s e line to the t o p of the skull. T h e y neither included h e a d d r e s s e s , tall ears, raised a r m s , high reaching w e a p o n s etc. n o r b a s e s or other things o n t o w h i c h statues c o u l d b e placed.^^

Cf IvERSEN (1975), passim. '"^ CHASSINAT (1935), p. 39, lines 9 - 1 2 and p. 150, lines 3 - 5 ; CHASSINAT (1952), p. 15, lines 10-11 and p. 124, lines 7-9. I would reconstruct the text referring to the statue as "Hathor - solar disk of god - ; painted wood; height: 1 cubit 3 palms 1 digit". In my opinion the text referring to the naos is to be understood as: "stone - doors: firwood; height (of the doors): 1 cubit 1 palm - ; breadth: 1 cubit 2 palms; length: 2 cubits; height 1 cubit 3 palms 2 digits". Following IvERSEN (1975), pp. 75ff Taken from CHASSINAT (1952), pi. CCCCXLIII. At least in the Greek period during which the Dendera temple was built.

F. Hoffmann

118

F i g u r e 11

F i g u r e 12

H e r e I should m e n t i o n s o m e e x t r e m e l y exceptional instances, w h e r e o n e d e p i c t i o n is p r o v i d e d w i t h two heights. A t D e n d e r a these are o n l y figures of falcons w i t h c r o w n s , like a representation of the g o d S o m t u s as a r e c u m b e n t falcon with a d o u b l e feather c r o w n (Figure 12).^' T h e text^** runs as follows: " S o m t u s ; g o l d (and) p a i n t e d w o o d ; height: 3 p a l m s ; height: 1 cubit 3 p a l m s 2 digits". In solving this riddle w e m u s t resort to actual statues o f lying falcons w i t h s u c h a c r o w n - the relief representations are n o t really reliable, b e c a u s e t h e y exhibit m u c h shorter feathers. T h e actual figures, however, m a k e u s realize that the p r o p o r t i o n of the b o d y to the c r o w n varies extremely b e c a u s e the feathers c a n b e o f v e r y different height. T h e c r o w n c a n take u p e v e n nearly 7 0 % of the overall height o f the c r o w n e d statue.^^ I n light o f this information I w o u l d suggest to interpret the m e a s u r e s as follows: 3 p a l m s , the height g i v e n first, is - as u s u a l - the height o f the b o d y only. T h e s e c o n d one, the 1 cubit 3 p a l m s 2 digits, is the total height including the c r o w n , w h i c h in this case w o u l d constitute 7 1 % of the total height - a n i u n b e r that fits quite well, I think. W h a t I w a n t e d to d e m o n s t r a t e in discussing?*this exceptional case is that e v e n here m y idea, that the Egyptians c o n s i d e r e d the height o f the b o d y as the all important m e a s u r e , w o r k s . It is only w h e n the c r o w n is higher t h a n the b o d y itself, that a s e c o n d w a y o f m e a s i u i n g was s o m e t i m e s c o n s i d e r e d necessary. T o c o n c l u d e , I w o u l d like to stiess that in c o n n e c t i o n with statues, the E g y p t i a n s w o r k e d with a structural tallness of the body o f the o n e d e p i c t e d a n d n o t w i t h a simply physical height o f the statue. C r o w n s , w e a p o n s , and b a s e s , a n d e v e n ears, a r m s etc. are a sort of addition to the statue p r o p e r . A r t historians should take this into a c c o u n t w h e n dealing with E g y p t i a n sculptiu"e. T h i s is not o n l y a vital p o i n t for

"

Taken from CHASSINAT (1934), pi. C L . CHASSINAT (1934), p. 194, lines 1-2.

25

Cf e.g. the falcon Cairo RT 18.11.24.46 (SALEHand SOUROUZIAN (1986), no. 268).

Measuring Egyptian Statues

119

a correct u n d e r s t a n d i n g o f Egyptian art, the m o r e so as a c c o r d i n g to the texts o f the late t e m p l e s the heights o f E g y p t i a n statues are n o t accidental b u t s e e m to b e t h e m s e l v e s e l e m e n t s o f a close-knit theological system, as their c o n n e c t i o n b y multiples s h o w s .

References Sylvie. 1 9 9 7 . Le temple (Bibliothèque d'Étude 117-119). Orientale

CAUVILLE,

CHASSINAT,

de Dendara: les chapelles osiriennes. C a i r o : Institut F r a n ç a i s d'Archéologie

Emile. 1934. Le temple de Dendara.

d ' A r c h é o l o g i e Orientale C H A S S I N A T , Emile. 1935. Le temple de Dendara. d ' A r c h é o l o g i e Orientale C H A S S I N A T , Emile. 1947. Le temple de Dendara.

V o l u m e 2. C a i r o : Institut Français V o l u m e 3 . C a i r o : Institut F r a n ç a i s V o l u m e 5 (Plates). C a i r o : Institut

F r a n ç a i s d ' A r c h é o l o g i e Orientale C H A S S I N A T , Emile. 1952. Le temple de Dendara. V o l u m e 5 ( T e x t ) . C a i r o : Institut Français d ' A r c h é o l o g i e Orientale É P R O N , L u c i e i m e et al. 1 9 3 9 . Le tombeau de 77. F a s e . 1. C a i r o : Institut F r a n ç a i s d ' A r c h é o l o g i e Orientale H E L C K , W o f g a n g . 1 9 5 8 . Urkunden der 18. Dynastie. N o . 2 1 . Berlin: A k a d e m i e Verlag I V E R S E N , Erik. 1 9 7 5 . Canon and Proportions in Egyptian Art. 2"" edition. W a r m i n s t e r : A r i s a n d Phillips N A V I L L E , E d o u a r d . 1 8 8 8 . The Shrine of Shaft el Henneh and the Land of Goshen (1885). L o n d o n : T r u b n e r P E E T , T o m a s Eric. 1 9 2 3 . The Rhind Mathematical Papyrus. British Museum 10057 and 10058. L o n d o n : H o d d e r a n d S t o u g h t o n Limited M o h a m e d a n d S 0 U R 0 U Z L \ N , H o u r i g . 1986. Die Hauptwerke

SALEH,

im Agyptischen

Museum Kairo. M a i n z : Philipp v o n Z a b e m SEIPEL, Wilfried. 1 9 8 9 . Agypten. Cotter, Graber und die Kunst. 4000 Jahre Jenseitsglaube. V o l i u n e 1. Linz: L a n d e s m u s e i m i Linz Vos, R e n é L. 1 9 9 3 . The Apis Embalming Ritual. P. Vindob. 3873. Leuven: Uitgeverij P e e t e r s a n d D é p a r t e m e n t Orientalistiek WILD, Henri. 1 9 5 3 . Le tombeau de Ti. Fase. 2 . C a i r o : Institut F r a n ç a i s d ' A r c h é o l o g i e Orientale

How to Educate a Kapo or Reflections on the Absence of a Culture of Mathematical Problems in Ur III Jens Heyrup,

Roskilde

To B o b (Englund) and Hans (Neumann)

Personal prehistories Firstly: S o m e fifteen years a g o , w h e n R o b e r t E n g l u n d h a d recently c h a n g e d his dissertation t h e m e from " U r III-Fischerei" to " O r g a n i s a t i o n u n d Verwaltimg der U r IIIFischerei", h e told m e (halfway jestingly) to b e alarmed, discovering h i m s e l f to h a v e b e c o m e a Stalinist: n a m e l y b e c a u s e analysis of the soiu-ces o b l i g e d h i m to c o n c l u d e that U r III w a s a slave society. T h i s agreed better than h e liked with the o r t h o d o x Stalinist version o f historical materialism, a c c o r d i n g to w h i c h "slave s o c i e t y " follows after "priinitive c o m m i m i s m " , without a n y intervening "Asiatic m o d e o f p r o d u c t i o n " . I s u g g e s t e d h e should not w o r r y t o o m u c h . In the present case, the o r t h o d o x y h a d little to d o with the contents or style o f Stalinist policies, apart from its b e i n g made a n orthodoxy. Stalin h a d j u s t h a p p e n e d to listen to the b e s t authority o n the question he could find: V. V. Struve, w h o , w h e n b e c o m i n g ciu-ator o f the H e r m i t a g e cimeiform collection, h a d started r e a d i n g U r III accounts a n d h a d b e e n led to the same c o n c l u s i o n as later E n g l u n d . ' Further on, E n g l u n d sharpened his views considerably: in v i e w o f the sfrict r e g i m e to w h i c h n o t only the workers but also the overseer-scribes w e r e s u b m i t t e d h e w o u l d n o w s p e a k of a ''Kapo e c o n o m y " - a Kapo b e i n g (in case a n y b o d y should not h a v e h e a r d about the system) the K Z prisoner responsible for the w o r k or organization o f a g r o u p o f fellow prisoners, in constant d a n g e r o f b e i n g r e d u c e d to the status of an ordinary p r i s o n e r as s o o n as he did not ftilfil his tasks in a w a y that contented the S S . A n d , as formulated in the c o n c l u d i n g w o r d s o f the p u b l i s h e d version of the dissertation,^ the imderstanding of w o r k i n g conditions c o n v e y e d b y the adminisfrative texts

'

See DiAKONOFF ( 1969), p. 5.

2

ENGLUND

(1990), p. 316.

122

J. Heyrup

k a n n vielleicht helfen, sich in den historischen D a r s t e l l u n g e n des 3 . J a h r t a u s e n d s v. Chr. die K o s t e n der b a b y l o n i s c h e n Palaste u n d Statuen plastischer vorzustellen. In a v o l u m e p u b l i s h e d in the same year^ w e find a m o r e detailed formulation in a c o m m e n t a r y to the a c c u m u l a t e d deficit in the yearly b a l a n c e of a n overseer-scribe: A u s a n d e r e n Texten w i s s e n wir, w e l c h e e m s t e n K o n s e q u e n z e n solch eine liickenlose U b e r w a c h u n g der a n w a c h s e n d e n F e h l b e t i a g e fiir d e n Aufseher u n d seinen Haushalt mit sich b r a c h t e . D i e F e h l b e t r a g e muBten offenbar u m j e d e n Preis b e g l i c h e n w e r d e n . Verstarb ein Aufseher, so w u r d e sein Nachlafi h e r a n g e z o g e n , die Schuld z u Tilgen. D a s b e d e u t e t e in der Regel, dafi die v e r b l e i b e n d e n H a u s h a l t s m i t g l i e d e r selbst in die staatlichen A r b e i t e r t m p p s eingegliedert w u r d e n , die die v o n d e n A u f s e h e m iiberwachten A r b e i t e n z u verrichten hatten. D a s w a r e n die A r b e i t s b e d i n g u n g e n der Aufseher, iiber die in d e n Verwaltimgstexten allein B u c h gefiihrt w u r d e . U b e r das Schicksal der Arbeiteriimen u n d Arbeiter sind d a g e g e n k a u m I n f o r m a t i o n e n iiberliefert. Sogenaimte " M u s t e n m g s t e x t e " b e r i c h t e n regelmaBig iiber in grofier Z a h l entflohene Arbeiter. M a n kaim sich angesichts der totalen U b e r w a c h u n g aller Leistungen der T r u p p s , in die sie eingegliedert waren, die Griinde leicht a u s m a l e n .

Secondly: A l r e a d y before I started m y conceptual analysis of the O l d B a b y l o n i a n " a l g e b r a i c " texts in 1982 I s u s p e c t e d these to represent a new g e n r e , irrespective o f current opinions. In H o y r u p ( 1 9 8 0 ) , p p . 2 0 - 2 2 I h a d thus written that'* the influence o f the school o n S u m e r i a n m a t h e m a t i c s w a s (as far as w e k n o w it firom p u b l i s h e d material^) restricted to the systematization applied

mathematics.

of

[...] Sumerian m a t h e m a t i c a l texts are c o n c e r n e d

w i t h real " r e a l - w o r l d - p r o b l e m s " ; this d o e s not imply that t h e y are always realistic: O n e school text fi-om the Sargonic epoch^ deals w i t h a

NissEN, D A M E R O W and E N G L U N D ( 1 9 9 0 ) , p. by Peter Damerow and Robert Englund. ^

89,

cf id.

(1994),

p.

54

- this chapter signed

* I change the numbering of notes and the format for the references but leave my original text intact (though truncated, and with omission of a number of notes) in other respects. Evidently, I would now formulate much of what is said in different terms (not least avoiding the notion of an Old Babylonian "pure mathematics"). To the extent one can distinguish substance from formulation, I believe most substantial points remain valid. ^ [...] comparison with the distribution of literary texts seems to indicate that Sumerian precursors of later Babylonian pure mathematics are really non-existent: Indeed, even if most literary texts are known only from later versions although they have Sumerian or Sargonic origins, a reasonable number of literary tablets are known illustrating the tradition all the way back to 2 5 0 0 B.C. H A L L O ( 1 9 7 6 ) , passim; A L S T E R ( 1 9 7 4 ) , p. 7. No such precursors for Babylonian pure mathematics exist. ^

See P O W E L L ( 1 9 7 6 ) , pp. 4 2 8 f

HOW TO EDUCATE A KAPO

field

123

as long as 1297.444 k m (given to that precision [...]). [...] In

historical retrospect, this characteristic is typical of the teaching o f e v e n practically oriented m a t h e m a t i c s w h e n this t e a c h i n g h a s b e e n institutionalized

a n d t h e r e b y has b e c o m e the task of a partly c l o s e d

milieu. [...]. T h e practical e v e n if s o m e t i m e s

abstract character o f

Sumerian

matheinatics is in perfect h a r m o n y with w h a t little w e k n o w a b o u t the c i u r i c u l u m o f the U r III school: It was p i u e l y utilitarian, a n d h a d n o r o o m for / 'art pour I 'art^. Babylonian culmination T h e practical fixation o f Siunerian m a t h e m a t i c s a n d m a t h e m a t i c s teaching m a y p e r h a p s b e r e g a r d e d as a c o n s e q u e n c e o f the integration of the Siunerian school a n d the scribal profession in the state administiation. W e m a y guess that the u n e q u i v o c a l p u b l i c a t t a c h m e n t of the scribal fimction m a y h a v e restricted the ideological a u t o n o m y o f the scribal school a n d t h e r e b y its institutional i n d e p e n d e n c e . T h i s is o n l y speculation, a n d w e m a y leave it as it stands. [...]. I n d i v i d u a l i s m in the O l d B a b y l o n i a n society w a s n o t confined to the c o m m e r c i a l sphere. [...]. I n this situation, the scribal profession s e e m s to h a v e b e c o m e m o r e i n d e p e n d e n t as a social b o d y ; at least, it b e c a m e less u n e q u i v o c a l l y attached to the p u b l i c authority a n d fimction. [...]. A t the s a m e time*, a g e n u i n e p u r e m a t h e m a t i c s was d e v e l o p e d [...], b a s e d m p a r t o n methods w i t h n o relevance for d o w n - t o - e a r t h practical tasks - derived, truly, fi:om p r a c t i t i o n e r s ' m e t h o d s , b u t tiansformed a n d d e v e l o p e d b y the contact w i t h the theoretically generated p r o b l e m s . A s I e n g a g e d afterwards in m y close r e a d i n g of the " a l g e b r a i c " texts, I found confirmation in the p r e d o m i n a n c e o f A k k a d i a n t e r m i n o l o g y ; in the A k k a d i a n language structure e v e n o f almost exclusively l o g o g r a p h i c texts; and, n o t least, in the fact that the o n l y t e r m s that occur exclusively or almost exclusively in S u m e r i a n (or as l o a n w o r d s p r o v i d e d with A k k a d i a n declination) are t h o s e that b e l o n g

' [Cross reference to an earher note, in which is found:] According to hymns made in the name of king §ulgi, the curriculum of the Ur III school contained writing, arithmetic, accounting, field measuring, agriculture, construction, and a few subjects the names of which are not understood; cf SJÔBERG ( 1 9 7 6 ) , pp. 1 7 3 f * The precise chronology of the process is at least for the moment not to be known. Still, the decisive steps must be placed in the earlier part of the period, since a number of characteristic texts dating from c. 1 8 0 0 B.C. have been found in Tell Harmal [...]. A limit post quern is obtained from the observation that the problem type most characteristic of the new pure mathematics- the "equation" of the second d e g r e e - is intertwined with the use of the full potentiality of the sexagesimal number system; it seems [...] that this type of mathematics can only have been developed after the use of sexagesimals had been generalized. [...] The language was Akkadian, the language of the new literary creativity.

124

J. Hoyrup

u n a m b i g u o u s l y to the sphere of practical c o m p u t a t i o n : u s , sag, a.sà, igi, igi.gub, ib/ba.sig, a.râ. Late in the 1980s I got the h u n c h - b y then still built o n very t e n u o u s e v i d e n c e that a tradition of s u r v e y o r s ' recreational riddles might b e " o l d e r than - p e r h a p s e v e n a source for - O l d B a b y l o n i a n scribe school ' a l g e b r a ' " , ^ b u t invested n o m o r e than this single line in the h y p o t h e s i s . ' " I h a d n o suspicion b y then that the various strands of w h a t p r e c e d e s m i g h t e n d u p b e i n g intertwined. T h a t they are only struck m e at m y third or fourth r e t u r n to the general structure of the O l d B a b y l o n i a n m a t h e m a t i c a l v o c a b u l a r y in 1998. T h e present p a p e r is m e a n t to tell that story.

The Old Babylonian vocabulary

mathematical

operations

and

their

T h e (more or less) technical terms u s e d in Old B a b y l o n i a n m a t h e m a t i c a l texts fall in two m a i n g r o u p s : (1) terms for operations, a n d (2) the m e t a l a n g u a g e w h i c h allows the formulation o f p r o b l e m s a n d the higher-level explanation o f the p r o c e d u r e s b y w h i c h they are solved. T h e former g r o u p c a n b e subdivided thus: Additive operations T w o operations b e l o n g to this group. O n e is the "identity-conserving" a d d i t i o n in w h i c h an extra p i e c e is j o i n e d to a quantity; it is always concretely meaningful. It is designated b y the A k k a d i a n t e r m wasâbum, with the S u m e r o g r a m dah, for b o t h o f w h i c h I shall use the standard translation "to a p p e n d " . " In a g r e e m e n t w i t h the "identity-conserving" character of the operation, n o particular t e r m for the c o r r e s p o n d i n g " s t u n " a p p e a r s to exist. T h e other o p e r a t i o n is symmetric, and d o e s not p r e s u p p o s e the addition to b e concretely meaningful. It h e a p s or " a c c i u n u l a t e s " (the m e a s u r i n g n u m b e r s of) t w o or m o r e a d d e n d s , connecting t h e m with the w o r d u ( " a n d " ) , and m a y thus b e r e g a r d e d as a genuine arithmetical operation. T h e m a i n A k k a d i a n t e r m is kamàrum, to w h i c h c o r r e s p o n d the S u m e r i a n t e r m gar.gar and the u n e x p l a i n e d l o g o g r a m UL.GAR. T h e c o r r e s p o n d i n g s u m h a s a n a m e : kumurrûm, with l o g o g r a m s gar. gar a n d UL.GAR. Occasionally, other n o u n s derived from kamàrum are u s e d for the s u m , m o s t

'

H 0 Y R U P ( 1 9 9 O a ) , p . 275.

'° The manuscript for the publication in question was finished in 1987; in H O Y R U P (1990b), pp. 79 f, written in 1989, the argument is elaborated and some supplementary evidence is cited. ' ' For convenience, I make use of a system of fixed "standard translations" in the following: it is easier to insert the conjugated forms of "append" in an English phrase than those of wasâbum; for non-Assyriologist readers it may also be easier to connect them to the common root. The ones 1 use here differ slightly from the set used in my (1990a) but coincide with those of H 0 Y R U P (2002a). For the reasons that lead to the interpretation of the terms I shall only refer to my earlier publications, in particular to H 0 Y R U P (1990a) and (2002a).

How to Educate a Kapo

r e m a r k a b l e a m o n g w h i c h is the plural kimrâtum,

125

apparently a reference to the s u m as

consisting o f still identifiable constituents. O n s o m e o c c a s i o n s , u alone serves in the same function; in o n e text ( B M 8 5 2 0 0 + V A T 6 5 9 9 ) an a b b r e v i a t e d form of the t e r m u s e d for the s u m total in accounting (NIGIN) turns u p twice; in b o t h c a s e s , t w o numbers

- viz a pair b e l o n g i n g together in the table o f reciprocals - are a d d e d .

Subtractive operations T w o " s u b t r a c t i o n s " occur in the texts, removal a n d comparison. B o t h are always concrete. Removal m a y b e s e e n as the inverse o f a p p e n d i n g . T h e m a i n t e r m is nasâhum, "to tear out", w i t h S u m e r o g r a m zi. It c a n o n l y b e u s e d w h e n the s u b t r a h e n d is part of the entity from w h i c h it is subtracted. M o s t texts firom early eighteenth-century E s h n i m n a a n d s o m e from G o e t z e ' s equally early " g r o u p 1"'^ prefer the A k k a d i a n harâsum, "to cut off', with n o p r o p e r S u m e r o g r a m ; ' ^ in confrast, zi is u s e d frequently e v e n in p r e d o m i n a n t l y syllabic texts.'^* Comparison indicates h o w m u c h o n e m a g n i t u d e A e x c e e d s a n o t h e r m a g n i t u d e B which it does not contain. It is n o inverse of kamàrum, a n d caimot b e the reversal o f any addition (since the s u m always contains the a d d e n d s ) . ' ^ T h e p h r a s e in u s e states that A eli B d itter/Tter, "A o v e r B, d it g o e s / w e n t b e y o n d " (from eli... watârum, " g o b e y o n d " , " b e ( c o m e ) / m a k e greater t h a n " ) , with the S u m e r o g r a p h i c equivalent A u g u 5^dirig.'^ "Multiplications" F o u r different operations c a n in s o m e w a y b e u n d e r s t o o d as " m u l t i p l i c a t i o n s " . O n e is the multiplication of n u m b e r b y niunber as found in the tables of multiplication, to w h i c h c o r r e s p o n d s the S u m e r o g r a m a.râ (from R Â , " "to g o " ) a n d

Goetze, in MCT, pp. 146-151. For the chronology of the group, see H0YRUP (2000), p. 149; when speaking in what follows of text groups I refer to the new delimitations of Goetze's original groups established in this latter publication. kud, used in the late Old Babylonian TMS XXVI, may stand for harasum but also for nakasum, "to cut down", or (}asâbum, "to break off" - or, not to be excluded, for nasâlfum. For the use of tabâlum, "to withdraw", and sutbûm, "to make leave", for specific removals, see H o y r u p (1993). Because of the symmetric character of accumulation, its actual inverse is the splitting into or singling out of components {bêrum, with no Sumerographic counterpart in the mathematical texts). It may be no accident that the term is used in AO 8862, the very text that speaks of the accumulation as a plural kimrâtum. The relatively rare comparison made the other way round, the statement of how much B falls short of A (using the verb matûm, "to be(come) small(er)", with Sumerogram lai), is discussed in H o y r u p ( 1993). The verb takes on a number of different forms depending on grammatical number and aspect (SLa, §268): gen and du (singular perfective and durative), re? and sug.b (plural ditto). Since both gen and du are written d u = RÀ, it is convenient to write the verb as r A in order to keep present the relation with the term a.râ (the pronunciation of which is certified by the loanword arum).

126

J. Hoyrup

the p h r a s e a a.râ b, m e a n i n g " a steps o f b ". T h e t e r m h a s n o A k k a d i a n c o u n t e r p a r t but gives rise to the l o a n w o r d

arûm<*ara-um}^

T h e s e c o n d is the d e t e r m i n a t i o n of a c o n c r e t e m a g n i t u d e b y m e a n s o f a multiplicative operation. It is u s e d in multiplications b y technical constants a n d m e t r o l o g i c a ! c o n v e r s i o n s ; it serves w h e n v o l u m e s are d e t e r m i n e d from b a s e a n d height and in the calculation o f areas that are n o t i m p l i e d b y the c o n s t r u c t i o n o f a rectangle (that is, areas of w h i c h are already

friangles,

of frapezia a n d

frapezoids,

a n d o f rectangles

there). T h e core t e r m is nasûm, " t o r a i s e " , with the S i u n e r o g r a m

il. A n o t h e r S u m e r o g r a m u s e d in the s a m e ftmction is n i m l o g o g r a p h i c a l l y c o n n e c t e d to elûm a n d saqûm a n d their various derivations ( b o t h "to b e / b e c o m e / m a k e h i g h " ) . ' ' T h e third multiplicative o p e r a t i o n is the repetition o f a c o n c r e t e m a g n i t u d e an integer n u m b e r o f times ("until n",2
9). It is s p o k e n o f in A k k a d i a n as

esëpum,

"to d o u b l e " , with the Simierographic equivalent t a b ( w h o s e original s e m a t i c r a n g e is wider, for w h i c h r e a s o n Seleucid m a t h e m a t i c a l texts c o u l d r e a d o p t it as a l o g o g r a m for the identity-conserving addition). T h e fourth o p e r a t i o n is better dealt with imder the following h e a d i n g . Rectangularization, squaring and "square root" T h i s operation, indeed, consists primarily in the " b u i l d i n g " {banûm)

of a r e c t a n g l e

a n d is only a multiplication in so far as the c o m p u t a t i o n of the a p p u r t e n a n t area is freated

as inherent in or implied b y the construction. T h e cenfral t e r m is

sutakûlum,

"to m a k e [two s e g m e n t s a a n d b] h o l d e a c h o t h e r " {viz as sides o f a r e c t a n g l e - at times g r a m m a t i c a l constructions are u s e d w h i c h rather i m p l y that a t o g e t h e r w i t h b contain or " h o l d " the rectangle), i.guy.guy (from gu?, " t o eat") is u s e d as a l o g o g r a m , p r o b a b l y b e c a u s e o f the p h o n e t i c near-identity b e t w e e n sutâkulum, other", a n d

" t o m a k e eat e a c h

sutakûlum.

Admittedly, the metaphor of "repeated going" expressed in Akkadian (alâkum) and used in a general way (for multiplication as well as repeated "appending") is found in various Old Babylonian texts from Susa. The pattern of thought behind the term a.râ thus found expression in Akkadian; but it did not produce an Akkadian equivalent of the term. " The original mathematical use of the term is connected to the determination of volumes. In these, indeed, the base is invariably "raised" to the height; in all other cases, the order of the factors is random from a mathematical point of view, and depends first of all on stylistic criteria - as a rule, it is the quantity that has been computed in the preceding operation that is "raised" to the other factor, irrespective of its meaning or role in the computation - cf H0YRUP(1992),pp. 351 f As is well known, volumes were measured in area units, which implies that these were thought of as provided with a virtual standard thickness of one kù§ (cubit). In consequence, determination of a prismatic volume meant that this virtual thickness was "raised" to the real height. Raising multiplications are thus operations of proportionality, and may hence be said to be "category-conserving". Traditionally, most workers (Thureau-Dangin being the chief exception), have taken the logogram as an argument that the term should be read sutâkulum, "to make [a and b] eat each other", which fits the cuneiform orthography just as well. However, certain texts refer to a segment that has been submitted to the operation with the relative clause "which you have made hold/eat", while others use a noun takïltum with precisely the same function; the latter term can only derive from kullum and mean "which is made hold". Of course, rebus- or pun-

How to Educate a Kapo

127

B e y o n d i.gU7.gU7 a n d the abbreviation ì.ga^, several other l o g o g r a m s are used: ( p r o b a b l y to b e r e a d duy.duy, for nitkupum, "to butt e a c h o t h e r " ) ; U R . U R ; a n d N I G I N , w h i c h c a n b e interpreted as a contracted L A G A B . L A G A B , In all four cases, the reduplication is p r o b a b l y not to b e u n d e r s t o o d as a genuine S u m e r i a n g r a m m a t i c a l form b u t rather as a w a y to r e n d e r in p s e u d o - S u m e r i a n the reciprocity o f the p r o c e s s that the A k k a d i a n language presents b y m e a n s o f the â t a n d Gt-stems {sutakûlum a n d nitkupum, respectively). Several texts p l a y with r e p e a t e d numerical multiplication, b u t n o specific t e r m for p o w e r s of numbers exists. Squaring, as a specific p r o c e s s , is geometric squaring. T h e A k k a d i a n t e r m for the square configuration is mithartum, a t e r m w h i c h refers to a confi:ontation of equals {viz, equal sides), a n d w h i c h w h e n e x p r e s s e d as a n u m b e r coincides w i t h the side of the square.^' T h e w o r d derives fi*om the v e r b mahârum, w h o s e (causative-reflexive) St-stem sutamhurum, "to m a k e s confi^ont i t s e l f , designates the construction o f a square w i t h side s. Often, h o w e v e r , o n e of the terms for rectangularization is u s e d instead. W i t h respect to o n e side, the other side m e e t i n g it in a c o m m o n c o m e r is c o n s i d e r e d its mehrum or " c o u n t e r p a r t " , s o m e t i m e s r e p l a c e d b y the S u m e r o g r a m gaba.^^ S o m e texts u s e L A G A B a n d N I G I N as ideograms (not necessarily l o g o g r a m s , the sign L A G A B being a square) in the same fimctions as sutamhurum and/or mithartum. S o m e series texts u s e ib.sig in the same fimction;^^ however, the n o r m a l fimction of this i r t p o r t a n t t e r m is different. UL.UL

O r i g m a l l y it is a S u m e r i a n finite v e r b form, a n d it often occiurs as a v e r b in p h r a s e s " ^ . e s ib.sig", w h e r e Q a n d s are n u m b e r s , s = •Jq . / . e / is the " e r g a t i v e " or "locative-terminative" suffix, sig a v e r b s t e m m e a n m g " b e i n g e q u a l " ; / i b . / c o m b m e s a m a r k o f finiteness / 1 / w i t h the "inanimate p r o n o m i n a l e l e m e n t " fbl - s, m d e e d , is n o p e r s o n . I n total, the p h r a s e therefore has to b e translated " b y Q, s is equal".^** T h e m e a n i n g is that w h e n the area Q is laid out as a square, it is flanked b y s as side of this square (equal of coiu-se to the other sides). like substitutions like that of sutâkulum for sutakûlum constitute the very fundament for the cuneiform writing system and should not surprise us. Its primary reference is thus the square frame, parametrized by the side, not as with Euclid that (area) which is contained by the boundary. meffrum is derived from the same verb as mithartum etc. gaba, on its part, is wholly independent, meaning rather "identical copy". Anybody who has tried to translate a technical terminology from one language into another will have observed that etymologically related terms often end up having unrelated translations, whereas it is a rare luck to find etymologically related translations for original terms which though semantically related are etymologically unrelated. Anticipating what is to follow we may conclude that the mitlfartum/sutamhurum/meffrum-tçrmmology is likely to have originated in Akkadian. A single procedure text, moreover (YBC 6504), uses ib.sig in parallel with du^.du^, in a construction where it might function logographically for sutaml}urum. In BM 15285, a catalogue text concerned with the subdivision of squares into other geometric figures, it alternates with mitl}artum as a term for the square configuration. In earlier publications I have blindly accepted the interpretation of .e as an ergative suffix, which would give the phrase the meaning "(caused) by Q, s is made equal". This was originally proposed by Thureau-Dangin, and was indeed the only possibility within the arithmetical reading- the number 9 cannot meaningfully be claimed to be close to the number 81.

128

J. Heyrup

O t h e r texts h a v e left b e h i n d the etymology, and u s e ib.sig as a noim, the n a m e for this side (with J o r a n Friberg w e m a y call it "the equalside") - so to s p e a k the g e o m e t r i c a l equivalent o f a square root.^^ Q u i t e a few o f the texts that use the t e r m as a n o u n e m p l o y the h o m o p h o n i c (unorthographic) writing ib.si. O t h e r texts use ba.si or ba.sig (the latter form also occiu-s as a v e r b ) ; the shift sig > si simply s h o w s that the e t y m o l o g y is n o longer thought of, b u t the pronimciation still Sumerian;^^ /ba/ is an alternative prefix w h i c h p r o b a b l y contains a locative element /a/, indicating that the p r o c e s s takes p l a c e "(out) t h e r e " . ^ ' A t times, the n o i m a p p e a r s as a l o a n w o r d basûm, further p r o o f that the p r o n u n c i a t i o n r e m a i n e d Sumerian.^^ T h e function o f the ib/ba.sig is not restiicted to the case of squares. It m a y also refer to the side of a c u b e , w h i c h should not disturb u s - e v e n the sides of a c u b e are equal a n d close to the voliune they contain. B u t true generalizations also occur, s o m e w h a t similar to o u r reference to the " r o o t ( s ) " of a n equation, d e r i v e d in tangled w a y s from the c o n c e p t o f a square root.

D i v i s i o n , p a r t s a n d igi Division, as w e u s e the w o r d , is b o t h a problem

- to solve the e q u a t i o n bx = a - a n d

an operation. T h e familiar assertion that division d o e s not exist in B a b y l o n i a n m a t h e m a t i c s refers to the a b s e n c e of division as a n arithmetical o p e r a t i o n o f its o w n . A s is well k n o w n , the w a y a division problem

is dealt with d e p e n d s o n w h e t h e r

the divisor b is regular or not, that is, w h e t h e r its reciprocal has a finite e x p r e s s i o n in the sexagesimal system. T h e reciprocal of such a n u m b e r n is called igi n gal.bi ("[of 1,] its igi « gal"^'), often shortened to igi n gal or igi n. W h e n a division b y such a n u m b e r n is to b e performed, the texts ask for igi n to b e " d e t a c h e d "

{patârumIAui),

after w h i c h the d i v i d e n d is "raised t o " the igi. In m o d e m terms, division b y n is thus p e r f o r m e d as multiplication b y J/ . A couple o f m a t h e m a t i c a l texts from E s h n i m n a

Even when no declination elements allow us to distinguish cases, the word order shows whether a verb or a noun is meant, both Sumerian and Akkadian being verb-final, en.nam ib.sig thus means "what is equal?", whereas ib.sig en.nam means "the equalside, what?". Thureau-Dangin, in contrast, believed ib.sig to be a logogram that was pronounced mithartum; in a few cases {viz when the reference is the square configuration) this was certainly the case, but mostiy not. The prefix /ba./ is often used regularly with the identity-conserving subtraction zi, "to tear out"; in contrast, the identity-conserving addition dah, "to append", is often preceded by lb\.l, which suggests a "here", a closer contact. Some publications use the transliteration îb.sâ instead of ib.sig. Later lexical lists, indeed, render the pronunciation of the Sumerian verb as sa-a; the preference of most editions of mathematical texts for sig is supported by the occasional homophonic shifts to ib.si and ib.si, which however might as well be ib.se and ib.sé. Since we also find the syllabic writing ba.se.e (which can not be ba.si.i) for ba.sii (IM 52301 rev. 7, 9), the real phonetic value might be between -se and -sa. Strictiy speaking, early tables of reciprocals show that the meaning is "[of T,] its igi n gal". These tables, indeed, list /^, /^, X etc. of sixty = V - see STEINKELLER (1979), p. 187. In later times, genuine reciprocals in our sense may have been thought of

How to Educate a Kapo

replace igi w i t h pa-ni,

"in front o f w h i c h

129

suggests a reference to the table o f

reciprocals, w h e r e igi n is indeed p r e s e n t " i n front o f ' n - a folk e t y m o l o g y close at h a n d , not least b e c a u s e gal m e a n s " t o be/place ( s o m e w h e r e ) " , " t o b e at disposition". H o w e v e r , since texts from L a g a s from c. 2 4 0 0 BCE s p e a k o f ^ » X

X

igi n

gal, it is clearly n o t h i n g but a s e c o n d a r y folk etymology, the o p i n i o n o f E . M . Bruins notwithstanding.^' W e also fmd an A k k a d i a n l o a n w o r d igûm. If the divisor is not (or carmot easily b e seen to b e ) regular (a recurrent situation in m a t h e m a t i c a l p r o c e d i u e texts t h o u g h hardly in practical c o m p u t a t i o n , all relevant technical coefficients b e i n g c h o s e n to b e regular), the text takes n o t e of the n o n existence o f the igi a n d t h e n formulates the division as a p r o b l e m - " w h a t m a y I posit (sakânum/gar)

to b w h i c h gives (nadânum/sum)

immediately, "posit p; raise pio

b,a

m e a" - a n d states the a n s w e r

it gives to y o u " (or s o m e slight variation on this

pattern). M a t h e m a t i c a l p r o b l e m s b e i n g constructed b a c k w a r d s from the solution, this could always b e d o n e without difficulty. Bisection W h e n dividing b y a niunber which, so to speak, is 2 only " b y accident", o u r texts find "igi 2 " = 3 0 ' a n d " r a i s e s " to that number. B e y o n d that, a particular sign a n d a c o r r e s p o n d i n g set of w o r d s (ww/i/m/su.ri.a) for the half exist. T h e y are u s e d , for instance w h e n the w i d t h o f a rectangle is said to b e half its length; if o n e entity is said to e x c e e d a n o t h e r b y its half; or to m d i c a t e a m e a s u r e ("half a b a r l e y c o m " ) . S u c h relations a n d m e a s u r e s are "accidental": they m i g h t as well h a v e b e e n slightiy different. B e s i d e s this accidental half, however, a different, " n e c e s s a r y " h a l f (invariably half of something, n e v e r a n u m b e r - n a m e l y of s o m e t h i n g w h i c h naturally or due to the nature o f the case falls in t w o halves) is u s e d in the texts, the bâmtum (no S u m e r o g r a m s e e m s to exist^^). It occurs in p l a c e s w h e r e s o m e t h i n g is b i s e c t e d into two necessarily e q u a l p a r t s : for instance w h e n the b a s e o f a friangle is bisected for the p u r p o s e of a n area calculation; w h e n the s a m e is d o n e to the s u m o f o p p o s i n g sides in a frapezium; a n d w h e n the radius is found from a cfrcular diameter. T h e " n e c e s s a r y h a l f is invariably found b y " b r e a k i n g " (hepûmfgaz). This verb, o n the other h a n d , has n o other function in the m a t h e m a t i c a l texts; it always g o e s ^° For instance Haddad 1 0 4 . The lexical form pa-nu occurs in a text from Sippar ( B M 9 6 9 5 7 + V A T 6 5 9 8 , in R O B S O N ( 1 9 9 9 ) , p. 2 3 1 ) - but as an interlinear gloss, which suggests that it may have been intended as an indication of pronunciation and nothing more. ^'

BRUINS ( 1971

), p. 2 4 0 , and elsewhere.

" Non-presence of a term in the extant text material is in general a weak argument. But absence from texts where it should have occurred is strong. Such texts are Str 3 6 6 , 3 6 7 , 3 6 8 , all of which insist on writing everything with Sumerograms (except grammatical particles that do not exist in Sumerian), although in clearly Akkadian sentence structures: they eschew the word (Str 3 6 6 ) , or they use the number sign '/2. This is also done in VAT 7 5 3 2 and VAT 7 5 3 5 . The text YBC 6 5 0 4 (already mentioned for its unique use of ib.sig in the function of sutam(}urum, "to make confront itself) has recourse to Su.ri.a. B A and B A . A , used in the function of bâmtum, are taken in (MKT) to be Sumerograms; actually, we are confronted with elliptic writings or, more likely, with irregular assimilations to the pronominal suffix -su, bâssu < *bâmat-su; the noun *bûm (MCT, p. 1 6 1 ) , constructed backwards from similar forms, should also be bâmtum.

130

J. Heyrup

together w i t h bâmtum (or with Vz or su.ri.a in the rare texts w h e r e these are u s e d logographically for bâmtum). In the m a t h e m a t i c a l texts (but only h e r e ) , " b r e a k i n g " is thus the s a m e as bisection.

Structure and metalanguage T h e s e m a t h e m a t i c a l operations are u s e d within texts that are o r g a n i z e d with a particular higher-order structure hypothetical-deductive

enunciation a n d exposition o f the p r o c e d u r e ,

arrangements,

relation

between

"true"

entity

and

representative - a n d w h i c h m order to express that structure m a k e u s e o f w h a t w e m a y call a m e t a l a n g u a g e - n a m e s for u n k n o w n quantities, t e r m s that indicate equality, t e r m s that delimit expressions ( c o r r e s p o n d i n g to the p a r e n t h e s e s of o u r algebraic expressions), terms that annoimce r e s u h s , etc. O f i m p o r t a n c e for the p r e s e n t argiunent are the following categories. Structuration T h e m a t h e m a t i c a l procediure texts are a r r a n g e d into statement a n d prescription, with a c o r r e s p o n d i n g distribution o f grammatical p e r s o n a n d tense. A t face value, the structure c o r r e s p o n d s to the scene of the scribe school as k n o w n from other soiu-ces: the m a s t e r states a p r o b l e m ("I h a v e d o n e so a n d s o " ) , n e x t the instructor or "big b r o t h e r " explains in the p r e s e n t tense or the imperative w h a t " y o u " (the student) should d o , referring occasionally to w h a t was said b y " h i m " (the m a s t e r ) . A g r o u p o f texts from E s h n u i m a from the early eighteenth century BCE, however, start b y asking "If s o m e b o d y has a s k e d y o u thus, T h a v e d o n e so a n d s o ' " , suggesting instead that the format b e l o n g s originally with mathematical riddles. Early texts from the south, on the other h a n d - b o t h m a n y o f those w h i c h b e l o n g to g r o u p 1^^ a n d t h o s e from early O l d B a b y l o n i a n U r - deviate from the p a t t e r n that prevails e l s e w h e r e . A l l in all, the format thus s e e m s n o t to h a v e originated as a porfrait o f the scribe s c h o o l b u t to h a v e b e e n i m p o r t e d into it; b e l o w (text aroimd n o t e 55) this c o n c l u s i o n will find fiirther confirmation. Within this format, the logical structure o f the texts a n d the w a y to c o n p o s e fairly u n a m b i g u o u s m a t h e m a t i c a l expressions is also standardized, t h o u g h not uniformly in all text g r o u p s . Occasionally, the hypothetical-deductive structure o f the p r o b l e m is m a d e explicit b y a n infroductory summa, " i f - also familiar as the o p e n i n g of the p r o t a s e s of o m i n a ("if the liver l o o k e d so a n d s o " , etc.). Since it is found in m o s t o f those texts from E s h n u n n a that d o not carry the fiill "if s o m e b o d y has a s k e d y o u t h u s " ( a n d in this fimction a l m o s t exclusively in texts s t e m m i n g from the s a m e n o r t h e m p a r t of B a b y l o n i a ) , the initial summa m a y originally h a v e b e e n a vestige o f that p h r a s e . In one text from Eshnurma ( H a d d a d 104), however, as in certain other texts, summa is used to mfroduce variations o f an e x e n p l a r , carrying thus the m e a n i n g "if (instead the situation is as follows:)", summa m a y also serve to infroduce a smaller p i e c e of deductive r e a s o n inside the prescription from a l r e a d y established foundations ("if (as you h a v e n o w established) . . . " ) , or to o p e n the proof; it c a n b e found in either

Thus AO 6770, AO 8862, YBC 7997, YBC 9856, YBC 9874.

How to Educate a Kapo

131

function b o t h in texts firom Eshnuima and other localities in the p e r i p h e r y and in texts from the ft)rmer S u m e r i a n heartland.^'* inuma, " a s " , is u s e d in a couple of texts firom the p e r i p h e r y to m a r k a p i e c e of deductive r e a s o n i n g o n already established foundations, assum, " s i n c e " , has the s a m e function in s o m e texts from the p e r i p h e r y and in s o m e firom the core (in the p e r i p h e r y s p e c i m e n s it m a i n l y serves as the o p e n i n g p h r a s e o f the p r o c e d u r e prescription).^^ It m a y also b e u s e d in connection with quotations from the statement w h i c h serve to justify the pertinency of single steps in the p r o c e d u r e . T h e w h o l e q u o t a t i o n is contained in the p h r a s e assum ... qabûku/iqbû, "since it is said to y o u " / " s i n c e h e h a s said". In certain cases, the statement falls into two sections, the first of w h i c h contains general information - the value of a technical constant to b e u s e d , the rent to b e p a i d p e r bur of a field, etc. - w h e r e a s the second presents the actual p r o b l e m . T h e s e c o n d section m a y then b e introduced b y the w o r d inanna, " n o w " . E v e n the prescription m a y b e divided explicitly into subsections. S u c h divisions m a y b e m a r k e d b y the verbs sahârum " t o turn aroimd", târum, " t o t u r n b a c k " , or nigin(.na), found in the late g r o u p 6 (fi-om Sippar) and p r o b a b l y m e a n t as a l o g o g r a m for târum (this t e r m occurs in related texts b u t sahârum not). A t least in the text A O 8 8 6 2 , sahârum a p p e a r s to b e used in the statement a b o u t a quite c o n c r e t e walk a r o u n d a field w h i c h h a s j u s t b e e n m a r k e d out, and târum a b o u t a return to the starting point, târum is u s e d in a similar w a y in texts firom several text g r o u p s ; it s e e m s n o u n r e a s o n a b l e guess that this concrete application m a y h a v e b e e n the origin of the abstract u s a g e as a textual delimiter in the prescriptions. M a n y p r o b l e m s - particularly those o f " a l g e b r a i c " character - are shaped as equations, statements that (the m e a s u r e of) a m o r e or less c o m p l e x quantity equals a n u m b e r ; also p r e s e n t t h o u g h less c o m m o n are equations that declare that the (measure of) o n e quantity equals (the m e a s u r e of) another quantity. In the former case, the equality is mostly implied b y the enclitic particle -ma o n the verb. In the latter case, the t e r m kïma, "as m u c h a s " , m a y b e found; in the series text b u t n o w h e r e else it is written logographically as gin7.(nam), the S u m e r i a n equative suffix. A t e r m for equality that m a y serve as a kind of b r a c k e t w h e n c o m p l e x quantities are constructed verbally is mala, " s o m u c h a s " . It is found in the e x p r e s s i o n " s o m u c h as a over b g o e s b e y o n d " , m e a n i n g {a-b). In the series texts it is replaced b y the S u m e r i a n mterrogative p r o n o u n a.na. T h e n u m e r i c a l value of a quantity Q m a y b e asked for in t w o w a y s , either b y the question mînûm Q, " w h a t (is) Q", with logographic equivalent e n . n a m , or b y the question Q kï masi, " Q c o r r e s p o n d i n g to what?".^^ Collective questions for each o f several values m a y b e asked for with the question kiyâ, " h o w m u c h e a c h " . A division of the text corpus into "northem" and "southern" types was originally undertaken by Goetze, who based himself on orthographic criteria (almost ail mathematical texts known in 1945 came from illegal diggings and were thus of unknown origin). In my (2000) the inclusion of the text groups fi-om Eshnunna and Susa led me to replace Goetze's division by a distinction between the (former) "Sumerian core" and its "periphery". Not, however, in texts beginning "if somebody In the division question, the accusative mïnâm is used, "what may I posit to Similarly when a geometric square root is asked for by the verbal construction, "by Q, what is

132

J. Heyrup

"Variables" W h a t p e r m i t s us to s p e a k of an Old B a b y l o n i a n " a l g e b r a " is the existence o f a standard representation, a mathematically structured d o m a i n onto w h i c h p r o b l e m s dealing w i t h entities b e l o n g i n g to other d o m a i n s b u t entering in m a t h e m a t i c a l relations that are h o m o - or i s o m o r p h i c with t h o s e of the standard d o m a i n m a y b e m a p p e d a n d then solved b y analytic p r o c e d u r e s (implying that in p r a c t i c e the standard d o m a i n is r e d u c e d to its m a t h e m a t i c a l structure, a n d that it is thus functionally abstract). In the O l d B a b y l o n i a n case, the standard d o m a i n is that of squares and r e c t a n g l e s " with m e a s u r e d or measiu-able sides and c o r r e s p o n d i n g areas. In a niunber of p r o b l e m s , these sides represent prices (actually inverse prices, quantity p e r m o n e t a r y unit), w o r k e r s a n d w o r k i n g days, n u m b e r s , or e v e n areas or volumes as in m o r e recent school algebra, however, m o s t extant p r o b l e m s deal with the b a s i c representation itself T h e n a m e s of the entities b e l o n g i n g to the standard d o m a i n are, for rectangles, us ("length"), sag ( " w i d t h " ) , a.§à ("surface"), a n d for squares mithartum ("equalside") and a.sà ("surface"). Outside their function as standard representation, u s a n d sag are l o g o g r a m s w h i c h m a y b e r e p l a c e d b y the syllabic equivalents siddum a n d pûtum (and, in other coimections, b y other w o r d s - sag thus by rësum, " h e a d " ) . W i t h e x c e e d i n g l y few exceptions, this d o e s not h a p p e n w h e n they serve within this function. Similarly, a.sà is n e v e r r e p l a c e d b y the syllabic eqlum; in this case, however, the l o g o g r a m m a y b e p r o v i d e d with A k k a d i a n p h o n e t i c elements. I n the case of square configurations a n d sides, the situation is different ( t h e n areas are still a.sà). T h e configuration may b e referred to as LAGAB or ib.sig, b u t mithartum occurs quite as often. A s to the value o f the side itself it m a y also b e s p o k e n of as mithartum; as LAGAB p r o v i d e d w i t h a p h o n e t i c c o m p l e m e n t that identifies the A k k a d i a n pronunciation; or it m a y said that s "confronts i t s e l f (imtahhar).^^ A set o f curiously related catalogue texts from E s h n u n n a , S u s a a n d elsewhere'*" refer to the side of the configuration as us, b u t with a p h o n e t i c

equal". When the "equalside" is regarded as a noun, it may be asked for by the nominative mînûm, or the student is asked to make it come up", A basâsu suli, or to "take" it {laqûm). Certain problems dealing with prismatic excavations are of the third degree. However, as their constituent elements never serve to represent entities of other kinds, they do not belong to the standard domain; instead, those excavation problems which are of the second or first degree are represented themselves within the standard representation. A distinction between "true width" and "width" in the Susa text TMS XVI could also mean that the width 20 of a real field {viz 20 nindan) is represented by a standard-domain width 20 {viz 20' nindan, fit for the dimensions of the school yard). But this remains a hypothetical interpretation of a passage which seems not to be explainable in any other way. In general, there is no positive evidence that the distinction between "standard-domain" and "real" fields, fairly well-respected in the practice revealed by the texts, was also formulated as a principle. Or, instead, asked "how much, each, confronts itself, with the particular interrogative particle làyâ. IM 52916, IM 52685+52304 (Eshnunna), TMS V, TMS VI (Susa), CBS 154+921 (unknown but certainly different provenience).

How to Educate a Kapo

133

c o m p l e m e n t that identifies it as siddum. T h r e e texts'*' in contrast, s p e a k of the sag or pat, plural " w i d t h s " , of square configurations, but at least the first t w o m a y b e argued to s p e a k of " r e a l " , not standard fields. In m o d e m letter-based algebraic c o m p u t a t i o n s it is often m n e m o t e c h n i c a l l y convenient a n d therefore c u s t o m a r y to label derived variables w h i c h s o m e h o w fiilfil the s a m e fimction as the initial ones b y s o m e kind of m a r k i n g - x for x, etc. T h e B a b y l o n i a n s u s e d several similar tricks - distinguishing for instance b e t w e e n entity and " t m e " (kïnum/gi.m) entity or b e t w e e n entity a n d "false" (sarrum/hû) entity, or referring to the w a y the n e w entity is p r o d u c e d . E v e n within g r o u p s of closely related texts, however, n o imiform p a t t e m for d o i n g so c a n b e identified, n o r are " t m e " a n d " f a l s e " u s e d exclusively in this fimction within the texts. All devices o f the k i n d were clearly as m u c h ad hoc as o i u p i c k i n g of X, ^ , U O T X for " n e w x ". Recording N i u n b e r s o c c u r as data in the statement, as intermediate o u t c o m e of calculations, a n d as final results. Several terms a n d phrases m a y b e u s e d in this coimection. O f particular i m p o r t a n c e is the v e r b sakânum, "to posit", w i t h l o g o g r a m gar. T h e t e r m a p p e a r s to designate various kinds of material r e c o r d i n g - "putting d o w n " in a c o m p u t a t i o n a l s c h e m e , writing the value of a length or a n area into a d i a g r a m , etc. It is m a i n l y u s e d in two fimctions: to take note o f data in the b e g i n n i n g o f the prescription, a n d thus to p r e p a r e their u s e ; and in the formulation o f the division p r o b l e m " w h a t m a y I posit to b w h i c h gives m e a", with the a n s w e r " p o s i t p; raise p to b, a it gives to y o u " (with m i n o r variations). In the latter case, insertion into a c o m p u t a t i o n a l s c h e m e is likely to b e meant."*^ S o m e texts also " p o s i t " a n " e q u a l s i d e " and its "counterpart", i.e., t w o sides of a square; the s a m e p r o c e s s is indicated in other texts b y the verbs lapâtum, "to inscribe", or nadûm, " t o lay d o w n " . In the g e o m e t i i c text B M 15285 there is n o d o u b t that the actual sense of the latter t e r m is to lay d o w n in drawing; m o s t likely, its general m e a n i n g in m a t h e m a t i c a l texts is " t o lay d o w n in writing or d r a w i n g " , lapâtum is also u s e d regularly a b o u t n u m b e r s that afterwards serve in additive a n d subtiactive operations; since these s e e m to h a v e b e e n p e r f o r m e d o n a counting b o a r d a n d not in clay,'*^ it m a y also h a v e referred to r e c o r d i n g o n such a device, w h i c h is still in a g r e e m e n t with the general m e a n i n g o f the term, " t o grasp/take hold o f ; "inscription" of coefficients, o n the other h a n d , m o s t likely refers to their b e i n g written d o w n on a tablet for r o u g h work.'*'* A specific p h r a s e for r e c o r d i n g a n (invariably intermediate) result is rëska likïl, " m a y your h e a d h o l d (it)". It seems to b e reserved for n u m b e r s that are not to b e inserted in a fixed s c h e m e a n d therefore are not "posited". T h e a p p e a r a n c e of a result m a y b e a n n o u n c e d in several w a y s . It m a y b e said that a n u m b e r illiakkum, " c o m e s u p for you", fi-om elûm, or that a calculation " g i v e s " a certain r e s u h {nadânum, s u m ) ; a h e m a t i v e l y , the text m a y state that " y o u s e e " the result (tammar, from amârum, "to s e e " ) . Very often, the calculation is simply

UET 864, from early Old Babylonian Ur; BM 13901, #23; and CBS 19761, from Nippur. In YBC 6504, gar is used to take note of both intermediate and final results. H0YRUP (2002b). Cf

ROBSON

(1999), p. 30.

134

J. Hoyrup

followed b y the enclitic particle -ma a n d the number, or b y n o t h i n g b u t the n a k e d number.

Third-millennium terminology We h a v e few m a t h e m a t i c a l texts from the third millenniiun, a n d o i u direct e v i d e n c e for the c o r r e s p o n d i n g technical terminology is c o r r e s p o n d i n g l y w e a k . W e k n o w that the v e r b sig w a s u s e d at least since c. 2 6 0 0 BCE to express that a s e g m e n t X w a s the side of the c o r r e s p o n d i n g square area; us, "length", sag, " w i d t h " a n d aSàs ( = G À N = IKU), " s i u f a c e " , c a n b e followed b a c k to 2 4 0 0 BCE;'*^ the u s e o f the p h r a s e igi n gal is also d o c u m e n t e d for n = 3 , 4 a n d 6 since c. 2 4 0 0 BCE ( c f a b o v e ) . T h e o n l y m a t h e m a t i c a l d o c u m e n t s from the U r III p e r i o d that contain t e r m s for m a t h e m a t i c a l operations are the tables of reciprocals a n d o f multiplication, o f w h i c h the former u s e igi n gal and the latter a.râ, "steps o f T h e stable a n d invariably S u m e r i a n t e r m i n o l o g y o f the tables of square a n d c u b e roots, ib.sig a n d ba.sig, allows us to c o n c l u d e that these terms, too, will h a v e b e e n u s e d already in U r III.'*^ T h e other m a t h e m a t i c a l d o c u m e n t s from the e p o c h , accoimts a n d m o d e l d o c u m e n t s , only give results, and tell neither the details of calculations n o r the t e r m i n o l o g y in w h i c h these w e r e s p o k e n about. A h y m n in the praise of K i n g Sulgi relates that the scribe school is a p l a c e w h e r e zi.zi.i gâ. g â are l e a m t together with sid, " c o u n t i n g " , a n d nig.sid, "accounting".'*^ T h e use o f the r e d u p l i c a t e d forms gâ.gâ a n d zi.zi.i m a y p e r h a p s d e p e n d on the context (description of a habitual practice a n d not of the single operation), gâ.gâ is the marû ( a p p r o x i m a t e l y = imperfective/durative) stem of gar, " t o place",'*' later u s e d logographically for sakânum, "to posif'; in g o o d a g r e e m e n t w i t h the m e a n i n g o f kamàrum for w h i c h gar.gar is u s e d logographically in the O l d B a b y l o n i a n age ( n a m e l y " t o p l a c e in layers, to a c c u m u l a t e " ) , gâ.gâ m a y thus b e u n d e r s t o o d as " o n g o i n g p l a c i n g " - b u t also as "habitual placing", zi.zi is the marti s t e m o f zi, " t o rise, to stand up",^° a n d m a y p e r h a p s b e i m d e r s t o o d as "take u p from" - n o t t o o far r e m o v e d from nasâhum, "to tear out", for w h i c h zi is u s e d logographically in O l d

Texts in A L L O T T E D E L A F U Y E (1915), pp. 124-132. For non-rectangular fields, these surveying texts distinguish us and us 2.kam, "2nd length", and sag an.na and sag ki.ta, "upper" and "lower width". The equality of (e.g.) lengths is expressed u§ sig. a.§à is used about the area in Sargonic texts (Whiting (1984), p. 69). Until very recently it was impossible to establish with certainty that any of the extant specimens were really of Ur III date; I have now been told by Eleanor Robson (personal communication) that at least tables of reciprocals have been found in dated UR-III contexts. The aberrant use of ba.sig in Eshnunna and Ur, it is true, could suggest that the distinction which all other text groups uphold between ib.sig and ba.sig is a secondary development, and perhaps (namely if we believe that early Old Babylonian Ur is a better witness of Ur-lII usages than Eshnunna, only submitted to Ur III until 2025) that the form originally connected with the function as a verb was ba.sig. In Eshnunna, it might then have displaced ib.sig even when used as a noun; elsewhere, the term of the tables might have got the upper hand. *^ "èulgi-Hymn B", 1.17, ed. SLa, §305. *° SLa, §322.

CASTELLINO

(1972), p. 32.

How to Educate a Kapo

13 5

B a b y l o n i a n t e x t s / ' n o r h o w e v e r t o o close, g a r o r g â . g â a n d zi o r zi.zi m a y therefore h a v e b e e n t h e s t a n d a r d t e r m s for a d d i t i o n a n d subtraction in the U r III s c h o o l ; it is not to b e e x c l u d e d , h o w e v e r , that the reference is m o r e specifically to " p l a c i n g " o n the coimting b o a r d a n d " t a k i n g u p " firom it. O t h e r t e r m s a r e n o t m e n t i o n e d in the text, n o t e v e n a t e r m for multiplication, e v e n t h o u g h m u l t i p l i c a t i o n w a s certainly a c o r n e r s t o n e in the a c c o i m t i n g system. H o w e v e r , the relation b e t w e e n syllabic a n d l o g o g r a p h i c writings o f t e c h n i c a l t e r m s in O l d B a b y l o n i a n m a t h e m a t i c a l texts m a y p e r m i t u s to a p p r o a c h the q u e s t i o n from tìie o p p o s i t e side. First there

are the t e r m s that are written

invariably

(or a l m o s t

so)

with

S u m e r o g r a m s : u s , sag (including sag a n . n a a n d sag ki.ta) a n d a.§à, w h e n the " l e n g t h s " , " w i d t h s " and " s u r f a c e s " o f q u a d r a n g u l a r a n d triangular fields are meant;^^ and a.râ. To these c o m e those w h i c h o c c i u a l t e m a t i n g l y as S m n e r o g r a m s a n d as A k k a d i a n l o a n w o r d s ( w h i c h indicates that t h e y w e r e really s p o k e n with the S i u n e r i a n p h o n e t i c v a l u e ) : igi/igûm

(with the c o g n a t e s igi.hi/igibûm

and

igi.guh/igigubbûm);

a n d ib.sÌ8/ib.si/ba.SÌ8/ba.se.e/.../òfl5z2m w h e n n o t u s e d l o g o g r a p h i c a l l y for the s q u a r e configuration. B o t h of these categories, t h o u g h u s e d in the a l g e b r a i c texts, h a v e their r o o t s in a m u c h simpler a n d m u c h m o r e utilitarian calculational p r a c t i c e . T h e n there are t e r m s that m a y a p p e a r in syllabic A k k a d i a n as w e l l as in logographic

writing:

wasabum/dah,

"to

append";

/rawârww/gar.gar/UL.GAR,

a c c u m u l a t e " , with kumurrûm/kimrâtum/gar.gar/VL.GAK, zi, " t o tear out"; eli... r a i s e " ; esëpum/tab, hold

each

watârum/uga

"to

nasâhum/

... dirig, " o v e r ... g o b e y o n d " ; nasûm/iVmm,

"to

"to d o u b l e " ; 5MTOA:M/Mm/i.gU7.gU7/DU7.DU7/UR.UR/NIGIN, " t o m a k e

other";

sutamhurum/mGlN,

"to

miY^arrM/n/LAGAB/NLGLN/ib.sig; patârum/dus, posit", w i t h nadânum/sum, mislum/su.n.a,

"accumulation";

make

confiront

itself,

with

"to d e t a c h " ( a n igi); sakânum/gar,

"to

"to g i v e " , in the a n s w e r to the " d i v i s i o n q u e s t i o n " ;

" a h a l f (together with other t e r m s for simple

"to b r e a k " , t o g e t h e r witii bâmtum,

fi-actions);

hepûm/gaz,

the "natural h a l f , for w h i c h the l o g o g r a m for '/2

or su.ri.a a p p e a r in a small n u m b e r o f texts that insist o n writing as m u c h as at all p o s s i b l e with l o g o g r a m s . Finally there is a cluster o f t e r m s that o c c u r o n l y in A k k a d i a n , o r logograms

are either

late

or a r g u a b l y n o t S u m e r o g r a m s

proper but

where

artificial

constructions w h i c h m a y h a v e b e e n r e c o g n i z e d as s u c h at the t i m e . O n l y o n e t e r m for a n o p e r a t i o n b e l o n g s to this cluster, n a m e l y harasum, o f f . It is m o r e c o m m o n t h a n nasâhum,

" t o cut

" t o tear out", in the early A k k a d i a n g r o u p s

( " g r o u p 1" a n d the E s h n u i m a g r o u p ) b u t a p p e a r s t o h a v e fallen in disuse afterwards, p e r h a p s b e c a u s e it o c c u p i e d the s a m e c o n c e p t u a l n i c h e as nasâhum

w h i c h w a s well

p r o v i d e d w i t h a s t a n d a r d l o g o g r a m (zi). I n contrast, m o s t t e r m s b e l o n g i n g t o the m e t a l a n g u a g e a r e o f this k i n d (those designating variables not, as d i s c u s s e d a b o v e ) . A c c o r d i n g to O l d Babyloitian lexical lists, summa,

"if,

could be replaced by

HT.da ("at t h e d a y w h e n " ) or tukun.bi, o f w h i c h at least the former is n o m o r e

zi.zi is used in the unpublished text IM 121613 (courtesy of Joran Friberg and Farouk alRawi) but does not appear in any published Old Babylonian text. In contrast, siddum may replace u§ when the length of a wall or a carrying distance is meant; similarly, sag may occur as rësum, "head", when an initial value is intended, and is mostly substituted by the latter when an intermediate result is to be kept in memory. This excludes that the logogram is used simply because it is more easily written.

136

J. Hoyrup

difficult to write t h a n the syllabic writing; but in all its mathematical functions, the w o r d always remains syllabic. inuma, " a s " , m i g h t also h a v e b e e n replaced b y U4.da; assum m i g h t h a v e b e e n m u , as it actually h a p p e n s in the Late B a b y l o n i a n b u t pre-Seleucid m a t h e m a t i c a l text W 2 3 2 9 1 ; b u t m O l d B a b y l o n i a n mathematical texts neither is ever written logographically. inanna, " n o w " , u s e d to divide general from specific information in the statement, could h a v e b e e n i.ne.sè - b u t it always remains syllabic. S o d o the terms that serve to demarcate divisions within a prescription (sahârum, "to t u m aroimd", and târum, "to turn b a c k " ) , except in the late texts from Sippar. Lexical lists give nigin as well as nigin for either. -ma, s o m e t i m e s u s e d to form " e q u a t i o n s " w h e r e the right-hand side is a number, has n o p r o p e r S u m e r i a n equivalent, and often a p p e a r s within texts that are otherwise written in m o r e or less grammatical Sumerian. kîma, "as m u c h a s " , the indication that t w o n o n - n u m e r i c a l quantities are equally great, is replaced m the late series texts but n o w h e r e else b y gin7.(nam), the Sumerian equative suffix. A l s o m the series texts but n o w h e r e else, the "algebraic b r a c k e t " mala, " s o m u c h a s " , a p p e a r s as a.na. A m o n g the interrogatives, mînûm, "what", is written logographically as e n . n a m in the early U r texts U E T V, 1 2 1 , 859 and 864; in several later texts ( g r o u p s 3 and 6, and s o m e scattered texts) the same equivalence is used; it is absent from Eshnimna. F o r the accusative mïnâm, U E T 859 uses a.na.àm o n c e , and so d o e s I M 5 5 3 5 7 from Eshnuima (asking for " w h a t is equal", that is, for the side of a square), a . n a . à m is also given b y lexical lists, and is regular S u m e r i a n m e a n i n g " w h a t is i f (SLa, § 1 2 0 quotes it from Inanna's Descent); there is hence n o n e e d to postulate any direct link b e t w e e n the two a p p e a r a n c e s , e n . n a m as far as I h a v e b e e n able to find out, has n o antecedents before the O l d B a b y l o n i a n e p o c h ; moreover, ( A H w , p . 6 5 6 a ) only r e c o r d s it as u s e d in mathematical texts. All in all, it s e e m s to b e a n ad hoc construction m a d e in the context o f early O l d B a b y l o n i a n m a t h e m a t i c s teaching, p r o b a b l y in Ur.^^ kï masi has n o logographic equivalent in the m a t h e m a t i c a l texts, even t h o u g h the smiilar phrase mala masi u s e d in legal texts ( A H w , 62 Ib)^'* looks like a n A k k a d i a n franslation o f a.na.àm. kiyâ, the question for several v a l u e s , is similarly deprived o f logographic equivalent (but s e e m s to b e so in general, n o t only within the m a t h e m a t i c a l corpus). Since the semantic b o u n d a r i e s b e t w e e n sakânum, lapâtum and nadûm is difficult to establish, the a p p a r e n t absence of logographic writings for the last t w o terms m a y not b e significant - w e cannot exclude that gar is u s e d m o r e b r o a d l y than sakânum (though n o positive evidence suggests it to b e ) . M o r e interesting are the t e r m s that a n n o u n c e results. O f these, nadânum, "to give", w a s already m e n t i o n e d for h a v i n g the s u m in coimection with the "division question". In this function, the t e r m all text g r o u p s , either in logographic or in syllabic writing; as a general results it is only found in g r o u p 3 and (as nadânum) in A O 6 7 7 0 ( g r o u p 1) two texts Y B C 4 6 6 2 and 4 6 6 3 , in which s o m e slips indicate that it

logogram is u s e d in marker of a n d in the has been

Since it occurs together with a.na.àm in UET V, 859 but in a distinct function, it is unlikely to be a mere phonetic writing of that term. It is also found (in the subjunctive form mala masû) in the mathematical text VAT 7531.

How to Educate a Kapo

137

introduced as a substitute for a p h r a s e tammar u s e d in a n original firom w h i c h they w e r e c o p i e d with terminological revisions. In m y ( 2 0 0 0 ) , p . 1 6 5 , I s i u n m e d u p the conclusions that c a n b e d r a w n from this disfribution as follows: T h e origin of nadânum as a m a r k e r o f results is with multiplications in the sexagesimal system (and thus with the post-Sulgi fradition); t h e m a i n u s e o f the t e r m in texts w h e r e it d o e s n o t serve for results in general is in the division question; s e c o n d c o m e s the indication o f the result o f " r a i s i n g " multiplications; thirdly that o f final results. It is p r o b a b l y significant that e v e n g r o u p 3 tends to u s e the syllabic forms in this original d o m a i n . It is also significant that the u s e o f nadânum goes together w i t h the only systematic b r e a k i n g of the n o r m a l switch b e t w e e n g r a m m a t i c a l p e r s o n s . ^ ' T h i s switch b e t w e e n the statement formulated b y a n " I " a n d a prescription of w h a t " y o u " should d o (at times involving q u o t e s from w h a t " h e " h a s said) b e c o m e s a general characteristic o f m a t u r e O l d B a b y l o n i a n m a t h e m a t i c s , the only texts that h a v e s o m e difficulty w i t h the principle b e i n g indeed those o f g r o u p 1. B u t if it is systematically b r o k e n in c o n n e c t i o n with a t e r m pointing t o w a r d s t h e post-Sulgi school fradition, it m u s t c o m e from elsewhere - and t h e format o f [ m a n y of the Eshnuima] texts shows h o w it fits the riddle fradition: T h e " h e " o f the prescription is indeed the " s o m e o n e " w h o h a s p o s e d the p r o b l e m and the " I " of the statement, w h e n c e different from the speaking p e r s o n o f the prescription w h o explains the p r o c e d u r e . elûm, " t o c o m e u p " , h a s n o S u m e r o g r a p h i c equivalent at all w h e n u s e d a b o u t results ( w h e n u s e d instead of nasûm, " t o raise", it is usually written nim.^^ T h i s is quite sfriking: in Eshnuima, w h e r e it is u s e d altematingly with tammar, it is i n d e e d p r e d o m i n a n t l y linked with s u c h c o m p u t a t i o n s as p o i n t t o w a r d the Ur-III scribal fradition. I n v i e w o f the invariably A k k a d i a n writing w e c a n b e fairly certain that the i d i o m of results "coining u p " is a p o s t - U r III confribution to the scribal computational fradition - p r o b a b l y m a d e in the p e r i p h e r y or the n o r t h e m p a r t of the core ( N i p p u r ) since the t e r m is absent from all texts from the core area with the e x c e p t i o n of three group-1 texts and the N i p p u r texts. tammar, " y o u s e e " , may o n rare occasions a p p e a r in logographic writings: pad ("to s e e " ) in early Ur, igi.dii in I M 5 5 3 5 7 , igi.dug ("to look a f ) in t h e series texts Y B C 4 6 6 9 a n d 4 6 7 3 , igi in VAT 6 7 2 . A p a r t from the a p p e a r a n c e in U r a n d the apparent slips in Y B C 4 6 6 2 it is totally absent from the core area w h e r e a s it d o m i n a t e s the p e r i p h e r y (in E s h n i m n a u s e d alongside with " c o m i n g u p " , " s e e i n g " b e l o n g i n g p r e d o m i n a n t l y in p r o b l e m s o f riddle character, " c o m i n g u p " a s stated a b o v e rather with the fraditional utilitarian scribal type). T h e a b s e n c e from the core is noteworthy, since the p a r l a n c e c a n b e seen to h a v e b e e n k n o w n e v e n there: Sfr 3 6 6

[Namely the question "what may / posit to b which gives me a", with answer "posit p; raise pto b,a it gives to you"]. The only Old Babylonian text which has a Sumerographic writing of "coming up" is the Nippur text CBM 12648, which appears to construct Sumerograms for everything, and which has ib.sig X Cn.dè, "make the equalside of x come up".

138

J. Hoyrup

and VAT 7 5 3 1 , b o t h from g r o u p 3 (the g r o u p in w h i c h the " g i v i n g " o f results w a s a d o p t e d generally) refer to it obliquely in phrases " m order that I see [as r e s u l t ] " , how much "do I see". T h e fluctuation o f the logographic equivalents for tammar is best e x p l a i n e d if w e suppose the A k k a d i a n p h r a s e to b e p r i m a r y a n d the l o g o g r a m s to b e franslations from this. O n e thing m u s t b e observed, h o w e v e r : results are a l r e a d y " s e e n " in a g r o u p of Sargonic school texts about rectangles a n d s q u a r e s . " In these, the w o r d is pad. This is n o p r o o f (and hardly circumstantial evidence) that the use o f pad is continuous: pad s i m p l y h a p p e n s to b e the m o s t precise S u m e r i a n franslation o f the A k k a d i a n v e r b amârum;^^ m o r e o v e r , i / p à d h a d b e e n carried b y the fradition, there w o u l d h a v e b e e n n o r e a s o n to invent a S u m e r i a n substitute in Eshnuima. B u t it is fairly hard e v i d e n c e that the A k k a d i a n a n d the O l d B a b y l o n i a n p a r l a n c e is linked in one o f the t w o l a n g u a g e s , a n d thus p r o b a b l y t h r o u g h an A k k a d i a n ( w h e n c e necessarily non-scribal) fradition - w h i c h fits the a p p a r e n t b o n d b e t w e e n " s e e i n g " and the riddle fradition perfectly well.

Educating the Kapo I shall not repeat the a r g u m e n t s I h a v e p r e s e n t e d e l s e w h e r e in favour o f the thesis that the particular p a t t e m of O l d B a b y l o n i a n m a t h e m a t i c s arose from a synthesis b e t w e e n a ttadition o f scribal c o m p u t a t i o n deriving from the Ur-III (and, only rarely to b e distmguished, pre-Ur-III^') scribal c o m p u t a t i o n a n d o n e or m o r e fraditions of non-scribal p r a c t i t i o n e r s ' m a t h e m a t i c s - the latter confributing w i t h m a t h e m a t i c a l riddles that might serve the display of professional proficiency, in a g r e e m e n t w i t h the ideals of O l d B a b y l o n i a n scribal " h u m a n i s m " (nam-lu.ulti). T h e a b o v e p r e s e n t s o n l y a small part of the e v i d e n c e ; for fiulher elaboration o f the argiunent I shall o n l y refer to m y ( 2 0 0 0 ; 2 0 0 1 ; 2 0 0 2 a ) . Instead I shall concenfrate o n what vindicates R o b e r t E n g l u n d ' s u n p l e a s a n t v i e w o f U r III - a view so u n p l e a s a n t that m a n y Assyriologists prefer not to believe in its veracity. Let us first look at w h a t h a p p e n e d outside U r w h e n the p a t t e m o f O l d B a b y l o n i a n m a t h e m a t i c s was formed. A p a r t from s o m e m o d e s t vacillation in E s h n i m n a it a p p e a r s to h a v e b e e n a c c e p t e d b y e v e r y b o d y that the s t a n d a r d m e n s u r a t i o n a l t e r m s (us, sag, including " u p p e r " a n d " l o w e r " , and a.sà) and the c o r e v o c a b u l a r y for using the p l a c e value s y s t e m (a.râ, igi, igi.gub) referred to legitimately S u m e r i a n p r a c t i c e s a n d should h e n c e b e e x p r e s s e d m Sumerian; even m E s h n u n n a , m o s t deviations from the n o r m s prevailing e l s e w h e r e take the shape of u n o r t h o g r a p h i c S u m e r i a n or l o a n w o r d s , syllabic A k k a d i a n is m u c h rarer.

"

See

W H I T I N G ( 1 9 8 4 ) , p. 5 9 ,

n.2.

By the same argument, the unorthographic igi.du, the corresponding orthographic igi.dug, and the ellipsis igi - less adequate and all presumably from the northem periphery - are more likely to be connected. As I have argued elsewhere, the use of ib/ba.sig as a verb is likely to derive from pre-UrIII usages, while the treatment of the term as a noun reflects its appearance as an "object" in the tables that became increasingly important after the Ur-III introduction of the place value system.

How to Educate a Kapo

139

A s far as m a t h e m a t i c a l operations are c o n c e r n e d , it d e p e n d s o n the g e n e r a l style of the text w h e t h e r they are s p o k e n of in syllabic A k k a d i a n or l o g o g r a m s . In a n u m b e r o f cases, l o g o g r a m s a p p e a r to b e fresh inventions, either b a s e d o n p i m s (e.g., i.gU7.gU7), o n g e o m e t r i c (e.g., L A G A B ) or s e m a n t i c - g r a m m a t i c a l similarity (reduplication of the sign r e n d e r i n g reciprocity); in others, traditional Siunerian n a m e s for o p e r a t i o n s w e r e e m p l o y e d , e v e n t h o u g h t h e n non-technical m e a n i n g did not c o i n c i d e p r e c i s e l y w i t h that of the A k k a d i a n t e r m s for w h i c h t h e y served as l o g o g r a m s , at times in spite of d i s a g r e e m e n t with the preferences of lexical lists (zi for nasâhum, gar.gar for kamàrum). In s o m e cases, l o g o g r a m s w e r e c h o s e n for w h i c h w e h a v e n o e v i d e n c e (nor, however, counter-evidence) that they h a d b e e n u s e d in earlier t u n e s as m a t h e m a t i c a l terms, but w h e r e the c h o i c e is semantically meaningful ( d a h for wasâbum, tab for esëpum, g a z for hepûm^\ Some schools may h a v e h a d syllabic writing as their stylistic preference a n d others m a y h a v e favoured l o g o g r a p h i c writing. T h e i r w a y to speak of m a t h e m a t i c a l o p e r a t i o n s w a s then g o v e r n e d b y this general c a n o n of style; the texts b e t r a y n o t e n d e n c y overriding general preferences to favour or to a v o i d referring to the o p e r a t i o n s in S u m e r i a n . W h e n fuming to the m e t a l a n g u a g e w e e n c o u n t e r a w h o l l y different situation. N o n e of the "logical o p e r a t o r s " summa, inuma a n d assum are ever written logographically; of the terms that a l l o w to formulate a specific q u e s t i o n o n the b a c k g r o u n d o f general information or to s t m c t u r e a c o m p l e x p r o c e d u r e {inanna, sahârum a n d târum), only the last is r e p l a c e d b y a l o g o g r a m , a n d this o n l y h a p p e n s in the late g r o u p 6. Similarly, those w h i c h allow the formulation o f an e q u a t i o n {-ma, kïma a n d mala) always r e m a i n syllabic outside the late g r o u p o f series texts, written in v e r y c o m p a c t p s e u d o - S u m e r i a n . A m o n g the interrogatives, kï /mala masi a n d kiyâ always r e m a i n syllabic, a n d mînûm always so except for o n e a p p e a r a n c e o f a . n a . à m in E s h n u i m a a n d from its r e p l a c e m e n t in the late texts from g r o u p 6 b y w h a t m a y at the time h a v e b e e n r e c o g n i z e d as a specifically m a t h e m a t i c a l p s e u d o - S u m e r o g r a m . Finally, a m o n g the terms that a n n o u n c e a result, only the o n e cormected to p l a c e value multiplications (that is, to a practice with a clear S u m e r i a n history) is ever written logographically, except for the utterly scarce a p p e a r a n c e s o f igi.dug for tammar, all b u t o n e of which, m o r e o v e r , are late. Since l o g o g r a m s for m o s t o f these terms were available m lexical lists ( w h e n c e fainiliar to the authors a n d users o f the m a t h e m a t i c a l texts), w e m u s t c o n c l u d e that they w e r e c o n s c i o u s l y e v a d e d . B u t these are the terms that are n e e d e d if m a t h e m a t i c s teaching is to g o b e y o n d the inculcation o f routines, that is, if it is to b e b a s e d on p r o b l e m s . W e m u s t conclude that the authors o f the m a t h e m a t i c a l texts c h o s e to d e m o n s t r a t e t h r o u g h the w a y they formulated p r o b l e m s that problems were a new genre a n d n o inheritance from the S u m e r i a n scribe school. A n d p r o b l e m s , as is well k n o w n , constituted the core a n d m o s t of the b o d y o f O l d B a b y l o n i a n m a t h e m a t i c s . S o far I h a v e left out early O l d B a b y l o n i a n U r from this analysis. Ur, the former q u e e n o f U r III, o f course w a s not likely to e m p h a s i z e the b r e a k with the past, e v e n t h o u g h the sparse p r o b l e m texts that h a v e b e e n found d o suggest that they c o m e from a b o r r o w e d tradition.^' But lexical lists also give the equivalences gaz~nasâl}um, tah~wasâbum, which is no less meaningful. In UET V, 864, everything is in Sumerian except the key technical terms dakâsum and dikistum; the topic was thus something which could not be discussed in Sumerian, obviously

140

J. Heyrup

If is therefore n o t a s t o u n d i n g that the m e t a m a t h e m a t i c a l v o c a b u l a r y o f these texts is in Sumerian, b u t n o t e w o r t h y that it is m e a g r e a n d n o t v e r y consistent: e n . n a m ( p r o b a b l y a n e w p s e u d o - S u m e r o g r a m , c f a b o v e ) is u s e d in U E T V, 1 2 1 , 8 6 4 (in varying spellings) a n d 8 5 9 , a n d a.na.àm once in U E T V, 8 5 9 . R e s u l t s are " s e e n " (pad)

in U E T V, 8 5 9 a n d 8 6 4 , w h e r e a s they carry the Siunerian enclitic c o p u l a - à m

("it i s " " ) in U E T V, 8 5 8 . T h e h y p e r o r t h o g r a p h i c ( w h e n c e not traditional - ib.tag4 d o e s not o c c u r in the perfectly parallel form i.ib.tag4) p r o s p e c t i v e prefix ù . u b . , " a n d then", a p p e a r s to b e a fresh translation of the A k k a d i a n suffix -ma. T h i s certainly d o e s n o t l o o k like a continuation of a stable tradition, rather as a tentative exploration o f an u n k n o w n terrain. Before d r a w i n g the final conclusions w e m i g h t ask w h e t h e r O l d B a b y l o n i a n m a t h e m a t i c s t e a c h e r s are really likely to have e x p r e s s e d a n y ideas or ideals t h r o u g h their shaping a n d u s e o f a technical terminology. T h e a n s w e r is e m p h a t i c a l l y affirmative - e x a m p l e s o f v e r y c o n s c i o u s a d o p t i o n o f a characteristic t e r m i n o l o g i c a l c a n o n a n d o f elimination of u n w a n t e d terms a b o u n d . O n e e x a m p l e w a s m e n t i o n e d a b o v e : the elimination o f the familiar p h r a s e tammar firom g r o u p 3 a n d Y B C 4 6 6 2 7 , 4663

and

its

replacement

by

nadânumisum

(elsewhere

exclusively

used

in

c o n n e c t i o n with s e x a g e s i m a l multiplication). W e m a y also m e n t i o n A O 8 8 6 2 from g r o u p 1, w h i c h asks questions a b o u t the d i m e n s i o n s of fields b y mînûm, w h e r e a s kî masi is u s e d w h e n b r i c k calculations are c o n c e m e d . M o s t striking o f all is p e r h a p s a g r o u p of 9 tablets found in a single r o o m in Tell H a r m a l , E s h n u i m a , w h i c h invariably c o u p l e results " c o m i n g u p " with the question mînûm and the " s e e i n g " o f results with kî masi (a s y s t e m w h i c h is found n o w h e r e else a n d e v e n b r o k e n in a tablet c o m i n g from the n e i g h b o u r i n g room^^). Deliberate

shaping

o f the v o c a b u l a r y

beyond

what

was

requned

by

the

matheinatics it w a s m e a n t to express w a s thus n o t alien to the m i n d - s e t o f O l d B a b y l o n i a n m a t h e m a t i c s teachers, a n d w e m a y trust that t h e y h a d a r e a s o n for d o i n g so w h e n requiring o n e c a t e g o r y of t e r m s to b e written invariably with S u m e r i a n l o g o g r a m s , allowing others to a p p e a r in syllabic or l o g o g r a p h i c writing a c c o r d i n g to the local stylistic c a n o n , a n d still others to b e written always in syllabic A k k a d i a n . Seeing that those w h i c h are invariably in S u m e r i a n are legitimately s o fi-om the historical p o i n t o f view, w e m a y c o n c l u d e that those w h i c h are n o t a l l o w e d to a p p e a r in S u m e r i a n - t h o s e that a l l o w the formulation a n d solution o f problems

-

were

s u p p o s e d to r e p r e s e n t s o m e t h i n g n e w ; the m e t a - t e r m i n o l o g i c a l vacillations o f the p r o b l e m texts from U r indicate that the m a t h e m a t i c s teachers of the r e m a i n i n g O l d B a b y l o n i a n orbit w e r e n o t m i s t a k e n o n this account. T h e w h o l e discourse problems

of

m u s t h a v e b e e n absent fi-om the legacy left b y the U r III school.

M a t h e m a t i c a l p r o b l e m s , of course, w e r e n o t a fi-esh invention o f the Babylonian

school.

Students

of the O l d A k k a d i a n

schools h a d

leamed

Old their

because it had not been done before; UET V, 858 presents us with the problem of a bisected trapezium, but in a trite version that suggests that the author had heard that the transversal 13 of a trapezium with parallel sides 17 and 7 divides the height in ratio 2:3 but either did not know that or did not know why this transversal bisects the area. That it did was known in Sargonic surveying and constituted stock knowledge in Old Babylonian mathematics in general; it can only have been found out by somebody who also knew the an argument why. Never used elsewhere in this function in Old Babylonian mathematics! "

See H0YRUP (2000), p. 122, n.9 and p. 126.

How to Educate a Kapo

141

mathematics from p r o b l e m s , a n d " s a w " results; p r o b l e m s are also to b e ft)imd in the school texts from Shuruppak. W h a t will h a v e b e e n n e w is instead a m a t h e m a t i c s teaching not b a s e d o n p r o b l e m solution, a m a t h e m a t i c s teaching deliberately eliminating e v e n the slightest a p p e a l to i n d e p e n d e n t thought o n the p a r t o f the students. It is also imique in history: e v e n the school o f the T h i r d R e i c h w e n t n o fiirther in its confrol o f the s t u d e n t s ' m i n d s than t o having p r o b l e m s deal with "artillery frajectories, fighter-to-bomber ratios a n d b u d g e t deficits accruing from the d e m o c r a t i c p a m p e r i n g o f hereditarily diseased families".'^'* In that school w h i c h c a m e out o f a n adminisfrative reform w h i c h c o m p l e t e d a military reform, U r III s e e m s to h a v e a i m e d at exactly w h a t O R W E L L ' S ( 1 9 5 4 , p . 2 4 1 ) Newspeak w a s m e a n t to effectuate: " t o m a k e all [unauthorized] m o d e s o f t h o u g h t i m p o s s i b l e " - at least a m o n g those miserable a n d terrible K a p o - o v e r s e e r - s c r i b e s w h o constituted the " O u t e r P a r t y " o f king Sulgi's statai system.

G R U N B E R G E R ( 1 9 7 4 ) , p. 3 6 7 .

142

J. Hoyrup

List of tablets referred to A O 6 7 7 0 . P u b l i s h e d in M K T I I , p . 37f., cf. I l l , p p . 6 2 ff. A O 8 8 6 2 . P u b l i s h e d in M K T I , p p . 1 0 8 - 1 1 3 ; I I , p i . 3 5 - 3 8 ; I I I , p . 5 3 . B M 1 3 9 0 1 . P u b l i s h e d in T H U R E A U - D A N G I N ( 1 9 3 6 ) . B M 1 5 2 8 5 . P u b l i s h e d in M K T I , p . 137; with a n additional fragment i n S A G G S ( 1 9 6 0 ) ; with yet another in R O B S O N ( 1 9 9 9 ) , p p . 2 0 8 - 2 1 7 . B M 8 5 2 0 0 + V A T 6 5 9 9 . P u b l i s h e d in M K T I , p p . 193 ff.; I I , p i . 7 - 8 (photo), pi. 3 9 - 4 0 ( h a n d c o p y ) . B M 9 6 9 5 7 + V A T 6 5 9 8 . PubUshed in R O B S O N ( 1 9 9 9 ) , p p . 2 3 1 - 2 4 4 . C B M 1 2 6 4 8 . P u b l i s h e d in M K T I , p . 2 3 4 f, m u c h i m p r o v e d in FRIBERG ( 2 0 0 1 ) , p . 149. C B S 1 5 4 + 9 2 1 . P u b l i s h e d in R O B S O N ( 2 0 0 0 ) , p . 4 0 . C B S 1 9 7 6 1 . P u b l i s h e d in R O B S O N ( 2 0 0 0 ) , p . 3 6 f I M 5 2 3 0 1 . P u b l i s h e d in B A Q I R ( 1 9 5 0 b ) . I M 5 2 9 1 6 + 5 2 6 8 5 + 5 2 3 0 4 . P u b l i s h e d in G O E T Z E ( 1 9 5 1 ) . I M 5 5 3 5 7 . P u b h s h e d in B A Q I R ( 1 9 5 0 a ) . I M 1 2 1 6 1 3 . Preliminary publication in FRIBERG a n d A L - R A W I ( 1 9 9 4 ) . S t r 3 6 6 . P u b l i s h e d in M K T I , p . 2 5 7 ; I I I , p . 5 6 , h a n d c o p y F R A N K ( 1 9 2 8 ) , # 9 = p i vii. s t r 3 6 7 . P u b l i s h e d in M K T I , p . 2 5 9 f, h a n d c o p y F R A N K ( 1 9 2 8 ) , # 1 0 = p i viii. S t r 3 6 8 . P u b h s h e d in M K T I , p . 3 1 1 , h a n d c o p y F R A N K ( 1928), # 11 = p i ix. T M S V. P u b l i s h e d in T M S 3 5 - 4 9 , p i 7 - 1 0 . T M S V I . P u b l i s h e d in T M S 4 9 - 5 1 , p i 1 1 - 1 3 . T M S X V I . PubUshed in T M S 91 f, p i 2 5 . T M S X X V I . PubUshed in T M S 124 f, p i 3 9 . U E T V, 1 2 1 . P u b l i s h e d in V A J M A N ( 1 9 6 1 ) , p . 2 4 8 , c f FRIBERG ( 2 0 0 0 ) , p . 3 5 f U E T V, 8 5 8 . P u b l i s h e d in V A J M A N ( 1 9 6 1 ) , p . 2 5 1 , c f FRIBERG ( 2 0 0 0 ) , p . 3 8 f U E T V, 8 5 9 . P u b l i s h e d in V A J M A N ( 1 9 6 1 ) , p . 2 5 4 f, c f MuROi ( 1 9 9 8 ) , p . 2 0 1 f a n d FRIBERG ( 2 0 0 0 ) , p . 3 9 .

U E T V, 8 6 4 . P u b l i s h e d in V A J M A N ( 1 9 6 1 ) , p . 2 5 7 f, c f M U R O I ( 1 9 9 8 ) , p . 2 0 0 a n d FRIBERG ( 2 0 0 0 ) , p . 3 7 .

VAT 6 7 2 . PubUshed in M K T I, p . 2 6 7 ; I I , pi. 4 3 . VAT 7 5 3 1 . P u b l i s h e d in M K T I , p . 2 8 9 f ; II, p i . 4 6 ; I I I , p . 5 8 . VAT 7 5 3 2 . P u b l i s h e d in M K T I , p . 2 9 4 f ; I I , pi. 4 6 ; I I I , p . 5 8 . VAT 7 5 3 5 . PubUshed in M K T I , p p . 3 0 3 - 3 0 5 ; I I , p i . 4 7 . W 2 3 2 9 1 . PubUshed in FRIBERG ( 1 9 9 7 ) . Y B C 4 6 6 2 . P u b l i s h e d in M C T , p . 71 f, p i . 8. Y B C 4 6 6 3 . P u b l i s h e d in M C T , p . 6 9 , p i 7. Y B C 4 6 6 9 . PubUshed in M K T I , p . 5 1 4 ; I I I , p . 2 7 f, p i . 3 . Y B C 4 6 7 3 . P u b l i s h e d in M K T I , p . 5 0 7 f ; I I , p . 5 0 8 ; I I I , p p . 2 9 - 3 1 , p i . 3 . Y B C 6 5 0 4 . P u b l i s h e d in M K T I I I , p . 2 2 f, pi. 6. Y B C 7 9 9 7 . P u b l i s h e d in M C T , p . 9 8 , p i 2 3 . Y B C 9 8 5 6 . P u b l i s h e d in M C T , p . 9 9 , p i 4 . Y B C 9 8 7 4 . P u b l i s h e d in M C T , p . 9 0 , p i 1 1 .

How to Educate a Kapo

143

Abbreviations A H w = VON SODEN (1965-1981) M C T = NEUGEBAUER and SACHS (1945) M K T = NEUGEBAUER (1935-1937) SLa = T H O M S E N (1984) T M S = BRUINS and RUTTEN (1961)

References ALLOTTE D E LA F U Y E , F r a n ç o i s - M a u r i c e .

1 9 1 5 . "Mesiu-es agraires et formules

d ' a r p e n t a g e à l ' é p o q u e p r é s a r g o n i q u e " . Revue d'Assyriologie ALSTER,

B e n d t . 1974. The Instructions

12: 1 1 7 - 1 4 6 .

ofSuruppak. A Sumerian

Proverb

Collection

( M e s o p o t a m i a : C o p e n h a g e n Studies in Assyriology, 2 ) . C o p e n h a g e n : A k a d e m i s k Forlag B A Q I R , Taha. 1950a. " A n Important M a t h e m a t i c a l P r o b l e m Text from Tell H a r m a l " . Sumer 6: 3 9 - 5 4 . — 1 9 5 0 b . " A n o t h e r I m p o r t a n t M a t h e m a t i c a l Text from Tell H a r m a l " . Sumer

6:

130-148. B R U I N S , E v e r t M . 1 9 7 1 . " C o n f u t a t i o n in t h e O l d B a b y l o n i a n P e r i o d " . Janus 5 8 : 222-267. — a n d M a r g u e r i t e R U T T E N . 1 9 6 1 . Textes mathématiques

de Suse. ( M é m o i r e s d e la

M i s s i o n A r c h é o l o g i q u e e n Iran, X X X I V ) . Paris: P a u l G e u t h n e r D I A K O N O F F , Igor M . (éd.). 1969. Ancient Mesopotamia, Collection

of Studies

by Soviet

Scholars.

Socio-Economie

History. A

M o s k v a : « N a u k a » Publishing H o u s e ,

Cenfral D e p a r t m e n t o f Oriental Literatiue. E N G L U N D , R o b e r t K . 1 9 9 0 . Organisation

und Verwaltung

der Ur

III-Fischerei

( B e r l m e r Beifrâge z u m Vorderen Orient, 10). Berlin: Dietrich Reimer. F R A N K , Carl. 1928. Strafiburger Sprache

Keilschrifttexte

in sumerischer

und

babylonischer

(Schriften d e r Strafiburger Wissenschaftlichen Gesellschaft i n H e i d e l

b e r g , N e u e F o l g e , Heft 9 ) . Berlin & Leipzig: Walter d e Gruyter. FRIBERG, Joran. 1997. " ' S e e d a n d R e e d s C o n t i n u e d ' . A n o t h e r M e f r o - M a t h e m a t i c a l T o p i c Text from Late B a b y l o n i a n U r u k " . Baghdader

Mitteilungen

28: 251-365,

pi. 4 5 - 4 6 . — 2 0 0 0 . " M a t h e m a t i c s a t U r in t h e O l d B a b y l o n i a n P e r i o d " . Department Mathematics, Preprint

Chalmers

University

of Technology,

The University

of

of

Goteborg.

2000/25.

— 2 0 0 1 . " B r i c k s a n d M u d in Mefro-Mathematical H 0 Y R U P & Peter D A M E R O W (eds), Changing Mathematics

C u n e i f o r m T e x t s " . In: J e n s

Views on Ancient

(Berliner Beifrâge z u m Vorderen Orient,

Near

Eastern

19): 6 1 - 1 5 4 . Berlin:

Diefrich Reimer. — a n d A L - R A W I , F a r o u k . 1994. " E q u a t i o n s for R e c t a n g l e s a n d M e t h o d s o f False Value in O l d B a b y l o n i a n Geomefric-Algebraic P r o b l e m texts". Manuscript progress,

in

courtesy o f the authors.

G O E T Z E , Albrecht. 1 9 5 1 . " A M a t h e m a t i c a l C o m p e n d i u m from Tell H a r m a l " . 7: 1 2 6 - 1 5 5 .

Sumer

144

J. Hoyrup

G R U N B E R G E R , R i c h a r d . 1974. A Social History

of the Third Reich.

Harmondsworth,

Middlesex: Penguin. HALLO,

William

W.

Sumerological

1976.

"Toward

a History

of

Sumerian

Literature".

In:

Studies in H o n o r of T h o r k i l d J a c o b s e n o n his S e v e n t i e t h Birthday,

June 7, 1974 ( T h e Oriental Institute of the University o f C h i c a g o , A s s y r i o l o g i c a l Studies 2 0 ) : 1 8 1 - 2 0 3 . C h i c a g o & L o n d o n : U n i v e r s i t y of C h i c a g o P r e s s . H 0 Y R U P , Jens. 1980. "Influences of Institutionalized M a t h e m a t i c s T e a c h i n g o n the D e v e l o p m e n t a n d Organization of M a t h e m a t i c a l T h o u g h t in the P r e - M o d e m Period. Investigations into an A s p e c t of the A n t h r o p o l o g y o f M a t h e m a t i c s " . Materialien Bielefeld

und Studien. m

Institut fur Didaktik

der Mathematik

der

Universitat

7-137.

— 1990a. " A l g e b r a a n d N a i v e Geometry. A n Investigation o f S o m e B a s i c A s p e c t s of O l d B a b y l o n i a n M a t h e m a t i c a l T h o u g h t " . Altorientalische

Forschungen

17:

27-69, 262-354. — 1990b. "Sub-Scientific M a t h e m a t i c s . O b s e r v a t i o n s o n a P r e - M o d e m P h e n o m e n o n " . History

of Science

28: 63-86.

— 1 9 9 2 . " T h e B a b y l o n i a n Cellar Text B M 8 5 2 0 0 + VAT 6 5 9 9 . R e t r a n s l a t i o n and A n a l y s i s " . In: Sergei S. D E M I D O V et al. (eds.). Amphora. Wussingzu

seinem

— 2000. "The

65. Geburtstag:

Finer

Stmcture

Festschrift

fiir

Hans

3 1 5 - 3 5 8 . B a s e l etc.: Birkhauser.

o f the O l d

Babylonian

Mathematical

Corpus.

E l e m e n t s of Classification, with s o m e R e s u l t s " . In: J o a c h i m M A R Z A H N a n d H a n s N E U M A N N (eds.), Assyriologica

et Semitica.

Festschrift fiir J o a c h i m O e l s n e r

anlaBlich seines 6 5 . G e b i u t s t a g e s a m 18. F e b m a r 1997 (Alter O r i e n t i m d Altes Testament, 2 5 2 ) : 1 1 7 - 1 7 7 . Miinster: Ugarit Verlag. — 2 0 0 1 . " O n a C o l l e c t i o n of G e o m e t r i c a l Riddles a n d T h e i r R o l e in the S h a p i n g of F o u r to Six ' A l g e b r a s ' " . Science — 2 0 0 2 a . Lengths,

in Context

Widths, Surfaces.

14: 8 5 - 1 3 1 .

An Examination

of Old Babylonian

Algebra

and its Kin. N e w York etc.: Springer Verlag. — 2002b.

"A Note

Mathematica

on Old

Babylonian

Computational

Techniques".

Historia

29: 193-198.

M U R O I , K a z u o . 1 9 9 8 . " E a r l y O l d B a b y l o n i a n M a t h e m a t i c a l P r o b l e m s W r i t t e n in S u m e r i a n " . Historia NEUGEBAUER,

Otto.

Scientiarum,

S e c o n d Series 7 : 3 : 1 9 9 - 2 0 3 .

1935-1937.

Mathematische

Keilschrift-Texte,

Vol.

I-III.

( Q u e l l e n u n d Studien zur Geschichte der M a t h e m a t i k , A s t t o n o m i e i m d Physik. Abteilimg A : Quellen. 3 . B a n d , e r s t e r - d r i t t e r Teil). Berlin: Julius Springer. N E U G E B A U E R , O t t o a n d S A C H S , A b r a h a m . 1 9 4 5 . Mathematical

Cuneiform

Texts

( A m e r i c a n Oriental Series, vol. 2 9 ) . N e w H a v e n , Connecticut: A m e r i c a n Oriental Society. N I S S E N , H a n s J.; D A M E R O W , Peter, a n d E N G L U N D , R o b e r t . 1990. Friihe Schrift Techniken

der

speicherung — 1994. Archaic

Wirtschaftsverwaltung

und -verarbeitung Bookkeeping:

im alten

Vorderen

Orient.

und

Informations-

vor 5000 Jahren. B a d Salzdetfiirth: Franzbecker. Writing and Techniques

of Economic

Administration

in the Ancient Near East. C h i c a g o : C h i c a g o University Press. ORWELL,

George.

Penguin.

1954.

Nineteen

Eighty-Four.

Harmondsworth,

Middlesex:

How to Educate a Kapo

R O B S O N , Eleanor. 1 9 9 9 . Mesopotamian Constants

in Bureaucracy

145

Mathematics

and Education

2100-1600

BC.

Technical

(Oxford Editions o f C i m e i f o r m Texts,

1 4 ) . Oxford: C l a r e n d o n Press. R O B S O N , Eleanor. 2 0 0 0 . " M a t h e m a t i c a l C u n e i f o r m Tablets in P h i l a d e l p h i a . P a r t 1 : P r o b l e m s a n d Calculations". SCIAMVS

1: 1 1 - 4 8 .

S A G G S , H . W . R 1 9 6 0 . " A B a b y l o n i a n G e o m e t r i c a l T e x f . Revue d'Assyriologie

54:

131-146.

SJÔBERG, Â k e W. 1 9 7 6 . " T h e O l d B a b y l o n i a n E d u b a " . In: Sumerological

Studies in

H o n o r o f T h o r k i l d J a c o b s e n o n his Seventieth Birthday, J u n e 7 , 1 9 7 4 ( T h e Oriental Institute o f t h e University o f C h i c a g o , Assyriological Studies, 2 0 ) : 1 5 9 - 1 7 9 . C h i c a g o & L o n d o n : University o f C h i c a g o Press. V O N S O D E N , Wolfram. 1 9 6 5 - 1 9 8 1 . Akkadisches

Handworterbuch.

W i e s b a d e n : Otto

Harrassowitz. T H O M S E N , M a r i e - L o u i s e . 1 9 8 4 . The Sumerian History

and Grammatical

Structure

Language.

An Introduction

to its

(Mesopotamia, 1 0 ) . Kobenhavn: Akademisk

Forlag. STEINKELLER, Piotr. 1 9 7 9 . " A l l e g e d G U R . D A = ugula-gés-da a n d t h e R e a d i n g o f the Sumerian Archaologie

Numeral

6 0 " . Zeitschrift

fiir

Assyriologie

und

Vorderasiatische

69: 176-187.

THUREAU-DANGIN,

François.

1 9 3 6 . "L'Équation

d u deuxième

degré

d a n s la

m a t h é m a t i q u e b a b y l o n i e n n e d ' a p r è s u n e tablette inédite d u British M u s e u m " . Revue d'Assyriologie

33: 27-48.

V A J M A N , A b r a m A . 1 9 6 1 . Sumero-vavilonskaja

matematika.

III-I Tysjaceletija

do n.

e. M o s k v a : I z d a t e l ' s t v o Vostocnoj Literatury. W H I T I N G , R o b e r t M . 1 9 8 4 . " M o r e E v i d e n c e for S e x a g e s i m a l C a l c u l a t i o n s in the Third

Milleimium

Archaologie

B . C . " Zeitschrift

74: 5 9 - 6 6 .

fiir

Assyriologie

und

Vorderasiatische

The Algorithmic Structure of the Egyptian Mathematical Problem Texts Annette Imhausen,

Cambridge

(Mass.)

1. Introduction T h e reputation o f E g y p t i a n science in ( m o d e m ) history o f science is rather m o d e s t . ' U n d e r t h e designation " E g y p t i a n S c i e n c e " tliree subjects are usually c o m b i n e d : m a t h e m a t i c s , astronomy, a n d medicine.^ T h e c o m b i n a t i o n o f these three subjects as constituting " E g y p t i a n s c i e n c e " is, h o w e v e r , not derivable from the available s o i u c e material. O n the b a s i s o f the extant p a p y r i , m e d i c i n e a n d m a t h e m a t i c s c o u l d b e g r o u p e d together. I n b o t h cases so-called "procediure t e x t s " w h i c h s h o w a n u m b e r o f formal similarities constitute the core o f the written s o i u c e s . T h e so-called asfronomical texts differ significantly from them.^ T h i s p a p e r focuses o n the E g y p t i a n m a t h e m a t i c a l p r o b l e m texts. After a brief description o f their historiographical freatment, the formal characteristic features will b e p o i n t e d out. O n e o f these featiues p r o v i d e s u s with a n e w a p p r o a c h to describe a n d analyze their m a t h e m a t i c a l content. T h i s a p p r o a c h is a n historiographically m o r e a d e q u a t e m e t h o d o f w o r k i n g w i t h ancient m a t h e m a t i c a l texts than w h a t h a s b e e n u s e d so far. A c o m p l e t e historical accoimt o f E g y p t i a n matheinatics s h o u l d also consider the context in w h i c h the d e s c r i b e d m a t h e m a t i c a l techniques w e r e d e v e l o p e d a n d used. T h e r e are several possibilities to contextualize E g y p t i a n m a t h e m a t i c s as I h a v e s h o w n previously.'* T h i s p a p e r focuses o n l y o n the i n t e m a l m a t h e m a t i c a l content.

For comments on an earlier draft of this paper I wish to thank Jens H0YRUP and John Steele. My work has profited for several years from the advice of James Ritter to whom I am sincerely indebted. ' Cf Otto Neugebauer: "Egypt provides us with the exceptional case of a highly sophisticated civilization which flourished for many centuries without making a single contribution to the development of the exact sciences." N E U G E B A U E R ( 1 9 7 5 ) , p. 5 5 9 . ^ E.g. Marshall Clagett's series of books on Egyptian Science: The first volume contains those texts that are meant to constitute the intellectual background, the second volume combines documents relating to calendars, clocks, and astronomy and the third volume includes all mathematical texts. The planned fourth volume is meant to treat medicine. C L A G E T T ( 1 9 8 9 , 1 9 9 5 , and

1999).

number of astronomical texts have been edited in N E U G E B A U E R and recent study on Egyptian astronomy can be found in V O N L I E V E N (20ÔO). ^

A

IMHAUSEN ( 1 9 9 9 )

and (in prep. a).

PARKER ( 1 9 6 0 ) .

A

148

A. Imhausen

2. Historiography T h e available p r i m a r y sources for Egyptian m a t h e m a t i c s are unfortunately quite limited.^ T h e m a i n source texts w e r e p u b l i s h e d b y 1930.^ F r o m then on, o n l y two fragmented ostraca h a v e b e e n a d d e d to the available sources.^ N e v e r t h e l e s s , e v e n w i t h this shortage of source material, there has b e e n a steady flow o f literature o n Egyptian m a t h e m a t i c s . M a n y contributions, h o w e v e r (especially in r e g a r d to m a t h e m a t i c a l content), o n l y r e w r o t e or discussed w h a t h a d b e e n written before. T h e r e a s o n for this stagnation is twofold: neither h a v e the available p r i m a r y sources significantly increased, n o r have a p p r o a c h e s to the various p r o b l e m s changed. T h e literature has b e e n written b y scholars from t w o c o m m u n i t i e s , history of m a t h e m a t i c s a n d E g y p t o l o g y . It often seems that o n e c o m m u n i t y d o e s n o t k n o w of the others r e c e n t results. This has lately b e e n explicitly stated from the Egyptological side in a r e v i e w of M a r s h a l l C l a g e t t ' s b o o k s o n E g y p t i a n science.^ T h e lack of n e w sources has led scholars to include administrative texts involving calculations in their presentations of m a t h e m a t i c a l texts.^ W h i l e it is certainly true that adminisfrative sources c a n a n d should b e u s e d to enlarge our k n o w l e d g e of E g y p t i a n m a t h e m a t i c a l practice, it m u s t b e kept in m i n d that t h e y are not m a t h e m a t i c a l texts.'*'

3. Source Material and its Characteristic Features In the extant E g y p t i a n texts, t w o kinds of m a t h e m a t i c a l texts c a n b e differentiated: p r o b l e m texts a n d table texts. T a b l e texts are collections of m a t h e m a t i c a l data, structured in the form o f lists, that are used in solving m a t h e m a t i c a l p r o b l e m s . T h e extant m a t h e m a t i c a l table texts include tables that h a v e b e e n u s e d in fraction r e c k o n i n g " and tables for conversions of m e a s u r e s . ' ^ P r o b l e m texts, o n the other h a n d , state a m a t h e m a t i c a l p r o b l e m a n d then indicate the solution o f that p r o b l e m . T o d a y w e k n o w o f a r o u n d o n e h u n d r e d p r o b l e m s from the extant texts, m o s t of

^

See

RITTER

(2000), pp. 115-116.

^ In order of their publication (if edited several times, all editions are listed): pRhind (BM 10057 and 58): E I S E N L O H R (1877); P E E T (1923a); C H A C E , B U L L , M A N N I N G , and A R C H I B A L D (1927-1929). lUahun mathematical fragments: GRIFFITH (1898). pBerlin 6619: S C H A C K S C H A C K E N B U R G (1900 and 1902). Cairo wooden boards (CG 25367 and 25368): D A R E S S Y (1901) and P E E T (1923b). pMoscow (E4676): S T R U V E (1930). Mathematical leatherroll (BM 10250): GLANVILLE(1927). ^

In order of their publication: Ostracon Senmut 153: H A Y E S (1942). Ostracon Turin 57170: (1980). For OSenmut 153 see also F O W L E R (1999), p. 269.

LOPEZ *

ALLEN

'

E.g.

(1992 and 1996);

GILLINGS

SPALINGER

( 1972) and

CLAGETT

(1997).

( 1999).

Mathematical texts are not all texts that reflect mathematical knowledge, but only those that have been written about mathematics. Cf R O B S O N (1999), p. 7. " E.g. the 2:n table (preserved in the Rhind papyrus and in one of the Illahun fragments), the table in pRhind, No. 61, and the table in the mathematical leatherroll
The Algorithmic Structure of the Egyptian Mathematical Problem Texts

149

w h i c h are found in the R h i n d a n d M o s c o w papyri. T h e s e p r o b l e m s c o v e r a variety of topics w h i c h are often b a s e d o n daily life.'^ In m y s o o n to b e p u b l i s h e d thesis I h a v e r e w o r k e d the E g y p t i a n m a t h e m a t i c a l p r o b l e m texts, starting with a n e w edition of the texts, w h i c h w e r e last edited seventy years ago.''* F u r t h e r m o r e I p r o p o s e d a classification of the different types of p r o b l e m s , a n d I e m b e d d e d the g r o u p s of practical p r o b l e m s in their b a c k g r o u n d , w h i c h enables a d e e p e r imderstanding o f the m a t h e m a t i c a l texts. B a s e d o n m y edition of the texts, I h a v e analyzed the individual p r o b l e m s t h r o u g h a n e w a p p r o a c h that will b e e x p l a i n e d in detail in the following sections.'^ Figure 1 s h o w s the text of p r o b l e m 2 6 from the R h i n d p a p y r u s , w i t h m y franscription into hieroglyphic text a n d franslation g i v e n b e l o w . A s c a n b e seen from this e x a m p l e , the E g y p t i a n p r o b l e m texts c a n b e characterized, as h a s b e e n d o n e b y J a m e s Ritter,'^ as numeric, rhetorical, a n d algorithmic. " N u m e r i c " d e n o t e s the practice of always using concrete n u m b e r s . T h e texts are " r h e t o r i c a l " m the sense that all operations are formulated without the u s e of symbolism. T h e y c a n b e n a m e d "algorithmic", since the solution is g i v e n as a s e q u e n c e o f instructions. T h e t e r m " a l g o r i t h m " is very general. It is defined as a s e q u e n c e o f instructions, starting from g i v e n p r e m i s e s , that result in the solution of a g i v e n p r o b l e m . E x a m p l e s o f algorithms include c o o k m g recipes, c o m p u t e r p r o g r a m s or technical instructions. D e s p i t e this generality, it has to b e k e p t in m i n d that one vital feature of the algorithm is that it is a defined sequence of steps. In the case of E g y p t i a n m a t h e m a t i c s , the individual instructions usually consist o f single arithmetic operations: addition, subfraction, multiplication, division, a n d halving. A l t h o u g h the algorithmic character of the p r o b l e m texts has often b e e n stated as a characteristic feature, this fimdamental quality has as yet n e v e r b e e n u s e d to analyse the m a t h e m a t i c s p r e s e n t e d within this framework. Instead m a t h e m a t i c a l contents h a v e usually b e e n represented as a s u p p o s e d l y " e q u i v a l e n t " equation.

E.g. problems 39 and 40 from the Rhind papyrus in which rations are calculated, and problem 41 to 46 from the same text which deal with the calculation of granary volumes. A large group of problems, found in both, the Rhind and the Moscow papyrus deal with calculations related to baking and brewing. I M H A U S E N (in prep. b). I would like to thank the British Museum, especially Richard Parkinson, for enabling me to work with the Rhind papyrus; the Petrie Museum, especially Stephen Quirke, for making available the mathematical fragments of the Illahun papyri; the Brooklyn Museum, especially Edward Bleiberg, for granting me access to the fragments of pRhind; the Agyptisches Museum und Papyrussammlung, SMPK Berlin, especially Ingeborg MuUer, for supplying me with photographs of pBerlin 6619; and last but not least, Walter Reineke for providing me with photographs of the Moscow papyrus that are much better than those in the published edition of the text.

This approach had already been suggested by James Ritter for working with Mesopotamian and Egyptian problem texts some years ago, see RJTTER (1989) (English translation in R I T T E R (1995)) and RITTER (1998). Both articles present the method of algorithmic analysis using selective examples. In my thesis, I applied this method to the complete corpus of Egyptian problem texts. This description is given in RITTER (1989), p. 44 for Egyptian as well as Mesopotamian problem texts. They are valid for the majority of Egyptian problems, however, the last "problems" of pRhind (Nos. 82-84) have to be excluded.

A. Imhausen

150

I I I

^^Xl I I Û T

./3

11

I I I

iirii 11 Û f 12 I I

—««X 13

T

i 9 Di 1

lO 11

'n.

II

\t

1 i<=> .10

I lO

sic!

IIII/8

X 1 1 I in<£

I I I

.6

111

9

I I I

117

Translation 1

A q u a n t i t y , its 4 to it, it results as 15. Calculate with 4 ! T h e n y o u calculate its 4 as 1. Total 5.

2

D i v i d e 15 b y 5!

3-4

[calculation]

5

T h e n 3 results. Multiply 3 by 4!

6-8

[calculation]

9

T h e n 12 results.

10-11

[calculation o f the p a r t of the quantity]

12-13

[verification]

Figure 1. p R h i n d , n o . 2 6 , text (Copyright T h e British M u s e u m ) , transcription and translation.'^

hieroglyphic

" Rubra are rendered as underlined in the hieroglyphic text and as bold print in the translation.

1

The Algorithmic Structure of the Egyptian Mathematical Problem Texts

151

Twenty-five years a g o , the historian Sabetai U n g u r u p o i n t e d o u t t h e d a n g e r s a n d pitfalls o f a p p l y i n g m o d e m m a t h e m a t i c a l s y m b o l i s m s a n d t e r m i n o l o g y in the h i s t o r i o g r a p h y o f ancient mathematics.*^ H i s article c a u s e d a scandal at the t i m e , for U n g i u i i h a d d a r e d to accuse famous scholars (for e x a n ç l e O t t o N e u g e b a u e r ) o f having made serious historiographical errors in t h e n a p p r o a c h . * ' This g r o u n d b r e a k i n g article s h o w s v e r y clearly that the existence o f a n " e q u i v a l e n t e q u a t i o n " m u s t n o t b e t a k e n for granted. C o m i n g b a c k t o the text o f p r o b l e m 2 6 : within the text o f t h e p r o b l e m , several parts c a n b e distinguished. First the type o f p r o b l e m is indicated. T h i s , as the b e g i n n i n g o f a n e w p r o b l e m is m a r k e d b y t h e use o f r e d ink. T h e n the g i v e n entities are n a m e d , followed b y the entity that is to b e calculated. T h e style o f t h e text c h a n g e s w i t h the b e g i n n i n g o f the n e x t section, in w h i c h the m s t m c t i o n s to solve the p r o b l e m are listed, s o m e t i m e s followed b y the result o f the individual arithmetic step. T h i s p a r t o f the p r o b l e m is grammatically m a r k e d b y the u s e o f imperatives (in the i n s t m c t i o n s ) a n d the u s e o f the v e r b form sdm.hr-f (in t h e i n s t m c t i o n s a n d intermediate results). B e s i d e s the m a t h e m a t i c a l texts this v e r b form is m a i n l y found m m e d i c a l texts w h e r e it h a s the s a m e function. T h i s function c a n b e s t b e d e s c r i b e d as expressing a n e c e s s a r y s e q u e n c e w h i c h results from the p r e v i o u s l y stated conditions.^'' A s for annoimcing the result o f a m a t h e m a t i c a l operation, the v e r b form sdm.hr-f expresses " m a t h e m a t i c a l facts" - if y o u a d d 2 a n d 2, the result will necessarily b e 4 . T h e u s e o f the sdm.hr-f in the m s t m c t i o n s t h e m s e l v e s c a n b e interpreted a s underlining their algorithmic aspect - g i v e n the stated m a t h e m a t i c a l p r o b l e m , it is n e c e s s a r y to execute those particular m a t h e m a t i c a l o p e r a t i o n s t o find a solution. Traditionally, the sdm.hr-f in the m a t h e m a t i c a l texts h a s b e e n r e n d e r e d in t w o w a y s - as a n imperative w h e n u s e d w i t h m the i n s t m c t i o n s a n d in t h e p r e s e n t tense w h e n u s e d within the a i m o u n c e m e n t o f the results. T h i s h o w e v e r ignores t h e fact that it is o n e v e r b form that is u s e d in b o t h contexts in the s a m e function n a m e d a b o v e . Therefore a u n i f o r m translation o f t h e v e r b form in its t w o u s e s is preferable.^* T h e calculations themselves c a n b e n o t e d explicitly after t h e b o d y o f i n s t m c t i o n s or, as in p r o b l e m 2 6 , directly following e a c h instmction, although s o m e t i m e s they m a y n o t b e n o t e d at all. In s o m e o f the p r o b l e m s a r o u g h d r a w i n g is a d d e d to illustiate the p r o b l e m . T h e s e sections o f the m d i v i d u a l p r o b l e m texts are further s e p a r a t e d b y the u s e o f stereotyped formulas.^^

UNGURU (1975).

Reactions (most of them rather sarcastic in tone and some of them obviously without an understanding of Unguru's criticism) are V A N D E R W A E R D E N ( 1 9 7 6 ) ; F R E U D E N T A L ( 1 9 7 7 ) ; and W E I L ( 1 9 7 8 ) . Unguru expanded his argument further in U N G U R U ( 1 9 7 9 ) ; U N G U R U and R O W E ( 1 9 8 1 and 1 9 8 2 ) ; and most recently in U N G U R U and F R I E D ( 2 0 0 1 ) . ^°

Cf

M A L A I S E and W I N A N D ( 1 9 9 9 ) , pp.

388-389.

This is rendered in my translations by "then" (dependency on condition) and the verb in present-tense (expression o f unch^geable truth}. An analysis of these formulas can be found in

IMHAUSEN

(in prep, b),

§4.3.1.

152

A. Imhausen

4. Algorithmic Analysis T h e following section gives a detailed description o f the algorithmic analysis as a tool to imderstand a n d interpret the Egyptian p r o b l e m texts.^^ T h e first p a r t explains the p r o c e d u r e , while the s e c o n d part illustrates its u s e , focusing o n the g r o u p o f 'h'p r o b l e m s . Translations a n d algorithms o f the p r o b l e m s discussed in the s e c o n d part c a n b e foimd in the a p p e n d i x of this article. 4.1 G e n e r a l D e s c r i p t i o n T h e algorithmic analysis is b a s e d o n establishing t w o forms o f algorithms from the rhetorical a l g o r i t h m g i v e n in a p r o b l e m text: a n u m e r i c a l a n d a symbolical algorithm. T h e n u m e r i c a l algorithm preserves the n u m b e r s u s e d in the individual e x a m p l e s ; the symbolical form of the algorithm consists only o f p l a c e - h o l d e r s . W h i l e the n u m e r i c a l version thus remains o n e step closer to the original (and enables o n e to follow the choice of the numerical values given in the source), the symbolic a l g o r i t h m p r o v i d e s the structure o f the p r o b l e m a n d m a k e s it easily c o m p a r a b l e to the structure o f other p r o b l e m s . Translation '

Numerical Algorithm

A quantity, its 4 to it. it results as 15. Calculate with 4 ! T h e n you calculate its 4 as 1. T o t a l 5.

^

D i v i d e 15 b y 5!

Symbolical Algorithm Di

4 15 [1:

4]=4

(1)

[1:D,]

4- 4= 1 4+1=5 15 : 5 = 3

(2) (3)

D,(l) ( l ) + (2)

(4)

D2:(3)

3-4=12

(5)

(4)-(l)

[calculation] ^

T h e n 3 results. Multiply 3 by 4! [calculation]

'

T h e n 12 results. '

'^"'^

[calculation o f the part] [verification]

Figure 2. p R h i n d , n o . 2 6 , translation and algorithm. A s has b e e n said in the description of the style of the p r o b l e m texts ( § 3 ) , the p r o b l e m s usually start with a n indication of the type of p r o b l e m followed b y a list of the g i v e n entities. T h e s e are the " d a t a " o f the algorithm. I n t h e e x a m p l e of p r o b l e m 2 6 of p R h i n d ( c f Figure 2) t w o data are m e n t i o n e d , n a m e l y 4 a n d 15. In the symbolic a l g o r i t h m the data are assigned to D i a n d D2. N e x t the section o f the instructions has to b e

tiansformed.

O n e instruction a n d the a p p r o p r i a t e

result

constitute o n e step in the algorithm. This is n o t e d b y its m o d e m n o t a t i o n o f the appropriate arithmetic operation.

My approach is based on the work done by James Ritter, cf (1998).

RITTER

RITTER

(1995) and

The Algorithmic Structure of the Egyptian Mathematical Problem Texts

153

In the symbolic form o f the algorithm, the smgle steps are n u m b e r e d , begirming with step (1). Results of p r e v i o u s steps, if u s e d in a later step, are referred to b y this n u m b e r . W h i l e the niuneric algorithm translates the individual rhetorical instructions into a s e q u e n c e of arithmetic operations in m o d e m notation, the s y m b o l i c a l g o r i t h m consists o f the identical series o f arithmetic operations using the assigned s y m b o l s (i.e. Di for the g i v e n data or n u m b e r s for the result of a step in the a l g o r i t h m that is referred to). Figure 2 s h o w s the translation a n d the t w o forms of the a l g o r i t h m for p r o b l e m 2 6 of the R h i n d p a p y m s . F r o m this e x a m p l e , a few initial observations c a n b e m a d e : •

T h e basic s t m c t u r e o f the instmctions is sequential. T h a t m e a n s , the result o b t a i n e d in o n e step will usually b e u s e d in the i n s t m c t i o n of the following step.^'*

•

T h e given data c a n enter at various p o i n t s . O n e or m o r e are naturally u s e d in the first step.

A s c a n b e s e e n from fiu-ther e x a m p l e s , constants m a y a p p e a r as further quantities b e s i d e s data a n d intermediate results in s o m e o f the p r o b l e m s . I n other, m o r e c o m p l e x e x a m p l e s further distinctions m u s t b e m a d e . B e s i d e s the s i m p l e s e q u e n c e s , parallel a n d iterative steps c a n b e found:^^ • parallel steps are m a r k e d b y identical operations w h i c h are not founded o n e a c h other. T h e results obtained within t h e m are often u s e d in the step following the series o f parallel steps. • iterative steps are identical operations, the o b t a i n e d result of w h i c h is u s e d in the following step. Ideally the d e s c r i b e d p r o c e d u r e leads to the algorithm in a form in w h i c h it c a n b e easily c o m p a r e d to the algorithms o f other p r o b l e m s . It is then p o s s i b l e to frace variation o f algorithms within o n e g r o u p of p r o b l e m s as well as to c o m p a r e algorithms of p r o b l e m s c o m i n g from different g r o u p s , or finding the e m b e d d i n g of one algorithm in another o n e . T h e r e b y the analysis of the p r o b l e m s a c c o r d i n g to their algorithms enables a m o d e m r e a d e r to give a detailed description of the m a t h e m a t i c a l procediu-es u s e d a n d to c o m p a r e solutions o f p r o b l e m s with e a c h other. It m u s t b e m e n t i o n e d , however, that the m e t h o d d e s c r i b e d a b o v e is n o t always as sfraightforward a n d s m o o t h as it m i g h t appear. T h i s is partly d u e to the scarcity of the soiu-ces. A s a c o n s e q u e n c e o f this situation, a n u m b e r o f algorithms o c c u r only o n c e t h r o u g h o u t the extant mathematical p r o b l e m texts. I w o u l d e x p e c t tìiat e a c h o f the algorithms h a d a g r o u p o f p r o b l e m s that were solved a c c o r d i n g to it. T h e existing texts s h o w three further features that are p r o b l e m a t i c for the analysis o f their algorithms. All o f t h e m are due to the familiarity the a u t h o r s o f the m a t h e m a t i c a l p a p y r i m u s t h a v e h a d with the m d i v i d u a l p r o b l e m s a n d their algorithms:

Exceptions to this rule indicate a problem in the source text. One such example is pRhind, problem 69. A discussion of this example can be found in I M H A U S E N (in prep, b), §7.2.1.1. In my analysis of the problem texts parallel steps are rendered by using identical numbers with alphabetical differentiation, like (la), (lb), (Ic) etc. while iterative steps are represented by identical numbers and a series of apostrophes, like (1'), (1"), (1"') etc.

154

A. Imhausen

( 1 ) S o m e o f the d a t a are n o t explicitly n a m e d . (2) S o m e o f the instructions are missing. (3) O n e g i v e n instruction contains m o r e t h a n o n e arithmetical o p e r a t i o n . D a t a that h a v e b e e n forgotten in the introduction o f the p r o b l e m , a n d are t h e n u s e d in the instructions, c a n b e r e c o g n i z e d during the establishing of the s y m b o l i c algoritlun, since t h e y d o n o t result from one of the p r e v i o u s steps. I n the case o f the m i s s i n g instructions, in s o m e instances at least, a calculation h a s b e e n n o t e d from w h i c h it is possible to r e c o n s t r u c t the instruction. If not, h o w e v e r , the step has to b e filled in. T h i s is m a r k e d in the a l g o r i t h m as a reconstruction. H o w e v e r , especially if m o r e t h a n a single arithmetic o p e r a t i o n is left out, there is often m o r e than o n e w a y to reconstruct it. T h e s a m e k i n d o f p r o b l e m is caused, if o n e g i v e n instruction contains m o r e than o n e arithmetical operation. T h e s e m u s t then b e s e p a r a t e d first. T h e a b o v e m e n t i o n e d p r o b l e m s are n o t inherent t o the algorithmic analysis; rather they s h o w the limits of information the s o u r c e text supplies. 4.2 T h e A l g o r i t h m i c A n a l y s i s A p p l i e d to t h e '^^'^-Problems I will n o w p r e s e n t the algorithmic analysis a p p l i e d t o the p r o b l e m s of o n e g r o u p , n a m e l y the *^/i^-problems.^^ O n e e x a m p l e of t h e m p a p y r u s R h i n d , n o . 2 6 h a s a l r e a d y b e e n d i s c u s s e d in the p r e v i o u s sections ( § 3 a n d §4.1). G i v e n is a n u n k n o w n quantity, called ^ifi^ w h i c h is the E g y p t i a n t e r m designating an a m o i m t or a quantity. Several fractions or, as in other e x a m p l e s , multiples o f this quantity, are a d d e d to it a n d the r e s u h of this is given. In thefr past treatment, '^/I'^-problems h a v e b e e n interpreted as the E g y p t i a n equivalents of algebraic equations. R e p e a t e d l y the t e r m '^h' h a s b e e n identified in those interpretations w i t h the m o d e m variable x o f a n algebraic e q u a t i o n . H o w e v e r , a l r e a d y O t t o N e u g e b a u e r r e m a r k e d that this simple e q u i v a l e n c e is n o t correct: „DaJi es sich offenbar um zwei gesuchte Grofien handelt, scheint mir besonders beachtenswert und Jur den Unterschied zwischen moderner und agyptischer Betrachtungsweise charakteristisch. Der modernen Auffassung geniigt es, die eine Unbekannte x zu bestimmen, welche der vorgelegten linearen Gleichung geniigt; der Àgypter dagegen sucht n a c h d e n e i n z e l n e n S u m m a n d e n , aus denen sich die gegebene rechte Seite aufbauen soil, und nennt sie demgemafi einzeln itn Résultat. " In the interpretations of the E g y p t i a n p r o b l e m texts as p r e c u r s o r s o f algebraic freatments, the '^-problems h a v e r e c e i v e d special attention from m o d e m r e a d e r s . T h e individual p r o b l e m h a s t h e n usually b e e n r e n d e r e d as algebraic e q u a t i o n a n d b e e n t a k e n as a p r e c u r s o r of algebraic p r o b l e m s . T h e scholarly opinio communis e n d s , h o w e v e r , as s o o n as the m e t h o d u s e d for their solution is c o n c e m e d : the first editor o f the R h i n d p a p y m s , A u g u s t Eisenlohr, interpreted the m s t m c t i o n s as

The translations and algorithms of these problems can be found in the appendix of this article. N E U G E B A U E R ( 1931 ), p.

308.

The Algorithmic Structure of the Egyptian Mathematical Problem Texts

155

equivalents to the transformation of equations.^^ T h o m a s Eric P e e t stated in his edition o f the s a m e text that the instructions are without d o u b t to b e interpreted as the " m e t h o d o f false p o s i t i o n " . ^ ' T h e m e t h o d o f false p o s i t i o n to solve a p r o b l e m a s s m n e s in its first step a n arbitrary ( a n d therefore p r o b a b l y w r o n g ) " s o l u t i o n " that e n a b l e s a n e a s y calculation o f the left side o f the given equation. W i t h this " s o l u t i o n " the v a l u e o f the right p a r t of the e q u a t i o n is d e t e r m i n e d . T h e correct solution is t h e n o b t a i n e d b y d e t e r m i n i n g the c o r r e s p o n d i n g "scaling factor" a c c o r d i n g to = c •c

X • X

^ • -^wrong

^ • wrong •

W i t h the p u b l i c a t i o n of the M o s c o w p a p y r u s and the B e r l i n m a t h e m a t i c a l fragments, fiuther e x a m p l e s o f '^/I'^-problems b e c a m e k n o w n ; h o w e v e r , the p r o b l e m o f their m e t h o d o f solution r e m a i n e d unsolved. In o r d e r to a v o i d d i s a p p o i n t m e n t , let m e say first that I will n o t e n d that a r g u m e n t b y p r o v i n g that o n e of the p o s i t i o n s is correct. Instead o f r e d u c i n g the available c o r p u s to the a f o r e m e n t i o n e d modem interpretation, I will first d e s c r i b e the individual p r o b l e m s b y m e a n s o f their algorithms, and, b a s e d o n this description a d d a few further r e m a r k s c o n c e m i n g their solution. T h e available c o r p u s consists of fifteen p r o b l e m s that are found in the R h i n d , M o s c o w , Illahun a n d B e r l i n papyri. T h e a p p e n d i x s h o w s m y franslation a n d the a l g o r i t h m of e a c h p r o b l e m . A c c o r d i n g to t h e n algorithms, these p r o b l e m s c a n b e assigned to five different g r o u p s : T h e first g r o u p consists of p R h i n d , n o s . 2 4 to 26.^' T h e y are the first ''/i'^p r o b l e m s in the R h i n d p a p y m s . T h e foiu p r o b l e m s are all similar t o e a c h other in their s t m c t u r e : a quantity ("h') a n d a g i v e n fraction ( D i ) of that quantity shall b e a d d e d a n d result in a g i v e n a m o u n t (D2). B e s i d e s the s a m e arithmetical o p e r a t i o n s to d e t e r m i n e '^h', all o f t h e m calculate the initially n a m e d fraction o f '^h' as well. This is explicitly stated m the text of p R h i n d , n o . 2 6 , a n d i n c l u d e d in the verifications of all the other p r o b l e m s o f this g r o u p . T h e a l g o r i t h m itself u s e s an auxiliary n u m b e r that is d e t e r m i n e d m the first step as the inverse o f the first d a t u m . T h i s a l g o r i t h m is g i v e n as a series o f i n s t m c t i o n s only in p R h i n d , n o . 2 6 . In all o f the other e x a m p l e s it c a n b e reconstiiicted b a s e d o n the calculations that h a v e b e e n noted. T h e first step, the d e t e r m i n a t i o n o f the

E I S E N L O H R (1877), p. 6 0 - 6 1 . This was criticized first by Léon Rodet who introduced the "method of false position" into the discussion, see R O D E T ( 1881 ), p. 188. PEET

(1923a), p 60.

Kurt Vogel gives the following explanation with pRhind, No. 24 ("A quantity, its 7 to it, it results as 19"): "Fiir die Unbekannte wird eine 'Versuchszahl' angenommen. Die Zahl ist nicht vollig willkUrlich sondern mufi, wenn sie ihren Zweck gut erfullen soil, durch 7 teilbar sein. Diese Versuchszahl wird in die linke Seite der Gleichung eingesetzt. Erhalt man nun bei angenommener Versuchszahl als Résultat die Zahl 8 statt, wie es wirklich sein solite, die Zahl 19, so zeigt sich, dafi das richtige Résultat ein Vielfaches des 'Versuchsresultates ' ist. Die zahlenmafiige Bestimmung dieses Proportionalitatsfaktors geschieht durch die Division 19:8= 2 1 " . Schliefilich mufi, da das richtige Resultai 21- mal so grofi ist als das Versuchsresultat, auch die Versuchszahl selbst mit diesem Faktor 2 - multipliziert werden. " V O G E L (1930), p. 137. The hieratic text can be found in

ROBINS

and S H U T E ( 1 9 8 7 ) , pi. 10-11.

156

A. Imhausen

auxiliary n u m b e r , is n o t explicitly stated in a n y o f the p r o b l e m s . p R h i n d , n o . 2 6 starts with the instruction "Calculate with 4 ! " , w h i c h n a m e s the auxiliary n u m b e r u s e d in this case b u t n o t the calculation of it. T h e reconstruction o f the first step in this algorithm is b a s e d o n the g i v e n data and the auxiliary n u m b e r that is u s e d in the s e c o n d step. T h e s e c o n d g r o u p , w h i c h is formed b y p R h i n d , n o s . 2 8 a n d 2 9 s h o w s a m o r e c o m p l e x p r o b l e m to solve.^^ T h e texts of b o t h p r o b l e m s are not without difficulties. C a u s e d p r o b a b l y b y the shnilarity o f values u s e d in them, the t w o p r o b l e m s h a v e b e e n written d o w n as if they were o n l y one.^^ T h e text starts with a i m o u n c i n g the p r o b l e m of N o . 2 8 a n d the instructions to solve it. F o l l o w i n g the s e q u e n c e of instructions of N o . 2 8 the calculations of N o . 29 are noted. F r o m t h o s e calculations it has b e e n possible to reconstruct the p r o b l e m of N o . 2 9 , as well as the last instruction of the solution a n d the following verification. P r o b l e m 2 8 n a m e s t w o firactions ( 3 = D] a n d 3 = D2) of a n u n k n o w n quantity C/I'^. T h e first is a d d e d to it. F r o m this sum, the s e c o n d fraction is t a k e n a w a y . T h e result of these operations is given as the last d a t u m ( 1 0 = D 3 ) . T h e solution o f this p r o b l e m is distinctively different fi-om the p r o b l e m s before: the t w o instructions to solve it are g i v e n in o n e line of text, the r e m a i n i n g text constitutes foiu instructions of the verification. This solution requires a specific structure o f the p r o b l e m that is i n d e e d g i v e n in this case: the s u m o f the "coefficients" of the u n k n o w n quantity ("h') c a n b e written in m o d e m notation as — . T h e multiplicative factor that is u s e d in the first step c a n b e d e t e r m i n e d as the inverse o f ( m + 1 ) . P r o b l e m 2 9 d o e s not fiilfiU the conditions for this kind o f algorithm. N e u g e b a u e r a s s u m e d a solution like the o n e s in p R h i n d , n o s . 24-27.^^ T h i s is p o s s i b l e ; h o w e v e r , this d o e s n o t m a k e u s e o f the last instmction, w h i c h is the only o n e that is p r e s e r v e d in this case. T h e r e f o r e , I w o u l d rather p r o p o s e a solution that u s e s a n auxiliary n u m b e r (27) to multiply the individual fractions a n d fractions o f fractions. T h e n these are a d d e d or subfracted as required b y the p r o b l e m . Finally, the auxiliary n u m b e r is d i v i d e d b y the result of this p r o c e d u r e . T h e result of this step is multiplied with the last d a t u m (D3 = 1 0 ) w h i c h constitutes the o n l y r e c o n s t m c t e d ( b a s e d o n the calculation that w a s n o t e d ) instmction. Following this, five fiirther calculations w e r e n o t e d that represent the verification. C o m p a r e d to the verification of p r o b l e m 2 8 , three steps are equal. H o w e v e r , due to the lack of text, a c o m p a r i s o n b e t w e e n the s e q u e n c e s o f i n s t m c t i o n s is not possible. T h e third g r o u p consists of p R h i n d , nos. 3 0 to 3 4 and one p r o b l e m from the M o s c o w p a p y m s (no. 25).^^ Thefr solution is basically a two-step a l g o r i t h m formed

It is remarkable, that they could have been solved by applying the algorithm of pRhind, No. 24-27 twice. This, however, is not used. "

Cf

ROBINS

and S H U T E ( 1 9 8 7 ) , pi. 10-11.

Peet and Neugebauer assume that the given instructions are caused by the fact that the scribe knew the answer to the problem beforehand, cf P E E T (1923a), p. 63 and N E U G E B A U E R , (1931), p. 343. "

^

NEUGEBAUER

(1931), p. 343.

For the hieratic text of these problems cf S T R U V E (1930), pi. 10, col. X L V .

ROBINS

and

SHUTE

(1987), pi. 10-Î3 and

The Algorithmic Structure of the Egyptian Mathematical Problem Texts

157

b y a n addition followed b y a division. T h i s g r o u p c a n b e divided, a c c o r d i n g to the c o m p l e x i t y o f the p r o b l e m s , into three s u b g r o u p s . T h e first s u b g r o u p contains only p r o b l e m 3 0 o f the R h i n d p a p y r u s . It is a " p r e c u r s o r " o f the other p r o b l e m s , since the first step o f the general algorithm, the addition, is missing. G i v e n is a fraction (D2 = 3 10 ) of an u n k n o w n quantity ('^h') w h i c h shall b e a g i v e n quantity (Di = 10). T h e s e c o n d s u b g r o u p also contains only o n e p r o b l e m , p r o b l e m 2 5 of the M o s c o w p a p y r u s . G i v e n is a n u n k n o w n quantity^' to w h i c h a m u l t i p l e (Di = 2) o f it shall b e a d d e d . T h e result of this is g i v e n ( D 3 = 9 ) . T h e a l g o r i t h m d e t e r m i n e s the s u m of the multiples o f the quantity, a n d then divides the g i v e n result ( D 3 ) b y this siun. T h e third s u b g r o u p consists o f four p r o b l e m s s h o w i n g a similar p r o b l e m : a n i m k n o w n quantity {'^h') is n a m e d , a n d a niunber o f fractions o f that quantity that shall b e a d d e d to it. T h e result of this addition is given. T h e a l g o r i t h m to solve these four p r o b l e m s consists o f t w o steps: a n addition followed b y a division. A certain variation within this s u b g r o u p c a n b e n o t e d on the arithmetical level o f executing the instructions: C a u s e d b y the d e p e n d a n c y of the division o n the n u m e r i c values chosen, the division is executed in a n u m b e r o f s t e p s . T h e instructions o f the algorithms h a d to b e reconstructed for all o f the p r o b l e m s b a s e d o n the calculations. T h e fourth g r o u p includes one e x a m p l e from p M o s c o w (no. 19) a n d o n e from the Illahun fragments ( p U C 3 2 1 3 4 = p K a h u n L V . 3 ) . T h e text o f the Illahun e x a m p l e h a d to b e partly restored, a n d only s o m e o f the data a n d the instructions are extant. P r o b l e m 19 o f the M o s c o w p a p y r u s differs from all other '^/i''-problems: Instead o f a fraction of the u n k n o w n quantity, a multiplicative factor is g i v e n ( D i = 1 2 ) . T h e n another n u m b e r (D2 = 4) is a d d e d , the result o f this is g i v e n as the third d a t u m (D3= 10). In p U C 3 2 1 3 4 a n i m k n o w n quantity is given. F r o m this quantity a g i v e n fraction shall b e subfracted, a n d the result is given. A close c o m p a r i s o n of the algorithms reveals that their similarity is n o t only a formally identical s e q u e n c e o f arithmetical operations. In b o t h p r o b l e m s , the u n k n o w n quantity is infroduced as a p r o d u c t with a given multiplicative factor ( D i ) . ^ ' In p U C 3 2 1 3 4 a n o t h e r p r o d u c t of the u n k n o w n quantity with a multiplicative factor is subfracted from the first p r o d u c t . T o d e t e r m i n e the u n k n o w n quantity, the result of the multiplications is n e e d e d . T h i s is g i v e n m p U C 3 2 1 3 4 as the thfrd d a t u m . In p M o s c o w , n o . 19 it has to b e d e t e r m i n e d from t w o data (D2 a n d D 3 ) , w h i c h is d o n e in the first step of its algorithm. T h e t w o final steps o f the a l g o r i t h m d e t e r m i n e the inverse o f the multiplicative factor a n d multiply this m v e r s e with the result of the m e n t i o n e d operations. T h e t w o p r o b l e m s differ in the ffrst step. W h i l e the subtraction o f p M o s c o w , no. 19 results in the multiplicative factor o f the u n k n o w n quantity, the subfraction o f p U C 3 2 1 3 4 calculates the result of the indicated operations. T h e fifth a n d last " g r o u p " s h o w s o n l y one representative, w h i c h is o u t s t a n d i n g b e c a u s e it is the o n l y e x a m p l e that involves s q u a r m g a n d the exfraction o f square " The mentioning of 'h" is included within the data of this problem ( D 2 = 1) since the algorithm makes use of it in its first step. All of these divisions involve fractions and are executed in more than one step. The calculation o f pRhind, no. 31 has been misplaced within the text of another problem. In the case of pUC32134 the multiplicative factor is 1.

158

A. Imhausen

roots. A p a r t from that pecuUarity, the algorithm is similar to those fr)und in the third group. In this p r o b l e m , it is not the multiples o f the u n k n o w n quantity ('^h') that are subject to the indicated operations, b u t the squares o f these multiples instead. Unfortunately the soiu-ce text is rather fragmentary a n d h a d to b e r e c o n s t r u c t e d to a certain extent. T h e instructions g i v e n s e e m to indicate that, imlike the other p r o b l e m s , the a l g o r i t h m w a s not a n n o i m c e d sfraightforwardly, b u t included recapitulations, r e p e a t i n g information that h a d b e e n g i v e n at the b e g i n n i n g o f the problem. T o s u m m a r i z e , the analysis of the ^-problems s h o w s that there are several distinct sfrategies for thefr solution. T h e difficulties of interpreting t h e m as algebraic equations w a s m e n t i o n e d in the b e g i n m n g , b u t the description of the g i v e n instructions as fransformations of equations is equally p r o b l e m a t i c . T h e instructions g i v e n to solve the p r o b l e m s of the first g r o u p are similar to the m e t h o d o f false position. H o w e v e r , the "false v a l u e " is n e v e r explicitly m e n t i o n e d as such. M o r e o v e r , it is n o t arbifrarily c h o s e n b u t d e t e r m i n e d in the first step as the inverse of one o f the g i v e n data. T h e solutions o f the other p r o b l e m s c a n h a r d l y b e interpreted as u s i n g the m e t h o d o f false position. Finally I w o u l d like to r e m a r k that the a r r a n g e m e n t o f the ''/i'^-problems m the R h i n d p a p y r u s g r o u p s t h o s e p r o b l e m s with identical or similar a l g o r i t h m s neatly together. T h e ^/i*^-problems t e a c h h o w to h a n d l e relations o f u n k n o w n quantities. D e p e n d i n g o n the n a t i n e of the given p r o b l e m the appropriate a l g o r i t h m h a s to b e chosen.

5. Conclusions T h e algorithmic analysis o f the mathematical p r o b l e m texts allows a d e s c r i p t i o n of E g y p t i a n m a t h e m a t i c a l techniques that goes b e y o n d r e n d e r i n g a m a t h e m a t i c a l p r o b l e m as a n algebraic e q u a t i o n w i t h one i m k n o w n . It enables a m o d e m r e a d e r to follow the m d i c a t e d solution a n d to c o m p a r e the m d i v i d u a l p r o b l e m s w i t h e a c h other. F u r t h e r m o r e , a c o m p a r i s o n to other algorithmic m a t h e m a t i c a l texts is in principle possible a n d s e e m s p r o m i s i n g to m e . T h e s e are, o n the o n e h a n d (staying within the E g y p t i a n fradition) the D e m o t i c p r o b l e m texts. A c o m p a r i s o n o f the algorithms found in the hieratic mathematical texts, m a i n l y assigned to the M i d d l e K i n g d o m with algorithms found in the D e m o t i c m a t h e m a t i c a l texts m i g h t p r o v i d e clues to the d e v e l o p m e n t o f m a t h e m a t i c a l techniques over time. O n the other h a n d a c o m p a r i s o n with other mathematical cultures is p o s s i b l e as well, since the transmission o f m a t h e m a t i c a l k n o w l e d g e in forms o f algorithms is found m n u m e r o u s cultures. F o r instance, M e s o p o t a m i a n as well as C h i n e s e m a t h e m a t i c a l texts s h o w sùnilar features.'*" A c o m p a r i s o n of E g y p t i a n w i t h M e s o p o t a m i a n a n d Chinese algorithms m a y result in r e v e a l m g sunilarities a n d differences b e t w e e n the individual m a t h e m a t i c a l cultures. A l t h o u g h the p r o b l e m s that are faced in these texts are occasionally similar, their w a y of solution m a y differ considerably. T h i s has

For the algorithms of the Mesopotamian texts see Chinese algorithms see C H E M L A (1990 and 1991).

RITTER

(1989 and 1998). For the

The Algorithmic Structure of the Egyptian Mathematical Problem Texts

159

a l r e a d y b e e n s h o w n b y J a m e s Ritter for s o m e e x a m p l e s of E g y p t i a n a n d Mesopotamian problems/* T h e available sources o n the E g y p t i a n side are, as I stated initially, v e r y m e a g e r . H o w e v e r , t h e y are as yet far from b e i n g exhaustively studied.

6. References A L L E N , J A M E S P . 1992. " R e v i e w : M a r s h a l l Clagett, A n c i e n t E g y p t i a n Science. V o l i u n e I: K n o w l e d g e a n d Order, Philadelphia: A m e r i c a n P h i l o s o p h i c a l Society 1989"./5W 8 3 : 117. — 1 9 9 6 . " R e v i e w : M a r s h a l l Clagett, A n c i e n t E g y p t i a n Science. V o l u m e II: C a l e n d a r s , C l o c k s , a n d Asfronomy, Philadelphia: A m e r i c a n P h i l o s o p h i c a l Society 1 9 9 5 " . Isis 8 7 : 3 4 3 - 3 4 4 . B E N O I T , P a u l ; C H E M L A , K a r i n e ; a n d RITTER, J a m e s . 1992. Histoire de fractions, fractions d'histoire. B a s e l , B o s t o n , B e r l m : B i r k h a u s e r V e r l a g . C H A C E , A m o l d B . ; B U L L , L u d l o w ; M A N N I N G , H e n r y P . ; and A R C H I B A L D , R a y m o n d C. 1927 a n d 1929. The Rhind Mathematical Papyrus: British Museum 10057 and 10058 (2 B d e . ) . O b e r l i n ( O h i o ) : M a t h e m a t i c a l A s s o c i a t i o n o f A m e r i c a . C H E M L A , K a r i n e . 1990. " D e l ' a l g o r i t h m e c o m m e liste d ' o p é r a t i o n s " . Extrême Orient, Extrême Occident 12: 7 9 - 9 4 . — 1 9 9 1 . " T h e o r e t i c a l A s p e c t s o f the Chinese A l g o r i t h m i c T r a d i t i o n (First to Third C e n t u r y ) " . Historia Scientiarum 4 2 : 7 5 - 9 8 . CLAGETT, M A R S H A L L . 1989. Ancient Egyptian Science. Volume I: Knowledge Order. Philadelphia: A m e r i c a n Philosophical Society. — 1 9 9 5 . Ancient Egyptian Science. Volume II: Calendars, Philadelphia: A m e r i c a n Philosophical Society.

Clocks,

and

and

Astronomy.

— 1 9 9 9 . Ancient Egyptian Science. Volume III: Ancient Egyptian Mathematics. Philadelphia: A m e r i c a n Philosophical Society. D A R E S S Y , G e o r g e s . 1 9 0 1 . Catalogue général des antiquités égyptiennes du Musée du Caire: Nos 25001-25385: Ostraca. Kafro: Institut français d ' a r c h é o l o g i e orientale d u Cafre. EISENLOHR, A u g u s t . 1877. Ein mathematisches Handbuch der alten Àgypter. Leipzig: H i m i c h s (reprint W a l l u f bei W i e s b a d e n : Dr. M a r t i n S à n d i g o H G 1972). F O W L E R , D a v i d . 1999. T h e M a t h e m a t i c s of P l a t o ' s A c a d e m y . A N e w R e c o n s t r u c t i o n ( S e c o n d Edition). Oxford: C l a r e n d o n Press. F R E U D E N T H A L , H a n s . 1977. " W h a t is A l g e b r a a n d W h a t H a s B e e n its H i s t o r y ? " . Archive for History of Exact Sciences 16: 1 8 9 - 2 0 0 . GILLINGS, R i c h a r d . 1972. Mathematics in the Time of the Pharaohs. Cambridge, M a s s . : M I T Press. GLANVILLE, S t e p h e n R.K. 1927. " T h e M a t h e m a t i c a l L e a t h e r Roll m the British M u s e u m " . Journal of Egyptian Archaeology 13: 2 3 2 - 2 3 9 . GRIFFITH, F r a n c i s L. 1898. Hieratic Papyri from Kahun and Gurob. London: Quaritch. H A Y E S , W i l l i a m C. 1942. Ostraca and Name Stones from the Tomb of Sen-Mut (No. 71) at Thebes. N e w Y o r k : Mefropolitan M u s e u m o f Art.

RITTER

(1989 and 1998).

160

A. Imhausen

I M H A U S E N , A n n e t t e . 1999. " D i e M a t h e m a t i s i e r u n g v o n B r o t i m d B i e r i m A l t e n Agypten". In: D a n y B E C K E R S , Katja P E T E R S a n d C a r s t e n V O L L M E R S (eds), 9. Novembertagung zur Geschichte der Mathematik, Nijmegen, 29.X-1.XI1998: 1 2 - 2 1 . N i j m e g e n : n o publisher. — in p r e p . a. " E g y p t i a n M a t h e m a t i c a l Texts a n d their C o n t e x t s " . T o a p p e a r in: Science in Context. — in p r e p . b . Àgyptische Algorithmen. Eine Untersuchung zu den mittelagyptischen mathematischen Aufgabentexten. Wiesbaden: Harrassowitz. V O N L I E V E N , A l e x a n d r a . 2 0 0 0 . Der Himmel iiber Esna: Eine Fallstudie zur religiosen Astronomie in Agypten am Beispiel der kosmologischen Decken- und Architravinschriften im Tempel von Esna ( A g y p t o l o g i s c h e A b h a n d l u n g e n 64). Wiesbaden: Otto Harrassowitz. LOPEZ, Jesus. 1980. Ostraca Ieratici. N. 57093-57319: Catalogo del Museo Egizio di Torino, Serie Seconda - Collezioni, Vol. Ill, Fasciola 2. M i l a n o : Istituto Editoriale Cisalpino-LaGoliardica. M A L A I S E , M i c h e l a n d W I N A N D , Jean. 1999. Grammaire raisonnée de l'égyptien classique ( i ï g y p t i a c a Leodiensia 6). Liège: Centre Informatique d e P h i l o s o p h i e et Lettres. N E U G E B A U E R , O t t o . 1 9 3 1 . "Arithmetik u n d R e c h e n t e c h n i k der À g y p t e r " . In: Otto N E U G E B A U E R , Julius STENZEL, Otto TOEPLITZ, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, AbteilungB: Studien, Bd. I: 3 0 1 - 3 8 0 . Berlin: Julius Springer. — 1 9 7 5 . A History of Ancient Mathematical Astronomy. Part Two. N e w Y o r k , H e i d e l b e r g , Berlin: Springer Verlag. — a n d P A R K E R , R i c h a r d . 1960. Egyptian Astronomical Texts ( B r o w n E g y p t o l o g i c a l Studies 3 ) . P r o v i d e n c e : B r o w n University P r e s s a n d L o n d o n : L. H u m p h r i e s . P E E T , T h o m a s E. 1923a. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. Introduction, Transcription, Translation and Commentary. London: H o d d e r & S t o u g h t o n Limited (reprint: N e n d e l n (Liechtenstein): K r a u s R e p r i n t 1970). — 1923b. "Arithmetic in the M i d d l e K i n g d o m " . Journal of Egyptian Archaeology 9: 9 1 - 9 5 . R I T T E R , J a m e s . 1989. " C h a c u n sa vérité: les m a t h é m a t i q u e s e n Egypte et e n M é s o p o t a m i e " . In: M i c h e l S E R R E S (éd.), Éléments d'histoire des sciences: 3 9 - 6 1 . Paris: B o r d a s . — 1 9 9 0 . " L e s p r a t i q u e s de la raison e n M é s o p o t a m i e " . In: J e a n - F r a n ç o i s M A T T É I (éd.), L a naissance de la raison e n G r è c e . A c t e s d u congrès d e N i c e , M a i 1987: 9 9 - 1 1 0 . Paris: P r e s s e s U n i v e r s i t a k e s de F r a n c e . — 1 9 9 5 . " M e a s u r e for M e a s u r e : M a t h e m a t i c s in E g y p t a n d M e s o p o t a m i a " . In: M i c h e l S E R R E S (ed.), A History of Scientific Thought. Elements of a History of Science: 4 4 - 7 2 . Oxford a n d C a m b r i d g e , M a s s . : B l a c k w e l l P u b l i s h e r s Inc. — 1 9 9 8 . Reading Strasbourg 368: A Thrice-Told fur Wissenschaftsgeschichte 103). B e r l m : schaftsgeschichte.

Tale (Preprint M a x - P l a n c k - I n s t i t u t Max-Planck-Institut fiir W i s s e n -

— 2 0 0 0 . " E g y p t i a n M a t h e m a t i c s " . In: Helaine SELIN (ed.), Mathematics Across Cultures. The History of Non-Western Mathematics: 115-136. Dordrecht/ Boston/London: Kluwer.

The Algorithmic Structure of the Egyptian Mathematical Problem Texts

161

R O B I N S , G a y and S H U T E , Charles. 1987. The Rhind Mathematical Papyrus. An Ancient Egyptian Text. L o n d o n : British M u s e u m . R O B S O N , Eleanor. 1999. Mesopotamian Mathematics, 2100 - 1600 BC. Technical Constants in Bureaucracy and Education. Oxford: C l a r e n d o n P r e s s . R O D E T , L é o n . 1 8 8 1 . " L e s p r é t e n d u s p r o b l è m e s d ' a l g è b r e du m a n u e l d u calculateur égyptien ( P a p y r u s R h i n d ) " . Journal Asiatique

( S e p t i è m e Série) 18: 1 8 4 - 2 3 2 i m d

390-459. S C H A C K - S C H A C K E N B U R G , H a n s . 1900. " D e r Berliner P a p y r u s 6 6 1 9 (mit 1 Tafel)". Zeitschrift fur Àgyptische Sprache und Altertumskunde 38: 135-140. — 1 9 0 2 . " D a s kleinere F r a g m e n t des Berliner P a p y r u s 6 6 1 9 " . Zeitschrift fur Àgyptische Sprache und Altertumskunde 40: 6 5 - 6 6 . SPALINGER, Antìiony. 1997. " R e v i e w : M a r s h a l l Clagett, A n c i e n t E g y p t i a n Science. V o l u m e II: C a l e n d a r s , C l o c k s , a n d A s t r o n o m y , Philadelphia: A m e r i c a n P h i l o s o p h i c a l Society 1 9 9 5 " . Bibliotheca Orientalis 5 4 : 6 7 7 - 6 8 4 . S T R U V E , W a s i l i W . 1930. Mathematischer Papyrus des staatlichen Museums der schônen Kùnste in Moskau (Otto N E U G E B A U E R , Julius STENZEL, O t t o TOEPLiTZ (eds.), Q u e l l e n i m d Studien zur Geschichte der M a t h e m a t i k , A s t r o n o m i e i m d Physik, A b t e i l i m g A : Quellen, B d . 1), Berlin: Julius Springer. U N G U R U , Sabetai. 1 9 7 5 . " O n the N e e d to R e w r i t e the H i s t o r y of G r e e k M a t h e m a t i c s " . Archive for History of Exact Sciences 15: 6 7 - 1 1 4 . — 1 9 7 9 . " H i s t o r y o f A n c i e n t M a t h e m a t i c s : S o m e Reflections o n the State o f the A r t . " Isis 7 0 : 5 5 5 - 5 6 5 . — a n d RowE, D a v i d E . 1 9 8 1 . " D o e s the Quadratic E q u a t i o n h a v e G r e e k R o o t s ? A Study of 'Geometric Algebra', 'Application of Areas', and Related Problems (Part 1)". Libertas Mathematica 1: 1 ^ 9 . — a n d RowE, D a v i d E. 1982. " D o e s the Quadratic E q u a t i o n h a v e G r e e k R o o t s ? A Study of ' G e o m e t r i c A l g e b r a ' , ' A p p l i c a t i o n of A r e a s ' , a n d R e l a t e d P r o b l e m s (Part 2 ) " . Libertas Mathematica 2: 1-62. — a n d F R I E D , M i c h a e l . 2 0 0 1 . Apollonius ofPerga's Conica: Text, Context, Subtext. Leiden, B o s t o n : Brill. V A N D E R W A E R D E N , B a r t e l L. 1976. "Defense o f a ' s h o c k i n g ' p o i n t o f v i e w " . Archive for History of Exact Sciences 15: 1 9 - 2 1 0 . V O G E L , Kurt. 1930. " D i e A l g e b r a der À g y p t e r des M i t t l e r e n R e i c h e s " . Archeion 12: 126-162. W E I L , A n d r é . 1978. " W h o B e t r a y e d E u c l i d ? " . Archive for History of Exact Sciences 19: 9 1 - 9 3 .

A. Imhausen

162

Appendix: Translations and Algorithms of the *^^*'-Problems pRhind, n o . 24 Translation ' A quantity, its 7 to it, it results as 19.

Algorithm

Symbolical

Algorithm

D,

7

D2

19 [ 1 : 7 = 7]

(1)

[1:D,]

7-7 = 1

(2)

Dr(l)

7+1=8

(3)

( l ) + (2)

[calculation]

19 : 8 = 2 4 8

(4)

D2:(3)

[calculation]

248 • 7 = 1628

(5)

(4)-(l)

[calculation] '^'^

Numerical

[verification]

p R h i n d , n o . 25 Translation ' A q u a n t i t y , its 2 to it, it results as 16. [calculation]

Numerical

Algorithm

Symbolical

2

D,

16

D2

[1: 2 = 2 ]

(1)

[1:D.]

2-2=1

(2)

D,(l)

Algorithm

2 + 1=3

(3)

( l ) + (2)

[calculation]

16:3=53

(4)

D2:(3)

[calculation]

5 3 - 2 = 103

(5)

(4)-(l)

[verification]

p R h i n d , n o . 26 Translation ' A quantity, its 4 t o it. it results as 15.

Numerical

Algorithm

Symbolical

Algorithm

D,

4

D2

15

Calculate with 4 !

[1: 4 ] = 4

(1)

[l:Di]

T h e n y o u calculate its 4

4 -4= 1

(2)

Dr(l)

4+1=5

(3)

( l ) + (2)

15 : 5 = 3

(4)

D2:(3)

3 - 4 = 12

(5)

( 4 ) • (1)

4-12 = 3

(6)

D,(5)

as 1. Total 5. ^ D i v i d e 15 b y 5 ! •^"•^

[calculation]

^ T h e n 3 results. Multiply 3 by 4! [calculation] ^ T h e n 12 results. '""''[calculation of part] [verification]

163

The Algorithmic Structure of the Egyptian Mathematical Problem Texts

oRhind, no. 27 Translation ' A q u a n t i t y , its 5 to it. it results as 2 1 .

Numerical

Algorithm

Symbolical

Algorithm

D,

5

D2

21 [1:5=5]

(1)

[1:D,]

(2)

D,-(l) ( l ) + (2)

^"^

[calculation]

5-5 = 1 5 + 1=6

(3)

'^'^

[calculation]

21 : 6 = 3 2

(4)

D2:(3)

[calculation]

3 2 - 5 = 172

(5)

(4)-(l)

^'''^

[verification]

pRhind, no. 28 Translation '

Numerical

Algorithm

Symbolical

Algorithm

3 as a d d e d

3

D,

3 as s u b t r a c t e d

3

D2

10

D3

10areleft._ ^ Calculate 10 o f this 10.

10-10 = 1

(1)

10 - D j

10- 1 =9

(2)

D3-(l)

T h e n 1 results, so that the r e m a i n d e r is 9. ^

Its

3

(PI)

D,-(2)

9 + 6 = 15

(P2)

(2) + ( P I )

3-15 = 5

(P3)

D2(P2)

1 5 - 5 = 10

(P4)

(P2)-(P3)

3

as 6

is a d d e d to it.

-9 = 6

so that the s u m is 15 Its 3 is 5. It is 5 that h a s b e e n subtracted. so that the r e m a i n d e r is 10. Calculation as it results.

pRhind, no. 29 Translation

Numerical

Algorithm

Symbolical

3

Di

3

D2

3

D3

10

D4

[...]

_ y (x+1)

(x)-D4

(PI)

D , • (x+1)

13 2 +9 = 222

(P2)

(x+1) + ( P I )

1> -222

=12

(P3)

D2 • ( P 2 )

22 2 + 7 2

=30

(P4)

(P2) + (P3)

(P5)

D3

' ^

[calculation]

1410

^

[calculation]

3

[calculation]

7-8

[calculation]

Algorithm

-132

1 0 = 13 2 =9

3 - 3 0 = 10

• (P4)

164

A. Imhausen

)Rhind, no. 30 Translation

Numerical

Algorithm

' If a scribe tells you: 10

Symbolical

10

resulted as 3 10 of w h a t ?

3

Algorithm D,

D2

10

H e shall hear: T h e n y o u divide 10 b y 3 10 .

10: 3 Î Ô

=1323

(1)

D,:D2

(PI)

(1)-D2=D,

[calculation] [calculation]

1323 •

3ÏÔ

= 10

) R h i n d , n o . 31 Translation '

Numerical

Algorithm

Symbolical

A quantity,

1

D,

its 3 ,

3

D2

its 2

2

D3

its 7 are a d d e d to it

7

D4

so that 33 results.

33

D5

[calculation]

Algorithm

[1+3 + 2 + 7 = 1 3 2 7 ]

(1)

D1+D2 +

33 : 1 3 2 7

(2)

D5:(l)

D3+D4

pRhind, no. 32 Translation '

Numerical

A quantity.

Algorithm

Symbolical

Algorithm

1

D,

its 3 a n d

3

D2

its 4 to it

4

D3

so that 2 results.

2

D4

^"^•^

D2 +

1 + 3 + 4 = 1 3 4

(1)

D,+

[calculation]

2 : 1 3 4 = 1 6 12 114 2 2 8

(2)

D4:(l)

[verification]

1 3 4 - 1 6 12 114 2 2 8 = 2

(P)

(1)(2)=D4

D3

pRhind, no. 33 Translation '

Numerical

Symbolical

Algorithm

Algorithm

A quantity.

1

D,

its

3,

3

D2

its 2 a n d

2

D3

its 7 to it

7

D4

so that 3 7 results.

37 1+3 + 2 + 7 = 1 3

'•^•^^

D5 2

7

(1)

D1+D2 +

D3+D4

[calculation]

37:13

2 7 = 1656 573 776

(2)

D5:(l)

[verification]

1 3 2 7 - 1 6 5 6 573 7 7 6 = 3 7

(P)

( 1 ) ( 2 ) = D5

165

The Algorithmic Structure of the Egyptian Mathematical Problem Texts

p R h i n d , no. 34 Numerical

Translation '

Symbolical

Algorithm

A quantity,

1

D.

its 2 a n d

2

D2

its 4 to it

4

D3

is 10.

10 1+2 +

^'^

Algorithm

D4

4 = 1 2 4

(1)

D , + D2+D3

[calculation]

1 0 : 1 2 4 = 5 2 7 14

(2)

D4:(l)

[verification]

5 2 7 14 • 1 2 4 = 10

(P)

( 2 ) ( 1 ) = D4

p M o s c o w , n o . 19 Translation

Symbolical Algorithm

Numerical Algorithm

' M e t h o d o f calculating a quantity, 12

D,

together w i t h ^ 4

4

D2

and it h a s c o m e to 10.

10

D3

calculated 1 2 times

W h a t is the quantity that says it? ^ T h e n y o u calculate the difference of these 10

10-4 = 6

(1)

D3-D2

1 : 12=

(2)

1 : D,

(3)

(2)-(l)

to these 4. T h e n 6 results. T h e n y o u divide 1 b y 1 2 . T h e n 3 results. ^

T h e n y o u calculate

3 o f these 6 T h e n 4

3

-6 = 4

3

results. B e h o l d it is 4 (the quantity that) ^ says it. W h a t has b e e n found b y y o u is correct.

pMoscow, no. 25 Translation ' M e t h o d o f calculating a quantity. calculated 2 times

Numerical Algorithm

Symbolical Algorithm

1

D,

2

D2

9

D3

together w i t h (the quantity) c o m e to 9. ^ W h a t is the quantity that says it? T h e y o u calculate the s u m of this

D2

1+2 = 3

(1)

D, +

9:3 = 3

(2)

D3:(l)

quantity together with these 2 . ^ T h e n 3 results. T h e n you divide 9 b y 3 . T h e n 3 times result. B e h o l d it is 3 that says it. W h a t has b e e n found b y y o u is correct.

A. Imhausen

166

P U C 3 2 1 3 4 (= K a h u n L V . 3 ) Translation

Symbolical Algorithm

Numerical Algorithm 1

' A quantity, its 2 4 subtracted,

2 4

D2

so that 5 is left.

5

D3

^ W h o says it? T h e n y o u calculate [1] ^ m i n u s 2 4 .

1 - 2 4 = 4

(1)

D,-D2

1:4=4

(2)

1:(1)

5 • 4 = 20

(3)

D3-(2)

T h a t w h a t results is 4 . T h e n y o u divide

1 by 4 .

T h a t w h a t results is 4. ^ T h e n y o u calculate 5 times 4. T h a t w h a t results is 2 0 . ^ 2 0 says it.

pBerlin 6619, no. 1 Translation

Numerical Algorithm

Symbolical Algorithm

' A n o t h e r [...] If you are told []

[100]

D,

[] as a quantity

[1] [24]

D2

^ [] the quantity to the other quantity

D3

Y o u shall let m e k n o w the quantity [] 2

[] ^ S q u a r i n g the rectangle as 1 together with the calculation of 2 4 of

4 - 1 = 2 4

(1)

D 3 D 2

1^=1

(2a)

(D2)'

( 2 4 ) ^ = 2 16

(2b)

(1)'

the one [ ] ^

2 4 of the o n e quantity to the other.

T h e n 2 4 results. T h e n y o u calculate it [ ] ^ T h e o n e quantity is 1, the other is 2 4 . T h e n y o u calculate each one [] ^ T h e n 1 2 { ^ } 16 results. T h e n y o u calculate its square-root.

1 + 2 Î 6 = 1 2 16

(3)

r—

Vl216 = 1 4

(4a)

(2a) + (2b)

V(3)

T h e n 1 4 results. T h e n you calculate [] / T h e n [ ] results.

VlOO = 1 0

(4b)

D i v i s i o n of this 10 b y this 1 4 .

10 : 1 4 = 8

(5)

( 4 b ) : (4a)

2 4-8 = 6

(6)

D3(5)

VD,

T h e n [] 8 results. [] * [] this 8. [] results

Babylonian Lunar Theory in Roman Egypt: Two New Texts Alexander

Jones,

Toronto

In 1988 O . N e u g e b a u e r p u b l i s h e d a partial transcription and d i s c u s s i o n o f a firstcentury A . D . G r e e k p a p y r u s fragment in a private collection {P. Colker), w h i c h as h e s h o w e d contains a r u n o f values o f the function G r e p r e s e n t i n g a fnst a p p r o x i m a t i o n to the d u r a t i o n of consecutive synodic m o n t h s in the B a b y l o n i a n S y s t e m B limar theory. * Before this discovery, n o o n e w o u l d h a v e e x p e c t e d to find o n e of the k e y fimctions of the B a b y l o n i a n lunar theories turning u p in essentially immodifled form in a G r e c o - R o m a n p a p y r u s . A few of the s e v e n t y or so astronomical p a p y r i k n o w n u p to that time s h o w e d the influence o f B a b y l o n i a n m a t h e m a t i c a l astronomy, b u t n o n e could b e r e a s o n a b l y d e s c r i b e d as a n " A C T text in G r e e k " (where A C T refers b y a c r o n y m to the k i n d s of a s t r o n o m i c a l tablets included in N e u g e b a u e r ' s Astronomical Cuneiform Texts). P. Colker d e m o n s t r a t e d that part of the tradition o f a s t r o n o m i c a l c o m p u t a t i o n practiced b y the a s t r o l o g e r s — a t least, w e p r e s u m e that such w e r e the p e o p l e w h o p r o d u c e d m o s t o f the R o m a n - p e r i o d a s t r o n o m i c a l p a p y r i — w a s so close to the B a b y l o n i a n m o d e l s that the t r a n s m i s s i o n of m e t h o d s m u s t h a v e b e e n quite direct: not, for e x a m p l e , t h r o u g h the m e d i u m o f theoretical h a n d b o o k s or treatises such as those of G e m i n u s a n d P t o l e m y . Since the a p p e a r a n c e of N e u g e b a u e r ' s article, m o r e has b e e n l e a m e d a b o u t the practice of A C T - s t y l e m e t h o d s in R o m a n Egypt. First, from a closer i n s p e c t i o n o f P. Colker it b e c a m e clear that the very p o o r l y p r e s e r v e d c o l u m n o f n u m e r a l s to the left of the c o l u m n for G m u s t contain a function with a periodicity of r o u g h l y twelve synodic m o n t h s ( w h e r e a s G ' s periodicity is r o u g h l y fourteen m o n t h s ) . T h e m o s t plausible c a n d i d a t e for this function is J, representing a c o r r e c t i o n to G with a n annual p e r i o d , that is, d e p e n d e n t o n the s u n ' s longitude. T h i s confirms that w e are dealing w i t h a m u l t i - c o l u m n table like the B a b y l o n i a n S y s t e m B tablets, n o t s o m e other tabular s t m c t u r e that treats G m isolation; it also points to a p o s s i b l e c o n n e c t i o n o f P. Colker with a particular type o f S y s t e m B tablet in w h i c h J is tabulated to the left of G m s t e a d o f o c c u p y i n g its usual p o s i t i o n t w o c o l u m n s to the right o f G. Secondly, m a n y m o r e astronomical tables c a m e to light a m o n g the p a p y r i e x c a v a t e d b y B . P . Grenfell a n d A . S. H u n t at O x y r h y n c h u s , n o w k e p t at the A s h m o l e a n M u s e u m at Oxford. T h e edition of these p a p y r i that I p u b l i s h e d in 1999 includes several containing A C T - s t y l e tables for p l a n e t a r y p h e n o m e n a , m o s t o f t h e m c o n f o r m i n g to k n o w n B a b y l o n i a n arithmetical models.^ H e n c e w e k n o w n o w that '

NEUGEBAUER (1988).

2

JONES ( T 9 9 7 ) .

3

JONES ( 1 9 9 9 ) ;

see also

JONES ( 1 9 9 8 ) .

168

A.Jones

the transmission included a large part of the B a b y l o n i a n planetary theories as well as p a r t of the S y s t e m B limar theory. H o w e v e r , n o n e o f the tables in m y p u b l i c a t i o n could b e certainly identified as pertaining to either o f the B a b y l o n i a n lunar theories. It w a s , in fact, j u s t as the edition was g o i n g to press that a n u n i n v e n t o r i e d S y s t e m B lunar table c a m e to light a m o n g the u n p u b l i s h e d O x y r h y n c h u s p a p y r i . T h e text has the inventory n u m b e r P. Oxy. ined. 2 3 3 B 1/0(1-4)c, a n d will in d u e course b e p u b l i s h e d formally with a n assigned text n u m b e r in the Oxyrhynchus Papyri series. It c o n ^ r i s e s three fi-agments: a larger o n e (3 c m w i d e b y 3.5 c m tall) that p r e s e r v e s the b o t t o m eleven lines o f the table with part of the lower m a r g i n b e l o w the table and v a c a n t m a r g i n to the left o f the table, a n d t w o smaller fi-agments b e l o n g i n g a b o v e the m a i n one, so that fi-om the t o p partially p r e s e r v e d line to the b o t t o m w o u l d h a v e b e e n twenty lines. T h e r e are s o m e slight ttaces o f writing, apparently a d o c u m e n t , o n the back. T h e h a n d o f the table a p p e a r s to b e l o n g to the s e c o n d or third century A . D . ; unfortimately, the information surviving m the table caimot b e dated asttonomically. In the following provisional tianslation n u m e r a l s ; all restorations are certain. I

[29 14 19 [28 5 6 19 [28 3 8 19 [28 2 0 19 [28 18 5 9 [28 3 6 5 9 [28 5 4 5 9 [29 12 5 9 [29 3 0 5 9 [29 4 8 5]9 [2]9 [ 5 6 5]8 [2]9 3 [ 8 5]8 2 9 2 0 58 29 2 58 2 8 4 4 58 28 26 58 2 8 12 2 1 2 8 3 0 21 2 8 4 8 21 6 21 29

o f the table, brackets e n c l o s e restored

III

II

40] 40] 40] 40] 40] 40] 40] 40] 40] 40 20 20 20 20 20 20 [0] 0 0 0

[Pisces] [Arie] s [Ta]urus [Gem]ini [Cjancer [L]e[o] [Vi]rgo [Libra] [Scorpio] [Sagittarius] [Capricorn] [Aquarius] Pisces Aries Taums Taurus Gemini Cancer Leo Virgo

[1]4 13 11 9 8 6 5 5

[4] [0] 38 59 18 [55] [50]

2 0 [20] 4 0 [0] 59 40 19 2 0 19 0 18 4 0 18 2 0

[3 1]8 [0] [4 3 4 1 ] 7 [ 4 0 ] [4 2 3 17 2 0 ] [4 2 0 15 4 0 ] [3 5 9 1]4 [0] 3 2 0 12 2 0 2 2 3 10 4 0 8 9 0 1 7 [20] 2 9 35 27 47 28 20 2 6 17 4 9 [20] 6 10 2 0 25 2 4 12 31 2 0

C o l u m n i is a r u n o f v a l u e s o f S y s t e m B fimction A , representing the p r o g r e s s o f the m o o n in longitude since the p r e c e d m g syzygy, a n d c o l u m n s ii a n d iii give fimction B , the longitude o f the m o o n at the syzygy calculated as the running total o f c o l u m n i. Since A is smallest w h e n the c o r r e s p o n d i n g longitude is in G e m i n i or C a n c e r , the tabulated p h e n o m e n a are conjunctions rather than fiill m o o n s . ( T h e ideal m a x i m a a n d m i n i m a of the fimction h a v e b e e n set to c o r r e s p o n d to longitudes s u c h that the m i d p o i n t o f the shortest possible synodic arc is very n e a r G e m i n i 11°, a n d the m i d p o i n t o f the longest possible arc is very near Sagittarius 11°.) T h e defining p a r a m e t e r s of fimction A are the u n a b b r e v i a t e d o n e s ( e n ^ l o y i n g three s e x a g e s h n a l

Babylonian Lunar Theory in Roman Egypt: Two New Texts

169

fractional p l a c e s ) characteristic o f m o s t o f the A C T S y s t e m B tablets from B a b y l o n . In the c i m e i f o r m tables, a c o l u m n p r o v i d i n g the year a n d c a l e n d a r m o n t h o f the s y z y g y regularly a p p e a r s to the left o f A ; w e m a y s u p p o s e that the c o r r e s p o n d i n g information in the p a p y r u s table w a s p u t together with the c o m p u t e d d a y a n d t i m e o f the syzygy, s o m e w h e r e in the lost c o l u m n s to the right. A m i n o r difference o f format in c o m p a r i s o n to the cimeiform tables is that the z o d i a c a l signs o f the longitudes are p l a c e d , a c c o r d i n g to the n o r m a l G r e e k c o n v e n t i o n , to the left o f the degrees. B o t h this O x y r h y n c h u s p a p y r u s a n d P. Colker are e a s y to identify a s p e r t a i n i n g to the B a b y l o n i a n lunar S y s t e m B b e c a u s e they p r e s e r v e substantial p a r t s o f actual c o l u m n s o f data. T h e r e m a i n i n g p a p y r u s to b e d i s c u s s e d in this article p r e s e n t s greater difficulties. It w a s originally m t e n d e d to a p p e a r in v o l u m e 15 o f t h e Papiri della Società Italiana, the p u b l i c a t i o n o f w h i c h h a s unfortunately b e e n l o n g d e l a y e d ; the p a p y r u s is a s s i g n e d the p u b l i c a t i o n n u m b e r PSI1491. N e u g e b a u e r , w h o assisted in p r e p a r i n g a draft text a n d c o m m e n t a r y in 1964, kindly p r o v i d e d m e with a c o p y of his n o t e s a n d a p h o t o g r a p h o f the p a p y r u s in 1 9 8 2 . M o r e r e c e n t l y I h a v e b e e n able to s t u d y a n excellent c o l o u r p h o t o g r a p h t h r o u g h the k i n d n e s s o f P r o f G. B a s t i a n i n i . T h e fragment h a s d i m e n s i o n s o f a p p r o x i m a t e l y 7 x 1 6 c m , a n d p r e s e r v e s p a r t o f a single c o l u m n o f text with a b o t t o m m a r g i n o f a b o u t 4 . 5 c m . T h e t o p a n d b o t h e d g e s of the c o l u m n are b r o k e n o f f T h e r e is n o text o n the other side o f t h e p a p y r u s . I w o u l d estimate that the h a n d b e l o n g s t o the s e c o n d c e n t u r y A.D.'* T h e following is a franscription o f the p a p y r u s with the m i n i m a l restorations that c a n b e m a d e without u n d e r s t a n d i n g the subject o f the text: cleXifÔiv

] column [

\{c.5Uttm]\'K[ OJIOCTTINA *

x]8...[

[

] interval 0 . . . [

] ôeûxepov C£(WÔiv

] second column [

]T|v ènicwayouërvnv

5

tpitov 1 ceWÔiv Ç(!)5{û)Iv TÉTOPTJOV ceXiSiv ou ôpoc

third] column OF zodiacal signs ( [

fourth] column, OF which l i m i t . . . [

I oc ^OI(pcûv) Ica x l e e T t t i ylòp

] . . . 21 degrees. For it is placed [

Km\o. TÒV Kpeiòv m l t ò v Zvfyòv ] ^oKpwv) ^ èXaxicTTiv o ù k è[ ] jipocea4Kxipecic * ç '

] accumulated [

at] Aries and Libra [ 10

[

] . . . 8 degrees. Least n o t . . . [ ] increment/decrement 0 6 . . . [

I'e' JiéfiJtxov ceXiSiv ov [

] . . . fifth column, of which [

I MéyicTÓc c c T i v X'p'

] . . . greatest is 3 2 . . . [

[

t i e i e t a i 7àp icatà xfiv xov A'iyfóiceplo) Kox KapKivoD Mo((pav) rf f

] For It is placed at Capricorn 15

] and Cancer, 8 degrees [

The hand is said in N E U G E B A U E R ( 1 9 6 2 ) , p. 3 8 8 to be of the late first or second century, and in N E U G E B A U E R ( 1 9 7 5 ) , p. 9 4 6 to be second century; it is similar in many respects to P. Bodmer ii (TURNER ( 1 9 8 7 ) , no. 6 3 ) , which has been variously dated to the second or eariy third centuries.

170

A.Jones

ëKxolv ceXiSiv ànò napefévoD

sixth] column from Virgo [

itpoceia<|Kxlpecic wc 'Ixeûcov [

incrcmcnt/]dccrcmcnt to' Pisces

ëpÔol^ov CEXÌSIV 6pó|xoc ce[

] seventh column, course ... [

]T|c ou ôpoc F K'E' K'C 1 ]tiv (lèv èXaxicttiv n i

] ... of which limit 4 25 27 [ 20

] ... least... [

T h e text is obviously a description of a table containing at least s e v e n c o l u m n s , w h i c h are m e n t i o n e d in order, along with pertinent data including z o d i a c a l signs, degrees, and other niunbers. N e u g e b a u e r t h o u g h t that the character o f the descriptions was suggestive of a table with c o l u m n s for several different h e a v e n l y b o d i e s : p e r h a p s the sun, m o o n , and five planets. H e fiuther p r o p o s e d identifying the C a n c e r a n d C a p r i c o r n o f c o l u m n 5 with the p e r i g e e and a p o g e e o f M a r s , the V i r g o and Pisces of c o l u m n 6 with the a p o g e e a n d perigee o f Jupiter, and the A r i e s of c o l u m n 4 with the p e r i g e e o f Mercury. Since he w a s u n a b l e to r e c o v e r any c o n t m u o u s sense firom the p r e s e r v e d text, h o w e v e r , h e r e m a i n e d very c i r c u m s p e c t a b o u t the interpretation.^ T h e starting p o i n t for understanding the p a p y r u s is the very strong i m p r e s s i o n that only a little text has b e e n lost b e t w e e n consecutive lines. In particular, lines 810 are paralleled in construction b y lines 14-15, and there s e e m s to b e nothing missing except for the c o m p l e t i o n o f partially p r e s e r v e d w o r d s . A l l o w i n g the m a r g i n s implied b y these restorations to guide us elsewhere, o n e c a n o b t a i n a nearly u n b r o k e n text o v e r several lines. T h e k e y to u n d e r s t a n d i n g the nature of the table m i g h t b e e x p e c t e d to lie in the r e m a r k s o n c o l u m n s four, five, and seven. T h e fourth c o l u m n is said to h a v e a greatest (?) limit o f 2 1 ° , and this (?) is s u p p o s e d to b e p l a c e d at the diametrically opposite longitudes, Aries 8° and Libra 8°. Similarly, the fifith c o l u m n has a greatest limit of 3 2 units o f s o m e kind, p e r h a p s followed b y a fraction, at the diametrically opposite points, C a p r i c o m 8° and C a n c e r 8°. T h e s e four longitudes are respectively the equinoctial p o m t s and the solstitial points a c c o r d i n g to the B a b y l o n i a n S y s t e m B n o r m ( w h i c h is well attested also m G r e c o - R o m a n a s t t o n o m y ) . T h e y h a v e n o relevance for the m o t i o n of the planets, w h i c h casts d o u b t u p o n N e u g e b a u e r ' s hypothesis o f a p l a n e t a r y table. M o r e o v e r , the seventh c o l u m n ' s " l i m i f o f 4 ; 2 5 , 2 7 , [ . . . ? ] looks suspiciously like a m i s c o p y i n g of 4 ; 2 9 , 2 7 , 5 , which is the u p p e r b o u n d of the function G in a S y s t e m B lunar table; the p r o b l e m a t i c digit is a n epsilon (5), often m i s t a k e n for theta (9) in G r e e k capital scripts. If w e accept this hypothesis, m o s t of the other data in the p a p y r u s quickly fall into place. It is n o w obvious that the text is a description o f a variety o f S y s t e m B table specifically c o n c e m e d with the p r e d i c t i o n o f full m o o n s .

'

by

NEUGEBAUER JONES

(1984).

(1962), p. 388 and (1975), p. 946; further (wrong-headed) embellishment

Babylonian Lunar Theory in Roman Egypt: Two New Texts

171

T h i s is w h a t the text n o w looks like: jipdnov c]EXi|5tv l.l

il'ri'

first] column [ V]S..,[

Kf

] Interval 0 ... [

SlidcTima * 1

1 Seûtepov ce[Xi6iv

ëxei

T|TIV

] Second column

èmcuvoyojiélvnv.

5

has] the running total.

tpitov] ceX{6iv Ç(p5i(olv.

Third] column is for the zodiacal signs.

tétaptlov ceW8tv oh ôpoc ìi\é-

Fourth] column, of which the maximum

yicjtoc laoi(pwv) ica. t(0etai ylap

limit is 21 degrees. For it is placed

tòv Kpeiòv

Katlèt

at] Aries and Libra

K U I TÒV Z U -

YÒ]y ^oi(pâ)v) îv èXxxxictnv o-ùic èXei.] n:pocea(|)a(pecic » ç'

10

X'

Increment/decrement 0;6,4'[x

l'e'. JtÉM^ntov ceXiSiv ou fô-

] . . . Fifth column, of which

pojc liéyictoc èctiv X'P' Kl'r\'.

the maximum limit is 32;2'[8.

t(81etm yòtp m t à xr\v xov fiìv([óKeplto Ktti KapKxvov |Aoi(pav) rf. fëKto]v

8 degrees. It docs not have a minimum.

For it is placed at Capricorn 15

ceXiôiv ànb nape[é-

and Cancer 8 degrees. Sixth column, from Virgo

vouj à<jia{pecic ax; 'IxBxxàv. [ë-

"subtraction" until Pisces.

pSoJnov ceXvSiv 6pó|X0C ce-

Seventh column, course of the moon,

Xr|v|Tic ou Ôpoc 'S k'e k'Ç le • iTlv

^lèv èXaxicTiiv r|X[

of which limit 4;25',27,[5 20

] . . . minimum ...[

W e m a y b e g i n the discussion o f details w i t h the fourth c o l u m n (lines 7 - 1 2 ) . T h i s c o l u m n is said in the text to h a v e a m a x i m u m o f 2 1 " d e g r e e s " at A r i e s S° a n d L i b r a 8°, n o m i n i m u m (i.e. a m i n i m u m of z e r o ) , a n d a n i n c r e m e n t / d e c r e m e n t o f 0^6,.... T h i s s o u n d s v e r y m u c h like the specification o f a Imear z i g z a g fimction. In a S y s t e m B table the fimction H , w h i c h represents the m o n t h - t o - m o n t h c h a n g e in the c o m p o n e n t o f the d u r a t i o n of the synodic m o n t h d e p e n d e n t o n the s u n ' s longitude, is a z i g z a g fimction, with m a x i m u m 2 1 t i m e - d e g r e e s , m i n i m u m 0, a n d i n c r e m e n t / d e c r e m e n t 6;47,30 t i m e - d e g r e e s ( = 0;6,47,30 "large h o u r s " ) . T h e p e r i o d o f H is half a sidereal year, s o that the m a x i m u m a n d m i n i m u m will b e a s s o c i a t e d w i t h diametrically o p p o s i t e solar (or lunar) longitudes. T h e r e c a n b e n o d o u b t that the fourth c o l u m n referred to in the p a p y r u s w a s H . B u t w h a t of the statement that the m a x i m a are p l a c e d at A r i e s a n d L i b r a 8 ° ? O n theoretical g r o u n d s H s h o u l d b e s y n c h r o n i z e d with the fimction A (the s y n o d i c arcs, i.e. p r o g r e s s in longitude from o n e conjunction or full m o o n t o the n e x t ) , in such a w a y that the z e r o p o i n t s of H coincide with b o t h the m a x i m a a n d m i n i m a o f A.*^ Since A is a m e a s u r e o f the a v e r a g e solar velocity during the s y n o d i c m o n t h , its e x t r e m a c o r r e s p o n d to the situations w h e r e the s u n ' s p r o g r e s s d u r i n g the m o n t h straddles the solar apsidal line. H e n c e the s u n ' s longitude at the e n d o f a m o n t h with * In fact perfect synchronization is not possible because the periodicities of Columns A and H are slightly different.

172

A. Jones

a m a x i m u m or m i n i m u m synodic arc (wliich is the longitude that w o u l d b e r e c o r d e d as the running total B o n the same line as the m a x i m u m or m i n i m u m v a l u e o f A ) w o u l d b e about 15° p a s t the apsidal line. In the B a b y l o n i a n S y s t e m B tables these points w e r e generally p l a c e d near the beginning of C a n c e r a n d C a p r i c o m , a n d likewise the m a x i m a of H are near the begiiming o f Aries and L i b r a . ' T h e r e is n o e v i d e n c e that the B a b y l o n i a n s ever assimilated t h e m to the solstitial a n d equinoctial points at 8 ° , but it is not surprising that s o m e o n e should h a v e d o n e so. Lines 12-15 o f the p a p y m s describe the fifth c o l u m n as h a v i n g m a x i m a o f 3 2 , . . . at C a p r i c o m a n d C a n c e r 8°. W e h a v e seen that the fourth c o l u m n w a s H , a n d the only conceivable r e a s o n for having H is in order to derive from its r u n n i n g totals the fimction J, the c o m p o n e n t d e p e n d e n t o n the s u n ' s longitude of the d i u a t i o n o f the synodic m o n t h . T h i s agrees excellently with w h a t the p a p y m s says. J is c o m p u t e d like a zigzag fimction, except that the i n c r e m e n t / d e c r e m e n t s are t a k e n fi^om H instead o f b e i n g constant. T h e m a x i m u m of J in the B a b y l o n i a n tables is either 3 2 ; 2 8 , 6 t i m e - d e g r e e s or s i n g l y 32;28 time-degrees. T h e m i n i m i m i is the n e g a t i v e of the m a x i m u m ; or, equivalently, one m a y consider the m i n i m u m to b e z e r o , with a l t e m a t e cycles o f J t a k e n as additive and subtractive. F o r the oscillations o f J to w o r k properly, it is essential that its zeros coincide as nearly as p o s s i b l e w i t h the m a x i m a o f H , so that if H has its m a x i m a at Aries a n d Libra 8°, the positive a n d negative e x t r e m a o f J should b e at C a p r i c o m a n d C a n c e r 8 ° . T h e p a p y m s g o e s o n to characterize the sixth c o l u m n s h n p l y as " s u b t i a c t i o n fi-om V i r g o until P i s c e s " . All this m e a n s is that w e reserve a c o l u m n n e x t to C o l u m n J in w h i c h w e n o t e w h i c h entiies are to b e u n d e r s t o o d as subtractive. T h i s is j u s t w h a t w e find in the B a b y l o n i a n tables, except that a n a p p r o p r i a t e i d e o g r a m is written for b o t h additive a n d subtiactive entiies. " F r o m V i r g o until P i s c e s " h a s to b e u n d e r s t o o d as m e a n i n g that J is subtiactive b e t w e e n its crossing o f z e r o , w h i c h c o m e s after the s y z y g y with a longitude in V i r g o , a n d its n e x t crossing o f z e r o , w h i c h c o m e s after the syzygy with a longitude in Pisces. T h e elliptical u s e o f m e r e sign n a m e s without d e g r e e s to specify the z o n e o f slow or fast solar m o t i o n in the ecliptic is exactly paralleled in B a b y l o n i a n p r o c e d i n e texts.* In fact J will b e additive w h e n the s u n is in this half of the ecliptic, so the longitudes in q u e s t i o n m u s t b e those of the m o o n at o p p o s i t i o n — o u r only indication that the table is for full m o o n s rather t h a n n e w m o o n s . T h e seventh c o l u m n , unlike the foregoing, is given a n a m e : " c o u r s e o f the m o o n " . T h i s is not, as o n e might expect, the m o o n ' s daily velocity (fimction F of a S y s t e m B table), b u t G, the c o m p o n e n t of the duration of the synodic m o n t h d e p e n d e n t o n the m o o n ' s anomaly. O n l y the m a x i m u m of this linear z i g z a g fimction is specified in the surviving part of our text. A s n o t e d a b o v e , the correct p a r a m e t e r , 4,29;27,5 t i m e - d e g r e e s , has b e e n c o r m p t e d t h r o u g h the v e r y c o m m o n m i s r e a d i n g o f theta as epsilon. T h e first three c o l u m n s o f the table pretty well h a v e to c o r r e s p o n d to A a n d B of the B a b y l o n i a n S y s t e m B tables. T h e description o f the first c o l u m n is v e r y p o o r l y p r e s e r v e d , but it s e e m s at least possible that the n u m b e r in line 2 w a s the ^ For example in ACT 1 2 2 the extrema of A are near Cancer and Capricom 4 ° , and those of H near Aries 5 ° and Libra 1°. The procedure text ACT 2 1 0 , section 5 , prescribes that the extrema of Column A should be in Cancer and Capricom. *

See, e.g., ACT

( 1 9 7 5 ) , p.

192.

204

§7

with Neugebauer's commentary, and

AABOE

and

HENDERSON

Babylonian Lunar Theory in Roman Egypt: Two New Texts

173

i n c r e m e n t / d e c r e m e n t of the zigzag fimction for A , 0;18° p e r synodic m o n t h . T h e s e c o n d c o l u n m is clearly B , i.e. the longitude o f the full m o o n , c o m p u t e d as the ruiming total of A . Just as in the B a b y l o n i a n tables, b u t unlike the u s u a l G r e e k practice, the z o d i a c a l sign of the longitude w a s written to the right of the d e g r e e s , in what the p a p y r u s calls the third coliunn. T h e table, then, c o m p r i s e d C o l u m n s A , B , H, J, a n d G in that order. M o s t o f the B a b y l o n i a n S y s t e m B tables that h a v e these c o l u m n s p l a c e H a n d J after G, a n d insert other c o l u m n s b e t w e e n B a n d G; b u t the o r d e r H J G d o e s o c c u r in A C T 119, a n d w e h a v e a k e a d y seen that P. Colker apparently h a d J to the left o f G. M o r e o v e r , there are several cimeiform tables, w h i c h N e u g e b a u e r calls " a u x i l i a r y " tables ( A C T 170-174), in w h i c h the only c o l u m n s are T (the year a n d lunar m o n t h ) , B , J, G, K ( = J + G, the d u r a t i o n of the synodic m o n t h over 2 9 days in t i m e - d e g r e e s ) , a n d L (the date of syzygy, as the running total of K ) . E x c e p t that it incorporates the c o l u m n s ( A and H ) from w h i c h B a n d H are derived, the table of the p a p y r u s follows this p a t t e m so far as it g o e s . T h e p u r p o s e o f such a table w o u l d b e to calculate the exact longitudes, dates, a n d times of opposition, without fiuther c o n c e m for eclipse or visibility p h e n o m e n a such as are p r e d i c t e d in the fuller S y s t e m B tables. A text such as the p r e s e n t p a p y m s w o u l d definitely not h a v e p r o v i d e d e n o u g h information to teach o n e h o w to generate a S y s t e m B table of full m o o n s ; it m i g h t h a v e b e e n a n aide-mémoire for a n e x p e r i e n c e d scribe. T w o fiuther p o i n t s d e s e r v e to b e m e n t i o n e d . O n e is the conservative character o f the tradition. T h e S y s t e m B tables surviving in cimeiform fall within a n interval o f t w o centuries, from about 2 5 0 B . C . to a b o u t 50 B . C . N o w after a d o c u m e n t a r y g a p of t w o h u n d r e d years or m o r e , in a different r e g i o n a n d a different l a n g u a g e , w e e n c o u n t e r tables n o t only b a s e d o n the s a m e astronomical t h e o r y a n d m a t h e m a t i c a l m e t h o d s , b u t e v e n p r e s e r v i n g such details of format as p l a c i n g the n a m e s of z o d i a c a l signs following the d e g r e e s a n d the indications o f subtractive quantities following the quantities. T h i s fact d e m o n s t r a t e s , overriding all the h i s t o r i a n ' s a priori a s s u m p t i o n s a b o u t the difficulty o f d e e p cultural contacts b e t w e e n M e s o p o t a m i a a n d the classical world, that a n u n b r o k e n practical tradition at a h i g h technical level led from the scribes o f B a b y l o n a n d U m k to those of provincial Egypt. Secondly, w e h a v e to confess that w e d o not k n o w w h y these tables w e r e b e i n g p r o d u c e d . T o b e sure, our position with respect to the B a b y l o n i a n lunar tables is not m u c h better, b u t there at least w e c a n appeal v a g u e l y to the use o f a lunar calendar, the o m i n o u s significance o f sightings o f the n e w a n d full m o o n , a n d t h e — i t s e l f not a d e q u a t e l y e x p l a i n e d — p r a c t i c e of regularly m e a s u r i n g the t u n e intervals b e t w e e n m o o n r i s e or m o o n s e t a n d simrise or sunset close to the syzygies. In the G r e c o E g y p t i a n context there is n o o b v i o u s application for precisely c o m p u t e d syzygies, except p e r h a p s in the c o m p i l i n g o f eclipse c a n o n s as part of general asfrology.

174

A.Jones

Abbreviations ACT

= NEUGEBAUER ( 1 9 5 5 ) .

References A A B O E , Asger, a n d H E N D E R S O N , Janice A. 1 9 7 5 . " T h e B a b y l o n i a n T h e o r y o f L u n a r Latitude

and

d'histoire

des sciences

Eclipses

According

to

S y s t e m A " . Archives

internationales

25: 1 8 8 - 2 2 2 .

J O N E S , A l e x a n d e r . 1 9 8 4 . " A G r e e k Saturn T a b l e " . Centaurus

27: 3 1 1 - 3 1 7 .

— 1 9 9 7 . " A G r e e k P a p y r u s Containing B a b y l o n i a n L u n a r T h e o r y " . Zeitschrift Papyrologie —1998.

und Epigraphik

" S t u d i e s in the A s t r o n o m y of the R o m a n P e r i o d . III. P l a n e t a r y E p o c h

T a b l e s " . Centaurus —1999.

fiir

1 1 9 : 1 6 7 - 1 7 2 a n d plate II.

Astronomical

40: 1 - 4 1 . Papyri

from

Oxyrhynchus

(Memoirs

o f the

American

Philosophical Society 2 3 3 ) . Philadelphia: A m e r i c a n P h i l o s o p h i c a l Society. N E U G E B A U E R , O t t o . 1 9 5 5 . Astronomical

Cuneiform

Texts. 3 vols. L o n d o n : L u n d

Hunqjhries ( R e p r i n t N e w Y o r k : Springer 1 9 8 3 ) . — 1 9 6 2 . " A s t r o n o m i c a l P a p y r i and Ostraca: B i b l i o g r a p h i c a l N o t e s " . Proceedings the American

Philosophical

Society

of

106: 3 8 3 - 3 9 1 .

— 1 9 8 8 . " A B a b y l o n i a n L u n a r E p h e m e r i s from R o m a n E g y p t " . In: E r i e LEICHTY, M a r i a d e J. ELLIS, a n d P a m e l a G E R A R D I (eds.), A Scientific Humanist: Studies in Memory of Abraham Sachs ( O c c a s i o n a l Publications of the S a m u e l N o a h K r a m e r F u n d 9 ) : 3 0 1 - 3 0 4 . Philadelphia: S a m u e l N o a h K r a m e r Fimd. T U R N E R , Eric G. 1 9 8 7 . Greek Manuscripts of the Ancient World. R e v i s e d edition. L o n d o n : University o f L o n d o n .

Early Babylonian Observations of Saturn: Astronomical Considerations Teije de Jong, Amsterdam

and

Utrecht

Introduction In recent years a large amoimt of textual material o n ancient a s t r o n o m y a n d celestial divination originating from M e s o p o t a m i a h a s b e e n m a d e available to the n o n specialist student b y the publication of n e w editions a n d franslations o f c u n e i f o r m texts, hitherto o n l y accessible to the philological expert. In addition the subject matter has b e e n s u m m a r i z e d a n d r e v i e w e d in several recent conference p r o c e e d i n g s a n d m o n o g r a p h s . ' T h i s d e v e l o p m e n t p r o v i d e s r e s e a r c h opportunities for a n e w generation o f students in a field that has u p to n o w b e e n d o m i n a t e d b y a small c o m m u n i t y o f assyriologists a n d historians o f science. B r o a d e n i n g the r e s e a r c h participation m a y t u m out to b e beneficial to a field o f r e s e a r c h w h i c h b y definition is multi-disciplinary in nature. Since the practice of systematically carrying out asfronomical o b s e r v a t i o n s s e e m s to h a v e b e e n first d e v e l o p e d in Babylonia, the c o m p a r a t i v e l y well-preserved r e c o r d s of this activity p r o v i d e a u n i q u e database, b o t h for the study of B a b y l o n i a n asfronomy in itself as well as for those p a r a m e t e r s o f solar s y s t e m d y n a m i c s w h o s e rate o f c h a n g e o n a time-scale of a few t h o u s a n d years is n o t well k n o w n or understood.^ A t the basis o f M e s o p o t a m i a n o b s e r v i n g practice is the o b s e r v a t i o n o f heliacal risings and settings o f the brighter stars a n d planets. D u e to the rotation o f the earth stars rise in the east a n d set in the west. T h e y are only visible at night w h e n the S u n is b e l o w the h o r i z o n . A s the S u n m o v e s tlu-ough the ecliptic (its p a t h o n the sky against the b a c k g r o u n d o f stars) in the course of o n e year, stars e x p e r i e n c e a p e r i o d of invisibility w h e n the S u n is t o o close. T h e first d a y that a star b e c o m e s visible again at d a v m o n the e a s t e m h o r i z o n a b o u t h a l f a n h o u r before sunrise after h a v i n g b e e n invisible for days, w e e k s or e v e n m o n t h s ( d e p e n d i n g o n its b r i g h m e s s a n d p o s i t i o n in the sky) is called the date o f heliacal rising or o f first ( m o m i n g ) visibility. O n that d a y the star r e m a i n s visible for a b o u t 5 to 10 m i n u t e s before it disappears in the brightening m o m i n g sky. A s the S u n m o v e s o n from d a y to d a y the star rises earlier e a c h night thereby extending its p e r i o d o f visibility d u r i n g the night. Eventually the S u n starts c a t c h m g u p with it from b e h m d so that the nightly p e r i o d o f visibility d e c r e a s e s again until finally the d a y o f last visibility is r e a c h e d . O n that date the star a p p e a r s for the last time for about 5 to 10 m i n u t e s at d u s k j u s t after sunset on the w e s t e m horizon. T h e cycle repeats itself e v e r y year. T o a p p r e c i a t e the '

Cf

^

For example

SWERDLOW (

1999);

HUNGER

STEPHENSON (

and

1997).

PINGREE (

1999); B R O W N (2000a).

176

T. de Jong

excitement of e x p e c t i n g a n d then indeed o b s e r v i n g the first visibility of a star in the m o m i n g sky and the g n a w i n g uncertainty o f p e r h a p s h a v i n g m i s s e d it in the slowly d a r k e n i n g evening s k y I r e c o m m e n d a n y student o f B a b y l o n i a n a s t r o n o m y to o n c e s p e n d a few days carrying out such observations h i m s e l f A t the g e o g r a p h i c a l latitude of B a b y l o n the c o i n s e of events d e s c r i b e d a b o v e h o l d s for the majority (about 9 0 % ) o f all stars visible fi-om B a b y l o n . T h e r e m a i n i n g fi-action n e v e r sets a n d is thus always visible at night b e c a u s e of its l o c a t i o n close to the north celestial p o l e (the so-called c k c u m p o l a r stars). Heliacal r i s m g a n d setting of the outer p l a n e t s ( M a r s , Jupiter a n d Saturn) h a p p e n s in the s a m e w a y as d e s c r i b e d a b o v e for the stars w i t h the additional c o m p l i c a t i o n that the planets themselves also m o v e so that the rising a n d setting occurs a g a m s t a different b a c k g r o i m d o f stars e a c h t u n e while the p e r i o d from one rising (or setting) to the n e x t d o e s n o t last exactly o n e year (as for the stars) b u t varies fi-om thirteen m o n t h s (for Saturn) to o v e r t w o years (for M a r s ) . T h e p a t t e m o f the helical rismgs a n d settmgs o f tìie inner p l a n e t s is e v e n m o r e c o m p l i c a t e d b e c a u s e the s p e e d at w h i c h they tiaverse the sky m a y e x c e e d that o f the S u n so that they (like the M o o n ) experience in addition a last m o m i n g visibility a n d a first evening visibility. I n addition to the dates of fnst a n d last visibility the B a b y l o n i a n s also r e c o r d e d the dates at w h i c h tìie planets r e a c h e d their so-called stationary p o i n t s , i.e. w h e n they r e a c h e d the e x t t e m e points m their a p p a r e n t orbit w h e r e t h e y c h a n g e t h e n direction o f m o t i o n w i t h respect to the stars (from p r o g r a d e to r e t t o g r a d e a n d vice versa). T h e t u n i n g o f these latter events is quite m a c c u r a t e b e c a u s e ( m particular for the outer planets) it takes quite s o m e time (days to w e e k s ) before the r e v e r s i o n of m o t i o n b e c o m e s clearly noticeable. T h e oldest k n o w n r e c o r d s of p l a n e t a r y observations are t h o s e o f V e n u s d m i n g the reign o f A m m i s a d u q a (17th century B C ) r e c o r d e d o n tablet 6 3 o f the o m e n series Enuma Anu EnliÛ T h e early B a b y l o n i a n k n o w l e d g e of the r i s m g a n d setting p a t t e m s o f stars a n d p l a n e t s is s u m m a r i z e d in the collection o f tablets k n o w n as MUL.APIN.'* T h e heliacal observations o f planets eventually resulted m the d e t e r m i n a t i o n o f p l a n e t a r y p e r i o d s w h i c h form the basis of the later d e v e l o p m e n t o f a m o r e exact B a b y l o n i a n p l a n e t a r y theory.^ T h e heliacal observations o f stars w e r e o f i m p o r t a n c e for the synchronization o f the lunar calendar w i t h the solar year b y intercalation.^ T h e o b s e r v a t i o n o f the first visibility of the M o o n w a s of c e n t t a l m p o r t a n c e in B a b y l o n i a in v i e w o f its role iii' fixing the fnst d a y o f the m o n t h in the (limar) calendar. T h i s p r e o c c u p a t i o n with the M o o n p o s e d a formidable p r o b l e m to B a b y l o n i a n scholars t r y m g to p r e d i c t its fnst a p p e a r a n c e in v i e w o f the rather irregular m o t i o n o f the M o o n a n d the relatively large tilt o f 5° o f its p a t h w i t h respect to the ecliptic. T h i s eventually lead to the d e v e l o p m e n t o f t w o quite sophisticated lunar theories, s y s t e m A a n d s y s t e m B m the 5th c e n t u r y B C . T h e

and P I N G R E E (1975).

^

REINER

"

HUNGER

^

For example BM 41004 studied by

H U N G E R and

^

and P I N G R E E ( 1989). PINGREE ( I W ) ,

p.

NEUGEBAUER

and

SACHS

(1967), p. 200; see also

203.

How this was done in practice is described in

HUNGER

and P I N G R E E ( 1987), pp. 89-91.

Early Babylonian Observations of Saturn: Astronomical Considerations

computational

methods

developed

for

these

theories

were

177

also u s e d

for

the

c o n t e m p o r a n e o u s d e v e l o p m e n t of t h e p l a n e t a r y theories. T o b e able t o date a n d to interpret the ancient o b s e r v a t i o n s o f risings a n d settings of stars a n d p l a n e t s o n e w o u l d like t o p r e d i c t the rising a n d setting d a t e s in the Julian

and

the

Babylonian

calendar

using

physical

principles

and

modem

m a t h e m a t i c a l m e t h o d s . A t p r e s e n t these p r e d i c t i o n s are usually still m a d e u s i n g the m e t h o d first d e s c r i b e d b y P t o l e m y in the 2 n d century A D b a s e d o n the u s e o f the socalled " a r c u s v i s i o n i s " , the m m i m a l difference in altitude at suiu-ise/simset b e t w e e n the star o r p l a n e t w i t h the S u n at w h i c h the star or p l a n e t is just/still visible.'' T h e a r c u s visionis d e p e n d s o n the m a g n i t u d e of t h e star o r p l a n e t (e.g. for S a t u m the A l m a g e s t gives 11°). H o w e v e r , the a r c u s visionis m e t h o d , t h o u g h m a t h e m a t i c a l l y s h n p l e a n d t h u s convenient, is a rather primitive m e t h o d b e c a u s e it d o e s n o t take accoimt o f variations in t h e p l a n e t a r y m a g n i t u d e a n d of a t m o s p h e r i c effects w h i c h m a y introduce a n u n c e r t a i n t y o f u p to five days in t h e p r e d i c t e d dates. T o r e m e d y t h e s e s h o r t c o m i n g s w e h a v e d e v e l o p e d a m o r e p h y s i c a l m e t h o d to c o m p u t e the d a t e s o f heliacal rising a n d setting o f stars a n d p l a n e t s . A c c o r d i n g to this m e t h o d stellar a n d p l a n e t a r y visibilities are calculated u s i n g t h e e x p e r i m e n t a l l y d e t e r m i n e d r e s p o n s e function o f the h u m a n eye in twilight c o n d i t i o n s , t h e k n o w n m a g n i t u d e of tìie star or planet, the m e a s u r e d b r i g h t a e s s o f the twilight sky a n d a t m o s p h e r i c e x t i n c t i o n a n d refraction.^ A similar m e t h o d h a s b e e n m d e p e n d e n t l y developed by SCHAEFER (1987). In this p a p e r I will a p p l y this n e w a p p r o a c h to the analysis a n d interpretation of t w o sets o f early B a b y l o n i a n o b s e r v a t i o n s o f the p l a n e t S a t u m dating f r o m the 6 t h a n d 7 t h centuries B C . A s it h a p p e n s these t w o sets o f o b s e r v a t i o n s a r e a b o u t 6 0 years a p a r t so that t h e y p r o v i d e

the oldest k n o w n material

from

which

the

B a b y l o n i a n s c o u l d h a v e d e r i v e d the p e r i o d i c i t y in the S a t u m p h e n o m e n a o f a few d a y s less t h a n 59 y e a r s , w h i c h they later u s e d extensively in t h e c o m p u t a t i o n o f p l a n e t a r y p o s i t i o n s in " g o a l - y e a r " texts. In c l o s m g t h e s e i n t i o d u c t o r y r e m a r k s I w a n t to e m p h a s i z e a n o t h e r i m p o r t a n t aspect o f a n y a t t e m p t t o interpret B a b y l o n i a n r e c o r d s of heliacal p h e n o m e n a , n a m e l y t h e fact that t h e y a r e o b s e r v e d at the h o r i z o n d u r i n g twilight. T h i s i m p l i e s that usually only a few o f the brightest stars are visible a n d that o n e therefore o u g h t to c h e c k whetiier reference in the text to stars and/or stellar constellations a r e b a s e d o n those few stars that are actually visible o r o n the general k n o w l e d g e o f the B a b y l o n i a n o b s e r v e r o f that r e g i o n o f the sky as it w o u l d a p p e a r if it w e r e o b s e r v e d at o t h e r times o f the y e a r o r e l s e w h e r e in the sky. A s I will illustrate in this p a p e r , i m a g e p r o c e s s i n g is a n i m p o r t a n t m o d e m tool to g r a p h i c a l l y illustrate the actual sky as it a p p e a r e d to t h e B a b y l o n i a n o b s e r v e r at t h e time o f o b s e r v a t i o n .

The Observations T h e o b s e r v a t i o n a l material of S a t u m d i s c u s s e d h e r e dates from the 7 * a n d t h e 6**" century B C .

'

Almagest XIII, 7;

^

INKLAAR(1989).

'

See

HUNGER

TOOMER

( 1984), p. 639.

and P I N G R E E ( 1999), pp. 167-173.

178

T. de Jong

B M 7 6 7 3 8 + 7 6 8 1 3 ( p r e s u m a b l y originating from B a b y l o n ) w a s recently fransliterated, franslated a n d discussed b y W A L K E R ( 1 9 9 9 ) . It is a j o i n e d fragment o f a tablet containing a collection o f observations o f the p l a n e t S a t u m m a d e d i n i n g t h e first 1 4 years o f t h e reign o f K a n d a l a n u ( 6 4 7 - 6 2 7 B C ) . It is o n e o f t h e earliest collections o f p l a n e t a r y data presently k n o w n . Collections like t h e o n e a n a l y s e d here w e r e p r e s u m a b l y gathered from asfronomical diaries, o f w h i c h t h e oldest k n o w n s p e c i m e n dates from 6 5 2 B C . ' ^ B M 7 6 7 3 8 + 7 6 8 1 3 contains 2 8 records o f first a n d last a p p e a r a n c e s (heliacal risings a n d settings) o f t h e planet S a t u m o b s e r v e d in B a b y l o n . F o r a m o r e detailed discussion t h e r e a d e r is referred t o the p a p e r b y W A L K E R ( 1 9 9 9 ) . O f these 2 8 r e c o r d s 7 r e c o r d s a r e u n r e a d a b l e or incomplete b e c a u s e o f textual d a m a g e , while 6 r e c o r d s a r e imreliable a c c o r d i n g to t h e professional aimotations o f the o b s e r v e r ("not o b s e r v e d " , " c o m p u t e d " or "high", i.e. visibility o c c u r r e d a few d a y s late, p r e s u m a b l y b e c a u s e o f c l o u d y skies o n the e x p e c t e d d a y o f first visibility). T h e r e m a i n i n g 1 5 reliable observations ( o f w h i c h 8 refer to heliacal risings a n d 7 to heliacal settings) a r e listed m T a b l e 1 . C o l u m n 1 contains t h e y e a r o f t h e r e i g n o f K a n d a l a n u , c o l u n m 2 lists t h e date in t h e B a b y l o n i a n c a l e n d a r g i v e n as lunar m o n t h / d a y , c o l u m n 3 gives t h e nature o f the event, c o l u m n s 5 a n d 6 give t h e dates a c c o r d i n g to t h e Julian calendar. T h e s e dates w e r e d e r i v e d from W a l k e r ' s r e c o n s t m c t i o n o f t h e lunar calendar during t h e r e i g n o f K a n d a l a n u . I h a v e c h e c k e d W a l k e r ' s r e c o n s t m c t i o n b y calculating the first d a y o f t h e m o n t h s (first a p p e a r a n c e of lunar crescent i n t h e w e s t e m e v e n m g sky) u s i n g a m o d e m e p h e m e r i s o f the M o o n (see b e l o w ) a n d u s i n g t h e visibility criterion o f M A U N D E R ( 1 9 1 1 ) . R e f m e m e n t s o f this criterion h a v e b e e n suggested b y ILYAS ( 1 9 8 4 ) b u t t h e m a t t e r is sufficientiy uncertain that I d i d n o t fmd it w o r t h t h e effort t o try to h n p l e m e n t t h o s e . F o r the m o n t h s in w h i c h t h e r e c o r d e d observations o c c u r m y calculations c o n f i r m W a l k e r ' s reconstmction. T h e other set o f observations o f S a t u m is t a k e n from T a b l e t 1 7 1 in t h e collection SBTU

I V , p u b h s h e d b y voN W E I H E R ( 1 9 9 3 ) a n d d i s c u s s e d b y H U N G E R ( 1 9 9 9 ) . It

contains 1 1 observations o f t h e dates o f fnst a n d last a p p e a r a n c e s a n d o f r e a c h i n g the stationary p o i n t s o f S a t u m during t h e years 2 8 - 3 1 o f t h e r e i g n o f N e b u c h a d n e z z a r II ( 6 0 4 - 5 6 2 B C ) . T h e tablet w a s u n c o v e r e d in U m k . F o r t h e p u r p o s e o f this p a p e r o n l y t h e 6 d a t e d observations o f fnst a n d last a p p e a r a n c e s ( 3 o f e a c h ) are relevant. T h e y a r e listed m T a b l e 2 w h i c h h a s t h e same format as T a b l e 1 . T h e dates in t h e Julian calendar are taken from H i m g e r w h o u s e d t h e tables o f P A R K E R / D U B B E R S T E I N ( 1 9 5 6 ) , apart from tiie date o f t h e first visibility in 5 7 7 B C w h i c h falls o n e d a y later a c c o r d i n g to m y o w n calculations. N o t e that t h e dates g i v e n b y H i m g e r for first visibility h a v e to b e increased b y o n e d a y to give t h e civil dates ( c o u n t e d from m i d n i g h t to midnight) listed in T a b l e 2 .

'°

S A C H S and H U N G E R ( 1 9 8 8 ) .

Early Babylonian Observations of Saturn: Astronomical Considerations

Reign o f Kandalanu Year

Date

Obs

IV/24

first

179

Julian calendar Year

Date

Loc Time

k(ext)

V(mag)

V(app)

AppAlt

Arc Vis

Duration

18-Jul

4:07

0.35

0.29

4.4

4.3

13.2

13 min

7-Jul

19:40

0.27

0.67

3.2

5.7

12.2

11 min

I-Aug

19:34

0.24

1.17

3.4

5.8

12.5

12 min

-646

1

-644

3 IV/7

last -642

5 V/23

last -641

6 V/20

last

16-Aug

19:21

0,20

0.98

3.2

4.5

11.1

9 min

VI/22

first

18-Sep

4:58

0.23

0.90

4.4

3.0

12.1

12 min

VII/15

first

29-Sep

5:09

0.24

0.78

4.4

3.0

12.0

11 min

7-Sep 8-Oct

18:55 5:20

0.24

0.74

3.2

9 min

0.67

3.9

5.1 1.6

11.7

0.15

9.5

9 min

last

30-Sep

18:26

0.26

0.60

3.3

5.0

11.9

8 min

VIII/23 first

3-Nov

5:36

0.27

0.56

4.4

3.2

12.3

12 min

last

11-Oct

18:10

0.25

0.56

3.2

5.0

11.6

9 min

VIII/15 first

13-Nov

5:46

0.26

0.53

4.3

3.2

12.1

12 min

22-Nov

5:55

0.17

0.51

4.2

1.6

10.1

14 min

7-Dec

6:05

0.31

0.52

4.5

3.7

13.1

14 min

15-Nov

17:37

0.24

0.53

3.1

4.8

11.5

11 min

-640

7

-639

8 VI/5 VII/5

last first -637

10 VII/20

-636

11 VII/13

-635

12 D(/5

first -634

13 X/1

first -633

14 VIII/20 last

T a b l e 1. O b s e r v e d a n d c o m p u t e d risings a n d settings of S a t u m o n tablet B M 7 6 7 3 8 + 76813

Nebucadnezzar II Year

Julian calendar Duration

Date

Loc time

k(ext)

V(mag)

V(app)

AppAlt

Arc Vis

last

23-Oct

17:58

0.29

0.54

3.4

5.3

12.4

8 min

K/4

first

29-Nov

5:54

0.41

0.53

4.7

5.1

15.0

14 min

VII/21

last

4-Nov

17:46

0.27

0.53

3.2

5.3

12.2

11 min

VIII/27

first

lO-Dec

6:05

0.39

0.53

4.6

4.6

14.5

12 min

VIII/14

last

16-Nov

17:38

0.26

0.53

3.1

5.2

12.1

12 min

D(/20

first

22-Dec

6:13

0.39

0.55

4.6

5.0

14.5

12 min

Date

Obs

VII/28

Year -576

28

-575

29

-574

30

T a b l e 2. O b s e r v e d a n d c o m p u t e d risings and settings of S a t u m o n tablet S B T U IV # 1 7 1

180

T. de Jong

Comparison of Observed and Calculated Dates of the First and Last Visibility of Saturn A s m e n t i o n e d in the introduction I h a v e calculated the dates o f first a n d last visibility o f S a t u m using a n e w m e t h o d b a s e d o n physical principles instead o f o n the rather primitive arcus visionis m e t h o d . T h i s m e t h o d has the following feahires: 1. It e m p l o y s recent accinate e p h e m e r i d e s of the M o o n b y C H A P R O N T TOUZÉ/CHAPRONT (1991) and of the Sun and planets by B R E T A G N O N / S I M O N ( 1 9 8 6 ) . T h e s e e p h e m e r i d e s are accurate to the a r c m i n u t e level for dates b a c k to about 4 0 0 0 B C . 2.

It uses a c o n v e r s i o n of Universal T i m e to E p h e m e r i s T i m e b a s e d o n the most recent determination.' '

3.

M a g n i t u d e s o f the planets are calculated u s i n g algorithms g i v e n in the Explanatory

4.

Supplement

to the Astronomical

Almanac}^

T h e obliquity o f the ecliptic a n d p r e c e s s i o n corrections are c o m p u t e d u s i n g the formulae g i v e n b y S I M O N et al. ( 1 9 9 4 ) w h i c h are accurate to the a r c m i n u t e level for dates b a c k to about 4 0 0 0 B C .

5.

Stellar m a g n i t u d e s a n d p r o p e r m o t i o n corrections are t a k e n from the Yale Bright Star

6.

Catalogue}^

Stellar a n d planetary visibilities are calculated u s m g the experimentally d e t e r m i n e d r e s p o n s e frinction of the h u m a n eye in twilight c o n d i t i o n s , the m e a s i u e d b r i g h t a e s s of the twilight sky a n d a t m o s p h e r i c extinction a n d refraction.''*

P o i n t s (1) t h r o u g h (5) m a k e sure that up-to-date astronomical data a n d m o d e m techniques are u s e d in the calculations. Point (6) forms the m o s t innovative aspect of this n e w m e t h o d . B y freating the extinction as the m a m i n d e p e n d e n t v a r i a b l e in the calculation it allows its determination at the t u n e o f o b s e r v a t i o n from the c o n d i t i o n that the m o d e l r e p r o d u c e s the o b s e r v e d dates o f first or last visibility. T h e results o f fitting the o b s e r v e d dates are s h o w n in c o l u m n s 6 t h r o u g h 12 of T a b l e s 1 a n d 2. Calculations are carried out for the g e o g r a p h i c a l location o f B a b y l o n ( 3 2 ° 3 3 ' N o r t h , 4 4 ° 2 5 ' East) a n d of U r u k (31° 18' N o r t h , 4 5 ° 4 0 ' East), respectively. In coliunn 6 w e give the Local T i m e of first a p p e a r a n c e of S a t u m (in hrs:min), c o l u m n 7 lists the d e r i v e d visual extinction coefficient, c o l u n m 8 gives the visual m a g n i t u d e o f S a t u m at the e p o c h of o b s e r v a t i o n a n d c o l u m n 9 its a p p a r e n t m a g n i t u d e at first a p p e a r a n c e , c o l u m n 10 lists the a p p a r e n t altitude (in d e g r e e s ) o f S a t u m a b o v e the h o r i z o n at first a p p e a r a n c e , c o l u m n 11 gives the d e r i v e d v a l u e o f the arcus visionis (in degrees) a n d finally in c o l u m n 12 w e list the d u r a t i o n o f the visibility o f S a t u m after its first a p p e a r a n c e in the m o m i n g / e v e n i n g sky. T h e results s u m m a r i z e d in T a b l e s 1 a n d 2 n e e d s o m e fiirther d i s c u s s i o n a n d clarification. First, I discuss the atmospheric extinction ( c o l u m n 7). T h e extinction coefficient k(ext) is defined as the increase in m a g n i t u d e (an increase in m a g r u t u d e c o r r e s p o n d s to a decrease in brightaess!) at zenith w h e r e the a t m o s p h e r i c p a t h "

STEPHENSON ( 1 9 9 7 ) . SEIDELMANN ( 1 9 9 2 ) . HOFFLElT(1982). INKLAAR(1989).

Early Babylonian Observations of Satum: Astronomical Considerations

181

traversed b y the light of a star or planet o n its w a y to the o b s e r v e r is m i n i m a l . T h e a t m o s p h e r i c p a t h length at zenith is defined as o n e " a i r m a s s " . F r o m the data in T a b l e s 1 a n d 2 I find an average extinction coefficient in B a b y l o n / U m k o f k(ext) = 0.27 ± 0.07 p e r airmass, c o r r e s p o n d i n g to a transparency of the a t m o s p h e r e at zenith jQ-0.4k(ext)^Q7g

D u e to the c i n v a t u r e o f the a t m o s p h e r e at altitude h a b o v e the h o r i z o n the a t m o s p h e r i c p a t h length is (to first a p p r o x i m a t i o n ) increased to airmass =scosec(h). T h i s implies that at an altitude of a b o u t 4", w h e r e S a t u m first a p p e a r s at its heliacal rising, the airmass » 15 so that the extinction a m o u n t s to about 4 m a g n i t u d e s , consistent with the values of the apparent m a g n i t u d e o f S a t u m listed in c o l u m n 9 of T a b l e s 1 a n d 2 ( o n a v e r a g e about 4 m a g n i t u d e s larger than V ( m a g ) ) . N o t e that the apparent altitude at first a p p e a r a n c e of S a t u m correlates with extinction as expected: the larger the extinction coefficient the higher the a p p a r e n t altitude. T h e a t m o s p h e r e also causes refraction o f the light o f a star or p l a n e t (the b e n d i n g of its light rays) so that it a p p e a r s closer to the h o r i z o n than its actual p o s i t i o n in the sky. T h e m a g n i t u d e of this effect also d e p e n d s o n airmass a n d a m o u n t s to a few tenth of a d e g r e e at the rising/setting altitude of S a t u m . T h e effect of refraction is included in the v a l u e s o f the a p p a r e n t a h i t u d e o f S a t i u n at first a p p e a r a n c e in c o l u m n 10 of T a b l e s 1 and 2.

0.11-0.15

0.16-0.20 0.21-0.25 0.26-0.30 0.31-0.35

0.36-0.40 0.41-0.45

Visual extinction (magnitude per airmass) F i g i u e 1. F r e q u e n c y disfribution of atmospheric extinction in B a b y l o n i a aroimd 6 0 0 B C as derived from fitting B a b y l o n i a n a n d U r u k o b s e r v a t i o n s o f S a t u m . T h e s h a p e o f this distribution is quite similar to that o f m o d e m observations at p r e s e n t - d a y asfronomical o b s e r v i n g sites (e.g. G U E R R E R O et al. ( 1 9 9 8 ) ) . In figure 1 I s h o w the disfribution o f the extinction values for B a b y l o n a n d U r u k a n d in T a b l e 3 I list the average values for B a b y l o n a n d U r u k w i t h thefr standard deviations. F o r c o m p a r i s o n I also list values derived from observations at L e i d e n Observatory,'^ typical for g o o d photomefric nights in a h u m i d climate at sea level, a n d at the La P a l m a O b s e r v a t o r y , ' ^ typical for conditions at a m o d e m asfronomical 5

nights in November-December

2850

nights during

1984-1998;

1961;

see

see

KWEE

GUERRERO

and

et al.

VANG£NDÊREN< 1963).

(1998).

T. de Jong

182

o b s e r v a t o r y in a d r y climate at altitudes a b o v e 2 5 0 0 meter. T h e fact that the extinction at L a P a l m a is a b o u t four tunes smaller t h a n in L e i d e n illustrates the e x t r e m e sensitivity o f the extinction t o the relative hiunidity o f the air. T h e s h a p e of the distribution in figine 1 is v e r y similar to that found at the L a P a l m a O b s e r v a t o r y w h i c h gives confidence in the reality o f the results for B a b y l o n / U r u k .

Location

Country

Altitude

Climate

Babylon

Iraq

S ea 1 evd

D es eit/Subtropi c

0.25 +/-0.05

Uruk

Iraq

S ea 1 evd

D es ert/Subtr opi c

0.34 +/-0.07

Leiden

the Netherlands

Sea level

Sea dimate

043+/-0.11

La Palma

Canary Islands

2600 meter Subtropic/Sea dimate

0.11+/-0.02

T a b l e 3 . A t m o s p h e r i c extinction m B a b y l o n , U r u k , L e i d e n a n d L a P a l m a T h e s o m e w h a t larger values of the extinction d e r i v e d for the U r u k o b s e r v a t i o n s c o m p a r e d t o those for B a b y l o n m a y b e attributed t o a seasonal effect s m c e the U r u k o b s e r v a t i o n s w e r e d o n e in winter t i m e . T h i s is illustrated in figure 2 w h e r e the extinction values in T a b l e s 1 a n d 2 are plotted against the time o f year. T h e error b a r s indicate the r a n g e o f extinction values that p r o v i d e a fit t o the o b s e r v a t i o n s . T h e c i n v e d r a w n is a s e c o n d - o r d e r p o l y n o m i a l fit to the data. A l t h o u g h the n u m b e r o f data points is small the data suggest that extinction is largest m s u m m e r a n d w i n t e r a n d smallest in the fall. T h i s is consistent with s e a s o n a l p a t t e m s . In s u m m e r the tiansparency o f the air in s o u t h e m M e s o p o t a m i a is affected b y o c c a s i o n a l h e a v y s a n d storms a n d b y the relatively h i g h h u m i d i t y o f the air c a u s e d b y e v a p o t r a n s p n a t i o n in the s u m m e r h e a t ( a b o u t 2 0 0 m m w a t e r e v a p o r a t e s p e r m o n t h in July a n d A u g u s t ) , while m winter the hiunidity of the air is e n h a n c e d d u e to o c c a s i o n a l ramfall ( a b o u t 1 0 - 2 0 n m i p e r m o n t h in N o v e m b e r - D e c e m b e r ' ' ) . A c c o r d m g to its definition the values of the arcus visionis listed m c o l u n m 11 of T a b l e s 1 a n d 2 h a v e b e e n calculated b y taking the difference b e t w e e n the real altitude o f S a t u m a b o v e a n d o f the S u n b e l o w the h o r i z o n . T h e data in T a b l e s 1 a n d 2 s h o w that the d e r i v e d v a l u e of the arcus visionis varies from a b o u t 9 ° to 15° d e g r e e s . T h e v a l u e s listed i m p l y an altitude o f the S u n o f a b o u t 8° - 10° b e l o w the h o r i z o n at the first a p p e a r a n c e o f S a t u m in the m o m i n g s k y o n the d a t e o f first visibility a n d o f a b o u t 6° - 7° b e l o w the h o r i z o n at first a p p e a r a n c e o f S a t u m in the e v e n i n g sky o n the date of last visibility. T h e m e a n v a l u e of the arcus visionis a v e r a g e d o v e r all o b s e r v a t i o n s in T a b l e s 1 a n d 2 equals 12.3° ± 1.3°. T h i s is intermediate b e t w e e n the t w o values a d v o c a t e d b y P t o l e m y ( 1 1 ° in the Almagest a n d 13° in the Handy

Tables^\

Cf ZUIDERVtlET(1981)p. 13. See

NEUGEBAUER

(1975), p. 1017.

183

Early Babylonian Observations of Satum: Astronomical Considerations

1-jul

31-Jul

31-aug

1-okt

31-okt

1-dec

1-jan

Time of year Figure 2, V a r i a t i o n of the a t m o s p h e r i c extinction in B a b y l o n i a d i u i n g the year as d e r i v e d from fitting B a b y l o n i a n a n d U r u k observations from S a t u m aroimd 6 0 0 B C . Error b a r s represent the uncertainty in the d e t e r m i n a t i o n o f the extmction. T h e curve d r a w n represents a s e c o n d o r d e r p o l y n o m i a l fit to the data p o i n t s . T h e data suggest that extinction is highest in s u m m e r a n d winter a n d lowest in the fall.

16,0

n

15,0 14,0 w

13,0 12,0 -

'>

11,0 -

to

3

<

10,0 9,0 8,0 0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

Visual extinction (magnitude per airmass) Figure 3 . D e p e n d e n c e o f the arcus visionis o n a t m o s p h e r i c extinction as d e r i v e d fi-om fitting B a b y l o n i a n observations of S a t u m a r o u n d 6 0 0 B C . E r r o r b a r s indicate the uncertainty in the determination o f the arcus visionis. T h e Ime d r a w n represents a linear regression t h r o u g h t h e data p o i n t s . A d o p t i n g a fixed value of 12° w o u l d result in heliacal rismg/settmg dates o f S a t u m w h i c h c a n b e off b y u p to 3 d a y s in e i t h e r direction.

T. de Jong

184

B o t h the variable atmospheric extinction a n d the variation in m a g n i t u d e of S a t u m affect the v a l u e o f the arcus visionis. In figure 3 I s h o w the d e p e n d e n c e o f the d e r i v e d value o f the arcus visionis o n tìie visual extinction u s i n g data t a k e n from c o l m n n s 7 and 11 of T a b l e s 1 and 2. T h e error b a r s reflect the uncertainty in the determination o f the arcus visionis d u e to the r a n g e of extinction v a l u e s that fit the o b s e r v e d dates. T h e line d r a w n represents a linear regression t h r o u g h the data points. A d o p t i n g a fixed value of 12° w o u l d result in heliacal rising/setting dates of S a t u m w h i c h c a n b e off b y u p to 3 d a y s in either direction since 1° difference in the arcus visionis c o r r e s p o n d s to a shift o f about o n e d a y in the c o m p u t e d d a t e . T h e derived value of the arcus visionis also d e p e n d s o n the b r i g h t a e s s o f S a t u m . T h e data in coliunn 8 of T a b l e 1 s h o w that the visual m a g n i t a d e of S a t u m varies from a b o u t 0.3 to 1.2. T h i s variation is m u c h larger taan for the other outer planets ( M a r s a n d Jupiter). T h i s is d u e to the fact that S a t a m is the o n l y p l a n e t that is s u r r o u n d e d b y bright rings. T h e s e rings consist of small dust particles w h i c h are exfremely effective in reflecting the sunlight. T h e o b s e r v e d variation in m a g n i t u d e of S a t u m is d u e to the variation in the aspect angle o f its rings with r e s p e c t to the terresfrial observer. A t the m a x i m u m tilt of 2 9 ° S a t u m ' s rings are a b o u t three times brighter t h a n the p l a n e t a r y disk so that they d o m i n a t e S a t u m ' s visual m a g n i t a d e (the m a g n i t a d e scale is logarithmic; one m a g n i t a d e c o r r e s p o n d s to a factor 2.5 in b r i g h t a e s s ) . T h e confribution from the rings to the b r i g h t a e s s o f S a t u m varies w i t h a p e r i o d o f about 2 9 years, the orbital period o f S a t u m .

0,40

0,60

0,80

1,00

1,40

Visual magnitude of Satum Figure 4 . D e p e n d e n c e o f the arcus visionis o n p l a n e t a r y m a g n i t a d e as c o m p u t e d for a fixed visual extinction k(ext) = 0.25 m a g n i t a d e s p e r a i r m a s s for the p l a n e t S a t u m in B a b y l o n during 6 4 7 - 6 3 4 B C . Error b a r s indicate the e s t h n a t e d uncertainty in the d e t e r m m a t i o n o f the arcus visionis. T h e Ime d r a w n represents a linear regression through the data points. T h e b r i g h t a e s s o f S a t u m varies b y m o r e than o n e inagnitade w i t h a p e r i o d o f a b o u t 2 9 y e a r s (its orbital p e r i o d a r o u n d the Sim) d u e to the varying tilt o f its rings with respect to the terrestrial observer. A d o p t i n g a fixed value of 12° for the arcus visionis w o u l d result ui heliacaf rising/setting dates o f S a t a m w h i c h c a n b e off b y 1 d a y in either direction.

Early Babylonian Observations of Satum: Astronomical Considerations

185

T h e d e p e n d e n c e o f the arcus visionis o n m a g n i t u d e is illustrated in figure 4 w h e r e I plot data that h a v e b e e n c o m p u t e d for the helical rising a n d setting o f S a t u m in 6 4 7 6 3 4 B C , a d o p t i n g a fixed value o f the visual extinction o f k(ext) = 0.25 m a g n i t u d e s p e r a u m a s s . T h e line d r a w n represents a linear regression t h r o u g h the d a t a p o i n t s . A d o p t i n g a fixed value o f 12° for the arcus visionis w o u l d result in heliacal rising/setting dates o f S a t u m w h i c h c a n b e off b y u p to 1 d a y in either direction. T h u s , it is clear that using the arcus visionis m e t h o d to c o m p u t e dates o f heliacal rising a n d setting o f stars a n d planets has its limitations for the interpretation o f Babylonian observations.

Watching the Heavens T h e sky is e m i n e n t l y suitable to b e illustrated. In this p a p e r I shall a t t e m p t to d o this for t w o o b s e r v a t i o n s : the heliacal rising o f S a t i u n in N o v e m b e r 6 3 7 B C o b s e r v e d in B a b y l o n a n d the reaching o f its E a s t e m stationary p o i n t in July 5 7 5 B C o b s e r v e d m Uruk. Figiu-e 5 s h o w s the E a s t - S o u t h - E a s t e m s k y m the m o m i n g o f 13 N o v e m b e r 6 3 7 B C at 5:53 hrs L o c a l T i m e in B a b y l o n , a b o u t 10 m i n u t e s after the m o m e n t that S a t u m h a d a p p e a r e d again after having b e e n invisible for 3 3 d a y s . It illustrates line 22 o f tablet B M 7 6 7 3 8 + 7 6 8 1 3 , franslated b y W A L K E R ( 1 9 9 9 ) as follows: " Y e a r 1 1 , m o n t h 8, d a y 15, a b o v e Lisi 6 '/2 U S , first a p p e a r a n c e ; w i t h reference to Lisi a little in front" T h e u p p e r p a n e l o f figiu^e 5 s h o w s S a t i u n a n d all visible stars d o w n to m a g n i t u d e 5.0 m a n area o f s k y o f a b o u t 5 0 x 3 0 s q u a r e d e g r e e s . T h e constellations S c o r p i o ( G I R . T A B ) a n d Libra (zibânîtu) are rising. Lines c o n n e c t i n g the brightest stars a n d delineating the i m a g e s of these t w o constellation are s h o w n . Satiim is the brightest a n d Lisi ( = a S c o ) is the next brigthtest object close to the h o r i z o n , e a c h o n a different side o f the ecliptic (thin line). F o r the p a r a m e t e r s of S a t u m see T a b l e 1. T h e l o w e r p a n e l s h o w s the sky as actually o b s e r v e d b y the B a b y l o n i a n s ; o n l y S a t u m a n d 10 stars are bright e n o u g h to b e visible in the twilight sky. A s n o t e d b y B R O W N ( 2 0 0 0 b ) this observation is o f interest b e c a u s e the t e r m uS is u s e d h e r e to indicate a n angular m e a s u r e while usually U § is e m p l o y e d as a unit o f time ( o f 4 m i n u t e s , equivalent to the rising o f 1° at the h o r i z o n ) . T h e text suggests that m e a s u r e m e n t s w e r e t a k e n w i t h respect to the ecliptic as a n (mvisible) reference system. T o further analyse the situation w e list the c o o r d i n a t e s o f S a t i u n a n d of a Sco: Sattun:

X = 212.85°

/3=

Lisi(aSco):

X = 213.18°

/3=-4.23°

2.01°

M e a s u r e d m ecliptic coordinates the position difference is AX = - 0.33° a n d Ap = 6.33°. T h e B a b y l o n i a n observer could h a v e h a d a rather g o o d idea o f the l o c a t i o n of the ecliptic b y c o n n e c t i n g the t w o " N o r m a l S t a r s " , ' ' ô S c o ( " m i d d l e star o f h e a d o f

Cf HUNGER and PINGREE ( 1 9 9 9 ) , p. 1 4 8 - 1 5 1 .

186

1

T. de Jong

/ / \

^—"^"

•

/ / .

•

«

\

/

•

1

/

•

••

• •

\ •

/

. • • / V/.•

••

• •

•

• • • (

E

SE

•/ / / / ;

/

• •

/

• • /

• / •

•

/

E

SE

Figure 5, T h e E a s t - S o u t h - E a s t e m sky m the m o m i n g of 13 N o v e m b e r 6 3 7 B C at 5:53 hrs L o c a l T i m e in B a b y l o n , a b o u t 10 m i n u t e s after the m o m e n t that Satirni h a d r e a p p e a r e d again after h a v i n g b e e n invisible for 3 3 d a y s . T h e u p p e r p a n e l s h o w s S a t u m a n d all visible stars d o w n to m a g n i t u d e 5.0. T h e constellations S c o r p i o ( G I R . T A B ) and L i b r a (zibânîtu) are rising. F i g u r e s delineated b y the brightest stars m e a c h constellation are s h o w n . S a t u m is the brightest a n d Lisi ( = OL Sco) is the next brigthtest object close to the h o r i z o n , e a c h o n a different side o f the ecliptic (thin line). F o r the p a r a m e t e r s o f S a t u m see T a b l e 1. T h e lower p a n e l shows the sky as actually o b s e r v e d b y the B a b y l o n i a n s ; o n l y S a t u m a n d 10 stars are bright e n o u g h to b e visible m the m o m i n g twilight sky. [Adapted from C y b e r s k y 3 . 2 . 1 , c o p y r i g h t © 1999 S t e p h e n M . Schimpf]

Early Babylonian Observations of Satum: Astronomical Considerations

187

S c o r p i o n " ) a n d a L i b (the central star of the Scales delineated in figine 5 a n d k n o w n b y the B a b y l o n i a n s as " s o u t h e m part of the Scales") w h i c h are b o t h visible in the twilight s k y a c c o r d i n g to figure 5. A s far as the statement " a b o v e Lisi 6 Vi U S " is c o n c e m e d , this implies that 1 Ui§ equals about 1° (6 Vi U § = 6.33°), as p o i n t e d out before b y B R O W N ( 2 0 0 0 b ) . N o t e that the statement "with reference to Lisi a little in front" is inconsistent if it refers to a measiu-ement a l o n g the ecliptic in the direction of increasing longitude. B u t it is consistent if the m e a s u r e m e n t is m a d e in the direction of the attitude o f the scorpion, i.e. from tail to head. A distance o f 0.33° a h e a d of Lisi c o r r e s p o n d s to about 2/3 of the diameter o f the M o o n a n d is certainly significant in v i e w of the a c c u r a c y o f B a b y l o n i a n m e a s u r e m e n t s . A direction o f m e a s u r e m e n t o p p o s i t e to the direction o f increasing ecliptic longitude in the 1^ century B C d o e s n o t h a v e to w o r r y u s b e c a u s e there are other e x a m p l e s of this in the early diaries,^° while it is k n o w n that the s c h é m a t i s a t i o n of the z o d i a c in 12 signs o f 3 0 ° with longitude increasing from W e s t to E a s t is a later invention ( p r o b a b l y j u s t p r e c e d i n g the d e v e l o p m e n t of lunar s y s t e m A for w h i c h it is a prerequisite). T h e other text that I w a n t to illusfrate asfronomically is t a k e n from tablet

SBTU

I V # 1 7 1 . B a s e d o n the G e r m a n franslation of H U N G E R ( 1 9 9 9 ) lines 1 1 - 1 3 read: " V I I 2 0 , 5/6 K Ù â u n d e r the little star of P a b i l s a g , 3 K Ù è kalakku, stationary, a n d w e n t b a c k to the E a s t "

above

N o w y e a r 3 0 , m o n t h VII, d a y 2 0 c o r r e s p o n d s to Julian date 2 3 O c t o b e r 5 7 5 B C w h i c h is o v e r t w o m o n t h s later than the asfronomically e x p e c t e d date 11 A u g u s t 5 7 5 B C . I n s p e c t i o n o f the text shows that the m o n t h n a m e § U (Du'Qzu, m o n t h I V ) m a y easily b e m i s r e a d as D U e (TeSrïtu, m o n t h V I I ) so that the actual date m a y well h a v e b e e n I V / 2 0 c o r r e s p o n d i n g to 2 6 July 575 B C w h i c h w o u l d fit m u c h better. A difference o f t w o w e e k s is quite a c c e p t a b l e in v i e w of the difficulty to acciu-ately d e t e r m i n e the date at w h i c h S a t u m reaches its stationary p o i n t b e c a u s e it m o v e s only a b o u t 1 arc m i n u t e in four days in the fortitight a r o u n d the reversal o f its dfrection o f motion. In figme 6 I s h o w the W e s t - S o u t h - W e s t e m sky m U m k at m i d n i g h t o n 2 6 / 2 7 J u l y 5 7 5 B C . In o r d e r n o t to m a k e the picture o v e r c r o w d e d o n l y stars o f the 4th m a g n i t u d e a n d brighter are s h o w n . T h e ecliptic is a g a i n r e p r e s e n t e d b y a thin line. S a t u m (the brightest object located close to the ecliptic) is in b e t w e e n the tip o f the " b o w " o f P a b i l s a g (Sagittarius a n d part of O p h i u c h u s ) a n d the "little star" m e n t i o n e d in the text, w h i c h is located j u s t right a n d a b o v e S a t u m . K a l a k k u is franslated b y V O N W E I H E R ( 1 9 9 3 ) a n d H U N G E R ( 1 9 9 9 ) as " B e h a l t e r " w h i c h m e a n s s o m e t h i n g like "container". Figiu^e 6 suggests that k a l a k k u is the star X Sag in the tip o f the " b o w " o f P a b i l s a g while the "little star"is to b e identified with the star /n Sag. T o b e able to analyse the situation m o r e quantitatively I list tiie ecliptic c o o r d i n a t e s o f S a t u m a n d the t w o reference stars b e l o w : little star ( / i S a g ) :

X = 237.47°

0=

2.68°

Sattun:

X = 237.85°

0=

1.29°

k a l a k k u (X Sag):

X = 240.60°

/3 = -

1.40°

SACHS and HLJNGER, ( 1984).

T. de Jong

188

Figure 6. T h e W e s t - S o u t h - W e s t e m sky in U r u k at m i d n i g h t o n 2 7 J u l y 5 7 5 B C . In order n o t to m a k e the p i c t i u e o v e r c r o w d e d only stars brighter than m a g n i t u d e 4.0 are s h o w n . T h e ecliptic is r e p r e s e n t e d b y the thin line. S a t u m (the brightest object located close to the ecliptic) has r e a c h e d its E a s t e m stationary point. T h e " b o w a n d a r r o w " o f P a b i l s a g (Sagitarius a n d p a r t o f O p h i u c h u s ) are delineated b y the brightest stars. T h e "little star" m e n t i o n e d in the text is located j u s t right a n d a b o v e S a t u m . [ A d a p t e d from C y b e r s k y 3 . 2 . 1 , copyright © 1999 S t e p h e n M . Schhnpf] T h e o n l y w a y in w h i c h the distances 5/6 K Ù § a n d 3 K Ù S m a k e n u m e r i c a l sense is if they refer to the a n g u l a r distances (/x Sag - S a t u m ) a n d (X Sag - S a t u m ) ; if so 1 K Ù â « 1 . 7 ° . T h u s in this early observation there is n o indication that the t e r m s " a b o v e " a n d " b e l o w " should b e interpreted as referring to angular distance measiu-ed p e r p e n d i c u l a r to the ecliptic (ecliptic latitude) as a r g u e d b y G R A S H O F F ( 1 9 9 9 ) from the analysis o f a large n u m b e r of diary citations o f the p o s i t i o n o f the M o o n relative to " N o r m a l " stars. M o r e o v e r GraChoff derives a v a l u e o f 1 K Ù S « 2 . 4 ° , significantly larger than the v a l u e resulting from the analysis o f the o b s e r v a t i o n d i s c u s s e d here. Since the large majority o f observations analysed b y Grafihoff dates from the 3rd century B C and later this could u n p l y that the m e t h o d to d e s c r i b e the relative positions o f the p l a n e t s relative to stars m a y h a v e b e c o m e m o r e formalised in the course of time a n d that the value o f the K Ù S also m a y h a v e e v o l v e d . T h e "little star" o f P a b i l s a g is not o n e o f the regular " N o r m a l S t a r s " listed b y HuNGER/PiNGREE ( 1 9 9 9 ) . H o w e v e r , R O U G H T O N / C A N Z E R O N I ( 1 9 9 2 ) h a v e p r o p o s e d a n identification o f the "four stars in the W e s t o f Sagittarius", o c c a s i o n a l l y q u o t e d m the Diaries, with a small tight g r o u p o f four w e a k to v e r y w e a k stars o f w h i c h /x Sag is the brightest. O u r text is as far as I k n o w the first time that this ( g r o u p of) star(s) is q u o t e d as such.

The 59-Year Period of Saturn A s already m e n t i o n e d in the infroduction it so h a p p e n s that the t w o texts analysed a n d discussed in this p a p e r are about 6 0 years apart, so that in p r i n c i p l e they c o u l d h a v e b e l o n g e d to the collection of observations u s e d during the N e o - B a b y l o n i a n

Early Babylonian Observations of Satum: Astronomical Considerations

189

e n t i r e to d e r i v e the 59-year periodicity o f the S a t u m p h e n o m e n a . It is n o t well k n o w n w h e n exactly the p l a n e t a r y p e r i o d s w e r e d i s c o v e r e d but the available e v i d e n c e suggests the 6th century B C as the m o s t p r o b a b l e time f r a m e . I n T a b l e 4 I list three pairs o f observations that could h a v e b e e n u s e d for this determination. T o imderstand h o w these periodicities m i g h t h a v e b e e n d i s c o v e r e d it h e l p s t o familiarize oneself w i t h the B a b y l o n i a n frame of m i n d w h i c h w a s p r e o c c u p i e d w i t h r e c o g n i z i n g p a t t e m s in p a s t a n d p r e s e n t celestial events in o r d e r to b e a b l e to p r e d i c t the ftiture. F u r t h e r m o r e , it is important to realize that the p l a n e t s are n o t the only celestial b o d i e s e x p e r i e n c m g this cyclic b e h a v i o u r . In fact, b e c a u s e the b a s i c unit o f time m e a s u r e m e n t is the lunar m o n t h , periodicities also exist for stars m the sense that after a certain n u m b e r o f m o n t h s the heliacal rising or setting of a star o c c u r s at 22

the s a m e d a y o f the m o n t h . O n e early text i n d e e d lists a 2 7 - y e a r p e r i o d for the brightest star Sirius (a C M a ) . T h e stellar cycle basically serves to s y n c h r o n i z e the lunar calendar to the solar year. E x a c t l y 2 7 years ( = 3 3 4 lunar m o n t h s ) later Sirius rises a n d sets heliacally again o n the s a m e d a y in the s a m e m o n t h (at least if intercalation h a d b e e n t a k e n care of p r o p e r l y ) . Therefore it d o e s n o t c o m e as a surprise that 2 7 y e a r s is the s u m of the lengths o f the t w o k n o w n m t e r c a l a t i o n cycles u s e d in M e s o p o t a m i a o f 8 a n d 19 years. I n fact, o n e m a y t u m the a r g u m e n t a r o i m d a n d w o n d e r w h e t h e r the 19-year cycle has b e e n d i s c o v e r e d in this w a y o n c e it h a d b e c o m e clear that the 8-year cycle s h o w e d severe s h o r t c o m i n g s . T h e c m c i a l o b s e r v a t i o n n e e d e d to d e t e r m i n e the p l a n e t a r y p e r i o d is to o b s e r v e the planet at its heliacal rising or setting against exactly the s a m e b a c k g r o i m d o f stars as at the b e g i i m i n g o f the cycle, i.e. at the s a m e p o s i t i o n in the sky a n d at the s a m e distance from the Sun. Since V e n u s has the shortest p e r i o d o f 8 y e a r s a n d since observations o f V e n u s a r e the oldest surviving, the p e r i o d i c i t y o f the V e n u s p h e n o m e n a m a y h a v e b e e n d i s c o v e r e d first. O n the other h a n d , the m o t i o n o f V e n u s is irregular a n d c o m p l i c a t e d so that a periodicity is m o r e difficult to r e c o g n i z e in the data. In c o m p a r i n g s k y m a p s for the 6 p a n s of observations o f S a t u m in T a b l e 4 I b e c a m e indeed i m p r e s s e d b y the pair-wise similarity o f the s k y at observation, all the m o r e striking b e c a u s e the sky is rather e m p t y at twilight. T h i s strong visual m i p r e s s i o n is not reflected in the rather dry professional c o m m e n t a r y o f the observers. H o w e v e r , the c o m m e n t s are consistent. F o r instance, m t h e 2 n d p a i r o f observations the B a b y l o n i a n o b s e r v e r r e c o r d s : "at the b e g i n n i n g o f P a b i l s a g " , w h i l e the U r u k o b s e r v e r writes: "at the e n d o f the a r r o w o f P a b i l s a g " (6 O p h ) (see figure 6). I n the 4 t h p a n the Babyloitian tablet gives: " i n the m i d d l e o f P a b i l s a g " w h i l e the U r u k tablet h a s : "before the k a l a k k u of P a b i l s a g " (X S a g ) . Clearly, the p o s i t i o n a l references are m o r e p r e c i s e in the later set of observations a n d the later o b s e r v a t i o n s also suggest that the n o t i o n of ecliptic longitude m c r e a s m g m the E a s t w a r d d n e c t i o n h a d b e e n introduced. F r o m the d a t a in T a b l e 4 I derive a B a b y l o n i a n p e r i o d o f 7 3 0 m o n t h s ( = 5 9 years) - 4 d a y s , w h i l e the attested p e r i o d from the oldest source (4th c e n t u r y B C ) is

HUNGER and PINGREE ( 1 9 9 9 ) , p. 2 0 3 . BM 4 5 7 2 8 ; HUNGER and PINGREE ( 1 9 9 9 ) , p. 2 0 3 .

T. de Jong

190

5 9 years - 6 days.^^ T h i s p e r i o d a n d the ones for the other p l a n e t s w e r e extensively u s e d for the calculation of p l a n e t a r y positions in the so-called " g o a l - y e a r " texts.^"*

Reign of Nebuchadnezzar II

Reign of Kandalanu Year

Date

Obs

Note

Year

12

Date

Obs

d(days)

VII/28

last

-6/-7

IX/4

first

28 VIII/5

last

IX/5

first

computed

13

-1

29 VIII/26

last

X/1

first

not

observed?

VIII/20

last

VIII/14

last

W20+?

first

IX/20

first

14

VII/21

last

-J

VIII/27

first

-3/-4

30 -6

<0

T a b l e 4. O b s e r v a t i o n s relevant for the determination o f S a t u m ' s p e r i o d i c i t y

Conclusions In this p a p e r I h a v e s h o w n that accepting the c o m p e t e n c e o f the ancient Babyloitian a n d U r u k o b s e r v e r s a n d c o m b i n i n g t h e n observational data with a p h y s i c a l t h e o r y o f heliacal p h e n o m e n a a n d m o d e m asfronomical k n o w l e d g e a n d t e c h n i q u e s allows a d e e p e r i m d e r s t a n d i n g o f tìie p h e n o m e n a o b s e r v e d , in particular o f the influence of a t m o s p h e r i c effects. F u r t h e r m o r e , it clearly illusfrates the limitations o f the u s a g e of the arcus visionis m e t h o d to interpret ancient o b s e r v a t i o n a l r e c o r d s o f ffrst a n d last a p p e a r a n c e s of stars a n d planets. I h a v e also s h o w n that for a p r o p e r mterpretation o f heliacal p h e n o m e n a it is essential to realize that o n l y few objects are visible in the twilight sky. T h i s m a y b e quite u n p o r t a n t for the interpretation o f m a n y texts w h e r e the first visibility o f constellations is r e p o r t e d b e c a u s e usually only o n e or at m o s t a few o f the stars o f that constellation are actually visible. F u r t h e r m o r e the ffrst visibility o f that star p r o b a b l y fixes the date o f the rising o f the constellation as a w h o l e . T o d o this p r o p e r l y it is c m c i a l to b e able to realistically s h n u l a t e the sky as o b s e r v e d . F u r t h e r m o r e , b y analysing a few observations in detail I h a v e s h o w n the potential o f this m e t h o d for a d e e p e r u n d e r s t a n d m g o f the " P l a n e t a r i u m Babylonicum".^^ F m a l l y , the data discussed here form the m o s t ancient d a t a b a s e to date from w h i c h the p l a n e t a r y p e r i o d o f S a t u m c o u l d h a v e b e e n d e r i v e d b y the M e s o p o t a m i a n asfronomers.

B M 4 1 0 0 4 ; NEUGEBAUER and SACHS ( 1 9 6 7 ) . See HUNGER andPiNGREE ( 1 9 9 9 ) ^ pp. 1 6 7 - 1 7 3 . "

GOSSMAN ( 1 9 5 0 ) ; PINGREE and WALKER ( 1 9 8 8 ) ; KOCH ( 1 9 9 2 ) .

Early Babylonian Observations of Satum: Astronomical Considerations

191

Acknowledgements I w o u l d like t o e x p r e s s m y sincere thanks t o D r D a v i d B r o w n for p r e s e n t m g m e with a reprint o f his C A J p a p e r w h i c h started this project, t o J o h n Steele a n d L i s B r a c k B e m s e n for a critical r e a d i n g o f this p a p e r , a n d to D r C h r i s t o p h e r W a l k e r for d r a w i n g m y attention t o t h e p a p e r o f P r o f H u n g e r a n d for hosting a v e r y p l e a s a n t a n d s t u n u l a t m g conference at the British M u s e u m .

References B R E T A G N O N , P i e r r e a n d S I M O N , Jean-Louis. 1986. Planetary Programs and Tables from -4000 to +2800. R i c h m o n d : W i l m a n n - B e l l . B R O W N , D a v i d . 2 0 0 0 a . Mesopotamian Planetary Astronomy-Astrology. Gronmgen: Styx. B R O W N , D a v i d . 2 0 0 0 b . " T h e C u n e i f o r m C o n c e p t i o n o f Celestial S p a c e a n d T i m e " . Cambridge ArchaeologicalJoumal 10: 1 0 3 - 1 2 2 . C H A P R O N T - T O U Z É , M i c h e l l e a n d C H A P R O N T , Jean. 1 9 9 1 . Lunar Tables and Programs from 4000 BC to AD 8000. R i c h m o n d : W i h n a n n - B e l l . GOSSMANN, Felix. 1950. Planetarium Babylonicum. Rome. GRABHOFF, G e r d . 1 9 9 9 . " N o r m a l Star O b s e r v a t i o n s i n Late B a b y l o n i a n A s t r o n o m i c a l D i a r i e s " . In: N o e l M . S W E R D L O W (ed.), Ancient Astronomy and Celestial Divination: 9 7 - 1 4 7 . C a m b r i d g e M A : M I T press. G U E R R E R O , M a r t i n A . ; G A R C L \ - L O P E Z , R . J.; C O R R A D I , R o m a n o L. M . ; J I M E N E Z ,

A n t o n i o ; FuENSALiDA, Jose M . ; R O D R I G U E Z - E S P I N O S A , J o s e M . ; A L O N S O , A n g e l ; CENTURION, M i r i a m a n d P R A D A , F r a n c i s c o . 1 9 9 8 . " E x t i n c t i o n O v e r t h e C a n a r i a n O b s e r v a t o r i e s : T h e Limited Influence o f S a h a r a n D u s t " , New Astronomy Reviews 4 2 : 5 2 9 - 5 3 2 . HOFFLEIT, Dorrit. 1982. The Bright Star Catalogue. N e w Haven: Yale University Observatory, H U N G E R , H e r m a i m , 1999. " S a t u m b e o b a c h t i m g e n aus der Zeit N e b u k a d n e z a r s II". I n J o a c h i m M A R Z A H N a n d H a n s N E U M A N N (eds.), Assyriologica et Semitica: Festschrift fiir Joachim Oelsner: 1 8 9 - 1 9 2 . Miinster: Ugarit V e r l a g . H U N G E R , H e r m a n n a n d PINGREE, D a v i d . 1 9 8 9 . MUL.APIN: An Astronomical Compendium in Cuneiform (Archiv fur Orientforschimg Beiheft 2 4 ) . H o m : Berger & Sohne. H U N G E R , H e r m a i m a n d PINGREE, D a v i d . 1999. Astral Sciences in Mesopotamia. L e i d e n / B o s t o n / K o l n : Brill. I L Y A S , M o h a m m a d . 1984. Islamic Calendar, Times and Qibla. K u a l a L u m p u r : Berita P u b l i s h i n g C o . INKLAAR, F r a n k . 1989, Een Nieuwe Méthode voor de Bereking van Heliakische Opkomsten. A m s t e r d a m : Universiteit v a n A m s t e r d a m . K O C H , J o h a n n e s . 1992, " D e r S t e m e n k a t a l o g B M 7 8 1 6 1 " . Welt des Orients 2 3 : 39-67. KWEE, K m g K . a n d V A N G E N D E R E N , A m a u d M . 1 9 6 3 . "Photo-electric O b s e r v a t i o n s o f 3 1 a n d 3 2 C y g n i d i n i n g N o v e m b e r a n d D e c e m b e r 1 9 6 1 " , Bulletin o f t h e A s t r o n o m i c a l Institute o f the N e t h e r l a n d s 17: 5 3 - 5 5 . M A U N D E R , E d w a r d W a l t e r 1 9 1 1 . " O n t h e Smallest Visible P h a s e o f the M o o n " . Journal of the British Astronomical Association 2 1 : 3 5 5 - 3 6 2 .

192

T. de Jong

N E U G E B A U E R , O t t o . 1 9 7 5 . A History of Ancient Mathematical Astronomy. Berlin: Springer V e r l a g . N E U G E B A U E R , O t t o a n d S A C H S , A b r a h a m J. 1967. " S o m e Atypical A s t r o n o m i c a l C u n e i f o r m T e x t s . I". Journal of Cuneiform Studies 2 1 : 1 8 3 - 2 1 7 . P A R K E R , R i c h a r d A . a n d D U B B E R S T E I N , W a l d o H . 1956. Babylonian Chronology 626 B.C. - A.D. 75. P r o v i d e n c e : B r o w n U n i v e r s i t y Press. PINGREE, D a v i d a n d W A L K E R , Christopher. 1988. " A B a b y l o n i a n S t a r - C a t a l o g u e : B M 7 8 1 6 1 " . I n Erie LEICHTY, M a r i a deJ. ELLIS, a n d P a m e l a G E R A R D I (eds.), A Scientific Humanist: Studies in Memory of Abraham Sachs: 313-322. Philadelphia: T h e S a m u e l N o a h K r a m e r Fimd. REINER, E r i c a a n d PINGREE, D a v i d . 1 9 7 5 . The Venus Tablet of Ammisaduqa (Bibliotheca M e s o p o t a m i c a 2/1). M a l i b u : U n d e n a Publications. R O U G H T O N , N o r b e r t A . a n d C A N Z E R O N I , G . L . 1992. " B a b y l o n i a n N o r m a l Stars i n Sagittarius", Journal for the History SIMON,

Jean-Louis;

BRETAGNON,

of Astronomy

Pierre;

23: 193-200.

CHAPRONT,

Jean;

CHAPRONT-TOUZÉ,

Michelle; FRANCOU, Gerard, and LASKAR, Jacques. 1994. "Numerical E x p r e s s i o n s for P r e c e s s i o n F o r m u l a e a n d M e a n E l e m e n t s for t h e M o o n a n d t h e P l a n e t s " . Astronomy and Astrophysics 282: 6 6 3 - 6 8 3 . S A C H S , A b r a h a m J. a n d H U N G E R , H e r m a i m . 1 9 8 8 . Astronomical Diaries and Related Texts from Babylonia Volume 1. W i e n : Osterreichische A k a d e m i e d e r Wissenschaften. S E I D E L M A N N , P . K e n n e t h . 1 9 9 2 . Explanatory Supplement to the Astronomical Almanac. M i l l Valley: University Science B o o k s . SCHAEFER, B r a d l e y E . 1 9 8 7 . " H e l i a c a l R i s e P h e n o m e n a " . Archeoastronomy 11:19. S T E P H E N S O N , F . R i c h a r d . 1 9 9 7 . Historical Eclipses and Earth's Rotation. C a m b r i d g e : C a m b r i d g e University P r e s s . S W E R D L O W , N o e l M . 1999. Ancient Astronomy and Celestial Divination. Cambridge M A : M I T Press. T O O M E R , G e r a l d J. 1984. Ptolemy's Almagest. L o n d o n : D u c k w o r t h . W A L K E R , C h r i s t o p h e r B . F . 1999. " B a b y l o n i a n O b s e r v a t i o n s o f S a t u m d u r i n g t h e R e i g n o f K a n d a l a n u " . I n N o e l M . S W E R D L O W (ed.). Ancient Astronomy and Celestial Divination: 6 1 - 7 6 . C a m b r i d g e M A : M I T P r e s s . V O N W E I H E R , Egbert. 1 9 9 3 . Spatbabylonische Texte aus Uruk IV. M a i n z a m R h e i n : Phillip v o n Z a b e m ZuiDERVLlET, A . C . 1 9 8 1 . Irak, Landendocumentatie nr.2. A m s t e r d a m : Koninklijk Instituut v o o r d e T r o p e n .

The Eye of Horus and the Planet Venus: Astronomical and Mythological References Rolf Krauss,

Berlin

The Eye of Horus as a Heavenly Body T h e e y e o f H o r n s is a v e r y c o m p l e x m y t h o l o g i c a l n o t i o n . ' O n e o f its aspects is c o s m o l o g i c a l . I n t h e H y m n s to t h e D i a d e m the e y e o f H o r u s is a d d r e s s e d as the W h i t e C r o w n a n d identified as a h e a v e n l y body:^ Praise t o the W h i t e C r o w n : Veneration to you, eye of Horns, white o n e , great o n e , at w h o s e b e a u t y the E i m e a d rejoices, w h e n it rises m the e a s t e m horizon. T h o s e w h o a r e in the lifting u p o f S h u praise thee, t h o s e w h o g o d o w n in the w e s t e m horizon, w h e n y o u s h e d light for t h o s e w h o are in the D u a t . In 1911 A d o l f E r m a n identified the e y e o f H o r u s here as the sun, w h i c h i n d e e d rises in the east a n d m a y shed light t o those in t h e netherworld after setting in t h e west. E r m a n b a s e d his interpretation o n t h e notion, w e l l - k n o w n i n ancient E g y p t , that t h e s u n a n d m o o n w e r e the t w o eyes o f the s k y g o d . Since E r m a n ' s d a y E g y p t o l o g i c a l scholarship h a s d e t e r m i n e d that this idea evolved v e r y late historically speaking.^ T h e H y m n s t o t h e D i a d e m date t o the M i d d l e K i n g d o m , a n d thus they caimot s h o w a n y trace o f the idea that t h e sim w a s o n e o f the eyes o f the s k y g o d . B u t e v e n in E r m a n ' s day, the "great, white e y e o f H o r n s " c o u l d h a v e b e e n u n d e r s t o o d alternatively as a star w h i c h rises in t h e east. S i m u l t a n e o u s w i t h its ascent, other stars set in the west. A s long as the star w h i c h e m b o d i e s t h e e y e o f H o r n s is visible, i.e. until simrise, it s h e d s its light for the b l e s s e d d e a d w h o t h e m s e l v e s a r e stars in a p a r t o f the s k y imderstood a s the D u a t . T h e issue is c o m p l i c a t e d b e c a u s e the e y e o f H o r n s is s o m e t i m e s d e s c r i b e d as t h e eye o f the s u n g o d R a , as m the following citations from the P y r a m i d T e x t s : (PT § 698 b-d. T.P.M.) the S e h e d star of N N will b e m a d e h i g h with R a ,

WESTENDORF (1980), p. 4 8 - 5 1 . ERMAN (1911), p. 22. SPELEERS (1934), p. 85f; RUDNITZKY (1956), p. 12f

194

R. KRAUSS

that N N travels t o a n d fro is in t h e F i e l d s o f Offering. N N is this e y e o f R a w h i c h s p e n d s the night and is c o n c e i v e d a n d b o m every day. V a r i a n t (§ 6 9 8 d. N ) : N N is this e y e o f H o r n s w h i c h s p e n d s the n i g h t a n d is c o n c e i v e d a n d b o m e a c h day. K u r t Sethe in his c o m m e n t a r y to this text e x p l a i n e d the eye o f R a a s s i m p l y a n o t h e r n a m e for the smi itself

H e d i d n o t attach m u c h m ç o r t a n c e to the variant " e y e o f

H o r n s " for " e y e o f R a " . Y e t Sethe h i m s e l f r e m a r k e d the confradiction inherent in his interpretation a c c o r d i n g to w h i c h the sim w o u l d fravel to a n d fro in the F i e l d s o f Offering, since the F i e l d s o f Offering a r e otherwise quite clearly a p a r t o f t h e starry night sky. B y confrast to Sethe, R u d o l f A n t h e s c o n s i d e r e d it quite p o s s i b l e t o e x p l a i n the e y e o f the s u n as a star o r at least a n entity differing from t h e s u n i t s e l f

An

a r g u m e n t m favour o f this p r o p o s a l is found in the w e l l - k n o w n m y t h a b o u t t h e e y e o f the s u n g o d w h i c h left its lord to search for his twin children. W h e n t h e e y e r e t u m e d to the s u n g o d , a n o t h e r e y e h a d g r o w n in its p l a c e . T h e original e y e b e c a m e e x c e e d i n g l y a n g r y a n d r a g e d against the s u n g o d . T o pacify the e y e the s u n g o d g a v e it a n o t h e r p l a c e o n his brow.^ If this m y t h reflects reality or n a t u r e , t h e n t h e e y e w h i c h leaves the s u n g o d a n d afterwards r e t u m s to h i m c a n n o t b e identical with the s u n disk itself

The Planet Venus as the Eye of Horus T h e p r i m e c a n d i d a t e for a h e a v e n l y b o d y w h i c h m o v e s a w a y from the s u n o n l y to r e t u m is t h e p l a n e t V e n u s . It c a n b e h y p o t h e s i z e d that V e n u s e m b o d i e s t h e e y e o f R a as well as t h e e y e o f H o r n s . T h e m y t h o l o g i c a l n o t i o n that there are t w o e y e s o f H o r n s c o u l d relate to V e n u s as the M o m i n g Star = H o r n s t h e Y o u n g e r a n d as the E v e n i n g Star = H o r n s the Elder. B o t h these forms o f the g o d H o r n s w e r e s o n s o f Osfris a n d Isis, T h e E l d e r H o r n s w a s c o n c e i v e d a n d b o m before Seth, the b r o t h e r o f Osiris a n d Isis, killed Osfris, T h e Y o u n g e r H o r n s w a s c o n c e i v e d p o s t h u m o u s l y , after t h e m u r d e r o f Osfris, Egyptologists k n o w that the g o d s a r o i m d Osiris e m b o d y stars a n d constellations. Osfris h i m s e l f a p p e a r s as the constellation Orion, w h e r e a s Isis is identical w i t h the fixed

star Sfrius, a n d Seth is the p l a n e t M e r c u r y . ' In a s t m g g l e , Seth g o u g e d out

H o m s ' s e y e . T h e w o u n d e d e y e w e n t missing, b u t the m o o n g o d T h o t h r e t u m e d it to H o r n s a n d m a d e it w h o l e again. B e c a u s e o f the p l a n e t a r y a s s o c i a t i o n attested for Seth a n d the p r o p o s e d p l a n e t a r y association o f H o r n s with V e n u s , it is feasible that

"

SETHE (1962), p. 277.

^

ANTHES (1961), p. 9.

^

OTTO(1975),p. 564f

'

NEUGEBAUER and PARKER ( 1969), p. 180.

The Eye of Horus and the Planet Venus: Astronomical and Mythological References

195

the m j ^ h a b o u t the eye, first w o i m d e d a n d afterwards h e a l e d , reflects a s t r o n o m i c a l o b s e r v a t i o n s c e n t e r i n g o n M e r c u r y , V e n u s a n d the moon.*

The Eye of Horus the Elder as Venus, the Evening Star, in the Calendar of Lucky and Unlucky Days T h e so-called c a l e n d a r of L u c k y a n d U n l u c k y D a y s ' contains i n f o r m a t i o n w h i c h reinforces the identification of Seth with the p l a n e t M e r c i n y a n d s h o w s that the eye of H o r u s the E l d e r e m b o d i e s the p l a n e t V e n u s as the E v e n i n g S t a r . ' " T h e c a l e n d a r contains p r o g n o s t i c a t i o n s for e a c h d a y w h i c h is t e r m e d either " l u c k y " o r " u n l u c k y " accordingly. In general, the p r e d i c t i o n s are b a s e d o n the activities o f g o d s . T h e calendar, w h i c h is p r e s e r v e d in t w o m a n u s c r i p t s dating from aroimd 1 2 0 0 B C E , n m s t h r o u g h o n e entire civil year from d a y 1 to d a y 3 6 5 . F o r the c o n v e n i e n c e o f n o n E g y p t o l o g i s t s in w h a t follows, I shall n o t refer to the d a y s m t e r m s o f the E g y p t i a n calendar, b u t c o u n t t h e m instead from 1 to 3 6 5 . H o r u s or his e y e is m e n t i o n e d firom d a y 18 to d a y 3 5 0 , i.e. d u r i n g a t i m e s p a n of 3 3 2 d a y s . If H o r u s or his eye d o e s m d e e d represent the p l a n e t V e n u s , the entries caimot relate to continual visibility as M o m i n g or E v e n i n g Star, b e c a u s e o n e s u c h p h a s e d o e s n o t last m o r e t h a n 2 7 0 days. T h e c a l e n d a r specifically m e n t i o n s H o r n s the E l d e r ' s eye o n d a y s 2 1 3 a n d 2 1 8 a n d a g a i n o n d a y s 3 1 1 a n d 3 1 5 . O n d a y 2 1 8 the eye of H o m s the E l d e r is said to b e i n c o m p l e t e , b e c a u s e its p a r t s are c o u n t e d as fractions w h o s e s u m a m o u n t s to less t h a n 1/2. B u t o n d a y 3 4 9 the eye o f H o m s is d e s c r i b e d as m a state o f particular c o m p l e t e n e s s : " T h i s eye o f H o m s h a s c o m e , b e i n g filled, b e i n g soimd, n o t h i n g is m i s s i n g in it." If this state o f affairs is the c o n t i n u a t i o n o f w h a t w a s d e s c r i b e d o n d a y 2 1 8 , t h e n w e are dealing w i t h the s a m e eye o f H o m s b e t w e e n d a y s 2 1 3 a n d 3 5 0 , that is, for a p e r i o d o f 137 d a y s or m o r e t h a n half a visibility p e r i o d o f 2 7 0 d a y s . T h e p e r i o d o f visibility m q u e s t i o n s h o u l d h a v e e n d e d o n d a y 3 5 0 + x a n d started 2 7 0 d a y s earlier o n d a y 8 0 + x. T h e r e f o r e H o m s or his eye w h i c h are m e n t i o n e d b e t w e e n d a y s 18 and 7 2 s h o u l d n o t relate t o the e y e o f H o m s the Elder, b u t rather to the eye o f H o m s the Y o u n g e r . It is p o s s i b l e to d e t e r m i n e w h i c h eye of H o m s relates t o the E v e n i n g Star or the M o m i n g Star, b e c a u s e the c a l e n d a r p r o v i d e s information a b o u t a p e r i o d o f mvisibility after d a y 7 2 : d a y 1 0 3 : A p p e a r a n c e of the W h i t e O n e d a y 111 : S e a r c h for die A k h e t eye d a y 168: S e a r c h for the A k h e t eye in L e t o p o l i s d a y 169 : F m d i n g / S e e i n g o f the g o d In the guise o f the W h i t e O n e , the eye o f H o m s a p p e a r e d o n d a y 1 0 3 . " T h e s e a r c h for the A k h e t eye is t o b e i m d e r s t o o d as a s e a r c h for the eye o f H o m s . ' ^ T h e fact that *

' '°

KRAUSS (1997), p. 287 ff.

LEITZ(1994). KRAUSS ( 1990), pp. 49-56; KRAUSS ( 1999), pp. 233-254.

' ' For White One as a synonym for the eastern eye of Horus, cf the Hymns to the Diadem, above.

196

R. Krauss

there was a search for the A k h e t eye o n d a y 111 c o u l d indicate that the eye w a s difficult to find or w a s n o t to b e foimd at all. Fifty-seven days later a s e a r c h for the A k h e t eye w a s m a d e in L e t o p o l i s , the cult centre o f H o r u s the Elder. T h e following day, the g o d w a s found or seen. T h e 57 d a y interval b e t w e e n the t w o s e a r c h e s c o r r e s p o n d s to the l o w e r r a n g e of the a p p r o x i m a t e l y 55 to 7 0 days o f invisibility of V e n u s b e t w e e n M o m i n g a n d E v e n i n g Star p h a s e s . ( T h e invisibility o f V e n u s b e t w e e n E v e n i n g a n d M o m i n g Star p h a s e a m o u n t s to a b o u t 19 d a y s at most.'^) T h i s indicates that the e y e o f H o m s the E l d e r is to b e identified as the E v e n i n g Star in the Calendar of Lucky and Unlucky Days. O n e d a y after t h e finding or s e e m g the god, the U d j a t eye, a n o t h e r d e s i g n a t i o n o f the e y e of H o m s , h a d m o v e d t o w a r d s tìie south a n d k e p t o n m o v i n g in this direction until, o n d a y 190, the eye w a s o n the m o v e t o w a r d s the north. d a y 170: T h e U d j a t eye m o v e s t o w a r d s the s o u t h d a y 188: F o l l o w i n g the eye t w o g o d s m o v e t o w a r d s the s o u t h d a y 190: T w o G o d s a c c o m p a n y the m i g h t y flame t o w a r d s the n o r t h In a s t r o n o m i c a l t e r m s , the m o v e m e n t o f the Udjat eye or the " m i g h t y

flame"

first

s o u t h w a r d s a n d t h e n n o r t h w a r d s indicates that the s o u t h e m declination o f V e n u s w a s fnst d e c r e a s i n g a n d t h e n i n c r e a s m g for 2 0 days at m o s t following fnst visibility. T h e d e s i g n a t i o n U d j a t e y e w a s u s e d indiscriminately for the right or left eye o f H o m s from the N e w K i n g d o m o n w a r d s . ' ' '

Astronomical Dating of the Calendar of Lucky and Unlucky Days If the interpretation s u g g e s t e d h e r e b e accepted, it s h o u l d b e p o s s i b l e to d e t e r m i n e m w h i c h particular y e a r V e n u s as the E v e n i n g Star b e c a m e visible after 5 8 d a y s or less of invisibility a r o u n d d a y 169 a n d w h e n the p l a n e t ' s s o u t h e m declination started to d e c r e a s e s o m e d a y s later. B e c a u s e the oldest m a n u s c r i p t o f the c a l e n d a r d a t e s t o the r e i g n o f M e m e p t a h at the latest, the y e a r in q u e s t i o n o u g h t to b e earlier t h a n 1204 B C E w h e n h e died. O n the a s s u m p t i o n that the c a l e n d a r o r i g i n a t e d in M e m p h i s or H e l i o p o l i s ' ^ a n d that the o b s e r v a t i o n s it r e c o r d s w e r e m a d e there, the asfronomical c o m p u t a t i o n s are c a l c u l a t e d for latitude 2 9 . 9 ° , u s i n g U r a n i a S t a r 1.1.'^ W h e t h e r the A k h e t eye alias V e n u s w a s found or s e e n o n d a y 111 is e q u i v o c a l ; therefore, it is u n c l e a r w h e t h e r invisibility lasted 5 8 d a y s or less. T h e invisibility of V e n u s c o n t m u e s for a b o u t 5 8 ± 3 days w h e n the u p p e r conjunction t a k e s p l a c e at solar longitude b e t w e e n 2 3 0 ° a n d 330*; thereafter, the onset o f e v e n i n g visibility o c c u r s 2 4 to 3 4 d a y s after u p p e r conjunction, i.e. at solar l o n g i t u d e o f ca. 254°

to

( 3 6 0 " +) 4«.'^ T h e s e c o n d i t i o n s imply tiie p e r i o d b e t w e e n 1700 a n d 1250 B C E w h e n

Akhet eye is a designation of the eye of Horus on days 295 and 309; cf below. VAN DER WAERDEN (1942), p. 50.

GRIFFITHS (1958), p. 182ff. LEITZ(1994), p. 8. PiETSCHNiGand VOLLMANN ( 1 9 9 5 ) .

VAN DER WAERDEN (1942), p. 50; HUBER (1982), p. 11.

The Eye of Horus and the Planet Venus: Astronomical and Mythological References

197

d a y 169, as the b e g i n n i n g o f evening visibility, c o r r e s p o n d e d to a solar l o n g i t u d e of ca. 2 5 4 * to ( 3 6 0 ° +) 4 ° . W i t h i n this p e r i o d the onset o f e v e n i n g visibility t o o k p l a c e in L o w e r E g y p t on d a y 169 only in the following years, e a c h separated from the n e x t b y an interval o f 8 years reflecting the short-term cyclic b e h a v i o u r o f V e n u s : -1305/04 -1289/88

-1273/72

-1297/96 -1281/80

-1265/64

Before - 1 3 0 5 / 0 4 V e n u s b e c a m e visible in L o w e r E g y p t o n d a y 170 or earlier; at that t u n e the s o u t h e m declination w a s decreasing, not increasing a r o u n d d a y 169. In years later t h a n - 1 2 6 5 / 6 4 V e n u s b e c a m e visible before d a y 169, at a time w h e n the s o u t h e m declination w a s increasing. F i g i n e

1 illustrates the m o v e m e n t o f the

E v e n i n g Star s o u t h w a r d s in - 1 3 0 5 / 0 4 , - 1 2 9 7 / 9 6 and - 1 2 6 5 / 6 4 . T h e situation in other cyclic years c a n b e interpolated. After - 1 3 0 5 / 0 4 the m o v e m e n t s o u t h w a r d s b e c a m e m o r e c o n s p i c u o u s a n d of longer duration.

200

190

180

170

h o r i z o n 238'

south

240'

242

F i g u r e 1. Positions o f V e n u s b e t w e e n d a y 169 a n d 2 0 0 , 30 minutes

after

sunset;

horizon

at

Memphis,

years

- 1 3 0 5 / 0 4 , - 1 2 9 7 / 9 6 and - 1 2 6 5 / 6 4 T a k i n g into c o n s i d e r a t i o n the times o f M e r c i n y ' s , alias S e t h ' s , visibility r e d u c e s the n u m b e r of possible a l t e m a t i v e s . T h e calendar r e c o r d s a case of visibility o n d a y 164: " D o not g o out o n it at the b e g i n n i n g o f d a w n . It is the d a y o f s e e i n g the r e b e l a n d S e t h ' s killing h i m in the b o w o f the great b a r q u e of the sim g o d . " P r e s u m i n g that

198

R. KRAUSS

M e r c u r y is Seth, the n o t i o n o f the g o d standing in the b o w o f the solar b a r q u e c a n b e imderstood as M e r c u r y seen at d a w n o n the e a s t e m horizon.'* F o r the ancient Egyptians w h o i m a g i n e d that the sun g o d travelled m a b a r q u e , it w a s evident that M e r c u r y - S e t h s t o o d m the p r o w o f the solar b a r q u e . B e t w e e n - 1 3 0 5 / 0 4 a n d - 1 2 6 5 / 6 4 the p l a n e t w a s visible o n d a y 164 only in the following years: -1305/04 -1297/96 -1265/64 T h e s e years fit the a s t r o n o m i c a l indications o f the calendar to a greater or lesser extent. F o r e x a m p l e - 1 2 6 5 / 6 4 fits the entry for d a y 3 4 0 to a lesser d e g r e e than the t w o other p o s s i b l e years: "It is the d a y o f the entering o f the eye o f R a into his [the sun g o d ' s ] h o r i z o n . " A s stated a b o v e , the eye o f the s u n g o d is identical w i t h the eye of H o m s . T h e r e f o r e the text s e e m s to say that the eye of H o m s the E l d e r ( V e n u s the E v e n i n g Star) set at the s a m e spot on the h o r i z o n w h e r e the sun h a d set earlier o n the same evening. In principle this situation pertains w h e n the declination o f V e n u s the E v e n i n g Star and the settmg sun is the same. T h e r e are uncertainties to consider, such as the ancient o b s e r v e r ' s definition of sunset a n d different v a l u e s o f refraction for the setting of the sun a n d o f V e n u s . T h e following table contains the differences m declinations o f V e n u s , m i n u s declinations o f the sun, b e t w e e n simset a n d the setting of V e n u s a r o u n d d a y 3 4 0 for the three years in question: day 335

-1296

336

-1304 +5' +11'

337

-12'

+30'

338

-36' -60'

-6'

339 340 341

-83'

-1264

+47'

-18' -41'

+127' +104'

342

+80

343 344

+56' +32'

345 346

+9' -15' -39'

347

T h e declinations o f the s u n a n d V e n u s d o not m a t c h o n d a y 3 4 0 in a n y o f the three altematives. In the years - 1 2 9 7 / 9 6 u n d - 1 3 0 5 / 0 4 the declinations w e r e m o r e or less identical v e r y shortly before d a y 3 4 0 , b u t not in - 1 2 6 5 / 6 4 . Therefore it is m o r e likely that tiie c a l e n d a r refers to - 1 2 9 7 / 9 6 or - 1 3 0 5 / 0 4 tiian to - 1 2 6 5 / 6 4 .

KRAUSS (1990), p. 52.

The Eye of Horus and the Planet Venus: Astronomical and Mythological References

199

Venus and Mercury at the Western Horizon in -1297/96 T h e r e follows a n analysis o f select c a l e n d a r entries: d a y 2 1 4 : G o d s a n d g o d d e s s e s are satisfied, w h e n they see the children of G e b resting on t h e n thrones. d a y 2 1 5 : Setting forth o f the majesty o f H o r u s . G e b ' s children o f d a y 2 1 4 are evidently identical with the children of G e b ' s s p o u s e N u t o f d a y 2 8 (see b e l o w ) . T h e situation described for d a y 2 1 4 s e e m s to b e a conjunction of M e r c u r y a n d V e n u s . In the p r e c e d i n g days V e n u s h a d gotten n e a r e r to Mercury. O n day 216 Venus passed by Mercury: Venus's movement on day 215 may have b e e n i m d e r s t o o d b y the observer as "setting forth". T h i s interpretation o f days 2 1 4 a n d 2 1 5 is o n l y p o s s i b l e in - 1 2 9 7 / 9 6 , b u t not in - 1 3 0 5 / 0 4 a n d - 1 2 6 5 / 6 4 , w h e n M e r c u r y w a s n o t visible in the evening a r o u n d d a y 2 1 4 . T h i s indicates that the c a l e n d a r originated in - 1 2 9 7 / 9 6 .

Mercury Venus

10"

horizon 240

250

260'

F i g u r e 2 . Conjimction of M e r c u r y a n d V e n u s o n d a y 2 1 4 in - 1 2 9 6 , 3 0 m i n u t e s after sunset; h o r i z o n at M e m p h i s . M o r e t h a n a m o n t h later, o n day 2 5 0 , the calendar r e c o r d s : "Setting forth o f the majesty [or: the W h i t e O n e ] o f the s k y g o i n g n o r t h . " " T h i s c o u l d relate to the fact that aroimd d a y 2 5 0 V e n u s m o v e d into the n o r t h w e s t e m part o f the sky, n o r t h from azimuth 2 7 0 ° . O n d a y 2 5 0 the p o s i t i o n of V e n u s w a s a b o u t 30* west o f O r i o n ; a m o n t h later V e n u s p a s s e d O r i o n w h o s e heliacal setting started a r o i m d that date in antiquity. T h e r e is a slight possibility that the calendar entry o f d a y 2 7 6 relates to V e n u s p a s s i n g b y Orion: "Setting forth b y H o r u s to repel w h a t is d o n e against his father a n d t a k m g a d v i c e with the followers o f his father O n n o f r i s . " A s F i g u r e 3 s h o w s , H o r u s , alias V e n u s , m e t Jupiter a n d S a t u m in close conjunction o n d a y 2 7 6 . U n d e r the g i v e n c i r c u m s t a n c e s Jupiter a n d S a t u m c o u l d h a v e b e e n d e s c r i b e d as followers o f Orion, KRAUSS ( 1 9 9 9 ) , p. 2 3 8 .

200

R. KRAUSS

alias Osiris, T h e v e i l e d reference to s o m e t h i n g impleasant that h a p p e n e d to O r i o n , alias O s k i s , could b e u n d e r s t o o d as the heliacal settmg o f the stars o f Orion, F i g u r e 3 illustrates the stars o f the G r e e k O r i o n for the r e a d e r ' s orientation, I follow L o c h e r in identifying the ' b e l t s t a r s ' o f the G r e e k O r i o n as the stars a b o v e the h e a d (the c r o w n ) o f the E g y p t i a n Orion,^°

Saturn

Sirius

\

^^^^^

J"P't«'

Orion 10" horizon 230*

250'

270*

290*

Figure 3 , Conjimction o f V e n u s , Jupiter and S a t u m , o n d a y 2 7 6 in - 1 2 9 6 , 3 0 m i n u t e s after simset; h o r i z o n at M e n p h i s . B e t w e e n d a y 3 0 3 a n d 3 2 4 the c a l e n d a r refers to a crisis w h i c h w a s f o r e s h a d o w e d o n day 2 9 5 , w h e n the A k h e t eye w a s described as satisfied; s o o n thereafter the w r a t h o f the e y e w o u l d e m p t . day 295: day 300:

T h e A k h e t eye is satisfied Setting forth o f Shu, H e will b r i n g b a c k the Udjat-eye, It is pacified o n this d a y b y T h o t h

day 303:

R a g i n g o f the majesty

day 305:

G o i n g o f the g o d d e s s to the p l a c e w h e n c e she c a m e

day 309:

P e a c e b e t w e e n R a a n d the A k h e t eye

d a y 3 1 1 : T h e e y e o f H o m s the Elder, r a g e s a g a m s t R a d a y 3 1 4 : R a g m g o f the eye o f H o m s the E l d e r d a y 3 2 4 : H e intends to m o v e b a c k t o w a r d s t h e south. T h e n u m b e r o f c a l e n d a r entries during the p e r i o d s h o w s that the o b s e r v e r w a s intrigued b y this p e r i o d of crisis, w h e n V e n u s s l o w e d d o w n o n the w a y n o r t h w a r d s a n d then started returning t o w a r d s the south. T h e r e t u m is inaugurated b y the

LOCHER ( 1 9 9 3 ) , pp. 2 7 9 - 2 8 1 .

The Eye of Horus and the Planet Venus: Astronomical and Mythological References

201

"setting forth o f S h u " , w h o shall b r i n g b a c k the eye, a n d b y the pacification o f the eye b y T h o t h . T h o t h a n d S h u a p p e a r here in their traditional roles vis-à-vis the Udjat eye.^' It is p o s s i b l e that the r e m a r k a b o u t Shu w a s inserted after die r e t u m o f the eye. O t h e r w i s e it is n e c e s s a r y to s u p p o s e that the observer a l r e a d y k n e w o n d a y 3 0 0 that the r e t u m o f the eye w a s i m m m e n t . It is feasible that the o b s e r v e r k n e w a b o u t the 8 year cycle o f V e n u s a n d that, o n the b a s i s o f w h a t h a d o c c u r r e d in - 1 3 0 5 / 0 4 h e anticipated w h a t w o u l d h a p p e n m - 1 2 9 7 / 9 6 . T h e e y e r a g e d o n d a y s 3 0 3 , 3 1 1 a n d 3 1 4 , b u t w a s peaceftil o n d a y 3 0 9 . It j u s t so h a p p e n s that these are the days w h e n V e n u s s l o w e d d o w n , b u t it r e m a i n s imclear h o w the o b s e r v e r u n d e r s t o o d raging a n d peacefulness. T h e r e m a r k o n d a y 3 2 4 a b o u t " h i s " intention to m o v e t o w a r d s the south c a n b e i m d e r s t o o d in t e r m s o f a s t r o n o m y . T h e o b s e r v e r c o u l d h a v e realized o n that d a y that V e n u s h a d m o v e d s o u t h w a r d s after standing still a r o u n d d a y 3 1 0 : A c c o r d i n g to conq)utation V e n u s set o n d a y 3 1 0 at a n a z i m u t h o f 3 0 0 * 8', o n d a y 3 2 4 o f 2 9 9 * 35'. T h e difference a m o u n t s to the d i a m e t e r o f a full m o o n w h i c h m a y well h a v e b e e n detectable w i t h the aid o f a p l u m b l i n e .

280'

Figure 4 . Positions of V e n u s after d a y 2 4 0 in - 1 2 9 6 , 3 0 m i n u t e s after simset; h o r i z o n at M e m p h i s . Actually it w a s the n o r t h e m declination of V e n u s w h i c h r e a c h e d its m a x u n u m d u r i n g these d a y s o f crisis, w h e r e a s the e l o n g a t i o n w a s still increasing. T h e e l o n g a t i o n r e a c h e d its m a x i m u m a r o u n d d a y 3 5 4 , w h e n the c a l e n d a r d o e s n o t m e n t i o n H o m s o r his eye. B u t it d o e s so 5 d a y s e a r h e r , o n d a y 3 4 9 , w h e n , for practical p u r p o s e s , the elongation h a d a k e a d y attained its m a x i m u m . O n d a y 3 4 9 the c a l e n d a r r e c o r d s that the eye o f H o m s h a d c o m e , w a s filled a n d c o n ç l e t e , without anything missing. T h i s

JUNKER ( 1 9 1 7 ) , pp.

124,151.

202

R. Krauss

assertion c o u l d relate to the state of m a x i m u m e l o n g a t i o n w h e n the brilliance o f the E v e n i n g Star is v e r y c o n s p i c u o u s . T h e situation o n d a y 3 4 9 is reminiscent o f w h a t the c a l e n d a r r e c o r d e d for d a y 2 1 3 or a b o u t 130 d a y s earlier: "Fight o f the great o n e s with the g o d d e s s

Wpyt

[uraeus at the b r o w ] . She c o u n t s what lies a h e a d of her. T h a t eye o f the E l d e r H o r u s c o m e s m t o b e m g as a lion. D a y 2 1 8 : T h e E i m e a d is in veneration, < w h e n t h e y > see that eye o f H o m s the E l d e r in its p l a c e , b e i n g c o i m t e d in all its p a r t s . 1/64, 1/128, 1/4, 1/8, 1/16 /// are w i t h it m all of its c o u n t i n g s . " T h e information that the eye o f H o m s the E l d e r u n d e r w e n t a c h a n g e , b e c o m i n g a lion o n d a y 2 1 3 , c o u l d relate astronomically t o t h o s e e v e n i n g s w h e n V e n u s asserts herself b y r e m a i n i n g visible longer m the evening a n d b e c o m i n g brighter. F i v e days later the fractions o f the eye o f H o m s are m e n t i o n e d . B e c a u s e o f the l a c u n a e in the text it is u n c l e a r w h e t h e r all fractions w e r e eniunerated. T h e a n o m a l o u s p r e s e n c e o f 1/128 a r g u e s against a r e c o n s t m c t i o n o f the c o m p l e t e series w h i c h e n d s w i t h 1/64. C o m p a r i n g the r e m a r k for d a y 3 4 9 w i t h that for d a y 2 1 8 , the latter p r o b a b l y m e a n s that the eye h a s n o t yet b e e n c o m p l e t e l y filled, i.e. that the p l a n e t h a d n o t yet attained its m a x i m u m brilliance.

Venus and Mercury at the Eastern Horizon in -1297/96 V e n u s the M o m i n g Star b e c a m e visible m o n t h s before d a y 1 o f the C a l e n d a r of L u c k y a n d U n l u c k y D a y s . A c c o r d i n g to c o m p u t a t i o n with U r a n i a S t a r 1.1 V e n u s r e m a i n e d visible in all o f E g y p t b e t w e e n d a y 1 a n d d a y 111 or d a y 112. T h e s e a r c h for the A k h e t eye o n d a y 1 1 1 , c o n s i d e r e d a b o v e , w o u l d h a v e t a k e n p l a c e o n the last d a y o f visibility or the last b u t o n e . T h e enfries for days 2 5 to 2 8 c o n c e m not o n l y V e n u s b u t M e r c i n y as well: D a y 2 5 : S a k h m e t g o e s to the east, to repel the confederates o f Seth. D a y 2 6 : F i g h t b e t w e e n H o m s a n d Seth in the D u a t / U n d e r w o r l d . Isis h e l p s Seth. H o m s t u m s against Isis. Isis n m s a w a y from H o m s . D a y 2 7 : P e a c e b e t w e e n H o m s a n d Seth. D a y 2 8 : C h i l d r e n o f N u t are peacefiil a n d content. S a k h m e t is a w e l l - k n o w n m y t h o l o g i c a l d e s i g n a t i o n for the eye of Homs.^^ T h e c a l e n d a r ' s assertion that S a k h m e t w e n t to the east o n d a y 2 5 c o u l d e x p r e s s the fact that the e y e of H o m s , alias S a k h m e t , w a s seen as M o m m g Star o n that day. B e c a u s e S e t h / M e r c u r y h a d b e c o m e visible in the east t w o d a y s earlier, the E g y p t i a n o b s e r v e r could

mterpret

the

situation a s - S a k h m e t

repelling

the

confederates

of

Seth,

s u p p o s e d l y the fixed stars in the n e i g h b o i u h o o d o f S e t h / M e r c u r y , w h i c h w o u l d b e in principle the stars in the ecliptical belt. T h e fight m the U n d e r w o r l d o n d a y 2 6 c a n b e i m d e r s t o o d as t a k i n g p l a c e before V e n u s , alias H o m s , a n d M e r c u r y , alias Seth, rose a b o v e the h o r i z o n . Isis w h o h e l p e d her b r o t h e r Setìi against h e r son H o m s e m b o d i e s Sirius a m o n g the fixed stars. F i g u r e 5 illusfrates the situation o n d a y 2 6 in - 1 2 9 7 at the m o m e n t w h e n V e n u s r o s e before the onset o f asfronomical d a w n : M e r c u r y is still b e l o w the h o r i z o n , Sfrius h a d risen

a b o u t 2 0 m i n u t e s earlier t h a n V e n u s . T h e E g y p t i a n o b s e r v e r interpreted the

rising of Sirius before V e n u s as Isis r u n n i n g a w a y from H o m s .

LEiTZ(1994),pp. 6 0 f

The Eye of Horus and the Planet Venus: Astronomical and Mythological References

203

Sirius

10* Venus

50'

horizon

/

/ •

70'

90"

110*

Mercurv

Figure 5. Sirius, V e n u s , a n d M e r c u r y o n d a y 2 6 in - 1 2 9 7 , ca. 3 h 1 5 m local time; h o r i z o n at M e m p h i s . T h e a s t r o n o m i c a l situation varied little b e t w e e n d a y s 2 6 a n d 2 8 : T h e ecliptical distance b e t w e e n the t w o planets d e c r e a s e d from 10*32' to 9*22'. T h e ancient E g y p t i a n o b s e r v e r c h o s e to describe the j o i n t a p p e a r a n c e o f M e r c m y a n d V e n u s o n d a y s 27 a n d 2 8 as p e a c e b e t w e e n Seth a n d H o r u s . If the a s t r o n o m i c a l interpretation of days 2 5 to 2 8 suggested h e r e is correct, then the C a l e n d a r of L u c k y and U n l u c k y D a y s defmitely refers to - 1 2 9 7 / 9 6 , b e c a u s e m - 1 3 0 5 / 0 4 a n d - 1 2 6 5 / 6 4 M e r c u r y w a s not visible on d a y s 2 5 to 2 8 . O n d a y 72 the c a l e n d a r r e c o r d s that the Udjat eye w a s p o s i t i o n e d a b o v e the h e a d o f the Sim g o d . T h e entry s e e m s to refer to the o b s e r v a t i o n that V e n u s w a s visible directly a b o v e the a p p r o a c h i n g s u n o n d a y 7 2 m - 1 2 9 7 . T h i s p h e n o m e n o n resulted from three factors: O n m o r n i n g s in a u t u m n the ecliptic h a s a steep a n g l e ; a r o u n d day 72 in - 1 2 9 7 V e n u s h a d a n o r t h e m latitude a n d a w e s t e m e l o n g a t i o n o f o n l y 15.8*. V e n u s w a s n e a r i n g a vertical p o s i t i o n a b o v e the rising s u n before d a y 7 2 as the following table s h o w s : d a y in - 1 2 9 7

difference in a z i m u t h b e t w e e n V e n u s a n d the s u n at sunrise

50

2.97*

54

2.29*

59

1.58*

64

1.00*

69

0.56*

72

0.35*

74

0.24*

79

0.02°

T h e situation o f d a y 72 is illustrated m F i g u r e 6. T h e p l a n e t c o n t i n u e d to o c c u p y this p o s i t i o n until the e n d o f visibility. W h y the calendar r e c o r d e d this configuration only for d a y 72 r e m a i n s open.

204

R. Krauss

Venus

/ a

1 •-•

10' horizon 80

T

sun

90

Figure 6. V e n u s at s u m i s e o n day 7 2 in - 1 2 9 7 , h o r i z o n at M e m p h i s .

Conclusions P l a n e t a r y Identities of S e t h and H o r u s P r o o f for the identity o f Seth and H o r u s (or his eye) w i t h M e r c u r y a n d V e n u s , respectively, follows from the c o r r e s p o n d e n c e b e t w e e n assertions in the c a l e n d a r a n d a s t r o n o m i c a l p h e n o m e n a o b s e r v a b l e in year - 1 2 9 7 / 9 6 : a) conjimction o f M e r c u r y a n d V e n u s a r o u n d d a y 2 6 (August 8, - 1 2 9 7 ) ; b ) p o s i t i o n of V e n u s a b o v e the h e a d of the sun g o d o n d a y 7 2 ( S e p t e m b e r 2 3 , - 1 2 9 7 ) ; c) visibility o f M e r c u r y o n the m o m i n g o f d a y 164 ( D e c e m b e r 2 4 , - 1 2 9 7 ) ; d) begiiming o f e v e n i n g visibility o f V e n u s o n d a y 169 ( D e c e m b e r 2 9 , - 1 2 9 7 ) ; e) m o v e m e n t o f V e n u s in a z i m u t h first s o u t h w a r d s b e g m n i n g o n d a y 170 ( D e c e m b e r 3 0 , - 1 2 9 7 ) , then n o r t h w a r d s at least since d a y 190 ( J a n u a r y 19, - 1 2 9 6 ) ; f) conjimction of M e r c u r y a n d V e n u s o n d a y 2 1 4 ( F e b m a r y 12, - 1 2 9 6 ) ; g) m o v e m e n t o f V e n u s to the north o f the s k y a r o u n d d a y 2 5 0 ( M a r c h 19, - 1 2 9 6 ) ; h) r e t u m o f V e n u s t o w a r d s the sun, b e g i n n i n g a r o u n d d a y 3 1 0 ( M a y 18, - 1 2 9 6 ) ; i) setting o f the sun and V e n u s in the s a m e spot a r o u n d d a y 3 4 0 (June 17, - 1 2 9 6 ) ; j ) c o m p l e t e filling o f the e y e of the Elder H o m s a r o u n d the t i m e of greatest brilliance o f V e n u s as the E v e n i n g Star a r o u n d d a y 3 4 9 ( J u n e 2 6 , - 1 2 9 6 ) . Additional indications supporting the identification o f H o m s or his eye with V e n u s are k n o w n fi-om sources other than the C a l e n d a r of L u c k y a n d U n l u c k y days.^^ Therefore the definitive identification o f H o m s the E l d e r or his e y e with V e n u s as the E v e n i n g Star should not c o m e as a surprise. T h e c a l e n d a r d o e s n o t describe the e a s t e m eye of H o m s or the eye of the Y o u n g e r H o m s explicitly as V e n u s the M o m i n g Star. B u t given the p r e m i s e that the ideas a b o u t the eyes of H o m s are consistent, the identity o f the e a s t e m e y e o f H o m s with V e n u s as the M o m i n g Star c a n b e inferred fi-om the identity o f the eye of the E l d e r H o m s and V e n u s as the E v e n i n g Star.

KRAUSS ( 1997), pp. 261 - 2 7 4 .

The Eye of Horus and the Planet Venus: Astronomical and Mythological References

205

T h e w o u n d e d e y e o f H o r u s is not m e n t i o n e d explicitly in the C a l e n d a r o f L u c k y a n d U n l u c k y D a y s . Rather, the calendar s p e a k s of filling the eye o f H o r u s , alias the E v e n i n g Star, o n d a y s 2 1 8 and 3 4 9 . T h e "filling" is i n c o m p l e t e o n d a y 2 1 8 , but c o m p l e t e o n d a y 3 4 9 . It is possible that the fillmg of the eye relates to the brilliance o f the E v e n i n g Star w h i c h r e a c h e s its m a x i m u m o n l y t o w a r d s the e n d o f a p e r i o d o f visibility. In the case of the M o m i n g Star, the p r o c e s s is reversed: m a x m i i u n brilliance is attained at the b e g i n n i n g o f the visibility p e r i o d a n d d e c r e a s e s t o w a r d s its end. T h e r e f o r e the M o m i n g Star is an unsuitable identification for the eye to b e filled. T h e C a l e n d a r o f L u c k y and U n l u c k y d a y s indicates h o w the E v e n i n g Star distances itself from the sun g o d a n d r e t u m s to him. T h e m y t h a b o u t the eye w h i c h leaves the sun g o d relates h o w the eye b e c a m e e m a g e d w h e n it r e t i u n e d to find a n o t h e r eye in its p l a c e . B u t in the C a l e n d a r o f L u c k y a n d U n l u c k y d a y s the e y e is said to b e c o m e a n g r y long before its r e t u m . P e r h a p s the m y t h reflects the fact that the E v e n i n g Star b e c o m e s visible only after a t w o - m o n t h s p e r i o d o f invisibility. It is feasible that this p e r i o d w a s interpreted as a state o f b e i n g lost. T h e m c r e a s e of elongation after the b e g i n n i n g of visibility c o u l d b e e x p l a i n e d as the d i s t a n c m g of the star from the s u n god. T h e P y r a m i d T e x t s , the Coffin T e x t s a n d the Celestial D i a g r a m all i n c l u d e b o t h H o m s A ^ e n u s a n d Seth/Mercury. T h e y o c c u r in these funerary contexts as h e l p e r s , or s o m e t i m e s e n e m i e s , o f the b l e s s e d d e a d w h o a p p e a r as stars in the sky. I n the r e a l m of political i d e o l o g y b o t h p l a n e t s w e r e important, b e c a u s e they w e r e manifest in the king. A s early as the b e g i n n i n g o f dynastic times H o m s s e e m s to b e identified with the p l a n e t V e n u s . T h e n a m e s o f five so-called royal v i n e y a r d s describe H o m s as a star.^^ T h e n a m e of D j o s e r ' s vineyard reveals that H o m s is a particular star "at the front o f the sky". T h e identification o f H o m s with V e n u s as k n o w n from the P y r a m i d T e x t s suggests itself If so, H o m s ' s partner S e t h m a y a l r e a d y h a v e b e e n e q u a t e d w i t h M e r c u r y at this early time. R o y a l ideology a n d ideas a b o u t the Hereafter s e e m to h a v e h a d c o s m o l o g i c a l a n d stellar foundations w h i c h m a y well g o b a c k to predynastic tunes. B e g i n n i n g of t h e C a l e n d r i c a l D a y T h e asfronomical analysis of the C a l e n d a r o f L u c k y a n d U n l u c k y D a y s coincidentally p r o v i d e s confirmation of M i c h e l M a l i n i n e ' s c o n c l u s i o n that the scribe c o n s i d e r e d the calendrical d a y to b e g i n before sunrise at dawn.^^ M a l i n i n e a n a l y z e d a p a s s a g e in the c a l e n d a r to s h o w that the calendrical d a y b e g a n at d a w n (Egyptian: hd-ti). O n II p e r e t 14 Seth a p p e a r s at hd-t3 in the b o w o f the solar b a r q u e . F u r t h e r m o r e , the 1st third of the calendar d a y II Peret 14 is d e s c r i b e d as u n l u c k y . T h e user o f the c a l e n d a r is advised n o t to leave his h o u s e . O n these p r e m i s e s M a l i n i n e c o n c l u d e d that the calendrical d a y b e g i n s before sunrise during hd-t3. M a l i n i n e did n o t take into consideration the identity o f Seth w i t h M e r c u r y . B e c a u s e M e r c u r y rises at d a w n a n d disappears from sight long before simrise, it

HELCK (1987), pp. 2 0 4 f; ZIBELÌUS(1978), pp. 2 0 4 ff.

MALININE (1935-1938), pp. 887f, 898.

206

R. Krauss

follows from t h e calendrical assertion a n d its asfronomical interpretation that t h e ancient E g y p t i a n c a l e n d a r d a y started at hd-tS. T h e m e a n i n g o f hd-ti

is d e d u c i b l e from t h e times w h e n t h e angular p o s i t i o n s o f

the B i g D i p p e r ( M e s e k h t i u ) w e r e observed, i.e. at t h e b e g i i m i n g a n d t h e m i d d l e o f the night a n d finally at ìid-tì^^

B e c a u s e t h e stars o f the B i g D i p p e r d i s a p p e a r

sight at t h e b e g i n n i n g o f civil d a w n , hd-ti

from

is e q u i v a l e n t t o a p o s i t i o n o f the s u n o f at

least a b o u t 1° b e l o w the horizon.^' A s t r o l o g y i n the R a m e s s l d e P e r i o d In general, asfrology w a s i m k n o w n i n ancient E g y p t before t h e m i d d l e o f t h e 1st m i l l e n i u m B C E , w h e n t h e influence o f B a b y l o n i a n asfrology is first n o t i c e a b l e . H o w e v e r , there a r e t w o earlier isolated instances. A n official inscription from t h e m i d d l e o f the 9 t h c e n t u r y B C E m e n t i o n s a c a s e o f political u p h e a v a l i n cormection with a lunar eclipse. I n t h e last d e c a d e s , m o s t scholars u n d e r s t o o d t h e text t o m e a n that political turmoil h a d occurred, although n o negative p o r t e n t s u c h a s a lunar eclipse h a d p r e d i c t e d it.^* A c c o r d i n g t o t h e o l d e r E g y p t o l o g i c a l interpretation o f the s a m e text w h i c h m a y b e g r a m m a t i c a l l y s o u n d after all, ' t h e political u p h e a v a l o c c u r r e d before a n actual eclipse o f the m o o n . R e g a r d l e s s o f w h i c h interpretation is correct, b o t h share the n o t i o n that a lunar eclipse w a s a b a d o m e n . A n allusion t o asfrology is found in a n m s c r i p t i o n o f K i n g M e m e p t a h . I n h i s 5 t h year ( 1 2 0 8 B C E ) M e m e p t a h r e c o r d e d h i s victory o v e r i n v a d m g L i b y a n s . A c c o r d i n g to

among

them

"interpreters o f p o r t e n t s w h o w a t c h thefr stars''.^" It is u n c l e a r w h e t h e r

t h e text, P h a r a o h ' s

victory

came

not imexpected

to some,

these

asfrologers w e r e n a t i v e E g y p t i a n s o r not. T h e earliest systematic B a b y l o n i a n o b s e r v a t i o n s o f V e n u s d a t e t o t h e t i m e o f A m m i s a d u k a . A n c i e n t E g y p t i a n observations o f V e n u s a n d M e r c u r y a s attested in the C a l e n d a r o f L u c k y a n d U n l u c k y days a r e several centuries y o u n g e r . I n B a b y l o n i a a n d E g y p t t h e stars w e r e o b s e r v e d for the p u r p o s e o f asfrological p r o g n o s t i c a t i o n . I n B a b y l o n i a t h e p r o g n o s t i c a t i o n s related t o t h e king, w h e r e a s t h e p r e d i c t i o n s listed in the c a l e n d a r o f l u c k y a n d u n l u c k y days applied t o e v e r y o n e . The Venus Year of Velikovsky T h e identification o f the e y e s o f H o m s with V e n u s a s E v e n i n g a n d M o m i n g Star h a s a b e a r m g o n t h e idea o f I m m a n u e l V e l i k o v s k y that Sothis a s t h e star o f Isis is n o t Sfrius, b u t t h e p l a n e t V e n u s mstead, a n d that t h e E g y p t i a n year o f 3 6 5 d a y s w a s a V e n u s year."" L y n n R o s e t o o k u p this idea in a n e m e n d e d version.^^ B o t h the original

NEUGEBAUER and PARKER ( 1969), p. 51. "

KRAUSS (in press).

CAMiNOS(1958),pp. 88f JANSEN-WINKELN ( 1994), p. 127. BRUNNER(1973),pp. 26f

VEUKOVSKY ( 1977), pp. 237-244. "

ROSE (1999), pp. 119-125.

The Eye of Horus and the Planet Venus: Astronomical and Mythological References

207

idea a n d t h e u p - d a t e a r e i n v a h d / ^ n o t only b e c a u s e elements o f t h e m a r e n o n s e n s i c a l a n d charlatanesque, b u t also b e c a u s e the eyes o f H o r u s , n o t Isis, a r e identical with the p l a n e t V e n u s .

References A N T H E S , R u d o l f 1 9 6 1 . " D a s S o i m e n a u g e in den P y r a m i d e n t e x t e n " . Zeitschrift Àgyptische BRUNNER,

Sprache

fiir

9)6: 1 - 2 1 .

H e l m u t . 1 9 7 3 . " Z e i c h e n d e u t u n g aus S t e m e n imd W i n d e n in Agypten".

In: H a r t m u t G E S E and H a n s P . R Û G E R (eds.). Wort und Geschichte. fur Karl Elliger

zum 70. Geburtstag:

Festschrift

2 5 - 3 0 . (Alter Orient und Altes T e s t a m e n t

18). K e v e l a e r : B u t z o n und B e r c k e r . CAMINOS,

Ricardo

A.

1 9 5 8 . The

Chronicle

of Prince

Osorkon.

(Analecta

Orientalia 3 7 ) . R o m : Pontificium Institutum B i b l i c u m . E R M A N , A d o l f 1 9 1 1 . Hymnen der

Sammlung

an das Diadem

Golenischeff

der Pharaonen

(Abhandlungen

der

aus einem

Koniglich

Papyrus

Preufiischen

A k a d e m i e der Wissenschaften, philosophisch-historische K l a s s e 1911/1). Berlin: Georg Reimer. GINZEL, K a r l F . 1906. Handbuch Das Zeitrechnungswesen

und technischen

Chronologie.

John G. 1 9 5 8 . " R e m a r k s o n the M y t h o l o g y o f the E y e s o f H o r u s " .

GRIFFITHS,

Chronique HELCK,

der mathematischen

der Volker. I. Leipzig: J. C . H i i u i c h s .

d'Egypte ZZ: 1 8 2 - 1 9 3 .

Wolfgang.

1 9 8 7 . Untersuchungen

zur

(Agyptologische

Thinitenzeit

Abhandlungen 45). Wiesbaden: Otto Harrassowitz. H U B E R , P e t e r J. 1982. Astronomical

Dating

of Babylon

I and Ur III ( M o n o g r a p h i c

Journals o f the N e a r East. O c c a s i o n a l P a p e r s o n the N e a r E a s t V o l u m e 1, Issue 4), M a l i b u : U n d e n a Publications. JUNKER,

Hermaim.

1 9 1 7 . Die

Onurislegende.

(Kaiserliche

Akademie

der

Wissenschaften in W i e n , philosophisch-historische Klasse, Denkschriften 5 9 , 1/2). W i e n : H o l d e r . ( R e p r m t : H i l d e s h e i m : 0 1 m s ( 1 9 8 8 ) ) KRAUSS, Rolf Kairener

1 9 9 0 . "Vorlaufige B e m e r k u n g e n z u S e t h und H o r u s / H o r u s a u g e i m Tagewahlkalender

nebst

K a l e n d e r t a g e s " . Bulletin de la Société —1997.

Astronomische

Pyramidentexten

Konzepte

(Agyptologische

Bemerkungen

d'Égyptologie und

zum

Genève

Jenseitsvorstellungen

Abhandlungen

59).

Anfang

des

14: 4 9 - 5 6 . in

den

Wiesbaden:

Otto

Harrassowitz. — 1 9 9 9 . " N â h e r e Mitteilungen uber S e t h / M e r k u r und H o r u s - H o r u s a u g e / V e n u s i m grossen T a g e w a h l k a l e n d e r " . Studien zur Altagyptischen

Kultur 2 7 : 2 3 3 - 2 5 4 .

— 2 0 0 1 . " L y n n E . R o s e , S u n , M o o n , and Sothis". Das Altertum

46: 306-307.

— in press. " A r g u m e n t s in favor o f a L o w C h r o n o l o g y for the M i d d l e and N e w K i n g d o m in E g y p t " . In: P r o c e e d i n g s S C I E M 2 0 0 0 - E u r o C o n f e r e n c e 2 n d - 7 t h o f M a y 2 0 0 1 - Haindorf, L o w e r Austria. LEITZ,

Christian.

1994.

Tagewahlerei

Wiesbaden: Otto Harrassowitz.

"

KRAUSS (2001 ), pp. 3 0 6 - 3 0 7 .

(Agyptologische

Abhandlungen

55).

208

R. Krauss

L O C H E R , Kurt. 1 9 9 3 . " N e w argiunents for the celestial location o f the d e c a n a l belt a n d for the origins o f the S3h-hieroglyph". In: COMITATO O R G A N I Z Z A T I V O D E L C O N G R E S S O (eds.), Sesto

Congresso

Intemazionale

di Egittologia.

Atti

II:

2 7 9 - 2 8 4 . T o r i n o : International Association o f Egyptologists. M A L I N I N E , M i c h e l . 1 9 3 5 - 3 8 . " N o u v e a u x fragments d u calendrier é g y p t i e n des j o u r s fastes et néfastes". In: Mélanges

Maspero

I, 2-3.

Orient

ancien:

879-899.

( M é m o i r e s p u b l i é s p a r les m e m b r e s de l'Institut français d ' a r c h é o l o g i e orientale d u Caire 6 6 ) . C a i r o : Institut Français d'Archéologie Orientale. N E U G E B A U E R , O t t o a n d P A R K E R , R i c h a r d A . 1969. Egyptian

Astronomical

Texts III.

London: Lund Hiunphries. O T T O , E b e r h a r d . 1 9 7 5 . " A u g e n s a g e n " . In: W o l f g a n g H E L C K a n d E b e r h a r d O T T O (eds.), Lexikon der Agyptologie

I: 5 6 2 - 5 6 7 . W i e s b a d e n : O t t o H a r r a s s o w i t z .

PlETSCHNIG M i c h a e l a n d VOLLMANN, W o l f g a n g . 1 9 9 5 . UraniaStar

Release

I.I.

V i e n n a : n o publisher. R O S E , Lyim E. 1999. Sun, Moon, Reforms

and Sothis. A Study of Calendars

and

Calendar

in Ancient Egypt. Deerfield B e a c h , F L : K r o n o s P r e s s .

R U D N I T Z K Y , Gunter, 1956. Die Aussage

titer

"das Auge

des Horus"

(Analecta

A e g y p t i a c a 5). C o p e n h a g e n : Ejnaar M u n k s g a a r d . SETHE, Kiut.

1 9 6 2 . Ubersetzung

und Kommentar

zu den

Pyramidentexten

III.

Gliickstadt: J, Augustin, SPELEERS, L o u i s . 1934, Comment faut-il

lire les Textes des Pyramides?

Brussels:

[nopubhsher], V E L I K O W S K Y , I m m a n u e l . 1977. Peoples

of the Sea. L o n d o n : S i d g w i c k & J a c k s o n .

V A N D E R W A E R D E N , Bartel L. 1942. " D i e B e r e c h n u n g der ersten u n d letzten Sichtbarkeit v o n M o n d i m d P l a n e t e n u n d die Venustafeln d e s A m m i s a d u q a " . Sachsische

Akademie

der

Wissenschaften

zu

Leipzig.

Berichte

der

m a t h e m a t i s c h - p h y s i k a l i s c h e n Klasse 9 4 : 2 3 - 5 6 , WESTENDORF; Wolfliart. 1980, " H o r u s a u g e " , In: W o l f g a n g HELCK a n d E b e r h a r d OTTO

(eds,), Lexikon der Agyptologie IT. 4 8 - 5 1 . W i e s b a d e n : O t t o H a r r a s s o w i t z ,

ZIBELIUS, Karola, (Beihefte

zum

1 9 7 8 . Agyptische Tubinger

Siedlungen

Atlas

des

nach

Texten

Vorderen

Geisteswissenschaften 19). W i e s b a d e n : O t t o H a r r a s s o w i t z .

des Alten

Orients,

Reiches

Reihe

B,

The Historicity Question in Mesopotamian Divination Daryn Lehoux, Halifax (Nova

Scotia)

In the m i d 1950s, Sollberger speculated that o n e o f the M a r i livers m a y h a v e b e e n a r e c o r d o f a specific set o f real historical events surrounding t h e e n d o f t h e r e i g n o f Shulgi. I n 1974 D o t t e r ò t o o k this o n e step further t o argue that all o f the o m e n s m a y h a v e b e e n derived from a set o f empirical observations o f the c o i n c i d e n c e o f events in t h e r e m o t e past. O p p e n h e i m s e e m s t o h a v e taken t h e p o s i t i o n that s o m e o f the o m e n s were historical a s well, b u t R e i n e r w a s cautious. Later, H u b e r p u t a real p o i n t o n these ideas b y arguing that the c o i n c i d e n c e o f specific important historical events with three particular eclipses in the third milleimium m a y h a v e m a r k e d the o r i g i n o f the astral o m e n tradition.' Since then, other scholars h a v e w e i g h e d m t o s u p p o r t o r criticize this ' h i s t o r i c a l ' position.^ A t its heart, t h e historical p o s i t i o n sees at least s o m e o f t h e o m e n s a s 'historical', that is, a s r e c o r d i n g a n d p r e s e r v i n g t h e factual c o i n c i d e n c e o f certain events in r e m o t e antiquity, w h i c h actually happened. Because of t h e literal facticity at t h e heart o f its claim, I will hereafter call this p o s i t i o n veridically historical,^ t o distinguish it from t h e relative historical a r g u m e n t that I will b e m a k i n g in this paper. B u i l d i n g o n w o r k b y R o c h b e r g a n d Finkelstein,'' I a m arguing i n this p a p e r that the question o f the facticity o f the o m e n s misses t h e point. I f w e a r e t o u n d e r s t a n d M e s o p o t a m i a n divination, w e will n o t d o s o b y i m p o s i n g o u r o w n m o d e m criteria o f historical facticity o n it. Rather, w e should look t o h o w t h e divinatory fraditions t h e m s e l v e s freated t h e o m e n s , taking w h a t R o c h b e r g called a n ' m t e m a l ' approach.^ D i d t h e d i v m e r s - t h e v e r y p e o p l e studying, teaching, a n d w o r k i n g w i t h t h e o m e n texts, fraditions, a n d p r a c t i c e s - d i d they t h e m s e l v e s think, o r at least a c t a n d talk as if, the o m e n s reflected a t m e a c c o u n t o f what h a d h a p p e n e d i n the p a s t ? ' SOLLBERGER (1954-1956), p. 22; BOTTÉRO (1974), p. 149 f; OPPENHEIM (1977), p. 210; REINER ( 1974), p. 261 ; HUBER ( 1987).

^ On the pro side: see HUNGER and PINGREE (1999), p. 5; and to some extent LARSEN (1987), p. 211 f ; HUNGER (1992), p. xiii (but contrast HUNGER (2000)). On the con side, see BROWN (2000), p. 109 f;

KOCH-WESTENHOLZ (1995), p. 15 f;

and KOCH-WESTENHOLZ

(2000), p. 11. It is interesting to note that none of Huber's detractors has actually addressed his arguments, they have instead merely rejected his conclusions. ^ This phrase was first used in this precise sense by A. Hayward in 1861 (reprinted in HAYWARD (1878)) referring to a mid-nineteenth century argument about the value of Homer as a reporter of true historical facts. In a slightly looser sense, it tums up in Urquhart and Motteux's 1653 translation of Rableais. Perhaps the Rabelais is more salient because, as the title of the book implies, oracles are one of the things discussed in the "veridical history". *

ROCHBERG ( 1999a), FINKELSTEIN ( 1963).

'

ROCHBERG (1999b). For a similar approach, if differently nuanced, see HALLO and

SIMPSON (1971), HALLO (1998).

210

D. Lehoux

The Arguments For and Against Veridical Historicity A s a preliminary, it will b e best to slcetch an outline of the historicity q u e s t i o n a n d h o w it h a s b e e n dealt with to date. M o s t o f the argiunents h a v e b e e n a d d u c e d to either support or criticize the veridical historical position. T h e r e are several arguments in favour of veridical historicity: (1) S o m e o m e n s are explicit about their historicity,^ e.g., w e find qualifications such as " o m e n of S a r g o n " in the astral o m e n s , a n d also fi-equently in the M a r i livers.^ So also, the M a r i livers are s o m e t i m e s inscribed with " w h e n [such and such an event] h a p p e n e d , [the liver] l o o k e d like this"* (2) H u b e r presents a statistical a r g u m e n t (with, a c c o r d i n g to H u b e r , a probability of error of less than 1%) s h o w i n g that three transitions o f reign in a 120-year p e r i o d in the U r III p e r i o d w e r e i m m e d i a t e l y p r e c e d e d b y eclipses. H e c o n c l u d e s that the repetition of such striking c o i n c i d e n c e s " m a y (represent) the birth of o m e n astrology."^ T h e a r g u m e n t s against veridical historicity are several. T h e m o s t detailed treatment is in B r o w n ' s Mesopotamian Planetary Astronomy-Astrology. Here B r o w n divides veridical historical claims into t w o classes: (a) n o n - c a u s a l ' ° or dilute claims, a n d (b) causal claims, w h e r e causal claims see a particular k i n d o f link b e t w e e n the signs a n d the things signified, such that the sign causes the effect, and n o n - c a u s a l c l a h n s see only the coincidence o f the sign a n d the signified, w i t h n o causal substrate hypothesized. B r o w n ' s causal class consists o f C i c e r o , B o t t é r o , Larsen, a n d H u b e r , but o n examination, I think e a c h o f these individuals m u s t b e m o v e d over into the non-causal class. Let us b e g i n with C i c e r o . B r o w n sets C i c e r o u p as a causalist, b u t I contest this strongly. B r o w n cites C i c e r o ' s De divinatione 1.109: In every field of enquiry great length o f time in c o n t m u e d o b s e r v a t i o n b e g e t s a n extraordinary fund of k n o w l e d g e ... since r e p e a t e d o b s e r v a t i o n m a k e s it clear w h a t effect follows any g i v e n c a u s e . T h e p r o b l e m here is o n e of translation. Falconer, w h o franslated the L o e b edition B r o w n is w o r k m g from, is solely responsible for the causal language h e r e , a n d w e cannot rnipute it to C i c e r o . Let u s look at the Latin: adfert autem vetustas omnibus in rebus longinqua observatione incredibilem scientiam; ... cum quid ex quoque eveniat et quid quamque rem significet crebra animadversione perspectum est.

^

This is Bottéro's argument, for example. See BOTTÉRO (1974), p. 148 f

See e.g., K3601+Rm. 103, r. 2-3 (in REINER and PINGREE (1998), p. 216); for examples in the Mari livers, see MEYER (1987), M1-M6 (p. 190, f ) , M8-9. For a complete list of historical references in the Mari livers, see KOCH-WESTENHOLZ (1995), p. 16. * See e.g., MEYER (1987), M7, MIO, M22 (p. 194, 196, 204). See also the list in KOCHWESTENHOLZ (1995), p. 16 for other similar examples. ' He does not consider the possibility that the omens simply were true predictions confirmed by the deaths of the kings, which of course would equally well account for the coincidence. '° Brown uses the phrase 'non-casuistic', but I prefer 'non-causal,' since 'casuistic' usually applies to a part of ethics, and/or to a style of reasoning (this is the sense in which LARSEN (1987) uses it) rather than implying causality. See BROWN (2000), p. 109.

The Historicity Question in Mesopotamian Divination

211

In everything, great age brings about a n exfraordinary k n o w l e d g e b y m e a n s o f continuous o b s e r v a t i o n . . . since w h a t (event) h a p p e n s after w h a t (event) is seen with repeated observation, a n d also w h a t (event) is a sign o f w h a t thing. W h e r e F a l c o n e r ' s translation has " w h a t effect follows a n y given c a u s e " , C i c e r o h a s m e r e l y said quid ex quoque eveniat, or, " w h a t (event) happens after w h a t ( e v e n t ) . " Indeed, C i c e r o is very carefiil all through De div. to distance his a r g u m e n t fi-om one that invokes causal c l a i m s . " F o r e x a m p l e , h e uses almost identical l a n g u a g e as here at 1.29: etenim dirce, sicut cetera auspicia, ut omnia, ut signa, non causas adferunt cur quid eveniat... " A n d indeed, portents, and also auspices, o m e n s , and signs are not causes of w h y something h a p p e n s after t h e m . . . " A s for the m o d e m scholars classed in the causal category, I w o u l d p o i n t out that Bottéro, a l t h o u g h h e is t e m p t e d b y the post hoc ergo propter hoc claim, opts for a m u c h softer claim, and I think m the end m u s t b e c o u n t e d a non-causalist. A b o u t the derivation o f a p r e d i c t e d apodosis (what h e calls le second terme) from an observation {le premier terme), B o t t é r o says: "...le s e c o n d t e r m e se ttouvait rattaché au p r e m i e r et, sinon causé, d u m o i n s annoncé par lui."'^ A s for H u b e r , I c a n find nothing m his 1987 article to justify calling h i m a causalist. H i s sfrongest c l a i m says merely: " w h o w o u l d n o t b e c o m e superstitious if t w o v e r y similar total lunar eclipses o c c u r shortly before the deaths of t w o consecutive kings ( M a n i s h t u s h u and N a r a m s i n ) , and then there is even a thfrd instance?"'^ B u t ' w h o w o u l d n o t b e c o m e superstitious?' is not equivalent to ' w h o w o u l d not make a causal assumption? ' I can, for e x a m p l e , believe that eclipses always and imfailingly p r e c e d e the deaths of kings without actually believing that eclipses cause those deaths, j u s t as I c a n believe that the whistle o n m y kettle always p r e c e d e s m y c u p of m o m i n g tea, b u t does not cause it. A s in this latter case, s o m e m o r e or less c o m p l e x causal c h a i n m a y b e linking in the b a c k g r o u n d in the divinatory instances, b u t it is b y n o m e a n s necessarily a case o f dfrect propter hoc causation. Lastly, L a r s e n offers us a c o n p l e x and n u a n c e d reading of the o m e n fradition that uses w o r d s like ' e m p i r i c a l ' to explain h o w h e sees the o m e n fradition as justified, although he m a y m e a n something j u s t short o f 'veridically empirical'.''* H e

" I have argued this point in another paper. Here a few examples from Cicero will suffice: atque his rerum prœsensionibus prognostica tua referta sunt, guis igitur elicere causas prœsensionum potest? (De div. 1.13); ilia vero cur eveniant quis probabiliter dixerit? (1.14); videmus hœc signa numquam fere mentientia nec tamen cur ita fiat videmus. (1.15); non quœro cur, quoniam quid eveniat intellego. (1.15); hoc sum contentus, quod etiamsi, quor quidque fiat ignorem, quid fiat intellego. (1.16); video, quod satis est; curpossit nescio. (1.16); rationem quam habeant non satis perspicio; vim et eventum agnosco scio approbo. (1.16); similiter quid fissum in extis quid fibra valeat accipio; quce causa sit nescio. (1.16); and many more. Contrast this with the tack taken by Aristotle in De div. per somn. 463a21 f BOTTÉRO (1974), p. 150, italics his. See also BOTTÉRO (1987), p. 162, where he misuses the phrase post hoc ergo propter hoc, throwing it into a veridical historical account of the origin of the omen tradition, without meaning to imply that the protasis actually causes the apodosis. HuBBR(1987),p. H. See, e.g.,

LARSEN

(1987), p.

IXl-lU.

212

D. Lehoux

argues that the o m e n tradition was founded o n a logic that coimected coincident events in the past with coincident events in the fiitiue, such that T h e j i u n p from the p h r a s e 'this is w h a t the liver l o o k e d l i k e , ' to 'this is w h a t the liver is going to l o o k l i k e , ' m a k e s explicit the principle b e h i n d the entire o p e r a t i o n . . . . T h e c o m p i l a t i o n s are built o n tiiis principle o f p r e c e d e n c e w h i c h places a n absolute m e a n i n g o n the relationships tìiat h a v e b e e n - i n t h e o r y - e m p i r i c a l l y established.'^ B u t w e should n o t e that L a r s e n d o e s not c l a u n that these empirically established relationships are specifically causal. If I a m one b y o n e m o v i n g p e o p l e out o f B r o w n ' s second, causal class into his first, non-causal class, this should alert u s to the following question: w h y s h o u l d the causal class b e a null set? T h a t is: is there anything in the M e s o p o t a m i a n divinatory texts that should lead u s to believe the diviners t h e m s e l v e s s a w the signs as signs only, a n d specifically not as causes? It will b e helpfiil here to outline the general possibilities for the w o r k i n g o f a predictive sign: (1) T h e sign c a u s e s the event predicted. F o r e x a m p l e , the p r e s e n c e o f M a r s in Scorpio causes m e to b r e a k out in boils. T h i s is the m o d e l that s o m e G r e c o R o m a n asfrologers o p e r a t e with. (2) T h e sign m e r e l y m d i c a t e s the event predicted, b u t does not cause it. E.g., m y d o c t o r tells m e I h a v e six m o n t h s to live, b u t she is not the c a u s e o f m y death. N o w , class (2) c a n w o r k in several different w a y s : ( 2 a ) T h e sign indicates the event b e c a u s e the g o d s (or s o m e o n e ) s e n d the sign to wam me. (2b) T h e sign indicates the e v e n t b e c a u s e the universe is structured such that signs o f such k i n d a l w a y s or often p r e c e d e such events (these are the k i n d s o f signs m y d o c t o r is frained to interpret). (2c) Signs of this k i n d h a v e b e e n s e e n in the p a s t to p r e c e d e events o f this kind, but n o third-party causation (such as w e find in (2a) a n d (2b)) is s u p p o s e d . (2d) (Just for c o m p l e t e n e s s ' sake) T h e sign indicates the event b e c a u s e the event causes the sign. E.g., s m o k e is a sign o f f n e , or a n o p e n d o o r indicates that a colleague is in h e r office. This is not a possibility I seriously entertain with respect to M e s o p o t a m i a n divination as it w o u l d s e e m to imply that t u n e runs backwards. B r o w n has argued that (1) is u n t e n a b l e for M e s o p o t a m i a n divination, a n d I think h e m a y well b e right, at least with respect to extispicy a n d early astiai o m e n s . It is h a r d to imagine h o w the diviners could h a v e c o n c e i v e d of a causal coimection b e t w e e n a spot o n a liver a n d the death of a king or the success o f a c a m p a i g n . A n d in the asfral o m e n s , w e find n o n - c o m m i t t a l language, w h i c h at least d o e s n o t a l l o w u s to m a k e a causal claim: b e s i d e s the ubiquitous summa clauses, w h i c h are n o n - c o m m i t t a l as to causality, w e find s o m e other non-specific relationships b e t w e e n p r o t a s e s and a p o d o s e s . F o r e x a m p l e in EAE 50, w e have: " T h e star o f the sunset is for (ana)

LARSEN ( 1 9 8 7 ) , p. 2 1 3 .

The Historicity Question in Mesopotamian Divination

213

raining."'^ In MUL.APIN w e find " F r o m t h e first o f A d d a r u imtil t h e 3 0 * o f Ajaru the s u n s t a n d s in t h e p a t h o f the A n u stars; w i n d a n d w e a t h e r . " ' ' I a m less c o n v i n c e d o f the mvalidity o f (1), h o w e v e r , w h e n it c o m e s t o t h e later astrological tradition. Certainly in t h e G r e c o - R o m a n tradition, a n d i n early E g y p t i a n stellar a s t r o l o g y w e find causal l a n g u a g e repeatedly.'* A n d i n M e s o p o t a m i a n a s t r o l o g y there a r e external considerations that m a y b e relevant. F o r e x a m p l e , in a m a g i c o - r e l i g i o u s context w e s e e a deification o f t h e stars, a n d it is clear firom p r a y e r s , rituals, a n d incantations that t h e stars were s e e n t o exert a n influence o n earthly affairs a n d objects.'^ A l l this b y w a y o f saying that w e still n e e d t o d o s o m e carefitl w o r k t o figure o u t which, if a n y , o f these possibilities apply, a n d if s o , exactly w h e r e a n d h o w tiiey apply. B u t t o r e t u m t o m y s u m m a r y o f t h e a r g u m e n t s against veridical historicity. A different tack than B r o w n ' s is t a k e n b y K o c h - W e s t e n h o l z in h e r attack o n t h e veridical historical position. H e r a r g u m e n t s are b a s e d o n h e r r e a d i n g o f p h i l o s o p h e r s of science like H a n s o n , P o p p e r , a n d K u h n . H e r m a i n c l a i m is that It is generally a g r e e d b y m o d e m p h i l o s o p h e r s o f science that k n o w l e d g e a b o u t t h e w o r l d is rarely obtained b y p i n e l y enq)irical observation, w i t h o u t s o m e pre-existing t h e o r y t o integrate t h e o b s e r v e d data. In other w o r d s , t h e "circumstantial a s s o c i a t i o n " assiuned t o b e the foimtamhead o f t h e historical o m e n s , is in itself unlikely.^" B u t n o n e o f t h e supporters o f the veridical history c l a i m h a s said that t h e earliest formulators o f divination w e r e s o m e sort o f p u r e , ideal, theory-firee, ultta-rationalist a n d non-hypothesis-fingo-ing uberscientific tabulée rasce w h o h a d no assiunptions a b o u t anything before t h e y m a d e the c h a i n o f o b s e r v a t i o n s that l e d t h e m t o formulate a n early divinatory system. Q u i t e tiie contiary, c o m p l e x sets o f a s s u m p t i o n s , beliefs, a n d w o r l d - v i e w s m u s t h a v e b e e n at p l a y right firom t h e start. B u t since a n y o f the scholars against w h o m she is a r g u i n g w o u l d p r o b a b l y c o n c e d e this, t h e a r g u m e n t itself falls flat. M o r e o v e r , a d e e p e r p r o b l e m with this a r g u m e n t c a n b e seen firom a simple reversal, such that w e c o u l d j u s t a s well p o s e the following a r g u m e n t : It is g e n e r a l l y a g r e e d b y m o d e m p h i l o s o p h e r s o f science that k n o w l e d g e a b o u t t h e w o r l d is rarely o b t a i n e d b y piurely theoretical speculation, without s o m e b a s i c observations aroimd w h i c h t h e t h e o r y g r o u p s itself, a n d for w h i c h it is s u p p o s e d t o account.^' I n o t h e r w o r d s , t h e "mystical speculation"^^ a s s u m e d t o b e t h e fountainhead o f the o m e n ttadition, is in itself unlikely.

EAE 50,1.17. MULAPIN.

In REINER and PINGREE (1975). II.Gap A. 1-2. In HUNGER and PINGREE (1989).

See LEHOUX (2000). See REINER (1995). KOCH-WESTENHOLZ (1995), p. 15.

HACKING (1983), PUTNAM (1962), KITCHER (1993), among many, many others, KOCH-WESTENHOLZ (1995), p. 19.

214

D. Lehoux

A n d this a r g u m e n t is n o m o r e convincing than the first. T o b e fair, h o w e v e r , I note that K o c h - W e s t e n h o l z d o e s qualify h e r attack. S h e says: " t h e ' h i s t o r i c a l ' o m e n s are n o m o r e historical than the rest o f the O l d B a b y l o n i a n historical tradition, a n d they must b e a c c o u n t e d for in other ways."^^ I w h o l e h e a r t e d l y agree, a n d I a m here arguing for o n e o f those other w a y s , o n e that, instead o f seeing this equality as a fault, glories in the fact that t h e historical o m e n s are exactly as historical as the rest o f the O B historical tradition, a n d in m a n y o f the s a m e w a y s , a n d I a d d that this is a legitimate sense of "historical", a n d i n d e e d o n e that is m o r e usefiil, a n d less p r o b l e m a t i c , than the veridical a p p r o a c h .

Rochberg's Argument T h e m o s t sophisticated recent treatment of s o m e o f the issues aroimd the historicity question is t o b e found m a 1999 p a p e r b y Rochherg.^"* H e r e she looks at the b r o a d e r question o f e m p i r i c i s m rather than addressing historicity specifically, b u t s o m e o f her observations will b e o f u s e to the present inquiry. Similar to m y o w n c l a i m here, R o c h b e r g argues for a m o r e relative a p p r o a c h to the idea o f empiricism. H e r p a p e r addresses the q u e s t i o n o f observables in the o m e n texts. It h a s long b e e n n o t e d that the o m e n series fi-equently contain o m e n protases, s u c h as If [there is a n eclipse] o n tiie 21** d a y [of the m o n t h ] . . .

that are physically impossible. Since the M e s o p o t a m i a n c a l e n d a r s w e r e all observationally lunar, t h e only possible eclipse dates cluster a r o u n d m i d m o n t h , that is, t h e 1 4 * , 1 5 * a n d 16*. R o c h b e r g argues that even t h o u g h w e m a y k n o w a n eclipse to b e i m p o s s i b l e o n the given date, n o n e t h e l e s s , w e still n e e d to c o n c e d e that the d i v m e r s w h o u s e d the o m e n series s e e m t o h a v e thought the eclipse t o h a v e b e e n a n o b s e r v a b l e possibility in s o m e sense. S h e argues that the category o f " o b s e r v a b l e " s h o u l d b e s e e n as " n o t lunited to m a t e r i a l things s e e n m the world, b u t [to include] u n o b s e r v e d entities that nonetheless could b e imagined b y extension or extrapolation from o b s e r v e d p h e n o m e n a . " T h i s strikes m e as correct, although I w o u l d caution s t i o n g l y against r e a d i n g this as a genealogical account, that is, as claiming that the ' m i p o s s i b l e ' o m e n s were in fact derived by extension or extrapolation from observed phenomena, since that w o u l d p u t u s in the v e r y situation R o c h b e r g is tiying to get u s to avoid: the epistemologica! bias that divides p h e n o m e n a into e n ç i r i c a l , o n t h e o n e hand, a n d rationally extrapolated o n the other, such that w h a t w e w o u l d b e seeing h e r e is a core o f o b s e r v e d , true, o m e n s initially c a t a l o g u e d a n d t h e n c o n s c i o u s l y (or unconsciously, if w e a c c e p t Britton a n d W a l k e r ' s scribal error hypothesis)^^ e x p a n d e d o n for c o m p l e t e n e s s . T o s o m e extent, though, e v e n R o c h b e r g is u n a b l e to avoid this mire, a n d late in the p a p e r she tries t o s h o w that there m a y n o t b e a n actiial o b s e r v a t i o n at the r o o t o f the association b e t w e e n t h e protasis a n d the a p o d o s i s . H e r e she follows Starr a n d others in arguing for a linguistic, o r m o r e

KOCH-WESTENHOLZ (1995), p. 16. ROCHBERG (1999a).

"

See, e.g., EAE 22, part t, in ROCHBERG-HALTON (1988). BRITTON and WALKER ( 1996), pp.

42^3.

/ The Historicity Question in Mesopotamian Divination

215

generally, p s y c h o l o g i c a l l y associative derivation.^' B u t this analysis r e q u i r e s as its staring p o i n t the v e r y distinction she is arguing against. H e r a t t e m p t to p r e s e r v e h e r initial p o s i t i o n b y subsiuning this psychologically associative association u n d e r the rubric " e m p n i c a l " is insufficient. Instead, in order to save t h e p o s i t i o n she is w o r k i n g t o w a r d s , w e h a v e to simply dismiss a n y question o f the actual d e r i v a t i o n o f the o m e n s , a n d treat t h e m instead in terms o f h o w the M e s o p o t a m i a n s s a w t h e m as b e m g derived. T h a t is, w e n e e d to give u p o n tiie quest for the veridical o r i g m s o f the v a r i o u s o m e n s , a n d stop asking b o t h fiiiitless a n d i m a n s w e r a b l e questions a b o u t w h e t h e r or in w h a t sense this or that particular o m e n is literally or veridically "historical", a n d w h e t h e r or in w h a t sense this o r that particular o m e n is d e r i v e d b y e x t e n s i o n firom this or that other o m e n . B u t there are e v e n d e e p e r p r o b l e m s than the historiographie o n e s I a m flagging here. W e also n m into logical a n d epistemologica! p r o b l e m s w h e n w e say, for e x a m p l e , that a n eclipse is " i m p o s s i b l e " o n a particular date.

Bigger Problems with the Impossibility Claim W h a t d o e s it e v e n m e a n to say that eclipses are n o t possible o n certain dates? Certainly we k n o w they were n o t possible o n those dates, a n d certainly, w e are told, later r e a d e r s a n d transcribers o f EAE k n e w eclipses w e r e h n p o s s i b l e o n certain dates.^* B u t this c l a i m contains t w o major p r o b l e m s . O n e is that t h e p r e d i c a t e ' n o t p o s s i b l e ' is epistemologically indistinguishable firom the p r e d i c a t e ' n o t possible a c c o r d m g t o a specific set o f rules for w h a t constitutes possibility,'^' a n d s o w e n e e d to b e clear o n w h o it is d o i n g the p e r c e i v i n g in o r d e r t o avoid the confiision o f i n p u t i n g the facts implied b y o f oiur o w n world v i e w o n t o the p o o r imsuspecting M e s o p o t a m i a n s . Specifically: ' w e k n o w they w e r e n o t p o s s i b l e o n those d a t e s ' c a n m e a n n o t h i n g m o r e t h a n ' w e k n o w they w e r e n o t possible according to our own rules of possibility o n those d a t e s . ' B u t o f c o i u s e o u r rules o f possibility a n d the rules o f p o s s i b i h t y for the diviners t h e m s e l v e s are t w o v e r y different things. T h e first part o f the impossibility c l a i m thus m e a n s that for u s , a n eclipse is n o t p o s s i b l e o n the g i v e n dates. B u t what about for t h e m ? I will retimi t o w h a t this implies after a look at the s e c o n d p a r t o f the question, w h i c h claims that later r e a d e r s a n d transcribers o f EAE k n e w eclipses w e r e impossible o n certain dates. D i d they necessarily? I m p o s s i b l e is a s t i o n g w o r d , a n d implies a k n o w l e d g e o f s o m e factor, w h i c h m a k e s the possibility o f o c c u r r e n c e nil. F o r u s , this factor is tiie physical fact o f the m o t i o n o f the m o o n about the earth a n d the e a r t h ' s relation to the sun, c o m b m e d w i t h the historical fact o f the b e g m n m g o f the M e s o p o t a m i a n lunar m o n t h with the observation o f the n e w m o o n crescent. W h a t m i g h t it b e for the M e s o p o t a m i a n s ? Certaiidy a n eclipse possibility is s o m e t h i n g g e n e r a t e d o u t o f a set of m a t h e m a t i c o - a s t r o n o m i c a l tables. A n a s t i o n o m e r simply calculates o r looks u p w h e t h e r a n eclipse is possible o n a certain date. B u t w h a t g e n e r a t e s a n eclipse impossibility? T o s a y that eclipses a r e k n o w n to b e i n ^ o s s i b l e o n all other dates is

ROCHBERG (1999a), p. 567. See, e.g., ROCHBERG (1999a), p. 563.

Because the meaning and force of a modal is entirely dependent on the logical and epîstêmôtôgïcal framework (world viewy in which it is employed f See^ QUBME ( 1 9 5 3 ) , GOODMAN ( 1 9 7 8 , 1 9 7 9 ) ) . As Michael Hymers put it: "All possibility is co/w-possibility."

216

D. Lehoux

to say that the criteria p r e s e r v e d in the tablet are k n o w n to represent 1 0 0 % o f the criteria for possibility. S u c h a c l a i m miplies that the a s t r o n o m e r s t h o u g h t they h a d gotten h o l d of n o t only a certain k n o w l e d g e o f w h e n eclipses c o u l d o c c u r , b u t a certain k n o w l e d g e of w h e n all eclipses could o c c i n , a n d this is a different kettle o f fish entnely.^" I n d e e d , it requires that they h a d found a n d e x h a u s t e d the very possibility conditions imder w h i c h the g o d s t h e m s e l v e s c o u l d o p e r a t e . A g a m s t this view, I p o i n t to the fact that they piously preserve the v e r y o m e n s in question, w h i c h w o u l d s e e m to indicate that they did not think the tables e x h a u s t e d all possibility conditions. If there are other w a y s a n eclipse could h a p p e n (let us call these, for lack o f a better term, ' a b n o r m a l possibilities') then eclipses cease to b e i m p o s s i b l e o n other dates. S u c h an a b n o r m a l possibility might b e s o m e t h m g like the interposition of s o m e ( u n c o m m o n ) divine or magical force onto the n o r m a l order o f things. T h u s , w h a t w e are faced w i t h is the fact that these t r o u b l e s o m e dates do t u m u p as eclipse possibilities in the o m e n texts. F r o m this it b e h o o v e s us to assiune that the p e r s o n or p e o p l e w h o originally c o m p o s e d the o m e n s , a n d also t h o s e w h o tianscribed a n d p a s s e d these texts o n to fiiture generations of diviners, t h o u g h t that eclipses were p o s s i b l e (either n o r m a l l y or a b n o r m a l l y ) o n the tioublesome dates. T o b e sure a n d a v o i d the trap p o s e d b y the possibility question, w h a t w e n e e d to d o here is set aside any question o f w h i c h o m e n s are ' p o s s i b l e ' a n d w h i c h ' i m p o s s i b l e ' . It is clear from the v e r y fact of t h e n p r e s e r v a t i o n in canortical series that all the o m e n s w e r e seen as possible b y the M e s o p o t a m i a n s . T h i s is w h a t I see as the real t h m s t o f w h a t R o c h b e r g is getting at, a n d it n e e d s to b e e m p h a s i z e d sttongly. T h e r e is n o such thing as an ' i m p o s s i b l e ' o m e n . B u t I e m p h a s i z e that I a m m a k m g a stionger c l a i m than that o f R o c h b e r g . S h e is arguing that " o b s e r v a b l e " m e a n s " w o r t h y o f o b s e r v a t i o n " and n o t sttictiy " w h a t c o u l d b e s e e n " ( p p . 5 6 8 - 5 6 9 ) . B u t this does n o t quite get u s c o n p l e t e l y a r o i m d the p r o b l e m . T h e o n l y w a y to d o that is to h o l d that the w h o l e set o f o m e n s , the " p o s s i b l e " a n d " i m p o s s i b l e " alike, b y t h e n v e r y inclusion in the series, are in fact of a class of things that " c o u l d b e s e e n " in every applicable sense o f this p h r a s e .

That the Rules are Always Already Given B y a v o i d i n g veridical historical questions a n d instead h y i n g to s e e the o m e n s in the framework in w h i c h as the M e s o p o t a m i a n d i v m e r s t h e m s e l v e s w o u l d h a v e s e e n t h e m , w e thus see the o m e n t e x t s ' a s a collected set of m l e s for the mterpretation of possible observations o f the natural world b r o a d l y speaking. P e r h a p s the m o s t u n p o r t a n t feature o f this v i e w is that, in the historical r e c o r d as it h a s b e e n p r e s e r v e d , the m l e s o f divination p r e s e n t t h e m s e l v e s to the diviners as always already given.^^ T h a t is: w e n e v e r get a clear statement in the historical r e c o r d that the diviners t h o u g h t s o m e o n e j u s t m a d e u p the m l e s , or d e r i v e d t h e m from observation, or whatever. Instead, w e see a set o f m l e s that are a l w a y s c o u c h e d in To be sure, there is one letter (PARPOLA (1993), no. 71) that Parpola reads as saying that "[Eclipses] cannot occur [dur]ing certain periods", but given the state of the tablet, with several lines destroyed just before this sentence, the restoration of the subject ("eclipses") is extremely uncertain. They manifest themselves, as Heidegger would say, as 'zuhanden'. See HEIDEGGER (1927).

The Historicity Question in Mesopotamian Divination

217

t e r m s that m a k e t h e m a p p e a r to b e , simply, true facts a b o u t the u n i v e r s e , a n d m o s t importantly, facts that s e e m to h a v e always b e e n true. E v e n w h e n the tradition claims to b e r e v e a l e d b y the g o d s at s o m e particular p o i n t in the past, as s o m e texts c l a i m it to be,^^ this revelation s e e m s m o r e a p a s s i n g o n o f truths t h a n a c a n o n i z a t i o n o f t h e m . T h e g o d s are p a s s i n g the rules on, rather t h a n m a k i n g t h e m u p o n the spot. W h a t this m e a n s is that the diviners s e e m to h a v e t a k e n the truth o f divination as a given, a n d furthermore s e e m n o t to h a v e a s k e d questions a b o u t the g e n e a l o g y o f that truthfulness. D i v m a t i o n is n o t s o m e t h i n g that c a m e t o b e true at a n y particular p o i n t in the past, b u t s o m e t h i n g that w a s g i v e n to the diviners of e a c h successive g e n e r a t i o n as always h a v i n g b e e n true. A n d this aspect o f the truth o f divination, its a l w a y s - h a v i n g - b e e n - n e s s , requires s o m e set of historical claims to b a c k it u p . N o w , there are several w a y s in w h i c h this legitimization t h r o u g h historicity m i g h t w o r k . T o e x p l o r e these, let us look at the s o m e w h a t b r o a d e r question of e p i s t e m i c validity in general.

Epistemic Validity So far, the analysis has s h o w n that questions o f veridical historicity will n e c e s s a r i l y involve the tacit assvunption that our o w n e p i s t e m o l o g i c a ! frame of reference is the s t a n d a r d b y w h i c h all such questions c o u l d a n d should b e j u d g e d , an a s s u m p t i o n I a m n o t willing to m a k e . B u t e v e n m o r e importantly s u c h a n a p p r o a c h will also necessarily b a r u s from fully u n d e r s t a n d i n g why the v e r y claims to historicity w e r e b e i n g m a d e b y s o m e of the o m e n texts in the first p l a c e . B u t reframing the issue allows us to b e g i n to a n s w e r questions o f the following type: H o w , a n d to w h a t d e g r e e , did the diviners see the history of their discipline as contributing to the e p i s t e m o l o g i c a ! validity o f divination? M o r e generally, w e c a n a s k w h a t it w a s a b o u t the o m e n texts a n d the traditions of divinatory p r a c t i c e s that g a v e a n y g u a r a n t e e o f certainty, in the eyes o f the diviners, to the p r e d i c t i o n s . I see a few possibilities right off the bat, s o m e c o m b i n a t i o n of w h i c h m u s t certainly underlie M e s o p o t a m i a n divination: (1) Divine sanction. S o m e t h i n g in the relationsliip b e t w e e n p e o p l e a n d the g o d s g u a r a n t e e d that the g o d s s p o k e truly to those w h o c o u l d p r o p e r l y i m d e r s t a n d them.^'* N o historical c l a i m is necessary. (2)

The structure of the universe. S o m e t h i n g in the w a y the universe w a s constructed m e a n t that the g o d s ' intentions, and/or fiiture events c o u l d b e d i s c e r n e d t h r o u g h the appropriate techniques a p p l i e d within the a p p r o p r i a t e theoretical framework.

(3) Divine revelation of the tradition.^^ R e v e l a t i o n c o u l d w o r k in c o n c e r t with categories (1) and/or (2), d e p e n d i n g o n h o w the truths r e v e a l e d are s u p p o s e d to function. L i k e (1) a n d (2), this c a t e g o r y can o p e r a t e with n o c l a i m that the o m e n s t h e m s e l v e s are historical, b u t it does require the historical c l a i m of revelation itself "

SeeKocH-WESTENHOLZ(1995),p. 7 4 f This list is not necessarily exhaustive.

See, e.g.. HUNGER (1992), p. xiii: "It was believed that the gods send messages announcing future events". "

See e.g., KOCH-WESTENHOLZ (1995), p. 18-9, 21, 74 f

218

D. Lehoux

(4) Induction. O b s e r v a t i o n s o f protases a n d a p o d o s e s occurring t o g e t h e r in the past p r o v i d e the basis for predictions o f the futme, either (a) w i t h a direct o r thirdparty causal substrate h y p o t h e s i z e d or p r e s i u n e d , o r (b) with n o direct o r thirdp a r t y causation implied or supposed. In either o f these c a s e s , for induction to w o r k as legitunation w o u l d require historical claims o f the sort m a d e in the historical o m e n s , although with the a d d e d assiunption that the particular o m e n s were ' d i s c o v e r e d ' b y s o m e o n e at a specific p o m t in the past. (5) The age of the tradition. T h e simple fact that these coimected p r o t a s e s a n d a p o d o s e s h a d b e e n h a n d e d d o w n t h r o u g h the generations lent t h e m s o m e o f their credibility. A s witii ( 4 ) , it m a y b e that the historical o m e n s w e r e m e a n t t o imderline this aspect o f the tradition, a l t h o u g h the type o f historical c l a i m n e c e s s a r y h e r e is different fi-om the type n e c e s s a r y for ( 4 ) , insofar a s here w e only n e e d a ' g i v e r m e s s ' o f the tradition, rather t h a n implying discovery. (6) Precedence. S o m e o r all o f the o m e n s w e r e seen b y the diviners a s a set o f p r e c e d e n t s t h r o u g h w h i c h fiiture signs c o u l d b e interpreted (fiiture decisions j u d g e d ) . T h i s h a s as its basis a relative historical c l a i m insofar as if the o m e n s were s e e n b y the diviners as p r e c e d e n t s , t h e n they w e r e also s e e n as containing historical information. T h e historical o m e n s c o u l d b e fimctioning t o legitimate the o m e n tiadition in o n e or a c o m b i n a t i o n o f these last three possibilities. Indeed, s o m e sort o f historical claim w o u l d b e necessary for a n y o f (3), ( 4 ) , or (5). Let u s look at the last category, p r e c e d e n c e , to see w h a t it will reveal.

Speculations On Precedence L a r s e n a n d Ritter h a v e p o i n t e d out a very tantalizing possibility:^^ the o m e n texts m a y b e m o r e t h a n superficially related to m e d i c a l a n d legal texts. T h e o m e n texts m a y actually share s o m e o f the s a m e c o n c e p t u a l imderpiimings a n d e p i s t e m o l o g i c a ! foimdations as, say, the Codex Hammurapi. T h r o u g h a close structural analysis o f verbal tense shifts in several different genres (I should s a y modem genres) of M e s o p o t a m i a n text (mathematical, divinatory, m e d i c a l , a n d legal), Ritter h a s a r g u e d that these foiu types o f text b e t i a y important structural similarities that m a y indicate other kinds o f otherwise n o n - o b v i o u s sunilarities b e t w e e n them. .. .the simple fact o f t h e existence o f a g r o u p o f texts c o m i n g fi-om a restticted n u m b e r o f O l d B a b y l o n i a n intellectual d o m a i n s a n d sharing a c o m m o n distinctive formal structure, different from a n y other in the c u n e i f o r m literature o f the p e r i o d . . . signals that w e o u g h t to l o o k carefiilly at w h a t are the s i m i l a r i t i e s - a n d the d i f f e r e n c e s - a m o n g the d o m a i n s thus singled out.^' Specifically, h e h a s u s e d this structural analysis o f verbal tenses t o s h o w that there are s o m e important parallels b e t w e e n h o w p r e c e d e n c e w o r k s in the M e s o p o t a m i a n legal texts, a n d h o w ' e m p i r i c i s m ' or 'recipe-following'^* w o r k in M e s o p o t a m i a n See LARSEN (1987), p. 211-12; RITTER (1995), p. 43; and especially RITTER (1998). " 38

RITTER (1998), p. 25. See RITTER (1998), p. 26.

/ The Historicity Question in Mesopotamian Divination

219

m a t h e m a t i c a l texts. T o b u i l d o n his analysis, I w o u l d a d d that the structural similarities b e t w e e n divinatory texts a n d legal texts^' m a y b e e v e n m o r e p r o n o u n c e d than w i t h the m a t h e m a t i c a l texts, insofar as the legal a n d divinatory geiu-es, at p e r h a p s a n e v e n m o r e basic level than that o f verbal tense shifts, b o t h consist a l m o s t entirely of summa clauses.'*^ L a r s e n has also argued that the role of p r e c e d e n c e in the law c o d e s is parallel to the role o f p r e c e d e n t in the o m e n s . A n o t h e r c o n s i d e r a t i o n c o m e s out o f the c o s m o l o g i c a l context w i t h i n w h i c h divination w a s s u p p o s e d to work. Jeyes reports that the chain o f c o m m i m i c a t i o n in an extispicy b e g i n s , at the top, with the puhrum, or divine coimcil.'*' B u t the w o r d puhrum d o e s also h a v e a legal implication, referring to the a s s e m b l y in a j u d i c i a l p r o c e e d i n g . F m t h e r m o r e , the extispicy itself is called in s o m e texts b y a w o r d that can m e a n "legal c a s e " (awàtum).^^ Lastly, the scribal contexts of the legal and divinatory texts s h o w s o m e i m p o r t a n t similarities a n d overlaps that also lend s u p p o r t to this interpretation.'*^ G i v e n this analysis, o n e set o f questions that w o u l d b e w o r t h asking of the legal context, for the possible light it m a y shed on the legitimation of extispicy, is this: Are

the laws of Hammurapi valid Hammurapi? Or, were they decided by Hammurapi

because

they

were

because

they are

decided

by

valid?

W h i l e these t w o possibilities are n o t necessarily m u t u a l l y exclusive,'*^ I think the contrast d o e s p o i n t u p an interesting pair o f a p p r o a c h e s that m a y e n d us u p w i t h t w o different c o n c l u s i o n s . O n the o n e side, w e h a v e the legitimacy of the laws s t e m m i n g from the authority o f H a m m i u a p i as a historical king, a n d o n the other, the legitimacy o f the laws is a p r e - g i v e n fact of the universe w h i c h guides or c a u s e s H a m m u r a p i ' s (v^se or inspired) decisions. Likewise, w e n e e d to k e e p these t w o possibilities in m i n d as w e look at possible types o f legitimization for divination, in order to w a t c h for clues, a n d to carefiilly consider the types o f e v i d e n c e that w o u l d allow u s to d e c i d e the question one w a y or the other. W h e t h e r or not this p r o v e s to b e a fi-uitfiil line o f inquiry, the v e r y possibility o f it is revealing of s o m e interesting facets of m o d e m scholarship. O n e of the real p r o b l e m s this a p p r o a c h points to is this: w e as a scholarly c o n m i i m i t y are v e r y comfortable w i t h a c c e p t i n g the fact that the " c o d e " o f H a m m u r a p i is really the " c o l l e c t i o n " o f H a m m u r a p i , that is, a collection o f legal decisions m a d e b y H a m m i n a p i m e a n t to fimction as legal p r e c e d e n t s to guide fiiture decisions.'*^ B u t faced with a s t m c t i u a l l y almost identical evidential set in the d i v i n a t o r y t e x t s I omit medical texts here merely for the sake of concision, not because they are irrelevant. The similarities and differences between these and the two I deal with here are certainly deserving of their own study. If, following Reiner's practice in BPO 3, we read the DI§ and/or BE signs as merely paragraph markers, then we cannot adduce this 'further weight' argument. Instead, we would have to fall back on Ritter's analysis alone, which 1 still think makes the point sufficiently. JEYES ( 1 9 8 9 ) , p. 17. JEYES ( 1 9 8 9 ) , pp.

17-19.

See WESTBROOK ( 1 9 8 5 ) , BOTTÉRO ( 1 9 8 2 ) , KRAUS ( 1 9 6 0 ) .

'*'* Although bdievtng both at once would admittedly 4î€ circular. See, e.g.. ROTH ( 1 9 9 5 ) .

220

D. Lehoux

indeed o n e w h i c h at points m a k e s e v e n stronger historical claims than the legal c o l l e c t i o n s - w e refuse to accept the idea o f historicity of a n y type, let alone precedence, in m a n y cases, a n d a r g u e about it in a few. Certainly w e d o not u n a n i m o u s l y agree o n any. B u t ultimately this only reflects oiu: o w n m o d e m bias that legal decisions such as H a m m i u a p i ' s are plausible, a n d births of octuplets, or r e e d s walking a b o u t the countryside, or eclipses o n the 21*' are i m p o s s i b l e . P e r h a p s they are for us. B u t equally obviously, they w e r e neither impossible, n o r implausible to the M e s o p o t a m i a n s .

Conclusion In the end, m y real point is that w e m u s t b e careful to a p p r o a c h o m e n texts in a w a y that is logically, epistemologically, a n d historiographically sound. A b o v e all, the biggest danger is of allowing our o w n categories of facticity, historicity, a n d possibility to distort o u r understanding of the admittedly v e r y foreign p r a c t i c e s o f divination, at least insofar as these categories h a m p e r u s firom getting as ' i n s i d e ' a v i e w as is possible across so m a n y centuries. A s a case study in this distortion, I h a v e singled out a n u m b e r of issues surrounding the question of veridical historicity. In p l a c e o f the q u e s t i o n o f veridical historicity, I p r o p o s e an a p p r o a c h that takes the (relative) historicity o f the o m e n s for granted, a n d that takes the claims to historicity foimd in the o m e n traditions very seriously, m o r d e r to then b e g i n asking questions a b o u t w h a t the historical claims m a d e b y the o m e n traditions m i g h t m e a n for those traditions t h e m s e l v e s , a n d for those engaging in t h o s e traditions (diviners, kings, a n d so on), so that w e m a y get a better grasp o n divination in its o w n t e r m s . In m a n y w a y s m y c l a i m that the v e r y debate over the veridical historicity o f the o m e n fraditions misses the point, is not a n e w o n e . B u t w e n e e d to fmally take this a r g u m e n t seriously if w e are to avoid m i m i n g in circles trying to find a n s w e r s to i m a n s w e r a b l e questions, a n d instead focus our attentions o n lines o f inquiry that will further o u r u n d e r s t a n d m g of divination in t e r m s o f its w o r k i n g s , cultural roles, epistemological claims, a n d its relationships w i t h other types o f text.

References ARISTOTLE. 1 8 3 7 . Opera (ed. I m m a n u e l B e k k e r ) . Oxford: Oxford U n i v e r s i t y P r e s s . B O T T É R O , Jean. 1 9 7 4 . " S y m p t ô m e s , signes, écritures en M é s o p o t a m i e a n c i e n n e " . In: J.P. V E R N A N T et al. (eds.). Divination et rationalité: 7 0 - 1 9 7 . Paris: Éditions d u Seuil. — 1 9 8 2 . " L e ' c o d e ' d e H a m m u r a b i " . Annali della scuole normale 12:

superiore

di Pisa,

409^4.

— 1 9 8 7 . Mésopotamie: L'écriture, la raison, et les dieux. Paris: Éditions G a l l i m a r d . B R I T T O N , J o h n P . a n d W A L K E R , Christopher. 1 9 9 6 . "Asfronomy a n d Asfrology in M e s o p o t a m i a " . In: Christopher W A L K E R (ed.). Astronomy Before the Telescope: 4 2 - 6 7 . L o n d o n : British M u s e u m P r e s s . B R O W N , David. 2 0 0 0 . Mesopotamian Planetary Astronomy-Astrology. Groningen: Styx. CICERO, M . Tullius. 1 9 4 9 . De divinatione (ed. W . A x ) . Leipzig: B . G . T e u b n e r . FALCONER, W i l H a m A . 1923. Cicero: de seneetute, de amicHia, de divinatione. C a m b r i d g e , M A : H a r v a r d University Press.

The Historicity Question in Mesopotamian Divination

221

FINKELSTEIN, J a c o b J. 1 9 6 3 . " M e s o p o t a m i a n Historiography'*. In: Cuneiform Studies and the History of Civilization, Proceedings of the American Philosophical Society Held at Philadelphia for the Promotion of Useful Knowledge, 107.6: 4 6 1 - 4 7 2 . Philadelphia: A m e r i c a n P h i l o s o p h i c a l Society. G O O D M A N , N e l s o n . 1978. Ways of Worldmaking. Indianapolis: H a c k e t t . — 1979. Fact, Fiction, and Forecast. C a m b r i d g e , M A : H a r v a r d U n i v e r s i t y Press, H A C K I N G , Ian. 1 9 8 3 . Representing and Intervening. Cambridge: Cambridge University P r e s s . H A L L O , W i l l i a m W . 1998. " N e w Directions in H i s t o r i o g r a p h y ( M e s o p o t a m i a a n d Israel)". In: Manfried DIETRICH a n d O s w a l d L O R E T Z (eds.), dubsar anta-men, Studien zur Altorientalistik: Festschrift fiir Willem H.Ph. Romer: 109-128. Munich: Ugarit-Verlag. — a n d W i l l i a m K. S I M P S O N . 1 9 7 1 . The Ancient Near East, A History. N e w Y o r k : Harcourt Brace Jovanovich. H A Y W A R D , A b r a h a m . 1 8 7 8 . Selected Essays. L o n d o n : L o n g m a n ' s , G r e e n . HEIDEGGER, Martin. 1927. " S e i n u n d Z e i f . Jahrbuch fiir Phdnomenologie und phanomenologische Forschung 8: 1 - 4 3 8 . H U B E R , Peter J. 1987. " D a t i n g b y L u n a r Eclipse O m i n a , with S p e c u l a t i o n s o n the B i r t h of O m e n A s t r o l o g y " . In: J. L e n n a r t B E R G G R E N a n d B e r n a r d R. G O L D S T E I N (eds.). From Ancient Omens to Statistical Mechanics: 3 - 1 3 . Copenhagen: U n i v e r s i t y Library. H U N G E R , H e r m a i m . 1992. Astrological Reports to Assyrian Kings. Helsinki: Helsinki University P r e s s . — 2 0 0 0 . " U s e s of E n u m a A n u Enlil for C h r o n o l o g y " . Akkadica 1 1 9 - 1 2 0 : 1 5 5 - 1 5 8 . — a n d D a v i d PINGREE. 1989. MUL.APIN.Cuneiform. H o m : F . B e r g e r & S o h n e .

An

Astronomical

Compendium

in

— a n d D a v i d P i n g r e e . 1999. Astral Sciences in Mesopotamia. L e i d e n : Brill. J E Y E S , UUa. 1989. Old Babylonian Extispicy. Istanbul: N e d e r l a n d s H i s t o r i s c h A r c h a e o l o g i s c h Instituut. KiTCHER, Philip. 1 9 9 3 . The Advancement of Science. Oxford: O x f o r d University Press. K O C H - W E S T E N H O L Z , UUa. 1995. Mesopotamian Astrology. Copenhagen: Carsten N i e b u h r Institute. — 2 0 0 0 . Babylonian Liver Omens. C o p e n h a g e n : C a r s t e n N i e b u h r Institute. K R A U S , Fritz R u d o l f 1960. " E i n zentrales P r o b l e m des a l t m e s o p o t a m i s c h e n R e c h t s : W a s ist d e r C o d e x H a m m u - r a b i ? " . Geneva 8: 2 8 3 - 9 6 . L A R S E N , M o r g e n s TroUe. 1987. " T h e M e s o p o t a m i a n Lidcewarm M i n d , Reflections o n Science, Divination, a n d Literacy". In: F r a n c e s c a R O C H B E R G - H A L T O N (ed.). Language, Literature, and History: Philological and Historical Studies Presented to Erica Reiner: 2 0 3 - 2 5 . N e w H a v e n : A m e r i c a n O r i e n t a l Society. L E H O U X , D a r y n . 2 0 0 0 . Parapegmata, or, Astrology, Weather, and Calendars in the Ancient World, Being an Examination of the Interplay between the Heavens and the Earth in the Classical and Near-Eastern Cultures of Antiquity, with particular reference to the Regulation of Agricultural Practice, and to the Signs and Causes of Storms, Tempests & c. (Dissertation, University of T o r o n t o ) . M E Y E R , J a n - W a a l k e . 1987. Untersuchungen zu den Tonlebermodellen aus dem alten Orient. K e v e l a e r ; B u t z o n & Bercker.

222

D. Lehoux

OPPENHEIM, A . L e o . 1977. Ancient Mesopotamia, r e v . ed. with Erica REINER. C h i c a g o : University o f C h i c a g o Press. P A R P O L A , Srnio. 1 9 9 3 . Letters from Assyrian and Babylonian Scholars. Helsinki: Helsinki U n i v e r s i t y Press. P U T N A M , H i l a r y . 1 9 6 2 . " W h a t T h e o r i e s are N o t " . In: E . N A G E L , P . S U P P E S , a n d A .

T A R S K I (eds.), Logic, Methodology, and Philosophy of Science: 215-227. Stanford: Stanford University Press. Q U I N E , Willard V . O . 1953. " T w o D o g m a s o f E m p i r i c i s m " . In: W i l l a r d V . O . Q U I N E , From a Logical Point of View: 2 0 - 4 6 . C a m b r i d g e , M A : H a r v a r d U n i v e r s i t y Press. REINER, Erica. 1 9 7 4 . " N e w Light o n s o m e Historical O m e n s " . In: K . BiTTEL, P h . H . J . HouwiNK T E N C A T E , a n d Erica R E I N E R (eds.), Anatolian Studies Presented to Hans Joseph Giiterbock on the Occasion of his 65'^ Birthday: 2 5 7 - 2 6 1 . Istanbul: N e d e r l a n d s Historisch-Archaeologisch Instituut. — 1 9 9 5 . Astral Magic in Babylonia (Transactions o f the A m e r i c a n P h i l o s o p h i c a l Society 85). Philadlephia: A m e r i c a n Philosophical Society. — a n d PINGREE, D a v i d . 1975. Babylonian Planetary Omens, Part Two. M a l i b u : Undena. — a n d PINGREE, D a v i d . G r o n i n g e n : Styx.

1 9 9 8 . Babylonian

Planetary

Omens,

Part

Three.

RITTER, J a m e s . 1 9 9 5 . " B a b y l o n - 1 8 0 0 " . In: M i c h e l S E R R E S (ed.), A History Scientific Thought: 1 7 - 4 3 . L o n d o n : Blackwell.

of

— 1 9 9 8 . Reading Strasbourg 368: a Thrice Told Tale (Max-Plank-Institut fur Wissenschaftsgeschichte Preprint 103). R O C H B E R G , F r a n c e s c a . 1999a. " E m p i r i c i s m i n B a b y l o n i a n O m e n T e x t s a n d t h e Classification o f M e s o p o t a m i a n Divination a s S c i e n c e " . Journal of the American Oriental Society 119: 5 5 9 - 5 6 9 . — 1999b. " C o n t i n u i t y a n d C h a n g e in O m e n Literature". In: B a r b a r a BÓCK, E v a C A N C I K - K I R S C H B A U M , a n d T h o m a s RICHTER (eds.), Munuscula Mesopotamica: Festschrift fiir Johannes Renger (Alter O r i e n t u n d Altes T e s t a m e n t 2 6 7 ) : 4 1 5 - 4 2 5 . M u n i c h : Ugarit-Verlag. R O C H B E R G - H A L T O N , Francesca. 1988. Aspects of Babylonian Celestial Divination: The Lunar Eclipse Tablets of Enuma Anu Enlil. H o m : F . B e r g e r & S o h n e . R O T H , M a r t h a T . 1 9 9 5 . Law Collections from Mesopotamia and Asia Minor. Atlanta: Scholars P r e s s . SOLLBERGER, E d m o n d . 1 9 5 4 - 1 9 5 6 , " S u r la c h r o n o l o g i e d e s rois d ' U r et q u e l q u e s p r o b l è m e s c o i m e x e s " . Archivfiir Orientforschung 17: 1 0 - 4 8 . U R Q U H A R T , T h o m a s , a n d P e t e r A . M O T T E U X . 1 6 5 3 . The Works of Francis Rabelais, Doctor in Physick: Containing Five Books of the Lives, Heroick Deeds, and Sayings of Gargantua, and his Sonne Pantagruel, Together with the Pantagrueline Prognostication, the Oracle of the Divine Bacbuc, and Response of the Bottle. L o n d o n : R i c h a r d B a d d e l e y . W E S T B R O O K , R a y m o n d . 1 9 8 5 . "Biblical a n d C u n e i f o r m L a w C o d e s " . Revue Biblique 9 2 : 2 4 7 - 2 6 4

Gnosis and Astrology 'Book IV' of the Pistis Sophia^ Alexandra

von Lieven,

Berlin

T h e so-called Pistis Sophia is a Christian Gnostic treatise.' Unfortunately it is not exactly clear w h e n it was written. T h e only k n o w n m a n u s c r i p t attesting it, the C o d e x A s k e w i a n u s , is d a t e d a p p r o x i m a t e l y to the 4th cent. C E . It is k e p t in the British M u s e u m . T h e text itself is b e l i e v e d to h a v e b e e n c o m p o s e d m o s t l y m the late 3rd cent. C E . It s e e m s to h a v e b e e n written first in G r e e k a n d later translated into Coptic t h o u g h others a r g u e m support of a Coptic original. I n any case it is certain that the tieatise w a s c o m p o s e d in E g y p t b y s o m e o n e familiar w i t h the fraditional religion.^ T h e text in the o n l y existing m a n u s c r i p t consists of foiu: b o o k s in the usual count t h o u g h there are g o o d r e a s o n s to believe that there w e r e in fact m o r e t h a n four. T h e s u p p o s e d fourth is b e l i e v e d to b e not c o n t e m p o r a r y with the other three b u t rather older t h a n t h e m , p r o b a b l y dating to the early 3rd cent. C E . It is quite clear that the Pistis S o p h i a as it stands is not a coherent c o m p o s i t i o n o f one author but a compilation of at least t w o different b l o c k s m t o one w o r k . B o o k I V is the m o s t different from the rest but e v e n the other three contain s o m e inconsistencies. B e s i d e s the textual quality o f the p r e s e r v e d text is rather p o o r in all four b o o k s . T h e w h o l e text deals witii the teachings Jesus s u p p o s e d l y g a v e to his disciples after the resurrection. T h e first t w o b o o k s consist mostiy o f a report o n the m i s a d v e n t u r e s o f the Pistis Sophia interspersed w i t h her h y m n s of r e p e n t a n c e a n d their interpretation in coimection with O l d T e s t a m e n t P s a l m s . T h e y s e e m to b e rather self-contamed. T h e r e the asfral sphere is only i n v o k e d in the b e g i n n i n g of the first book. T h i s is followed b y one chapter, w h i c h in fact forms the e n d of a n o t h e r lost b o o k and w h i c h is then followed b y the third b o o k p r o p e r . T h e t h e m e s o f this b o o k are sins a n d their puitishment. T h e r e is a considerable a m o u n t of u n d e r l y m g asfrological theories. F o r e x a m p l e there is a n explanation of m a n ' s birth u n d e r the influence of asfral forces (called archons a n d the like). All his m i s d e e d s in life are credited to these m a l i g n influences. O f special interest is the description of twelve p l a c e s of p u i u s h m e n t s p r e s i d e d over b y animal h e a d e d a r c h o n s a l r e a d y i m d e r s t o o d b y

I wish to thank Mr. Paul O. Ford for correcting my English. ' Edition: SCHMIDT (1925), reprinted with English translation in SCHMIDT and MACDERMOT (1978), German translation: SCHMIDT and TILL, (1962^), XVI-XXV, pp. 1-254. Book IV contains chapters 136-148 of the whole text as it stands now. ^

KÂKOSY (1970), pp. 244-246 on the Pistis Sophia.

224

A. von Lieven

W i l h e l m G u n d e l t o represent a n adaptation o f the astrological feature k n o w n as dodekaoros? A p a r t from this, the m o s t interesting section o f the Pistis S o p h i a in v i e w o f asfrology"* is the so-called fourth b o o k . It is rather short a n d unfortunately s o m e leaves are missing. A g a i n this b o o k is c o n c e m e d with sins a n d their p u n i s h m e n t s . A t its v e r y b e g m n i n g , in chapters 1 3 6 - 1 3 7 , while Jesus is d o i n g s o m e m a g i c ritual, s u n a n d m o o n are seen in the form o f disks a c c o m p a n i e d b y m y t h o l o g i c a l creatures. O b v i o u s l y Jesus a n d his disciples h a v e risen b y that rite to s o m e elevated p l a c e in the sky. It is characterized as a n "airy p l a c e o n the W a y o f the M i d d l e w h i c h is b e l o w t h e s p h a e r a . " W h e n the disciples ask w h e r e they are, Jesus tells t h e m h o w the p l a c e s o f the W a y o f the M i d d l e c a m e into b e i n g w h e n there w a s a separation b e t w e e n g o o d a n d evil archons. T h e g o o d o n e s b e l i e v e d in the Light a n d p r a c t i c e d its mysteries while the evil o n e s w e r e e n g a g e d in sexual activity. F r o m their intercourse

other

archons,

archangels,

angels,

leitoingoi

and

decans

were

e n g e n d e r e d . This p a s s a g e is at first sight s o m e w h a t p u z z l i n g as n o r m a l l y in asfrological literature the d e c a n s are m o r e important than the leitoingoi.^ T h e d e c a n s are originally 3 6 star constellations rising a n d setting e a c h in ten d a y mtervals, w h i c h separate t h e m from their i m m e d i a t e followers. Witti the infroduction o f the z o d i a c from M e s o p o t a m i a into E g y p t the d e c a n s were associated with its 3 6 0 ° b y giving e a c h d e c a n 10°. T h i s also m e a n t thefr association w i t h the ecliptic while

ttaditionally

the decans were constellations in the s o u t h e m sky. N o r m a l l y the leitourgoi are other entities, e a c h linked to a third o f a d e c a n m a k i n g t h e m three p e r decan. H e r e the t e r m i n o l o g y is m o s t p r o b a b l y j u s t reversed. E q u a l l y sfrange is the fiirther mentioning o f archangels a n d angels a b o v e the d e c a n s . N o r m a l l y these are j u s t epithets for t h e d e c a n s . T h e m o s t p r o b a b l e solution is that these are to b e t a k e n in the literal sense o f angeloi

as m e s s e n g e r s . A s I h a v e

demonsfrated elsewhere the d e c a n s w e r e closely associated with m e s s e n g e r d e m o n s in

fraditional

P a g a n E g y p t i a n religion.^ T h e s e v e n d e c a n s n o t visible at a n y g i v e n

time w e r e b e l i e v e d to b e "Slaughtering D e m o n s " carrying out the p u n i s h m e n t s the g o d s h a d inflicted o n siimers. T h u s I p r o p o s e interpreting the a r c h a n g e l s a n d angels here as the m e s s e n g e r d e m o n s associated with the d e c a n s a n d leitourgoi. Since a r c h o n is only a n u m b r e l l a t e r m for different b e i n g s , they s h o u l d p e r h a p s n o t b e t a k e n to represent s o m e t h i n g m o r e specific. If n o t , they are m o s t likely t o b e

^

BOLL, BEZOLD, and GUNDEL ( 1977^), pp. 187-191.

'* Though quite aged the best introduction to antique astrology is still BOUCHÉLECLERQ (1899), while a fine survey on astrological literature is provided by GUNDEL and GUNDEL (1966). Unfortunately, the bulk of Egyptian astrological sources in Demotic script and language is still unpublished, thus considerably distorting the picture of the role Egypt played in the history of astrology. ^ On the decans see NEUGEBAUER and PARKER (1960), NEUGEBAUER and PARKER (1969), LEITZ (1995), pp. 3 - 1 1 6 , KAKOSY (1982), VON LIEVEN (2000a), GUNDEL (1936), QUACK (in

preparation) among others. Strangely most researchers take the decans in Gnostic texts as the military term not related in any way to astrology, though this connection is difficult to overlook. The same holds ttue for the leiturgoi, see e.g. MALICH (1990). For a critique of this see QUACK (in preparation).

^

VON LIEVEN (2000b), pp. 50-55, p. 188f and passim.

Gnosis and Astrology. 'Book IV' of the Pistis Sophia

225

u n d e r s t o o d as the p r e s i d i n g genii of the twelve z o d i a w h o are called a r c h o n s later in the text. W h i l e the g o o d a r c h o n s were elevated to a p u r e p l a c e , the evil o n e s w e r e b o u n d in the " S p h a e r a o f N e c e s s i t y (heimarmene)". T h e r e w e r e 1800 a r c h o n s b o u n d in every aion, o f w h i c h there are twelve. T h e y were p r e s i d e d o v e r b y 3 6 0 others w h o were again g o v e r n e d b y five others. T h e n a m e s o f these five are g i v e n as K r o n o s , A r e s , H e r m e s , A p h r o d i t e a n d Z e u s . A s is i m m e d i a t e l y evident they are the five planets S a t u m , M a r s , M e r c u r y , V e n u s a n d Jupiter. T h e i r e n i u n e r a t ì o n d o e s not follow the m o s t u s u a l s e q u e n c e of antique astrology w h i c h is S a t u m , Jupiter, M a r s , M e r c u r y a n d V e n u s . ' T h e r e a s o n for this will b e c o m e clear later on. H a v i n g identified the five a r c h o n s as the planets, the other 3 6 0 archons are to b e u n d e r s t o o d as the 3 6 0 d e g r e e s o f the z o d i a c a n d the 1800 as its individual m i n u t e s . E a c h aion is one sign of the z o d i a c . I n addition the text tells that half o f the z o d i a c is p r e s i d e d o v e r b y a g o o d a r c h o n called Jabraoth, the other o n e b y a b a d one called S a b a o t h the A d a m a s . T h i s division is due to the astrological c o n c e p t o f daily a n d nightly parts o f the z o d i a c : starting w i t h L e o as solar h o u s e there are the d a y h o u s e s o f all the planets u p to S a t u m w h i l e in the other direction starting w i t h C a n c e r as lunar h o u s e there are all the p l a n e t ' s night h o u s e s (Figiure 1). O f c o i u s e d a y a n d its light is g o o d in G n o s t i c b e l i e f while night is d a r k a n d thus evil. T h e n J e s u s tells his disciples that forces w e r e d r a w n from different g o d s a n d b o i m d to the p l a n e t s . T h u s they t o o k o n the qualities o f the respective g o d s . S a t u m , M a r s a n d M e r c u r y r e c e i v e d p o w e r s of g o d s w h o s e n a m e s are c o i m e c t e d w i t h d a r k n e s s while V e n u s r e c e i v e d p o w e r from the Pistis Sophia a n d Jupiter from Little S a b a o t h the G o o d . N o w the s e q u e n c e of the planets b e c o m e s u n d e r s t a n d a b l e : first there are the b a d o n e s , then the g o o d ones follow. A s Jupiter w a s e n d o w e d with the greatest a m o u n t o f g o o d n e s s h e w a s c h o s e n to b e the o v e r l o r d o f the others. It was d e c i d e d that h e should s p e n d 13 m o n t h s in e a c h aion so that the evil archons could b e cleansed b y h i m . T h u s , in total Jupiter s p e n d s 13 years travelling t h r o u g h the c o m p l e t e z o d i a c . O f course this is not astronomically correct as Jupiter n e e d s only 11.86 years for o n e revolution. E v e n if a c e r t a m inaccuracy of the antique data o n this m o v e m e n t is p e r m i t t e d the G n o s t i c text is far off the track. T h e r e are several p o s s i b l e explanations for this. O n e w o u l d b e that the n u m b e r is simply a c o r m p t i o n for t w e l v e . If written with G r e e k letters for n u m b e r s , s u c h a c o r m p t i o n s e e m s not too i m p l a u s i b l e a n d twelve w o u l d b e a r o u n d figine n o t too far a w a y from the correct date. T h e p r o b l e m with this h o w e v e r is that the text d o e s not use letters for the n u m b e r 13 b u t writes it as w o r d s . A n o t h e r possibility w o u l d b e that there is a symbolical v a l u e b e h i n d the figure thirteen for w h i c h the author sacrificed a s t t o n o m i c a l accuracy. T h i s w o u l d b e c o m p a r a b l e to the thirteenth aion a b o v e the twelve others that is s o m e t i m e s m e n t i o n e d as a special p l a c e not subject to heimarmene w h i c h g o v e m s the other twelve. Lastly there is the possibility o f simple ignorance o f the author. B u t as the rest o f the text s e e m s quite intelligently a r r a n g e d a c c o r d i n g to astiological theory. '

BOUCHÉ-LECLERQ

(1899),

p.

107,

NEUGEBAUER

(1975),

pp.

690-693.

There

is

nevertheless one Pagan Egyptian source that attests the same order as the Pistis Sophia, see SPÏÉGELBÊRG ( 1 9 0 2 ) ^ (re-edited by NEUGEBAUER-(1943), pp. 4 2 1 - f 22>. This text obviously reflects some Mesopotamian influence, see QUACK ( 2 0 0 1 ) , p. 3 4 2 .

226

A. von Lieven

this is the least likely explanation. I think the s e c o n d possibility is the m o s t plausible one, the r e a s o n for w h i c h will b e seen later. F u r t h e r m o r e Jesus says that Jupiter w a s given " t w o aions as dwellings facing the ones o f H e r m e s . " L o o k i n g at the m o d e l of the z o d i a c as it w a s d e s c r i b e d a b o v e , the m e a n i n g o f this b e c o m e s perfectly clear (Figure 2 ) . T h e aions as dwellings c a n only b e the d a y and night h o u s e s of Jupiter, i.e. Sagittarius a n d P i s c e s . In fact they perfectly face G e m i n i a n d V i r g o the night and d a y h o u s e s o f M e r c i u y . C o i m e c t e d b y lines, all four form a square, i.e., a square aspect astrologically s p e a k i n g . S u c h a square aspect was a n important and powerful configuration (schematismos). C o n c l u d i n g this section, Jesus gives the " i m p e r i s h a b l e " n a m e s o f the planets such as M i m i c h u n a p h o r for M a r s a n d the like. Unfortimately their e t y m o l o g y is at the m o m e n t o b s c u r e to m e p r o b a b l y due to s o m e c o r r u p t i o n in the c o u r s e o f tradition. T h u s these first t w o chapters give a r e m a r k a b l y detailed account o f the z o d i a c a n d related astrological p h e n o m e n a . A t this point the p u p i l s b e g Jesus to explain to t h e m the nature o f the W a y s o f the M i d d l e as it relates to the p u n i s h m e n t of the siimers. After a lenghty p r o m i s e to reveal all mysteries o f o b v i o u s astral-magical character u n t o them, J e s u s finally answers their question. H e tells t h e m that there w e r e 3 6 0 other a r c h o n s b e l o n g i n g to unbelieving A d a m a s b o u n d into the sphaera. F i v e other great a r c h o n s o f the o n e s o f the W a y o f the M i d d l e w e r e set over them. A g a i n s o m e serious p r o b l e m s arise c o n c e m i n g the u n d e r s t a n d i n g o f this section. T h e n u m b e r s 3 6 0 a n d 5 are o f course i m m e d i a t e l y suspicious to r e p r e s e n t the d e g r e e s of the z o d i a c a n d the planets respectively. A s the W a y o f the M i d d l e is usually the r o a d a l o n g w h i c h entities that are to b e identified as the planets m o v e , it is quite clear that this should b e the Gnostic t e r m for the ecliptic (from G r e e k ho dia ton meson). A s this is b e y o n d doubt it should reinforce the identification o f the five archons with the planets. Unfortunately, from the following descriptions of these five it b e c o m e s evident that they are unlikely to b e the planets. Jesus relates that e a c h o f these five presides over a special r a n k o f d e m o n s w h o first incite m e n to w i c k e d n e s s and later p u n i s h their victims. T h e y are released after a certain time w h e n Jupiter a n d V e n u s are s t a n d m g in certain aions, i.e. zodia. A s the t w o g o o d planets are explicitly m e n t i o n e d h e r e they are unlikely to b e also a m o n g the d e m o n i c a r c h o n s . B e s i d e s , the d e m o n i c archons are said to b e "other five" in the text itself and t h e y h a v e their o w n n a m e s , three of w h i c h are female while a m o n g the planets there is o n l y o n e female one. A n o t h e r a r g u m e n t against the identification o f these five a r c h o n s as planets will b e s e e n later. If one collects the information about the five ranks Jesus p r o v i d e s in his descriptions, an interesting s t m c t u r e emerges ( T a b l e 1). M o s t o b v i o u s is the sigitificance of the positions o f Jupiter a n d V e n u s in their respective zodia. In e a c h r a n k they are standing in diagonal aspect (opposition) to e a c h other w h i c h is considered to b e especially effective m astrology (Figure 3 ) . In addition, o n e o f t h e m is p l a c e d in another u n p o r t a n t position: either m o n e o f his h o u s e s or in his h y p s o m a . O n l y rank 5 is a n exception, since tiiere Jupiter is not d n e c t l y in o n e o f his h o u s e s . Instead he is standing in tiiangular aspect to Libra, V e n u s ' s d a y h o u s e . A s the tiiangular aspect to a n important sign is considered nearly as powerful as the position m that sign itself this is a m i n o r flaw. It b e c a m e necessitated b e c a u s e o f the strict c o n c e p t o f h a v i n g Jupiter a n d V e n u s s t a n d i n g e v e r y \ d i e r e i n diagonal a s p e c t to e a c h other.

227

Gnosis and Astrology. 'Book IV' of the Pistis Sophia

a G

E

|1

S

a

H

2 ^ ON

IS

2

1^2 O

A

dì >>

o

oo m

«

H

--^ o o

3

9 E

S

>

>

C

e C

2

.2

'5

ES

(M

® v;

o

C O ••O

g*

3 «3

S

'ór

•c

E O

I _

O

dû o

Ì

Il I ^ !-: i3

ON

•C 2

c ^

^

S

3 O

s

>

B O

on

^ OO la û û ' S O O t« ' S B .S

\^

et

2

^

S

O

-A

c/a c/3 c/a

^

^

8

O

O

I

*; A

O

O

sa o

E

. l i

Cd J3

•a

IX,

h. S

o

A O

E O

= C

.2

II

u

S

O

2

il

S

C O

JS tj u

< a et ça

fN IT)

H

f2

228

A. von Lieven

L o o k i n g at the r e p r e s e n t e d signs o f the z o d i a c all are p r e s e n t e x c e p t for t w o . T h e t w o missing o n e s are V i r g o a n d Pisces. A s the latter o n e is the h y p s o m a o f V e n u s it s e e m s u n p e r a t i v e t o restore a missing sixth r a n k w h i c h should b e dissolved u p o n Jupiter reaching V i r g o a n d V e n u s reaching Pisces simultaneously ( T a b l e 2 ) . B u t if a n original sixth r a n k is missing, t h e n it also should h a v e h a d t h e other categories filled in w h i c h m e a n s that there also h a s to b e a sixth a r c h o n p r e s i d i n g over it. O f c o i n s e if there are six archons they caimot b e the five p l a n e t s . T h i s is the last a n d m o s t important a r g u m e n t already s p o k e n o f S o a g a i n the q u e s t i o n arises what these a r c h o n s actually are. A s the p o s i t i o n s o f the planets are astrologically well structured, it is o f c o u r s e t e m p t i n g to relate the other information p r o v i d e d also to astral p h e n o m e n a . T h e description o f the first a r c h o n as a w o m a n w i t h long hair a n d o f the s e c o n d o n e as a b l a c k w o m a n recalls s o m e w h a t the descriptions o f d e c a n s in astiological tieatises. T h e q u e s t i o n arises if these a r c h o n s c o u l d b e related to the decans o f the respective zodia. T h e n the n u m b e r 6 w o u l d m a k e s o m e sense, since 3 6 is o f course six times six. Unfortimately there are different tiaditions c o n c e m i n g the i c o n o g r a p h y o f the d e c a n s a n d e v e n within o n e tiadition there are considerable variants b e t w e e n different sources. If m y idea is correct the p r e s e n t list should m o s t p r o b a b l y b e l o n g to w h a t I call " D r i t t e R e i h e " i.e. the tiadition that is found in s o m e H e r m e t i c m a n u s c r i p t s , o n e n g r a v e d g e m s , the T a b u l a B i a n c h i n i a n d o n the ivory tablets from o

G r a n d . D i s a p p o i n t i n g l y w h e n c o m p a r i n g the Dritte R e i h e with the Pistis S o p h i a list, the d e c a n s o f the z o d i a for Jupiter a n d V e n u s d o n o t r e s e m b l e m u c h the descriptions of the archons with o n e notable exception. This is the fourth r a n k w h e r e the a r c h o n is called T y p h o n a n d the respective aions are Sagittarius a n d G e m i n i . I n d e e d the i c o n o g r a p h y for t h e first d e c a n o f G e m i n i is a figure w i t h a d o n k e y h e a d . A s T y p h o n is t h e G r e e k n a m e for the Egyptian g o d Seth w h o s e animal is the d o n k e y everything fits perfectiy. I d o u b t that this is p u r e coincidence b u t u p to n o w I c a n n o t give a n e x p l a n a t i o n w h y t h e rest d o e s n o t fit. P r o b a b l y the significance o f the six archons w a s n o t a n y m o r e i m d e r s t o o d b y a later scribe w h o also associated the archons o f the W a y o f the M i d d l e w i t h the planets a n d thus e m e n d e d the sixth r a n k away. E v e n m o r e p r o b l e m a t i c than the a r c h o n s ' identity is the n u m b e r o f d e m o n s g i v e n for e a c h rank. O n e possibility w o u l d b e to relate the n u m b e r o f d e m o n s to the n u m b e r o f stars o f a g i v e n constellation. Unfortunately the ancient asfrological authors d o n o t a g r e e o n the n u m b e r o f stars e a c h z o d i o n c o m p r i s e s . A s a b a s i s I t o o k 9

t w o authors o f the 2 n d cent. C.E., C l a u d i o s P t o l e m a i o s a n d Vettius V a l e n s o f w h o m is particularly interesting t h o u g h i n c o m p l e t e l y h a n d e d d o w n .

^

ABRY ( 1993), VON LIEVEN (2000a).

^

MANTTIDS and NEUGEBAUER ( 1 9 6 3 ) , pp. 4 3 - 5 4 .

'°

PINGREE ( 1 9 8 6 ) , pp. 5 - 1 3 .

10

the last

229

Gnosis and Astrology. 'Book IV' of the Pistis Sophia

S

« SS JU O V. o .5 " fi "P

5

(N II

Z

N «8

5

fi

cÂ V O fN

fN

00

oo

00

h. o

z

di

sI

«M

o

.

Ou ^

a

oc

s ta

-M

fN

fN

II

fN

+

o PQ

fN

+

o

+

«.^

OS

S

O ew s

s

o

«<- 1-^

•*

il (Si

00

fN

fN

«- « .S »—c

s -2 o z g^

II

II

fN

II

^1

fi O

s

fN

fN

fN

fi

2

VO

fl

H

230

A. von Lieven

B o t h give the n u m b e r o f stars b e l o n g i n g to a z o d i o n , plus other stars a r o u n d that are not t a k e n as b e l o n g i n g to that constellation p r o p e r . T a b u l a t i n g the n u m b e r s given for e a c h sign Jupiter a n d V e n u s are said to r e a c h against the n u m b e r o f d e m o n s of that particular rank, s o m e suspicious similarities c a n b e o b s e r v e d ( T a b l e 3). It is evident that there is neither a c o m p l e t e c o r r e s p o n d e n c e b e t w e e n the two authors n o r one b e t w e e n star n u m b e r s given b y t h e m a n d the n u m b e r s o f d e m o n s in the G n o s t i c text. N e v e r t h e l e s s at least for C a n c e r in the s e c o n d r a n k (if the e m e n d a t i o n of B o l l is a c c e p t e d ) a n d for T a u r u s in the third r a n k the n u m b e r s of Vettius V a l e n s are identical with the n m n b e r o f d e m o n s . M o r e o v e r in the f o m t h r a n k the 31 stars g i v e n b y P t o l e m a i o s for Sagittarius are not that far a w a y f r o m the 32 d e m o n s o f the Pistis Sophia. A s rank six is missing a n d in rank five w e are o n l y told that there is " a great n i u n b e r " o f d e m o n s , n o certainty c a n b e gained. B u t at least the n u m b e r s 4 2 or e v e n 4 5 for A q u a r i u s are the highest figures given b y P t o l e m a i o s for a z o d i o n ' s star n u m b e r . Interestingly s o m e t i m e s the z o d i o n for Jupiter a n d s o m e t m i e s the o n e for V e n u s fits better w i t h the d e m o n n u m b e r s a n d this exactly c o r r e s p o n d s with the sign w h i c h is a special one, e.g. a h o u s e of the respective p l a n e t or s o m e t h i n g o f the like. I h a v e highlighted this b y setting these in b o l d type. T h e r e r e m a m s the first order. A s Libra is the d a y h o u s e o f V e n u s it is s o m e w h a t disappointing to see that its 8 or 17 stars are far a w a y from the 2 5 d e m o n s . B u t there is s o m e t h i n g that should b e recalled. N o t all a s t i o n o m e r s c o n s i d e r e d Libra to b e a separate sign. Especially the G r e e k ttadition saw instead only the claws o f the following scorpion. In fact w h a t I call L i b r a in the table a b o v e is i n d e e d labelled " C l a w s of the S c o r p i o n " b y P t o l e m a i o s h i m s e l f T h a t d o e s not m a k e die n u m b e r fit better, of course. B u t looking at the n i u n b e r o f signs of Scorpio itself, 21 or (including the " e x t t a stars") 2 4 stars are given, w h i c h are n o t far from the 25 required. A s there is n o exact c o r r e s p o n d e n c e this is j u s t a suggestion, but I think it is o n e w o r t h considering. A n o t h e r p r o b l e m are the sins associated with e a c h rank. N o r m a l l y astiological p r o g n o s e s d o not s i m p l y d e p e n d u p o n the sign of the z o d i a c b u t u p o n the p l a n e t standing in this sign (and other factors). A s Jupiter a n d V e n u s are the b e s t p l a n e t s , the c h a n c e for a b a d p r o g n o s i s is rather small. M o r e o v e r , since in the fourth rank the a r c h o n T y p h o n a n d his d e m o n s are said to incite p e o p l e to lust a n d sexual m i s b e h a v i o u r , w h i c h w o u l d b e quite appropriate for the E g y p t i a n g o d Seth I w o n d e r w h e t h e r the suis c o u l d b e c o n n e c t e d to the d e c a n s . T h e y were s o m e t i m e s g i v e n p r o g n o s e s o f their o w n . B u t this w o u l d d e p e n d o n the identification o f the a r c h o n s as d e c a n s w h i c h w o r k s for the fourth r a n k but not the others. So I h a v e to leave this o p e n at the m o m e n t . Finally the sttange time intervals g i v e n for the d u r a t i o n of a sirmers' stay in e a c h rank are to b e considered. A s astiai p h e n o m e n a are p r o m i n e n t l y i n v o l v e d in the w h o l e text, the idea that also these time intervals h a v e a n a s t t o n o m i c a l significance is suggesting itself I reflected o n several possibilities. Since Jupiter is the m o s t iirç)ortant planet in the text with a g i v e n p e r i o d of 13 years, I fried to divide the intervals b y this p e r i o d . R e g a r d l e s s if it is set at 11,86, 12 or the 13 years of the text the division gives neither significant n o r even n u m b e r s . N e x t I fried to find out if it could b e the time Jupiter a n d V e n u s n e e d to stand again in the s a m e signs. A g a i n this did n o t lead to any satisfying result. Finally I h a d the idea that the starting p o i n t should b e t h e very t h i n g to look for, t h a t w o u l d m e a n e.g., finding o u t w h e r e Jupiter was standing 133 years a n d 9 m o n t h s before its entry into A r i e s . T o get this

Gnosis and Astrology. 'Book I V of the Pistis Sophia

231

information I c h a n g e d the years a n d m o n t h s into p u r e years (e.g. 133 years a n d nine m o n t h s to 133.75 years), d i v i d e d t h e m b y the p e r i o d of Jupiter, eliminated the full year n u m b e r in front o f the d e c i m a l p o i n t a n d muUiplied the figines after the d e c i m a l p o i n t b y twelve to get the appropriate sign o f the z o d i a c . A s the entry into a sign is the important thing, all that w a s after the d e c i m a l p o i n t o f m y result w a s t a k e n as one further sign. T h u s I got the n u m b e r of signs to g o b a c k starting from the sign g i v e n as that w h i c h will trigger the dissolution of a specific r a n k w h i c h in t u m g a v e m e the sign of the z o d i a c m w h i c h Jupiter s t o o d at the begiiming o f the cycle. T h e first r a n k is g i v e n here as an e x a m p l e : 133.75 : 13 - 10.288 ^ 0 . 2 8 8 X 12 = 3.456 -> 4 signs b a c k w a r d s from Aries - > Sagittarius Rank

J u p i t e r starts in

1

Sagittarius

A s t r o l o g i c a l Significance d a y h o u s e of Jupiter

2

Libra

day house of Venus

3

Virgo

day house of Merciuy

4

Aries

h y p s o m a of S u n

5

Gemini

night h o u s e o f M e r c u r y

Table 4 L i k e w i s e I got the signs for all the other r a n k s ( T a b l e 4 ) . I calculated these data with 11,86, 12 a n d 13 years as the p e r i o d for Jupiter. T h e results for 13 w e r e the m o s t satisfactory, since n o t o n l y each rank starts with a g o o d position b u t the starting signs of e a c h r a n k are all standing in diagonal aspect to one o f the other r a n k s (Figure 4 ) : Sagittarius ( r a n k 1) is p a i r e d with G e m i n i (rank 5), a n d Libra (rank 2) w i t h Aries (rank 4 ) . O n l y V i r g o (rank 3) is single b u t o f c o u r s e the sixth r a n k is missing. Restoring it as the sign in diagonal aspect to V i r g o w o u l d b e the m o s t p l a u s i b l e . Rank

J u p i t e r starts in

Astrological Significance

6

Pisces

night h o u s e of Jupiter

Table 5 T h u s it should b e Pisces w h i c h is n o t h i n g else than the n i g h t h o u s e o f Jupiter ( T a b l e 5). Interestingly, Pisces, G e m i n i , V i r g o a n d Sagittarius are j u s t the four cardinal p o i n t s of the square of the h o u s e s o f Jupiter a n d M e r c u r y m e n t i o n e d in the text itself w h e n the 13 years as p e r i o d for Jupiter are given. A s there are six ranks one p a n c o u l d n o t b e fitted into this square. It consists of the d a y h o u s e of V e n u s for o n e part, a n d since the signs should stand in diagonal aspect, the night h o u s e o f V e n u s w a s n o t available as the other part. B u t the h y p s o m a o f the s u n is o f c o u r s e also a powerful constellation. T h e i m p o r t a n c e o f the diagonal aspect h e r e parallels the situation for the signs g i v e n for the dissolution o f the ranks before. T h e perfection w i t h w h i c h all these dates fit p r o v e s the textual c o r r e c m e s s of the figure 13 for the p e r i o d o f Jupiter despite its a s t t o n o m i e m e a n i n g l e s s n e s s . In c o m p a r i s o n w i t h t h e ^ n t i r e effort p u t ^nto t h e a p p l i c a t i o n o f a s t r o l o g i c a l theories i n this text, s u c h a n inaccuracy might s e e m surprising a n d confradictory. B u t it s h o u l d

232

A. von Lieven

b e recalled that the C h u r c h Father A u g u s t i n e , w h o w a s v e r y m u c h interested in astrology as a yoimg m a n , felt r e p u l s e d from M a n i c h a e i s m after a w h i l e for the v e r y r e a s o n that he d i s c o v e r e d that the Gnostic kind of asfrology w a s " i m s c i e n t i f i c " , ' ' that is to say religiously adapted. After this lenghty explanation o f the ranks o f p i m i s h m e n t J e s u s ' s disciples are afraid a n d b e g for h e l p a n d forgiveness. M o r e teachings follow w h e r e asfrology is not as p r o m i n e n t as before. T h e n there is a gap of several p a g e s . T h e text starts again in the m i d d l e of a long discourse on sins a n d p u n i s h m e n t s w h i c h also gives time intervals. It differs quite r e m a r k a b l y from the o n e before, thus I w o u l d p r o p o s e taking it as the r e m n a n t s of another b o o k . T h e r e , precise astrological data is lacking, p l a c i n g it outside the s c o p e o f this paper. In conclusion, w h a t lessons are to b e l e a m e d from the case of the Pistis S o p h i a ? First it b e c o m e s clear h o w important antique asfrology is for a correct u n d e r s t a n d i n g o f Gnostic ideology. T h o u g h it has long b e e n k n o w n that the G n o s t i c s w e r e influenced b y asfrology this is n o r m a l l y not t a k e n into accoimt as sfrongly as it should b e in interpreting thefr scriptures. W h e n this k n o w l e d g e is p r o p e r l y applied, these texts, w h i c h at first sight s e e m so a b s t m s e , reveal t h e m s e l v e s to b e n o t only quite c o m p r e h e n s i b l e b u t also very well stmctured. S e c o n d l y it is to b e r e m e m b e r e d that there is a n o n g o i n g d i s c u s s i o n a b o u t h o w far G n o s t i c i s m is influenced b y fraditional E g y p t i a n religion. U s u a l l y studies o n this subject t u m to N e w K i n g d o m or e v e n older theological texts or the like a n d speculate about p o s s i b l e connections regardless o f the g a p in time b e t w e e n t h e m a n d the Gnostic material. T h e r e is indeed a long fradition of various religious texts from older e p o c h s far into the R o m a n period. N e v e r t h e l e s s it is j u s t the asfrological lore, the latest d e v e l o p m e n t o f Religious A s t r o n o m y that is the m o s t i m p o r t a n t a n d clearly d i s c e m a b l e element o f the P a g a n w o r l d v i e w i n c o r p o r a t e d into the G n o s t i c belief 12

system. O f course this is n o t resfricted to Egypt. B u t it is consistent w i t h the p l a c e o f asfrology in c o n t e m p o r a r y P a g a n Egyptian religion as reflected in n u m e r o u s t e m p l e reliefs, p a p y r i from t e m p l e libraries a n d private m o n u m e n t s as e.g., e n g r a v e d g e m s a n d coffins. E v e n if these display a considerable d e g r e e o f iimovations t h e y are also to b e v i e w e d as standing in a n d evolving from the fradition of Religious Asfronomy, already present in the first religious texts p r e s e r v e d from Egypt. T h e characteristic of Religious A s f r o n o m y is the ultimate c o n n e c t i o n o f scientific interest in asfral p h e n o m e n a with m y t h o l o g i c a l and, religious interpretations. T h i s attitude so typical o f A n c i e n t E g y p t is exactly w h a t the Gnostics also held. T h u s in the asfrological elements o f Gnostic texts like the Pistis S o p h i a there is i n d e e d E g y p t i a n influence to b e detected. In fact these elements c a n e v e n t h r o w an interesting light o n late P a g a n b e l i e f F o r that reason, such texts d o - or at least should - m o v e a w a y from a rather marginal area o f interest within E g y p t o l o g y to a m o r e favourable p l a c e .

''

Confessions V, 3 - 7 (LABRIOLLE ( 1950), pp. 94-102).

QUACK (1995), pp. 9 7 - 2 2 discusses an interesting case of the adaptation of decan metofhesfa, a concept thai is quite prorhineht in Rórnàn period astral magic but has much older antecedents.

Gnosis and Astrology. 'Book IV' of the Pistis Sophia

233

Figure 1 : Z o d i a c w i t h d a r k half (left, night h o u s e s starting with M o o n u p to S a t u m ) a n d bright half (right, starting with S u n u p to S a t u m ) . ( D r a w n b y the author)

Figure 2: S q u a r e aspect o f h o u s e s of Jupiter and M e r c u r y . ( D r a w n b y the author)

234

A. von Lieven

Figure 3 : Positions of Jupiter and V e n u s triggering the dissolution o f the five ranks o f p u n i s h m e n t . ( D r a w n b y the author)

F i g u r e 4 : Starting positions for Jupiter including restored sixth rank. T h e outer ring o f p l a n e t s y m b o l s signifies their exaltations ( h y p s o m a t a ) . ( D r a w n b y the author)

Gnosis and Astrology. 'Book IV' of the Pistis Sophia

235

References A B R Y , J o s è p h e - H e n r i e t t e (ed.). 1 9 9 3 . Les tablettes

astrologiques

de Grand

(Vosges).

Paris: Diffusion de B o c c a r d . B O L L , F r a n z ; B E Z O L D , Cari, a n d G U N D E L , W i l h e l m . "^1977. Sternglaube und Sterndeutung. Stuttgart: T e u b n e r . B O U C H É - L E C L E R Q , A u g u s t e . 1899. L'astrologie grecque. Paris: E . L e r o u x (Reprint A a l e n : Scientia V e r l a g 1979). G U N D E L , W i l h e l m . 1936. Dekane und Dekansternbilder. Gliickstadt u n d H a m b u r g : J.J. A u g u s t i n (Reprint: D a r m s t a d t : Wissenschaftliche Buchgesellschaft ^1969). G U N D E L , W i l h e l m , a n d G U N D E L , H a n s G e o r g . 1966. Astrologumena. Die astrologische Literatur in der Antike und ihre Geschichte (Sudhoffs A r c h i v Beihefte 6). W i e s b a d e n : Steiner. K À K O S Y , L a s z l o . 1970. " G n o s i s u n d àgyptische R e l i g i o n " . In: U g o B I A N C H I (ed.), Le Origini dello Gnosticismo: IZ^-lAl. Leiden: E.J. Brill. K Â K O S Y , L a s z l o . 1982. " D e c a n s in L a t e - E g y p t i a n R e l i g i o n " . Oikumene 3 : 1 6 3 - 1 9 1 . DE LABRIOLLE, Pierre. 1950. Saint Augustin, Confessions. Livres I-VIIL T o m e I. Paris: Société d ' É d i t i o n s " L e s belles Lettres". LEITZ, Christian. 1995. Altàgyptische Sternuhren (Orientalia Lovaniensia A n a l e c t a 6 2 ) . L e u v e n : Uitgeverij Peeters e n D é p a r t e m e n t Orientalistiek. V O N L I E V E N , A l e x a n d r a . 2 0 0 0 a . " D i e dritte R e i h e d e r D e k a n e oder T r a d i t i o n u n d Iimovation in der spâtàgyptischen Religion". Archiv fiir Religionsgeschichte 2: 21-36. V O N L I E V E N , A l e x a n d r a . 2 0 0 0 b . Der Himmel Uber Esna: Eine Fallstudie zur Religiosen Astronomie in Agypten am Beispiel der kosmologischen Decken- und Architravinschriften im Tempel von Esna ( A g y p t o l o g i s c h e A b h a n d l u n g e n 6 4 ) . Wiesbaden: Harrassowitz. M A L I C H , B u r k h a r d . 1990. " D i e Stellung der Liturgen in d e n k o p t i s c h - g n o s t i s c h e n Schriften". In: Peter N A G E L (éd.), Carl-Schmidt-Colloquium an der MartinLuther-Universitât Halle-Wittenberg 1988: 2 1 3 - 2 2 0 . H a l l e (Saale): W i s s e n schaftspublizistik der Martin-Luther-Universitàt H a l l e - W i t t e n b e r g . M A N I T I U S , Karl, a n d N E U G E B A U E R , Otto. 1 9 6 3 . Ptolemàus Handbuch der Astronomie II. Leipzig: B . G . T e u b n e r . N E U G E B A U E R , O t t o . 1 9 4 3 . " D e m o t i c H o r o s c o p e s " . Journal

of

the

American

Oriental Society 6 3 : 1 1 5 - 1 2 8 . N E U G E B A U E R , O t t o . 1 9 7 5 . A History of Ancient Mathematical Astronomy II (Studies in the H i s t o r y of M a t h e m a t i c s a n d Physical S c i e n c e s 1). B e r l i n / H e i d e l b e r g / N e w Y o r k : Springer. N E U G E B A U E R , O t t o , a n d P A R K E R , R i c h a r d A . 1960. Egyptian Astronomical Texts 1. The Early Decans. L o n d o n : L. H u m p h r i e s . N E U G E B A U E R , O t t o , a n d P A R K E R , R i c h a r d A . 1969. Egyptian Astronomical Texts III. Decans, Planets, Constellations and Zodiacs. L o n d o n : L. H u m p h r i e s . PINGREE, D a v i d . 1986. Vettii Valentis Antiocheni Anthologiarum libri novem. Leipzig: B . G . T e u b n e r . Q U A C K , J o a c h i m F . 1995. " D e k a n e u n d G l i e d e r v e r g o t t u n g . Altàgyptische T r a d i t i o n e n i m A p o k r y p h o n J o h a n n i s " . Jahrbuch fiir Antike und Christentum 3 8 : 97-122.

236

A . von Lieven

Q U A C K , J o a c h i m F . 2 0 0 1 . " Z u m ersten astrologischen Lapidar i m S t e i n b u c h des D a m i g e r o n imd E v a x " . Philologus 1 4 5 : 3 3 7 - 3 4 4 . Q U A C K , J o a c h i m F . In preparation, Beitràge zu den agyptischen

Dekanen

und

ihrer

Rezeption in der griechisch-romischen Welt. S C H M I D T , Carl. 1 9 2 5 . Pistis Sophia, (Coptica 2 ) . H a u n i a e : G y l d e n d a l , N o r d i s k Forlag. S C H M I D T , Carl, a n d M A C D E R M O T , Violet. 1 9 7 8 . Pistis Sophia ( N a g H a m m a d i Studies 9 ) . L e i d e n : Brill. S C H M I D T , Carl, a n d TILL, W a l t e r . M 9 6 2 . Koptisch-gnostische Schriften I. Berlin: Akademie Verlag. SPIEGELBERG, W i l h e l m . 1 9 0 2 . " E i n agyptisches V e r z e i c h n i s der P l a n e t e n Tierkreisbilder". Orientalistische Literaturzeitung 5: 6 - 9 .

und

Ration Computations at Fara: Multiplication or Repeated Addition? Duncan J. Melville,

Canton (New

York)

Introduction In the mid-third milleimium, F a r a (ancient S u r u p p a k ) w a s a substantial city with extensive e c o n o m i c ties to s u r r o i m d m g cities and a h i g h l y - d e v e l o p e d b u r e a u c r a c y . ' O n e of its major c o n c e r n s was the distribution o f rations. In m o s t cases, a tablet r e c o r d s n a m e d individual officials receiving various quantities o f grain a n d s o m e t i m e s a total disbursement. H o w e v e r , in a few cases, there are r e c o r d s o f fixed quantities b e i n g g i v e n to n u m b e r s o f workers. O u r c o n c e m in this p a p e r is to d e t e r m i n e h o w the relevant calculations w e r e carried out a n d to u s e the results of this analysis to s h e d light o n w i d e r questions of c o n c e p t d e v e l o p m e n t in mid-third millenniiun m a t h e m a t i c s . T h e analysis d e p e n d s heavily u p o n certain features o f the metrological systems in u s e at F a r a during this p e r i o d ; the salient issues are summarized below.

' According to MARTIN (1988), pp. 12, 127, the walled area of the city probably encompassed some 220 hectares, with an inhabited area in EDIIIa (c. 2500 BCE) of around 100 hectares and a population estimated at 15,000-30,000 people. VISICATO (2000), pp. 22ff. identifies a hundred individuals as various types of scribes and gives a detailed analysis of the economic structure of Fara including a typology of officialdom, so far as is currently known, based upon his and Pomponio's re-editing of the economic documents (see especially VISICATO (1995), Chapter 3). The so-called Hexapolis, comprising Adab, Lagal, Nippur, âuruppak, Umma and Uruk, has been recently argued for by POMPONIO and VISICATO (1994) (EDATà),pp. 10-20. Fara has been excavated twice, each time quite briefly. It was also visited by Hilprecht in the spring of 1900. The disposition of his finds is unclear (MARTIN (1988), p. 15). In 1902-3, the Deutsche Orient-Gesellschaft excavated for seven and a half months, and in 1931 the University of Pennsylvania sponsored a three-month expedition. Approximately 1000 tablets and fragments were found, more than a third of them at one single location, the 'Tablet House' (MARTIN (1988), p. 86). However, publication of the tablets was slow. Tablets from the D.O.G. excavations were divided between Istanbul and Berlin. The Berlin tablets were published in the 1920's by DEIMEL (1923) (SF) and DEIMEL (1924) (WF), and the Istanbul tablets in the 1930's and 1950's by JESTIN (1937) (TSS) and JESTIN (1957) (NTS§). The tablets from the University of Pennsylvania excavations were described briefly by KRAMER (1932),^ but are just now being published (MARTIN et al. (2001)). This latter work has not been available to me. Fara tablets discussed in this paper are referred to ^yorTgiliâl publication number. A concordance of tablet numbers is given below.

238

D.J. Melville

Ration Exercises O v e r a h u n d r e d tablets r e c o r d the disbursement o f rations to w o r k e r s a n d officials, usually o f barley, occasionally of flour.^ H e n c e , it is not siuprising that there are also several tablets r e c o r d i n g exercises to d o with ration c o m p u t a t i o n s . W e discuss three such school tablets here before locating these p r o b l e m s in the c o n t e x t o f c o n t e m p o r a r y administrative archives. T h e three tablets are T S § 8 1 , T S § 5 0 a n d T S â 6 7 1 . O f these, the latter t w o h a v e rightly attracted considerable attention as crucial to any interpretation of mid-third m i l l e i m i u m m a t h e m a t i c s , b u t the former has b e e n less noticed. H o w e v e r , analysis of T S S 81 s h o w s that current a s s u m p t i o n s about F a r a c o m p u t a t i o n a l techniques are imtenable. T S à 81 is a small r o u n d e d tablet o f the sort u s e d for exercises. It contains a p r o b l e m stated o n the o b v e r s e with a solution g i v e n on the reverse. It m a y b e a m a t h e m a t i c a l exercise, it m i g h t have b e e n used for a c o m p u t a t i o n pertaining to s o m e larger tablet, or it m a y represent a particular e c o n o m i c tiansaction.^ T h e tablet r e a d s as follows: Transliteration O. 40 dumu sidim zi-ba flour 2 ( b â n ) su ba-ti R. 3 lid-ga zi l ( b a r i g a ) 2(bân)

Translation 4 0 c a r p e n t e r s ' sons gift each received 2(bân) 3 lid-ga l ( b a r i g a ) 2(bân) flour

T h e tablet includes m o r e information than w o u l d s e e m n e c e s s a r y for a ' s c r a t c h - p a d ' c o m p u t a t i o n and this interpretation is unlikely. T h e r e are n o p e r s o n a l n a m e s or titles o n the tablet, w h i c h is u n u s u a l for a n e c o n o m i c d o c u m e n t a l t h o u g h not u n i q u e ( c o m p a r e , for e x a m p l e T S S 5 5 4 ) . T h e given ration quantity is realistic, since it is the same as that in the e c o n o m i c d o c u m e n t W F 9 8 , discussed b e l o w . N o findspots are k n o w n for any of the T S è tablets'* so w e caimot give this tablet a context relating to other tablets. T h e r e is n o question stated, as o n e w o u l d expect in a formal exercise in later p e r i o d s , b u t there is n o question in the exercise tablet T S S 5 0 either. T h e implied question is o b v i o u s firom context. If it w e r e an exercise, the q u e s t i o n w o u l d p r e s u m a b l y have b e e n supplied orally. The o b v e r s e o f the tablet contains j u s t the right information for setting u p a p r o b l e m for a student. O n b a l a n c e , it s e e m s m o s t likely that this tablet d o e s r e c o r d an exercise. T h e result is correct. T h e question is h o w tiie result w a s obtained. T h e m o d e m a p p r o a c h w o u l d b e to multiply 2(bân) b y 4 0 to get 8 0 ( b â n ) , a n d t h e n c o n v e r t this quantity into the correct metiological units. H o w e v e r , the intermediate step of 80(bân) is impossible to write in Fara m e t i o l o g i c a l notation, and it is doubtfiil that

^

E D A T S , p. 3.

^ POWELL (1976), p. 436, n. 19 includes it in his list of mathematical tablets but does not discuss it further, EDZARD (1976), p. 171 merely refers to it as a 'Notiz' and KREBERNIK (1998), p. 350 identifies it as an economic document. The tablet is not included in either EDATè or Bureaucracy of Suruppak, presumably because there are no personal names, so it is not clear whether Pomponio and Visicato regard it as a true economic document. (But see EDAT§,p. 16,n.l7.) '

MARTIN (1988), p. 98.

Ration Computations at Fara: Multiplication or Rqjeated Addition?

239

the c o m p u t a t i o n w a s p e r f o r m e d in this way. T o see the p r o b l e m w e s u m m a r i z e briefly F a r a metrological notation.

Fara Metrology O v e r the c o i n s e o f the third m i l l e n n i u m the c o m p l e x archaic collection o f n a r r o w l y specialized metrological systems w a s gradually simplified.^ H e r e , w e n e e d b e c o n c e m e d with o n l y t w o systems: the discrete s y s t e m u s e d for c o u n t i n g m o s t discrete items a n d registering large quantities in other systems, a n d the capacity system u s e d for m e a s u r i n g grain a n d grain p r o d u c t s . The Discrete System T h e s y s t e m in u s e at F a r a in the mid-third m i l l e n n i u m for m e a s u r i n g discrete items, such as, in the cases w e consider, p e o p l e , w a s d e r i v e d from the similar archaic system. It b e g a n w i t h a b a s e unit. E a c h higher imit w a s t h e n a l t e m a t e l y 10 or 6 times the p r e v i o u s unit. In terms o f factor d i a g r a m s , the c o m p l e t e attested s y s t e m is g i v e n as:

sam

sar

gesù

ges

u

dis

T h e s a m thus signified 3 6 , 0 0 0 of the b a s e unit a n d w a s the largest n u m b e r s y m b o l attested in this period. P r o b l e m s involving discrete items, w h e t h e r school exercises or actual administrative questions, w o u l d b e written a n d calculated u s i n g this system. U s i n g this notation, scribes could write n u m b e r s u p to 2 1 6 , 0 0 0 (the hypothetical next unit in the system). T h i s b o u n d is w o r t h n o t i n g w i t h reference to the tablet T S § 50 discussed b e l o w , although it should p e r h a p s n o t b e t a k e n too seriously.^ The Capacity System T h e Fara c a p a c i t y s y s t e m as is c o m m o n for s u c h metrological systems, built u p o n a b a s e unit a n d c o n t a m e d a n u m b e r of intermediate-sized units u p to a largest n a m e d unit. M u l t i p l e s o f the large unit were written using the discrete s y s t e m d e s c r i b e d a b o v e . T h e r e is o n e slight c o m p l i c a t i o n at F a r a in that t w o different m a x i m a l units were in use, the lid-ga a n d the g u r - m a h ; these will b e discussed b e l o w . T h e b a s e unit w a s the sila. Quantities of sila u p to the next unit w e r e written using multiples o f the standard (i.e., discrete) symbol for 1, with the qualification sila. In a list o f quantities of similar size, the sila sign w a s usually o n l y u s e d to

'

NissEN, DAMEROW and ENGLUND ( 1 9 9 3 ) , pp. 1 3 8 - 1 4 0 .

^ H0YRUP ( 1 9 8 2 ) , p. 3 3 , n.8 has an amusing footnote discussing this very aspect of the discrete system. Commenting on the relative frequencies of the different symbols, he notes that they 'fit beautifully to a sttaight line in a log-log diagram. By exfrapolation, the next numeral of the series ( " 2 1 6 0 0 0 " ) should be expected to occur on approximately 1/2 to 1 % of all tabîëts, if it existed. On the other hand, the t)nly a r g u m e n t ^ the use of l«g-log^nalysis is^ that it works between " 1 " and " 3 6 0 0 0 " . '

240

D.J. Melville

qualify the first entry. T h e r e has b e e n a great deal o f discussion as to the absolute size of the sila. It m a y well h a v e varied in s p a c e a n d time and it is not clear h o w standardized third-millennium capacity m e a s u r e m e n t s were.^ T h e n e x t t w o units in the Fara capacity s y s t e m were the b a n o f 10 sila a n d the bariga of 6 ban. T h e t w o large units were the lid-ga o f 4 b a r i g a a n d the gur (gurm a h ) of 8 bariga. B o t h the lid-ga a n d the g i n w e r e d e n o t e d b y the standard n u m e r i c a l signs followed b y a unit qualification (or s o m e t i m e s j u s t a n i m p l i e d unit). T h e t w o large units n e v e r o c c u r together. It is n o t clear exactly w h a t d e t e r m i n e s their usage.^ H e n c e , w e c a n s u m m a r i z e the Fara capacity systems as: lid - ga <—^— bariga <—^— ban <—^-^— sila and gur - mah <—^— bariga <—^— ban

— sila.

F o r our p u r p o s e s , the m o s t important of the capacity notations are t h o s e quantities of ban. A m o u n t s fi"om l ( b â n ) to 5(bân) w e r e written as follows:

for

T h i s s y s t e m d o e s n o t lend itself to the r e c o r d i n g o f arbittary quantities o f b a n . T h e abstiact sexagesimal p l a c e value s y s t e m characteristic o f later p e r i o d s w a s still several h u n d r e d years in the fiiture. T h e r e w a s not yet the c o m p l e t e separation of quantity a n d quality seen in abstiact n u m b e r systems. All calculations were p e r f o r m e d within the particular metiological systems relevant to the items b e i n g r e c k o n e d with. C o m p u t a t i o n a l linkages b e t w e e n different systems a n d c o n v e r s i o n s

' POWELL (1990), p. 495 opts for a Fara sila of approximately 1 litre, basing his estimate on comparison with other periods, sowing rates and the intemal sttucture of the mettological system. Others have argued for smaller sizes. MARTIN (1988), p. 47 suggests a sila of approximately 0.83 litre and Pomponio and Visicato ( E D A T S , p. 32, n.4, including references to the literature), after reviewing various suggestions conclude that the Fara sila was only about 0.5 litre. However, their argument is partly based upon the ration disbursement of TS§ 50, which is not an economic document but a mathematical exercise, as discussed below. Arguing for absolute capacity sizes from rations, especially when comparing across periods, is very difficult as it is not at all clear that rations were always intended to provide the same level of support and sustenance. Nor is it certain how many individuals they were meant to support, especially when the rations were being given to higher-ranking officials. ^ POWELL (1990), p. 485 has suggested that the lid-ga corresponds to the amount of seed needed for 1 bur of land. He also notes that this unit first appears in the Fara tablets and comments on its success, 'this gur (i.e., lid-ga) was the most widely used large capacity unit down into the Akkad period, when it was replaced by the Akkad gur' (of 5 bariga). Pomponio and Visicato ( E D A T S , p. 183), on the other hand, claim that the lid-ga was 'generally limited to texts conceming occasional allocations and to those regarding sales.' They continue, 'it should be added that its use seems to be more ancient than that of the gur-mah which, at the time of our texts, was replacing it, at least in the practice of the central administration'.

Ration Computations at Fara: Multiplication or Repeated Addition?

241

b e t w e e n t h e m w e r e difficuh a n d important p r o b l e m s that scribes in the b u r e a u c r a c y h a d to h a v e skill in solving o n a n e v e r y d a y b a s i s .

Repeated Addition in Place of Multiplication T h e p r o b l e m on T S § 81 required finding the total quantity of flour d i s b u r s e d w h e n 4 0 p e o p l e r e c e i v e d 2 ( b a n ) each. Interpreting this as a m o d e m multiplication exercise necessitated the intermediate quantity 8 0 ( b â n ) , which, as w e s a w a b o v e , w a s a quantity i m p o s s i b l e to write in Fara capacity notation. T h e n e e d to write s u c h notation o n l y arises b y considering the p r o b l e m as o n e of finding the total quantity o f rations as a multiplication p r o b l e m . If the p r o b l e m is c o n c e i v e d o f as a p r o b l e m o f r e p e a t e d addition, the m e a s i n i n g out o f 2 ( b â n ) 4 0 times, then there is n o necessity ever to write arbitiary quantities o f b a n . T h e same result could b e a c h i e v e d b y a simple series o f c o r r e s p o n d e n c e s , a r r a n g e d in a table, or list, a n d requiring only simple addition within the g i v e n metrological system. In the case o f this particular calculation, the scribe could h a v e c o n s t m c t e d something similar to the following table: 2(bân) 4(bân) 1 (bariga) 2(bariga) 3 (bariga) 1 lid-ga 2 lid-ga 3 lid-ga

<"> <—>

<—> <—^

Ison 2 sons 3 sons 6 sons 9 sons 12 sons 2 4 sons 3 6 sons

* *

*

T h e n 4 0 = 36 + 3 + 1 sons r e c e i v e d 3 lid-ga, 1 (bariga), 2(bân) total flour. T h e c o m p u t a t i o n follows easily fi-om the c o r r e s p o n d e n c e s . T h e r e is n o n e e d at a n y p o i n t to think of the impossible 80 b a n .

Repeated Addition in Place of Division T h e p r o b l e m on T S S 81 is of the form, 'if x w o r k e r s receive y rations, w h a t is the total d i s b u r s e d ? ' Scribes m u s t h a v e m e t sùnilar p r o b l e m s frequently in actual administrative p r a c t i c e as well as in school exercises. H o w e v e r , the m v e r s e p r o b l e m ' G i v e n a total d i s b i n s e m e n t and the m d i v i d u a l rations, d e t e r m i n e the n u m b e r o f w o r k e r s ' , is clearly a n exercise. E x a c t l y such a p r o b l e m occurs o n t w o tablets, T S â 5 0 a n d T S § 6 7 1 . Since their p u b l i c a t i o n b y Jestin, these t w o versions o f the s a m e p r o b l e m h a v e attracted a considerable quantity of interest a n d d e b a t e . A s only the statement a n d solution of the p r o b l e m are written o n the tablets, the focus o f the v a r i o u s analyses h a s b e e n to explain h o w the c o m p u t a t i o n s m i g h t h a v e b e e n performed. T h e interest arises b e c a u s e , if the standard ration p r o b l e m is v i e w e d as o n e of multiplication, t h e n the inverse p r o b l e m m u s t b e o n e of division. T h e tablets T S § 5 0 a n d T S S 6 7 1 h a v e b e e n interpreted as p r o v i d i n g the m o s t compelling e v i d e n c e for division in mid-third milleimium mathematics.

242

DJ. Melville

T S S 50 r e a d s as follows. Transliteration^ O I 1 gur? §e sila 7 1 1Ù §u ba-ti lù-bi O II 4(§aru) 5(§ar) 4(ge§u) 2(ges) 5(u) 1 It sila su-tag4 3

Translation A granary of barley 7 sila 1 man received its men 4(saru) 5(sar) 4(gesu) 2(ges) 5(u) 1 3 sila of grain remains

O n l y the o b v e r s e is inscribed. T S S 6 7 1 , in contrast, b e g i n s o n the reverse, a c c o r d i n g to Jestin. Transliteration Translation R. se sila 7 gur? A granary of grain, 7 sila 1 1Ù §u ba-ti 1 man received O. guruS workers 4(§aru)'° 4(saru) 5(sar) 5(§ar) 3(geSu) 6(ge§) 3(gesu) 6(ges) In his discussion o f the m e t r o l o g y o f Fara, L a m b e r t called T S S 5 0 ' u n r e m a r q u a b l e p r o b l è m e de simple arithmétique, m a i s qui m o n t r e à quelle virtuosité les maîtres d ' é c o l e étaient p a r v e n u s . ' ' ' H o w e v e r , b e y o n d s h o w i n g that the result w a s correct, L a m b e r t did not offer any interpretation as to h o w the c o m p u t a t i o n s m i g h t h a v e b e e n performed. T h i s challenge was first taken u p b y G e n e v i è v e Guitel.'^ N o t i n g that the g r a n a r y contained 1 152 0 0 0 sila, Guitel o b s e r v e d that in the Fara p e r i o d discrete system, 1,152,000 = 32 X 36000 c o u l d b e written as 3 2 ( s a r u ) . Guitel did not explicitly state that the units were sila, b u t she c o n c e i v e d o f the c o m p u t a t i o n as b e i n g p e r f o r m e d in a n abstract system after a c o n v e r s i o n into a b a s e imit o f sila. W h i l e this is certainly h o w the p r o b l e m m a y h a v e b e e n tackled in O l d B a b y l o n i a n a n d m o d e m times, w e h a v e argued for a greater sensitivity to the ancient m e t i o l o g i c a l systems. T h e r e is n o evidence at F a r a for a n y o n e ever writing d o w n a large quantity o f sila in the discrete system. O n c e Guitel h a d the total quantity written as 3 2 ( s a m ) sila, she p r o c e e d e d to divide b y 7, a ' d i v i s i o n frès simple q u ' i l exécutait peut-êtte d e tête, o u m ê m e p a r simple sousttactions successives','^ thus obtaining a result of 4 ( s a m ) [ m e n ] w i t h a r e m a i n d e r o f 4 ( s a m ) [sila]. A s Guitel w a s w o r k i n g in a n abstiact notational s y s t e m she feh n o n e e d to specify the appropriate units. F o r the n e x t step, Guitel c o m m e n t e d , 'il est naturel d e c o n v e r t n dans l'unité i m m é d i a t e m e n t inférieure.' W e h a v e b e e n arguing precisely that this c o n v e r s i o n is not natural at all. P r o c e e d i n g w i t h the conversion, Guitel wrote 4 0 (§ar) [sila] a n d r e p e a t e d the simple division '

The transliteration is essentially that of HOYRUP ( 1 9 8 2 ) .

'°

The 4 (§aru) is written in area notation by mistake. The §aru for the discrete system is

rare, and may have been less familiar to the student than the area §aru. "

LAMBERT ( 1 9 5 3 ) , p. 2 0 6 . GUITEL ( 1 9 6 3 ) , pp.

145-150.

GUITEL ( 1 9 6 3 ) , p. 1 4 6 .

Ration Computations at Fara: Multiplication or Repeated Addition?

243

Step. T h e p r o c e s s t h e n naturally c o n t m u e d with the c o n v e r s i o n s t a k i n g p l a c e at e a c h step. A t n o t i m e d o gur, b a n , or bariga m a k e an a p p e a r a n c e . S i n c e the result r e c o r d e d o n the tablet is correct, a n d Guitel w a s following a c o r r e c t arithmetical p r o c e d i n e , she n o t e d n o discrepancy. W h i l e the c o m p u t a t i o n s o n T S § 5 0 are correct, P o w e l l realized the i m p o r t a n c e o f T S â 6 7 1 , w h e r e the s a m e p r o b l e m is attempted, b u t the c o m p u t a t i o n s are incorrectly carried out. E r r o r s m a y tell u s far m o r e a b o u t h o w c o m p u t a t i o n s m a y h a v e b e e n p e r f o r m e d t h a n c o r r e c t results. D r a w i n g attention t o T S S 6 7 1 , P o w e l l c o n m i e n t e d that it w a s ' w r i t t e n b y a b u n g l e r w h o did n o t k n o w the front from the b a c k of his tablet, did n o t k n o w the difference b e t w e e n standard n u m e r i c a l n o t a t i o n a n d area notation, a n d s u c c e e d e d in m a k i n g half a d o z e n writing errors in as m a n y lines.'''* T a k i n g issue with G u i t e l ' s interpretation, P o w e l l argued, 'she p o s i t s a m e t h o d o f solution " a b s o l u t e l y a n a l o g o u s to m o d e m p r a c t i c e . " It is, h o w e v e r , p r e c i s e l y this close c o r r e s p o n d e n c e to m o d e m practice that m a k e s the solution suspect. If m o d e m long division h a d b e e n u s e d in the F a r a p e r i o d , it is virtually certain that it w o u l d a p p e a r s o m e w h e r e in O l d B a b y l o n i a n inathematical texts, w h i c h is n o t the c a s e . ' ' ^ T h i s is a significant criticism a n d quite apart from the mefrological n o t a t i o n a l p o i n t s we commented on above. P o w e l l w a s m u c h s t m c k b y the fact that e a c h m a n r e c e i v e d 7 sila of grain. U s i n g s t a n d a r d s e x a g e s i m a l notation, P o w e l l w r o t e , ' A silo (gum) in this p e r i o d c o n t a i n e d 4 0 , 0 g u r , ' ^ e a c h o f w h i c h c o n t a i n e d 8,0 sila. T h u s , the n u m b e r b e i n g " d i v i d e d " b y 7 is 5,20,0,0. S e v e n is the only integer b e t w e e n 1 a n d 10 that will n o t p r o d u c e a n e v e n result, therefore, g i v e n this fact and the fact that t w o exercises dealing w i t h the s a m e p r o b l e m h a v e survived, the c h o i c e of 7 c a n hardly b e c o i n c i d e n t a l . ' ' ^ W h i l e the m e r e fact that 7 p r o d u c e s a c o n p l i c a t e d a n s w e r w o u l d s e e m sufficient m o t i v a t i o n for its c h o i c e in a n arithmetic p r o b l e m , P o w e l l c o n t i n u e d that 7 h a s ' n o material e x p l a n a t i o n ' a n d n e v e r a p p e a r s as a divisor in third m i l l e i m i u m e c o n o m i c d o c u m e n t s . T o P o w e l l , then, 'the c h o i c e of 7 c a n h a r d l y b e m o t i v a t e d b y a n y other cause than that it is an irregular n u m b e r . M o r e o v e r , the t w o different a n s w e r s suggest that the object is an exercise in using the reciprocal o f a n irregular n u m b e r . ' ' ^ In a t t e m p t i n g to v i e w this p r o b l e m as a n ' a n t e c e d e n t ' o f O l d B a b y l o n i a n m a t h e m a t i c s a n d a c c e p t i n g it as a division p r o b l e m P o w e l l is forced to a s s u m e that it w o u l d b e c o m p u t e d in the s a m e w a y a n O l d B a b y l o n i a n p r o b l e m in absfract s e x a g e s i m a l p l a c e v a l u e n o t a t i o n w o u l d h a v e b e e n , n a m e l y , b y multiplication b y the reciprocal. F o r the correct calculation, P o w e l l p r o p o s e d that the quantity 5,20,0,0 sila b e multiplied b y the r e c i p r o c a l of 7, a p p r o x i m a t e d to four s e x a g e s i m a l p l a c e s as 0 ; 8 , 3 4 , 1 7 , 8 . T h e fractions o f m e n are d i s c a r d e d a n d the result c o m p a r e d to the original quantity t o find the r e m a i n d e r o f 3 sila. P o w e l l a p p e a r s to h a v e b e l i e v e d that the c o m p u t a t i o n s w e r e all carried o u t in a n absfract s e x a g e s i m a l s y s t e m with n o

POWELL (1976), p. 432. "

POWELL (1976), p. 432.

The identification of the size of the granary or silo derives solely from the assumption that the computations are correctly carried out on TS§ 50. There is no independent archaeological or textual evidence for such a metrological unit during the Fara period. "

PowELL(1976>,p.433. POWELL (1976), p. 433.

244

D.J. Melville

reference t o the u n d e r l y i n g m e t r o l o g y . T o e x p l a i n the incorrect c a l c u l a t i o n in T S § 6 7 1 , P o w e l l s u p p o s e d the student u s e d the a p p r o x i m a t i o n 0;8,33 as the r e c i p r o c a l of 7. R e c o g n i z i n g a w e a k n e s s in his interpretation, P o w e l l c o m m e n t e d , ' h o w the p u p i l arrived at the c h o i c e of 0;8,33 for the reciprocal of 7 , 1 h a v e n o i d e a ' . ' ^ W h i l e P o w e l l d i s c u s s e d the tablets T S § 5 0 a n d 671 only briefly in the c o n t e x t of a m o r e general investigation of third m i l l e i m i u m m a t h e m a t i c s , H 0 Y R U P ( 1 9 8 2 ) subjected t h e m to a detailed analysis. H e r e p u b l i s h e d the tablets in p h o t o g r a p h a n d c o p y , correcting s o m e of J e s t i n ' s c o p y (they w e r e n o t p u b l i s h e d in p h o t o g r a p h in T S â ) , as well as p r o v i d i n g i m p r o v e d transliteration a n d translation. H e d i s c u s s e d the m e t r o l o g y a n d the n u m e r a t i o n systems before t u m i n g to a n analysis of the m a t h e m a t i c s , w h i c h he o p e n e d b y c o m m e n t i n g that, 'the p r o b l e m dealt w i t h o n the tablets is therefore a formal division problem' (italics in the original).^° T h i s is the p o i n t w e w i s h to c h a l l e n g e . After discussing G u i t e l ' s a n d P o w e l l ' s interpretation of the p r o b l e m , H o y r u p p r o p o s e d a n alternative p r o c e d i n e , w h i c h , as h e suggested, n o t o n l y a c c o m m o d a t e s the error o n T S S 6 7 1 , b u t ' i m p h e s that this error is o n e of the m o s t o b v i o u s o f all errors'.^* H o y m p w a s sensitive to the u n d e r l y i n g m e t r o l o g y o f the p r o b l e m a n d u s e d that as his g u i d e . Briefly, H o y r u p ' s interpretation is as follows. T h e student c o n c e i v e d of the g r a n a r y as 4 ( g e s u ) gur a n d d i v i d e d b y 7 to o b t a i n 5,42 a n d a r e m a i n d e r o f 6 gur. Importantly, H o y r u p c l a i m e d that this d i v i s i o n w a s a natural p r o c e d u r e b e c a u s e quantities o f gur w e r e r e c o r d e d in the s t a n d a r d discrete system, while quantities less t h a n a gur w e r e r e c o r d e d using the different m e t r o l o g i c a l notation. N e x t , if the resultant 5,42 (neglecting the r e m a i n d e r ) w a s m u l t i p l i e d b y 8,0, the n u m b e r o f sila in a gur, t h e n the student w o u l d o b t a i n 4 5 , 3 6 , 0 , p r e c i s e l y the result r e c o r d e d o n T S § 6 7 1 . In fact, o n e could c o n s i d e r this tablet as n o t s o m u c h an error o n the part o f the student, b u t rather a n i n c o m p l e t e result. A l t h o u g h s o m e w h a t h a m p e r e d b y a n insistence o n using sexagesimal notation t h r o u g h o u t , H o y r u p g a v e a detailed a c c o u n t o f h o w a student m i g h t a p p r o a c h the p r o b l e m a n d c o n c e i v e o f the intermediate steps. Still, his analysis left the result o f the first d i v i s i o n in a sort o f c o n c e p t u a l l i m b o . If 4 0 , 0 gur are d i v i d e d b y 7, the r e s u h is 5,42, b u t 5,42 w h a t ? Is this n u m b e r s u p p o s e d t o represent m e n , or g r a i n ? Since w e are a t t e m p t i n g to c o m p u t e a n u m b e r of m e n a n d are a b o u t to multiply this n u m b e r b y 8,0, w e s h o u l d s u p p o s e it to m e a n m e n . P e r h a p s the easiest w a y to m a k e this explicit is t o a s s u m e the student is first m a k i n g the a s s u n p t i o n that e a c h m a n r e c e i v e s 7 g u r o f grain. T h e n in the s e c o n d step, h e u s e s the fact that there are 8,0 sila in a gur to d e t e r m i n e the correct (partial) n u m b e r o f m e n . D e s p i t e H o y r u p ' s p r o t e s t a t i o n that, ' w h a t s e n s e w o u l d b e left to s u c h [mefrological] systems if a unit c o u l d n o t b e c o n v e r t e d to a smaller one?'^^ n o w h e r e is there a n y e v i d e n c e that a scribe in F a r a t h o u g h t in the sexagesimally i m p l i e d t e r m s o f 8,0 sila to a gur. T o m a k e this a s s u m p t i o n is to i m p o s e a n absfract a p p r o a c h to n u m e r a t i o n that is b e l i e d b y the mefrology. H o y r u p did n o t see a n y e v i d e n c e for p l a c e v a l u e in the p r o b l e m , b u t did see a n e e d for division a n d a relatively absfract form of multiplication. A n y a l t e m a t i v e analysis in t e r m s of c o r r e s p o n d e n c e s m u s t also suggest a p l a u s i b l e m e t h o d for a

POWELL (1976), p. 433.

2'

H0YRUP(1982),p.23. H0YRUP(1982),p. 27. H0YRUP(1982),p. 29.

Ration Computations at Fara: Multiplication or Repeated Addition?

245

scribe to obtain the incorrect result o n T S â 6 7 1 , and also deal with the c h o i c e o f 7 as ' d i v i s o r ' . First, it is w o r t h considering h o w the p r o b l e m w o u l d l o o k if a different ration w e r e chosen. S u p p o s e e a c h m a n were to receive 8 sila, a n d c o n s i d e r the resulting c o r r e s p o n d e n c e s . W e take the structine of the p r o b l e m as our g u i d e . T h a t is, w e are g i v e n a particular a m o u n t of g r a i n a n d are to d e t e r m i n e the n i u n b e r of m e n that c a n r e c e i v e the correct ration, with the r e m a i n d e r of so m a n y sila. W e construct the table to reflect this p a t t e m . 8 sila l(bân) 2(bân) 3(bân) 4(bân) 5(bân) 1 (bariga) 2(bariga) 3 (bariga) 4(bariga) 5 (bariga) 6(bariga) 7(bariga) 1 gur

<">
1 worker 1 worker, r e m a i n d e r 2 sila 2 w o r k e r s , r e m a i n d e r 4 sila 3 w o r k e r s , r e m a i n d e r 6 sila 5 workers 6 w o r k e r s , r e m a i n d e r 2 sila 7 w o r k e r s , r e m a i n d e r 4 sila 15 w o r k e r s 22 w o r k e r s , r e m a i n d e r 4 sila 3 0 workers 37 w o r k e r s , r e m a i n d e r 4 sila 45 workers 52 w o r k e r s , r e m a i n d e r 4 sila 60 workers

Since there w e r e , as H e y m p has it, 8,0 sila in a gur, if o n e w o r k e r r e c e i v e d 8 sila, then 6 0 m e n w o u l d receive 1 gur. N o w , quantities a b o v e a gur, a n d quantities of m e n are b o t h m e a s u r e d in the same discrete notation, a n d the parallel n o t a t i o n w o u l d m a k e the c o m p u t a t i o n s of higher quantities particularly simple. A g r a n a r y o f 4 ( g e s u ) gur w o u l d exactly feed 4 ( s a m ) w o r k e r s . S u c h a simple p r o b l e m w o u l d p r o v i d e practice in the n u m e r a t i o n s y s t e m b u t little else. T o m a k e the p r o b l e m m o r e challenging, w e m u s t c h a n g e the ration. A s n o t e d a b o v e , P o w e l l p o i n t e d out that 7 is the only ration b e t w e e n 1 a n d 10 sila that d o e s not give a simple answer. T h e fact that 7 is not a sexagesimally regular n u m b e r is related, b u t d o e s not m e a n that the p r o b l e m w a s intended as a n exercise in sexagesimal a p p r o x i m a t i o n o f irregular reciprocals. C o n s t m c t i n g a table of e q u i v a l e n c e s for a ration of 7 sila is o n l y a little m o r e c o m p l i c a t e d than that for a ration of 8 sila. 7 sila l(bàn) 2(bân)

1 worker 1 worker, r e m a i n d e r 3 sila 2 w o r k e r s , r e m a i n d e r 6 sila

It has been almost universally assumed by commentators that TSS 50 and 671 are exercise texts. Pomponio and Visicato, in a discussion of the size of the Fara sila, note, 'there are two school texts from Fara which indicate that 7 sila was the daily barley ration received by the hypothetical beneficiaries' (EDATS, p. 32, n.4). Visicato later goes further, suggesting that, 'TS§ 50 and 671, hitherto considered to be only scholastic texts, could have quite a different meaning. In fact, what has been recorded in the two texts appears to be a realistic and not a theoretical evaluation of the amount of food needed for the sustenance not only of the military contingent engaged in war but also of the population of the city' (VISICATO (1995), p. 69, n.24). However, ft is not clear fix)m Visicato'^ discussion w h y » f a t i © f t ^ 7 ^ a should have been chosen as appropriate for feeding either civilians or soldiers.

246

D.J. MELVILLE

3(bân) 4(bân) 5(bân) 1 (bariga) 2(bariga) 3 (bariga) 4(bariga) 5 (bariga) 6(bariga) 7(bariga) 1 gur

4 w o r k e r s , r e m a i n d e r 2 sila 5 w o r k e r s , r e m a i n d e r 5 sila 7 w o r k e r s , r e m a i n d e r 1 sila 8 w o r k e r s , r e m a i n d e r 4 sila 17 w o r k e r s , r e m a i n d e r 1 sila 25 w o r k e r s , r e m a i n d e r 5 sila 3 4 w o r k e r s , r e m a i n d e r 2 sila 4 2 w o r k e r s , r e m a i n d e r 6 sila 51 w o r k e r s , r e m a i n d e r 3 sila 60 workers 6 8 w o r k e r s , r e m a i n d e r 4 sila

It is certainly also possible that the scribes u s e d the basic metrological relationship that a ration of x sila to 1 w o r k e r implied a ration o f x b a r i g a to 6 0 w o r k e r s . Continuing in this vein, it w o u l d b e relatively straightforward for a scribe to determine the n u m b e r o f w o r k e r s c o r r e s p o n d i n g to 10 gin, to 6 0 gur, to 6 0 0 gur, a n d finally to 4 x 6 0 0 , or 4 ( g e s u ) gur. In fact, once b o t h sides o f the c o r r e s p o n d e n c e s are in discrete notation, it is a simple matter to m o v e u p e a c h stage o f the factor d i a g r a m without p e r f o r m i n g all the intermediate c o m p u t a t i o n s that are r e q u i r e d w h e n establishing the c o r r e s p o n d e n c e s b e t w e e n different m e t r o l o g i c a l systems. It is instructive to p e r f o r m these calculations in the F a r a notation. If all the arithmetic is c o m p l e t e d correctly, the correct solution will o f c o u r s e b e obtained. B u t w h a t if the arithmetic is not c o m p l e t e d correctly? T h a t is, d o e s this p r o c e d u r e a l l o w a p l a u s i b l e m e a n s b y w h i c h the incorrect solution given o n T S S 6 7 1 could b e o b t a m e d ? O u r hypothetical b u n g l e r n e e d s to m a k e o n e relatively simple m i s t a k e . H e h a s 1 gur c o r r e s p o n d i n g to l ( g e s ) 8 m e n , with 4 sila r e m a i n i n g . T h e n l ( u ) g u r c o r r e s p o n d to l ( g e s u ) l ( g e s ) 2(u) m e n a n d from the r e m a i n d e r o f 4 sila, h e a d d s 4 m e n . T h e rest of the c o m p u t a t i o n s are then correct. 1 gur l(u)gur l ( g e s ) gur l ( g e s u ) gur 4 (gesù) gur

l ( g e s ) 8 w o r k e r s , r e m a i n d e r 4 sila l ( g e s u ) l ( g e s ) 2(u) 4 w o r k e r s l ( s a r ) 8(ges) 2(u) 4 w o r k e r s l(§aru) l ( s a r ) 2(ge§u) 4(ges) w o r k e r s 4(§aru) 5(sar) 3(gesu) 6(ges) w o r k e r s

Part of the force of H o y r u p ' s interpretation of the e r r o n e o u s c o m p u t a t i o n w a s that all it required w a s neglect o f a remainder. H e r e , too, all that is r e q u i r e d is a m i s h a n d l i n g of a r e m a i n d e r . H e n c e , it is entfrely p o s s i b l e for a scribe to o b t a i n b o t h the correct a n d incorrect answers using only the tools o f lists or tables a n d addition. T o the m o d e m eye, the p r o b l e m stated in T S S 5 0 a n d 6 7 1 is a division p r o b l e m . T h e c o m p u t a t i o n a l techniques p o s i t e d for Fara scribes b y Guitel, P o w e l l a n d H o y m p all flow from this basic a s s u n ^ t i o n . T h e ' t o p - d o w n ' division p r o c e d u r e s d o n o t fit comfortably with F a r a mefrological notation, w h e r e a s a ' b o t t o m - u p ' s y s t e m of correspondences does.

Ration Computations T h e tablets T S § 8 1 , 5 0 a n d 671 carry exercises. H o w e v e r , t h o s e exercises w e r e similar to the routine c o m p u t a t i o n s required of scribes in the p e r f o r m a n c e o f their

Ration Computations at Fara: Multiplication or Repeated Addition?

247

daily duties. In this section, for c o m p a r i s o n , w e c o n s i d e r the c o m p u t a t i o n s i n v o l v e d in s o m e e c o n o m i c dociunents. N i u n e r o u s F a r a tablets r e c o r d d i s b u r s e m e n t s o f rations to w o r k e r s and officials. T h e s e rations usually consist of barley, or occasionally of floin.^'* I n m o s t cases the tablets r e c o r d issuing fairly standard quantities of grain to n a m e d officials. H e n c e , the standard form o f the tablet is a list of 'quantity, n a m e ( p r o f e s s i o n ) ' , with s o m e t i m e s a total at the e n d of the list. F o r e x a m p l e , W F 7 3 begms^^ 1 (bariga) 1 (bariga) 1 (bariga) 2(bariga) [ujnug''

2(bân) 2(bân) 2(bân) 4(bân)

inuig nin-me-duio me-é-zi-ta bilx-â-nu-kùs

and so o n t h r o u g h 16 entries. T h e r e m a y b e over 2 0 0 such entries o n a single tablet^^ a n d the totals w h e r e g i v e n are usually correct or v e r y close. P o w e l l claimed, ' S m n e r i a n b o o k k e e p m g reveals a rigorous attention to detail a n d carefiil h a n d l i n g of figures,' a n d noted, ' m y o w n e x p e r i e n c e in c h e c k i n g h i m d r e d s o f calculations in the P r e s a r g o n i c p e r i o d is that a c c u r a c y is very high, p e r h a p s 9 0 % or better'.^^ W e d o not k n o w h o w these totals w e r e c o m p u t e d , b u t it is clear that the scribes w e r e highly c o m p e t e n t at addition. R a t i o n d i s b u r s e m e n t s w e r e designed for bureaucratic c o n v e n i e n c e rather than necessarily b e i n g s i m p l y b a s e d o n w h a t a p e r s o n (or g r o u p ) w o u l d n e e d for sustenance. S t a n d a r d allocations for officials of v a r i o u s ranks i n c l u d e d l ( b a r i g a ) , 2 ( b â n ) (as in the e x a m p l e a b o v e ) , so that 3 officials w o u l d r e c e i v e 1 lid-ga, or 6 w o u l d receive 1 gur. O t h e r allocations included 2(bariga), 4 ( b â n ) (so that 3 officials w o u l d receive 1 g u r ) , or Vz gur, or multiples o f a gur. T h e s e are p r e s m n a b l y m o n t h l y rations, a l t h o u g h the p e r i o d the rations are m e a n t to c o v e r is n e v e r stated. If so, they are certainly not simple multiples of a daily ration. T h e rations are d e s i g n e d so that easy multiples of t h e m fit the standard large units o f lid-ga a n d gur.^^ T h a t is, the rations are c h o s e n to facilitate c o m p u t a t i o n . It is n o c o i n c i d e n c e that the c o r r e s p o n d e n c e s foimd m the ration tablets a n d exercises are so simple. M o s t of the F a r a b a r l e y tablets r e c o r d quantities of grain g i v e n to individual officials of various ranks. H o w e v e r , there are also several tablets d e a l i n g with g r o u p s of i m n a m e d gurus (workers). It m a y b e that s o m e of these w o r k e r s w e r e b e i n g a s s e m b l e d for a f o r t h c o m m g conflict. M a r t i n referred to these tablets as ' g u r u s m u s t e r lists', a n d Visicato called t h e m ' r e c m i t m e n t texts'.^^ A p a r t from s o m e

There are 109 such tablets in Visicato's typology (EDAT§, p. 3). We follow Pomponio and Visicato's transliteration (EDATâ, p. 56). WF 77 has 219 entties. POWELL (1972), p.

181.

Comparing the ration quantities in WF 98 and WF 99, Visicato suggests that 'it is probable that in the case of [WF 99] there a weekly allocation was registered while in [WF 98] this allocation was monthly' (VISICATO (1995), p. 50, n. 10). In the absence of any other third-millennium evidence for a four-week month, this suggestion may be discarded. ^ MARTIN ( 1988), p. 88 and VISICATO (1995), chapter 1. According to MARTIN (1988), p. 89 93% of all barley ration tablets were found in the 'Tablet House' XVh.

248

DJ. Melville

general s u m m a r y tablets, these gurus tablets follow similar formats to other e c o n o m i c d o c u m e n t s . T h a t is, each entry r e c o r d s the n u m b e r o f g u r u s assigned to specific n a m e d officials. F r o m a bureaucratic p e r s p e c t i v e , there is n o difference b e t w e e n assignment o f w o r k e r s and assignment o f b a r l e y or d o n k e y s . B u t w o r k e r s d o h a v e to eat. A m o n g these gurus tablets, there are t w o , W F 9 8 a n d W F 9 9 , found near e a c h other in a building o n the very n o r t h e m e d g e of the site.^° T h e contents of the two tablets are v e r y similar: b o t h record a n u m b e r o f gurus either b e l o n g i n g to, or assigned to, v a r i o u s officials. W F 9 8 has 22 entries a n d states the (correct) total of 82 gurus. W F 9 9 h a s 2 0 entries with a total of 7 9 , written 8 0 la 1. In B e i m e l ' s transliteration this total is correct; Visicato gives slightly different n u m b e r s for s o m e o f the entries a n d so records the total as incorrect.^' I h a v e not b e e n able to collate the original tablet, b u t j u d g i n g from the p h o t o g r a p h s o n the V o r d e r a s i a t i s c h e s M u s e u m website,^^ I see n o r e a s o n to d o u b t D e i m e l ' s readings o f the n u m b e r s although his r e a d i n g s o f other signs are not always the same as m o d e m a c c e p t e d values. W h a t m a k e s these t w o tablets stand out is that after the total o f w o r k e r s is g i v e n there is a notation that e a c h w o r k e r received a certain quantity of flour. It is u n u s u a l for w o r k e r s to receive flour rather than grain a n d n o explanation is g i v e n as to w h y the w o r k e r s receive flour. In W F 9 8 , each w o r k e r r e c e i v e d 2(bân) of flour; in W F 9 9 the quantity was 5 sila. Following is the (correct) total quantity o f flour disbursed. A s in the case of T S § 8 1 , w e are p r e s e n t e d with w h a t to m o d e m eyes is a 'multiplication' p r o b l e m . Let u s consider the case of W F 9 9 first. T h e m o d e m (counting O l d B a b y l o n i a n as m o d e m ) p r o c e d u r e w o u l d b e to note that e a c h of 7 9 w o r k e r s r e c e i v e s 5 sila of flour for a total o f 7 9 x 5 = 3 9 5 sila, w h i c h c o u l d then b e c o n v e r t e d into the appropriate units as 6(bariga) 3(bân) 5 sila. W h i l e the O l d B a b y l o n i a n scribe w o u l d p e r f o r m the multiplication in the abstract sexagesimal system, w e again confi-ont the question o f h o w the scribe in F a r a w o u l d h a v e written, or c o n c e i v e d , the intermediate r e s u h o f 3 9 5 sila. Since quantities o f sila are written in the discrete system, the quantity could h a v e b e e n written as 6(ges) 3(u) 5 sila. H o w e v e r , such a notation, r e c o r d i n g a quantity o f sila greater t h a n 9, is n o w h e r e attested in the c o r p u s of Fara tablets a n d o u r a r g u m e n t is that such a notation is n e v e r written. T h e n e e d to write such notation only arises b y considering the p r o b l e m of finding the total quantity o f rations as a multiplication p r o b l e m . If the p r o b l e m is c o n c e i v e d as a p r o b l e m o f repeated additions, the m e a s u r i n g out o f 5 sila 7 9 times, t h e n there is n o necessity o f ever writing arbitrarily large quantities o f sila.^^ A s in the exercise tablets discussed above, the same result could b e a c h i e v e d b y a simple WF98 = VAT 12455 has findspot F.1702 in location XVIIc; WF 99 = VAT 12537 has findspot F.1669 in the same location. See MARTIN (1988), p. 98 and KREBERNIK (1998), p. 390. VISICATO (1995), p. 10-11. "

The URL

for WF 99 is < h t t p : / / c d l i . m p i w g - b e r l i n . m p g . d e / D l W A M / F A R A / H T M L l / P 0 1 1 0 5 7 . H T M > .

The evolution of technical mathematical vocabulary and the shades of meaning that lie under different Sumerian and Akkadian terms that we translate as synonyms is a field that awaits detailed study. With regard to the various concepts we lump together as 'multiplication', the discussion in HGYRUP (1990), p. 4 6 - 4 9 provides a valuable start. Although Hoyrup discusses etymology, he does not indicate development in usage before the Old Babylonian period.

Ration Computations at Fara: Multiplication or Repeated Addition?

249

series o f c o r r e s p o n d e n c e s . In the case of this particular calculation, the scribe c o u l d have constructed s o m e t h i n g similar to the following table. 5 sila l(bân) 2(bân) 3(bân) 4(bân) 5(bân) 1 (bariga) 2(bariga) 3 (bariga) 4(bariga) 5 (bariga) 6(bariga) 7(bariga) 1 gur

1 worker 2 workers 4 workers 6 workers 8 workers 10 w o r k e r s 12 workers 2 4 workers 3 6 workers 4 8 workers 60 workers 7 2 workers 84 w o r k e r s 9 6 workers

*

Since 7 9 w o r k e r s is 7 2 + 6 + 1 w o r k e r s , the required quantity of flour is 6(bariga) 3(bân) 5 sila, exactly as r e c o r d e d o n the tablet.^'* N e x t , w e c o n s i d e r W F 9 8 , w h i c h contains a similar c o m p u t a t i o n , b u t presents s o m e difficuhies in interpretation. A s m e n t i o n e d a b o v e , the total n u m b e r o f g m u s listed in W F 9 8 is 82 a n d each r e c e i v e d 2(bân) of floin. C o m p u t i n g the total disbursement via a multiplication w o u l d require the intermediate n o t a t i o n of 82 X 2 = 164 b a n before its conversion into correct metrological imits. H e r e w e confront the implausibility of such a c o m p u t a t i o n e v e n m o r e starkly t h a n in the previous p r o b l e m considered. T h e notation for quantities from one to five ban prohibits r e c o r d i n g a n y quantity of b a n greater than five. A table o f c o r r e s p o n d e n c e s natinally r e m o v e s this p r o b l e m . If 1 w o r k e r receives 2 ban of flour, then 2 w o r k e r s receive 4 b a n a n d 3 w o r k e r s receive 1 bariga. H e n c e a suitable table o f c o r r e s p o n d e n c e s could b e g i n as follows. 2(bân) 4(bân) 1 (bariga)

1 worker 2 workers 3 workers

B u t here s o m e t h i n g sfrange h a p p e n s . T h e scribe in this instance s e e m s to h a v e taken the b a r i g a as a m a x i m a l unit, in effect as a type o f gin, a n d r e c o r d e d the total using gur notation. T h a t is, the fmal total is written as 2 7 gur x u 2 ( b â n ) , w h e n clearly what is m e a n t is 2 7 bariga, 2 b a n . It is not obvious w h e t h e r w e are dealing here with a m o r e flexible n o t i o n of gur than w e h a d u n d e r s t o o d , or with a scribe struggling against the confines of his mefrological system.^^

It was in fact this tablet, WF 99, that LAMBERT (1953), p. 205 cited to prove the relative size of the bariga. " DEIMEL (1924), p. 14 considered this as notation representing a gur of 1 bariga, noting the occurrence of the sign also in WF 138. Visicato also discusses this point briefly {(1991), p. 54, n.3).

250

D.J. Melville

Conclusions In e a c h o f the e x a m p l e s o f ration c o m p u t a t i o n s w e h a v e studied, w h e t h e r direct c o m p u t a t i o n s involving a total quantity o f rations g i v e n t o a k n o w n n u m b e r o f recipients, o r t h e inverse exercises o f d e t e r m i n i n g t h e n u m b e r o f recipients w h o c o u l d receive distributions from a g i v e n quantity o f grain, w e h a v e s h o w n that the results c o u l d h a v e b e e n o b t a i n e d b y a simple p r o c e d u r e involving r e p e a t e d addition. It is entirely p o s s i b l e t o solve these p r o b l e m s w i t h o u t r e c o u r s e t o t h e m o d e m notions of multiplication a n d division. I n particular, o u r m o d e m p e r c e p t i o n o f the p r o b l e m in T S S 5 0 a n d 6 7 1 a s a division p r o b l e m that n e e d s t o b e solved from t h e t o p (the granary) d o w n ( t o the sila) necessitates positing arithmetical t e c h n i q u e s for w h i c h there is n o direct e v i d e n c e . If the p r o b l e m is v i e w e d from t h e b o t t o m u p , these t e c h n i q u e s are n o l o n g e r required. If w e a s s u m e that m e t r o l o g i c a l c o m p u t a t i o n s w e r e carried o u t in t h e correct notation, then certain intermediate r e s u h s r e q u i r e d b y a m o d e m v i e w p o i n t are i m p o s s i b l e t o write. T h e n o t a t i o n for b a n is c m c i a l in this r e g a r d . T h e s e p r o b l e m s are a v o i d e d in a b o t t o m u p a p p r o a c h t o the c o m p u t a t i o n s . W e a r e n o t asserting that ration c o m p u t a t i o n s necessarily w e r e p e r f o r m e d in exactly t h e w a y d e s c r i b e d h e r e . T h e e v i d e n c e is t o o w e a k to s u p p o r t s u c h a strong c o n c l u s i o n . H o w e v e r , b y d e m o n s t r a t i n g p r o c e d u r e s that d o n o t r e q u i r e k n o w l e d g e o f either multiplication o r division, the o n u s o f p r o o f p a s s e s t o those w h o w i s h to project b a c k t o earlier p e r i o d s t e c h n i q u e s u s e d in later o n e s . T h e r e is a natural t e n d e n c y a m o n g historians o f ideas to look for a n t e c e d e n t s , t o p u s h ideas a n d p r e l i m i n a r y attempts at ideas as far b a c k a s p o s s i b l e . T h i s a p p r o a c h l e a d s t o a narrative o f long, s l o w d e v e l o p m e n t o f ideas. A n a l t e m a t i v e v i e w is that s o m e p e r i o d s m a y e x p e r i e n c e r a p i d d e v e l o p m e n t for r e a s o n s that m a y o r m a y n o t b e clear. A c o m b i n a t i o n o f these v i e w s is that the p r o d u c t o f a long, s l o w e v o l u t i o n m a y e n a b l e r a p i d d e v e l o p m e n t o f a p p a r e n t l y unrelated n o t i o n s . Fara stands o n t h e c u s p o f the d e v e l o p m e n t o f the abstraction o f quantity a n d quality. T h i s d e v e l o p m e n t w a s a long, s l o w p r o c e s s that led ultimately t o the s e x a g e s i m a l p l a c e v a l u e s y s t e m that is clearly attested b y t h e U r III p e r i o d .

Abbreviations B â = VISICATO (1995) E D A T S = POMPONIO and VISICATO (1994) M C T = N E U G E B A U E R and SACHS (1945) M K T = NEUGEBAUER (1935-1937)

N T S â = JESTIN ( 1 9 5 7 ) SF =

DEIMEL(1923)

T S â = JESTIN ( 1 9 3 7 ) W F = DEIMEL(1924)

Text Concordance T h e tablets d i s c u s s e d o r m e n t i o n e d in this p a p e r are: T S § 5 0 , T S § 8 1 , T S § 5 5 4 , T S § 6 7 1 , W F 7 3 , W F 7 7 , W F 9 3 , W F 9 8 , W F 9 9 a n d W F 138. T h e S F a n d W F tablets p u b l i s h e d b y D e i m e l also h a v e V A T m u s e u m n u m b e r s . S o m e o f t h e F a r a tablets

Ration Computations at Fara: Multiplication or Repeated Addition?

251

h a v e also b e e n r e p u b l i s h e d in E D A T S a n d Bureaucracy of Suruppak. P u b l i c a t i o n niunbers for these t w o v o l u m e s are consistent and are d e n o t e d b e l o w b y B â . T a b l e t s B â 1-180 a p p e a r e d m E D A T â , tablets 181-217 were p u b l i s h e d m B § . T S § 50 TSâ81 T S § 554 T S § 671 W F 73 WF77 W F 93 W F 98 W F 99 W F 138

B à 203 VAT VAT VAT VAT VAT VAT

12431 9109 12590 12455 12537 9121

BS5 B § 19 BS200 B § 192 B § 182

References D E I M E L , A n t o n . 1923. Die Inschriften von Fara, II: Schultexte aus Fara (Wissenschaftliche Veròffentlichungen der D e u t s c h e n O r i e n t Gesellschaft 4 3 ) . Leipzig: H i m i c h s . —1924. Die Inschriften von Fara, III: Wirtschaftstexte aus Fara (Wissenschaftliche Veròffentlichungen der D e u t s c h e n Orient Gesellschaft 4 5 ) . Leipzig: Hinrichs. E D Z A R D , Dietz O . 1976. "Fara u n d A b u Salabikh. D i e ' W i r t s c h a f t s t e x t e ' " , Zeitschrift fiir Assyriologie und Vorderasiatische Archaologie 66: 1 5 6 - 1 9 5 . E N G L U N D , R o b e r t K . 1996. Proto-Cuneiform Texts from Diverse Collections (Materialien z u d e n fkiihen Schriftzeugnissen des V o r d e r e n Orients 4 ) , Berlin: Gebriider M a n n V e r l a g GUITEL, G e n e v i è v e . 1 9 6 3 , "Signification m a t h é m a t i q u e d ' u n e tablette s u m é r i e n n e " . Revue d'Assyriologie 57: 145-150. H0YRUP, Jens. 1982. "Investigations o f a n E a r l y S u m e r i a n D i v i s i o n P r o b l e m , c. 2 5 0 0 B . C . " . Historia Mathematica 9: 1 9 - 3 6 . — 1990. " A l g e b r a and N a ï v e G e o m e t r y . A n Investigation of S o m e A s p e c t s of O l d B a b y l o n i a n M a t h e m a t i c a l T h o u g h f . Altorientalische

Forschungen

17: 2 7 - 6 9 ,

262-354. JESTIN, R a y m o n d . 1937. Tablettes sumériennes de Suruppak conservées au Musée de Stamboul ( M é m o i r e s d e l'Institut français d ' a r c h e o l o g i e d e S t a m b o u l 3). Paris: E. d e B o c c a r d . — 1 9 5 7 . Nouvelles tablettes sumériennes de Suruppak au Musée d'Istanbul ( B i b l i o t h è q u e A r c h é o l o g i q u e et Historique d e l'Institut F r a n ç a i s d ' A r c h é o l o g i e de S t a m b o u l 2). Paris: A . M a i s o i m e u v e . K R A M E R , S a m u e l N . 1932. " N e w T a b l e t s from F a r a " . Journal of the American Oriental Society 5 2 : 1 1 0 - 1 3 2 . K R E B E R N I K , Manfred. 1998. " D i e T e x t e aus Fara u n d Tell A b u Salabildi". In: J o s e f B A U E R , R o b e r t K . E N G L U N D a n d M a n f r e d K R E B E R N I K (eds.), Mesopotamien: Spaturuk-Zeit und Friihdynastische Zeit (Orbis Biblicus et Orientalis 160/1): 2 3 7 - 4 2 7 . F r e i b u r g ( S c h w e i z ) : Uifrversitatsverlag. L A M B E R T , M a u r i c e . 1 9 5 3 . " L a vie é c o n o m i q u e à S h u r u p p a k " . Sumer 9: 1 9 8 - 2 1 3 .

252

D.J. Melville

M A R T I N , Harriet P . 1988. Fara: A reconstruction of Suruppak. B i r m i n g h a m : Chris Martin.

of the ancient Mesopotamian

city

— and P O M P O N I O , F r a n c e s c o ; VISICATO, G i u s e p p e , a n d W E S T E N H O L Z , A a g e . 2 0 0 1 . The Fara tablets in the University of Pennsylvania Museum of Archaeology and Anthropology, B e t h e s d a , M D : C D L Press. NEUGEBAUER, Otto. Julius Springer.

1935-1937.

Mathematische

Keilschrifttexte

I-III.

Berlin:

— and S A C H S , A b r a h a m . 1945. Mathematical Cuneiform Texts. N e w H a v e n : A m e r i c a n Oriental Society and A m e r i c a n S c h o o l s o f Oriental R e s e a r c h . N I S S E N , H a n s J.; D A M E R O W , Peter, and E N G L U N D , R o b e r t K. 1 9 9 3 . Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East. C h i c a g o : C h i c a g o University Press. POMPONIO, Francesco and VISICATO, Giuseppe. 1994. Early Dynastic administrative texts of Suruppak. N a p o l i : Istituto Universitario Orientale. POWELL, M a r v i n A . 1972. " S u m e r i a n A r e a M e a s u r e s and the A l l e g e d D e c i m a l Substratum". Zeitschrift fur Assyriologie und Vorderasiatische Archeologie 62: 165-221. — 1976. " T h e A n t e c e d e n t s of O l d B a b y l o n i a n P l a c e N o t a t i o n a n d the Early H i s t o r y of B a b y l o n i a n M a t h e m a t i c s " . Historia Mathematica 3: 4 1 7 - 4 3 9 . — 1 9 9 0 , " M a s s e u n d G e w i c h t e " . In: Dietz O. E D Z A R D (ed.), Reallexikon der Assyriologie, V o l . 7: 4 5 7 - 5 1 6 . Berlin: W a l t e r d e Gruyter. VISICATO, G i u s e p p e . 1 9 9 1 . " U n i t à di misura di capacità a Fara: b a r i g a e g u r - m a h " . Nouvelles

assyriologiques

— 1995. The Bureaucracy — 2 0 0 0 , The Power

brèves et utilitaires ofÈuruppak.

1991/81: 5 3 - 5 4 .

Munster: Ugarit-Verlag.

and the Writing. B e t h e s d a , M D : C D L P r e s s .

Square Tablets in the Yale Babylonian Collection Karen R. Nemet-Nejat,

New

York

T h e r e are t w e n t y - t w o s q u a r e - s h a p e d tablets a m o n g a p p r o x i m a t e l y 180 m a t h e m a t i c a l tablets in the Y a l e B a b y l o n i a n Collection.' T h e s e tablets are primarily c o n n e c t e d b y their c o n m i o n shape. In addition, the majority o f s q u a r e tablets h a v e d r a w i n g s a n d schematic d i a g r a m s o n t h e m . Square tablets are a n u n u s u a l shape a m o n g school tablets, as m o s t school tablets are either rectangular or lenticular. T h e Y a l e square tablets date to the O l d B a b y l o n i a n period, b u t their p r o v e n a n c e is uncertain. T w e l v e o f the square-shaped tablets deal with systematic o p e r a t i o n s with reciprocals o f regular a n d irregular n u m b e r s . T e n p r o b l e m s are a c c o m p a n i e d b y schematic a n d d i a g r a m m a t i c drawings. T h e r e m a i n i n g s q u a r e - s h a p e d tablets deal with subjects s u c h as the derivation of reciprocals from t w o pairs o f numbers,^ inheritance, h e r d growth, transportation of bricks, canals a n d work. T h e s e p r o b l e m s a p p e a r also o n rectangular a n d lenticular tablets. Since practice w a s essential, the scribes devised exercises for l e a m i n g operations w i t h reciprocals. T h e tablets in the Y a l e B a b y l o n i a n Collection s h o w that square tablets formed one aspect o f the teaching of m a t h e m a t i c s in the O l d B a b y l o n i a n period.^ T h e Y a l e B a b y l o n i a n Collection is n o t e w o r t h y b e c a u s e o f the n u m b e r o f d r a w i n g s and schematic d i a g r a m s on square tablets and the n m n b e r of tablets o f that shape in o n e collection. T h e following is a brief s u m m a r y o f the s q u a r e - s h a p e d tablets in the Y a l e B a b y l o n i a n Collection. Since m a n y of these tablets h a v e p r e v i o u s l y o n l y b e e n p u b l i s h e d in transliteration, p h o t o g r a p h s of all b u t o n e are g i v e n at the e n d of the article.'* T e x t 1: Y B C 7 2 3 4 T h e tablet m e a s u r e s 91 X 81 X 3 3 m m and is in fair condition w i t h s o m e small pieces missing. T h e o b v e r s e contains m a n y erasures a n d the n u m b e r s in c o l u m n iv are lightly impressed. T h e reverse is uninscribed. T h e tablet is p u b l i s h e d a n d discussed in M C T p . 17.

' A version of this paper was first presented at the MELAMMU symposium in October 2 0 0 0 . 1 am greatful to Prof W. W. Hallo, Curator of the Yale Babylonian Collection, for permission to publish this imformation. This research was undertaken as part of an NEH grant for cataloging the Yale mathematical texts. ^

SACHS ( 1 9 4 7 ) , p. 2 3 3 .

^

Provenanced texts from Nippur help support this conclusion. See VELDHUIS ( 2 0 0 0 ) , p.

2 3 8 andn. 2 1 ; R O B S O N { Ì 9 9 9 ) , pp. 1 If

"

and275-277.

YBC 7 2 9 0 was identified as square tablet too late to be photographed.

K.R. Nemet-Nejat

254

T h e tablet deals with operations of reciprocals o f regular n u m b e r s . T h e r e are six entries such that c o l u m n ii = reciprocal of c o l u m n i; c o l u m n iii = c o l u m n ii X 1,10; c o l u m n iv = constant factor c = 1,10; the n u m b e r in the u p p e r right c o m e r = addition o f all n u m b e r s in c o l u m n iii. T e x t s 2 - 5 are similar tablets containing operations with reciprocals of regular numbers.^

1 30 20 15 12 10

1,10 35 23,20 17,30 14 11,40

2,51,[30] 1,10 1,10

WO] 1,10 1,10 1,10

T e x t 2: Y B C 7354 T h e tablet m e a s u r e s 91 X 86 X 3 0 m m . It is in fair condition with the right c o m e r and other pieces lost; several further pieces h a v e b e e n rejoined. T h e tablet has b e e n e r a s e d a n d re-used. In particular there are m a n y erasures in c o l u m n iv a n d the n u m b e r s r e m a i n i n g near the b r o k e n right c o m e r are written u p s i d e d o w n . T h e reverse in uninscribed. T h e tablet is published a n d discussed in M C T p . 17. T h e tablet deals w i t h operations with reciprocals of regular n u m b e r s . T h e r e are 6 entries such that c o l u m n ii = reciprocal of c o l u m n i; c o l u m n iii = c o l u m n ii X 3 0 ; c o l u m n iv = constant factor c = 3 0 ; the n u m b e r in the u p p e r right c o m e r = addition o f all n u m b e r s in c o l u m n iii.

1 30 20 15 12 10

30 15 10 7,30 6 5

1,13,[30] 30 30 30 30 30 [30]

^ Text 6 also concerns operations with reciprocals of regular numbers, but is somewhat different. Texts 7-8 concern operations with reciprocals of irregular numbers.

Square Tablets in the Yale Babylonian Collection

255

Text 3 : Y B C 11127 T h e tablet m e a s u r e s 91 X 79 X 3 2 m m . It is in fair condition with small pieces m i s s i n g from the u p p e r left c o m e r . T h e reverse is uninscribed. T h e tablet is p u b l i s h e d a n d discussed in M C T p . 17. T h e tablet deals with operations with reciprocals of regular n u m b e r s . T h e r e are six enfries such that c o l u m n ii = reciprocal of c o l u m n i; c o l u m n i n = c o l u n m ii X 2; c o l u m n iv = constant factor c = 2 ; the n u m b e r in the u p p e r right c o m e r = addition o f all n u m b e r s in c o l u m n iii.

1 30 20 15 12 10

[1] 2 3 4 5 6

2 1 40 30 24 20

4,54

2 2 2 2 2 2

Text 4: Y B C 7355 T h e tablet m e a s m e s 97 X 87 X 31 m m and is in p o o r condition; it is a b r a d e d a n d has a small crack. T h e reverse is uninscribed. T h e tablet is p u b l i s h e d a n d discussed in M C T p . 17. T h e tablet deals with operations with reciprocals o f regular n u m b e r s . T h e r e are six entries such that coliunn ii = reciprocal of c o l u m n i; c o l u m n iii = c o l u m n ii X 4 0 ; c o l u m n iv = constant factor c = 4 0 ; the n u m b e r in the u p p e r right c o m e r = addition of all n u m b e r s in c o l u i n n iii.

1,38 1 [2] [3] [4] [5] 6

1 30 [2]0 [15] [12] 10

40 20 13,20 10 8 6,40

40 40 40 40 40 40

K.R. Nemet-Nejat

256

Text 5: Y B C 7358 T h e tablet m e a s u r e s 9 2 X 82 X 2 9 m m a n d is in fair condition; it is slightly e r o d e d with small pieces missing. T h e reverse is uninscribed. T h e tablet is p u b l i s h e d a n d discussed in M C T p . 17. T h e tablet deals with operations with reciprocals o f regular n u m b e r s . T h e r e are five entries such that c o l u m n ii = reciprocal of c o l u m n i; c o l u m n iii = c o l u m n ii X 4 5 ; c o l u m n iv = constant factor c = 4 5 ; the n u m b e r in the u p p e r right c o m e r = addition o f all n u m b e r s in c o l u m n iii.

1,42,45 1 2 3 4

1 30 20 15 [12]

[5]

45 22,30 15 11,15 9

45 45 45 45 45

Text 6: Y B C 7235 T h e tablet m e a s u r e s 80 X 84 X 27 m m and is in fair condition w i t h small p i e c e s missing from the e d g e s a n d c o m e r s . T h e o b v e r s e h a s b e e n erased a n d re-used. T h e n u m b e r s o n the reverse of this tablet are partly e r a s e d a n d h a v e n o c o n n e c t i o n to the calculations o n the o b v e r s e . T h e tablet w a s p u b l i s h e d in M C T p . 17; since t h e n it has b e e n b a k e d a n d the reverse has b e e n collated. Obv. 1 3 4

1,40 5 6,40

Rev. 1 2 3 4 5 6

1 20 15

1, 40 13,20 10

3,20 1,6,40 1,6,40 1,6,40 40

35 3,30 37,15,4 10

6 28,4(!),30

(4 is written over a sign)

T h e obverse o f the tablet presents a different p a t t e m in dealing with o p e r a t i o n s of reciprocals of regular n u m b e r s than that s h o w n in T e x t s 1-5. T h e r e are three entries such that c o l u m n i = 1,40 X c o l u m n ii;

Square Tablets in the Yale Babylonian Collection

257

c o l u m n iii = reciprocal o f c o l u m n ii; c o l u m n iv = coliunn iii x 4 0 ; column V = 4 0 X 1 , 4 0 = 1 , 6 , 4 0 ;

the n u m b e r in the u p p e r right c o m e r = addition o f all the n u m b e r s in c o l u i n n iv. T h e constant factor c = 4 0 is written at the b o t t o m o f c o l u m n v. Text 7: Y B C 7353 T h e tablet m e a s u r e s 9 6 x 8 7 x 3 8 m m a n d is in fair condition with several p i e c e s rejoined. T h e reverse is uninscribed. T h e tablet is p u b l i s h e d a n d d i s c u s s e d in M C T p. 1 7 . T h e tablet deals with operations with reciprocals o f irregular n u m b e r s ; there are 4 entries such that c o l u m n i X c o l u m n ii = 16,41;*^ c o l u m n iii = 3 0 x c o l u m n ii; c o l u m n iv = 3 0 X 1 6 , 4 1 = 8 , 2 0 , 3 0 ;

the n u m b e r m the u p p e r right c o m e r = s u m o f the n u m b e r s in c o l u m n iii.

3,11,15 7

2,23

1,11,30

8,20,30

11

1,31

45,30

8,20,30

[13]

1,17

38,20

8,20,30

14

1,11,30

35,45

8,20,30

B o t h T e x t 7 a n d T e x t 8 ( b e l o w ) , b e l o n g to a g r o u p o f tablets with the n u m b e r s 7 , 1 1 , 1 3 a n d 1 4 . B o t h are exercises for operations w i t h reciprocals o f these irregular numbers.^ T e x t s : Y B C 11125 T h e tablet m e a s u r e s 9 5 x 8 8 x 3 0 a n d is in fair c o n d i t i o n w i t h small p i e c e s m i s s i n g and s o m e cracks; the l o w e r right c o m e r is b r o k e n a n d h a s b e e n repaired. T h e r e a r e m a n y erasures a n d lightly impressed n u m b e r s in c o l u m n s iii a n d iv o f the o b v e r s e . T h e tablet is p u b l i s h e d in M C T p p . 1 7 - 1 9 ; see also N E M E T - N E J A T ( 2 0 0 0 ) , p p . 6 6 - 6 9 .

T h e tablet deals with operations with reciprocals o f irregular n u m b e r s . O n the o b v e r s e there are four entries such that c o l u n m i X c o l u m n ii = 1 6 , 4 1 ;^ col iii = 4 0 X col ii;

^

I.e. the least common multiple of the numbers of the first column.

'

NEMET-NEJAT (2000), p. 6 1 .

* I.e. the product of the prime numbers 7, 1 1 , and 1 3 found in the first three rows of column i.

258

I

K.R. Nemet-Nejat

f 1

col iv = 4 0 x 1 6 , 4 1 ; the n u m b e r in the u p p e r right c o m e r = addition of all the n u m b e r s in col iii. A b l a n k space is u s e d to indicate a m e d i a l e m p t y p l a c e in c o l u m n iii: 1,40 should b e therefore b e u n d e r s t o o d as 1,0,40. Obv. 7 11 13 14

2,23 1,31 1,17 1,11,30

1,35,20 1,40 51,20 47,40

4,15 11,7,20 11,7,20 11,7,20 11,7,20

T h e reverse repeats parts of the first a n d last lines a n d the n u m b e r in the u p p e r right c o m e r (that is, the s u m o f n u m b e r s in c o l u m n iii) of T e x t 7. T h e first and last lines are separated b y a b l a n k space. Rev. 3,11,[15] 7

2,2 I 3 (sic)

1,11,30 (sic)

14 I 1,11,3[0] B o t h Text 7 and T e x t 8 b e l o n g to a g r o u p of tablets with the n u m b e r s 7, 1 1 , 13 a n d 14. B o t h are exercises for operations with reciprocals of these irregular numbers.^ Text 9: Y B C 7356 T h e tablet m e a s u r e s 91 x 9 9 x 27 m m a n d is in p o o r condition; it is e r o d e d , c r a c k e d and repaired. Small p i e c e s are missing. T h e reverse is uninscribed. T h e tablet is p u b l i s h e d in N E M E T - N E J A T ( 1 9 9 5 ) , p p . 2 5 3 a n d 2 5 6 figure 6. T h e tablet deals with operations with reciprocals. O n the o b v e r s e operations with reciprocals of regular n u m b e r s are m a p p e d o n a d r a w i n g of a rectangle d i v i d e d into 4 parts with inscribed n u m b e r s . T h e n u m b e r s within the b o x e s are r e c i p r o c a l pairs; the reciprocals are multiplied b y the constant 4 8 written at the b o t t o m o f the diagram, the resulting p r o d u c t is written at the t o p left of each b o x . T h e n u m b e r 1,40 written to the left o f the d r a w i n g is the s u m o f the n u m b e r s at the t o p o f e a c h b o x , i.e., 4 8 , 2 4 , 16 a n d 12. This tablet appears to b e a m a t h e m a t i c a l g a m e for students to practice reciprocals. 24

^4^8

46^

^r,40 1

1

48

NEMET-NEJAT (2000).

2 48

30

3 48

20

4 4^8^

'15'

Square Tablets in the Yale Babylonian Collection

259

Text 10: Y B C 7291 T h e tablet m e a s u r e s 88 x 74 x 17 m m a n d is in fair condition; the left c o m e r is missing, a n d t w o pieces h a v e b e e n rejoined. It is p r e v i o u s l y u n p u b l i s h e d . T h e o b v e r s e of the tablet deals with operations with r e c i p r o c a l s o f regular n u m b e r s m a p p e d o n a drawing o f a rectangle divided into 3 p a r t s with inscribed n u m b e r s . A similar d i a g r a m is partly d r a w n on the reverse. Obv. 40

20

1,20

40 1 {20}

2

1

30

13,20

13,7œ

T h e p r o b l e m is o b v i o u s l y incomplete. T h e n u m b e r s within the b o x e s are r e c i p r o c a l pairs. F o r the first b o x , multiplying the reciprocal 1 b y 13,20 yields 13,20. A further multiplication b y a constant factor of 3 gives the n u m b e r 4 0 , 0 found a b o v e the b o x . T h e s e c o n d b o x d o e s not agree, h o w e v e r : 0;30 x 13,20 = 6,40 a n d 6,40 x 3 = 2 0 , 0 ( w h i c h is actually to b e found to the right of s e c o n d b o x ) . If the m i e from the first b o x is w h a t w a s m t e n d e d t h e n the drawing should h a v e b e e n inscribed as follows:

1,13,20

13,20

13,20

13,20

T e x t l l : M L C 651 T h e tablet m e a s u r e s 9 4 x 8 5 x 2 8 m m a n d is in fair condition, t h o u g h m u c h erased a n d s o m e w h a t a b r a d e d . It is p u b l i s h e d in S A C H S ( 1 9 4 7 ) , p . 2 3 3 .

Obverse

Lower Edge Reverse

1 2 3 4 5 6 7 8

Number 1,20,54,81,6,40 12,8,10,40 18,1[2,1]6 [1,8,16] 30-[...]6,37,50,20^ [...]1(?),5,20,9^ 40'* 1,20,54,31,6,40

9

igi-bi 44,29,37,50,1-5,20^

' sic, error for 3,45 ^sic, error for 4,16 3 sic, 16

Reciprocal 91,30 3,45 51,25,37,30' [3,45]

'* sic, error for 16 ^sic, error for 4 4 , 2 9 , 4 0 , 3 9 , 5 0 , 3 7 , 3 0

260

K.R. Nemet-Nejat

T h e tablet contains a table s h o w i n g the p r o c e d u r e for calculating a pair of reciprocals b y d o u b l i n g a n d halving the left a n d right m e m b e r s respectively. T h i s p r o c e d u r e was n a m e d b y Sachs " T h e T e c h n i q u e " . T h e tablet b e g i n s with 1,20,52,31,6,40 a n d e n d s with the its reciprocal 4 4 , 2 9 , 3 7 , 5 0 , 1 5 , 2 0 (sic), a pair d e r i v e d from twenty-third pair o f 2,5 a n d 2 8 , 4 8 . ' ° T h e tablet contains six lines on the obverse, o n e o n the lower e d g e a n d t w o o n the reverse. O n the reverse a line separates 1,20,54,32,6,40 from "its reciprocal 4 4 , 2 9 , 4 0 , 3 0 , 5 0 , 1 ' 5 , 2 0 " . A n error w a s m a d e in lines 3 a n d 4 so that 4 4 , 2 9 , 3 7 , 5 0 , 1 ' 5 , 2 0 was listed as the reciprocal instead o f the correct 4 4 , 2 9 , 4 0 , 5 0 , 3 7 , 3 0 . T h e e n d of the exercise is signified b y a d o u b l e line d r a w n after the reciprocal. T e x t 12: Y B C 1 0 8 0 2 T h e tablet m e a s u r e s 64 X 64 X 2 2 m m a n d is in p o o r condition; the l o w e r e d g e is lost a n d the right c o m e r is b r o k e n . It is p u b l i s h e d a n d discussed b y S A C H S ( 1 9 4 7 ) , p . 2 2 3 . T w o lines only r e m a i n o n the tablet: 2,22,13,20 [

]

25,12(sic),42(sic)[

],

T h e s e lines are followed b y a n area w h i c h h a s b e e n erased. It a p p e a r s that this is a tablet similar to T e x t 11 containing a table for calculating reciprocals from the twelfth pair of 2,5 a n d its reciprocal 2 8 , 4 7 b y d o u b l i n g a n d halving the left a n d right m e m b e r s (i.e., u s i n g " T h e T e c h n i q u e " ) . T h e t w o lines r e m a i n i n g o n the tablet are p r o b a b l y to b e t a k e n as 2 , 2 2 , 1 3 , 2 0 a n d its reciprocal 2 5 , 1 8 , 4 5 . Text 13: N B C 8082 T h e tablet m e a s u r e s 73 X 7 3 X 2 9 m m and is in fair condition with the l o w e r right c o m e r lost a n d the reverse is completely eroded. It is p u b l i s h e d a n d d i s c u s s e d in M C T p . 10 ( T , T r ) a n d C M T p p . 14, 2 5 3 . T h e tablet b e l o n g s to a g r o u p of o n e - p r o b l e m m e t r o l o g i c a l tablets in w h i c h a short text is m a r k e d off, usually in the lower right c o m e r of the tablet; the r e m a i n i n g surface of the tablet is either uninscribed, erased, or s h o w s calculations w h i c h have little to d o with either the p r o b l e m ' ' or the unit c o n v e r s i o n part of the p r o b l e m . In such tablets the p r o b l e m is s t a t e d a n d the a n s w e r is given fully c o n v e r t e d to the c u s t o m a r y units o f m e a s u r e m e n t . T h e m e t h o d o f solution is not given. T h i s e x a m p l e is a metrological p r o b l e m for the area of a s q u a r e . T h e tablet contains n u m e r i c a l calculations, with the length a n d area e x p r e s s e d in standard metrological units. O n the o b v e r s e the p r o b l e m a p p e a r s at the b o t t o m of the tablet and is separated from the calculations as follows: '°

Sachs ( 1 9 4 7 ) , pp. 2 2 8 - 2 3 3 .

"

See

N B C 8 0 8 2 ( M C T , p. 1 0 ) ; N C B T 1 9 1 3 ( M C T , p. 1 0 ) ; 2 N - T 4 7 2 (NEUGEBAUER

and

SACHS ( 1 9 8 4 ) , pp. 2 4 6 - 2 4 7 ) , 2 N - T 1 1 6 (NEUGEBAUER and SACHS ( 1 9 8 4 ) , p. 2 4 7 ) ; 2 N - T 3 0 (NEUEGBAUER and (1984),

pp.

SACHS ( 1 9 8 4 ) , pp. 2 4 7 - 2 4 8 ) ; U M 2 9 - 1 5 - 1 9 2 (NEUGEBAUER and

2 4 8 and

251); U M B

1 6 / 2 2 5 , pi.

7 ; cf

YBC

7284 (MCT,

mathematical problem to determine the weight of a brick of type 1. C M T , p. 14; NEMET-NEJAT ( 1 9 9 5 ) , pp. 2 5 9 - 2 6 0 .

pp.

SACHS

97-98),

a

Square Tablets in the Yale Babylonian Collection

20

4 1,20

1,20 1,20 1,46,40 16 1 NINDA A.SÀ.B1 A.SÀ.BI

4 KUS EN.[NAM] 1 2/3 [SAR

261

IB.SIg 6 2/3 GIN]

T h e p r o b l e m is solved as follows: 2 0 x 4 = 1,20 x 4^ = 16 x ( l , 2 0 f

= 1 x 46,40.

Or, p u t another w a y , the n u m b e r s that r e m a i n o n the left side indicate 4 x 0;20 = 1;20; therefore (1;20)^ w a s derived from 4^ x (0;20)^. T h e metrological p r o b l e m c a n b e easily solved b y using a metrological multiplication table for squares. ^ Text 14: Y B C 10722 T h e tablet m e a s u r e s 86 x 85 x 2 9 rrmi a n d is in p o o r condition; it is b r o k e n , c r a c k e d and largely erased with large pieces missing from the u p p e r right a n d u p p e r left c o m e r s . It is p u b l i s h e d a n d discussed in M C T , p . 9 8 ; see also C M T p . 3 4 , N E M E T NEJAT (2000) and H 0 Y R U P (1993).

T h e tablet deals with the fransportation of b r i c k s . T h e o b v e r s e is uninscribed. T h e reverse contains a p r o b l e m w h i c h is o n e of a g r o u p involving the transportation o f 9,0 ( = 5 4 0 ) bricks a distance o f 3 0 ninda (rods). Line 4 states that four m e n are hired. T h e niunbers 7, 11, 13, a n d 14, in lines 5-8, m a y refer to the w a g e s p a i d in line 3 . T h e n u m b e r s 7, 11, 13, a n d 14 form a special series found in o p e r a t i o n s with reciprocals ( s e e , for e x a m p l e . T e x t s 7 and 8 above) a n d mahïru "price e q u i v a l e n c y " p r o b l e m s , b r i c k s , h e r d growth, a n d original quantities. Text 1 5 : Y B C 7 2 9 0 T h e tablet m e a s u r e s 7 2 x 78 x 27 m m a n d is in g o o d condition w i t h small p i e c e s lost from the l o w e r left c o m e r . It is published a n d discussed in M C T p . 4 4 . O n b o t h the obverse a n d the reverse there is a drawing o f a frapezoid. T h e frapezoid o n the o b v e r s e h a s inscribed n u m b e r s , w h e r e a s the frapezoid o n the reverse is a d r a w n without inscribed n u m b e r s . T h e n u m b e r s relating to the frapezoid on the o b v e r s e are calculated according to the formula A = < ^ . h .

Text 16: Y B C 7359 T h e tablet m e a s u r e s 9 2 x 8 2 x 2 9 m m a n d is in fair condition; t h e o b v e r s e is a b r a d e d and small p i e c e s missing from b o t h sides o f the tablets. It is p r e v i o u s l y u n p u b l i s h e d , b u t see K I L M E R ( 1 9 6 4 ) , p p . 1 4 0 - 1 4 6 a n d S A G G S ( 1 9 6 0 ) , p p . 1 3 1 , 1 3 4 , T e x t B .

O n b o t h the o b v e r s e a n d the reverse w e find the s a m e d r a w i n g o f a rectangle within a rectangle with inscribed n u m b e r s . T h e o b v e r s e is carefiilly d r a w n , a n d m a y be the t e a c h e r ' s copy, while the reverse is very sloppy a n d m a y b e a s t u d e n t ' s copy.

See MCT pp. 8-9; NEMET-NEJAT (1995), p. 260 and fig. 9.

262

K.R. NEMET-NEJAT

Obv. 1,40 6,30 3,30 9

1,31 10

3,30

[...] Rev. '1,40^ 6,30 3,30 22,45 [...]

1,31 3,30

10

9 [...]7 T h e only relation I c o u l d find b e t w e e n the calculations is as follows: 1,31 x 9 x 1,40 = 3,30 x 6,30 2 2 , 4 5 = 1,31 x 9 X 1,40 = 3 , 3 0 x 6,30 T e x t 17 = Y B C 1 0 8 0 1 T h e tablet m e a s u r e s 62 x 81 x 3 0 m m and is in fair condition, a l t h o u g h slightly abraded. It is p u b l i s h e d a n d discussed in M C T p . 3 5 . T h e reverse is uninscribed. O n the obverse a line is d r a w n p a r a l l e l to the shorter side and slightly left of center; three n u m b e r s written in a large, c l e a r script. 4,35 21,25 4,35 T h e tablet calculates the square of 4 , 3 5 . A m e d i a l e m p t y p l a c e is n o t i n d i c a t e d since (4,35)^ = 2 1 , 0 , 2 5 a n d this is written as 2 1 , 2 5 . T e x t 18 = Y B C 7 2 9 4 T h e tablet m e a s u r e s 67 X 57 X 2 4 m m and is in g o o d condition w i t h a c r a c k in it. It is p u b l i s h e d and discussed in M C T p . 35 and is similar to T e x t 17 above.''*

A ROUND TABLET CONTAINING THE SQUARE OF A NUMBER IS PUBLISHED IN M K T

III, P. 5 1 .

263

Square Tablets in the Yale Babylonian Collection

T h e reverse is uninscribed. O n the o b v e r s e a line is d r a w n slightly left of center, and three large-sized n u m b e r s h a v e b e e n clearly inscribed. T h e tablet calculates the square o f 2 , 3 0 = 6,15,0. 2,30 6,15 2,30 T e x t 19 = Y B C 7 2 7 3 T h e tablet m e a s u r e s 86 x 85 x 2 9 m m . S m c e its p u b h c a t i o n in b e e n b a k e d a n d collated b y the author. T h e obverse contains a d i a g r a m with calculations indicating sheep folds. 1,20 lambs are p r o d u c e d from 1,40 sheep in the lambs are p r o d u c e d from 1,40 sheep in the second fold. T h u s the and 7 0 % of the original sheep in e a c h of the two f o l d s . ' '

M C T p . 131 it has h e r d g r o w t h in t w o first fold a n d 1,10 lambs number 8 0 %

I c a n m a k e n o sense o f the information in the third a n d fourth c o l u n m s o f the diagram. T h e reverse h a s b e e n erased except for t w o p e r s o n a l n a m e s (Ib-qu-sa a n d i-li-'^xx''), p e r h a p s referrmg to the shepherds responsible for each fold. 1,40 1,20

[38,20] 48

26 5,40

8,20 8

1,40

{30} 42

{3}4 8 A'

4,10

1,10

{} e r a s e d ?

Text 20: Y B C 7326 T h e tablet m e a s i n e s 6S x 77 x 25 m m a n d is in fair c o n d i t i o n with s o m e small pieces m i s s i n g a n d two pieces rejoined. T h e o b v e r s e is s o m e w h a t e r o d e d . It is published a n d discussed in M C T , p p . 1 3 0 - 1 3 1 ; see also C M T , p p . 80, 2 5 8 . T h e o b v e r s e sets out a p r o b l e m c o n c e m i n g h e r d growth. T h e p r o b l e m is to fmd the original n u m b e r o f sheep a n d the n u m b e r if lambs in t w o folds. T h e lambs n u m b e r 8 0 % a n d 7 0 % , respectively, of the original sheep in each of the t w o folds, as

IM 54472, published by BRUINS (1954), p. 56, contains a problem on how to find the side of a square when the area is known. Squaring a number as part of the calculations in solving a problem occurs throughout the mathematical corpus. See ROBSON (1999), pp. 251-252. Cf VAT 8522 problem #3 (MKT, pp. 367ff.). The tablet is broken but calculations indicating herd growth in a sheep fold are preserved on the tablet. The lambs number 80% of the original sheep in the fold. Also, YBC 4669 problem #B7 and #B8 (MKT III, pp. 27-29) deal with herd growth. Problem #B7 concerns finding the number if sheep and the number of Iambs when the number of sheep (ug-udu-hi-a = sénu) added to the number of lambs (sila4gub = lillidu) is 9,0 (= 540). Problem #B8 deals with finding the number of sheep and the number of lambs when a number of sheep subtracted from a number of sheep is 60 and a number of lambs subtracted from a number of lambs is a number no longer preserved on the tablet. Other numbers in this problem are not completely clear. The problem assumes a known relation exists betweefl the number of sheep and the munber of lambs. See CMT, p. 80.

K.R. Nemet-Nejat

264

in Text 19 and the related texts discussed a b o v e . T h e obverse d o e s not state the p r o b l e m fully b u t o n the reverse there is a d i a g r a m w h i c h c o m p l e t e s and clarifies the problem. Rev. [1,4]0 1,20

50 40

8,20

1,40 1,10

41,40 29,10

10,50

Text 2 1 : Y B C 9874 T h e tablet m e a s u r e s 80 X 87 X 30 m m and is in fair condition w i t h s o m e p i e c e s missing from the reverse and two pieces rejoined. It is p u b l i s h e d in M C T , p p . 9 0 9 1 ; see also C M T , p p . 4 2 , 2 5 9 . T h e tablet deals w i t h t w o p r o b l e m s about canals and irrigation.'^ T h e o b v e r s e a n d reverse of the tablet m a y e a c h contain independent, b u t similar p r o b l e m s . O n the obverse there is a schematic drawing of the rectangular cross-section o f a canal, w h i c h is enlarged. 1,40

5

1,40

3,20

3,36

1

12

40

20

T h e n u m b e r s inscribed o n the d i a g r a m indicate the d i m e n s i o n s of the c a n a l : ' ' width o f the sides o f the canal w = 1,40 = 0;1,40 N I N D A ("rods") length 1 = 5 = 5,0 N I N D A ("rods") height h = 0;40 k ù s ("cubits") the unsilted interior width b = 1 k ù s ("cubit") T h e n u m b e r s written at the side of the d i a g r a m o b e y the relationship 3,20 X 3,36 = 12.'^ 3,20 is twice the w i d t h w, but it is not clear w h y this is multiplied b y 3,36. 2 0 should p e r h a p s b e interpreted as 0;20 sar. 0;20 sar is m e n t i o n e d o n the reverse as the v o l u m e of earth e x c a v a t e d b y one m a n y in one d a y at the u p p e r level of the canal.'^

There are numerous other examples of problems concerning excavating or enlarging a canal. For example: BM 85196 problem #15 (MKT II, pp. 43-59), Erm 15073 problems #4 and #5 (VAIMAN (1961), pp. 232-244), VAT 7528 problems #1-15 (MKT I, pp. 508-513) and YBC 7164 problems # 8 - 1 6 (MCT, pp. 81-90). Drawings of canals are rare. MCT, p. 90. MCT, p. 90. MCT, p. 90.

Square Tablets in the Yale Babylonian Collection

265

Text 22: Y B C 9856 T h e tablet m e a s u r e s 12 x 12 x 26 m m a n d is in fair condition w i t h small p i e c e s missing including the l o w e r right c o m e r . It contains t w o m a t h e m a t i c a l p r o b l e m texts o n the o b v e r s e . E a c h line is m l e d a n d a double m l i n g a p p e a r s at t h e e n d o f p r o b l e m s # 1 a n d # 2 . T h e reverse is uninscribed. T h e tablet is p u b l i s h e d in M C T , p p . 9 9 - 1 0 0 ; see a l s o C M T p p . 6 7 - 6 9 , 8 8 , 2 5 9 .

P r o b l e m # 1 c o n c e m s finding the parts o f a w o r k d a y spent in p e r f o r m i n g three kinds o f work, w h i c h are each executed at different speeds.^" T h e w o r k is d e s c r i b e d as iskarum, " t h e daily assignment p e r m a n " . A l t h o u g h the m e a n i n g s o f several w o r d s in the p r o b l e m are n o t k n o w n , the general m e t h o d c a n b e u n d e r s t o o d . See M C T p. 1 0 0 . P r o b l e m # 2 deals with inheritance, that is, finding the a m o u n t o f e a c h share w h e n 1 m a - n a o f silver is divided a m o n g five brothers.^' Generally, in the m a t h e m a t i c a l p r o b l e m texts b e t w e e n t w o a n d ten p e o p l e receive shares. T h e smallness o f the inheritance often m a k e s p r o p e r t y division s e e m theoretical b u t administrative texts o f the p e r i o d confirm tiiis information.^^

Text Concordance M L C 6 5 1 : Text 1 1

Y B C 7 3 2 6 : Text 2 0

Y B C 9 8 7 4 : Text 2 1

N B C 8 0 8 2 : Text 1 3

Y B C 7 3 5 3 : Text 7

Y B C 1 0 7 2 2 : Text 1 4

Y B C 7 2 3 4 : Text 1

Y B C 7 3 5 4 : Text 2

Y B C 1 0 8 0 1 : Text 1 7

Y B C 7 2 3 5 : Text 6

Y B C 7 3 5 5 : Text 4

Y B C 1 0 8 0 2 : Text 1 2

Y B C 7 2 7 3 : Text 1 9

Y B C 7 3 5 6 : Text 9

Y B C 1 1 1 2 5 : Text 8

Y B C 7 2 9 0 : Text 1 5

Y B C 7 3 5 8 : Text 5

Y B C 1 1 1 2 7 : Text 3

Y B C 7 2 9 1 : Text 1 0

Y B C 7 3 5 9 : Text 1 6

Y B C 7 2 9 4 : Text 1 8

Y B C 9 8 5 6 : Text 2 2

Abbreviations CMT

= NEMET-NEJAT ( 1 9 9 3 )

MCT

= N E U G E B A U E R and SACHS ( 1 9 4 5 )

M K T = NEUGEBAUER ( 1 9 3 5 - 1 9 3 7 ) TMS

= B R U I N S and R U T T E N ( 1 9 6 1 )

See CMT, pp. 87-89. Other problems dealing with work of an unspecified nature include YBC 4669 problem #B1 (MKT I, pp. 514-516) and YBC 4673 problems #17-21 (MKT III, pp. 29-34). The shares may form an arithmetic progression as in IM 31210 problem #1 (BRUINS (1954), pp. 57-58), Strssbg 362 problem #1 (MKT 1, pp. 239fF.), UET 5 no. 121 problem #1 (VAIMAN (1961), pp. 248ff.) and VAT 6597 problem #2 (MKT I, pp. 274ff.). When money is divided, the mathematical problems deal with amounts adding up to 1 ma-na of silver in the following texts: IM 31210 problem #5 (BRUINS (1954), pp. 57-58), TMS no. 10 (TMS, pp. 69-74), VAT 6597 problem #4 (MKT I, pp. 275 ff.) and the present text. 22

See CMT, p. 69.

266

K.R. Nemet-Nejat

References B A Q I R , Tahir. 1 9 5 1 . " S o m e M o r e M a t h e m a t i c a l T e x t s " . Sumer 7: 2 8 - 4 5 . B R U I N S , Evert M . 1954. " S o m e M a t h e m a t i c a l T e x t s " . Sumer 10: 5 5 - 6 1 . — and R U T T E N , M a r g u e r i t e . 1 9 6 1 . Textes mathmatiques de Suse ( M D P 3 4 ) . Paris: Librairie orientaliste Paul Geuthner. H 0 Y R U P , Jens. 1 9 9 3 . " R e m a r k a b l e N u m b e r s in O l d B a b y l o n i a n M a t h e m a t i c a l Texts: A N o t e o n the P s y c h o l o g y of N u m b e r s " . Journal of Near Eastern Studies 52: 2 8 3 - 2 8 6 . KILMER, A n n D . 1964. " T h e U s e o f A k k a d i a n dks in O l d B a b y l o n i a n G e o m e t r y T e x t s " . In: F r o m the W o r k s h o p o f the C h i c a g o A s s y r i a n Dictionary: Studies Presented to A. Leo Oppenheim: 1 4 0 - 1 4 6 . C h i c a g o : University o f C h i c a g o . N E M E T - N E J A T , K a r e n R. 1993. Cuneiform Mathematical Texts as a Reflection of Everyday Life in Mesopotamia ( A m e r i c a n Oriental Series 7 5 ) . N e w H a v e n : A m e r i c a n Oriental Society. — 1995. " S y s t e m s for L e a r n i n g M a t h e m a t i c s in M e s o p o t a m i a n Scribal S c h o o l s " . Journal of Near Eastern Studies 5 4 : 2 4 1 - 2 6 0 . — 2 0 0 0 . " L u c k y T h i r t e e n " . Bibliotheca

Orientalis

58: 5 9 - 7 2 .

N E U G E B A U E R , O t t o . 1 9 3 5 - 1 9 3 7 . Mathematische Keilschrifttexte (Quellen u n d Studien zur G e s c h i c h t e der Mathematik, A s t r o n o m i e u n d Physik). 3 vols. Berlin: Julius Springer. — and S A C H S , A b r a h a m J. 1945. Mathematical Cuneiform Texts ( A O S 2 9 ) . N e w H a v e n : A m e r i c a n Oriental Society and the A m e r i c a n S c h o o l s o f Oriental Research. — and S A C H S , A b r a h a m J. 1984. " M a t h e m a t i c a l a n d M e t r o l o g i c a l T e x t s " . Journal of Cuneiform Studies 3 6 : 2 4 3 - 2 5 1 . R O B S O N , Eleanor. 1999. Mesopotamian Mathematics, 2100-1600 EC: Technical Constants in Bureaucracy and Education ( O E C T 14). Oxford: C l a r e n d o n P r e s s . S A C H S , A b r a h a m J. 1947. " B a b y l o n i a n M a t h e m a t i c a l T e x t s I". Journal of Cuneiform Studies 1: 2 1 9 - 2 4 0 . S A G G S . H e n r y W . F . 1960. " A

Babylonian

Geometrical

Text".

Revue

d'Assyriologie 54: 1 3 1 - 1 4 5 . V A I M A N , A b r a m A . 1 9 6 1 . Sumero-vavilonskaya matematika, HI I tysyaceletiya do n.e. ( S u m e r o - B a b y l o n i a n mathematics, third to first millennium, B . C . ) . M o s c o w : Izdatel'stvo Vostocnoj Literatury. V E L D H U I S , Niek. 2 0 0 0 . "Sumeriail P r o v e r b s in T h e i r Curricular C o n t e x t " . Journal of the American Oriental Society 120: 3 8 3 - 3 9 9 .

Square Tablets in the Yale Babylonian Collection

T e x t 1. Y B C 7 2 3 4 (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 2. Y B C 7 3 5 4 (Copyright Y a l e B a b y l o n i a n Collection)

267

268

K.R. Nemet-Nejat

T e x t 3 . Y B C 11127 (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 4 . Y B C 7 3 5 5 (Copyright Y a l e B a b y l o n i a n Collection)

Square Tablets in the Yale Babylonian Collection

T e x t 5. Y B C 7 3 5 8 (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 6. Y B C 7 2 3 5 (Copyright Y a l e B a b y l o n i a n Collection)

269

270

K.R. Nemet-Nejat

T e x t 7. Y B C 7 3 5 3 (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 8a. Y B C 11125 (obverse) (Copyright Y a l e B a b y l o n i a n Collection)

Square Tablets in the Yale Babylonian Collection

T e x t 8b. Y B C 11125 (reverse) (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 9. Y B C 7 3 5 6 (Copyright Y a l e B a b y l o n i a n Collection)

271

272

K.R. Nemet-Nejat

T e x t 10a. Y B C 7 2 9 1 (obverse) (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 10b. Y B C 7 2 9 1 (reverse) (Copyright Y a l e B a b y l o n i a n Collection)

Square Tablets in the Yale Babylonian Collection

T e x t 1 l a . M L C 6 5 1 ( o b v e r s e ) (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 1 l b . M L C 651 (reverse) (Copyright Y a l e B a b y l o n i a n Collection)

273

274

K.R. Nemet-Nejat

T e x t 1 2 . Y B C 10802 (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 13. N B C 8 0 8 2 (Copyright Y a l e B a b y l o n i a n Collection)

Square Tablets in the Yale Babylonian Collection

T e x t 14. N B C 10722 (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 16a. Y B C 7 3 5 9 (obverse) (Copyright Y a l e B a b y l o n i a n Collection)

275

276

K.R. Nemet-Nejat

T e x t 16b. Y B C 7 3 5 9 (reverse) (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 17. Y B C 10801 (Copyright Y a l e B a b y l o n i a n Collection)

Square Tablets in the Yale Babylonian Collection

T e x t 18. Y B C 7 2 9 4 (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 19a. Y B C 7 2 7 3 (obverse) (Copyright Y a l e B a b y l o n i a n Collection)

277

278

K.R. Nemet-Nejat

T e x t 19b. Y B C 7 2 7 3 (reverse) (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 20a. Y B C 7 3 2 6 (obverse) (Copyright Y a l e B a b y l o n i a n Collection)

Square Tablets in the Yale Babylonian Collection

T e x t 2 0 b . Y B C 7 3 2 6 (reverse) (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 2 1 a . Y B C 9 8 7 4 (obverse) (Copyright Y a l e B a b y l o n i a n Collection)

279

280

K.R. Nemet-Nejat

T e x t 2 1 b . Y B C 9 8 7 4 (reverse) (Copyright Y a l e B a b y l o n i a n Collection)

T e x t 22a. Y B C 9 8 5 6 (obverse) (Copyright Y a l e B a b y l o n i a n Collection)

Square Tablets in the Yale Babylonian Collection

T e x t 2 2 b . Y B C 9 8 5 6 (reverse) (Copyright Y a l e B a b y l o n i a n Collection)

281

A Goddess Rising 10,000 Cubits into the Air... Or Only One Cubit, One Finger? Joachim

Friedrich

Quack,

Berlin

M y starting p o i n t will b e a hieroglyphic inscription from the t e m p l e of P h i l a e . T h i s temple is situated in the v e r y south of Egypt, near the first cataract o f A s w a n . T h e temple was built a n d decorated m a m l y during the P t o l e m a i c period, in the Hellenistic age. N a t m a l l y , tiiis d o e s not p r e c l u d e the use of older texts as m o d e l s for the inscriptions. H o w e v e r , I will p o s t p o n e questions of dating a n d first p r e s e n t the actual inscription. T h e inscription is placed in the great hypostyle hall of the t e m p l e . It is n o t part of a n offering scene, b u t stands as a so-called m o n o g r a p h . ' T h e hieroglyphic text is:

^

O Ci

C~2

J 0 W •

Û

BE'

^

0

wu (J O W

* = Û <^ W ^

O A ^

'•^.

'<m.

y/M

' A text giving fundamental information about the specific theology and mythology of a locality.

284

J.F. Quack

Translated, it runs as follows:^ " ( T h i s d i s t r i c t . . . ) , it is called Per-Mereret. It is the p l a c e w h e r e Shu and Tefnut stopped w h e n they c a m e n o r t h w a r d s from B u g e m while a great flame w a s a r o u n d her. She b u r n e d the e n e m i e s of her father R e . She w e n t u p into the sky 10,000 cubits a n d i m m e d i a t e l y b e c a m e peaceful. T h e n Shu said to Tefhut, w h e n he h a d seen her m a k i n g a great m a s s a c r e a m o n g the rebels: ' M a y you remain, m a y y o u r e m a i n there,^ m a y you b e risen, m a y y o u b e risen there, m a y you r e m a i n in Elephantine, m a y y o u r e m a i n there, m a y y o u b e risen in B i g g e , you w h o h a v e kindled"* a torch against all e n e m i e s of P t a h in M e m p h i s at the [first] t i m e . ' " O n the w h o l e , the translation d o e s not p r e s e n t t o o m a n y difficulties. H o w e v e r , there is one particular p o i n t w h i c h should b e further c o m m e n t e d on. A b o v e , I h a v e translated the m e a s u r e of distance as 10,000 cubits. Unfortunately, the o r t h o g r a p h y is susceptible to t w o different interpretations. T h e hieroglyphic s i g n of the finger j is the ordinary n u m e r a l for 10,000. A t the same time, it c a n b e r e a d as a m e a s u r e , i.e. the w i d t h of a finger. So, the g o d d e s s could rise either 10,000 cubits into the sky or j u s t one cubit a n d o n e finger. T h e last reading has n e v e r b e e n c o n s i d e r e d before b e c a u s e at first glance it d o e s not s e e m to m a k e m u c h sense. H o w e v e r , it c a n n o t b e e x c l u d e d offhand. A s a matter of fact, to express a c o m p a r a t i v e l y large distance j u s t in cubits w o u l d b e like giving the distance from L o n d o n to O x f o r d in y a r d s instead of miles. T h e E g y p t i a n s w o u l d h a v e h a d larger units of m e a s u r e m e n t at h a n d , especially the " r o d " , if they h a d w i s h e d to give larger distances. Therefore, I w o u l d like to consider b o t h possibilities. Several questions h a v e to b e a n s w e r e d at this point. W h o is this g o d d e s s a n d w h a t d o e s it m e a n that she is rising into the sky? C a n such an image b e c o n n e c t e d with any kind of o b s e r v a b l e astronomical p h e n o m e n o n ? D o e s a n astronomical interpretation o f the underlying m y t h give any h e l p t o w a r d s establishing w h i c h interpretation of the n u m b e r is correct?

^ The text was printed and translated in JUNKER ( 1917), p. 96. A more recent translation can be found in INCONNUE-BOCQUILLON (2001), p. 53. ^ The r in the text is an obvious scribal error for /m, as was recognised by JUNKER (1917), p. 96 n. 3. Surprisingly, INCONNUE-BOCQUILLON (2001), p. 53, while noting that the text is corrupt, still tries to translate it as it stands ("reste, pour y paraître"). This is not acceptable, for the construction of preposition plus stative is impossible in Egyptian grammar. In my translation, I have replaced the Egyptian abbreviation "twice" by really repeating the text intended to be read twice. " Following Junker, I interpret sti.t as a participle. The position of Inconnue-Bocquillon that this is an infinitive serving as heading of a ritual text, while syntactically possible, does not make much sense in this context.

I

A Goddess Rising 10,000 Cubits into the Air... or Only One Cubit, One Finger?

285

T h e g o d d e s s in this inscription is called Tefhut. T o g e t h e r with h e r b r o t h e r S h u w h o is also m e n t i o n e d in the text, she is the child o f the p r i m e v a l g o d Atiun, a manifestation o f t h e Egyptian solar g o d . T h e m y t h o l o g y coimected with h e r includes o n e highly relevant point for the inscription in question: T h e g o d d e s s has b e c o m e separated firom h e r father a n d dwells far a w a y in the southeast in the coimtry called B u g e m . H e r b r o t h e r Shu, together with T h o t , the g o d o f w i s d o m , tries to b r i n g h e r b a c k to E g y p t a n d to h e r father. After m a n y delays a n d p r o b l e m s , they m a n a g e to d o this. T h e text I a m discussing here describes o n e stage o f this r e t u m j o i u n e y . T h e final m e e t i n g o f the g o d d e s s with h e r father takes p l a c e in M e m p h i s , w h i c h is also m e n t i o n e d in o m inscription. W h i l e s o m e texts simply dwell o n the h a p p y r e u n i o n o f the family, otìiers a d d a m o r e dramatic point. T h e children o f the sim-god are s u p p o s e d to fight against the e n e m i e s o f t h e n father. T h i s aspect is highlighted at two points in o u r text. O n the o n e hand, the killing o f the e n e m i e s takes p l a c e at B i g g e , j u s t south o f the E g y p t i a n b o r d e r , j u s t before the g o d d e s s r e a c h e s Egypt. O n the other h a n d , it is c o m b i n e d w i t h M e m p h i s , the p l a c e o f the final m e e t i n g w i t h the sun-god. T h i s c o m p l e x m y t h is normally designated, in E g y p t o l o g i c a l research, as the m y t h o f the far-away g o d d e s s , t h o u g h " h o m e - c o m i n g g o d d e s s " w o u l d b e n e a r e r to the t m t h . ' T h e m o s t c o m p l e t e narrative a b o u t it is foimd in a v e r y long D e m o t i c text k n o w n b y its m o d e m title " M y t h o f the E y e o f the S u n " , found in at least six different p a p y r i o f w h i c h several are still unpublished.*^ In this text, t h e protagonists are g i v e n as a n E t h i o p i a n cat a n d a little d o g - a p e . A l s o , m u c h additional information, m o s t l y in t h e form o f short statements a n d epithets, c a n b e found in H i e r o g l y p h i c t e m p l e inscriptions. Several p r e v i o u s scholars have speculated o n possible c o n n e c t i o n s b e t w e e n this m y t h a n d natural p h e n o m e n a . T h e first scholar to deal extensively with it, H e r m a n n Junker, w a s reluctant to admit a n y such notion. H e preferred to interpret it o n a very simplistic level as a sttaight folk s t o r y . ' If h e admitted a n y a s t t o n o m i c a l reality b e h m d it, h e o p t e d for a lunar interpretation, without g o i n g into a n y details.^ S u c h a n interpretation in coimection with the m o o n is still a c c e p t e d n o w a d a y s in s o m e q u a r t e r s , ' t h o u g h it c a n b e hardly c o n s i d e r e d as the consensus o f E g y p t o l o g y . E v e n before that, K u r t h Sethe p r o p o s e d that there w a s a m e t e o r o l o g i c a l b a c k g r o u n d to the story. H e thought that the m y t h w a s p r o m p t e d b y dark c l o u d s ' The most important studies on this subject are JUNKER (1911); JUNKER (1917); SETHE (1912); further bibliographical references are given by INCONNUE-BOCQUILLON (2001), p. 9f ; to be added are e.g. KEES (1931); STERNBERG (1985), pp. 224-228; ENDRÔDI (1992); for more, see below note 9, 13 and 15f ^ Text editions for pLeiden 1 384 see SPIEGELBERG (1917); DE CENIVAL (1988); for pLille dem. 31 see DE CENIVAL (1985), DE CENIVAL (1987); DE CENIVAL (1989); for pTebtunis Tait 8 see TAIT (1974); TAIT (1977), pp. 35-37, T. 3. For the Greek Version of pBM 274, see REITZENSTEIN (1923); WEST (1969); T o m (1985), pp. 168-182. A new German franslation of the whole narrative will be published in an anthology of Demotic literature by Friedhelm Hoffmann and the author of this communication. '

JUNKER (1917), p. 129f

*

JUNKER (1917), pp. 166-168.

^

Most explictely DERCHAIN (1971), p. 12f and pp. 36-44; DERCHAIN (1972), p. 45;

DERCHAIN (1978), p. 51, n. 2; VERHOEVEN (1986).

286

J.F. Quack

t e m p o r a r i l y c o v e r i n g t h e sun; t h e a b s e n c e o f t h e s u n w a s interpreted a s a b s e n c e o f t h e g o d d e s s . ' ° T h i s idea h a s b e e n so conclusively criticised b y o t h e r s " that it d o e s not need much more comment. T h e r e is, h o w e v e r , a n o t h e r astronomical interpretation that, a t least t e m p o r a r i l y , h a s g a i n e d a c c e p t a n c e . It g o e s b a c k t o a p a s s i n g idea offered b y E . S c h w a r z e a n d reported b y Wilhelm Spiegelberg.

A c c o r d i n g t o this theory, the a p p a r e n t m o v e m e n t

o f the solar orbit during t h e different seasons s h o u l d b e the c a u s e o f t h e religious m y t h o f the a b s e n c e o f the g o d d e s s . T h o u g h S p i e g e l b e r g n e v e r e l a b o r a t e d t h e details of this interpretation a n d d i d n o t stake his r e p u t a t i o n o n it, s o m e h o w it b e c a m e the s t a n d a r d o p i n i o n o f Egyptologists'^ a n d got i n c l u d e d in t h e m o s t i m p o r t a n t reference tools." A n a l t e m a t i v e solution c o n n e c t e d the return o f the g o d d e s s with t h e arrival o f the N i l e inundation i n Egypt.'^ T h i s h a s n o t b e e n e l a b o r a t e d i n m u c h detail, b u t t h e m o s t interesting fact a b o u t it is that several scholars h a v e m e n t i o n e d connection

between

t h e heliacal

rising

o f Sirius

t h e intimate

a n d the b e g i n n i n g

of the

inundation.'^ H e r e , I w o u l d like t o recall m y o w n p r o p o s a l offered s o m e y e a r s a g o . I p o i n t e d o u t that it m a d e s e n s e to u n d e r s t a n d the r e t u r n o f the g o d d e s s a s a mythical interpretation o f the heliacal a p p e a r a n c e o f t h e star Sirius after

its p e r i o d o f

invisibility.'^ I referred t o a n inscription indicating that t h e g o d d e s s a p p e a r e d in t h e h o r i z o n b e h i n d O r i o n , ' * b u t h a d t o p o s t p o n e a detailed d i s c u s s i o n for ftirther studies. M e a n w h i l e , a n i n ^ o r t a n t step forward h a s b e e n m a d e b y A l e x a n d r a v o n L i e v e n . Working

with

m y proposed

model,

she w a s able

to

develop

meaningftil

interpretations for several c o s m o l o g i c a l inscriptions in t h e t e m p l e of E s n a , w h i c h allude to t h e m y t h o f the r e t u r n o f the g o d d e s s . " A l t h o u g h it is i m p o s s i b l e t o give a c o m p l e t e d e m o n s t r a t i o n w i t h i n t h e limited frame o f this p u b l i c a t i o n , I w o u l d like t o indicate at least s o m e o f the m o r e relevant texts a n d a r g u m e n t s i n o r d e r t o p r o v i d e m y c a s e with a better foundation. S o m e general discussion o f earlier r e s e a r c h also s e e m s in order. T h e t h e o r y o f the inundation o b v i o u s l y c a n n o t e x p l a i n m a n y details o f the m y t h . M o s t relevantly, t h e v e r y text I a m discussing h e r e clearly states that t h e g o d d e s s is "

SETHE(1912),p.38.

"

S e e e.g. JUNKER (1917), p. 165f SPIEGELBERG (1915), p. 877, n. 1 ; SPIEGELBERG (1917), p. 2.

The most important recent proponents were BARTA (1969), p. 77; BRUNNERTRAUT(1980), p. 34; VAN DIJK (1983), p. 240; KURTH (1983), pp. 161-163 and n. 4; KURTH (1982-1983), p. 73; GRAEFE (1984), p. 903f ; STERNBERG (1985), pp. 224-228. E.g. BONNET ( 1952); OTTO (1975); SMITH ( 1984). SAUNERON (1962), p. 58f ; and especially DESROCHES-NOBLECOURT and KUENZ (1968),

p l l 5 f ; DESROCHES-NOBLECOURT (1993), pp. 28 and 42; accepted KRIÉGER (1976), p. 447f ; BARGUET( 1977), p. 20; GERMONT(1989), p. 227.

by

POSENER-

GERMOND (1989), pp. 194-196 and 224-233; NAGUIB (1990), pp. 37-48; see also KESSLER(1988),p. 183. QUACK (1995), p. 116, note k. "

QUACK (1995), p. 109, n. 14.

'^

VON LiEVEN (2000), p. 75; VON LIEVEN (2001 ).

A Goddess Rising 10,000 Cubits into the Air ... or Only One Cubit, One Finger?

287

rising in the sky. T h i s d o e s not m a k e sense with the water of the N i l e inundation. T h i s m i g h t rise 16 cubits a b o v e zero-level, but neither 10,000 cubits n o r 1 cubit, 1 finger into the sky. I believe that the inundafion-theory mistakes an incidental part of the w h o l e situation for the basic core. A s the g o d d e s s is always designated as the d a u g h t e r o f the sim-god to w h o m she r e t u m s , any interpretation identifying the g o d d e s s herself w i t h the s u n runs into serious trouble: one sim w o u l d h a v e to p l a y two roles at o n c e . T h e interpretation of the g o d d e s s as the m o o n faces a similar d i l e m m a . O n e of the t w o g o d s responsible for bringing her b a c k is, after all, the g o d T h o t , w h o is a typical lunar deity.^° B e s i d e s , in E g y p t the m o o n is always cormected w i t h m a l e gods.^' It c a n b e clearly seen in s o m e texts that specific points in the p h a s e s of the m o o n are significant for the r e t u m o f the g o d d e s s in an ideal schema,^^ b u t they n e v e r g o so far as to indicate that the g o d d e s s herself is the m o o n . If, h o w e v e r , the g o d d e s s is Sirius, everything fits together. T h e lunar figure is a c o m p a n i o n o f the g o d d e s s watching over her punctual arrival. T h e father is the sun, a n d his m e e t i n g with his daughter w o u l d b e a m o s t appropriate e x p r e s s i o n of the heliacal rising of the star. A m o r e detailed study o f this p r o b l e m w o u l d h a v e to include the astrological ideas a b o u t the h o r o s c o p e of the w o r l d as e x p o s e d in G r e e k a n d Latin texts^^ and, incidentally, d e p i c t e d o n the ceiling o f the hypostyle hall in Dendera.^'' In this m o d e l , the w o r l d is b o m w h e n the h o r o s c o p e (ascendant) is in Cancer, in the e l e v e n t h h o u r o f the night w h e n Sirius is at the horizon. T h e m o o n either stands also in C a n c e r (as indicated b y F i r m i c u s M a t e m u s ) , directly beside Sirius. T h e n h e c a n b e j u s t a small sickle b e c a u s e the sun, standing in L e o , is rather close by. Or, as s h o w n o n the ceiling of D e n d e r a , h e is in his hypsoma in T a u m s , three signs a w a y fi-om the sun w h i c h m e a n s that h e is a b o u t the sixth or seventh d a y of the E g y p t i a n lunar cycle. T h i s is all the m o r e m ^ o r t a n t as several E g y p t i a n sources coimect the festival of the r e t u m of the g o d d e s s with the feast of the sixth d a y o f the lunar m o n t h . ^ ' F o r a c o m p l e t e elucidation, finther studies will b e necessary. S u m m i n g u p , the essential situation of the m y t h shows clearly that s u n a n d m o o n h a v e to a s s u m e roles different fi-om the g o d d e s s herself, but their p r e s e n c e w o u l d tally very well with interpreting the g o d d e s s as Sirius. O n e additional detail in favour o f this interpretation is tiiat a c c o r d i n g to all clear-cut g e o g r a p h i c a l indications, the g o d d e s s is c o m i n g fi-om tiie south-east.^*^ T h i s is the direction in w h i c h the n e w l y rising Sirius b e c o m e s visible fi-om E g y p t while it w o u l d not really fit either the sun

^° All citations fi-om the demotic text given by JUNKER (1917), p. 167f in favour of his interpretation, as a matter of fact, connect the moon not with the goddess herself but with Thot or other male gods. This has clearly been stated by WESTENDORF (1989), p. 89 against the lunar interpretation of Tefnut by Verhoeven. The best cases can be found in the inscriptions from Kom Ombo. "

BOUCHÉ-LECLERCQ(1899).

A fuller discussion of this will be included in my forthcoming work on the Egyptian decans. "

The references are given by BARTA (1969), p. 75f JUNKER (1917), p.

80f

288

J.F. Quack

or the m o o n . F u r t h e r m o r e , it is frequently stated that the g o d d e s s c o m e s as a h e a d ornament, or a s n a k e o n t h e b r o w o f h e r father. S u c h a m o d e l , i m p l y i n g t h e p r e s e n c e o f the g o d d e s s j u s t before h e r father, c o r r e s p o n d s v e r y well to t h e rising o f Sirius j u s t before t h e sun. S o m e other texts, especially from the t e m p l e o f E s n a , s a y that t h e s u n - g o d is snatching t h e light o f the g o d d e s s , a n d such a m y t h o l o g i c a l statement is visibly cormected w i t h t h e rising s u n outshining t h e star. N o w I will give s o m e citations from E g y p t i a n texts to s u p p o r t this m o d e l e v e n further. I b e g i n w i t h t w o e x a m p l e s from the D e m o t i c text.^' T h i s is b y far t h e longest a n d m o s t coherent a c c o u n t o f the myth. T h e g o d d e s s says: " Y o u are a b a b o o n with his b o w , so that y o u are like Sothis w h o h a s created t h o s e w h o h a v e created u s . I a m t h e n o b l e m a l e vulture o f t h e lord o f T h e b e s , i.e. the vulture o n w h o s e b o d y n o m a l e h a p p e n s . " T h i s is c o m m e n t e d b y " s h e c o m p a r e s herself t o Neith, b e c a u s e she is t h e o n e w h o c a m e into b e i n g without a n y o n e creating h e r [...], t h e o n e w h o b r o u g h t herself into b e i n g , w h o is Sothis before w h o m all things are d o n e , a n d w h o , also, is t h e year. If p e o p l e a r e intending to m a k e t h e w o r d " y e a r " in writing they should u s e a vulture for it. It is h e r w h o h a s created t h e m o n t h b e c a u s e she is the p r i m o r d i a l g o d d e s s w h o h a s created everything o n earth. Everything c a m e forth from her. H e h a s arrived at m a k i n g t h e g o d d e s s a p p e a r in the form o f a n amulet for t h e year. A female b a b o o n is w h a t h e h a s u s e d for that. H e h u n g the b o w into t h e sky a n d t h e a r r o w s are h e r s t a r s " ( M y t h u s L e i d e n 9, 6-15). A t first glance, this p a s s a g e m i g h t s e e m like c o m p l e t e gibberish. H o w e v e r , it b e c o m e s m u c h clearer o n closer examination. T h e o n e trying to b r i n g t h e g o d d e s s b a c k to E g y p t is a form o f the g o d Thot, a n d T h o t often h a s t h e form o f a b a b o o n . T h e b a b o o n is, o f course, t h e d o g ' s - a p e . T h e r e is attestation, especially from G r e e k treatises o f the n o n - G r e e k constellations, that there w a s t h e constellation o f a n a p e or d o g - h e a d e d figure with a b o w b e l o n g i n g to t h e paranatellonta o f L e o . S o w e h a v e , indeed, a b a b o o n w i t h his b o w in the sky, rather close to Sirius. O n t h e other h a n d , the g o d d e s s Sothis, the E g y p t i a n form o f Sirius, c a n s o m e t i m e s b e r e p r e s e n t e d in the late p e r i o d as shooting arrows, m a i n l y b e c a u s e o f a c o n n e c t i o n with Satis. O b v i o u s l y , the p u r p o s e o f this p a s s a g e is to bring t h e t w o protagonists o f the d e m o t i c text in intimate cormection with figures of the starry sky, a n d at the s a m e time to create a m o d e l o f their close relationship. A n o t h e r p a s s a g e from t h e D e m o t i c text might s e e m e v e n m o r e straightforward. T h e little d o g - a p e offers a h y m n : " M a y the s k y carry a w i n d from the north a n d m a y it bring t h e scent o f P u n t u p with it. M a y the inundation flow before it. M a y the s u n a p p e a r in the m o m i n g .

Compare the editions by SPIEGLEBERG ( 1 9 1 7 ) and DE CENIVAL ( 1 9 8 5 ) . For reasons of

space, I have refrained from giving also the Demotic text.

A Goddess Rising 10,000 Cubits into the Air... or Only One Cubit, One Finger?

289

b e i n g a sim-disc with large flame. M a y his looks b e joyful a n d his rays e n d o w e d with life, w i t h o u t a n y c l o u d s o n the w a y of Sothis. M a y her rays b e within the c n c l e o f Egypt, m a y she t h r o w t h e m against the N u b i a n c o u n t r i e s " ( M y t h u s L e i d e n 16, 4-8). This d e s c r i b e s the ideal situation a r o u n d the date o f the heliacal rising o f Sirius. A p e r i o d o f p r o l o n g e d n o r t h w i n d is traditionally c o n n e c t e d with the b e g i i m i n g of the inimdation.^^ T h i s is the time w h e n Sirius, c o m i n g from the south-east, starts to b e c o m e visible a g a i n in Egypt. S o , m y interpretation that the m y t h of the r e t u m of the g o d d e s s is the m y t h i c a l picture of the heliacal rising of the Sirius fits v e r y well. L a c k of time p r e v e n t s m e from dwelling o n all aspects of the m y t h in the t e m p l e inscriptions. Still, I will discuss o n e specific text from the t e m p l e of P h i l a e . T h e r e , it is said a b o u t the d a n g e r o u s g o d d e s s S a k h m e t : wnn

Éhm.t

wsr.ti

m snm.t

énwh sbi.w m hh-à pri-é r p.t hpr rn=É pw n Épt.t

iir

" S a k h m e t is powerful in B i g g e

m nér.t

while b u m i n g the e n e m i e s with her flame. She c a m e forth

as

fire-serpent into the sky, a n d so her

name

'Sothis'

came

into

b e i n g " (Philae I, 6 9 , 6-9).^' T h e c o n n e c t i o n w i t h our starting p o i n t is o b v i o u s . A g a i n , w e h a v e the g o d d e s s furious a n d b i n n i n g the e n e m i e s in c o m b i n a t i o n w i t h a n ascent into the sky. I n addition, w e are g i v e n o n e further detail. T h e n a m e o f the g o d d e s s while rising into the sky is explicitly g i v e n as Sothis, a n d this is the E g y p t i a n n a m e for Sirius. B y n o w , it s h o u l d b e r e a s o n a b l y clear that the g o d d e s s rising into the sky is, indeed, a m y t h o l o g i c a l m o d e l c o n n e c t e d with the h e h a c a l rising o f Sirius. W e can, h o w e v e r , still p r o c e e d a bit fiirther. A characteristic i n c i d e n c e of the r e t u m o f the g o d d e s s is the fact that she b e c o m e s furious,^^ or m o r e specifically, b u m s the e n e m i e s of h e r father. H o w e v e r , afterwards her fiiry is q u e n c h e d a n d she is c o o l e d d o w n , or it is said that she b e c o m e s peaceful a n d j o y o u s . H o w w o u l d this fit with Sirius? In o r d e r to solve this p r o b l e m w e h a v e to take the symbolic m e a n i n g o f c o l o u r s in E g y p t into account. W r a t h a n d fury against e n e m i e s are always c o m b i n e d with the hot c o l o u r red, while the cool c o l o u r s g r e e n and b l u e are c o n n o t i n g p e a c e , j o y a n d h a r m o n y . H e r e , w e h a v e to r e m e m b e r an a s t r o n o m i c a l fact. W h e n rising directly in the h o r i z o n , Sirius a p p e a r s red, due m a i n l y to the influence of the a t m o s p h e r e a n d its d u s t . ' ' After c l i m b i n g u p a bit m o r e , it a s s u m e s its n o r m a l white-blue colour. G i v e n AuFRÈRE(1991),p. 248. Also translated by INCONNTUE-BOCQUILLON (2001 ), p. 83. ^*^ See, for this, e.g. the text given by JUNKER (1917), p. I l l with the interpretation given by QUACK (1993), p. 76f, but unfortunately overlooked by INCONNUE-BOCQUILLON (2001), p. 27. See, for this LEITZ (1993).

290

J.F. Quack

the attitude of t h e Egyptians t o w a r d s these colours, it is h a r d l y surprising that they should interpret this c h a n g e o f colour as turn from a ftirious to a peacefiil m i n d . S o , this detail about o u r inscription also fits within the framework o f m y general m o d e l . T h a t brings u s finally b a c k to the question o f the m e a s u r e m e n t . A s I h a v e m e n t i o n e d , the g o d d e s s rises u p into the sky to a height o f either 10,000 cubits or j u s t one cubit, o n e finger. T w o questions arise: d o e s this exact indication fit with the general m o d e l , a n d is a decision b e t w e e n the t w o options possible? If w e want to c o n n e c t the cubit-measurement in any w a y with s o m e sort of m e a s u r e m e n t in the sky, w e h a v e to look for other possible references in m o r e o r less astronomical contexts. T h e r e is one Egyptian testimony w h i c h m i g h t b e o f s o m e h e l p for applying m e a s u r e m e n t s to the sky. T h e n e t h e r w o r l d guide " A m d u a t " - the B o o k o f W h a t is in t h e N e t h e r w o r l d - describes the circuit o f the s u n within the twelve h o u r s of the night. F o r the s e c o n d a n d third hour, it is specifically indicated that the s u n traverses 3 0 9 I t e m within it. T h e I t e m is the largest distance m e a s u r e m e n t k n o w n in Egypt. It c o r r e s p o n d s to w h a t the G r e e k s called " S c h o i n o s " a n d it h a s a n o r m a l distance o f a b o u t 10.5 kilometers.^^ T h i s is m o s t p r o b a b l y the equivalent to 2 0 , 0 0 0 cubits. If w e s u p p o s e that this is the n o r m a l velocity o f the s u n a c c o r d i n g to Egyptian conceptions, w e c a n c o m p a r e it with o u r inscription. T h e result w o u l d b e quite devastating: the g o d d e s s rises only b y o n e 6 1 8 t h part of the distance that the s u n covers in o n e hour. Transferred in angular m e a s u r e m e n t s , the sun covers 15° in a n hour, so the g o d d e s s w o u l d rise b y only 1 m i n u t e a n d a b o u t 2 7 seconds of the a r c ; it w o u l d r e m a i n almost stable near the horizon. S u c h a result d o e s not m a k e sense. H o w c a n w e explain the failure? O n e solution w o u l d b e that the t w o different values are simply o n different levels a n d not m e a n t to b e c o m p a r e d o n the same scale. A n o t h e r solution is that the 10,000 cubits are not m e a n t to b e a n y specific value. T h e y m a y only indicate that the g o d d e s s rises quite a bit u p into the sky. Finally, there r e m a i n s the third possibility, a n d that is to explore p o s s i b l e m e a n i n g s o f the a l t e m a t i v e r e a d i n g 1 cubit 1 finger. At first sight, this s e e m s e v e n m o r e disappointing than the other v a l u e . S u c h a distance could easily b e m a n a g e d b y a n ordinary h u m a n j u m p i n g . Still, I w o u l d like to m e n t i o n s o m e t h i n g w h i c h seems all the m o r e appropriate as this is a j o i n t conference o n E g y p t a n d M e s o p o t a m i a . A s is well k n o w n , in M e s o p o t a m i a n a s t r o n o m y the cubit m e a s u r e w a s u s e d for m e a s u r i n g angular distances in the sky.''^ O n e cubit c o r r e s p o n d s to either 2 ° o r 2;30°. O n e finger is 0;05°. If w e a p p l y such a value to the E g y p t i a n indication, w e w o u l d g e t a b o u t 2;05° o r 2 ; 3 5 ° . Especially the last value seems n o t t o o b a d , as the m e a s u r e m e n t o f 2;30° for o n e cubit w a s far m o r e usual. S u c h a n angular distance d o e s n o t s e e m u n r e a s o n a b l e . It is a noticeable rise a b o v e the h o r i z o n b u t n o t very high u p into the sky. It w o u l d j u s t fit w i t h the short distance a star c a n c o v e r o n its heliacal rising before b e c o m i n g extinct b y the s u n ' s light.''*

S C H L O T T - S C H W A B ( 1 9 8 1 ) , pp.

See

most

recently

HUNGER

118-122.

and

FINGREE

(1999),

pp.

xiv

and

1 4 7 . Cf

also

GRASSHOFF ( 1 9 9 9 ) , pp. 1 3 7 - 1 3 9 and FATOOHI and STEPHENSON ( 1 9 9 7 - 1 9 9 8 ) , pp. 2 1 0 - 2 1 4 .

As a reaction to my original lecture. Dr. T. de Jong was kind enough to calculate Sirius risings for Memphis and Philae at the times around 3 0 0 , 6 0 0 , and 1 5 0 0 BC for four-yearsequences. The minimal altitude of Sirius, in those cases, was just 2 . 3 degrees, while 2 . 5 degrees (or 2 ; 3 0 ° ) is not infrequent. The mean altitude for Philae is mostly 2 . 6 degrees

A Goddess Rising 10,000 Cubits into the Air ... or Only One Cubit, One Finger?

291

T h e r e is one m o r e point. A s already said, Sirius c a n a p p e a r quite r e d a n d this w o u l d b e associated with the fury of the g o d d e s s . S u c h a r e d n e s s , a c c o r d m g to m o d e m observations, c a n h a p p e n u p to a b o u t 3° a b o v e the h o r i z o n . T h i s w o u l d n o t b e too far from the value w e get w h e n interpreting the E g y p t i a n inscription b y M e s o p o t a m i a n angular m e a s u r e m e n t s ; a n d thus, the p o i n t in the s k y w h e r e the g o d d e s s b e c o m e s peacefiil w o u l d b e about the p o i n t w h e r e Sirius c e a s e s to a p p e a r r e d d i s h to the observer. O f course, all this might b e j u s t a sfrange c o i n c i d e n c e , b u t it d o e s n o t s e e m useless to test it. N o w , it is time to sunmiarize. T h e m y t h of the r e t u m of the g o d d e s s h a s a distinct astronomical b a c k g r o u n d . It is a description in m y t h o l o g i c a l t e r m s o f natural p h e n o m e n a associated w i t h the heliacal rising of Sirius. S u c h a result s h o w s quite clearly that r e s e a r c h o n E g y p t i a n asfronomy caimot afford to concenfrate o n " p u r e " scientific texts while neglecting the religious b a c k g r o u n d . R a t h e r it is typical for the E g y p t i a n s that asfronomical b a c k g r o i m d is p r e s u p p o s e d in religious texts. F o r the particular inscription indicating h o w m u c h the g o d d e s s w e n t u p into the sky, w e are still facing t w o altematives. Either w e r e a d it as 10,000 cubits; t h e n w e c a n n o t p i n p o i n t a n y specific m e a n i n g for the n u m b e r . O r w e r e a d it as 1 cubit 1 fmger; then, b y applying m e a s u r i n g units otherwise o n l y attested for M e s o p o t a m i a , ' ^ w e c a n r e a c h a result w h i c h tallies r e a s o n a b l y well w i t h the facts about the heliacal rising o f Sirius. T h i s should m a k e it acceptable, at least as a w o r k i n g h y p o t h e s i s .

References AUFRÈRE,

Sydney.

1991.

L'univers

minerale

dans

la

pensée

égyptienne

(BibUothèque d ' É t u d e 105) Cairo: Institut français d ' a r c h é o l o g i e orientale du Caire. B A R G U E T , Pierre 1977. " L e cycle lunaire d ' a p r è s d e u x textes d ' E d f o u " . d'Égyptologie BARTA,

Winfried.

Agyptische

Revue

29: 1 4 - 2 0 . Sprache

1969.

"Zur

Bedeutung

und Altertumskunde

B O N N E T , H a n s . 1952. Reallexikon

des

jnwï-Festes".

Zeitschrift

fiir

95: 7 3 - 8 0 .

der agyptischen

Religionsgeschichte:

733-735

(s.v. S o n n e n a u g e ) . Berlin: D e G m y t e r B O U C H É - L E C L E R C Q , A u g u s t e . 1899. L'astrologie

grecque.

Paris: E. L e r o u x (Reprint

A a l e n : Scientia V e r l a g 1979). BRUNNER-TRAUT,

Emma.

1980. Altàgyptische

Tiergeschichte

und

Fabel.

6^

Edition. D a r m s t a d t : Wissenschaftliche Buchgesellschaft. DE C E N I V A L , F r a n ç o i s e . 1985. " L e s n o u v e a u x fragments du ' M y t h e d e l'œil d u s o l e i l ' de l'Institut d e P a p y r o l o g i e et d'Égyptologie de Lille". Cahier de Recherches l'Institut de Papyrologie

et dÉgyptologie

de

de Lille 7: 9 5 - 1 1 5 .

(around 300 BC it is 2.7) which is 2;36° - almost exactly the value of the Egyptian inscription if read as one cubit, one finger and interpreted by the Mesopotamian measuring conventions. The value for Memphis is quite similar, normally about 0.1 degrees higher. "

LEITZ (1993), p.

153.

In an unpublished astrological text from Tebtunis, a "cubit" is mentioned, but the context is not sufficiently well established to ascertain that the cubit is used as an angular measurement.

292

J.F. Quack

—1987.

"Transcription

hiéroglyphique

l'Université d e Lille". Cahier d'Égyptologie

d'un fragment

de Recherches

de Mythe

de l'Institut

conservé

à

de Papyrologie

et

de Lille 9: 5 5 - 7 0 .

— 1988. Le mythe

de l'œil

du soleil

( D e m o t i s c h e Studien 10). S o m m e r h a u s e n :

G. Zauzich. — 1989. " L e s titres d e s couplets d u M y t h e " . Cahier Papyrologie

et d'Égyptologie

de Recherches

de l'Institut

de

de Lille 1 1 : 1 4 1 - 1 4 4 , T . 16.

D E R C H A I N , P h i l i p p e . 1 9 7 1 . Les monuments

religieux

à l'entrée

de l'Ouadi

Hellal

( E l k a b 1). Bruxelles: F o n d a t i o n egyptologique R e i n e Elisabeth. — 1 9 7 2 . Hathor

quadrifrons.

Récherches

sur la syntaxe

d'un mythe

égyptien

(Publications d e l'Institut historique et a r c h e o l g i q u e d e S t a m b o u l 2 8 ) . Istanbul: N e d e r l a n d s H i s t o r i s c h - A r c h a e o l o g i s c h Instituut in h e t N a b i j e O o s t e n . — 1978. " E n l ' a n 3 6 3 d e Sa majesté le R o i d e H a u t e et B a s s e É y g p t e R â - H a r a k h t y vivant p a r d e l à le t e m p s et l ' e s p a c e " . Chronique DESROCHES-NOBLECOURT, Archeologica

Christiane.

d'Egypte

1993.

53: 48-56.

"Le

zodiaque

d'Abou

Simbel.

de

pharaon".

292: 21-45.

— a n d K U E N T Z , C h a r l e s . 1 9 6 8 . Le petit

temple

C a i r o : Centre d e

d o c u m e n t a t i o n et d ' é t u d e sur l ' a n c i e n n e Egypte. V A N DIJK,

Jacobus.

egyptische Documenten

godin

1983. "Hymnen Moet".

In:

uit h e t dagelijks

Klaas

R.

uit het oude nabije oosten vertaald

V e r h a n d e l i n g e n E x Oriente L u x 2 4 ) : IZZ-IA^.

tempelritueel

VEENHOF,

voor

Schrijvend

en toegelicht

de

verleden.

(Mededelingen En

Leiden: E x Oriente Lux.

E N D R Ó D I , Julia. 1 9 9 2 . " N â h e u n d F e m e . D e r Auftakt z u r ' O n u r i s l e g e n d e ' . In: U l r i c h L U F T ( e d ) . The Intellectual Kàkosy

by Friends

Heritage

and Colleagues

of Egypt.

Studies

on the Occasion

presented

to

of his 60th Birthday

Làszló (Studia

A e g y p t i a c a 14): 1 2 5 - 1 3 2 . B u d a p e s t : Innova P r e s s . FATOOHI,

Louay

J.

and

STEPHENSON,

F.Richard.

1997-1998.

"Angular

M e a s u r e m e n t s in B a b y l o n i a n A s t r o n o m y " . Archiv fur Orientforschung

44-45:

210-214. GERMONT,

Philippe.

1 9 8 9 . Sekhmet

et la protection

du monde

(Aegyptiaca

H e l v e t i c a 9). B a s e l / G e n e v e : Â g y p t o l o g i s c h e s S e m i n a r d e r Universitat B a s e l . G R A E F E , Erhart. 1984. " D a s Ritualgerat Sbt I wnèb I wt(\ In: Studien zu Sprache Religion Freunden

Àgyptens.

Zu Ehren

und Schtilern:

von Wolfhart

Westendorf

Uberreicht

von

und seinen

8 9 5 7 9 0 3 . Gottingen; S e m i n a r fur À g y p t o l o g i e u n d

K o p t o l o g i e d e r Georg-August-Universitat Gottingen. GRASSHOFF,

Gerd.

1999. "Normal

Star

Observations

in

Late

A s t r o n o m i c a l D i a r i e s " . In: N o e l M . S W E R D L O W (ed.). Ancient Celestial

Divination:

Babylonian

Astronomy

and

9 7 - 1 4 7 . Cambridge, M A : M I T Press.

H U N G E R , H e r m a i m a n d PINGREE, D a v i d . 1 9 9 9 . Astral

Sciences

in

Mesopotamia

( H a n d b u c h d e r Orientalistik 4 4 ) . L e i d e n / B o s t o n / K o l n : Brill. I N C O N N U E - B O C Q U I L L O N , Danielle. 2 0 0 1 . Le mythe de la Déesse

lointaine

à

Philae

(Bibliothèque d ' É t u d e 132). C a i r o : Institut français d ' a r c h é o l o g i e orientale d u Caire. J U N K E R , H e r m a t m . 1 9 1 1 . Der Auszug

der Hathor-Tefnut

aus Nubien

(Anhang zu

den Abhandlungen der Kôniglich Preussischen Akademie der Wissenschaften v o m J a h r e 1911). Berlin: V e r l a g d e r K ô n i g l i c h e n A k a d e m i e d e r Wissenschaften.

A Goddess Rising 10,000 Cubits into the Air ... or Only One Cubit, One Finger?

—1917.

Die

Onurislegende

(Denkschriften

d e r Kaiserlichen

293

Akademie

der

Wissenschaften in W i e n 5 9 , 1-2). W i e n : A . Holder. KEES,

Àgyptische

H e r m a n n . 1 9 3 1 . " D i e Besanftigung d e s R a u b t i e r e s " . Zeitschrift fiir

Sprache

und Altertumskunde 6 7 : 5 6 - 5 9 .

K E S S L E R , Dieter. 1 9 8 8 . " D e r satirisch-erotische P a p y r u s T u r i n u n d d a s ' V e r b r i n g e n des s c h ô n e n T a g e s ' " . Studien zur altagyptischen

Kultur

15: 1 7 1 - 1 9 6 .

K U R T H , Dieter. 1 9 8 2 - 1 9 8 3 . " D e r k o s m i s c h e Hintergrund d e s groBen H o r u s m y t h o s v o n Edfu". Revue d'Égyptologie

34: 7 3 - 7 5 .

— 1 9 8 3 . D i e D e k o r a t i o n d e r Saulen i m P r o n a o s d e s T e m p e l s v o n Edfu ( G o t t i n g e r Orientforschungen I V / 1 1 ) . W i e s b a d e n : Otto H a r r a s s o w i t z . LEITZ, Christian.

1 9 9 3 . " D i e N a c h t d e s K i n d e s in s e i n e m N e s t in D e n d e r a " .

Zeitschrift fiir Agyptische Sprache VON LIEVEN, Alexandra. religiosen

und Altertumskunde

2 0 0 0 . Der Himmel

Astronomie

in

Agypten

iiber

120: 1 3 6 - 1 6 5 a n d 1 8 1 . Esna.

Eine

(Agyptologische

Fallstudie

zur

Abhandlungen

64).

W i e s b a d e n : O t t o Harrassowitz. — 2 0 0 1 . " S c h e i b e n a m H i m m e l " . Studien zur altagyptischen N A G U I B , S a p h i n a z - A m a l . 1990. Le clergé féminin

Kultur 2 9 : 2 7 7 - 2 8 2 .

d'Amon

thébain

à la 21^

dynastie

(Orientalia L o v a n i e n s i a A n a l e c t a 3 8 ) . Leuven: Peeters P r e s s . O T T O , E b e r h a r d . 1 9 7 5 . " A u g e n s a g e n " . In: W o l f g a n g H E L C K a n d E b e r h a r d O T T O (eds.), Lexikon der Agyptologie I: 5 6 2 - 5 6 7 . W i e s b a d e n : Otto H a r r a s s o w i t z . P O S E N E R - K R I É G E R , P a u l e . 1976. Les archives Kakaï. Les papyrus

d'Abousir.

Traduction

du temple funéraire et commentaire

de

Néferirkarê-

(Bibliothèque d'Étude

65). Cairo: Institut français d ' a r c h é o l o g i e orientale d u Caire. QUACK,

Joachim

Aegyptia

Friedrich.

1 9 9 3 . " E i n altâgyptisches

Sprachtabu".

Lingua

3: 59-79.

— 1 9 9 5 . " M o n u m e n t a l d e m o t i s c h " . In: Louise G E S T E R M A N N , H e i k e S T E R N B E R G ELH O T A B I (eds.), Per aspera Geburtstag:

ad astra.

Wolfgang

Schenkel

zum

neunundfùnfzigsten

1 0 7 - 1 2 1 . Kassel: Selbstverlag Louise Gestermaim.

REITZENSTEIN, R i c h a r d . 1 9 2 3 . Die griechische

Tefnut-Legende

(Sitzungsberichte d e r

H e i d e l b e r g e r A k a d e m i e der Wissenschaften, P h i l o s o p h i s c h - H i s t o r i s c h e

Klasse

2). H e i d e l b e r g : C. Winter. SAUNERON,

Serge.

paganisme

1 9 6 2 . Les fêtes

religieuses

dEsna

au derniers

siècles

du

(Esna V ) . Cairo: Institut français d ' a r c h é o l o g i e orientale d u Caire.

S C H L O T T - S C H W A B , Adelheid. 1 9 8 1 . Die Ausmafie

Àgyptens

nach

altagyptischen

Texten ( À g y p t e n u n d Altes T e s t a m e n t 3). W i e s b a d e n : Otto H a r r a s s o w i t z . S E T H E , Kurt. 1 9 1 2 . Zur altagyptischen

Sage vom Sonnenauge,

das in der

Fremde

war ( U n t e r s u c h u n g e n zur Geschichte u n d A l t e r t m n s k u n d e À g y p t e n s 5, 3 . Heft). Leipzig: J. C. H i i u i c h s c h e B u c h h a n d l u n g . SMITH, M a r k . 1 9 8 4 . "Soiuienauge, d e m o t i s c h e r M y t h u s v o m " . In: W o l f g a n g H E L C K a n d E b e r h a r d O T T O (eds.), Lexikon

der Agyptologie

V: 1 0 8 2 - 1 0 8 7 . W i e s b a d e n :

Otto Harrassowitz. SPIEGELBERG, W i l h e l m . 1 9 1 5 . " D e r aegyptische M y t h u s v o m S o n n e n a u g e in e i n e m demotischen Preussischen

Papyrus Akademie

der

rômischen

der Wissenschaften

Kaiserzeit".

Sitzungsberichte

der

1 9 1 5 : 8 7 6 - 8 9 4 . Berlin: V e r l a g d e r

A k a d e m i e d e r Wissenschaften. — 1917. Der àgyptische

Mythus

"Kufi" nach dem Leidener

vom Sonnenauge.

demotischen

Papyrus

Der Papyrus

der

Tierfabeln,

/. 384. Strafîburg: R. Schultz.

294

J.F. Quack

S T E R N B E R G , H e i k e . 1985. Mythische Motive und Mythenbildung in den agyptischen Tempeln der griechisch-romischen Zeit (Gottinger Orientforschungen I V / 1 4 ) . W i e s b a d e n : O t t o Harrassowitz. T A I T , W i l l i a m John. 1974. " A Duplicate V e r s i o n of the D e m o t i c Kufi T e x t " . Acta Orientalia 3 6 : 2 3 - 3 7 . — 1977. Papyri from Tebtunis in Egyptian and Greek (Texts from E x c a v a t i o n s 3). L o n d o n : E g y p t E x p l o r a t i o n Society. TOTTI, M a r i a . 1985. Ausgewahlte Texte der Isisund Serapisreligion. Hildesheim/Ziirich/New Y o r k : G. 0 1 m s . V E R H O E V E N , Ursula. 1986. "Tefnut". In: W o l f g a n g H E L C K a n d E b e r h a r d O T T O (eds.), Lexikon der Àgyptologie VI: 2 9 6 - 3 0 4 . W i e s b a d e n : Otto H a r r a s s o w i t z . W E S T , Stephanie. 1969. " T h e G r e e k V e r s i o n of the L e g e n d of Tefnut". Journal of Egyptian Archaeology 55: 1 6 1 - 1 8 3 . W E S T E N D O R F , Wolfhart. 1989. Bemerkungen und Korrekturen zum Lexikon der Àgyptologie. Gottingen: Seminar flir À g y p t o l o g i e u n d K o p t o l o g i e der G e o r g August-Universitat Gottingen.

Aristarchos and the 'Babylonian' Month Dennis Rawlins,

Baltimore

Aristarchos o f S a m o s , the earhest scientist t o teach h e l i o c e n t r i s m w i d e l y (c.280 B C ) , lived s o m e w h a t before t h e attested a p p e a r a n c e ( o n B a b y l o n i a n cuneiform texts) o f the r e m a r k a b l y accurate S y s t e m B " B a b y l o n i a n " m e a n synodic monthlength: M A = 29*^31'50"08'"20""

(1)

T h e e n o r m o u s m a s s o f extant B a b y l o n i a n data h a s never e x p l a i n e d t h e origin o f M A B u t a tiny handful o f empirical a n d conventional G r e e k relations ( s o m e c o n n e c t a b l e to Aristarchos) lead right to M A , in a very few steps o f p l a i n arithmetic. P t o l e m y (Almajest 4 . 2 ) cites the 126007'' Ol*" eclipse cycle, 4267 m o n t h s , as M A ' S empirical basis. Since H i p p a r c h o s ' a n d P t o l e m y ' s e m p h a s e s , it h a s n o t b e e n m u c h appreciated that this cycle is r e m a r k a b l e for its nearly-invariant length, w h i c h w e k n o w stayed in the (relatively) very n a r r o w range 126007'' Ol'' ± l** t h r o u g h o u t the e r a of interest. T h e implied empirical m o n t h : M e = 1 2 6 0 0 7 ' ' 0 l ' ' / 4 2 6 7 = 3024169''/102408 = 29''31'50"08"'09""...

(2)

Since C o p e r n i c u s , it h a s often b e e n n o t e d that equations 1 a n d 2 d o n o t quite agree. H o w e v e r , (a) the d i s a g r e e m e n t is m e r e l y about o n e p a r t in 36 million; a n d (b) w e will s e e b e l o w (equation 4 ) that r e a s o n a b l e r o u n d i n g during a natural A r i st ar ch an c o m p u t a t i o n easily gives birth to e q u a t i o n 1 out o f equation 2 . B y p e r c e p t i v e l y noting that highly reliable equation 2 m a d e (virtually on-thenose) the familiar 223-month " S a r o s " (S) equal to 18 Kallippic years ( o f 365*^ 1 / 4 each) plus 1 0 2 / 3 degrees, Aristarchos - p r o p o s e r o f antiquity's largest universe could establish his typically-ambitious vast calendaric " G r e a t Y e a r " ( G Y ) o f 4868 years, c o m p r i s i n g exactly 2 7 0 Saros a n d exactly 1778037''. T h u s : ' S = 223 x M = ( 1 8 + 1 0 ° 2/3) X

2434/135 X

or 4868 X X K / 2 7 0

(3)

So in this G Y calendar, a m o n t h w a s ( c o m p u t i n g t o w a r d s a s e x a g e s i m a l l y convenient result, t h r o u g h a r o u n d i n g w h o s e smallness is nicely a p p r o p r i a t e t o the d a t a ' s accuracy):

MGY = (1778037'' / 223) / 270 ^ 7 9 7 3 " 06" 15*") / 270 = 29''3r50"08"'20'"' ( 4 ) w h i c h is equation 1 , as p r o m i s e d a b o v e . N e x t , since it w a s c o m m o n in antiquity to follow M e t o n ' s e q u a t i o n o f a tropical (civil) year with 2 3 5 / 1 9 m o n t h s , w e m a y find the tropical year KAI implicit in

' Equation 3 is explicitly attested in antiquity (Almajest 4.2), providing an independent check on our reconstruction.

296

D. Rawlins

e q u a t i o n 4 ' s p r o p o s a l that the precise value for M A originated in c o n n e x i o n with A r i s t a r c h o s ' G r e a t Year:^ 4 8 6 8 X YAÌ = 4 8 6 8 x ( 2 3 5 / 19) x A/^ = 1778021 '^.53 «1778022"^ = Kallippic G Y - 15"

(5)

So A r i s t a r c h o s ' tropical year w o u l d h a v e been: yAt= 1778022'*/4868 = 8 8 9 0 1 1 " / 2 4 3 4 = 365'' 1 / 4 - 1 5 / 4 8 6 8

(6)

P u t into c o n t i n u e d fraction format, this c a n b e w r i t t e n : ' YAt = 3 6 5 d +

-

-

(7)

20 + — 60 w h i c h perfectly m a t c h e s (digit for digit) the A r i s t a r c h a n y e a r l e n g t h e x p r e s s i o n listed o n V a t . g r . 381 fol. 163": A r i s t a r c h o s o f S a m o s : [365] Ì/4K'^0'=

[365] 1/4 2 0 ' 6 0 2 '

E q u a t i o n 8's luckily-surviving attestation of e q u a t i o n 7 p r o v i d e s a gratifying reality-check o n the foregoing development."*

(8) final

Acknowledgements T h e author is grateful for the assistance a n d - o r a d v i c e of C h r i s t o p h e r W a l k e r , A l e x a n d e r J o n e s , J o h n Britton, H u g h Thurston, a n d J o h n Steele.

References H E A T H , T h o m a s L. 1 9 1 3 . Aristarchos of Samos. Oxford: Oxford U n i v e r s i t y P r e s s . N E U G E B A U E R , O t t o . 1 9 7 5 . v4 History of Ancient Mathematical Astronomy. Berlin: Springer V e r l a g .

^ We note in passing that the span of Aristarchos' civil (though not Kallippic or sidereal) GY could have been half of 4868^ (as indicated by Censorinus, Tannery and Heath (1913), pp. 314ff; see Neugebauer (1975), p. 603), since halving equation 5 yields: 2434 tropical years = 889011** ^ To give due credit: it was Otto Neugebauer (1975), p. 602 who first suggested that the 60 could indicate sexagesimal expression. A substantially expanded version of these analyses appears in volume 11 of DIO: The International Journal of Scientific History. (All published volumes of DIO are freely downloadable from .)

Closing the Eye of Horus: The Rise and Fall of 'Horus-eye Fractions' Jim Ritter,

Paris

F o r nearly a century n o w students o f E g y p t o l o g y as well as o f the H i s t o r y o f Science h a v e p a u s e d in w o n d e r w h e n learning o f the tmique attitudes o f Egyptians "religious b e y o n d m e a s u r e , m o r e than a n y other p e o p l e " (Herodotiis, Histories, ch. 3 7 ) ; even that m o s t m i m d a n e o f subjects, weights a n d m e a s u r e s , w a s apparently i m b u e d b y t h e m with d e e p mystical purport. F o r part o f their s y s t e m o f capacity m e a s u r e s was apparently a p r o d u c t o f theological speculation, b a s e d o n o n e o f their m o s t p o t e n t mythological s y m b o l s , the E y e o f H o m s , d i s m e m b e r e d b y his evil b r o t h e r Seth a n d then restored to soundness {wdit) b y T h o t h . E a c h p a r t o f the b r o k e n E y e i n this perspective w a s assigned a dimidiated fractional v a l u e — i n d e s c e n d m g order: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, representing parts o f the basic capacity unit, t h e hqit. T h e definitive form o f this v i e w w a s given b y A l a n G a r d i n e r in the first edition o f his Egyptian Grammar: ' I . T h e corn-measure.*—The symbols employed in this, as shown in the accompanying cut, are derived from the ancient myth according to which the eye of the falcon-god Horus, often depicted on the monuments in the form was torn into fragments by the wicked god Seth." Later, the ibis-god Thoth miracujyr \ \ .

1

IA

'

lously 'filled' or 'completed' (w^) the eye, joining together the parts, whereby the eye regained its title

°

^

Mîk^lfe

«'fl^A • the sound e y e ' . In ^ accordance with this myth the sign < was used for \ , 0 for \, — for \, 1 » for for ^ and \ for These fractions together add up to H ; presumably the missing ^ was supplied magically by Thoth. This is the accoimt, a n d the image, that h a s continued to b e used, often without attiibution, in a h n o s t every m e n t i o n o f the H o m s - e y e - a s - c a p a c i t y - m e a s u r e story until very recent t u n e s . 1 shall argue that j u s t as the H o l y R o m a n E m p i r e was "neither holy n o r R o m a n nor a n empire,"^ so the " H o m s - e y e fractions" w e r e , m their origin at least, neither fi-actions nor associated with the E y e o f H o m s . I h a v e discussed e l s e w h e r e ' t h e first question a n d I shall thus avoid speaking a b o u t fractions in this c o n t e x t — I shall u s e '

GARDINER ( 1 9 2 7 ) , p. 1 9 7 .

^ "Ce corps qui s'appelait et qui s'appelle encore le saint empire romain n'était en aucune manière ni saint, ni romain, ni empire." Voltaire, Essai sur l'histoire générale et sur les mœurs et l'esprit des nations ( 1 7 5 6 ) , ch. 7 0 . ^

RITTER ( 1 9 9 2 ) .

298

J. Ritter

rather the t e r m ' c a p a c i t y system s u b m u h i p l e s ' ( C S M ) and, in the a b s e n c e of any e v i d e n c e for their phonetic values {pace the conjecture o f S E T H E ( 1 9 1 6 ) , p . 74), I shall refer to t h e m b y the u s e of quotation m a r k s a r o u n d their p u r p o r t e d fractional values. Since the question o f sign forms—hieratic a n d h i e r o g l y p h i c — a r e at the b a s e of the following discussion, it is important to note the differences in evidential weight that will b e a c c o r d e d to the signs in question. O f the six hieratic signs, t w o : 1 ("1/16") a n d ^ ("1/32") h a v e shapes w h i c h are diagnostic, that is, are sufficiently c o m p l e x a n d differentiated from other hieratic signs to enable one to m a p the r a n g e of synchronic variations a n d of diachronic d e v e l o p m e n t (the latter b e i n g of particular i m p o r t a n c e for us), while three: Û ("1/4"), / ("1/8") a n d + ("1/64") are t o o simple and/or too close to others to be of equal value for these p u r p o s e s . T h e third: ^ ("1/2"), falls s o m e w h e r e b e t w e e n the t w o classes o n the diagnostic scale."* Similarly for the p u r p o r t e d hieroglyphic equivalents, but here the diagnostic signs are < ("1/2"), i - ( " 1 / 1 6 " ) , ("1/32") and \ ( " 1 / 6 4 " ) ; while o ("1/4") and ~ ("1/8") still r e m a i n less helpful. If I a m right that there is n o solid evidence for the H o r u s - e y e interpretation of the C S M signs the q u e s t i o n arises j u s t w h y the story h a s h a d such an imaginative hold o n succeeding generations. This in t u m opens u p t w o questions, o n e specific to the E g y p t o l o g i c a l c o m m u n i t y a n d the other of b r o a d e r application. T h e first question c o n c e m s the n a t u r e o f Egyptian writing a n d h o w the E g y p t o l o g i c a l literature traditionally treats it. This will b e a constant c o n c e m o f ours during our e x a m i n a t i o n o f the relationships b e t w e e n the Eye o f H o r u s a n d the dimidiated capacity units a n d w e shall return to it explicitly at the e n d of our enquiry. T h e s e c o n d question c o n c e m s h o w conjectures, originally p u t forward in a tentative fashion, lose their provisional character o v e r time, e v e n in the a b s e n c e of a n y c h a n g e in the quantity or quality o f the e v i d e n c e in their favour. A s I h o p e to s h o w in w h a t follows, the case of the Eye o f H o r n s a n d capacity units p r o v i d e s a n e a r l y p a r a d i g m a t i c e x a m p l e . B u t to d o so requires us to g o b a c k to the origins a n d early history of the h y p o t h e s i s — o r rather the t w o h y p o t h e s e s — w h i c h lie at the basis of the story.

I. Origins and Reception In 1 9 1 1 , G e o r g M o l l e r at the Berlin M u s e u m w a s nearing complefion of his m o n u m e n t a l Hieratische Palaographie. T h e first two v o l u m e s h a d a l r e a d y b e e n p u b l i s h e d in 1909, c o v e r i n g the period from the O l d K i n g d o m to the e n d o f the N e w K i n g d o m . T h e structure of the work, with chronologically o r d e r e d e x a m p l e s o f hieratic script o r g a n i z e d b y hieroglyphic sign, necessitated a choice of hieroglyphic " o r i g i n a l " for each hieratic sign. T h e C S M hieratic signs p o s e d n o p r o b l e m , they w e r e k n o w n from a n u m b e r o f texts: accounting, m a t h e m a t i c a l and medical.^ T h e i r p l a c e in the c a p a c i t y s y s t e m h a d

* All the examples are drawn from New Kingdom sources, see Table I. ^ Sources in these categories used by Moller in the first two volumes of his Hieratische Palaographie are: Administration: various lllahun fragments, pBoulaq 18, pLouvre E 3226, pRollin, pHarris; Mathematics: pRhind; Medicine: pEbers, pBerlin 3038.

Closing the Eye of Horus: The Rise and Fall of'Horus-eye Fractions'

299

b e e n authoritatively s u m m e d u p t w e n t y years earlier b y Griffith.^ H o w e v e r the u n i q u e full set o f the hieroglyphic C S M signs k n o w n a t the t i m e — f r o m the festival c a l e n d a r o n the walls o f R a m e s s e s I l l ' s t e m p l e in M e d i n e t H a b u — w e r e available only

in t h e rather

mediocre

hand

copies

due to Johaimes

Diimichen,

and

t y p o g r a p h i c a l forms o f these h a d h a d t o serve in the first t w o v o l u m e s . ' B u t current w o r k o n p h o t o g r a p h s o f votive cubits b y H e i m i c h Schafer a n d , especially, L u d w i g Borchardt^ r e v e a l e d the u n e x p e c t e d p r e s e n c e o f hieratic C S M signs integrated into t h e hieroglyphic inscriptions o n the b a c k o f t h e c u b i t s ; a n d " o n the

same

c u b i t s " , t h e hieroglyphic

d e c o m p o s a b l e into H + ^

signs —-, <, o a n d f^.

Since

the last—

— w a s r e c o g n i z a b l e as t h e u n d e r p a r t o f the e y e o f H o r u s ,

the other signs c o u l d b e identified with the s a m e object. T h u s t h e e n i g m a t i c hieratic c a p a c i t y - s y s t e m signs a n d the imtìl n o w e q u a l l y enigmatic h i e r o g l y p h i c signs from M e d i n e t H a b u s e e m e d to b e intimately linked to the culturally sigitificant o f H o m s . T h e third voliune o f M o l l e r ' s Hieratische p u b l i s h e d with t h e n e w hieroglyphic

equivalents

Palaographie

wdSt-eye

could n o w be

in the ''Bruchteile

vom

section.' T o s u m u p r a p i d l y h e r e , M o l l e r p r e s e n t s essentially t w o a r g u m e n t s for t h e identification o f the C S M signs with parts o f the c o m p l e t e d right e y e o f H o m s : \ ° the p r o x i m i t y o f s o m e hieratic a n d s o m e hieroglyphic forms o f t h e signs o n t w o u n p u b l i s h e d v o t i v e cubits a n d 2° the visual similarities o f the r e m a i n i n g signs, w h i c h " c a n b e t r a c e d b a c k t o parts o f t h e ^ " . ' ° H e finishes t h e article b y a d d i n g a further t w o p o i n t s : 1° that t h e u n p u b l i s h e d O l d K i n g d o m B e r l i n p a p y m s

10500"

s h o w s t h e antiquity o f the s y s t e m a n d 2 ° that " d o u b t l e s s " , its origin is t o b e s o u g h t in t h e m y t h o l o g i c a l tradition o f the wdit-tyt.

M o l l e r t h e n r e c o n s t m c t s t h e outline o f

the m y t h from a n u m b e r of disparate s o u r c e s . In a final n o t e , M o l l e r u n d e r l i n e s the tentative nature o f this last: " W e a r e c o n s c i o u s that, in these last p a r t s o f o u r discussion, w e h a v e several times

ttodden

upon the domain of hypothesis. In so

d o i n g w e h a v e only i n t e n d e d t o u r g e that fiirther allusions i n t h e m y t h o l o g i c a l texts in question b e e x p l o r e d . " ' ^ H o w e v e r the e i r ^ h a s i s o n t h e a g e o f t h e r e l a t i o n s h i p b e t w e e n C S M a n d the E y e o f H o m s is, for Moller, certain a n d a further p r o o f o f his belief that " w e m u s t c o u n t t h e birth o f the greater p a r t o f t h e h i e r o g l y p h i c script as symbolic g a m e s (Spielereien)I

shall call this original a r g u m e n t , that t h e C S M -

H o m s - e y e e q u i v a l e n c e is early a n d reflective o f the d e v e l o p m e n t o f E g y p t i a n writing, t h e strong form o f the E y e o f H o m s fractions thesis.

* ^

GRJFFiTH(1892),p.426. MOLLER ( 1909 a) and ( 1909 b).

The Medinet Habu calendar forms were introduced in BRUGSCH (1883) and the printing fonts based on these were used in GRIFFITH (1892). The photographic edition of Medinet Habu was only to appear forty years later (EPIGRAPHIC SURVEY (1934)). * Borchardt's preliminary results were published in the same issue of ZAS as Moller's article (BORCHARDT (1911)). ' MOLLER (1912), sect D D c . '°

MOLLER (1911), p. 100.

" This Sixth Dynasty accounting text, the so-called 'Sharuna' papyrus, remains unpublished still. Its hieratic forms for the CSM were not published until 1936, in the posthumous Erganzungsheft to the Hieratische Palaographie due to Grapow and Przybylla. ("MOLLER" (1936), p. 1 s.v. Elephantine nos. 708, 710). MOLLER (1911), p. 101, n2.

300

J. Ritter

F o r s o m e d o z e n years after its publication, M o U e r ' s thesis stirred little interest; for this p e r i o d I h a v e b e e n able to locate only t w o m i n o r references. T h e first is that o f Kurt Sethe in his influential Von Zahlen und ZahlwortenP T h i s is limited to a p a s s i n g m e n t i o n ( p . 7 4 ) o f the H o r u s - e y e interpretation in the context o f the question o f the Egyptian n a m e s for the area a n d capacity submultiples. T h e only details g i v e n here, b o t h c o m i n g fi:om Moller, are the "strange Spielerei" at the origin o f this use'"* a n d a r é s u m é in n o t e o f the m y t h o f the E y e . T h e s e c o n d occurs in the last line o f a discussion b y B a t t i s c o m b e G u n n o f the c o n n e c t i o n in P y r a m i d Spell 3 5 9 b e t w e e n the E y e o f H o r u s a n d a finger-counting ritual; " o n e is involuntarily r e m i n d e d o f the m a t h e m a t i c a l properties o f the Wdit."^^ It w a s o n l y in 1923 that substantive c o m m e n t w a s forthcoming. T h e p u b l i c a t i o n in that year b y T . Eric P e e t o f the second, a n d best, o f the R h i n d m a t h e m a t i c a l p a p y r u s editions included a c o m m e n t a r y on the M o l l e r thesis.'^ M o l l e r ' s sketch is r e p r o d u c e d along with the various fractional values assigned to the eye parts. B u t this is followed b y a denial o f the antiquity o f such a n identification o n t w o g r o u n d s : 1° the hieratic forms " s h o w b u t slight r e s e m b l a n c e to the parts o f the e y e " ' ' a n d 2° the hieroglyphic forms are m u c h later, they d o n o t p r e d a t e D y n a s t y 18 or, for s o m e forms, D y n a s t y 2 0 . T h e frequent u s e o f the hieratic forms in the m e d i c a l texts leads P e e t to speculate that the H o r u s - e y e identification w a s " a n ingenious d i s c o v e r y of s o m e [medical] s c r i b e " since the sacred e y e w a s " s o closely c o n n e c t e d in E g y p t i a n eyes with healing p o w e r . " T h i s c o n c l u d e s P e e t ' s treatment a n d is the first a p p e a r a n c e in print o f w h a t I shall call the weak thesis: the hieratic C S M signs are early b u t b e a r n o relation to the E y e o f H o r u s ; the hieroglyphic versions d o b e a r s u c h a relation, b u t are o f N e w K i n g d o m date at the earliest a n d the result o f later scribal speculation.'* H o w e v e r , the year 1927 s a w the publication o f the first v o l u m e o f y e t another edition o f the R h i n d p a p y r u s , the w o r k o f A m o l d C h a c e , H e n r y M a i m i n g a n d R a y m o n d A r c h i b a l d . " T h e last t w o were m a t h e m a t i c i a n s , the principal author a b u s i n e s s m a n a n d chancellor o f B r o w n University; the w o r k w a s p u b l i s h e d b y a n association o f A m e r i c a n m a t h e m a t i c s teachers a n d w a s a i m e d explicitly at " m a t h e m a t i c i a n s a n d the general p u b l i c " ( p . Ill), rather than Egyptologists. T h e discussion o f capacity m e a s u r e s includes a simple reference to " ' H o r u s - e y e ' fractions" (p. 3 1 ) with reference given to M o l l e r ' s article. N o n e o f P e e t ' s reservations about the origins o f the C S M are e v e n m e n t i o n e d . T h e existence o f t w o n e a r - c o n t e m p o r a n e o u s editions o f the R h i n d P a p y r u s w a s to h a v e a n important i m p a c t o n the future of studies o f E g y p t i a n m a t h e m a t i c s . T h e existence in later years o f t w o distinct coinmunities in this field, Egyptologists a n d

SETHE (1916).

This is reinforced by a later (p. 80) mention of the Sixth Dynasty presence of the complete set of six CSM; an error, since no OK source known at the time contains the smallest two CSM. The references given in note by Sethe are in fact discussions of Ptolemaic evidence for small-end area submuhiples. GUNN (1922), p. 72. PEET (1923a).

"

All citations are from PEET (1923 a), p. 25-26. Peet repeated the weak thesis the same year in his discussion of the MK writing boards with metrological exercises from Akhmim, PEET (1923 b), p. 92. CHACE e/fl/. (1927).

I

'

Closing the Eye of Horus: The Rise and Fall o f Horus-eye Fractions*

301

historians o f m a t h e m a t i c s , each with its o w n ' R h i n d p a p y r u s ' h a s resulted in a n unfortunate split, t o w h i c h w e shall r e t u m at the e n d o f this section.^" It w a s in this s a m e year that t h e definitive form w a s t o b e g i v e n o f the strong thesis. It o p e n s the section o f A l a n G a r d i n e r ' s Egyptian Grammar d e d i c a t e d to weights a n d m e a s u r e s ( t h e initial subsection is r e p r o d u c e d at the b e g i n n i n g o f this article).^' F o r t h e first time, a n d without c o m m e n t a r y , t h e E y e is d r a w n i n r e v e r s e d o r i e n t a t i o n — c o n t r a M o l l e r , it is n o w a left QyeP N o n e o f M o l l e r ' s a r g u m e n t s are r e p r o d u c e d , only his m y t h o l o g i c a l r e c o n s t m c t i o n is t a k e n u p , constituting essentially the w h o l e o f the subsection. T h e s e t w o aspects, tiie left-orientation o f the E y e a n d a discussion limited t o t h e associated myth, a r e to b e found i n virtually e v e r y later m e n t i o n o f the H o m s - e y e thesis, w h e t h e r they explicitly a c k n o w l e d g e G a r d i n e r o r not." A t r o u g h l y t h e s a m e time, the E y e o f H o m s also m a k e s its entry into A d o l f E r m a n ' s work, i n tiie first v o l u m e o f the B e r l i n Worterbuch^^ and, t w o years later, in the fourth edition o f his Àgyptische Grammatik^^ r e p r e s e n t e d in b o t h b y a c o p y o f M o l l e r ' s sketch o f t h e E y e b u t , like G a r d i n e r ' s , r e v e r s e d in orientation ( a n d , exceptionally, without a n y m e n t i o n o f t h e myth).^^ T h e Worterbuch gives a N e o E g y p t i a n date for t h e metrological u s e o f t h e E y e o f H o m s a n d the Belegstelle^^ cites, b e s i d e t h e M o l l e r article, only t h e late c o m p o s i t i o n " T h e T e a c h i n g o f A m e n o p e " . T h i s unplicit e n d o r s e m e n t o f t h e w e a k thesis w a s t o receive further s u p p o r t a few years later. T h e p u b l i c a t i o n i n 1 9 3 0 o f the s e c o n d great m a t h e m a t i c a l p a p y m s , that o f Moscow,^^ w a s t h e o c c a s i o n for the editor, Vasilij S t m v e , t o give a b r i e f r e m i n d e r o f the H o m s - e y e nature o f the C S M (citing P e e t and G a r d i n e r ) a n d a s u g g e s t i o n that a n identification o f the E y e o f R a with the ypf-grain-measure {Amenope 1 8 , 2 3 ) w a s at the origin o f t h e identification o f tiie ifj^^r-grain-measine with t h e E y e o f H o m s . ^ ' F o r this last, S t m v e , following Peet, points t o a m e d i c a l p a p y m s ( p H e a r s t 14, 2 - 3 ) . If Stiiive's m e n t i o n o f a s o i n c e for t h e E y e signs o f t h e C S M i n a m e d i c a l p a p y m s o f the N e w K i n g d o m c a n b e u n d e r s t o o d a s implicit s u p p o r t for tiie w e a k thesis, it w a s Otto N e u g e b a u e r ' s article o f the s a m e year that p r o v i d e d t h e only detailed argiunents for this p o s i t i o n . ' " I n t h e first substantive discussion o f the s t m c t u r e o f t h e E g y p t i a n capacity m e a s u r e s since Griffith's classic 1 8 9 2 article, N e u g e b a u e r a r g u e s ( p . 4 6 ) , o n t h e basis o f calculational p r a c t i c e s i n the

See the discussion in IMHAUSEN, this volume and references therein. GARDINER (1927), § 266.

In addition to the change in orientation, Gardiner modifies the form of — and replacing Moller's outline drawings by line drawings for these two elements. A partial exception is Richard Gillings's influential 1972 work on Egyptian mathematics, where he retums, without argument, to a right Eye (GILLINGS (1972), p. 210-211); he nonetheless takes up other parts of Gardiner's modifications of Moller's original drawing and, like Gardiner, omits all but the mythological justification for the identification. ERMAN and GRAPOW ( 1926 a), p. 402. "

ERMAN (1928), p. 105.

To complete the three classic grammars, Gustave Lefebvre's Grammaire of 1940 follows Gardiner on both points, image of left Eye and mythical reference (LEFEBVRE (1940), p. 111). ^"^

ERMAN and GRAPOW (1926 b), p. 67. STRUVE (1930). STRUVE (1930), p. 51. NEUGEBAUER (1930).

302

J. Ritter

m a t h e m a t i c a l papyri, that there were originally t w o i n d e p e n d e n t systems o f capacity m e a s u r e : one b a s e d o n the hq?t (and hnw) a n d the other o n the ri. S o m e o f the H o r u s - e y e signs are to b e attached to the s e c o n d a n d not the first, i.e., they were originally multiples o f ri a n d n o t fi-actions o f a hqit. T h u s the hieratic equivalent of ^ is to b e u n d e r s t o o d not as 1/32 hq?t but as 10 r?, a n d ( p r o b a b l y ) the hieratic equivalent of ^ as 2 0 r3. W h e n the two systems w e r e put together, the large m e t r o l o g i c a l ' s p a c e ' b e t w e e n the rS a n d the hqit w a s filled with additional hieratic signs, p r e s u m a b l y d r a w n from a H o r u s - e y e interpretation, in a systematized dyadic structure into w h i c h the t w o earlier r? -multiples w e r e integrated. In a last section o n the p a l e o g r a p h y of the capacity signs, N e u g e b a u e r reinforces his a r g u m e n t b y p o i n t i n g out that the hieratic signs for " 1 / 1 6 " a n d " 1 / 3 2 " b e a r n o relation to their s u p p o s e d H o r u s - e y e equivalents but d o a p p e a r to r e s e m b l e r e c o n s t i n c t e d hieratic forms o f hieroglyphic n (20 [ri]) a n d 2> ( 1 0 r?) respectively. A d d i n g to this the fact that all " 1 / 2 " hieroglyphic forms can only b e correlated to their hieroglyphic H o r u s - e y e counterpart b y m e a n s of a reflection (right eye v e r s u s left eye), h e c o n c l u d e s that the introduction of the E y e of H o r u s in this context, like a n u m b e r of the fractions themselves, is of late date. T h e only r e a c t i o n to N e u g e b a u e r ' s assault o n M o l l e r ' s strong thesis I k n o w of w a s a b r i e f note in the following issue of the s a m e j o u m a l b y a n o t h e r historian of ancient m a t h e m a t i c s , K u r t Vogel.^' A n d e v e n h e r e the m a i n thrust of his defense of the traditional u n d e r s t a n d i n g is b a s e d o n a denial o f the i n d e p e n d e n c e o f separate hqit a n d ri systems; n o t i n g that the occasional writing of hieratic " 1 / 1 6 " a n d " 1 / 3 2 " w i t h superfluous dots a b o v e t h e m s h o w their essentially fractional n a t u r e . N e i t h e r h e n o r any later writer save one w o u l d ever r e s p o n d to the question of the validity of the r e s e m b l a n c e o f the H o r u s - e y e parts a n d the C S M signs. ^ T h u s b y the b e g i n n i n g of the nineteen-thirties, the H o r u s - e y e interpretation h a d entered the t w o canonical g r a m m a r s and, especially in the form g i v e n to it b y Gardiner, w a s to b e c o m e the standard interpretation for Egyptologists a n d historians o f science in the s u c c e e d i n g d e c a d e s . T h i s w a s true very early; i n d e e d neither Struve n o r N e u g e b a u e r n o r V o g e l cites M o l l e r ' s article but only Gardiner. M o r e o v e r , since the presentation in the g r a m m a r s w a s necessarily a b b r e v i a t e d with, at m o s t , a short recapitulation o f the p r o p o s e d mythical b a c k g r o u n d b u t without M o l l e r ' s a r g u m e n t s , these last d i s a p p e a r e d c o m p l e t e l y from the literature.^'' T h e authors o f the only substantive article o n capacity mefrology before the nineteen-sixties content t h e m s e l v e s with c o p y i n g b o t h G a r d i n e r ' s picture a n d part o f his text (p. 90-91).^"*

VOGEL ( 1 9 3 0 ) .

The only later reference to Neugebauer's publication is the final phrase of the article on the Eye of Horus in the Lexikon der Àgyptologie (MOLLER-WOLLERMANN (1985), col. 824) where the author defends the weak thesis, citing specifically the paleographical section. Later references to Moller are almost invariably "priority references"; citations of historical interest for purposes of acknowledging priority, but without use (or often real knowledge) of the original. The illustration, if there is one, and the accompanying text are almost invariably Gardiner's. LUCAS and ROWE ( 1 9 4 0 ) . The reason for the excursus by archeologist Alan Rowe on Horus-eye GSM signs in this article is not clear since, despite his statement on p. 9 1 , such signs do not appear on any of the inscribed vessels under discussion.

Closing the Eye of Horus: The Rise and Fall of 'Horus-eye Fractions'

303

If w e try to u n d e r s t a n d the d y n a m i c s of the vicissitudes of the H o m s - e y e thesis over this twenty-year period, w e are led, I feel, to t w o specificities of the d o m a i n . O n the o n e h a n d , the a b s e n c e of n e w data or e v e n the p u b l i c a t i o n o f the old, led to a freezing of the state o f the debate o n c e initial positions h a d b e e n staked out. T h e e n s h r i n e m e n t o f an a b b r e v i a t e d version of M o l l e r ' s strong thesis in the form o f a drawing a n d a little m y t h o l o g i c a l story in the necessarily confined c o n t e x t o f a g r a m m a r settled the q u e s t i o n for the great majority o f m a i n s t r e a m Egyptologists, particularly as, for m o s t o f t h e m their interest did n o t generally m n in this p e r i o d to accounting texts or the like. T h i s e b b of interest in matters metrological w a s a c c e n t u a t e d b y the rise of specialization within E g y p t o l o g y , w i t h matters of ' s c i e n c e ' (including m e t r o l o g y ) in E g y p t c o m i n g increasingly u n d e r the exclusive p u r v i e w of historians of science or of m e d i c i n e , often without first-hand k n o w l e d g e o f the E g y p t o l o g i c a l literatme. C o m i n g from a different d o m a i n , with questions a n d tools d e v e l o p e d e l s e w h e r e , the interests o f these last, imtil very recently, simply did n o t b e a r on m e t r o l o g i c a l questions, v i e w e d only as i m p e d i m e n t a to the study o f the ' s e r i o u s ' c o n c e p t u a l m a t h e m a t i c a l or m e d i c a l theories extractable from the texts. T h e m a t h e m a t i c i a n ' s R h i n d p a p y m s was that of C h a c e et al., P e e t ' s e d i t i o n — a n d his r e s e r v a t i o n s — r e m a i n i n g for the m o s t part i m k n o w n ; thus practically all treatments of E g y p t i a n m a t h e m a t i c s , if they m e n t i o n weights a n d m e a s u r e s at all, uncritically a c c e p t the strong H o m s - e y e thesis. In m e d i c i n e the situation is even simpler. A l m o s t n o n e of the m a j o r p u b l i c a t i o n s o n E g y p t i a n m e d i c i n e e v e n m e n t i o n metrological questions and t h o s e few w h o d o often m a k e n o m e n t i o n o f the E y e of H o m s . E v e n the m a s s i v e Grundriss der Medizin der alten Àgypter reserves its w h o l e discussion of capacity units to the final s u p p l e m e n t a r y v o l u m e ' ^ w h e r e it ignores c o m p l e t e l y the nature of the C S M signs, a b l a c k o u t e x t e n d e d to any m e n t i o n o f capacity units at all in the 2 5 0 p a g e introduction to the m o s t recent edition of m e d i c a l text translations.'^ T h e division o f the field a n d the delegation o f interest in the C S M signs to historians of science for nearly half a century h a d a n o t h e r effect. W h e n the old sources u s e d b y M o l l e r in his arguments w e r e finally p u b l i s h e d a n d n e w e v i d e n c e c o n c e m i n g the signs started to a p p e a r in m a i n s t t e a m E g y p t o l o g i c a l p u b l i c a t i o n s , it h a d a l m o s t n o impact. T h o s e w h o r e a d s u c h things did not h a v e m e t r o l o g i c a l questions in m i n d a n d historians of science, e v e n with the rebirth o f interest in the nineteen-eighties of questions c o n c e m i n g m e t r o l o g y a n d the d e v e l o p m e n t of writing, h a d n o accesss to this n e w information. It is thus to a s u r v e y o f this n e w material that w e n o w t u m .

II. New Evidence It w a s the p u b l i c a t i o n of n e w t e x t s — a n d the reinterpretation of p r e v i o u s l y - k n o w n d o c u m e n t s — i n the 1960s a n d 1970s w h i c h permits a n e w l o o k at the question. T h e s e n e w a p p r o a c h e s o p e n u p a g l i m p s e o f the situation in the third m i l l e n n i u m as well as access to s o m e o f the then-unpublished e v i d e n c e cited b y M o l l e r . W e shall see that the results o f b o t h tend to the s a m e end, a w e a k e n i n g o f the w h o l e idea of

"

VON DEINES et ai BARDINE? ( 1 9 9 5 ) .

( 1 9 7 3 ) , p.

1-16.

304

J. Ritter

" H o r u s - e y e fractions". I n this section w e shall r e s u m e t h e n e w e v i d e n c e , b o t h hieroglyphic a n d hieratic, s t e m m i n g from t h i r d - m i l l e n n i u m s o u r c e s . T h e d i s c o v e r y a n d publication^' o f the Fifth D y n a s t y t o m b o f N j - ^ n h - H n m a n d H n m - h t p at S a q q a r a , a particularly rich source o f mefrological information,^* g a v e the first secure g l i m p s e o f a hieroglyphic H o r u s - e y e fraction in the O l d K i n g d o m . I n particular, t h e w e l l - p r e s e r v e d m a r k e t scene o n t h e n o r t h wall o f t h e " g a t e r o o m " s h o w s a client, c a n y i n g a n u m b e r o f objects, a p p r o a c h i n g a fruit vendor:^^

T h e fruit seller states: " G i v e ( m e ) this ^

g r a i n - m e a s u r e s o that (I) m a y m e a s u r e o u t

this in it.". T h o u g h t h e detailed analysis o f t h e c u s t o m e r ' s r e p l y is unclear,'*" t h e d o u b l e p r e s e n c e o f a clearly m a r k e d " 1 / 2 " C S M sign following t h e g r a i n - m e a s u r e sign

in b o t h p a r t s o f the dialogue is certain.

P o s s i b l e e v i d e n c e for hieroglyphic C S M h a d b e e n in fact available e v e n earlier, in t h e p u b l i c a t i o n s o f A r c h a i c p e r i o d t o m b s at S a q q a r a a n d H e l w a n d u r i n g t h e interw a r p e r i o d . T h e s e r e v e a l e d S e c o n d D y n a s t y offering stelae o n w h i c h w e r e inscribed b r e a d lists o f a n e w kind, o n e in w h i c h capacity s y s t e m signs, i n c l u d i n g < a n d o, a p p a r e n t l y a p p e a r e d a s p a r t o f the n a m e o f the bread.'*' A s w a s first s u g g e s t e d b y S e l i m H a s s a n , a m o n g these t h e " 1 / 2 " C S M o c c a s i o n a l l y figures."*^ T h e earliest clear e x a m p l e o f this sign is o n t h e stela o f a S e c o n d D y n a s t y royal p r i n c e s s n a m e d S j hfnr:'.43

MOUSSA and ALTENMÛLLER ( 1977).

This includes not only capacity measures but also information on length and weight systems, see RITTER (forthcoming). MOUSSA and ALTENMÛLLER (1977), fig. 10 and pi. 24.

Moussa and Altenmiiller suggest (p. 84), rather unconvincingly, that the shopper replies: "Oh foreigner(?), should I give (yOu) the director's half grain measure?" The identification of breadstuffs by content in grain quantity recalls the p^vv-baking ratios in later mathematical texts and offering scenes. For the former, see now IMHAUSEN (forthcoming); the latter will be discussed in RITTER (forthcoming). HASSAN (1944), p. 95, n2. Followed in this by Winfried BARTA (1963), p. 16 and 27, and Jochem KAHL (1994), p. 443 s.v. D 11. Note that this interpretation is combated by Peter KAPLONY (1963), p. 248, who sees here a form of the sign to be read wcfb. However, as his argument is based precisely on the dissimilarity between the sign and its Horus-eye counterpart, it begs the question raised here. QuiBELL(1923),pl.27.

/

Closing the Eye of Horus: The Rise and Fall of'Horus-eye Fractions'

305

0 cSfi

T h e a p p e a r a n c e o f " 1 / 2 " in alternance with the clear c a p a c i t y - m e a s u r e sign o a n d the bread name

dp{(),

well-known

from

later sources, r e n d e r s this

interpretation

plausible at least. T h e difficulty is that this sort o f b r e a d n a m e d i s a p p e a r s early in the T h i r d D y n a s t y , so that later p h o n e t i c or other clues are unavailable. H o w e v e r if this interpretation is correct, there are four sets of the hieroglyphic form o f the " 1 / 2 " sign at a v e r y early stage.'*'* ( S e e T a b l e I). Thus

the

third-millenniiun

hieroglyphic

evidence—like

every

hieroglyphic

instance until D y n a s t y 1 9 — i n v o l v e s only the largest C S M . T h e mutability of f o r m e v e n in those cases w h e r e the s a m e text contains m o r e than o n e c o p y o f the sign (inscriptions o f Sj-hfiir and, especially, of Nj-'^nh-Hnm a n d H i u n - h t p ) s h o w s the kind of variability that n o r m a l l y is typical of attempts to r e n d e r in h i e r o g l y p h i c form signs c o m i n g from a hieratic metrological context.''^ B u t n o n e of the variants s h o w the b r o a d o p e n i n g a n d p o i n t e d tail, typical of the small white of the E y e o f H o r u s , <(, which, in the N e w - K i n g d o m e x a m p l e s o f this C S M sign, h a d so largely i m p r e s s e d M o l l e r a n d others. T i n n i n g to hieratic

C S M signs in the third milleimiiun, practically all the

e v i d e n c e is a g a i n c o m p a r a t i v e l y recently available. W e recall that the o n l y thirdm i l l e i m i u m e v i d e n c e k n o w n to M o l l e r w a s a n u n p u b l i s h e d sixth-dynasty accoimting p a p y r u s ( p B e r l i n 10500 A & B ) , e x a m p l e s of w h o s e C S M signs w e r e first p u b l i s h e d in 1936, in the Erganzungsheft

to the Hieratische

Palaographie^^

E x a m i n i n g the

e x a m p l e s d r a w n from these papyri"*' a n d s h o w n in T a b l e I, the diagnostic " 1 / 1 6 " sign,

is the o n l y one to s h o w a p o s s i b l y significant difference w i t h its later

counterparts, such as the D y n a s t y 12 form, \ P

T h a t this difference is i n d e e d

significant will b e c o m e apparent as w e n o w t u m to e v e n earlier forms.

^ These are the Second Dynasty stele of Sj-hfnr (QUIBELL (1923), pi. 27) and Jmtj (SMITH and SIMPSON (1981), pi. 32) from Saqqara and Dw3t from Helwan (SAAD (1957), pi. 6 and KÔHLER and BIRRELL (forthcoming)); and the Third Dynasty stela of Jry-stt / J'^-Hjpw (?) from Saqqara (KAPLONY (1966), pi. 4). Hassan's proposal for the presence of this sign on the "Bankfield" stela (GARDINER (1917), pi. 55) is not convincing. Compare the hieroglyphic forms of the area submultiples, e.g., the Old Kingdom examples in SCHAFER (1902), p. 34-36 and JACQUET-GORDON (1962), p. 3 n2 and passim. In general, see BAER (1956) and RITTER (forthcoming). Examples of these signs (for all but the smallest two CSM) have been republished in GOEDICKE (1988), p. 70-71. 1 should like to thank the Staatliche Museen zu Berlin for providing me with photographs of these papyri. I have been unable to locate an appearance of the sign "1/32" in this papyrus, apparently

306

J. Ritter

It w a s n o t until 1 9 6 8 that a n edition o f O l d K i n g d o m a c c o u n t i n g texts w a s p u b l i s h e d / ^ with n u m e r o u s fragments from the archives o f the m o r t u a r y t e m p l e o f the fifth-dynasty p h a r a o h Nfr-jr-k3-R*^ K 3 - k 3 j i n Abusir. O n l y t h e largest three o f the C S M s i g n s — t h e least d i a g n o s t i c — a r e r e p r e s e n t e d in these fragments a n d their essential identity with those o f the S h a n m a p a p y r i is clear. ^° T h e oldest k n o w n a c c o u n t i n g p a p y r i are those found at the site o f G e b e l e i n b y Giulio Farina in 1935 a n d d a t e d t o the F o u r t h Dynasty.^' T h o u g h still u n p u b l i s h e d , the p a l e o g r a p h y o f their capacity s y s t e m signs h a s b e e n t h e subject o f a n i m p o r t a n t article b y P a u l e Posener-Kriéger,^^ including a table o f sign forms from w h i c h the C S M entries for t h e F o u r t h D y n a s t y in T a b l e I h a v e b e e n taken. H e r e t h e fiiU range o f C S M signs is available, a situation w h i c h h a s n o parallel until t h e M i d d l e Kingdom. T h e information c o n t a i n e d i n these t h i r d - m i l l e n n i u m hieratic s o u r c e s — a n d especially in the G e b e l e i n p a p y r i — s e e m s t o m e decisive against t h e strong thesis o f M o l l e r . N o t o n l y d o these p a p y r i confirm clearly t h e line o f e v o l u t i o n o f the " 1 / 1 6 " sign, a l r e a d y s u g g e s t e d in t h e S h a n m a p a p y r u s , b u t they s h o w that t h e " 1 / 3 2 " a n d " 1 / 6 4 " t o o are quite different than w h a t w o u l d b e e x p e c t e d from t h e M i d d l e a n d N e w K i n g d o m instances. E v e n t h e non-diagnostic large C S M signs r e s e r v e s o m e surprises; the " 1 / 8 " s h o w s n o sign o f curvature, t h e " 1 / 4 " is m o r e r e m i n i s c e n t o f a solid d o t than t h e o p e n versions occurring later. A l l this points t o t h e fact that n o n e o f the early C S M hieratic s i g n s — w i t h t h e c o n c e i v a b l e e x c e p t i o n o f " 1 / 2 " — r e s e m b l e their putative hieroglyphic H o r u s - e y e counterparts. A n o t h e r p o s s i b l y usefiil a v e n u e o f c o m p a r i s o n might h a v e b e e n that b e t w e e n hieratic a n d hieroglyphic C S M signs a n d those representing parts o f the ( H o r u s ) e y e in n o n - m e t r o l o g i c a l t h i r d - m i l l e n n i u m contexts. H e r e h o w e v e r t h e r e is a difficulty; neither o n the h i e r o g l y p h i c side, w h e r e only t h e " 1 / 2 " sign ( D 11) c o m e s into question, n o r o n t h e hieratic, w h e r e all t h e eyeparts ( D 1 1 - D 16) a r e relevant, d o w e k n o w o f a n y non-mefrological u s e o f these signs before the M i d d l e K i n g d o m at t h e e a r l i e s t . " A n d this in itself is a difficulty for the sfrong thesis. If the u s e o f parts o f the H o r u s - e y e w e r e indeed, a s M o l l e r h a d it, "Spielereien", t h e n their total a b s e n c e from the early written r e c o r d in a n y other u s e t h a n a s C S M indicators r o b s t h e g a m e o f m u c h o f its point. T h e trend o f the e v i d e n c e o f third-milleimium sign forms that h a v e a c c u m u l a t e d since t h e e n d o f the d e b a t e in t h e nineteen-thirties is clear. T h e fiirther b a c k w e g o the fiirther the hieratic signs diverge from their s u p p o s e d H o r u s - e y e c o u n t e r p a r t s . T h e inconsistency b e t w e e n t h e hieratic signs for " 1 / 1 6 " a n d " 1 / 3 2 " a n d their

referred to in POSENER-KRIÉGER (1994), p. 270. Nor is the sign indicated for this papyrus either in "MOLLER" (1936) or in GOEDICKE (1988). POSENER-KRIÉGER and DE CENI VAL (1968), POSENER-KRIÉGER ( 1976).

^° POSENER-KRIÉGER (1994 b), p. 269, states that the R'-nfrf archives contain examples not only of these three but also of the next two, "1/16" and "1/32". Pending publication, nothing further can be said. No significant report was ever published by the excavator. See for now POSENER-KRIÉGER (1975) and DONADONI ROVERI (1993). I should like to take this opportunity to acknowledge my deep gratitude to the late Mme. Posener-Kriéger for her generosity in sharing her copies of the Gebelein papyri with me. POSENER-KRIÉGER ( 1994 b).

" For the hieroglyphic case, cf KAHL (1994), p. 443; GARDINER (1957), p. 451-452; for the hieratic, cf GOEDICKE (1988), p. 6.

Closing the Eye of Horus: The Rise and Fall of 'Horus-eye Fractions'

307

p u r p o r t e d liieroglypliic originals, a l r e a d y signaled b y N e u g e b a u e r (see last section), is e v e n greater for the original third-milleimium forms a n d the others r e v e a l similar difficulties. T h e hieroglyphic " 1 / 2 " signs v a r y w i d e l y a m o n g t h e m s e l v e s a n d all differ from their N e w K i n g d o m d e s c e n d a n t s . W e r e o n e to offer a p r e l m i i n a r y synthesis o f t h e third-milleimium e v i d e n c e , it w o u l d b e t o i m d e r s c o r e t h e n o n representational nature o f the hieratic signs a n d the fluidity a n d c o n s e q u e n t p r o b a b l e s e c o n d a r y nature o f the sole venture into creating a hieroglyphic equivalent o f this essentially absfract mefrological s t r u c t m e .

III. A New Look at Old Evidence If, a s I t h m k , t h e e v i d e n c e p r e s e n t e d m the last section disposes o f M o l l e r ' s sfrong thesis, it is still possible that the w e a k thesis, p r o p o s e d b y P e e t a n d N e u g e b a u e r , is true; that, t h o u g h t h e C S M signs a n d (parts of) t h e E y e o f H o r u s w e r e o f i n d e p e n d e n t thfrd-millenniiun origins, m a later p e r i o d ari identification w a s m a d e b e t w e e n t h e m . I n v i e w o f what h a s b e e n said i n t h e first section, all positive a r g u m e n t s for s u c h a n identification—at a n y p e r i o d o f E g y p t i a n h i s t o r y — g o b a c k to M o l l e r ' s original article. W e recall that t h e principal textual e v i d e n c e a d d u c e d b y M o l l e r for t h e identification o f the H o r u s - e y e parts a n d t h e C S M w a s a section o f the inscriptions foimd o n votive cubits, dating from the N e w K i n g d o m a n d later. T h e s e stone cubitlong objects w i t h a characteristic p o l y g o n a l cross section are inscribed w i t h a large n u m b e r o f texts r e l a t m g geographical, mefrological a n d cultic information: linear m e a s u r e m e n t s o f Egypt, s h a d o w - c l o c k a n d clepsydre tables, relations o f g o d s w i t h subdivisions o f the cubit, etc.^'* Since their m e n t i o n b y M o l l e r , a n u m b e r o f n e w cubits h a v e b e e n found, b u t thefr p u b l i c a t i o n h a s h a d t o await A d e l h e i d SchlottS c h w a b ' s thesis ( 1 9 7 3 , 1981) a n d s u b s e q u e n t pubhcations.^^ T h e section t o w h i c h M o l l e r referred is " T e x t j " in S c h l o t t - S c h w a b ' s terminology^^ a n d consists o f s o m e 14 cases (representing "finger" divisions o f the cubit), n u m b e r e d 7 - 2 0 b y her. T h e difficulty o f t h e section is such h o w e v e r that s h e r e n o u n c e s a n y a t t e m p t at a c o n n e c t e d analysis;^' i n d e e d that t h e section is a unity s e e m s doubtful — w e shall here o n l y consider the first nine cases, 7 - 1 5 , those o n w h i c h the c l a i m for the w e a k thesis rests. P h o t o g r a p h s o r copies o f the 11 more-or-less extant versions o f this s u b s e c t i o n c a n b e found i n T a b l e 11.^^ A s c a n b e seen there, e a c h case save 11 a n d t h e last contains o n e , t w o o r three hieroglyphs, generally d r a w n from parts o f t h e b o d y ( G a r d i n e r ' s Signlist Section D ) , at the t o p followed b y one o r m o r e n u m e r i c a l signs in the absfract s y s t e m — d i s c r e t e n u m b e r signs and fractional signs.^' BORCHARDT ( 1 9 2 0 ) , SCHLOTT-SCHWAB ( 1 9 8 1 ) .

The thesis was first published in 1 9 6 9 . The 1 9 8 1 réédition contains new photographic plates and a supplement to the bibliography (in the Vonvort) but is otherwise unchanged from the original publication. Subsequent publications that provide additional parts of Text j and are used here are GABRA ( 1 9 6 9 ) and ZIVIE ( 1 9 7 2 ) . "

"

SCHLOTT-SCHWAB ( 1 9 8 1 ) , p. 5 1 - 5 2 .

"Ich kann den Text nicht erklaren." SCHLOTT-SCHWAB ( 1 9 8 1 ), p. 5 2 . The last three cubit fragments (IX-XI) contain neither the hieratic signs of case 15 nor the hieroglyphic signs of 7-14 and will be considered here only when the question of numerical values arises. For the full section see SCHLOTT-SCHWAB ( 1 9 8 1 ) , p. 5 1 . For this system see RITTER ( 2 0 0 1 ), p. 1 2 1 .

308

J. Ritter

T h e h i e r o g l y p h s o f cases 7 - 1 0 are apparently in part identical with those u s e d for parts of the (left) E y e of H o r u s (though III has a right eye). T h e n o n - n u m e r i c a l h i e r o g l y p h s p r e s e n t are the lower half o f the E y e o f H o r u s (case 7), " 1 / 8 " (8), " 1 / 4 " (9), " 1 / 2 " a n d " 1 / 4 " (10). T h e following case, 1 1 , contains o n l y n u m e r i c a l signs (with the p o s s i b l e exception o f the m u c h d a m a g e d V I , w h i c h m a y c o n t a i n " 1 / 1 6 " ) . C a s e 12, written right to left, h a s a n apparently feminine w o r d written w i t h the hair sign, " ^ o . C a s e 13 r e s u m e s the left to right orientation with the (ordinary) eye ^ />(?), ' e y e ?; m a k i n g ? ' T h e next-to-last case o f o u r subsection, 14, contains rdj ^, ' h i g h l a n d s ? j o y ? ' C a s e 15 b r e a k s with this structure: jpt jrt mImV 1^ (var. -f in VIII) d'^rnw mi'^, ^jpt (-grain m e a s u r e ) m a d e of units (?) o f true e l e c t r u m ' w i t h a capacity quantity written using hieratic signs. T h e first thing that strikes one, in v i e w o f M o l l e r ' s insistence that the cubits give a clear c o n n e c t i o n b e t w e e n the hieratic a n d hieroglyphic C S M forms, is that the apparent hieratic signs for capacity m e a s u r e s are found in case 15, well r e m o v e d from those of the hieroglyphic candidates, cases 7 - 1 0 . M o r e o v e r the particular hieratic c o m b i n a t i o n in case 15, " 1 / 8 " + " 1 / 6 4 " , or its variant " 1 / 6 4 " (?), is n o t r e p r e s e n t e d in a n y o f the other c o m p a r t m e n t s . A n o t h e r p r o b l e m is p r e s e n t e d b y the sign in case 7; the sign a p p e a r i n g there is a separate a n d w e l l - k n o w n h i e r o g l y p h ( D 17), representing the b o t t o m h a l f o f the Eye o f H o r u s a n d attested since the M i d d l e K i n g d o m as a l o g o g r a m , tjt, or determinative in w o r d s c o n c e r n e d with ' i m a g e ' and, in m a t h e m a t i c a l p a p y r i , with 'fraction'. T h o u g h M o l l e r claims, n o d o u b t o n graphical g r o u n d s , that here the sign represents a ligature of " 1 / 3 2 + 1/64", in metrological contexts the t w o signs, w h e n a p p e a r i n g together, are n e v e r written in l i g a t u r e — c f M e d i n e t H a b u 3 7 4 , 3 8 6 , 4 4 7 , 985.^° Finally w e n o t e that, b a s e d on their n o n - n u m e r i c a l signs, cases 1 2 - 1 4 s e e m to separate themselves from those p r e c e d i n g : the r e v e r s e d orientation o f 12 a n d their use of c o m m o n signs, u n c o n n e c t e d with the wdit eye. T h o u g h i g n o r e d b y M o l l e r , they t o o are associated with n u m b e r s and s e e m to p l a y the s a m e fimctional role as the u n d o u b t e d H o r u s - e y e parts. T h e impression g i v e n b y the w h o l e s u b s e c t i o n of the votive cubit a p p e a r s to b e a n association of absfract n u m b e r s a n d h i e r o g l y p h s representing parts o f the head, but the relation o f such forms with case 15, the only possible reference to C S M , is far from obvious.^' T u m i n g n o w to the question o f the values o f the n u m b e r s a p p e a r i n g in cases 7 14, w e face difficulties at least equal to those p r e s e n t e d b y the n o n - n u m e r i c a l h i e r o g l y p h s . N u m e r i c a l variants a m o n g the different cubits, ambiguities of g r o u p i n g s of signs a n d of notatiónal technique (additive or multiplicative) p r e v e n t any i m m e d i a t e u n d e r s t a n d i n g . Let u s first deal with cases 1 2 - 1 4 . T h e y a p p e a r to form a unity; to the reasons a d d u c e d a b o v e o n the b a s i s of their n o n - n u m e r i c a l sign repertory c a n b e a d d e d the fact that they are the o n l y cases to contain high-end n u m e r i c a l signs (and only these); e v e n s u m s of the other cases leave us far from such large v a l u e s . U n l i k e cases 7 - 1 1 , the n u m e r i c a l values for these cases are the s a m e in n e a r l y all the votive

^'^ EPIGRAPHIC SURVEY ( 1 9 3 4 ) . Similarly, unknown. If the strange use of hieratic signs understood as a definition of the relation then the choice of non-matching values for

the ligature in II 10 for " 1 / 2 " + " 1 / 4 " is otherwise in an otherwise hieroglyphic context is to be between hieratic CSM and Horus-eye signs, why the two sides?

Closing the Eye of Horus: The Rise and Fall of ' Horus-eye Fractions '

309

cubits o n w h i c h they are c o m p l e t e l y (I, V I I ) or partially (VIII) e x t a n t — o n l y X I presents a variant reading. C a s e 12 w i t h 1 0 0 0 0 0 0 a n d 14 with a clear multiplicative writing o f 6 X 100 0 0 0 , that is 6 0 0 0 0 0 (except for X I which a p p a r e n t l y has a m i s r e a d i n g of T as ^

as well as only 5 strokes b e n e a t h '^f^.

C a s e 13 h o w e v e r

presents p r o b l e m s ; All texts h a v e 3 "^, e a c h followed b y a stroke, as well as a g r o u p i n g o f 4 strokes. T h e t e m p t a t i o n is strong to r e a d this also as a multiplicative niunber, 4 x 100 0 0 0 , for then 12 w o u l d b e the s u m o f 13 a n d 14. S u c h a r e a d i n g though, w i t h a plural

b e i n g multiplied, is contrary to standard practices.^'

B u t if 1 2 - 1 4 p o s e s o m e difficulties, 7 - 1 1 are nearly i m p e n e t r a b l e . First of all, the cubits s h o w n u m e r o u s variations; 7 alone offers various g r o u p i n g s o f the 6 unit strokes w h i c h yield 3 0 0 (^ + stroke) + 3 (part o f 1/13 ? = I, II); 3 0 0 ( 0 + 3 + 3 (part o f 1/13 ? = I V ) ; 3 0 0 ( 0 + 6 (part o f 1/16 ? = IX, X or part o f 1/10 + 1/6 = V ) — o r j u s t c o n c e i v a b l y in a multiplicative writing 3 (or 6) times 100 (written as a plural, see a b o v e ) . S e c o n d l y it is n o t clear h o w m a n y n u m b e r s are written in e a c h c a s e ; is 8 in II to b e r e a d 1040 — 22 or 1062 (I has 1000 + (2+n) x 10 — 10 + 2 ( + « ) x 1 or as t w o n u m b e r s , 1040 a n d 22 (II) or as 1000 + ( 2 + n ) x 10 + 12(+)? G i v e n so m u c h a m b i g u i t y it is difficult to a d v a n c e far in the c o m p r e h e n s i o n of the n u m e r i c a l s c h e m e . H o w e v e r certain suggestive points c a n b e discerned. T a k i n g seriously the division into t w o sets o f n u m b e r s for e a c h c o m p a r t m e n t , w e c a n r e a d 7 - 1 0 in the following way:^" 7: K , 1300 ( V , I, II, I X * , X*) — 1/10 1/6 ( V ) or 1/16 (X, I V * , I X * , I*, II*) 8: 1/8 1040(11,1) — 2 2 (II, IV*) 9: 1/4 2 1 0 0 (II, I, X * ) — 4 2 (II, I*, X*, XI*) 1 0 : 1 / 2 + 1/4 6 3 0 0 (II) — 1/8 (II) or 1/11 (XI, I*) W e concentrate h e r e o n the C S M signs h e a d i n g e a c h c o m p a r t m e n t a n d the first, largest n u m b e r in each. N o w the h i e r o g l y p h in 8 is 1/2 the C S M value 1/4 in 9; a n d i n d e e d the n u m b e r 1040 is quite close to one-half the value 2 1 0 0 . Similarly, the C S M hieroglyphic signs in 10, 1/2 + 1/4 = 3/4, i.e., 3 times the value o f the C S M sign m 9, p r e c e d e the n u m b e r 6 3 0 0 = 3 x 2 1 0 0 . Finally s u c h an i m d e r s t a n d i n g of the text implies that for the C S M sign in 7 to b e consistent w i t h 8, it m u s t b e a r the s a m e p r o p o r t i o n to 1/8 that 1300 d o e s to 1040, i.e., 1 3 0 0 / 1 0 4 0 = 5/4. T h u s = 5/4 X 1/8 = 5/32 = 4/32 + 1/32 = 1/8 + 1/32. I n the votive cubits then, the n u m e r i c a l hapax ^^ w o u l d represent the fraction 1/8 + 1/32—and not 1/32 + 1/64 as p r o p o s e d b y Moller. T h o u g h n o t w h a t o n e m i g h t expect, these are nonetheless dimidiated fractions. Finally, note that the n u m b e r given in 1 1 , 3 0 4 5 0 (I, V I I * ) , is not the s u m o f the p r o c e e d m g cases. H o w e v e r a n y s u c h attempt at a n u n d e r s t a n d i n g of this difficult section requires a careful choice of cubit a n d reading, selecting other c o m b i n a t i o n s of r e a d i n g s from I have not been able to collate this cubit for which, like Schlott-Schwab, I rely on Lepsius's old copy (LEPSIUS (1865), fig. Ill b). One might conjecture a reading along the lines of hfnw 4, 'hundred-thousands, 4'. The Roman numerals in parentheses following each number give the cubits which justify the reading; an asterisk following the numeral means "damaged but consistent with the reading." Note that the 2100 in 9 and 6300 in 10 are based on multiplicative readings, justified by the otherwise incomprehensible position of the unit strokes.

310

J. Ritter

other cubits w o u l d h a v e p r o d u c e d quite different results. T h e s e v e r y inconsistencies a m o n g the different copies are indicative of the lack o f clear c o m p r e h e n s i o n b y those w h o p r o d u c e d these objects o f the sense of the information c o n t a i n e d in them; a situation rather typical of esoteric material w h i c h has little m e a n i n g outside a restricted circle u n d e r g o i n g r e c o p y i n g over long p e r i o d s of time but not wholly c o n s o n a n t with a s u p p o s e d w i d e s p r e a d and general revision o f a n u n d e r s t a n d i n g of the nature of important elements of writing. T h u s , e v e n w e r e w e able to s h o w that the tantalizing b u t c o m p l e t e l y inconclusive e v i d e n c e of the votive cubits did represent a reinterpretation of the C S M signs as H o r u s - e y e c o m p o n e n t s s o m e t i m e in the N e w K i n g d o m , the question w o u l d still b e o p e n as to the u s e o f such a n e w interpretation in actual metrological contexts. T o e x a m i n e this question w e m u s t t u m to the e v i d e n c e from N e w K i n g d o m C S M u s e — w h i c h m e a n s essentially hieratic administrative and m e d i c a l texts a n d in hieroglyphic t e m p l e festival inscriptions. A s T a b l e I s h o w s , the objections to relations b e t w e e n the hieratic signs and their putative hieroglyphic equivalents r e m a i n in force; n o r e m o d e l i n g of the hieratic signs for the diagnostic " 1 / 1 6 " and " 1 / 3 2 " to m o r e closely m a t c h their hieroglyphic equivalents takes p l a c e . T h e only significant difference c o m e s in the change in " 1 / 8 " b e t w e e n the M i d d l e a n d N e w Kingdoms,^^ w h e n the the previously sfraight diagonal line d e v e l o p s a c u r v e or kink w h i c h m i g h t b e associated with an attempt to m i m i c the c o r r e s p o n d i n g non-linearity in the e y e b r o w hieroglyph. H o w e v e r , even in this case, n o attempt is m a d e to modify the vertical orientation to bring it in line with the strictly horizontal alignment of the E y e of H o m s e y e b r o w a n d the other n o n - m e t r o l o g i c a l hieratic representations o f the sign.^^ F u r t h e r m o r e , c h a n g e s in the hieroglyphic signs present fiarther e v i d e n c e against such a n iimovation. T h e r e p l a c e m e n t of the older C S M " 1 / 2 " sign b y the standard fractional ^ (gs) is seen for the first time in capacity imit inscriptions in precisely this p e r i o d — i n A m e n o p h i s I V ' s calendar at K a m a k ( D y n a s t y 18), R a m e s s e s I P s at A b y d o s ( D y n a s t y 19) a n d R a m e s s e s I V ' s wall stela at K a m a k ( D y n a s t y 2 0 ) . Further, the a p p a r e n t reversal o f orientation of the classic C S M " 1 / 2 " sign in R a m e s s e s I P s c a l e n d a r inscription at K a m a k m a y h a v e b e e n influenced b y j u s t such a n a l t e m a t i v e u s e o f « = . B u t neither the u s e n o r the influence o f this last is c o m p a t i b l e with an u n d e r s t a n d i n g b y the E g y p t i a n engraver of the C S M units as essential elements of the " s o u n d E y e " o f H o m s . A s far as the limited n u m b e r of published sources allows us to j u d g e , the u s e of the special C S M signs a p p e a r s to diminish in the Late a n d P t o l e m a i c p e r i o d s . T h e hieroglyphic forms s e e m not to h a v e b e e n used. T h e administiative p a p y r i s h o w n o essential c h a n g e in the sign forms,^' while the P t o l e m a i c m e d i c a l ( R u b e n s o h n ) fragments illustrate the virtually total r e p l a c e m e n t of the classic C S M signs b y their

" Or, more exactly. Dynasty 17, for the Rhind papyrus examples already show the New Kingdom form. ^ The unique diagonal instance, appearing in MOLLER (1909 a), no. 89 (Sinuhe B 58), is a conjectural identification of a problematic determinative in the word '^nw, which everywhere else is determined by or <^ (D7 or D8). He has not been followed in this by the dictionaries, nor shall we here do so. " The Dynasty 22 accounting papyrus now Berlin 23252+23253 (formerly Berlin 3081) is used by MOLLER (1912), nos. 708-709, 713, as the sole example of Late period C S M signs.

CLOSING THE EYE OF HORUS: THE RISE AND FALL OF 'HORUS-EYE FRACTIONS'

311

ordinary fraction equivalents except in c o m b i n a t i o n s of fractions.^* W h e n the hieratic C S M signs are used, h o w e v e r , they differ in n o w a y from their N e w K i n g d o m forms. T h e e v i d e n c e for the w e a k thesis, it s e e m s to m e , stands as follows. If it, unlike the strong thesis, caimot b e ruled out completely, the positive e v i d e n c e for it s e e m s to b e m e a g r e in the e x t r e m e . T h e difficulties and ambiguities o f the votive-cubit subsection p r o v i d e s o n l y suggestive e v i d e n c e for a possible attempt to u n d e r s t a n d the traditional hieratic C S M signs as parts of the E y e of H o r u s . T h e N e w K i n g d o m sign forms, as e v i d e n c e d in the festival inscriptions and the metrological a n d medical p a p y r i , p o i n t in rather a different direction. T h e p r o o f of w o r d p l a y at the origin of concepts e v e n in so unlikely a d o m a i n as that of capacity m e a s i n e s , that M o l l e r h a d seen in the close a n d early relation b e t w e e n the m y t h o f the E y e of H o r u s a n d capacity submultiples c a n certainly n o longer b e c o n s i d e r e d tenable; the early hieratic form o f the C S M signs, the derivative forms a n d fluctuations in the rare third-millennium h i e r o g l y p h i c equivalents for " 1 / 2 " , all rule the strong thesis out of court, as h a d a l r e a d y b e e n suspected b y a n u m b e r of critics in the nineteen-twenties, w h o s e a r g u m e n t s , ironically along with M o l l e r ' s original positive o n e s , w e r e for the m o s t p a r t later forgotten. T h e w e a k thesis, d e v e l o p e d b y these critics that, as a result o f later secondmillenniiun speculation, a c o n n e c t i o n w a s created b e t w e e n the E y e of H o r u s m y t h , with its h e a l m g or measiuring properties, a n d the dimidiated series o f the c a p a c i t y system. B u t e v e n h e r e w e h a v e seen there are serious p r o b l e m s . T h e b e g i n n i n g o f the r e p l a c e m e n t o f the special C S M signs b y their general fractional equivalents is to b e located j u s t at this p e r i o d w h e n , in the w e a k thesis, the idea that the classical C S M signs form a n important c o n c e p t u a l and religious unity b e g i n s its career. T h e votive cubits that are s u p p o s e d to enshrine this idea h a v e p r o v e n to b e m u c h m o r e a m b i g u o u s a n d imclear than foreseen, a n d this for the Egyptians t h e m s e l v e s as the innumerable variants or misunderstandings of sign disposition a n d content indicate. A n d n o w h e r e m o r e so than precisely in the section w h i c h contains, at s o m e distance, a possible t h o u g h strange hieratic writing of a particular c o m b i n a t i o n o f C S M units a n d a series o f hieroglyphs involving u n d o u b t e d parts o f the E y e o f H o r u s , t h o u g h not always those c o m b i n a t i o n s w h i c h o c c u r in metrological contexts, a l o n g with hieroglyphs c o m m g from quite different h o r i z o n s ; the w h o l e a c c o m p a n i e d b y abstract, non-volumetric, n u m b e r s with n o o b v i o u s ties to the capacity system. B u t e v e n if the cubits did represent a n attempt to reinterpret the C S M signs in terms of H o r u s - e y e parts, a speculation in the milieu w h i c h p r o d u c e d votive cubits d o e s not automatically m e a n that "the E g y p t i a n s " thought like that; for e x a m p l e , those E g y p t i a n s w h o s e task it w a s to e n g r a v e hieroglyphic inscriptions o n t e m p l e walls. T h e o l o g i c a l or any other constructs o f one c o m m u n i t y d o not necessarily p r o p a g a t e to e v e r y other; the Egyptians w e r e n o m o r e liable than a n y other p e o p l e to s p e a k with a single voice. T h e " H o r u s - e y e fractions" are a paradigmatic case o f the nature o f relations b e t w e e n writing a n d other areas of thought m Egypt. T h e linkage, feh to b e natural.

THE LARGE FRAGMENT HAS BEEN PUBLISHED BY WESTENDORF ( 1 9 7 4 ) WHILE THE C S M SIGNS IN THE SMALL FRAGMENT ARE LISTED IN MOLLER ( 1 9 1 2 ) , NOS. 7 0 8 , 7 1 1 - 7 1 3 . FOR EARLIER EXAMPLES OF THIS DEVELOPMENT, WITH A SOMEWHAT DIFFERENT INTERPRETATION, SEE VON DEINES et al. P. 4 - 6 .

(1973),

312

J. Ritter

especially in Egypt, b e t w e e n sign creation a n d other facts o f linguistic a n d religious experience, true for a n u m b e r o f signs, w a s natural in a n E g y p t o l o g i c a l r e s e a r c h c o m m u n i t y itself irrigated b y literary a n d theological studies. I n particular, t h e p r i m a c y o f w o r d p l a y a n d mythical speculation i n creating script a n d i n t h e form that script m i g h t take r e n d e r s the signs a n d c o n c e p t s involved i n n u m e r i c a l a n d metrological d e v e l o p m e n t s e c o n d a r y at best. T h i s , in t u m , is related t o what, in a n a l o g y with t h e situation in molecular biology, I shall call t h e "central d o g m a " o f o u r ideas about E g y p t a n writing: all hieratic signs g o b a c k ultimately t o p i c t o g r a p h i c signs.^^ T h e idea o f organizing hieratic signlists b y h i e r o g l y p h is o n e e x p r e s s i o n o f this and, a s w e h a v e seen, it was precisely in j u s t such a context that t h e search for the "hieroglyphic originals" o f the capacity m e t r o l o g i c a l signs l e d t o t h e H o m s - e y e hypothesis. B u t it is j u s t t h e metrological signs w h i c h here cast d o u b t o n this a p p r o a c h . R e c e n t w o r k o n t h e origins o f writing i n early M e s o p o t a m i a h a s s h o w n the i m p o r t a n c e o f treating n u m e r i c a l a n d n o n - n u m e r i c a l signs quite differently b e c a u s e they reflect quite different elements b o t h in t h e origin a n d in t h e d e v e l o p m e n t o f w r i t i n g . ' " T h e case o f the " H o m s - e y e fractions" h a s r e v e a l e d t h e s h o r t c o m i n g s o f a p p r o a c h e s w h i c h ignore this difference and u n d e r l i n e s t h e interest in globally r e e x a m i n i n g the E g y p t i a n case.

References B A E R , K l a u s . 1 9 5 6 . " A N o t e o n Egyptian Units o f A r e a in t h e O l d K i n g d o m " . Journal of Near Eastern Studies 1 5 : 1 1 3 - 1 1 7 . B A R D I N E T , Thierry. 1 9 9 5 . Les papyrus médicaux de l'Egypte pharonique. Paris: Fayard. B A R T A , Winfried. 1 9 6 3 . Die altagyptische Opferliste von der Friihzeit bis zur griechisch-romischen Epoche ( M û n c h n e r À g y p t o l o g i s c h e Studien 3 ) . Berlin: B . Hessling. BORCHARDT, Ludwig. 1 9 1 1 . "Altagyptische S o i m e n u h r e n " . Zeitschrift fur Agyptische

Sprache

und Altertumskunde

48: 9-17.

— 1 9 2 0 . Die altagyptische Zeitmessung. Berlin: d e G m y t e r . BRUGSCH, Heinrich. 1 8 8 3 . Thesaurus Inscriptionem Aegypticarum II: Kalendarische Inschriften altagyptischer Denkmaler. Leipzig: J. C . H i n r i c h s . C H A C E , A m o l d B . ; M A N N I N G , H e n r y P . ; a n d A R C H I B A L D , R a y m o n d C . 1 9 2 7 . The

Rhind Mathematical Papyrus. V o l u m e I: Free Translation and Commentary. O b e r l i n [ O H , U S A ] : M a t h e m a t i c a l Asssociation o f A m e r i c a . D A V I E S , N o r m a n d e G a r i e s . 1 9 2 2 . The Tomb of Puyemrê at Thebes. I: The Hall of Memories. N e w Y o r k : Mefropolitan M u s e u m o f Art. VON

DEINES,

Hildegard,

Erganzungen: Generalregister Verlag.

Hermann

GRAPOW

&

Wolfliart

WESTENDORF. 1 9 7 3 .

Drogenquanten, Sachgruppen, Nachtrage, Bibliographie, (Grundriss der M e d i z i n d e r alten Â g y p t e r 9 ) . Berlin: A k a d e m i e -

In its original biological context the reference is to the proposed rule that all proteins are expressions of genes encoded in nuclear DNA (genetic information flows in only one direction); see e.g., James Watson et al.. Molecular Biology of the Gene'*, 1987. In this field too, there are serious doubts concerning the universal validity of such a position. NissEN et al. ( 1994) and RITTER ( 1999).

Closing the Eye of Horus: The Rise and Fall of ' Horus-eye Fractions'

313

DONADONI ROVER], Anna Maria. 1993. "Gebelein". In Anna Maria D O N A D O N I R O V E R I et al. Il Museo egizio di Torino^: 2 4 3 - 2 5 0 . Turin: Institute g e o g r a p h i c o D e Agostini. EPIGRAPHIC S U R V E Y . 1934. Medinet Habu Ul. The Calender, the "Slaughterhouse," and Minor Records of Ramesses III (Orientai Institute Publications 2 3 ) . C h i c a g o : Oriental Institute. E R M A N , Adolf. 1 9 2 8 . Àgyptische Grammatik*. Berlin: R e u t h e r & R e i c h a r d . — and G R A P O W , H e r m a i m . 1926 a. Worterbuch J. C . H i n r i c h s .

der Agyptischen Sprache

I. Leipzig:

— a n d G R A P O W , H e r m a r m . 1926 b . Worterbuch der Agyptischen Sprache. Die Belegstellen I. Leipzig: J. C . H m r i c h s . G A B R A , Sami. 1 9 6 9 . " C o u d é e votive d e T o u n a el G e b e l H e r m o p o l i s Ouest, la Khemenow pa Meket des Égyptiens". Mitteilungen des Deutschen Archàologischen Instituts. A b t e i l u n g K a i r o 2 4 : 1 2 9 - 1 3 5 . G A R D I N E R , A l a n H . 1 9 1 7 . " A n A r c h a i c F u n e r a r y Stele". Journal of Egyptian Archœology 4 : 2 5 6 - 2 6 0 . — 1927. Egyptian Grammar^ Oxford: C l a r e n d o n P r e s s . GILLINGS, R i c h a r d J. 1 9 7 2 . Mathematics in the Time of the Pharaohs. Cambridge [US]: M I T P r e s s . G O E D I C K E , H a n s . 1988. Old Hieratic Paleography. Baltimore: Halgo. GRIFFITH, F . Llewellyn. 1 8 9 2 . " N o t e s o n E g y p t i a n W e i g h t s a n d M e a s u r e s " . Proceedings of the Society for Biblical Archœology 14: 4 0 3 - ^ 5 0 ( S u p p l e m e n t : PSBA 15 ( 1 8 9 3 ) 3 0 1 - 3 1 6 ) . G U N N , B a t t i s c o m b e . 1 9 2 2 . " ' F i n g e r - N u m b e r i n g in the P y r a m i d T e x t s " . Zeitschrift fiir Àgyptische Sprache und Altertumskunde 57: 71-72. H A S S A N , Selim. 1 9 4 4 . Excavations at Giza V : 1933-1934. Cairo: Government Press. I M H A U S E N , Aimette. ( F o r t h c o m i n g ) . Àgyptische Algorithmen (Agyptologische Abhandlimgen). J A C Q U E T - G O R D O N , H e l e n K. 1962. Les Noms des domaines funéraires sous l'Ancien Empire égyptien. C a i r o : Institut français d ' A r c h é o l o g i e orientale. J A M E S , T h o m a s G. H . 1962. The Hekanakhte Papers and Other Early Middle Kingdom Documents (Egyptian E x p e d i t i o n 19). N e w Y o r k : T h e M e t r o p o l i t a n M u s e u m o f Art. K A H L , J o c h e m . 1994. Das System der agyptischen Hieroglyphenschrift in der 0.-3. Dynastie (Gottinger Orientforschimgen IV. R e i h e À g y p t e n 2 9 ) . W i e s b a d e n : Harrassowitz. K A P L O N Y , Peter. 1 9 6 3 . Die Inschriften der agyptischen Friihzeit (Agyptologische A b h a n d l i m g e n 8). W i e s b a d e n : H a r r a s s o w i t z . — 1966. Kleine Beitràge zu den Inschriften der agyptischen Friihzeit (Agyptologische A b h a n d l u n g e n 15). W i e s b a d e n : H a r r a s s o w i t z . KITCHEN, K e n n e t h . 1 9 8 3 . " T h e W e s t - T h e b a n C a l e n d a r o f Festivals: M e d i n e t H a b u V e r s i o n - R a m e s s e u m F r a g m e n t s " . I n Ramesside Inscriptions. Historical and Biographical vol. V , p . 1 1 5 - 1 8 4 . Oxford: Oxford University P r e s s . K Ô H L E R , E. Christiana a n d BiRRELL, M i c h a e l . ( F o r t h c o m i n g ) . Helwan II. The Early Dynastic Stelae. L E F E B V R E , G u s t a v e . 1940. Grammaire de l'Égyptien classique^. C a i r o : Institut français d ' A r c h é o l o g i e orientale.

314

J. Ritter

L E P S I U S , Karl R i c h a r d . 1 8 6 5 . Ûber die altagyptische Elle und ihre Eintheilung. Berlin: K o n i g l i c h e A k a d e m i e der Wissenschaften. L U C A S , Alfi-ed a n d R O W E , Alan. 1940. " A n c i e n t E g y p t i a n M e a s u r e s o f C a p a c i t y " . Annales du Service des Antiquités de l'Egypte 4 0 : 6 9 - 9 2 . M E G A L L Y , M o u n i r . 1971 a. Considérations sur les variations et la transformation des formes hiératiques du papyrus E. 3226 du Louvre ( B i b l i o t h è q u e d ' É t u d e 4 9 ) . C a i r o : Institut F r a n ç a i s d ' A r c h é o l o g i e O r i e n t a l e . — 1 9 7 1 b . Le Papyrus hiératique comptable E. 3226 du Louvre (Bibliothèque d ' É t u d e 5 3 ) . C a i r o : Institut F r a n ç a i s d ' A r c h é o l o g i e Orieiltale. M O L L E R , G e o r g . 1909 a. Hieratische Palaographie. Die agyptische Buchschrift...!: Bis zum Beginn der achtzehnten Dynastie^. Leipzig: J. C . H i n r i c h s . — 1909 b . Hieratische Palaographie. Die agyptische Buchschrift...Il: Thutmosis' HI bis zum Ende der einundzwanzigsten Dynastie^. Hinrichs.

Von der Zeit L e i p z i g : J. C.

— 1 9 1 1 . " D i e Z e i c h e n fiir d i e Bruchteile d e s Hohlmafies u n d d a s U z a t a u g e " . Zeitschrift fur Agyptische Sprache und Altertumskunde 48: 99-101. — 1 9 1 2 . Hieratische zweiundzwanzigsten J. C . Hinrichs. "MOLLER,

Georg"

Palaographie. Die agyptische Buchschrift...Ill: Von der Dynastie bis zum dritten Jahrhundert nach Chr. \ Leipzig: [H.

GRAPOW

and

H.

PRZYBYLLA]. 1936.

Hieratische

Palaographie. Die agyptische Buchschrift...Ergànzungsheft zu Band I und IL Leipzig: J. C . H i n r i c h s . M O U S S A , A h m e d M . a n d A L T E N M Û L L E R , H a r t w i g . 1 9 7 7 . Das Grab des Nianchchnum und Chnumhotep ( A r c h â o l o g i s c h e Verôffentlichimgen d e s D e u t s c h e n A r c h à o l o g i s c h e n Instituts. A b t e i l u n g K a i r o 2 1 ) . M a i n z : v o n Z a b e m . M Û L L E R - W O L L E R M A N N , R e n a t e . 1 9 8 5 . " U d j a t a u g e " . In: W o l f g a n g H E L C K a n d Wolfhart W E S T E N D O R F (eds.), Lexikon der Àgyptologie: vol. VI: 824-826. Wiesbaden: Otto Harrassowitz. N E L S O N , H a r o l d H . a n d H O L S C H E R , EVO. 1934. Work in Western Thebes 1931-1933 ( O I C 18). C h i c a g o : U n i v e r s i t y o f C h i c a g o P r e s s . N E U G E B A U E R , O t t o . 1 9 3 0 . " U b e r d e n Scheffel u n d seine T e l l e " . Zeitschrift fur Agyptische Sprache und Altertumskunde 65: 42-48. NiSSEN, H a n s ; D A M E R O W , Peter, a n d E N G L U N D , R o b e r t K . 1 9 9 4 . Archaic Bookkeeping. Writing and Techniques of Economic Administration in the Ancient Near East. C h i c a g o : University o f C h i c a g o P r e s s . P E E T , T . Eric. 1923a. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. L o n d o n : H o d d e r & S t o u g h t o n (for L i v e r p o o l University P r e s s ) . — 1923b. "Arithmetic in t h e M i d d l e K i n g d o m " . Journal of Egyptian Archaeology 9:91-95. PETRIE, W . M . Flinders. 1926. Ancient Weights and Measures. L o n d o n : U n i v e r s i t y College. POSENER-KRIÉGER, Paule. 1 9 7 5 . " L e s P a p y r u s d e Gebelein. Remarques p r é l i m i n a i r e s " . Revue d'Égyptologie

27: 2 1 1 - 2 2 1 .

— 1 9 7 6 . Les Archives du temple funéraire de Néferirkarê-Kakaï (Bibliothèque d ' É t u d e 6 5 ; 2 vols.). C a i r o : Institut Français d ' A r c h é o l o g i e Orientale. — 1994. " L e s M e s u r e s d e grain d a n s les p a p y r u s d e G é b é l e i n " . I n C h r i s t o p h e r E Y R E et al (eds.), The Unbroken Reed. Studies in the Culture and Heritage of Ancient

Closing the Eye of Horus: The Rise and Fall of ' Horus-eye Fractions '

315

Egypt in Honour of A. F. Shore ( E E S O c c a s i o n a l P u b l i c a t i o n 11): 2 6 9 - 2 7 2 . L o n d o n : E g y p t E x p l o r a t i o n Society. — a n d D E C E N I V A L , J e a n Louis. 1 9 6 8 . The Abu Sir Papyri (Hieratic P a p y r i in the British M u s e u m . Fifth Series). L o n d o n : British M u s e u m . Q U I B E L L , J a m e s . 1 9 2 3 . Archaic Mastabas. (Service d e s Antiquités d e l'Egypte. E x c a v a t i o n s at S a q q a r a 6 ) . C a i r o : Institut Français d ' A r c h é o l o g i e Orientale. RITTER, J i m . 1 9 9 2 . " M e t r o l o g y a n d the Prehistory o f F r a c t i o n s " . In: P a u l B E N O I T , K a r i n e C H E M L A a n d J i m RITTER (eds.). Histoire des fractions, fractions d'histoire: 1 9 - 3 2 . B a s e l : B n k h a u s e r . — 1999. " M e t r o l o g y , W r i t i n g a n d M a t h e m a t i c s in M e s o p o t a m i a " . In: Jaroslav F O L T A (ed.). Calculi 1929-1999 ( F s N o v y ) : 2 1 5 - 2 4 1 . P r a g u e : P r a g u e Studies in the H i s t o r y o f Science a n d T e c h n o l o g y . — 2 0 0 1 . " E g y p t i a n M a t h e m a t i c s " . In: Helaine SELIN (ed.). Mathematics across Cultures: The History of Non-Western Mathematics: 115-136. Dordrecht: Kluwer. — ( F o r t h c o m i n g ) . Egyptian Metrology in the Third Millennium. R O B I N S , G a y a n d S H U T E , Charles. 1 9 8 7 . The Rhind Mathematical Papyrus: An Ancient Egyptian Text. L o n d o n : British M u s e u m . S A A D , R a m a d a n a n d M A N N I C H E , Lise. 1 9 7 1 . " A U n i q u e Offering List o f A m e n o p h i s I V R e c e n t l y F o u n d at K a m a k " . Journal of Egyptian Archaeology 57: 70-72. S A A D , Z a k y Y . 1957. Ceiling Stelae in Second Dynasty Tombs from the Excavations at Helwan ( A n n a l e s d u Service d e s antiquités d e l'Egypte S u p p l e m e n t , C a h i e r 21). C a i r o : Institut F r a n ç a i s d ' A r c h é o l o g i e Orientale. SCHÀFER, Heim-ich. 1902. Ein Bruchsttick altàgyptischer Annalen. Berlm: K ô n i g l i c h e A k a d e m i e der Wissenschaften. S C H L O T T - S C H W A B , A d e l h e i d . 1 9 7 3 . "Altàgyptische T e x t e iiber d i e A u s m a s s e À g y p t e n s " Mitteilungen des Deutschen Archàologischesn Instituts. Abteilung Kairo 2%: 1 0 9 - 1 1 3 . — 1 9 8 1 . Die Ausmasse Àgyptens nach altagyptischen Texten^ ( À g y p t e n u n d A l t e s Testament 3). Wiesbaden: Harrassowitz. S C O T T , N o r a E . 1 9 4 2 - 1 9 4 3 . " E g y p t i a n Cubit R o d s " , Bulletin Museum of Art {"^S) 1: 7 0 - 7 5 . S E T H E , Kurt. 1 9 0 6 - 1 9 0 9 . Urkunden Hinrichs.

der 18. Dynastie

of the

Metropolitan

(Heft 1-16). L e i p z i g : J. C .

— 1 9 1 6 . Von Zahlen und Zahlworten bei den alten Àgypten und was fiir andere Volker und Sprachen daraus zu lernen ist (Schriften d e r Wissenschaftlichen Gesellschaft StraBburg 2 5 ) . Strasbourg: T r u b n e r . SMITH, W . S t e p h e n s o n a n d S I M P S O N , W i l l i a m K. 1 9 8 1 . The Art and Architecture of Ancient Egypi. N e w H a v e n : Y a l e University P r e s s . S T R U V E , Vasilij Vasilevitch. 1 9 3 0 . Mathematischer Papyrus des Staatlichen Museums der schônen Kunst in Moskau ( Q u e l l e n i m d Studien A l ) . Berlin: Springer. V O G E L , Kurt. 1 9 3 0 . " Z u r F r a g e d e r Scheffelteile". Zeitschrift fur Agyptische Sprache und Altertumskunde 66: 3 3 - 3 5 . W E S T E N D O R F , Wolfhart. 1 9 7 4 . " P a p y r u s B e r l i n 10456. E i n F r a g m e n t d e s w i e d e r e n d e c k t e n m e d i z i n i s c h e n P a p y r u s R u b e n s o h n " . I n Festschrift zum

316

J. Ritter

ISOjàhrigen Bestehen des Berliner Agyptischen Museums: 2 4 7 - 2 5 4 . Berlin: Akademie-Verlag. ZIVIE, Alain-Pierre. 1972. " U n F r a g m e n t inédit d e c o u d é e v o t i v e " . Bulletin de I Institut Français d'Archéologie Orientale 7 1 : 1 8 1 - 1 8 8 .

317

Closing the Eye of Horus: The Rise and Fall of'Horus-eye Fractions'

2 «

«s tn

HT

«

£ VO

V c

l-H

41

^

3;

Q ^

r<

ro à

c e Q ^ <s

c

c >, fi

^

>. fi

ii ^

J. Ritter

318

i . . .

1 I-H

«5

1

•\ S S

C

t i

T3

0 «

0

»

I1

r \

a

n

on U

NI n n

00 «M

1.

^ ^

r<

tn

i

O ^

r4

f*^

Closing the Eye of Horus: The Rise and Fall of 'Horus-eye Fractions'

319

Legend to Table I F o r e a c h C S M typical sign forms, b o t h hieratic a n d hieroglyphic, a r e p r e s e n t e d in c h o m o l o g i c a l o r d e r b y dynasty. T h e hieratic forms a r e p r e s e n t e d o n t h e left a n d h i e r o g l y p h s o n t h e right o f e a c h column. T h e hieroglyphic signs h a v e h a d their original orientation c h a n g e d if n e c e s s a r y t o c o n f o r m t o a right t o left r e a d i n g d n e c t i o n , facilitating c o m p a r i s o n with the hieratic a n d permitting a d e t e r m i n a t i o n o f c h a n g e s in orientation over time. T h e A r a b i c niunbers are k e y e d t o the source list b e l o w . A n asterisk (*) indicates tìiat the sign f o r m is attested b u t is unpublished. W h e r e a p h o t o g r a p h w a s n o t available o r w a s o f insufficient c o n t t a s t I h a v e u s e d a published hand copy. Sources: Dynasty 2 1. Sjhfnr (Saqqara): QUIBELL ( 1 9 2 3 ) , pi. xxvi. 2 . Jmty (Saqqara): SMITH and SIMPSON ( 1 9 8 1 ), pi. 3 2 .

3 . Dw3t (Helwan): photograph by courtesy of Christiana Kohler. Dynasty 3 1. Jrj-Stt: KAPLONY ( 1 9 6 6 ) , Tf 4 ( 8 3 3 ) .

Dynasty 4 1. Gebelein papyri: POSENER-KRIÉGER(1994), fig. 1. Dynasty 5 1. Abusir papyri: POSENER-KRIÉGER and DE CENIVAL ( 1 9 6 8 ) , Pal. pi. xvi.

2 . Tomb of Nj-'^nh-Hnm & Hnm-htp (Saqqara): MOUSSA and ALTENMULLER ( 1 9 7 7 ) , Abb. 1 0 . Dynasty 6 1. pBerlin 1 0 5 0 0 A «fe B : GOEDICKE ( 1 9 8 8 ) , p. 7 0 - 7 1 .

Dynasty 11 1. Account papyrus: JAMES ( 1 9 6 2 ) , pi. 1 6 , IX, 1 9 .

Dynasty 1 2 1. Illahun papyri: MOLLER ( 1 9 0 9 a), nos. 7 0 8 - 7 1 3 .

Dynasty 1 7 1. pRhind 4 7 : 7 ( 2 , 8 ) ; 8 ( 4 ) ; 1 0 ( 1 6 , 3 2 , 6 4 ) : ROBINS and SHUTE ( 1 9 8 7 ) , pi. 1 5 .

Dynasty 1 8 1. pLouvre E 3 2 2 6 : MEGALLY ( 1 9 7 1 ) b, pi. LXXI, B r° V: 5 .

2 . Tuthmosis III campaign (Kamak): SETHE ( 1 9 0 6 - 1 9 0 9 ) , 7 5 6 , 8. 3 . Tuthmosis III calendar (Kamak): SETHE ( 1 9 0 6 - 1 9 0 9 ) , 7 6 2 , 6. 4 . Tomb of Pwj-m-R' (Tuthmosis III, Thebes): D A VIES ( 1 9 2 2 ) , pi. I, 3 6 . 5 . Amenophis IV Calendar (Kamak): SAAD and MANNICHE ( 1 9 7 1 ) , pi. xxi.

320

J. Ritter

Dynasty 19 1. pBerlin 3038 (medical): MOLLER (1909 b), nos. 708-713. 2. Ramesses II Festival calendar (Abydos): photographs by courtesy of Kenneth Kitchen, Z21-Z27. 3. Ramesses II Festival calendar (Kamak): NELSON and HÔLSCHER (1934), Fig. 11. (right 5) R24+26 (2, 4, 16, 32); KITCHEN 1983, R3.2 (8, 64). Dynasty 20 1. pRollin (Ramesses III): MOLLER (1909 b), nos. 708-713. 2. Ramesses III Festival calendar (Medinet Habu): KITCHEN (1983), MH 970 (2 4 8 16 32) MH 301 (64). 3. Ramsses IV wall-stela (Kamak): photograph courtesy of Centre Golenischeff, A XI

J

w 321

Closing the Eye of Horus: The Rise and Fall of'Horus-eye Fractions'

7

8

9

10

11

12

13

14

15

III

IV

VI

T a b l e II. V o t i v e Cubits.

322

J. RITTER

7

n

8

9

10

U

12

13

14

15

(X

III

T a b l e II. V o t i v e Cubits (continued).

Closing the Eye of Horus: The Rise and Fall of ' Horus-eye Fractions'

323

Legend to Table II All p h o t o g r a p h s are s h o w n in n e g a t i v e to increase contrast. N ° V I I I h a s b e e n r e v e r s e d m this printing t o c o i n c i d e with the direction o f writing o f t h e other votive cubits. T h e three cubits ( I X - X I ) for w h i c h I h a v e n o p h o t o g r a p h s h a v e b e e n p r i n t e d in h a n d c o p y t a k e n firom t h e n original publication. Sources (a royal name mentioned elsewhere on the votive cubit is included in parentheses following the museum number): I

BMA 71.38 +Cairo 31/12//22/1 (NectaneboII) BMA 71.38: photograph courtesy of the Brooklyn Museum of Art (Charles Edwin Wilbour Fund). Cairo 31/12//22/1: photograph courtesy of the Centre Golenischeff (Archive Lacau), 0124.

II

Cairo 31/12//22/2 (Osorkon N) photograph courtesy of the Centre Golenischeff (Archive Lacau), 01 24.

III

(Pefrie 2) photograph courtesy of the Petrie Museum of Egyptian Archaeology.

IV

Cairo31/12//22/5 photograph courtesy of the Centre Golenischeff (Archive Lacau), 0 1 23.

V

(Tuna el-Gebel) (Amenhotep III) GABRA (1969), pi. XVIII A.

VI

Cairo31/12//22/4 photograph courtesy of the Centre Golenischeff (Archive Lacau), 01 22.

VII Cairo31/12//22/3 photograph courtesy of the Centre Golenischeff (Archive Lacau), 0 1 24 VIII Turin Nr. Suppl. 2681 (Sheshonq I) photograph courtesy of the Centre Golenischeff (Archive Lacau) IX

MMA 25.7.41 SCOTT (1942-1943), p. 72

X

(Penie 1) PETRIE (1926), pi. XXIV. 1

XI

B M 3 6 656 LEPSIUS(1865), tf I l l b

More than Metrology: Mathematics Education in an Old Babylonian Scribal School Eleanor

Robson,

Oxford

1. Introduction 1.1 A p p r o a c h e s t o O B m a t h e m a t i c s a n d s c r i b a l e d u c a t i o n F o r m o s t o f the twentieth century the study of O l d B a b y l o n i a n ( O B ) m a t h e m a t i c s quite rightly focussed o n the r e c o v e r y o f k n o w l e d g e : what was k n o w n , a n d w h e r e a n d w h e n . T h e last d e c a d e has seen a m o v e t o w a r d s c o n c e p t u a l history: h o w m a t h e m a t i c a l language reflected the thought p r o c e s s e s b e h i n d the techniques. N e v e r t h e l e s s , the corpus o f O l d B a b y l o n i a n m a t h e m a t i c s w a s treated, m o r e or less, as a closed set o f d i s e m b o d i e d texts: there w e r e few attempts to p u b l i s h n e w sources, or to a c k n o w l e d g e that they w e r e r e c o r d e d o n physical objects w h i c h could b e located in time a n d space a n d fruittuUy related to other a r c h a e o l o g i c a l artefacts. T o b e fair, this w a s in large part due to the a c a d e m i c b a c k g r o u n d s of the small n u m b e r of core researchers c o n c e m e d , w h o w e r e almost exclusively trained in m a t h e m a t i c s or the history of science or ideas, a n d inevitably l a c k e d the technical skills i n v o l v e d in the p r i m a r y publication of cuneiform tablets or the r e c o n s t m c t i o n o f the archaeological record. Conversely, there w a s a c o n s p i c u o u s lack o f cuneiformists willing to d o so. It a p p e a r e d that the great p i o n e e r s o f c u n e i f o r m m a t h e m a t i c a l studies - N e u g e b a u e r , Sachs, B m i n s , a n d T h u r e a u - D a n g i n - h a d d o n e it all, a n d there w a s little left to d o b u t reanalyse their data. T h e contextual e v i d e n c e for the material t h e y h a d p u b l i s h e d was at best m e a g r e and m o r e c o m m o n l y n o n existent, while the m a t h e m a t i c a l tablets c o m i n g out o f m o r e r e c e n t e x c a v a t i o n s were invariably fiuther e x e m p l a r s o f the multiplication tables a n d m e t r o l o g i c a l lists that h a d b e e n so t h o r o u g h l y classified b y N e u g e b a u e r in the 3 0 s a n d 4 0 s . Since the m i d - 1 9 9 0 s , h o w e v e r , there has b e e n an increasing interest m the material culture o f scribal schooling, and a g r o w i n g realisation that there is a wealth of archaeological a n d artefactual data w h i c h c a n b e u s e d to c o u n t e r b a l a n c e the traditional sources of e v i d e n c e , the S u m e r i a n literary narratives a b o u t school. T h i s m o v e has g o n e h a n d in h a n d with a n increasingly sophisticated a p p r o a c h to textual evidence, w h i c h a c k n o w l e d g e s that authorial intention w a s often c o m p l e x a n d that literary text in particular caimot b e u s e d straightforwardly as a historical source. T h i s study is situated firmly within that r e s e a r c h tradition. It takes as its starting point o n e single architectural unit and the objects found within it to r e c o n s t m c t the role o f m e t r o l o g y , arithmefic, and m a t h e m a t i c s within the c u r r i c u l u m of an individual school. Its a i m is n o t to p r o d u c e a generalised scholastic framework for O l d B a b y l o n i a n m a t h e m a t i c s b u t rather, through c o m p a r i s o n with p i e c e s o f e v i d e n c e

326

E. Robson

from other contexts, to stress the variety o f a p p r o a c h e s to m a t h e m a t i c s e d u c a t i o n that existed in t h e early s e c o n d m i l l e n n i u m B C . Just as m o d e m s c h o l a r s h i p is c o n d u c t e d b y individuals w h o are c o n s t r a i n e d b y their e n v i r o n m e n t a n d e d u c a t i o n while free to m a k e p e r s o n a l c h o i c e s about the direction a n d c h a r a c t e r o f their w o r k so, w e shall see, w a s ancient e d u c a t i o n i m p a r t e d b y p e o p l e w i t h similar freedoms a n d constraints. 1.2 T h e h i s t o r y of H o u s e F H o u s e F w a s e x c a v a t e d in the first m o n t h s o f 1952 b y a t e a m of a r c h a e o l o g i s t s from t h e universities o f C h i c a g o a n d Permsylvania. It w a s their third field s e a s o n in the ancient s o u t h e m Iraqi city of N i p p u r a n d o n e of their e x p r e s s a i m s w a s t o find large n u m b e r s o f c u n e i f o r m t a b l e t s . ' F o r this r e a s o n t h e y h a d c h o s e n t w o sites o n the m o u n d k n o w n as T a b l e t Hill, b e c a u s e of the large n u m b e r of tablets that h a d b e e n found there in t h e late nineteenth century. T h o s e p r e v i o u s digs, t h o u g h , w h e n the d e v e l o p m e n t of r e c o r d e d stratigraphie a r c h a e o l o g y w a s still in its infancy, h a d necessarily b e e n little m o r e t h a n hunts for artefacts. T h e n e w g e n e r a t i o n of archaeologists labelled their e x c a v a t i o n areas T A a n d T B ( F i g u r e 1), deliberately siting T B right n e x t to the pits left b y their n i n e t e e n t h century p r e d e c e s s o r s . ^ N o w , h o w e v e r , they m a d e detailed a r c h a e o l o g i c a l r e c o r d s o f finds a n d findspots as a m a t t e r o f c o u r s e , so that w h e n they hit u p o n the large c a c h e o f tablets t h e y h a d b e e n h o p i n g for, the architectural context, sfratigraphic l o c a t i o n and p h y s i c a l d e s c r i p t i o n o f e v e r y o n e o f t h e m w a s noted.^ T h e tablets, o v e r 1400 of t h e m , w e r e in an u n r e m a r k a b l e l o o k i n g h o u s e in the c o m e r of A r e a T A , o n e o f eight m u d - b r i c k dwellings p a c k e d into the 2 0 X 4 0 m rectangle. O t h e r h o u s e s in T A a n d T B h a d yielded tablets, b u t in handfiils or d o z e n s , n o t in the t h o u s a n d s . T h e H o u s e F tablets w e r e not stored in j a r s or d i s c a r d e d o n the street as s o m e of the others h a d b e e n , b u t w e r e p a r t o f the v e r y fabric o f the h o u s e itself, built into the floors a n d w a l l s a n d fiimiture ( F i g u r e 2 ) . It q u i c k l y b e c a m e a p p a r e n t that the tablets w e r e n o t a n o r m a l h o u s e h o l d archive o f d o c u m e n t s relating to p r o p e r t y o w n e r s h i p , debt, a n d b u s i n e s s matters b u t c o m p r i s e d in the m o s t p a r t S u m e r i a n literary c o m p o s i t i o n s a n d p i e c e s of lexical lists, in n u m b e r s that h a d n e v e r before b e e n r e c o v e r e d from a c o n t r o l l e d excavation. T h e e x c a v a t o r s of H o u s e F h a d found a s c h o o l . W h i l e the h u g e n u m b e r o f school tablets w e r e n o t e n o u g h to confirm this at the time, h a v i n g b e e n found in s e c o n d a r y context u s e d as c o n s t m c t i o n m b b l e , the p r e s e n c e of large quantities of u n u s e d tablet clay and facilities for s o a k i n g and reusing tablets h a s since b e e n attested in other s c h o o l i n g e n v i r o n m e n t s a n d leaves little r o o m for doubt. T h e s c h o o l i n g t o o k p l a c e , it a p p e a r s , in the c o u r t y a r d s , loci 192 and 2 0 5 , w h e r e b e n c h e s a n d three r e c y c l i n g b i n s w e r e found. T h r e e small r o o m s to the n o r t h w e s t , 184, 189, a n d 1 9 1 , s e e m to h a v e b e e n p r i v a t e quarters (a b r e a d o v e n w a s d i s c o v e r e d in 1 9 1 , d o m e s t i c p o t t e r y in 184 a n d 2 0 5 , a n d d e c o r a t i v e p l a q u e s in 191 a n d 2 0 5 ) , while the partially e x c a v a t e d 2 0 3 m u s t h a v e s e r v e d as the entrance hall. T h e tablets, it s e e m s , w e r e laid d o w n

'

McCowN and HAINES ( 1967), p. viii.

^ GIBSON et al. (2001) give a thorough overview of the excavations at Nippur and their results. ^ Nevertheless, the 1950s field records still present major problems for researchers: see ZETTLER(1996), pp. 88-89.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

327

shortly after 1740 B C , the tenth regnal year of Samsu-ilima, k i n g o f B a b y l o n a n d son a n d successor of Hammurabi.'* M u d b r i c k structures like H o u s e F n e e d e d to b e rebuilt or extensively r e n o v a t e d e v e r y 25 years or so^ a n d i n d e e d H o u s e F a p p e a r s to h a v e u n d e r g o n e three or four such r e m o d e l l i n g s over the c o t n s e o f its hfe, from the late n i n e t e e n t h century to about 1721 B C . T h e u s e o f tablets as b u i l d i n g material s e e m s to m a r k the e n d o f the h o u s e ' s life as a school: w h e n it w a s later r e o c c u p i e d the n e w inhabitants a p p e a r to h a v e b e e n e n g a g e d in other activities.

+ 46 Ubtets?

Figure 1 : T o p o g r a p h i c m a p o f N i p p u r . A r e a T A is F i g u r e 2: P l a n o f H o u s e F , south o f I n a n a ' s t e m p l e ( G I B S O N et al. ( 2 0 0 1 ) , fig. 1). level 10 (after S T O N E ( 1 9 8 7 ) , pis. 1 7 - 1 9 ) . After excavation, the finds were disfributed b e t w e e n the Oriental Institute, C h i c a g o a n d the University M u s e i u n , Philadelphia. T h e Iraq M u s e u m m B a g h d a d also t o o k a share, while s e n d i n g p o r t i o n s of its allocation to C h i c a g o a n d P h i l a d e l p h i a o n longt e r m loan for publication purposes.^ T h e C h i c a g o loan w a s r e t u r n e d in the 1980s. T h e e x c a v a t i o n report o n T A a n d T B c a m e out fifteen years after the d i g ' a n d w a s later r e a n a l y s e d in the light o f information from a b o u t 100 h o u s e h o l d archival

The Middle Chronology, which puts Hammurabi's reign at 1 7 9 2 - 5 0 BC, is followed throughout this paper. ^

GASCHE and DEKIERE ( 1 9 9 1 ).

^ Chicago: 1 5 9 tablets plus 3 4 7 on long-term loan (now retumed; plastercasts of these fragments are kept in Chicago); Philadelphia: 2 0 0 tablets plus 5 3 3 on long-term loan (still in Philadelphia; HEIMERDINGER ( 1 9 7 9 ) ) ; Baghdad: 4 4 1 tablets (some plastercasts of which are in Chicago and Philadelphia) plus the 3 4 7 retumed from Chicago. Total 1 6 8 0 3 N - T tablets from TA, of which 1 4 2 5 are from House F, 2 0 9 from other houses, and 4 6 from undetermined locations in TA. ^

McCowN and HAINES ( 1 9 6 7 ) .

328

E . Robson

r e c o r d s from the t w o sites.* N e i t h e r w o r k treated in a n y detail the school tablets from H o u s e F or the rest of the site. M e a n w h i l e , m a n y of those tablets, identifiable b y the s i g l u m 3 N - T , were m a k i n g their w a y into critical editions of S u m e r i a n literary a n d lexical w o r k s , s o m e t i m e s contributing as m u c h as 2 5 % o f the sources. O n l y the fragments l o a n e d to Philadelphia w e r e systematically published,^ b u t e v e n this w o r k consisted solely of cuneiform copies with n o textual c o m m e n t a r y , edition, or discussion of the archaeological context. A s the tablets were p u b l i s h e d , it b e c a m e clear that fragments could b e j o i n e d to form larger p i e c e s o f tablets, b u t that often these j o i n s h a d to b e m a d e virtually, across the three collections. N e v e r t h e l e s s , the p r e s e r v a t i o n in Philadelphia of the original field n o t e b o o k s containing a c o m p l e t e r e c o r d o f the epigraphic finds as they were e x c a v a t e d , as well as a tablet c a t a l o g u e d r a w n u p in the 1970s, has m a d e it possible to attempt a r e c o n s t r u c t i o n o f the original tablet a s s e m b l a g e . A s B a g h d a d b e c o m e s m o r e accessible to the i n t e m a t i o n a l c o m m u n i t y , it is n o w also feasible to c h e c k those r e c o r d s against the tablets in the Iraq M u s e u m . A s this p r o c e s s h a s o n l y j u s t b e g u n , there are still s o m e g a p s a n d inconsistencies in the data, w h i c h should therefore not b e t a k e n as c o m p l e t e l y accurate. N e v e r t h e l e s s , it is already p o s s i b l e to m a k e s o m e interesting observations a n d d r a w s o m e preliminary conclusions a b o u t the fiinctioning o f the H o u s e F school in the m i d - e i g h t e e n t h century B C . Subject Mathematics O t h e r e l e m e n t a r y subjects S u m e r i a n literature Non-school Unidentified Total

Number of tablet

pieces

127 591 591 28 88

Percentage

of total

8.9 41.5 41.5 1.9 6.1

1425

F i g u r e 3 : Subject matter of the tablet pieces in H o u s e F . In total 1425 tablets w e r e r e c o v e r e d from Level 10 of H o u s e F (Figure 3), s o m e 9 p e r c e n t o f w h i c h are mathematical. All b u t four of these b e l o n g to the tablet typology of elementary schooling, w h i c h a c c o u n t altogether for 5 0 p e r c e n t o f the tablets found in the h o u s e . T h e position, content, style, a n d p u r p o s e o f m a t h e m a t i c a l instruction within the H o u s e F elementary c u r r i c u l u m will b e d i s c u s s e d in § § 2 - 3 b e l o w . T h e b u l k of the r e m a i n i n g tablets b e a r extracts from S u m e r i a n literary c o m p o s i t i o n s (42 p e r c e n t ) , s o m e of w h i c h m a y h e l p to shed light o n p o s t - e l e m e n t a r y m a t h e m a t i c a l training. This is the topic o f §4. T h e final 8 percent, h o u s e h o l d d o c u m e n t s a n d hitherto unidentified fragments, will not c o m e into the discussion.

^

STONE ( 1 9 8 7 ) .

'

HEIMERDINGER(1979).

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

329

1.3 C o m p a r a t i v e d a t a Several other a s s e m b l a g e s of O l d B a b y l o n i a n school tablets h a v e b e e n identified, all m u c h smaller than the H o u s e F corpus. T h e better p u b l i s h e d of t h e m include: • T h e 2 5 e l e m e n t a r y school tablets, including 7 m a t h e m a t i c a l o n e s , from a small r o o m in S î n - k â s i d ' s p a l a c e in U r u k , c.1860 BC,^° 120 years before H o u s e F . • T h e 3 8 0 - o d d identified tablets o f a reported 2 0 0 0 e x c a v a t e d from ' N o . 1 B r o a d Sfreet' in U r , including about 6 0 m a t h e m a t i c a l t a b l e t s . " A s in H o u s e F , the tablets h a d b e e n re-used as fill, but it is not clear w h e t h e r the h o u s e itself fimctioned as a school at any time. T h e tablets b e l o n g to t w o separate lots: o n e , almost exclusively of school d o c u m e n t s , found in one part o f the h o u s e , and another, p r e d o m i n a n t l y adminisfrative r e c o r d s covering the p e r i o d 2 0 2 0 - 1 7 8 7 B C , m i x e d with fiirther school tablets. T h e s e c o n d lot of school tablets, w e m i g h t therefore g u e s s , dates to a b o u t 1790 B C . ' ^ •

T h e 180 school tablets, including nine m a t h e m a t i c a l o n e s , from the so-called Scherbenloch or sherd-pit near the t e m p l e c o m p l e x in Uruk.'^ A l t h o u g h their original context h a s b e e n lost, they were found with a coherent g r o u p of letters a n d b u s i n e s s d o c u m e n t s dating to the 1780s ( R i m - S i n 3 1 - 4 2 ) , s o m e forty years before H o u s e F.

•

T h e 59 school tablets, of w h i c h s e v e n are m a t h e m a t i c a l , from the library' in Me-Turan.'"* This private h o u s e in a provincial t o w n in E s h n u n a also yielded 2 2 magico-liturgical tablets a n d 9 0 h o u s e h o l d d o c u m e n t s a n d letters. It a p p e a r s to h a v e b e e n desfroyed in c . 1 7 6 0 B C , years before H o u s e F .

•

T h e 4 6 school tablets from ' N o . 7 Quiet Sfreet' in Ur,'^ four o f w h i c h are mathemafical. T h e h o u s e b u m e d d o w n in or after 1740 B C ( S a m s u - i l u n a ' s 10th regnal year), m a k i n g it H o u s e F ' s exact c o n t e m p o r a r y . T h e 68 e l e m e n t a r y school tablets, eight of w h i c h are m a t h e m a t i c a l , found in and a r o u n d a recycling b i n in the m a i n courtyard o f a substantial h o u s e o w n e d b y a family of gala-mah priests in Sippir A m n a n u m . ' ^ T h e h o u s e w a s desfroyed b y fire in 1629 B C ( A m m i - s a d u q a ' s 18th regnal year), but the s c h o o l tablets were d e p o s i t e d in an earlier o c c u p a t i o n p h a s e , a r o u n d 1640 B C , a h u n d r e d y e a r s after the flomit o f H o u s e F .

•

'scholar's northem business about 20

T h e r e are also m o r e sparsely d o c i u n e n t e d finds of O l d B a b y l o n i a n school tablets, from N i p p u r , Larsa, Isin, Susa, a n d towns in the Diyala Valley. H o u s e F thus falls in the m i d d l e of the chronological and g e o g r a p h i c a l spread of these school corpora. M a t h e m a t i c s is p r e s e n t in all o f t h e m in p r o p o r t i o n s r a n g i n g from 5 p e r c e n t in the Scherbenloch to 16 p e r c e n t in N o . 1 B r o a d Stteet; the 9 p e r c e n t p r o p o r t i o n of m a t h e m a t i c s in H o u s e F c a n thus b e seen as relatively

C A VIGNEAUX ( 1 9 8 2 ) . C H A R P I N ( 1 9 8 6 ) , pp. 4 5 1 - 4 5 2 ; ROBSON ( 1 9 9 9 ) , pp. 2 4 5 - 2 7 2 .

The mathematics from Ur has recently been the subject of a study by FRIBERG ( 2 0 0 0 ) . CAviGNEAUx ( 1 9 9 6 ) ; VELDHUIS ( 1 9 9 7 - 1 9 9 8 ) . CAVIGNEAUX(1999). CHARPIN(1986). TANRET ( 1 9 8 2 ) ; GASCHE ( 1 9 8 9 ) ; TANRET ( 2 0 0 2 ) .

330

E. Robson

n o r m a l . " H o w e v e r , the contents and format o f the m a t h e m a t i c a l tablets in the s e v e n different a s s e m b l a g e s , as far as they c a n b e identified, v a r y quite r e m a r k a b l y .

2. Metrology in the elementary curriculum 2.1 T h e e l e m e n t a r y c u r r i c u l u m in H o u s e F F i v e types of tablet w e r e u s e d for elementary schooling in N i p p u r . T h i s classification w a s first u s e d to describe lexical lists — standardised lists o f signs a n d w o r d s — as follows: Type I refers to generally large tablets [...], with a full lexical list and a substantial part thereof and nothing else. Type II [...] tablets contain divergent material on each of [their] two sides. To the left of the flat side (Il/l) there is a carefully written lexical passage extracted from a fuller list, apparently the work of an instructor, while to the right the passage is copied by a student. On the convex side (11/2) of a Type II tablet, there is a multicolumn excerpt from a longer list. Type III tablets [...] contain just one column with material extracted from a longer list. Type IV are plano-convex (flat on one side and convex on the other) round (lenticular) tablets [...]. On the flat side, they have two to four lines wTitten by the instructor and copied underneath by the student. On some of them, the convex side gives the reading of the signs in syllabograms and/or their Akkadian translation.'^ A s N i e k V e l d h u i s h a s s h o w n , h o w e v e r , this tablet t y p o l o g y applies e q u a l l y to all e l e m e n t a r y school exercises including m a t h e m a t i c a l ones.*' T h e fifth tablet type, the p r i s m s , contain similar material to T y p e I tablets; w e shall refer to these as T y p e P . It h a p p e n s that n o m a t h e m a t i c s has survived o n T y p e I V or T y p e P tablets from H o u s e F;^° w e will thus b e dealing w i t h j u s t three types: the small T y p e I l l s a n d the larger T y p e I a n d lis, o f w h i c h it will b e i n ^ o r t a n t to distinguish o b v e r s e ( T y p e II/1) a n d reverse (II/2). B e c a u s e the T y p e II tablets contain different c o m p o s i t i o n s o n the o b v e r s e a n d reverse they c a n b e u s e d to reconstruct the curricular s e q u e n c e o f e l e m e n t a r y e d u c a t i o n in N i p p u r (see Figure 3). V e l d h u i s correlated the contents of o b v e r s e a n d reverse o n s o m e 1500 T y p e II tablets from N i p p u r , w o r k i n g from the h y p o t h e s i s that they h a d b e e n written b y the same student, w h o r e v i e w e d o n the r e v e r s e a n earlier p a r t of the s a m e c o m p o s i t i o n h e was learning o n the o b v e r s e , or long sections from

The school finds from Sîn-kâsid's palace and the gala-maljs' house are solely elementary exercises; mathematical tablets account for 28 percent and 12 percent respectively of those find-groups compared to 18 percent of the House F elementary tablets. CiviL(1995),p. 2308. VELDHUIS (1997), pp. 28-39. But it is known from other sites: for instance M D P 27: 61, a Type I V tablet from Susa with a 3 times multiplication table (VAN DER MEER (1935), p. 61) and AO 8865, a six-sided Type P prism from Larsa, dated 1749 BC, bearing standard lists of squares, inverse squares and inverse cubes (NEUGEBAUER (1935-1937), I, pp. 71-75). See also §4.4 below.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

Educational

Phase/Composition

First Phase: writing

Exercises in sign forms

1

Syllable Alphabet B

2

Lists of personal names ifinanatés)

4

5 6 7 8

Writing single and combined cuneiform wedges The proper formation of simple cuneiform signs The combination of signs into meaningful sense units

12

13 14 15 16

70 82

98

Division 1 : List of trees and wooden objects Division 2 : List of reeds, vessels, leather, and metal objects Division 3 : List of animals and meats Division 4 : List of stones, plants, fish, birds, and garments Division 5 : List of geographical names and terms, and stars Division 6: List of foodstuffs

28

Nigga } Proto-Kagal} order uncertain Proto-Izi } Proto-Lu

Proto-Ea Metrological lists and tables Multiplicafion and reciprocal tables Proto-Diri Fourth Phase: Sumerian

17

1

Second Phase: thematic vocabulary acquisition: realia

20

19 25 6 7

207

Third Phase: advanced lists

10

No. of tablets in House F'' 146

techniques

0

3

function

331

16

Ordered by key signs

11 30

Thematic vocabulary acquisition: titles and professions Sumerian readings of signs Weights and measures Number facts

22

The readings of compound signs

16

introductory

17 15 93

107

Model contracts

Simple Sumerian prose

54

Proverbs

Sumerian literary language

54

F i g u r e 4 : T h e elementary c u r r i c u l u m in H o u s e F .

22

Of all tablet types, not only Type II. The numbers in the column are not commensurate because of the co-occurrence of different compositions on the Type II tablets. This table excludes apparently extra-curricular compositions which are attested only once or twice in the house, such as OB Lu A and Proto-Aa, and which cannot easily be assigned a position in the curriculum; see too VELDHUIS ( 1 9 9 7 ) , p. 5 9 .

332

E. Robson

o n e h e h a d c o m p l e t e d earlier.^^ H i s results w e r e impressively consistent, a n d e n a b l e d h i m t o assign a b o u t twenty different c o m p o s i t i o n s t o four p h a s e s o f t h e e l e m e n t a r y curriculum: writing techniques, t h e m a t i c n o u n lists, a d v a n c e d lists, a n d i n t r o d u c t o r y S u m e r i a n . H e discussed t h e educational function o f e a c h p h a s e in t u m , s h o w i n g a steady p r o g r e s s i o n from first e x p o s u r e t o t h e physical f o r m o f c u n e i f o r m signs, t h e c o n s t m c t i o n o f w h o l e w o r d s , a n d t h e e x p l o r a t i o n o f the c o m p l e x i t i e s o f c u n e i f o r m writing, to t h e u s e o f w h o l e sentences o f g r a m m a t i c a l l y c o r r e c t S u m e r i a n . A l l p h a s e s i n v o l v e d t h e rote m e m o r i s a t i o n o f set texts, m a i n l y o f t h e sort traditionally characterised in t h e field as 'lexical t e x t s ' , n a m e l y , lists o f c u n e i f o r m signs o r S u m e r i a n w o r d s . B u t t h e c u r r i c u l u m also included m o d e l legal d o c u m e n t s , S u m e r i a n p r o v e r b s , a n d — m o s t importantly for o u r p u r p o s e s — l o n g s e q u e n c e s of multiplication tables a n d lists o f metrological units. It is i m p o s s i b l e , t h o u g h , o n p r e s e n t e v i d e n c e , t o estimate h o w long the student(s) h a d b e e n at school, o r h o w old t h e y w e r e , at this o r a n y other point in their educational careers. U s i n g the s a m e m e t h o d o l o g y o n t h e 2 5 0 o r s o T y p e II tablets from H o u s e F yields a similarly consistent picture (Figure 4 ) . It differs from V e l d h u i s ' general c o n c l u s i o n s only in t h e o m i s s i o n o f the syllable list tu-ta-ti from t h e first p h a s e a n d in the ordering o f t h e third p h a s e , w h e r e m a t h e m a t i c a l m a t t e r s a r e addressed.^'* W e i g h t s a n d m e a s u r e s w e r e l e a m e d systematically, b y m e a n s o f a s t a n d a r d series, t o w a r d s t h e e n d o f the third p h a s e o f t h e H o u s e F c u r r i c u l u m ( s e e §2.3). M u l t i p l i c a t i o n a n d division facts w e r e m e m o r i s e d i m m e d i a t e l y afterwards ( s e e § 3 ) . H o w e v e r , m e t r o l o g i c a l matters w e r e first a d d r e s s e d within t h e s e c o n d p h a s e , as s e q u e n c e s within t h e thematic n o u n list ( s e e §2.2) a n d later c o n t e x t u a l i s e d in t h e m o d e l legal contracts o f p h a s e four ( § 2 . 4 ) . L o o k i n g a t t h e n u m b e r s o f tablets attested, it is striking that t h e series o f divisions a n d multiplications is o n e o f the m o s t frequently o c c u r r i n g c o m p o s i t i o n s , while t h e m e t r o l o g i c a l s e q u e n c e is a m o n g t h e least represented. W h y this m i g h t b e , if it is n o t s i m p l y a n accident o f preservation, c a n n o t for the m o m e n t b e d e t e r m i n e d . 2.2 M e t r o l o g i c a l s e q u e n c e s i n the t h e m a t i c n o u n lists T h e s t u d e n t s ' first e x p o s u r e t o metrological n o t a t i o n w a s in t h e s e c o n d p h a s e o f e l e m e n t a r y education, as s u b - s e q u e n c e s within t h e six-part t h e m a t i c n o u n list. I n t h e first division, t h e list o f trees a n d w o o d e n objects, students m e t t h e m a i n c a p a c i t y m e a s u r e s in d e s c e n d i n g order. L a r g e r c a p a c i t y m e a s u r e s (c. 1 , 5 0 0 - 1 8 , 0 0 0 litres) w e r e c o n t e x t u a l i s e d as standard sizes o f b o a t within t h e 60-line section o n b o a t s (lines 2 6 1 - 3 2 0 ) . Smaller units ( c . 0 . 1 7 - 6 0 litres) w e r e treated later on, in a section o f their own:^^ 1.279 1.280 1.281 "

gi§-nna2-60-gur gi§-ma2-50-gur gi§-ma2-40-gur

Boat of 60 gur capacity (1 gur = c.300 litres) Boat of 50 gur capacity Boat of 40 gur capacity

VELDHUIS ( 1 9 9 7 ) , pp. 4 0 - 6 3 . ROBSON (forthcoming).

The list of trees and wooden objects also includes an obscure section, apparently on scribal apparatus (lines 1 4 2 - 1 5 9 ; VELDHUIS ( 1 9 9 7 ) , pp. 8 6 - 8 8 ) , which has attracted sporadic attention in the mathematical literature (e.g., WASCHKIES ( 1 9 8 9 ) , p. 8 7 ) . It includes gi§-as4lum "measuring stick" (line 1 4 2 ) and giS-âurum^-ma perhaps "accounting board" (line 1 5 0 ) but many of the other entries remain unexplained.

•"I

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

gi§-ma2-30-gur 1.282 gi§-ma2-20-gur 1.283 1.284 gi§-ma2-15-gur gi§-ma2-10-gur 1.285 1.286 giâ-ma2-5-gur gi§-ma2-tur 1.287 (After VELDUIS(1997),p. 157)

Boat of 30 gur capacity Boat of 20 gur capacity Boat of 15 gur capacity Boat of 10 gur capacity Boat of 5 gur capacity Small boat

gi§-lid2-ga 1.515 1.516 gi§-ba-rÌ2-ga 1.517 gi§-ba-an 1.518 gi§-ba-an-5-sila3 1.519 gi§-nig2-2-sila3 1.520 gi§-1 -silas 1.521 gi§-l/2-sila3 1.522 gi§-l/3-sila3 gi§-2/3-sila3 1.523 1.524 gi§-10-gin2 1.525 gi§-5-gin2 1.526 gi§-3-gin2 1.527 gi§-2-gin2 1.528 gi§-l-gin2 (After VELDHUIS (1997), p. 163)

Measuring Measuring Measuring Measuring Measuring Measuring Measuring Measuring Measuring Measuring Measuring Measuring Measuring Measuring

vat vat of vat of vat of vat of vat of vat of vat of vat of vat of vat of vat of vat of vat of

333

60 sila capacity (1 sila = c.l litre) 10 sila capacity 5 sila capacity 2 sila capacity 1 sila capacity 1/2 sila capacity 1/3 sila capacity 2/3 sila capacity 0;10 sila capacity 0;05 sila capacity 0;03 sila capacity 0;02 sila capacity 0;01 sila capacity

W e i g h t s w e r e treated v e r y briefly, within a five-lme section o n w e i g h i n g e q u i p m e n t , in the list of trees a n d w o o d e n objects, b u t w e r e c o v e r e d m o r e exhaustively, as stone w e i g h t s , as a section in the list o f stones (division four o f the t h e m a t i c n o u n list). 1.436 gi§-rin2 gi§-rin2-lib-lib-bi 1.437 gi§-rin2-l-gU2-un 1.438 1.439 gi§-rin2-ma-na 1.440 gi§-e2-rin2 1.441 gi§-dilim2-rin2 (After V E L D U I S ( 1997), p. 161)

Balance Balance Balance Balance Balance Balance

4.178 4.179 4.180 4.181 4.182 4.183 4.184 4.183 4.182 4.187 4.188 4.189 4.190 4.191 4.192 4.193 4.194 4.195

Weight Weight Weight Weight Weight Weight Weight Weight Weight Weight Weight Weight Weight Weight Weight Weight Weight Weight

na4-l-gU2 na4-50-ma-na na4-40-ma-na na4-30-ma-na na4-20-ma-na na4-15-ma-na na4-10-ma-na na4-5-ma-na na4-3-ma-na na4-2-ma-na na4-I-ma-na na4-2/3-ma-na na4-l/2-ma-na na4-1/3-ma-na na4-10-gin2 na4-5-gin2 na4-3-gin2 na4-2-gin2

arm' (Cf CAVIGNEAUX (1992)) for 1 talent (c.30 kg) for 1 mina (c.0.5 kg) box pan

of of of of of of of of of of of of of of of of of of

1 talent 50 minas 40 minas 30 minas 20 minas 15 minas 10 minas 5 minas 3 minas 2 minas 1 mina 2/3 mina 1/2 mina 1 /3 mina 10 shekels 5 shekels 3 shekels 2 shekels

334

4.196 4.197 4.198 4.199 4.200 4.201 4.202 4.203 4.204 4.205 4.206 (After

E. Robson

na4-l-gin2 na4-2/3-gin2 na4-l/2-gin2 na4-l/3-gin2 na4-igi-4-gal2 na4-igi-5-gal2 na4-22 1/2-se na4-20-se na4-15-se na4-10-§e na4-5-se L A N D S B E R G E R et al. (1970),

Weight of 1 shekel (c.8 g) Weight of 2/3 shekel Weight of 1/2 shekel Weight of 1/3 shekel Weight of a quarter (shekel) Weight of a fifth (shekel) Weight of 22 1/2 grains Weight of 20 grains Weight of 15 grains Weight of 10 grains Weight of 5 grains pp. 60-61)

A very few length measures were listed in a section on reed measuring rods in division two, but in general length and area metrology was not covered, presumably because little o f it could be related to the sizes o f material objects. 2.112 2.113 2.114 2.115 2.116 (After

gi-1 -ninda gi-l-kusj gi-1 /2-ku§3 gi-l/3-ku§3 gi-2/3-kus3 LANDSBERGER

Reed of 1 rod length (c.6 m) Reed of I cubit length (c.0.5 m) Reed of 112 cubit length Reed of 1/3 cubit length Reed of 2/3 cubit length (1959), pp. 191-192)

Later in the curricular sequence the names o f s o m e metrological units crop up in the more advanced list Proto-Ea, whose fimction w a s to list different Sumerian readings o f single signs. B e c a u s e the list is ordered b y the shapes o f the signs, the signs with metrological significance are scattered randomly throughout the 9 9 4 entries, amongst sequences dealing with the Sumerian values taken by individual cuneiform signs and their compounds. For instance the metrological unit ninda 'rod' is just one possible reading o f the sign GAR: 208 209 210 211 212 213

ni-ima ga2-ar in-da su-ku pa-ad ku-ru-um-ma

nig2 gar ninda §uku pad kurum^

The The The The The The

sign sign sign sign sign sign

GAR can GAR can GAR can sequence sequence sequence

be read as nig be read as gar be read as ninda U GAR can be read as suku U GAR can be read as pad U GAR can be read as kurum.

(After C I V I L et al. ( 1 9 7 9 ) , p. 4 0 )

Metrological units written with more than one sign, such as ma-na "mina" and su-si "finger" make n o appearance in Proto-Ea, and nor do measures such as b a n 2 , barig, and ese3, w h o s e units are implicit in the writing o f the numerical values. The metrological readings o f signs in Proto-Ea are as follows: 084 114 210 231 345 365 689

si-la bu-ru in-da ku-u§ gu-ur sa-ar se-e

sila3 bur3

ninda kus3 gur sar2 se

capacity measure, c. 1 litre area measure, c.6.5 ha length measure, c.6 m length measure, c.0.5 m capacity measure, c.300 litres area measure, c.3600 m^ multi-purpose small unit, 1/180 shekel

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

335

718 gi-im3 gin2 multi-purpose small unit, 1/60 of a sila3, ma-na, or sar 806 ku-ru gur? capacity measure, 3600 gur, c.l,080,000 litres (After CIVIL et al. (1979), pp. 30-63 passim) I m m e d i a t e l y after P r o t o - E a , it appears, the students o f H o u s e F m o v e d on to l e a m i n g the standard metrological series. T h e y h a d thus already a c q u i r e d s o m e systematic k n o w l e d g e of m e a s i n e s in context (in sub>sequences o f the t h e m a t i c n o u n list) a n d l e a m e d the contextualised readings o f m a n y o f the signs for m e t r o l o g i c a l units before they e n c o u n t e r e d the system as a w h o l e . 2.3 T h e s t a n d a r d m e t r o l o g i c a l s e r i e s V e r y little has b e e n studied o f the O l d B a b y l o n i a n metrological lists since N e u g e b a u e r a n d Sachs established the organisation of the four systems — length, area a n d v o l u m e , weight, a n d c a p a c i t y . I n d e e d , as s o i n c e s for the m e t r o l o g i c a l history of B a b y l o n i a they yield nothing o n c e the a p p r o x i m a t e sizes of the b a s i c units a n d the relationships b e t w e e n t h e m are k n o w n . H o w e v e r , w h e n v i e w e d as the p r o d u c t s o f scribal e d u c a t i o n they are potentially interesting o n c e m o r e . T h e standard m e t r o l o g i c a l series c o m p r i s e s sections o n capacity weights, areas (or v o l u m e s ) , a n d length in the following r a n g e s : gur

(5 X 60'* sila3)

Weight l/2se-100gun Area: 1/3 sar - 2 00 00 bur3 Length: 1 su-si - 1 00 danna (After FRIBERG (1987-90), p. 543)

(3 X 60" se) (60' sar) (3 X 60" su-si)

Capacity:

1/3 s i l a 3 -

10 danna 5 11 danna 6 30 12 danna 6 13 danna 6 30 14 danna 7 16 danna 7 30 17 danna 8 18 danna 8 30 19 danna 9 19 danna 9 30 20 danna 10 30 danna 15 40 danna 20 50 danna 25 1 danna 30

1 00 00

measure,

c.0.3 - 65 million litres c.0.05g-l,800kg c . l 2 m ^ - 4 7 , 0 0 0 ha c.l7 m m - 6 5 0 km

10 d a n n a = 5 00 00 (ninda) 11 danna = 5' 30 00 (ninda) 12 d a n n a = 6 00 00 (ninda) 13 danna= 6 30 00 (ninda) 14 d a n n a = 7 00 00 (ninda) 15' danna = 7 30 00 (ninda) 16' danna = 8 00 00 (ninda) 17' danna = 8 30 00 (ninda) 18' danna = 9 00 00 (ninda) 19 danna = 9 30 00 (ninda) 20 danna = 10 00 00 (ninda) 30 danna = 15 00 00 (ninda) 40 danna = 20 00 00 (ninda) 50 danna = 25 00 00 (ninda) 1 00 danna = 30 00 00 (ninda)

Figure 5: 3 N - T 3 1 6 = A 3 0 2 1 1 ( u n p u b l i s h e d ) . Detail o f r e v e r s e , s h o w i n g large length m e a s u r e s (1 ninda = 6 m ; 1 darma = 1800 ninda = 10.8 k m ) .

Exactly the same units were used for areas and volumes, volume units being defined as 1 (horizontal) area unit x 1 cubit height (NEUGEBAUER and SACHS (1945), pp. 4 - 6 ) .

336

E. Robson

Extracts from the series could b e written in the fr)rm o f lists — w i t h e a c h entry containing the standard notation for the m e a s u r e s o n l y — or as tables — w h e r e the standard writings were s u p p l e m e n t e d with their sexagesimal equivalents.^' For instance, the reverse o f 3 N - T 3 1 6 contains a n extract from the e n d o f the metrological table of lengths (Figure 5).

Capacity measures Weights Areas - volumes Lengths Total

Lists

Tables

Total

3

2

5

3

1 3

1 6

F i g u r e 6: M e t r o l o g i c a l exfracts o n tablets from H o u s e F . Fifteen tablets w i t h extracts from the standard m e t r o l o g i c a l series survive from H o u s e F . S o m e or all o f their contents, tablet type, a n d c o m p o s i t i o n a l format c a n b e d e t e r m i n e d for twelve o f t h e m so far (from catalogue r e c o r d s a n d p e r s o n a l inspection). A l m o s t all identifiable p i e c e s are T y p e II/2 tablets. O n their o b v e r s e s are a reciprocal table, sections o f Proto-Diri, m o d e l contracts, a n d S u m e r i a n p r o v e r b s : m e t r o l o g y thus p r e c e d e d these topics in the H o u s e F curriculum. O n e T y p e II/1 table of weights has an exfract from Proto-Izi on the reverse: m e t i o l o g y thus followed this c o m p o s i t i o n in the H o u s e F curriculum. T h e other fragments a p p e a r at this stage of r e s e a r c h to h a v e c o m e from T y p e I or T y p e II tablets; there is n o m e t r o l o g y surviving o n tablet types III or IV. O f the six tablets w h o s e contents a n d compositional format are identifiable, all b u t one are from the start o f the s e q u e n c e , but there is a n e v e n split b e t w e e n tabular a n d list format.^*

Figure 7: 3 N - T 5 9 4 = I M 5 8 5 7 3 . T h e obverse o f the T y p e II tablet (left) s h o w s a t e a c h e r ' s c o p y o f the list of reciprocals, with the s t u d e n t ' s c o p y to the right erased. T h e reverse (right) is a n extiact from the standard metrological list, with c a p a c i t y m e a s u r e s from 12 to 19 gur a n d 3 0 0 0 to 3 6 0 , 0 0 0 gur.

FRIBERG (1987-1990), pp. 542-543. Compare the six metrological tablets from the gala-mahs' house (§1.3): all are tables on fragments of Type I tablets. Three tabulate capacities only, one tabulates both capacities and weights, while two tabulate weights alone. It appears as though a single student had worked his way through the metrological series from the beginning to about the half-way point (TANRET (2002), pp. 100-112).

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

337

In short, o n present evidence little can b e said about metrology within H o u s e F, except that its position in the curriculum can be established, and that Type II/2 extracts from the begiiming o f the compositional sequence apparently predominate the meagre extant record. But it is impossible to determine whether the list and tabular formats had distinct pedagogical fimctions; neither is there much to b e deduced from comparative material (primarily because it is all under-published). However, w e can do a great deal more with the much more abundant remains from the standard arithmetical series which immediately followed it in the H o u s e F curriculum ( § 3 ) . First, though, w e will jump ahead to the end o f the elementary curriculum to examine the use o f metrology in model contracts. 2.4 M e t r o l o g y i n u s e : m o d e l c o n t r a c t s Towards the end o f elementary education in House F students were introduced to whole sentences in Sumerian for the first time, in the form o f model legal contracts. The genre as a whole, although apparently a relatively c o m m o n element in scribal schooling, has not yet been studied in depth.^^ The contracts from House F c o n c e m grain and silver loans, inheritance divisions, and sales o f slaves and houses. A l l o f them use metrological imits in quasi-realistic contexts, as the following t w o examples show: 1 (gur) se-gur

300 litres of grain

m a s 2 1 gur 1 (barig) 4 b a n 2 se-ta-am3

The interest is 100 litres of grain for every

si-ge4-de3 ki lugal-ezen-ta "'a-pil2-ku-ga-X [...] iti sig4-a [...]

To be removed' From Lugal-ezen to Apil-kuga-[ ] Brick-making month [

300 l i n e s

]

(3N-T 914.x = A 33446, unpublished) 1/3 s a r e2-du3-a da e2 digir-ga-mi-il

12 m^ built-up house, next to Dingir-gamil's house [2]5 s a r a-sa3 d u g a-hu-ni u§-a-du 900 m^ field, t h e ruin mound of Ahuni, digir-ga-mil bordering Dingir-gamil's (land) [1]2 1/2 sar gi§-kirÌ6 -§a3 id2 «lugal» 450 m^ date orchard of the royal waterway lugal zag gi§-kirÌ6 digir-ga-mil field next to Dingir-gamil's date orchard [I gi§-bansur-zag]-gu-la 1 giS-ga-nu-um-kas [1] large [offering table], 1 wooden pot stand for beer, [3 g i § - d i l i m 2 ] U3 nig2-gU2-[un]-a igi-4-gal2-bi [3 wooden spoons] and a quarter of its (i.e., the estate's) equipment, ha-la-ba a-pil2-Ì3-lÌ3-su Apil-ilishu's share. (3N-T342 = IM 58436, lines 10'-16', unpublished) In short, some aspects o f mefrology ran right through the elementary curriculum in House F. However, the focus appears to have been o n memorisation and contextual use; there is n o evidence that the House F students practised metrological conversions or calculations o f any kind (see §4.3).

For examples of edited model legal documents see CIVIL ( 1 9 7 5 ) , pp. 1 2 9 - 1 3 0 ; WILCKE ( 1 9 8 7 ) , pp. 1 0 4 - 1 0 7 ; ROTH ( 1 9 9 5 ) , pp. 4 6 - 5 4 ; VELDHUIS ( 2 0 0 0 ) , p. 3 8 6 , and B O D I N E ( 2 0 0 1 ) .

338

E. ROBSON

3. Arithmetic 3.1 T h e s t a n d a r d arithmetical series T h e s t a n d a r d h s t of multiplications w a s d e s c r i b e d l o n g a g o b y N e u g e b a u e r a n d Sachs, a n d is v e r y well known.^*^ N e v e r t h e l e s s , it is usefiil to s u m m a r i s e its salient features from a n e d u c a t i o n a l standpoint ( § 3 . 1 ) . Systematic differences in c o n t e n t and textual format a c r o s s tablet types reflect their p e d a g o g i c a l fiinction ( § 3 . 2 ) , while regular o m i s s i o n s from the standard list suggest o n e or t w o idiosyncrasies p a r t i c u l a r to H o u s e F ( § 3 . 3 ) . T h e series starts with a list of o n e - and t w o - p l a c e r e c i p r o c a l p a i r s , e n c o m p a s s i n g all the regular integers from 2 to 8 1 . It is followed b y multiplication ' t a b l e s ' for s e x a g e s i m a l l y regular h e a d n u m b e r s from 5 0 d o w n to 1 15 (see F i g u r e 11), with multiplicands 1-20, 3 0 , 4 0 , and 50.^' S o m e series also include the s q u a r e s a n d inverse s q u a r e s of e a c h h e a d n u m b e r . N e u g e b a u e r r e c o n s t r u c t e d t h e s t a n d a r d s e q u e n c e o n the b a s i s of w h a t he called ' c o m b i n e d multiplication t a b l e s ' — or in curricular t e r m i n o l o g y long exfracts o n tablet types I a n d II/2. H i s 'single multiplication t a b l e s ' t u m out to b e tablet t y p e s I I / l a n d III.

Figure 8: 3 N - T 2 6 1 = U M 5 5 - 2 1 - 2 8 9 ( o b v e r s e ) a v e r b o s e T y p e III m u l t i p l i c a t i o n table for 1;40 (left), a n d 3 N - T 6 0 8 = U M 5 5 - 2 1 - 3 6 0 ( o b v e r s e ) , a terse T y p e III multipUcation t a b l e for 3 (right). N e u g e b a u e r also identified three m a i n textual formats for m u l t i p l i c a t i o n s , a n d four less c o m m o n variants.^^ W e could call N e u g e b a u e r ' s T y p e s A a n d A ' verbose formats, in that t h e y r e p e a t the w o r d a-ra2 ' t i m e s ' in e v e r y line o f e a c h t a b l e {h a-raa 1 h, a-ra2 m hm "A t i m e s 1 is h, times m is hm"). H i s T y p e s B , B ' , B " , C, a n d C , h o w e v e r , are all terse, as a-ra2 ' t i m e s ' m a k e s at m o s t o n e a p p e a r a n c e in t h e first line; ^°

NEUGEBAUER ( 1 9 3 5 - 1 9 3 7 ) , I, PP. 3 2 - 6 7 AND NEUGEBAUER AND SACHS ( 1 9 4 5 ) , PP. 1 9 - 3 3 . IN THE FOLLOWING PARAGRAPH, I ABBREVIATE 'HEAD NUMBER' AS h AND 'MULTIPLICAND' AS m. I

HAVE PUT THE WORD 'TABLES' IN INVERTED COMMAS BECAUSE THESE TABLETS ARE NOT LAID OUT AS FORMAL TABLES WITH COLUMNAR DIVISIONS BUT AS LISTS LIKE THE BULK OF THE REST OF ELEMENTARY SCHOOL SUBJECT MATTER (SEE ROBSON ( 2 0 0 3 ) ) . "

NEUGEBAUER AND SACHS ( 1 9 4 5 ) , P. 2 0 .

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

339

thereafter the text is entirely n u m e r i c a l (m hm). In fact it t i u n s out that the formats so far attested in H o u s e F are all either T y p e A or T y p e C; for that r e a s o n t h e y will b e referred to s i m p l y as V e r b o s e a n d T e r s e formats, to p r e v e n t the confiising proliferation o f T y p e s in the discussion (Figure 8). A n a l o g o u s l y , the r e c i p r o c a l tables at the h e a d of the series m a y b e in V e r b o s e format (igi-n-gal2-bi 1/« "Its n t h p a r t is 1/n", e.g. F i g u r e 7) or T e r s e (n 1/n).^^ O f the 9 7 H o u s e F tablets currently k n o w n to c o n t a i n exttacts f r o m the s t a n d a r d multiplication s e q u e n c e , 32 c a n b e identified as T y p e III, 3 8 as T y p e II, a n d 10 as T y p e I ( F i g u r e 9 ) . N i n e fragments m a y b e from T y p e I or T y p e II tablets a n d the t y p o l o g y of the r e m a i n i n g s e v e n is u n k n o w n . E l e v e n of the twenty-five p r o b a b l e T y p e II/2 tablets h a v e identifiable c o m p o s i t i o n s o n their o b v e r s e s : eight are tables from t o w a r d s the e n d of the multiplication series (Figure 11), w h i l e there is o n e m o d e l confract, o n e s e q u e n c e o f S u m e r i a n p r o v e r b s , a n d o n e c o m p o s i t i o n yet t o b e distinguished. T h e r e are twenty-one multiplication tables o n T y p e II/1 tablets; apart from the eight multiplication reverses j u s t m e n t i o n e d , o n e e x e m p l a r e a c h o f the thematic n o u n list (division four), P r o t o - L u , Proto-Izi, a n d a mefrological list (Figure 7) h a v e b e e n identified. Tablet type

Number

of tablet pieces

10 38

I II II/l & 2 I I / l only II/2 o n l y

Percentage

of total

10.4 39.6 8 13 17

8.3 13.6 17.7

III l o r II Unknown

32 9 7

33.3 9.4

Total

96

WO

7.3

F i g u r e 9: T y p o l o g y o f the tablets b e a r i n g multiplication tables in H o u s e F . 3.2 T a b l e t f u n c t i o n s a n d t e x t u a l f o r m a t s W h y w e r e three different types o f tablet u s e d to r e c o r d the multiplication series in H o u s e F ? L o o k i n g first at the 3 4 tablets w h i c h b e a r j u s t o n e identifiable multiplication or r e c i p r o c a l table each, n a m e l y T y p e s I I / l a n d III ( F i g i u e 10), attested tables are scattered a p p a r e n t i y r a n d o m l y t h r o u g h the series: there are 9 tablets from the first quarter, 8 from the s e c o n d , 9 from the third, a n d 7 from the last, a n d there is little difference b e t w e e n the t w o tablet types. T h e p i c t u r e that e m e r g e s from the tablets containing longer exttacts from the series is v e r y different, h o w e v e r . O n b o t h the T y p e II/2 (Figure 11) a n d the T y p e I tablets (Figure 12) tiie attested tables are p r e d o m i n a n t l y from the first quarter o f the series, n a m e l y 18 of the 25 T y p e II/2 tablets a n d 6 o f the 10 T y p e Is ( 6 9 p e r c e n t in total). A l l b u t o n e o f tiie r e m a i n d e r are from the s e c o n d quarter, w h e r e it a p p e a r s that there w a s a formal section b r e a k b e t w e e n the tables for 2 0 and 18. T h i s disfribution is not peculiar to H o u s e F . A simple analysis o f the O l d B a b y l o n i a n ' c o m b i n e d multiplication t a b l e s ' p u b l i s h e d b y N e u g e b a u e r before

Cf NEUGEBAUER and SACHS ( 1 9 4 5 ) , p. 1 2 .

340

E. ROBSON

Head no.

Tablet Type II/1

Tablet Type III

Total

RECIPROCALS

1

2

50

1

1

1

1

3

48 45 44 26 40 40

1

1

1

4

36 30 25

3

24 22 30 2Ò 18

1

1

16 4 0

1

1

16

1

1

2

15 12 3 0

1

2

3

1

1

12 10 9 8 20 8 7 30

1

1

7 12 7 6 40

1

6

1 1

1

5 4 30

1

4

4

3

1

1

3 45

1

1

3 2Ó 3

1

1

2 30

1

1

2 24 2 15 2

1

1 40 1 30

1

1 20

1

1 15

Total

1 1

17

1 1 1

1

1

17

34

Figure 10: T a b l e s attested o n T y p e II/l a n d T y p e III tablets from H o u s e F.^'*

DOTTED LINES IN THIS AND FOLLOWING FIGURES ARE READING AIDS; THEY DO NOT MARK FORMAL DIVISIONS IN THE SERIES.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

IS

s

341

w

II ilecips 50 48 15 «4 26 40 40 (•)

36 )0

15

(•)

24 22 30

2Ó 18 16 40

(•)

16

(•)

15

(•)

12 30 12 10 J 8 20 8 Key

730 7 12 7 640

•

-

Attested

(•)

•

Not attested

=

Probably originally attested (tablet damaged or broken)

=

Tablet broken

5 45

J20" 230 2 24 2 15 40 30 20 15

I I

S

I

I I

IIÌ

1Î

1

Figure 11: Multiplication a n d division tables attested o n T y p e II/2 fragments in House F.

342

E. Robson

o

ON

o

1—1

OS

H

et

d o\ H

.£5

en o\ H

OS

0 \

o

OS

:^

1^ Recips 50 48 45 44 26 4 0 40 36 30 25 24 22 30 20 18 16 4 0 16 15 12 3 0 12 10 9 8 20

m

HI

ON

H I

m

(•) (•)

(-) (•) (-)

(-)

(

(•)

7 30 7 12 7 6 40 6 5 4 30 4 3 45 3 20 3 2 30 2 24 2 15 2 1 40 1 30 1 20 1 15 Figure 12: Multiplication and reciprocal tables attested on T y p e I tablets in H o u s e F.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

Tablet Type Start of sequence First quarter Second quarter Third quarter Fourth quarter Total

I and 11/2 House F 24 10

('combined ' tables) Neugebauer 51 6 8 5 70

I 35

II/l and III House F 9 8 9 7 34

343

('single ' tables) Neugebauer 56 44 34 25 159

Figure 13: Distribution of tablet types across the standard series o f multiplications. H o u s e F w a s e x c a v a t e d reveals a striking similarity. O f the 7 0 tablets h e listed, 51 o f t h e m (72 p e r c e n t ) apparently b e g i n their s e q u e n c e s o f multiplications in the first quarter o f the series, 6 in the s e c o n d quarter, 8 in the third, a n d 5 in the last (Figure 13).^^ O n the other hand, the n u m b e r of N e u g e b a u e r ' s 159 'single multiplicafion t a b l e s ' ^ ' decreases m o r e or less linearly across the series: 5 6 are firom the first quarter, 4 4 fi-om the second, 3 4 from the third, a n d 25 firom the last. W h i l e this p a t t e m of attestation d o e s not exactly m a t c h the e v e n distribution of tablet types II/l a n d III in H o u s e F , it is clearly distinct from the h e a v y s k e w t o w a r d s the begiiming of the series found in the ' c o m b i n e d multiplication t a b l e s ' ( T a b l e t T y p e s I and II/2) firom H o u s e F a n d elsewhere.^* N e u g e b a u e r highlighted the strong correlation b e t w e e n tablet type a n d textiaal format:^' s o m e 8 0 - 9 0 p e r c e n t of his ' c o m b i n e d ' multiplication tables ( d e p e n d i n g o n h o w o n e defines a n d coimts the tables) are in terse formats a n d the r e m a i n d e r are v e r b o s e . Conversely, a b o u t 7 0 - 8 0 p e r c e n t o f the ' s i n g l e ' multiplication tables are v e r b o s e a n d the rest terse. O n c e again w e find similar results m the H o u s e F c o r p u s , w h e r e formats can b e identified: 31 of the 35 T y p e I a n d T y p e II/2 tablets (89 p e r c e n t ) , b e a r tersely formatted tables, while 29 out o f the 3 4 T y p e I I / l a n d III tablets (85 p e r c e n t ) , are v e r b o s e . In sum, there are t w o clearly m a r k e d distinctions b e t w e e n the ' s i n g l e ' multiplication tables o n the one h a n d a n d the ' c o m b i n e d ' tables o n the other. O n the one h a n d , the single tables ( o n tablet T y p e s I I / l a n d III) are evenly distiibuted across the w h o l e series (but w i t h s o m e s k e w t o w a r d s the b e g i n n i n g in N e u g e b a u e r ' s s a m p l e ) a n d are p r e d o m i n a n t i y verbosely written, while the longer exttacts c o n t a m i n g s e q u e n c e s o f tables are v e r y heavily w e i g h t e d t o w a r d s the start o f the series a n d are generally terse. O n e c a n also m a k e a fiirther differentiation: it is generally t m e that the ' s i n g l e ' tables are written in a carefiil, calligraphic h a n d with clear line spacing, while the long exttacts c o m p r i s i n g m a n y tables a p p e a r to h a v e b e e n written with little regard for visual a p p e a r a n c e : there are generally n o line m l i n g s , for instance, and e v e n the coliunnar divisions are often difficult to m a k e out. "

NEUGEBAUER

(1935-1937),

I,

pp.

35;

II,

p.

37;

NEUGEBAUER

and

SACHS

(1945),

pp. 2 5 - 3 3 .

However, it is difficult to judge from the descriptions given by NEUGEBAUER and SACHS ( 1 9 4 5 ) , pp. 2 5 - 3 3 whether the tablets are fragments or not, and therefore whether complete sequences are attested on them. ^"^ NEUGEBAUER ( 1 9 3 5 - 1 9 3 7 ) , I, p. 3 4 ; II, p. 3 6 ; NEUGEBAUER and SACHS ( 1 9 4 5 ) , pp. 2 0 - 2 3 .

There is little comparative data fi-om known archaeological contexts (§ 1.3). The two Type III multiplication tables from Sîn-kâSid's palace in Uruk are for 4 5 and 2 2 3 0 (both terse). The five from the Uruk Scherbenloch are for 4 5 , 2 2 3 0 , 9 , 8 ; 2 0 , and 3 (all verbose). NEUGEBAUER ( 1 9 3 5 - 3 7 ) , I, pp. 6 2 - 6 4 .

344

E. Robson

T h e s e three factors c o m b i n e to suggest a clear p e d a g o g i c a l distinction b e t w e e n the well written, fully w o r d e d single tables o n the o n e h a n d a n d the hastily scribbled, terse s e q u e n c e s of tables on the other. W e h a v e already r e v i e w e d V e l d h u i s ' s hypothesis (§2.1) that T y p e II tablets h a d a dual function: on the o b v e r s e ( I I / l ) the student r e p e a t e d l y c o p i e d the t e a c h e r ' s m o d e l of a n extract (or table) that he w a s l e a m i n g for the first time, a n d then o n the reverse (II/2) w r o t e out a m u c h longer extract from earlier in that s a m e composition, or from one h e h a d a l r e a d y m a s t e r e d . T h e e v i d e n c e from the standard series of multiplication tables p r e s e n t e d here not only allows us to confirm that hypothesis but also to d r a w s o m e fiirther conclusions. First, it a p p e a r s that T y p e III tablets were also u s e d in the initial stages of l e a m i n g a n extiact, p r e s u m a b l y after the student h a d m e m o r i s e d it well e n o u g h to n o longer n e e d a m o d e l to c o p y in the T y p e I I / l p a t t e m . Equally, the T y p e I tablets a p p e a r to h a v e served a similar revision p u r p o s e to the T y p e II/2 tablets, on w h i c h students r e v i e w e d long stietches of material they w e r e n o longer actively w o r k i n g on, or p e r h a p s fitting their m o s t recent achievements into their p l a c e in the c o m p o s i t i o n a l s e q u e n c e . S e c o n d , and p e r h a p s m o r e interestingly, it seems that while students were given initial e x p o s u r e to the w h o l e of a composition, b y m e a n s of short extiacts o n tablet T y p e s I I / l a n d III, their revision o f that w o r k was m u c h less systematic, starting from the b e g i n n i n g again e a c h time a n d rarely reaching the end. This disfribution of tablet types across the series is found in other e l e m e n t a r y educational c o m p o s i t i o n s t o o . It is c o m p a r a b l e , for instance, to the survival p a t t e m s o f Old B a b y l o n i a n tablets from N i p p u r containing exttacts from division o n e o f the thematic n o u n list, the ttees a n d w o o d e n objects ( § § 2 . 1 - 2 ) . C o u n t i n g the n u m b e r o f sources for e a c h tablet type over the 707-line c o m p o s i t i o n in V e l d h u i s ' s edition,'*" the following p a t t e m e m e r g e s (Figure 14): there are a m e a n of 4 4 sources for e a c h o f the first five lines, 14 for lines 1 0 1 - 5 , four for lines 3 0 1 - 5 , three for lines 5 0 1 - 5 , a n d j u s t t w o for lines 7 0 1 - 5 . Dividing the tablet types into their functions o f 'first e x p o s u r e ' ( T y p e s I I / l , III, a n d p r o b a b l y IV; cf. §2.1) a n d ' r e v i s i o n ' ( T y p e s I, II/2, a n d P), w e see that t h e r l are never m o r e than four 'first e x p o s u r e s ' for a n y o n e o f the lines s a m p l e d b u t m o r e often o n e or n o n e . Conversely, the ' r e v i s i o n ' tablets are v e r y heavily w e i g h t e d i n d e e d t o w a r d s the b e g i i m i n g of the c o m p o s i t i o n (taking into a c c o u n t the c o m m o n l y occurring d a m a g e to t h é c o m e r s of tablets w h i c h has l o w e r e d the n u m b e r of attestations for the v e r y first t w o or three lines). In other w o r d s , this suggests that a l t h o u g h e l e m e n t a r y students in N i p p u r t e n d e d to b e taught c o m p o s i t i o n s in their entirety, from beginning to end, all revision in the e l e m e n t a r y c u r r i c u l u m w a s slanted t o w a r d s the o p e n i n g sections o f c o m p o s i t i o n s to the d e t i i m e n t o f their m i d d l e s a n d closing lines.

VELDHUIS (1997), pp. 191-252.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

Tablet Line 1 2 3 4 5

type

First II/l

exposure III IV

1

1 1 1 1 1

1 1

'idi 102 103 104 105

1 1 1 1 1

'301 302 303 304 305

2 1

'5'di 502 503 504 505

i 1

32 35 40 46 47

1

2

...2.

9. 4 2 1 3 4

1 1 1

Total 1 1 2 2

"

\ 1 1 1 1

1

7

18

34 37 42 49 50 •"Ï2"" 15 15 14 15

""Y"

1 1 1 1 1

4 2 4 5 2 3 2 4 4

1 1 3 3

1 1

15

Unclear P

" i 9 2 13 2 13 2 11

'7'di 702 703 704 705 Total

Revision I II/2

345

Ï 1

2 1 2 2

1 1 272

5

6

F i g u r e 14: Distribution of tablets over the O B r e s c e n s i o n o f the List o f trees a n d w o o d e n objects.

Nippur

3.3 M i s s i n g h e a d n u m b e r s R e t u m i n g to the standard series of multiplications as attested in H o u s e F , nine o f the forty k n o w n h e a d n u m b e r s — n a m e l y 4 8 , 4 4 2 6 4 0 , 2 0 , 7 12, 7, 5, 3 2 0 , 2 2 4 , and 2 1 5 — d o not siuvive o n k n o w n tablets. Should w e attribute these o m i s s i o n s to the accidents o f r e c o v e r y or to deliberate exclusion from the series? T h e p a t t e m s o f attestation m a k e it easier to m a k e definitive statements a b o u t the h i g h e r h e a d n u m b e r s than the lower. T h e h e a d n u m b e r 4 8 , for instance, is included in j u s t five o f the 71 ' c o m b i n e d ' tables catalogued b y N e u g e b a u e r (and j u s t t w o of t h o s e five are from N i p p u r ) , c o m p a r e d to twenty-three certain omissions. H e lists n o ' s i n g l e ' tables for 4 8 . Similarly, 2 15 occurs in t w o out of nine possible ' c o m b i n e d ' tables, neither of t h e m from N i p p u r , a n d in n o ' s i n g l e s ' . It is not surprismg, therefore, that the 4 8 a n d 2 15 times tables w e r e apparently not taught in H o u s e F . T h e exclusion o f 4 4 2 6 4 0 , is rather m o r e surprising: g i v e n its p l a c e near the start o f the standard series it is p r e s u m a b l y not srniply missing b y archaeological accident. O n the other h a n d n o n e of N e u g e b a u e r ' s ' c o m b i n e d ' tables a p p e a r to omit it, while h e lists three ' s i n g l e ' tables for 4 4 2 6 4 0 . This is a deliberate but idiosyncratic o m i s s i o n then.

346

E. Robson

particular to H o u s e F — t h o u g h p e r h a p s a j u d i c i o u s o n e ; n o n e o f t h e other h e a d n u m b e r s are three sexagesimal places long. It is p r o b a b l y best t o r e s e r v e j u d g e m e n t o n the r e m a i n i n g six ' m i s s i n g ' h e a d n u m b e r s .

4. Beyond elementary education 4.1 T h e S u m e r i a n l i t e r a r y c u r r i c u l u m i n H o u s e F A l t h o u g h , as w e h a v e seen (§1.2), the vast majority o f m a t h e m a t i c a l tablets in H o u s e F c a n b e assigned t o the elementary c u r r i c u l u m o n g r o u n d s o f content a n d tablet typology, there are four w h i c h caimot b e . T h r e e o f those tablets b e a r calculations, while the fourth contains a n extra-curricular table. A l t h o u g h the table is difficuh t o p l a c e p e d a g o g i c a l l y (§4.5), it is possible t o p o s i t i o n t h e calculations within t h e ' a d v a n c e d ' c u r r i c u l u m (§4.3), w h i c h in H o u s e F w a s d o m i n a t e d b y S u m e r i a n literature (§4.2), a n d to c o m p a r e this situation with calculations in other school c o r p o r a ( § 4 . 4 ) . First, however, w e n e e d to r e v i e w w h a t is k n o w n o f the S u m e r i a n literary c u r r i c u l u m in H o u s e F . O v e r eighty different literary w o r k s h a v e survived from t h e H o u s e , attested o n a r o u n d 6 0 0 different tablets. A l t h o u g h w e d o n o t h a v e a clear-cut tablet t y p o l o g y from w h i c h to d e d u c e a well defined a n d o r d e r e d curriculum, it is p o s s i b l e t o at least outline t h e contents o f that curriculum, b a s e d o n c o n t e m p o r a n e o u s literary catalogues a n d s o m e basic quantitative methods.'*' First, b y s i m p l y c o u n t i n g the n u m b e r o f (joined) sources for e a c h composition, it b e c o m e s clear that there is o n e ' m a i n s f r e a m ' g r o u p o f twenty-four literary w o r k s , e a c h with a m e a n o f 18 sources, c o m p a r e d t o t h e rest w h i c h h a v e o n average j u s t 3 attestations. S e c o n d , t e n o f those twenty-four ' m a i n s t i e a m ' w o r k s comprise a widely-attested curricular g r o u p i n g that Steve Tirmey h a s labelled the Decad."*^ T h e incipits o f the D e c a d m e m b e r s c o m p r i s e the first t e n enfries ( i n the s a m e order) o f three O l d B a b y l o n i a n literary catalogues, a n d h a v e a s t i o n g p r e s e n c e in four others. T h e r e m a i n i n g m e m b e r s o f that mainsfream g r o u p i n g , w h i c h I have called the H o u s e F Fourteen,'*^ a p p e a r o n three of those same catalogues, in a fixed order t h o u g h n o t clustered t o g e t h e r in a single b l o c k like the D e c a d (Figure 15).'*'' T h e r e m a i n i n g H o u s e F literature c a n b e r o u g h l y categorised into foiu" groups:'*^ m y t h s , epics a n d laments ( 1 3 w o r k s ) , h y m n s t o kings a n d deities ( 1 1 ) , school narratives, debates a n d dialogues ( 7 ) , a n d literary letters a n d related short p i e c e s (25). T h e r e is also at least o n e fi'agment o f extiacts from a l a w code"^ a n d o n e tablet containing a fragment o f a G i l g a m e s h myth in Akkadian."*'

TiNNEY ( 1 9 9 9 ) ; ROBSON (forthcoming). '2

TINNEY ( 1 9 9 9 ) , pp. 1 6 8 - 1 7 0 .

ROBSON (forthcoming).

Outside House F, the Decad members are found on an average of 4 1 Nippur tablets each and 3 5 non-Nippur tablets. For each of the Fourteen there are, on average, 3 0 Nippur tablets (outside House F) and 1 0 from beyond Nippur. ROBSON (forthcoming). ROTH ( 1 9 9 5 ) , p. 2 5 0 . CAVIGNEAUX and RENGER ( 2 0 0 0 ) .

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

No. of sources

Line number of catalogué*^

Literary composition

ETCSL no.

SHULGI HYMN A LIPIT-ESHTAF HYMN A THE SONG OF THE HOE INANA HYMN B ENLIL HYMN A KESH TEMPLE HYMN ENKI'S JOURNEY TO NIPPUR INANA AND EBIH NUNGAL HYMN A GILGAMESH AND HUWAWA ( A ) DEBATE BETWEEN SHEEP AND GRAIN CURSING OF AGADE DUMUZID'S DREAM GILGAMESH, ENKIDU AND THE NETHER WORLD INSTRUCTIONS OF SHURUPPAG DEBATE BETWEEN HOE AND PLOUGH SHULGI HYMN B EXPLOITS OF NINURTA UR LAMENT SCHOOLDAYS (Eduba COMPOSITION A ) Eduba COMPOSITION C Eduba DIALOGUE 1 FANNER'S INSTRUCTIONS Eduba COMPOSITION B

2.4.2.01 2.5.5.1

17 12

01 02

L [01] [02]

57 01 02

5.5.4

24

03

[03]

04

N2 DOL'" D02 D03 D04 D05 D06 D07 D08 D09 DIO FOI

F02 F03 F04

F05 F06 F07 FOB F09 FIG

FIL F12 F13 F14

347

Ul

U2

B4

Y2

—

04 05

07 08

01 02

09

03

08 16 23

03 10

04

28

24

^

— — —

36 24 22

04 05 06

[04] 05 06

03 05 06

1.1.4

9

07

07

07

1.3.2 4.28.1

18 19

08 09

08 09

08 09

10 18

13 14

02

1.8.1.5

21

10

10

R3

14

09

—

11

4.07.2 4.05.1 4.80.2

— — —

— — —

15

5.3.2

19

17

2.1.5

15

18

12

— —

17

— —

1.4.3

20

19

13

R4

26

— —

1.8.1.4

15

20

14

5.6.1

18

21

15

5.3.1

30

25

16

2.4.2.2 1.6.2

17 15

26

17 18

2.2.2

17

5.1.1

18

32 50

5.1.3

14

51

5.4.1

22

52

5.6.3

21

53

5.1.2

11

54

—

26

29?*°

— —

29?

19

—

18

— 01 — — — — 24?^' R9

— —

—

10

41 44 33?

— — — — 06?

24?

07?

24?

08?

22

— —

Figure 15: M a i n s t r e a m S u m e r i a n literary c o m p o s i t i o n s in H o u s e F.

35

22

07

— —

52

N2 (ETCSL 0.2.01) from Nippur; L (0.2.02) from Nippur?; SI (0.2.18) from Sippar; Ul (0.2.03), U2 (0.2.04) from Ur; B4 (0.2.11), Y2 (0.2.12) unprovenanced. R - reverse. DOl-10 = Decad; FOl-14 = House F Fourteen This entry, ud re-a ud sud-ta re-a, could be the incipit of either Gilgamesh, Enkidu and the Nether World or the Instructions of Shurrupag. This incipit, dumu e2'dub-ba-a, could belong to any one of Eduba A, Eduba C, Eduba F (ETCSL 5.1.a, unpublished), Eduba Dialogue 1, ox Eduba Dialogue 3 (ETCSL 5.4.3).

348

E. Robson

4.2 M a t h e m a t i c s in t h e S u m e r i a n literary c u r r i c u l u m References t o m a t h e m a t i c a l achievement a n d failure in S u m e r i a n literature h a v e b e e n collected before, usually i n a m i s g u i d e d attempt t o u s e literaiy w o r k s a s u n p r o b l e m a t i c sources o f historical evidence a b o u t ' S u m e r i a n s c h o o l ' . ^ H o w e v e r , o n c e w e r e c o g n i z e that those literary w o r k s were themselves e l e m e n t s o f a scribal curriculum, as for instance in H o u s e F , it b e c o m e s interesting a n d i m p o r t a n t t o study t h e m for the m e s s a g e s that they c o n v e y e d to t h e students a b o u t m a t h e m a t i c s a n d t h e s c r i b e s ' relationship t o it. M a t h e m a t i c a l a n d metrological elements a p p e a r in s o m e o f t h e h u m o r o u s narratives a n d dialogues about school life (the so-called eduba texts, n a m e d after the S u m e r i a n w o r d for school). A l t h o u g h w e c a n occasionally verify that particular details in the narratives are in s o m e sense ' t r u e ' in that they c o n c u r w i t h other e v i d e n c e , they a r e highly unlikely to h a v e b e e n straightforward d o c u m e n t a r y a c c o u n t s : after all, their intended audience, the scribal students, a l r e a d y k n e w exactly w h a t school w a s like. T h e narratives often m a k e u s e o f v e r y b r o a d h u m o u r to g e t their m e s s a g e across ( o r at least b r o a d h u m o u r is the o n l y type that w e , with our unsophisticated understanding o f Sumerian, c a n currently u n d e r s t a n d ) . It m a y b e that other elements o f h u m o u r lay in the contrast b e t w e e n school life a s d e p i c t e d a n d as e x p e r i e n c e d b y the students; in that case those apparently realistic details w o u l d h a v e served s i m p l y to a d d elements o f verisimilitude t o o t h e r w i s e highly fictionalised accounts.^'' In the m o s t famous o f these works, often k n o w n b y its m o d e m title ' S c h o o l d a y s ' , t h e teacher o f a n i n c o m p e t e n t scribal student is invited h o m e for dinner a n d bribery, in a n attempt to m a k e h i m ease u p o n t h e h a p l e s s child. T h e father flatters the s t e m teacher shamelessly, saying: 59-61 „jyjy ji^^jg fellow has opened (wide) his hand, (and) you made wisdom enter there; you showed him all the fine points of the scribal art; you (even) made him see the solutions of mathematical and arithmetical (problems)." (Eduba Composition A, after KRAMER (1963), p. 239)^^ A n earlier p a s s a g e in the narrative, however, m a k e s it clear that the teacher h a d s h o w e d h i m little except the business end o f his cane.

All literary compositions and ancient catalogues are published in the Electronic Text Corpus of Sumerian Literature (BLACK et al. 1998-) and cited according to their ETCSL titles and catalogue numbers. "

E.g. SJÔBERG (1975); NEMET-NEJAT (1993), pp. 5-10.

Compare Hogwarts, the boarding school for wizards in training, in the highly popular childrens' novels and film about Harry Potter. No child reader has ever set foot in an institution anything like Hogwarts, yet it is still recognisably a school. Its fascination and attraction lies in the fact the judicious combination of realism, fantasy, and humour with which the stories are constructed —just as in the Sumerian school narratives. This, of course, is where the similarity ends. No modem critical edition of this composition has ever been published, although there have been single-line composite texts and translations in the public domain for over half a century. I have not attempted to improve on Kramer's translation, apart from the addition of 'even' in the final line.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

349

In a gentler c o m p a n i o n p i e c e , s o m e t i m e s called ' S c r i b a l A c t i v i t i e s ' , a teacher quizzes a student o n w h a t he has l e a m e d , s o m e three m o n t h s before h e is d u e to leave school. T h e student lists everything h e has m a s t e r e d so far, m u c h o f w h i c h c a n b e m a t c h e d quite closely to the e v i d e n c e from the a r c h a e o l o g i c a l l y r e c o v e r e d e l e m e n t a r y tablets t h e m s e l v e s . (This is hardly surprising, as o n e a i m of the c o m p o s i t i o n m u s t h a v e b e e n to e n c o u r a g e identification with, a n d e m u l a t i o n of, this p a r a g o n o f l e a m m g . ) T h e standard metrological h s t s (§2.3) are as closely associated with the m o d e l contracts here are as they are in the e l e m e n t a r y c u r r i c u l u m itself ^''^'in the final reckoning, what I know of the scribal art will not be taken away! So now I am master of the meaning of tablets, of mathematics, of budgeting, of the whole scribal art. ... "^•"^I desire to start writing tablets (professionally): tablets of 1 gur of barley all the way to 600 gur; tablets of 1 shekel all the way to 20 minas. Also any marriage contracts they may bring; and partnership contracts. I can specify verified weights up to 1 talent, and also deeds for the sale of houses, gardens, slaves, financial guarantees, field hire contracts , palm growing contracts , adoption contracts — all those I can draw up. (Eduba Composition D, after V A N S T I P H O U T 1997: 592-3 and FRIBERG 1987-90: 543) A third p i e c e is often k n o w n as ' T h e D i a l o g u e b e t w e e n Girini-isag a n d E n k i m a n s h u m ' a l t h o u g h it is m o r e of a m m b u s t i o u s slanging m a t c h , in w h i c h the a d v a n c e d student Girini-isag belittles a n d humiliates his y o i m g e r c o l l e a g u e E n k i m a n s h u m ( w h o s e defences are often rather ineffectual): ^^'^^(Girini-isag speaks): "You wrote a tablet, but you cannot grasp its meaning. You wrote a letter, but that is the limit for you! Go to divide a plot, and you are not able to divide the plot; go to apportion a field, and you cannot even hold the tape and rod properly; the field pegs you are unable to place; you cannot figure out its shape, so that when wronged men have a quarrel you are not able to bring peace but you allow brother to attack brother. Among the scribes you (alone) are unfit for the clay. What are you fit for? Can anybody tell us?" ^^~^\Enki-manshum replies): "Why should I be good for nothing? When I go to divide a plot, I can divide it; when I go to apportion a field, I can apportion the pieces, so that when wronged me have a quarrel I soothe their hearts and [...]. Brother will be at peace with brother, their hearts [...]." (Following lines lost) (Eduba Dialogue 3, V A N S T I P H O U T (1997), p. 589) G i r i n i - i s a g ' s p o i n t is that accurate land surveys are n e e d e d for legal r e a s o n s — inheritance, sales, harvest confracts, for instance. If the surveyor c a n n o t p r o v i d e his services effectively h e will unwittingly cause disputes or p r e v e n t t h e m from b e i n g settled peacefiilly. F o r the scribal students in H o u s e F these three p a s s a g e s h e l p e d to define the role of m a t h e m a t i c a l ttaining within their education. T h e first exttact implies that a t m l y c o m p e t e n t teacher c a n h e l p e v e n the m o s t h o p e l e s s student u n d e r s t a n d difficult subjects like m a t h e m a t i c s . T h e s e c o n d outlines w h a t successful students c a n h o p e to achieve in the a p p r o p r i a t e application of m e t i o l o g i c a l k n o w l e d g e to legal d o c u m e n t s of various kinds, while the last w a m s of the humiliations of practical i n c o m p e t e n c e . It is not e n o u g h , Girini-isag implies, to h a v e l e a m e d y o u r school exercises well if you are physically incapable of putting t h e m into practice.^^ Eduba composition A is on 18 tablets from House F; it is the tenth member of the House F Fourteen. Eduba dialogue 3 is on 3 tablets. No House F sources have yet been identified for Eduba composition D but the whole composition is not yet in the public domain.

350

E. Robson

T w o royal p r a i s e p o e m s , widely u s e d in the early stages of the S u m e r i a n literary curriculum,^' cite m a t h e m a t i c a l achievement within the repertoire of a g o o d k i n g ' s a c c o m p l i s h m e n t s , b e s t o w e d o n h i m b y the g o d d e s s of scribalism N i s a b a - N a n i b g a l . T h e i r m e s s a g e to the students is that literacy a n d n u m e r a c y are highly desirable skills, v a l u e d so m u c h that e v e n kings b o a s t a b o u t acquiring t h e m . T h e following extract from a linguistically e l e m e n t a r y h y m n to Lipit-Eshtar of Isin (c. 1 9 3 4 - 2 4 B C ) addresses the king as o n e w h o is divinely a i d e d in his literacy a n d e n d o w e d with h o l y m e a s u r i n g equipment:^^ '^"^'*Nisaba, the woman radiant with joy, the true woman, the scribe, the lady who knows everything, guides your fingers on the clay: she makes them put beautiful wedges on the tablets and adorns them with a golden stylus. Nisaba generously bestowed upon you the measuring rod, the surveyor's gleaming line, the yardstick, and the tablets which confer wisdom. (Lipit-Eshtar hymn B, BLACK et al. (1998-), no. 2.5.5.2) In this exfract, b y confrast, the praise singer s p e a k s in the voice o f king Shulgi o f U r ( c . 2 0 9 4 - 4 7 B C ) , describing his p r o w e s s in school subjects:^^ '^"^°I, Shulgi the noble, have been blessed with a favourable destiny right from the womb. When I was small, I was at the academy, where I leamed the scribal art from the tablets of Sumer and Akkad. None of the nobles could write on clay as I could. There where people regularly went for tutelage in the scribal art, I qualified fully in subtraction, addition, reckoning and accounting. The fair Nanibgal, Nisaba, provided me amply with knowledge and comprehension. I am an experienced scribe who does not neglect a thing. (Shulgi hymn B, BLACK et al (1998-), no. 2.4.2.02) A praise p o e m in the voice o f king I s h m e - D a g a n o f Isin ( c . 1 9 5 3 - 1 9 3 5 B C ) e v e n self-referentially describes how his varied mathematical and scribal a c c o m p l i s h m e n t s h a v e b e e n set to song:^" scxxhdX art, in thc place of skilled craftsmanship, power; that I have solved calculation problems, counting and reckoning in all their depth and breadth, checking, coefficients, establishing the surface of a field, and laying out the reed measuring-pole; that I have on the podium, my chosen place; that I have learnt with my talented hands, my pure hands, to write the tablets of Sumer and Akkad; that I have lent lustre to the academy by completely mastering the reed stylus and the scribal art; [...] - all these things the scholars and the composers of my songs have put in my great songs and have declared in my hymns. (Ishme-Dagan hymn A+V, BLACK et al (1998-), no. 2.5.4.01) 359-366,375-377j|^^^

A few o f the S u m e r i a n literary w o r k s use mefrological concepts as a n essential part o f their narrative framework. F o r instance, a 33-line fictionalised letter^' from IshbiErra (first king o f the Isin dynasty, c . 2 0 1 7 - 1 9 8 5 B C ) to Ibbi-Suen, last king o f U r

"

VANSTIPHOUT (1979); TINNEY (1999), pp. 162-168. Attested on 3 tablets from House F (and on tablets from other sources). It is the seventh member of the House F Fourteen (and widely attested elsewhere).

^

Attested on 3 tablets from House F (and on tablets from other sources).

HUBER (2001) has shown convincingly that, on the grounds of grammatical, stylistic, and historical anachronisms the Royal Correspondence of Ur cannot be considered to be 'authentic'. If it ever had any 'historical core' it has been almost completely lost in fictional overlay and pedagogically-motivated accretions.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

351

( c . 2 0 2 8 - 2 0 0 4 B C ) , describes h o w h e , while still in the latter k i n g ' s service, has b e e n sent north to b u y grain in order to alleviate the famine in the south, b u t is h e l d b a c k b y incinsions o f n o m a d i c M a r t u people:^^ '"^Say to Ibbi-Suen, my lord: this is what Ishbi-Erra, your servant, says: ^'*You ordered me to travel to Isin and Kazallu to purchase grain. With grain reaching the exchange rate of 1 shekel of silver per gur, 20 talents of silver have been invested for the purchase of grain. ^•'^I heard news that the hostile Martu have entered inside your territories. I entered with 72,000 gur of grain — the entire amount of grain — inside Isin. Now I have let the Martu, all of them, penetrate inside the Land, and one by one I have seized all the fortifications therein. Because of the Martu, 1 am unable to hand over this grain for threshing. They are stronger than me, while I am condemned to sitting around. '^"'*Let my lord repair 600 barges of 120 gur draught each; 72 solid boats, 20 [40] rudders (?), 50 and 60 (?) boat doors on the boats (?), may he also ... (after B L A C K et al. (1998-), no. 3.1.17)

, 30 bows, all the boats.

T h e letter r e a d s suspiciously like a n O B school m a t h e m a t i c s p r o b l e m : the first p a r a g r a p h gives the silver-grain e x c h a n g e rate a n d the total a m o i m t o f silver available ( 7 2 , 0 0 0 shekels); in the s e c o n d the silver h a s b e e n correctly c o n v e r t e d into grain. N e x t that h u g e capacity m e a s u r e is divided equally a m o n g large b o a t s ( c f the contextualised large capacity m e a s u r e s in the list of ttees a n d w o o d e n objects, §2.2). A s is typical for school m a t h e m a t i c a l p r o b l e m s , the n u m b e r s are c o n s p i c u o u s l y r o u n d a n d e a s y to calculate with.^^ T h e n u m b e r s in the final, d a m a g e d p a r t o f the section q u o t e d are reminiscent of the final multiplicands o f a standard multiplication table ( § 3 . 1 ) or the sexagesimal fractions 1/3, 1/2, [2/3], 5/6.^^* T h e letter, at o n e level, is n o m o r e t h a n a p r e t e x t to s h o w simple m a t h e m a t i c s a n d m e t t o l o g y at w o r k in a quasi-realistic context. T h e longer c o m p o s i t i o n n o w k n o w n as ' T h e F a r m e r ' s Instructions'*^ u s e s school m a t h e m a t i c s in a v e r y different way. Ostensibly it is a description o f the agricultural year from irrigation to harvest, b u t it is h a r d l y pastoral in tone. C e n t t a l to its w h o l e rationale are the standard w o r k obligations b y w h i c h state institutions o f the twentyfirst c e n t u r y B C m e a s u r e d out agricultural labour to confract m a n a g e r s a n d their w o r k gangs.** A short exfract from the 111-line c o m p o s i t i o n is e n o u g h to c a t c h its flavour: ^^"^^The plough oxen will have back-up oxen. The attachments of ox to ox should be loose. Each plough will have a back-up plough. The assigned task for one plough is 180 iku (c.65

One attestation from House F; several other sources known. "

FRiBERG(1987-90),p. 539.

^ These numbers all refer to wooden objects that I suspect may turn out to be identifiable from the boats section of the list of trees and wooden objects (VELDHUIS (1997)). Only one known tablet, IM 44134, preserves the composition at this point. It is held in the Iraq Museum and was not available for collation. It is the thirteenth member of the House F Fourteen (and well attested elsewhere in Nippur). ^

CIVIL (1994), pp. 75-78, ROBSON (1999), pp. 138-166.

352

E. Robson

ha), but if you build the implement at 144 iku (c.2 ha), the work will be pleasantly performed for you. 180 (?) sila of grain (c. 180 litres) will be spent on each 18 iku area (c.6 1/2 ha). ^"•^^*After working one plough's area with a bardil plough, and after working the bardil plough's area with a tugsig plough, till it with the tuggur plough. Harrow once, twice, three times. When you flatten the stubborn spots with a heavy maul, the handle of your maul should be securely attached, otherwise it will not perform as needed. (BLACK et al. (1998), no. 5.6.3) T h e F a r m e r ' s Instructions is reminiscent o f a small g r o u p o f S u m e r i a n literary c o m p o s i t i o n s r e c e n t l y studied b y N i e k V e l d h u i s . ^ ' H e h a s h i g h l i g h t e d the intimate lexical a n d p e d a g o g i c a l relationship b e t w e e n the s t a n d a r d list o f fish a n d birds (division four o f the t h e m a t i c n o u n list, see §2.1) a n d t w o w o r k s n o w k n o w n as ' H o m e o f the Fish'^^ a n d ' N a n s h e a n d the Birds'.^^ B u t w h e r e a s t h e y p r o v i d e a literary f r a m e w o r k for n a m i n g a n d describing fish and b i r d s . T h e F a r m e r ' s Instructions sets out to sugar the bitter pill o f l e a m i n g agricultural w o r k rates. It w a s p r o b a b l y several h u n d r e d years b e h i n d c o n t e m p o r a r y scribal p r a c t i c e b y the time it w a s taught in H o u s e F , b u t so w a s m u c h of the other literature taught there (as can b e seen from the r e g n a l dates of the kings referred to in the extiacts q u o t e d in this section). 4 . 3 T h e g o o d , t h e b a d , a n d the u g l y : c a l c u l a t i o n s of r e c i p r o c a l s B y a g r e a t stioke o f fortune, o n e tablet has s u r v i v e d from H o u s e F that b e a r s b o t h S u m e r i a n literature a n d a m a t h e m a t i c a l calculation. T h e y are o n the s a m e sort of tablet as the e l e m e n t a r y T y p e III, w h i c h w a s c o m m o n l y u s e d to write s i n g l e - c o l u m n extiacts o f u p to 6 0 lines of literary w o r k s ( a n d for that r e a s o n called T y p e S in this c o n t e x t ) . ' " T h e literary extract is from the first lines o f a c o m p o s i t i o n n o w k n o w n as ' T h e A d v i c e o f a S u p e r v i s o r to a Y o u n g e r S c r i b e ' , o n e o f the curricular g r o u p i n g d i s c u s s e d a b o v e ( § 4 . 2 ) w h o s e fictionalised setting is the school a n d w h o s e a i m is to instil professional identity a n d p r i d e into tiainee scribes: ^~\The supervisor speaks:) "One-time member of the school, come here to me, and let me explain to you what my teacher revealed. ^"^"Like you, I was once a youth and had a mentor. The teacher assigned a task to me — it was man's work. Like a springing reed, I leapt up and put myself to work. I did not depart from my teacher's instructions, and I did not start doing things on my own initiative. My mentor was delighted with my work on the assignment. He rejoiced that I was humble before him and he spoke in my favour. '~'^"I just did whatever he outlined for me — everything was always in its place. Only a fool would have deviated from his instructions. He guided my hand on the clay and kept me on the right path. He made me eloquent with words and gave me advice. He focused my eyes on the rules which guide a man with a task: zeal is proper for a task, time-wasting is taboo; anyone who wastes time on his task is neglecting his task. '^^°"He did not vaunt his knowledge: his words were modest. If he had vaunted his knowledge, people would have frowned. Do not waste time, do not rest at night — get on "

VELDHUis(2001),esp. §3.2. BLACK et al (1998-), no. 5.9.1. VELDHUIS (2001), BLACK et al (1998-), no. 4.14.3.

™

TiNNEY(1999),p. 160.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

353

with that work! Do not reject the pleasurable company of a mentor or his assistant: once you have come into contact with such great brains, you will make your own words more worthy. ^'"^^"And another thing: you will never retum to your blinkered vision; that would be greatly to demean due deference, the decency of mankind." ( B L A C K et al. (1998-), no. 5.1.3. 3N-T 362 + 3N-T 366 = IM 58446 + 58447 (UM cast) obverse 1-19, reverse 1-3. Obverse unpublished; reverse ROBSON (2000), p. 22)

17 4 6 4 0

9

2 40

«2» 2 2 3 0

3 7 2 W

[2]

6 4[5] •"6 4 0 ^

9 8 5320 17 4 6

^4œ

F i g u r e 1 6 : 3 N - T 3 6 2 + 3 6 6 ( r e v ) . ( R O B S O N ( 2 0 0 0 ) , fig. 2 ) .

B e l o w this, t h e rest o f the r e v e r s e is t a k e n u p with a calculation o f r e g u l a r r e c i p r o c a l p a i r s ' ' u s i n g a m e t h o d that S A C H S ( 1 9 4 7 ) called " T h e T e c h n i q u e " ( F i g u r e 1 6 ) . O t h e r tablets w i t h similar a r r a n g e m e n t s o f n u m b e r s a r e k n o w n , as well as o n e v e r y d a m a g e d tablet o f i m k n o w n p r o v e n a n c e w h i c h originally c o n t a i n e d t w e l v e w o r k e d e x a m p l e s with instructions.'^ L i k e m u c h O l d B a b y l o n i a n m a t h e m a t i c s , a l t h o u g h it first a p p e a r s t o b e a n a l o g o u s to m o d e m algebraic o p e r a t i o n s it c a n i n fact b e b e s t u n d e r s t o o d in t e r m s o f very concrete m a n i p u l a t i o n s of lines a n d areas.'^ T h e best p r e s e r v e d o f the t w e l v e p r o b l e m s n m s as follows: ^2^ [13] 20 '\G\^-[bu-SU

EN.NAM]

[ZA.E] '•KIDX\TA.[ZU.DE3] ' i G r 3 20 DUg.A 18

[ta-mar]

8"' a-na 2 10 TUM2.A 3[9 1 DAH.HA 40

ta-mar]

[ta-mar]

IGI 40 DUg.A 1 30

[ta-mar]

1 3 0 a - « a 18 TUM2.''A'' 27 ta-mar [ki-a-am

27

\G\-'bu'-[su]

ne-pe^-sum]

What is the reciprocal of 2;[13] 20?'" [You, in your] working: Find the reciprocal of 0;03 20. [You will see] 11 Multiply 18 by 2; 10. [You will see] 39. Add l . [ Y o u will see] 40. Take the reciprocal of 40. [You will see] 1 30. Multiply 1 30 by 18. You will see 27. Your reciprocal is 27. [That is the method.]

(VAT 6505, II 8-16. NEUGEBAUER(1935-1937), I, pp. 270-273, II pis. 14,43; SACHS(1947), pp. 226-227)

''

ROBSON (2000), no. 2.

For the most recent discussion, see ROBSON (2000), p. 21. H0VRUP (1990) and (2002). 74

I have assigned arbitrary absolute sexagesimal value to the numbers in this problem and those in the following discussion.

E. Robson

354

17;4640

17:40

?

(a)

17:40

Ib)

(cj 1

2?9

0:0640

1

Oj03 2 2 3Û

1 •ì

!

1

. 1 '.

.

Figure 17: F i n d i n g sexagesimally regular r e c i p r o c a l s u s i n g T h e T e c h n i q u e . W e c a n p l u g the n u m b e r s from our H o u s e F tablet into this solution. T h e p r o d u c t of a n y reciprocal pair is, b y definition, 1. W e c a n therefore imagine 17;46 4 0 as the side of a rectangle w h o s e area is 1 (Figure 17(a)); the task is to find the length of the other side. W e c a n m e a s u r e off a part of the first side, so that it h a s a length that is in the standard reciprocal table — in this case 0;06 4 0 , w h o s e reciprocal is 9. W e c a n thus d r a w a n o t h e r rectangle with lines of these lengths, w h o s e area will also b e 1 (Figure 17(b)). T h i s gives us an L - s h a p e d figure. W e c a n fill it in to m a k e a rectangle b y multiplying the 9 b y 17;40, the p a r t o f the original length that w e h a v e n ' t u s e d yet — 2 3 9 (Figure 17(c)). A d d 1, the area of the 9 b y 0;06 4 0 rectangle. T h e total area is 2 4 0 . T h i s n e w large rectangle, 9 b y 17;46 4 0 , is 2 4 0 times b i g g e r than o u r original rectangle, with area 1. Therefore 9 is 2 4 0 times b i g g e r t h a n our m y s t e r y reciprocal. W e divide 9 b y 2 4 0 b y finding the inverse of 2 4 0 — 0;00 22 3 0 — a n d multiplying. T h e reciprocal w e w a n t e d to find is thus 9 X 0;00 2 2 3 0 = 0;03 2 2 3 0 (Figure 17(d)). T h i s is the n u m b e r in the m i d d l e o f tiie calculation. T h e scribe then c h e c k s his result b y w o r k i n g b a c k w a r d s from 0;03 22 3 0 to 17;46 4 0 again. T h e other t w o calculations identified so far on H o u s e F tablets are also attempts to find reciprocals, b u t c o n s p i c u o u s l y less successfiil than the first. T h e longest, written o n the b a c k o f a roughly m a d e , a p p r o x i m a t e l y square tablet, r e a d s :

20 50

16 40 16 W 16' 40 4 37 46 40 9 42 39 [ ]

4

W

Figure 18: 3 N - T 611 = A 3 0 2 7 9 (unpublished), reverse. A student's calculation.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

355

N o t h i n g r e m a i n s o n the o b v e r s e apart from a few apparently r a n d o m signs. T h e first part of the calculation is a squaring of the n u m b e r 16;40, set out in the u s u a l w a y w i t h the m u l t i p l i c a n d s a n d p r o d u c t aligned v e r t i c a l l y / ^ albeit with a n u n e x p l a i n e d extra c o p y of the 16;40.'* T h e a n s w e r is correctly g i v e n as 4 3 7 ; 4 6 4 0 , b u t the 9 written i m m e d i a t e l y to the left sttongly suggests that T h e T e c h n i q u e w a s t h e n u s e d to find its reciprocal. A s in o u r first e x a m p l e , the student has split 4 3 7 ; 4 6 4 0 into 4 3 7 ; 4 0 a n d 0;06 40. H e h a s appropriately t a k e n the reciprocal of the latter — 9 — a n d multiplied it b y the former, a d d m g 1 to the result. H o w e v e r , instead o f arriving at 4 1 3 9 + 1 = 4 1 40, OIU student h a s lost a sexagesimal p l a c e a n d found 4 1 ; 3 9 + 1 = 4 2 ; 3 9 . U n a b l e to g o fiirther with his calculation (for the n e x t stage is to find the r e c i p r o c a l of the niunber j u s t found, b u t his is c o p r h n e to 6 0 ) h e has a b a n d o n e d the exercise there. T h e correct answer w o u l d h a v e b e e n 0;00 12 57 3 6 . ' '

F i g u r e 19: 3 N - T 605 = U M 5 5 - 2 1 - 3 5 7 ( o b v ) , ROBSON ( 2 0 0 0 ) , n o . 1. A n a t t e m p t e d r e c i p r o c a l calculation. T h e last calculation of the three (Figure 19) is the m o s t pitiftil. W r i t i n g o n a T y p e S tablet like the first e x a m p l e , the student has got no further than: 4 26 40 igi-bi 2 13 2 0

4;26 4 0 Its reciprocal is 2 ; 13 2 0

T h e d o u b l e ruling u n d e m e a t h shows that h e thinks he has finished, a l t h o u g h h e has d o n e nothing m o r e than halve the first n u m b e r (Figure 19). T h e correct result is 0;13 3 0 . T w o o f the three n u m b e r s w h o s e reciprocals are to b e found c o m e from the standard school s e q u e n c e o f reciprocal pairs to w h i c h all other k n o w n e x e m p l a r s of

''

ROBSON ( 1999), pp. 250-252.

'* Three tablets from the early excavations at Nippur also bear squaring calculations in the same format: C B S 3551 (NEUGEBAUER and SACHS (1945), p. 36), H S 232 (FRIBERG (1983), p. 82), and N 3971 (ROBSON (1999), p. 275). " I do not yet have any explanation for the 20 and 50 written to the left of the calculation; presumably they relate to intermediate steps in the procedure. Compare similarly positioned auxiliary numbers in calculations from Ur, e.g. UET612 387 (ROBSON ( 1999), p. 249).

E. Robson

356

this exercise b e l o n g . ' ^ T h e s e q u e n c e is c o n s t r u c t e d b y successively d o u b l i n g / h a l v i n g a n initial p a i r 2 0 5 a n d 2 8 4 8 . O u r t w o are eighth (4 2 6 4 0 ) a n d tenth ( 1 7 4 6 4 0 ) respectively. O n t h e other hand, 4 3 7 4 6 4 0 d o e s not, as far as I c a n ascertain, fit the p a t t e m ; p r e s u m a b l y it w a s c h o s e n b e c a u s e , like the other t w o , it t e r m i n a t e s in the string 6 4 0 . O n e p o s s i b l e interpretation of this c o m m o n a l i t y is that three students w e r e set similar p r o b l e m s at the same time, u s i n g a c o m m o n m e t h o d a n d a c o m m o n starting p o i n t b u t r e q u i r i n g different n u m e r i c a l solutions. O n e o f three u s e d the m e t h o d correctly, p r o d u c i n g the right a n s w e r a n d c h e c k i n g his results; the s e c o n d c h o s e the a p p r o p r i a t e m e t h o d b u t c o u l d not a p p l y it satisfactorily, while the third h a d m i s s e d the p o i n t o f the exercise entirely. W e c a n find c o r r o b o r a t i o n for this h y p o t h e s i s in g r o u p i n g s o f other sorts o f calculations from t h e city o f U r . 4.4 C a l c u l a t i o n s in o t h e r curricula S o m e forty-five arithmetical calculations a r e k n o w n to h a v e c o m e f r o m ' N o . 1 B r o a d S t i e e t ' in O l d B a b y l o n i a n U r (§1.3), as I h a v e discussed e l s e w h e r e . ' ' O n e o f t h e m uses T h e T e c h n i q u e to find the r e c i p r o c a l 2 8 4 8 o f 2 0 5 , that is, the first p a i r in the s t a n d a r d s e q u e n c e j u s t discussed.^" A fiirther three calculate the s q u a r e s o f s e x a g e s i m a l l y regular n u m b e r s , * ' the first o f w h i c h is identical to the n u m b e r in 3 N - T 6 1 1 , n a m e l y 16;40. In all cases t h e visual layout o f the calculations is identical to those o n the H o u s e F tablets: the calculation p r o c e e d s d o w n w a r d s , with r e c i p r o c a l p a i r s written in h o r i z o n t a l a l i g n m e n t ( b u t with a l m o s t n o s p a c e s e p a r a t i n g t h e m ) a n d n u m b e r s t o b e s q u a r e d written twice in vertical a l i g n m e n t . P r o d u c t s are r e c o r d e d u n d e m e a t h m u l t i p l i c a n d s . N o m l i n g s , h o r i z o n t a l or vertical, are u s e d ( F i g u r e 2 0 ) .

F i g u r e 2 0 : U E T 6/2 2 9 5 a n d 2 1 1 (rev.). Calculations from N o . 1 B r o a d Stieet, U r . ( R O B S O N ( 1 9 9 9 ) , figs. A . 5 . 6 , A . 5 . 7 ) . T h e largest g r o u p of B r o a d Stieet calculations, h o w e v e r , is n o t p a r a l l e l e d in H o u s e F. O n s o m e 2 0 tablets, sequential multiplications are r e c o r d e d either side o f a

R0BS0N(1999),p. 23. ROBSON (1999), pp. 246-272. For an altemative interpretation of this material, see FRIBERG (2000). UET 612 295, ROBSON (1999), p. 250.

UET 6/2 2 1 1 , 2 2 2 , and 321, R0BS0N(1999), pp. 251-252.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

357

F i g u r e 21 : U E T 6/2 2 3 6 a n d 2 4 7 (rev). S e q u e n c e s of multiplications from N o . 1 B r o a d Street, U r . (ROBSON ( 1 9 9 9 ) , figs. A . 5 . 1 0 , A . 5 . 1 4 ) . vertical line.^^ Y e t j u s t as the three H o u s e F reciprocal calculations share a c o m m o n element in the n u m b e r 6 4 0 , t w o s u b g r o u p s of the B r o a d Streets multiplications h a v e multiplicands in c o m m o n . In five e x e m p l a r s (e.g.. Figure 2 1 , left) the third multiplicand is 3 a n d the fourth 0;06 (i.e., the divisor 10). In a n o t h e r five ( a n d a finther t w o p o s s i b l e d a m a g e d specimens) the fourth multiplicand is a l w a y s 6 4 0 (e.g.. Figure 2 1 , right). I c o n c l u d e d m y study o f those tablets fi^om U r with the following inferences, all b u t the last o f w h i c h w e c a n apply to the calculations in H o u s e F as well:^^ first, that students were set problems to solve, and that the mathematics education was not restricted to leaming arithmetical and metrological tables, and to copying out model solutions; •

second, that [small groups of] students were set problems of the same type but with different values for the variables — perhaps from the 'catalogue'-type lists [that teachers kept of appropriate whole-integer parameters — or from standard sequences such as the halved and doubled reciprocal pairs];

•

third, that they were taught to lay out their calculations in standard formats; fourth, that students were prone to both calculation (especially place value) errors and mistakes through misreading [or mis-remembering] coefficient lists and arithmetical tables;

• fifth, and most speculatively, that students knew the numerical results they were aiming for, and were not above fudging their calculations to fit.^" N e v e r t h e l e s s , h o w e v e r similar the content and organisation of the calculations from the t w o schools m i g h t b e , it is clear that they w e r e p e r f o r m e d in rather different ROBSON ( 1999), pp. 252-264. It was not possible to identify the exact problems that had been set for the Broad Street students to work on, but it has been attempted in one other instance. YBC 7289, an unprovenanced round tablet bearing a diagram and calculation of the length of the diagonal of a square, can confidently be linked to an entry in the coefficient list YBC 7243 and, more speculatively, to the set of geometrical problems about squares on BM 15285 (FOWLER and ROBSON (1998))

ROBSON (1999), pp. 263-264; further comments in square brackets.

358

E. Robson

curricular contexts. M o s t conspicuously, the m a t h e m a t i c a l w o r k from U r is n o t o n s q u a r e or T y p e S tablets, b u t o n r o u n d tablets identical to the N i p p u r e l e m e n t a r y T y p e I V ( § 2 . 1 ) . Further, while o n e H o u s e F e x e m p l a r is found o n the s a m e tablet as a S u m e r i a n s c h o o l narrative, o n the o b v e r s e of all b u t four o f the B r o a d Street e x a m p l e s are extracts from S u m e r i a n p r o v e r b collections^^ — w h i c h in N i p p u r b e l o n g t o the e n d of the e l e m e n t a r y curricular s e q u e n c e . It w o u l d b e t e m p t i n g to ascribe this difference to g e o g r a p h i c a l variation, if it were not for the existence o f a g r o u p of calculations from A r e a T B , not 100 m e t r e s from H o u s e F in N i p p u r (Figure 1). H o u s e B in A r e a T B ( L e v e l II) yielded s o m e 53 tablets in the s e a s o n before H o u s e F w a s e x c a v a t e d , all b u t a handfiil of w h i c h b o r e school-related subject matter.*^ It w a s a m u c h m o r e substantial h o u s e than F , with five r o o m s off a central courtyard. T h e majority of tablets w e r e found in that courtyard, a l t h o u g h there w e r e n o s c h o o l y a r d fittings s u c h as b e n c h e s or r e c y c l i n g b i n s as at H o u s e F or the galamahs' h o u s e in Sippir A m n a n u m ( § 1 . 3 ) . " S t o n e dates the school level to c . 1 8 7 0 1800, s o m e 6 0 - 1 3 0 years earlier t h a n H o u s e F b u t r o u g h l y c o n t e m p o r a r y with B r o a d Sfreet.** A t least n i n e of the school tablets h a v e b e e n c a t a l o g u e d as m a t h e m a t i c a l , a n d five h a v e b e e n p u b l i s h e d ; all the legible p i e c e s b e a r calculations.*' T w o s q u a r e s h a p e d tablets (like 3 N - T 6 1 1 a b o v e ) c a r r y calculafions a b o u t squares, b u t t h e y differ from the e x a m p l e s d i s c u s s e d earlier in that the p r o b l e m is r e c o r d e d o n t h e m t o o . In b o t h cases the task is to find the area o f a small s q u a r e — 2/3 cubit 9 fingers long ( c . 4 8 0 m m ) and 1/3 cubit 1/2 finger long ( c . l 7 5 m m ) respectively. T h e q u e s t i o n a n d answer are written o n the b o t t o m right c o m e r , and the calculation (without the answer) in s e x a g e s i m a l p l a c e value s y s t e m o n the top left. T h i s format of p r o b l e m is attested on five other square tablets, at least three of w h i c h are also from N i p p u r . ' ° M a y b e they are p r e c u r s o r s to, or variants on, the squarings from H o u s e F a n d B r o a d Street. H o w e v e r , w h e r e a s the H o u s e B e x a m p l e s all involve the c o n v e r s i o n of metrological units to their e q u i v a l e n t s in the s e x a g e s i m a l p l a c e value s y s t e m and b a c k a g a i n (as in the s t a n d a r d m e t r o l o g i c a l tables, §2.3), there is n o e v i d e n c e at all for m e t r o l o g i c a l e l e m e n t s in the H o u s e F or B r o a d Street calculations. M o s t interesting for our p r e s e n t discussion, though, are the three tablets b e a r i n g r e c i p r o c a l calculations. Like the three from H o u s e F , t w o of the tablets c a n b e

ALSTER (1997), pp. 306-328. The unprovenanced Type I V tablet YBC 7345 also bears a proverb on the obverse and calculations (sequential multiplications) on the reverse (ALSTER (1997), pi. 130). I did not include this house in the inventory of comparative data (§ 1.3) as almost nothing is published about its tablets. My information comes from matching excavation numbers of published tablets with unpublished excavation records kept at the University Museum, Philadelphia. See also VELDUIS (2000), pp. 387-388. STONE ( 1987), pp. 84-85, pis. 29-30; TANRET (2002), pp. 142-149. STONE(1987),p. 119. 2N-T 30 (squaring), 2N-T 115 (fragment), 2N-T 116 (squaring): NEUGEBAUER and SACHS (1984); 2N-T496 (reciprocals): AL-FOUADI (1979), no. 134; 2N-T500 (reciprocals): GORDON (1959), no. XXX, pi. 70. Further details in ROBSON (2000), p. 19, table 2. Listed in ROBSON ( 1999), p. 12.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

359

classified as T y p e S, a n d their textual fi^miat and contents are strikingly similar too. 2 N - T 5 0 0 u s e s the s a m e reciprocal p a i r as 3 N - T 3 6 2 + 3 6 6 (Figure 16), n a m e l y 17 4 6 4 0 a n d 3 22 3 0 , a n d the s a m e layout as 3 N - T 6 0 5 (Figure 19), w i t h a d o u b l e ruling u n d e m e a t h the statement of the p r o b l e m . In this case, t h o u g h , the student h a s solved the p r o b l e m correctly u n d e m e a t h , in a format identical t o the H o u s e F and B r o a d Street e x a m p l e s , after an abortive attempt o n the other side of the t a b l e t . ' ' O n 2 N - T 4 9 6 the r e c i p r o c a l to b e found is the a p p a r e n t l y the subject of the s q u a r i n g o n the H o u s e F tablet ( a n d o n e o f the B r o a d Street o n e s ) , n a m e l y 16 4 0 . A l l that survives, h o w e v e r , is the a n s w e r b e l o w — igi-bi 3 3 6 'Its r e c i p r o c a l is 3 3 6 ' — with n o w o r k i n g s s h o w n u n d e r the d o u b l e m l i n g , so w e caimot b e sure that the p r o b l e m w a s n o t incorrectly solved (as in 3 N - T 6 0 5 , the only other r e c i p r o c a l calculation with n o w o r k i n g s ) . T h e last tablet of the three, 2 N - T 1 1 5 , is an "irregularly s h a p e d fragment" b e a r i n g t w o d a m a g e d lines which, if correctly calculated, c a n b e restored as 9 2 8 5[3 2 0 ] / igi-bi 6 1[9 4 1 15]. L i k e the H o u s e F reciprocal p a i r s , these n u m b e r s all b e l o n g to the h a l v e d and d o u b l e d s e q u e n c e d e r i v e d from 2 05 a n d 2 8 4 8 — a n d all e n d in 6 4 0 (in o n c e case 3 2 0 ) as well. Is this s i m p l y c o i n c i d e n c e ? A s I h a v e d e s c r i b e d t h e m so far, the H o u s e B r e c i p r o c a l calculations s o u n d m u c h m o r e like those fi^om H o u s e F than the o n e s fi-om B r o a d Street. H o w e v e r , like the B r o a d Street tablets, t w o of t h e m b e a r S u m e r i a n p r o v e r b s rather t h a n l o n g e r literary extracts. O n the other h a n d , b o t h of those p r o v e r b s are a b o u t failures in scribal b e h a v i o u r , w h i c h situates t h e m rather closer to the school narrative o n 3 N - T 3 6 2 + 3 6 6 t h a n t h e y m i g h t otherwise appear: A foolish scribe: the most backward among his colleagues (2N-T 496, cf SP 2.42: ALSTER (1997), pp. 53,304. A chattering scribe: his guilt is very great (2N-T 500, SP 2.52: ALSTER (1997), p. 55) B o t h are from S u m e r i a n P r o v e r b C o l l e c t i o n 2 + 6 , w h i c h also h a p p e n s to b e the best attested of the S u m e r i a n p r o v e r b collections from H o u s e F.'^ T h e r e is a n interesting c o n t t a d i c t i o n here: o n the o n e h a n d , the H o u s e B reciprocals are o n the s a m e tablet types as H o u s e F, use the s a m e i n t t o d u c t o r y layout with statement a n d d o u b l e m l i n g , a n d u s e the s a m e reciprocals in 6 4 0 ; o n the other, they share tablets with S u m e r i a n p r o v e r b s like those from B r o a d Stteet in U r (but w h i c h are o n different a tablet type and d o not state the p r o b l e m ) . H o w e v e r , the s a m p l e at our disposal is i m d o u b t e d l y t o o small to confidently assign subject correlation to d i a c h r o n i c c h a n g e and tablet t y p o l o g y to g e o g r a p h i c a l variation. It is e n o u g h for the m o m e n t to see that while there are s o m e e x t t a o r d i n a r y consistencies in b o t h the b r o a d s w e e p a n d the detail of calculations taught in s c h o o l s , there w a s b y n o m e a n s a ' n a t i o n a l ' c i u r i c u l u m w h i c h all teachers followed.'^ ROBSON(2000), pp. 20-21. By contrast, of the 45 tablets from Broad Street that bear both proverbs and calculations, only 4 (9 percent) have extracts from the 'scribes' section of Sumerian Proverb Collection 2+6 (ROBSON (1999), p.

246).

House F has yielded no mathematical problem texts — that is, documents that set out a mathematical problem to be solved (ROBSON (1999), pp. 7-8). Discounting the Sîn-k3§id school tablets (§1.3) and those in the gala-mahs' house (solely elementary exercises, which would not therefore be expected to include mathematical problems), the only archaeologicaily-defined school corpora containing mathemafical problems are from the 'Scholar's House' in Mê-Turân with one tablet of 10 mixed problems (AL-RAWI and ROAF

E. Robson

360

4.5 A n e x t r a - c u r r i c u l a r t a b l e Finally, there is j u s t o n e mathematical tablet from H o u s e F w h i c h c a n n o t b e securely related to other elements of the scribal curriculum (Figure 2 2 ) : [ 1 ] . E 1 IB2.SIg [ 4 ] . E 2 IBz.SIg [9].E3lB2.Slg [ 1 6 ] . E 4 1B2.SI8 [ 2 5 ] . E 5 iBj.sig 3 6. E 6 iBj.sig 4 9 . E 7 iBs.Slg 1 0 4 . E 8 IB2.SI8 1 21.E9lB2.Sl8 1 4 0 . E 10lB2.SIg 2 Ol.E 11 iBa.Sig 2 2 4 . E 1 2 iB2.Sig [ 2 49].^El3^iB2.sig [ 3 16.E 1 4 ] IB2.SI8 [ 3 4 5 . E 1 5 iBj.sig]

[ 1 ] is the square of 1 [ 4 ] is the square of 2 [ 9 ] is the square of 3 [ 1 6 ] is the square of 4 [ 2 5 ] is the square of 5 3 6 is the square of 6 4 9 is the square of 7 1 0 4 is square of 8 1 2 1 is the square of 9 1 4 0 is the square of 1 0 2 0 1 is the square of 11 2 2 4 is the square of 1 2 [ 2 4 9 ] is the square of 1 3 [ 3 1 6 ] is the square of [ 1 4 ] [ 3 4 5 is the square of 1 5 ] [ 4 1 6 is the square of 1 6 ]

[ 4 16.E 1 6 lB2.SIg] [4 49.El7lB2.Sl8] [5 24.El8iB2.sig] [6 01.El9lB2.Slg] [ 6 4 0 . E 2 0 ] iB2.Sig [7 21.E21]lB2.Sl8 8 0 4 . ' E 2 2 " ' [iBs.Sig] [ 8 ] 4 9 . E 2 3 iB2.Sig [ 9 3 6 ] . E 2 4 iB2.sig [ 1 0 2 ] 5 . E 2 5 IB2.SI8 [ 1 1 1 ] 6 . E 2 6 iB2.sig [ 1 2 0 9 ] . E 2 7 IB2.SI8 [ 1 3 0 4 ] . E 2 8 iB2.Sig [ 1 4 0 1 ] . E 1 iB2.Sig [ 1 5 ] . E 1 IB2.SI8

[ 4 4 9 is the square of 1 7 ] [5 2 4 is the square of 1 8 ] [ 6 0 1 is the square of 1 9 ] [ 6 4 0 is] the square [of 2 0 ] [ 7 2 1 is] the square [of 2 1 ] 8 0 4 [is the square of 2 2 ] [ 8 ] 4 9 is the square of 2 3 [ 9 ] 3 6 is the square of 2 4 [ 1 0 2 ] 5 is the square of 2 5 [ 1 1 1 ] 6 is the square of 2 6 [ 1 2 0 9 ] is the square of 2 7 [ 1 3 0 4 ] is the square of 2 8 [ 1 4 0 1 ] is the square of 2 9 [ 1 5 0 0 ] is the square of 3 0

Figure 2 2 : 3 N - T 6 0 4 = U M 5 5 - 2 1 - 3 5 6 (unpublished). A n inverse list o f squares from H o u s e F . It bears a n inverse list o f squares, w h i c h it w o u l d p e r h a p s b e t e n d e n t i o u s to c o n n e c t with the squaring exercise o n 3 N - T 611 ( § 4 . 3 ) . It is a well attested table: N e u g e b a u e r and Sachs list eighteen other exemplars,'"* thirteen o f w h i c h are in this format; six of those thirteen are also from Nippur.'^

( 1 9 8 4 ) ) and Broad Street, from which probably six tablets contain mathematical problems (CHARPIN ( 1 9 8 6 ) , pp. 4 5 1 ^ 5 2 , 4 8 1 ^ 8 2 ) . We might also count the two squaring exercises from House B in Nippur TB (above). NEUGEBAUER ( 1 9 3 5 - 1 9 3 7 ) , I, pp. 7 0 - 7 1 ; NEUGEBAUER and SACHS ( 1 9 4 5 ) , pp. 3 3 - 3 4 .

Three of the four mathematical tablets from No. 7 Quiet Street in Ur are tables like this: 1 table of squares, I inverse table of squares, 1 inverse table of cubes ( § 1 . 3 ) .

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

361

5. Conclusions It turns out that a wealth o f interesting insights c a n b e gained from m a t h e m a t i c a l material that h a s traditionally b e e n dismissed as imimportant and trivial. A n awareness of archaeological a n d social context c a n illuminate the dullest o f texts. H o w e v e r , w e should b e careful not to blithely generalise the c o n c l u s i o n s r e a c h e d about H o u s e F to the w h o l e of M e s o p o t a m i a , or e v e n to B a b y l o n i a or N i p p i n : o n e of the m o s t striking o u t c o m e s of this study has b e e n to highlight b o t h the variations large a n d small b e t w e e n individual corpora of tablets, a n d the virtual impossibility o f ascribing those differences to diachronic change, g e o g r a p h i c a l variation or p e r s o n a l choice (although it appears that this last was m o r e p e r v a s i v e than w e m i g h t h a v e thought). T o s u m u p our findings a b o u t H o u s e F , a small scribal school operating in u r b a n N i p p u r in the m i d - e i g h t e e n t h century B C : W e c a n accurately attribute the m e m o r i s a t i o n o f standard m e t r o l o g i c a l a n d m a t h e m a t i c a l series to the third p h a s e of elementary e d u c a t i o n in H o u s e F . M e t r o l o g y w a s taught before multiplication b u t it w a s apparently less i m p o r t a n t (at least, m a n y fewer tablets survive); it is not yet clear w h y this is so. N o r is it yet possible to distinguish the didactic roles of metrological lists and tables; they cannot o b v i o u s l y b e assigned to 'first e x p o s u r e ' a n d ' r e v i s i o n ' ftmcfions. A m e t r o l o g i c a l thread r a n right t h r o u g h the curriculiun, from o r d e r e d lists o f metrologically-related objects in the s e c o n d - p h a s e thematic n o u n lists, t h r o u g h contextualised m e t r o l o g y in fourth-phase m o d e l contracts, to e n u m e r a t i o n s o f m e t r o l o g i c a l constants in the Siunerian literary c o m p o s i t i o n ' T h e F a r m e r ' s Instructions'. After m a s t e r i n g m e t i o l o g y , the students were p r o b a b l y taught the w h o l e of the multiplication series, in v e r b o s e form o n T y p e III a n d T y p e I I / l tablets; t h e y revised t h e m frequentiy, in terse form on tablet T y p e s I and T y p e II/2. ( T y p e I V tablets w e r e not u s e d for m a t h e m a t i c a l subjects in H o u s e F.) H o w e v e r , it was rare to revise m o r e than the first section or t w o (breaking b e t w e e n the tables for 2 0 a n d 18). ' T a b l e s ' is rather a m i s n o m e r for tiiis exercise, it t u m s out: rather, the students w e r e m e m o r i s i n g lists o f n u m b e r facts. In fact, the m a t h e m a t i c a l t h m s t o f the e l e m e n t a r y c u r r i c u l u m as a w h o l e c a n b e s u m m a r i s e d as the r e c o g n i t i o n o f n u m b e r s , weights, a n d m e a s u r e s in context, a n d their m e m o r i s a t i o n in s e q u e n c e . Calculations — active m a t h e m a t i c s — b e l o n g e d to the a d v a n c e d c u r r i c u l u m along with S u m e r i a n literature, s o m e of w h i c h h a d b e e n deliberately written or a d a p t e d for specifically mathematical aims, while rather m o r e of it w a s g e a r e d to instilling a sense o f professional p r i d e in n u m e r a c y a n d literacy in ttainee scribes. T h e few e x a m p l e s we have, o f finding squares a n d regular reciprocals, m i g h t suggest that students found arithmetic difficult and m a d e frequent m i s t a k e s . T h e r e is a similarity in subject matter a n d calculation format that extends b e y o n d this single school, to n e a r b y H o u s e B and to B r o a d Stteet in Ur. A t Ur, t h o u g h , calculations were practised o n T y p e IV tablets, while the students w e r e l e a m i n g S u m e r i a n p r o v e r b s . W e d o not yet k n o w the order of the c u r r i c u l u m in the U r s c h o o l - h o u s e s . In H o u s e B ( c o n t e m p o r a r y with B r o a d Stteet, older than H o u s e F ) , the tablets and m a t h e m a t i c a l exercises w e r e nearly identical to those in H o u s e F b u t a p p e a r e d in the same curricular context as B r o a d Stteet. H o u s e F p r o v i d e s n o e v i d e n c e , direct or indirect, for the use of m a t h e m a t i c a l p r o b l e m texts, or for a n y practice at all in additions and subttactions. A s w o r k o n the tablets from H o u s e F and its n e i g h b o u r s p r o g r e s s e s , h o w e v e r , these conclusions will u n d o u b t e d l y b e refined, corrected, and expanded.

362

E. Robson

Acknowledgments T h i s p a p e r arises from research ftinded b y a British A c a d e m y P o s t d o c t o r a l F e l l o w s h i p , 1 9 9 7 - 2 0 0 0 , w h i c h is p l a n n e d t o eventually a p p e a r a s a m o n o g r a p h , provisionally called The Tablet House. This r e s e a r c h project has necessitated trips t o the University M u s e u m , Philadelphia, the Oriental Institute, C h i c a g o , a n d t h e Iraq M u s e u m , B a g h d a d t o e x a m i n e tablets. I a m e n o r m o u s l y grateftil t o Professor J . A . B r i n k m a n , Professor E. Leichty, a n d Dr. N . A l - M u t a w a l l i for allowing m e t o access a n d p u b l i s h those tablets a n d for m a k i n g m y visits s o enjoyable a n d p r o d u c t i v e . It is also a p l e a s u r e t o t h a n k Dr. J a n v a n M a a n e n , for hosting a m o s t enjoyable seminar in t h e D e p a r t m e n t o f M a t h e m a t i c s at the U n i v e r s i t y o f G r o n i n g e n in J u n e 2 0 0 1 , w h e r e I gave a p r e l i m i n a r y version o f this paper. M i c h e l T a n r e t kindly g a v e m e a preprint o f his m a r v e l l o u s final publication o f the school tablets from the gala-mahs' h o u s e at Sippir A m n a n u m . J e r e m y Black, P a u l D e l n e r o , D u n c a n M e l v i l l e , J o n Taylor, L u k e T r e a d w e l l , a n d N i e k Veldhuis all g a v e g e n e r o u s l y o f their time a n d expertise t o m a k e this a m u c h better piece o f w o r k than it w o u l d o t h e r w i s e h a v e been. M i s t a k e s are m i n e alone.

References A L S T E R , B e n d t . 1997. Proverbs of Ancient Sumer: the World's Oldest Proverb Collections. B e t h e s d a : C D L Press. B L A C K , J e r e m y A . ; C U N N I N G H A M , G r a h a m G.; F L U C K I G E R - H A W K E R , Esther; R O B S O N , Eleanor, a n d ZÓLYOMI, Gabor. 1 9 9 8 - . The Electronic Text Corpus of Sumerian Literature . Oxford: T h e Oriental Institute. B O D I N E , W a l t e r R. 2 0 0 1 . " A M o d e l C o n t i a c t o f a n E x c h a n g e / S a l e T r a n s a c t i o n " . I n Tzvi

ABUSCH,

C. NOYES,

WilUam

W.

HALLO

and

Irene

WINTER

(eds.).

Historiography in the Cuneiform World ( P r o c e e d i n g s o f t h e R e n c o n t i e Assyriologique I n t e m a t i o n a l e 4 5 ) : I, 4 1 - 5 4 . B e t h e s d a : C D L P r e s s . C A V I G N E A U X , A n t o i n e . 1982. "Schultexte a u s W a r k a " . Baghdader Mitteilungen 1 3 : 21-30. — 1 9 9 2 . " L U L - b i = lib-bi". Nouvelles assyriologiques brèves et utilitaires 1992/109. — 1 9 9 6 . Vruk: altbabylonische Kopien von Adam Falkenstein Mainz: Von Zabera.

Texte aus dem Planquadrat Pe XVl-4/5 nach ( A u s g r a b u n g e n in U r u k - W a r k a . E n d b e r i c h t e 2 3 ) ,

— 1999. " A S c h o l a r ' s Library in Metiiran? W i t h a n Edition o f t h e T a b l e t H 7 2 (Textes d e Tell H a d d a d V I I ) " . In: Tzivi A B U S C H a n d K a r e l V A N D E R T O O R N (eds.), Mesopotamian Magic: Textual, Historical, and Interpretative Perspectives (Ancient M a g i c and Divination 1): 2 5 3 - 2 7 3 . G r o n i n g e n : Styx. — and R E N G E R , J o h a n n e s . 2 0 0 0 . " E i n altbabylonischer G i l g a m e s - T e x t aus N i p p u r " . In: A n d r e w R. G E O R G E a n d Irving L. FiNKEL (eds.). Wisdom, Gods and Literature: Studies in Assyriology in Honour ofW.G. Lambert: 9 1 - 1 0 5 . W i n o n a Lake: Eisenbrauns. C H A R P I N , D o m i n i q u e . 1986. L e clergé d'Ur au siècle d'Hammurabi (XIX'-XVIW siècles av. J.-C.) (Hautes Études Orientales, 2 2 ) . G e n e v a : Librarie D r o z . CIVIL, M i g u e l . 1 9 7 5 . " A p p e n d i x A : C u n e i f o r m T e x t s " . In: M c G u i r e G I B S O N (ed.). Excavations at Nippur: Eleventh Season: 1 2 5 - 1 4 2 . C h i c a g o : Oriental Institute.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

— 1 9 9 4 . The Farmer's Instructions. A Sumerian Agricultural Orientalis S u p p l e m e n t a 5 ) . B a r c e l o n a : Editorial A U S A .

Manual

363

(Aula

— 1 9 9 5 . " A n c i e n t M e s o p o t a m i a n lexicography". In: Jack M . S A S S O N Civilisations of the Ancient Near East: 2 3 0 5 - 2 3 1 4 . N e w York: Scribner.

(ed.),

— ; G R E E N , M a r g a r e t W . , a n d L A M B E R T , Wilfred G . 1 9 7 9 . Ea A = nâqu, Aa A = nâqu, with their Forerunners and Related Texts (Materials for t h e S u m e r i a n L e x i c o n 1 4 ) . R o m e : Biblical Institute P r e s s . A L - F O U A D I , A b d u l - H a d i . 1 9 7 9 . Lenticular Exercise School Texts, I ( T e x t s in t h e Iraq M u s e u m 1 0 / 1 ) . B a g h d a d : R e p u b l i c o f Iraq M i n i s t r y o f C u l t m e a n d A r t s . F O W L E R , D a v i d H . a n d R O B S O N , Eleanor. 1 9 9 8 . " S q u a r e R o o t A p p r o x i m a t i o n s in O l d B a b y l o n i a n M a t h e m a t i c s : Y B C 7 2 8 9 in context". Historia Mathematica 25: 366-378.

FRIBERG, Joran. 1 9 8 3 . " O n t h e B i g 6 - P l a c e T a b l e s o f R e c i p r o c a l s a n d S q u a r e s from Seleucid B a b y l o n a n d U r u k a n d their O l d B a b y l o n i a n a n d S u m e r i a n P r e d e c e s s o r s " . S'M/ner 4 2 : 8 1 - 8 7 . — 1 9 8 7 - 1 9 9 0 . " M a t h e m a t i k " . In: Dietz O . E D Z A R D (ed.), Reallexikon der Assyriologie

und vorderasiatischen

Archaologie

7: 5 3 1 - 5 8 5 . Berlin/New York:

W a l t e r d e Gruyter. — 2000.

"Mathematics

d'Assyriologie

at

Ur

in

the

Old

Babylonian

Period".

Revue

94: 9 7 - 1 8 8 .

G A S C H E , H e r m a i m . 1 9 8 9 . La Babylonie au IT siècle avant notre ère: approche archéologique, problèmes et perspectives (Mesopotamian History and E n v i r o n m e n t , Series II, M e m o i r s 1 ) . Ghent: T h e University o f G h e n t . — and D E K I E R E , L u c . 1 9 9 1 . " A p r o p o s d e la d i u é e d e v i e d ' u n e m a i s o n p a l é o b a b y l o n i e n n e e n b r i q u e s c r u e s " . Nouvelles assyriologiques brèves et utilitaires 1991/20.

G I B S O N , M c G u i r e ; H A N S E N , D o n a l d P . , a n d ZETTLER, R i c h a r d L. 2 0 0 1 . " N i p p u r B . A r c h â o l o g i s c h " . I n D i e t z O . E D Z A R D (éd.), Reallexikon der Assyriologie und vorderasiatischen Archaologie 9 : 5 4 6 - 5 6 5 . Berlin/New York: Walter de Gruyter. G O R D O N , E d m u n d I. 1 9 5 9 . Sumerian Proverbs: Glimpses of Everyday Life in Ancient Mesopotamia ( M u s e u m M o n o g r a p h s 1 9 ) . Philadelphia: U n i v e r s i t y Museum. HEIMERDINGER, J a n e W . 1 9 7 9 . Sumerian Literary Fragments from Nippur ( O c c a s i o n a l Publications o f t h e B a b y l o n i a n F u n d 4 ) . Philadelphia: U n i v e r s i t y Museum. H 0 Y R U P , J e n s . 1 9 9 0 . " A l g e b r a a n d N a i v e G e o m e t r y . A n Investigation o f S o m e Basic Aspects of O l d Babylonian Mathematical Thought". Altorientalische Forschungen

17:2 7 - 6 9 ,

262-354.

— 2 0 0 2 . Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and its Kin ( S o u r c e s a n d Studies in t h e History o f M a t h e m a t i c s a n d Physical Sciences). N e w Y o r k / B e r l i n : Springer. H U B E R , Fabierme. 2 0 0 1 . " L a c o r r e s p o n d a n c e Royale d ' U r , u n c o r p u s a p o c r y p h e " . Zeitschrift fiir Assyriologie 91:169-206. K R A M E R , S a m u e l N . 1 9 6 3 . The Sumerians: their History, Culture, and C h i c a g o : U n i v e r s i t y o f C h i c a g o Press. L A N D S B E R G E R , B e n n o . 1 9 5 9 . The Series HAR-ra = hubullu, Tablets (Materials for the S u m e r i a n L e x i c o n 7 ) . R o m e : Biblical Institute Press.

Character. VIII-XII

364

E. Robson

— ; REINER, Erica, a n d CIVIL, M i g u e l . 1970. The Series XVI, XVII, XIX, and Related Texts (Materials for R o m e : Biblical Institute Press. M C C O W N , D o n a l d E. a n d H A I N E S , R i c h a r d C. 1967. Scribal Quarter, and Soundings (Oriental Institute Oriental Institute.

HAR-ra = hubuUu, Tablets the S u m e r i a n L e x i c o n 10), Nippur I: Temple of Enlil, Publications 7 8 ) , C h i c a g o :

V A N DER M E E R , Petrus E. 1935. Textes scolaires de Suse ( M é m o i r e s d e la M i s s i o n A r c h é o l o g i q u e de Perse 2 7 ) . Paris: Librairie E r n e s t L e r o u x . N E M E T - N E J A T , K a r e n R. 1 9 9 3 . Cuneiform Mathematical Texts as a Reflection of Everyday Life in Mesopotamia ( A m e r i c a n Oriental Series 7 5 ) . N e w H a v e n : A m e r i c a n Oriental Society. N E U G E B A U E R , Otto. 1 9 3 5 - 1 9 3 7 . Mathematische Keilschrift-Texte, I-III (Quellen u n d Studien zur Geschichte der M a t h e m a t i k , A s t r o n o m i e u n d P h y s i k A 3 ) . Berlin: Springer V e r l a g . — a n d S A C H S , A b r a h a m . 1945. Mathematical Cuneiform Series 2 9 ) . N e w H a v e n : A m e r i c a n Oriental Society.

Texts ( A m e r i c a n Oriental

— and S A C H S , A b r a h a m . 1984. " M a t h e m a t i c a l a n d M e t r o l o g i c a l T e x t s " . Journal of Cuneiform Studies 3 6 : 2 4 3 - 2 5 1 . A L - R A W I , F a r o u k N . H . a n d R O A F , M i c h a e l . 1984. " T e n O l d B a b y l o n i a n M a t h e m a t i c a l P r o b l e m s from Tell H a d d a d , H i m r i n " . Sumer (1984): 175-218. R O B S O N , Eleanor. 1997. " T h r e e O l d B a b y l o n i a n M e t h o d s for D e a l i n g with ' P y t h a g o r e a n ' T r i a n g l e s " . Journal of Cuneiform Studies 4 9 : 5 1 - 7 2 . — 1 9 9 9 . Mesopotamian Mathematics, 2100-1600 BC: Technical Constants Bureaucracy and Education C l a r e n d o n Press.

in

(Oxford Editions o f C u n e i f o r m T e x t s 14). Oxford:

— 2000. "Mathematical Cuneiform Calculations". SCIAMVS—Sources 48.

Tablets in Philadelphia, I: P r o b l e m s and and Commentaries in Exact Sciences 1: 1 1 -

— 2 0 0 3 . " T a b l e s a n d T a b u l a r F o r m a t t i n g in A n c i e n t Sumer, B a b y l o n i a , and Assyria". T o a p p e a r in: M a r t i n C A M P B E L L - K E L L Y , M a r y C R O A R K E N , R a y m o n d G. F L O O D a n d E l e a n o r R O B S O N (eds.), From Sumer to Spreadsheets: the Curious History of Table-Making. Oxford: C l a r e n d o n Press. — (forthcoming). " T h e T a b l e t H o u s e : a N i p p u r Scribal S c h o o l , 1740 B C " . T o a p p e a r in Revue d'Assyriologie. R O T H , M a r t h a T. 1995. Law Collections from Mesopotamia and Asia Minor (Writings F r o m T h e A n c i e n t W o r l d 6). Atlanta: Scholars P r e s s . S A C H S , A b r a h a m . 1947. " B a b y l o n i a n M a t h e m a t i c a l T e x t s , I. R e c i p r o c a l s o f R e g u l a r S e x a g e s i m a l N u m b e r s " . Journal of Cuneiform Studies 1: 2 1 9 - 2 4 0 . SJOBERG, Â k e . 1 9 7 5 . " T h e O l d B a b y l o n i a n eduba". In: S t e p h e n J. LlEBERMAN (ed.), Sumerological Studies in Honor of Thorkild Jacobsen (Assyriological Studies 2 0 ) : 1 5 9 - 1 7 9 . C h i c a g o : University of C h i c a g o Press. S T O N E , Elizabeth C. 1987. Nippur Neighborhoods (Studies in A n c i e n t Oriental Civilization 4 4 ) . C h i c a g o : T h e Oriental Institute of the University of C h i c a g o . T A N R E T , M i c h e l . 1982. " L e s tablettes ' s c h o l a i r e s ' d é c o u v e r t e s à Tell e d - D ë r " . Akkadica 2 7 : 4 6 - 4 9 . — 2 0 0 2 . [Title to b e confirmed] ( M e s o p o t a m i a n H i s t o r y and E n v i r o n m e n t , Series III, T e x t s 3). G h e n t : T h e University of G h e n t . Cited a c c o r d i n g to draft o f July 2002.

More than Metrology: Mathematics Education in an Old Babylonian Scribal School

365

T I N N E Y , Steve. 1 9 9 9 . " O n the Curricular Setting o f S u m e r i a n Literature". Iraq 5 9 : 159-172.

V A N S T I P H O U T , H e r m a n L.J. 1 9 7 9 . " H o w D i d they L e a m S u m e r i a n ? " . Journal of Cuneiform Studies 3 1 : 1 1 8 - 1 2 6 . — 1 9 9 7 . " S u m e r i a n C a n o n i c a l C o m p o s i t i o n s . C. Individual F o c u s . 6 . S c h o o l D i a l o g u e s " . I n W i l U a m W . H A L L O (ed.), The Context of Scripture, I: Canonical Compositions from the Biblical World: 5 8 8 - 5 9 3 . L o n d o n / N e w Y o r k / K o l n : Brill. V E L D H U I S , Niek. 1 9 9 7 . " E l e m e n t a r y E d u c a t i o n at N i p p u r : T h e Lists of T r e e s a n d W o o d e n O b j e c t s " . U n p u b h s h e d doctoral thesis. University o f G r o n i n g e n . — 1 9 9 7 - 1 9 9 8 . " R e v i e w of C A V I G N E A U X ( 1 9 9 6 ) " . Archiv fur 45:

Orientforschung

4 4 -

360-363.

— 2 0 0 1 . Nanse and the Birds: Literature, Religion, Scholarship. Unpublished book manuscript, cited a c c o r d i n g to the p r e l i m i n a r y v e r s i o n o f O c t o b e r 2 0 0 1 . W A S C H K I E S , H a n s J. 1 9 8 9 . Anfdnge der Arithmetik im alten Orient und bei den Griechen. A m s t e r d a m : B . R . Griiner. W I L C K E , Claus. 1 9 8 7 . " D i e Inschriftenflinde d e r 7 . u n d 8 . K a m p a g n e n ( 1 9 8 3 u n d 1 9 8 4 ) " . I n B a r t h e l H R O U D A (ed.), Isin-Isân Bahrïyât III: Die Ergebnisse des Ausgrabungen I983-I984: 8 3 - 1 2 0 . Munich: Bayerische Akademie der Wissenschaften. ZETTLER, R i c h a r d L. 1 9 9 6 . " W r i t t e n D o c u m e n t s as E x c a v a t e d Artifacts a n d the Holistic Interpretation o f the M e s o p o t a m i a n A r c h a e o l o g i c a l R e c o r d " . In: Jerrold S. C O O P E R a n d G l e n n M . S C H W A R T Z (eds.). The study of the ancient Near East in the 21st century: the William Foxwell Albright Centennial Conference: 8 1 1 0 1 . Winona Lake: Eisenbrauns.

A Study of Babylonian Normal-Star Almanacs and Observational Texts Norbert

A. Roughton,

Denver

In this p a p e r I p r e s e n t t h e results o f analyses o f two tablets firom M e s o p o t a m i a . O n e of these is B M 3 2 2 4 7 , the latest k n o w n N o r m a l - S t a r ( N S ) A l m a n a c in t h e British M u s e u m a n d t h e other is B M 4 6 2 3 5 + , a n Observational T a b l e t for M e r c u r y a n d M a r s . E a c h o f these tablets h a s s o m e u n u s u a l features w h i c h will b e d i s c u s s e d in t h e appropriate sections o f the paper. N o attempt will b e m a d e in this p a p e r t o give a definitive explanation o f h o w the scribes of B a b y l o n w e r e able t o construct such impressive d o c u m e n t s . Instead, the reader should consider this p a p e r as a n attempt to p r o v i d e c o m p a r i s o n s b e t w e e n t h e data p r o d u c e d in antiquity a n d those that c a n b e calculated using m o d e m c o m p u t e r techniques. W e fmd that in m a n y c a s e s , the a g r e e m e n t b e t w e e n m o d e m a n d ancient data is excellent, b u t that is n o t always the case. I a m o f the opinion that a large a m o i m t o f data should b e available t o the workers w h o will eventually explain to all o f u s j u s t h o w the B a b y l o n i a n s c h e m e s w o r k e d . T h e r e h a v e b e e n giant strides m a d e in the u n d e r s t a n d i n g o f t h e so-called m a t h e m a t i c a l astronomical texts, b u t the techniques for p r o d u c i n g p r e d i c t i o n s for tablets like N S A l m a n a c s , ordinary A l m a n a c s , O b s e r v a t i o n a l texts, a n d D i a r i e s , ' a r e not as well imderstood.

B M 32247 T h i s tablet contains typical N S A l m a n a c data for Seleucid E r a ( S E ) year 2 3 4 . O n l y data for m o n t h s 9, 10, 1 1 , a n d 12 o f that year are well e n o u g h p r e s e r v e d o n the reverse side to b e o f u s e for a n analysis. T h e data o n t h e o b v e r s e side o f t h e tablet are t o o fragmentary to b e analyzed. T o the best o f m y k n o w l e d g e , n o p r e v i o u s edition o f this tablet h a s b e e n p r o d u c e d . A p h o t o g r a p h a n d m y c o m p u t e r g e n e r a t e d c o p y are s h o w n in figures 1 a n d 2, a n d m y translation o f the reverse is given b e l o w . Rev: 1 ' . . . [ M o n t h 9] [ 1 ] 4 * 4 " sunrise to m o o n s e t 15''' l ; 2 0 ^ . . 2 ' . . . [ 5 / ( 3 ] S c o r p n Vi cubit. N i g h t 12 after sunset 3 ' . . . 17? 1 8 * M e r c u r y in t h e west in C a p r i c o m 1 visibility.... 4 ' . . . [ y C a p r i c o m i ] 6 fingers. N i g h t 2 8 after sunset V e n u s a b o v e [Ô C a p ] . . . . 5 ' . . . [?] V e n u s A q u a r i u s reaches. 6 ' . . . [ M o n t h 1 0 ] . . . 5 ; 2 0 ° m o o n s e t to sunrise 15"' 6;50° m o o n r i s e to sunset sunrise to m o o n s e t 16"' 5;40° [sunset to m o o n r i s e ] . . .

For these text classifications, see SACHS ( 1 9 4 8 ) and HUNGER ( 1 9 9 9 ) .

7;40°

368

N . A . Roughton

7 ' . . .Solstice. 5 * Sirius acronychal rising. 8'*' M e r c u r y in the west in the b e g i n n i n g of A q u a r i u s [last v i s i b i l i t y ] . . . 8'...27'*' 15 kur. N i g h t 2 4 after sunset M a r s a b o v e a Tauri 3 c u b i t s . . . 9 ' . . . a f t e r sunset eclipse of the sun 5 m o n t h s after the last possibility will p a s s b y . Night 2 ? . . . . 1 0 ' . . 1*' M e r c u r y A q u a r i u s reaches. 1 5 * Jupiter Aries reaches. 22"** V e n u s Pisces reaches... 1 1 ' . . . [ M o n t h 11]...lO;?"" 14''' 15;20*' m o o n r i s e to sunset 1 4 * 1;40° m o o n s e t to sunrise 1 5 * ;34° sunset to m o o n r i s e 1 5 * [...].. 1 2 ' . . . 13? after sunset Jupiter b e l o w r| P i s c i u m 3 c u b i t s . . . 1 3 ' . . . e c l i p s e of t h e m o o n p a s s e d b y . Night 19 after sunset V e n u s b e l o w r\ P i s c i u m 1 4 ' . . . V e n u s b e l o w P Arietis 3 Vz cubits. After sunset M a r s b e l o w [p T a u r i ] . . . 1 5 ' . . . a f t e r sunset V e n u s b e l o w a Arietis 4 '/2 cubits. 2 8 * M e r c u r y [last visibility e a s t ] . . . 1 6 ' . . . 1 5 * M e r c u r y A q u a r i u s reaches. 1 6 * V e n u s Aries reaches. 1 7 ' . . . [ M o n t h 1 2 ] . . . 1 5 * [...] 1 5 * 1;10°? sunrise to m o o n s e t 16"^ 10;30° sunset to moonrise... 1 8 ' . [ . . . ] . . . n i g h t 5 after sunset M a r s a b o v e

Tauri [...]...

1 9 ' . . . . p Arietis 4 cubits. 1 0 * S a t u m in Scorpius stationary. N i g h t 1 7 . . . 2 0 ' . . .night 18 after sunset M a r s a b o v e r| G e m i n o r u m . . . 21'

4 fingers, [traces]

N S A l m a n a c s in general contain data about conjunctions o f planets with the N o r m a l Stars, first and last visibilities of the planets a n d the star Sirius, a c r o n y c h a l risings of the superior planets a n d Sirius, e a s t e m and w e s t e r n stationary points of t h e superior planets, lunar six data for the m o o n , dates of solstices and e q u i n o x e s , solar a n d lunar eclipse data, a n d if the tablet w a s written in Uruk,^ entries (reachings) o f the planets into the signs of the z o d i a c . Since the tablet presently u n d e r d i s c u s s i o n r e c o r d s reachings, it m a y b e from Uruk, although the r e m a i n i n g material in the British M u s e u m ' s 7 6 - 1 1 - 1 7 collection, w h i c h w a s a c q u i r e d b y p u r c h a s e , is a p p a r e n t l y from Babylon. W h i l e it is generally a s s u m e d that the data o n N S A l m a n a c s are all calculated, the present tablet h a s r e c o r d e d several items w h i c h h a v e exact m a t c h e s o n diaries for the same year. T h i s raises s o m e interesting possibilities: A r e the data o n b o t h the Diaries a n d this N S A l m a n a c all calculated, or h a s the N S A l m a n a c author obtained his data from the Diaries, or has the D i a r y author o b t a i n e d s o m e of his data from the N S Almanac? A n o t h e r u n u s u a l feature of the present tablet is the p l a c e m e n t of t h e lunar six data. In N S A l m a n a c s these usually appear in a separate section o n the left e d g e of the m o n t h l y data, or s o m e t i m e s o n the last r o w o f the m o n t h l y section. O n this tablet, the lunar sixes o c c u p y the first r o w of each m o n t h l y section.

See HUNGER and PINGREE ( 2 0 0 0 ) , p. 161 and SACHS ( 1 9 4 8 ) .

A Study of Babylonian Normal-Star Almanacs and Observational Texts

369

C o m p a r i n g the data o n B M 3 2 2 4 7 with that o n Diary^ B M 4 5 6 5 9 + 4 5 6 8 5 , N o . - 7 7 L B A T 525f: line i r , rev., 3 2 2 4 7 , M o n t h 11 " 1 4 * 1;40 m a t c h on line rev. 5 ' , 4 5 6 5 9 + 4 5 6 8 5 , M o n t h 11 " r 4 0 ' , clouds, I did not w a t c h "

m o o n s e t to sunrise " has a T h e 14*, m o o n s e t to s u m i s e :

line 1 3 ' , rev., 3 2 2 4 7 , M o n t h 11 " N i g h t 19 after sunset V e n u s b e l o w rj Piscium " has a m a t c h on line rev. 7 ' , 4 5 6 5 9 + 4 5 6 8 5 , M o n t h 11 " [Night] o f the 2 0 * , first part of the night, V e n u s w a s [nn] cubits b e l o w r| P i s c i u m

".

line 1 4 ' , rev., 3 2 2 4 7 , M o n t h 11 " V e n u s b e l o w (3 Arietis 3 Vz cubits " has a m a t c h on line rev. 9 ' , 4 5 6 5 9 + 4 5 6 8 5 , [.... N i g h t of the 2 5 * , first part of the night, V e n u s w a s ] 3 Vz cubits [below P] Ariefis

"

line 1 5 ' , rev., 3 2 2 4 7 , M o n t h 11 " after sunset V e n u s b e l o w a Arietis 4 Vz cubits " has a m a t c h on line rev. 1 0 ' , 4 5 6 5 9 + 4 5 6 8 5 , " [Night of the 2 ] 9 * , first part o f the night, V e n u s w a s [nn] cubits b e l o w a [Arietis]

"

line 1 5 ' , rev., 3 2 2 4 7 , M o n t h 11 " 2 8 * M e r c i u y [last visibility east] " has a m a t c h o n lines rev. 1 0 ' (and 1 2 ' ) , 4 5 6 5 9 + 4 5 6 8 5 , " a r o u n d the 2 8 * . M e r c u r y ' s last a p p e a r a n c e in the east in A q u a r i u s , clouds, I did not w a t c h " line 1 5 ' , rev., 3 2 2 4 7 , M o n t h 11 " 15* Mercury Aquarius reaches m a t c h o n line rev. 1 2 ' , 4 5 6 5 9 + 4 5 6 8 5 , " o n the 1 5 * M e r c u r y Aquarius " line 1 5 ' , rev., 3 2 2 4 7 , M o n t h 11 " o n l i n e rev. 1 2 ' , 4 5 6 5 9 + 4 5 6 8 5 , "

16th V e n u s Aries reaches [....] V e n u s r e a c h e d A q u a r i u s

" has a reached

" has a m a t c h "

A s c a n b e seen, there are a g r e e m e n t s and disagreements b e t w e e n the data o n the N S A l m a n a c a n d the Diary. Several events m e n t i o n clouds w h i c h p r e c l u d e d m e a s u r e m e n t s or observations, at least on the Diary, e v e n t h o u g h dates are given. H e r e w e h a v e a n a r g u m e n t that at least s o m e of the data o n the D i a r y are calculated. T h e conjimction data o n b o t h types of tablets s e e m to m a t c h quite well in angular separation b e t w e e n the planet and the star, e v e n t h o u g h dates d o not necessarily match. T h e single lunar six n u m b e r for time b e t w e e n m o o n s e t a n d sunrise o n the 1 4 * as 1° 4 0 ' a p p e a r s on both tablets. Since the precision m a t c h e s , the n u m b e r s on b o t h tablets m u s t h a v e a c o m m o n origin, either calculated or observed. T o p r o c e e d further, I n o w present results from m o d e m calculations of the first and last visibilities, conjunctions, and reachings for the m a t c h i n g events d e s c r i b e d a b o v e . T h e data in tabular form from these c o m p a r i s o n s c a n b e found in F i g u r e 3 . T a b l e s or data c o l u m n s w h i c h are labeled (R)eference contain data w h i c h w e r e extiacted from m y global tables of events and conjunctions c o m p u t e d for the years 6 0 1 B . C . to 75 A . D . T h o s e w h i c h are labeled (C)alculated are n e w l y calculated using the dates given o n the tablets themselves.

SACHS AND HUNGER ( 1 9 8 8 ) .

370

N.A. Roughton

T h e numerical data for the events were calculated using the following criteria: 1. 2.

3. 4.

5.

First a n d last visibilities: S c h o c h conditions pertain."* Stations: T h e m o m e n t s of these are t a k e n to b e w h e n the p l a n e t a c h i e v e s a m a x i m u m in longitude to 3 decimal places for an e a s t e m station, a n d a similar m i n i m u m in longitude for a w e s t e m station. A c r o n y c h a l risings (oppositions): the refracted i m a g e of the planet (or star) at first positive altitude a b o v e the e a s t e m h o r i z o n at the m o m e n t o f sunset. Conjunctions: T h e m o m e n t of m i n i m u m separation in longitude b e t w e e n the planet a n d the star, with the sun 10° b e l o w the horizon, m o m i n g or evening. Occasionally solar altitudes greater than - 1 0 ° are p e r m i t t e d if they are n e c e s s a r y to c o n f o r m to a statement about a conjunction on a tablet. R e a c h i n g s : T h e longitude of the planet at sunset o n the day g i v e n o n the tablet.

B M 46235+ Observational Text I n o w give the results of a similar analysis of B M 4 6 2 3 5 . H . H u n g e r labels this tablet as B M 3 5 3 3 9 with other parts B M * 4 6 2 3 5 + B M * 4 6 2 4 2 a n d p u b l i s h e s it as N o . 7 6 in Diaries vol. 5. T h i s tablet is nearly c o m p l e t e and has s o m e fascinating content. Information is written on the obverse, o n the right e d g e , o n the b o t t o m e d g e , a n d on the reverse o f the tablet. T h e tablet is written in the usual w a y s u c h that the obverse is c o m p l e t e d first and the tablet is rotated vertically a n d writing continues d o w n the reverse side. In this case, h o w e v e r , there are six lines c o n t i n u e d from the o b v e r s e a r o u n d the right edge, a n d then o n t o the reverse side after a horizontal rotation. Therefore the script w h i c h w a s continued from the o b v e r s e appears to b e inverted w h e n the reverse side of the tablet is v i e w e d in the n o r m a l way. N o w 4 6 2 3 5 + is m o r e than j u s t an observational text. It is also a partial goal-year text with data w h i c h c a n b e used to m a k e predictions using the goal-year p e r i o d techniques. T h e data a n d target events o n the tablet are g i v e n for: •

Conjunctions a n d planetary events for the years S E 8 3 , 84, a n d 85 for M e r c u r y . T h e s e c a n b e used to predict conjunctions a n d events for years S E 129, 130, a n d 131 respectively, i.e., 4 6 years later than the tablet dates.

•

Conjunctions o f M a r s for years SE 82, 8 3 , and 84, w h i c h c a n b e u s e d to predict conjunctions for the years S E 1 2 9 , 1 3 0 , a n d 1 3 1 , i.e., 4 7 years later than the tablet dates.

•

Planetary events of M a r s for the years S E 5 0 , 5 1 , a n d 52, w h i c h c a n b e u s e d to predict events for the years S E 129, 130, a n d 1 3 1 , i.e., 79 years later than the tablet dates.

M a t c h i n g events o n other tablets: Prediction: S E _ 8 3 , line 5 ' , " M e r c u r y ideally on the 4 * in the east in Libra last visibility." matches

LANGDON, FOTHERINGHAM, and SCHOCH ( 1 9 2 8 ) .

A Study of Babylonian Normal-Star Almanacs and Observational Texts

371

D i a r y - 1 8 2 ( S E _ 1 2 9 ) M o n t h V I I I line 1 2 ' , S H p 3 7 3 , " T h e 4 * ? M e r c u r y ' s [last a p p e a r a n c e ] in the east [in L i b r a . . . ] " Prediction: S E _ 8 3 , line 5 ' , " M e r c u r y ideally on the 4 * in the east in Libra last visibility" matches D i a r y - 1 8 2 ( S E _ 129) M o n t h 8 line 6 ' rev, "the 1'', rising of M e r c u r y to sunrise: 1 T ; aroimd the [ x ] + l t h . M e r c u r y ' s last a p p e a r a n c e in the east in L i b r a . " Prediction: S E _ 8 3 , line 7 ' , " . . . C a p r i c o m first visibility not s e e n " matches D i a r y - 1 8 2 ( S E _ 1 2 9 ) M o n t h X line 3 8 ' , " T h e 1 7 * M e r c u r y ' s first a p p e a r a n c e in the east in C a p r i c o m , 1 cubit in front of the m o o n to the west, it was bright, rising o f M e r c u r y to sunrise: 16°". Extraction: S E _ 8 5 , line 1 8 ' , " M o n t h 3 1 9 * M e r c u r y in the west in C a n c e r first visibility not seen", matches D i a r y - 2 2 6 ( S E _ 8 5 ) M o n t h 3 , " T h e 19*, M e r c u r y ' s first a p p e a r a n c e in the w e s t in Cancer; I did not w a t c h . " Extraction: S E _ 5 0 , line 2 7 ' , "23'*' M a r s at w e s t e m station...fingers b e h i n d p L e o n i s 1 cubit b e h i n d S a t u m station not s e e n . " matches D i a r y - 2 6 1 ( S E _ 5 0 ) " [ . . . w h e n M a r s b e c a m e stationary to the west, it b e c a m e stationary b e h i n d p Leonis, 1 cubit b e h i n d S a t [ u m . . . ] " . I give a b b r e v i a t e d data tables for this tablet in Figure 4 . O n e o f the goals o f this w o r k has b e e n the re-examination o f the c o n n e c t i o n b e t w e e n the G o a l - Y e a r texts a n d the N o r m a l Star A h n a n a c s . It will b e difficult to p r o v e that the N S A l m a n a c s w e r e actually p r o d u c e d b y goal-year p e r i o d techniques b e c a u s e o f the scarcity o f r e c o r d e d data. F o r e x a m p l e , there are 6 0 N S A l m a n a c s available, and 57 G o a l - Y e a r texts, b u t they overlap for only 15 years. A s demonsfrated i m m e d i a t e l y a b o v e , it is fairly easy to find exact exfractions of data from Diaries w h i c h h a v e b e e n inserted into the N S A l m a n a c s for matching years. C o n s i d e r a b l e w o r k r e g a r d i n g the origins of the data w h i c h a p p e a r in all types of B a b y l o n i a n astronomical tablets r e m a i n s to be d o n e .

Acknowledgements Let m e thank Professor H e r m a n n H u n g e r for providing various fransliterations a n d helpful suggestions, a n d Mr. Christopher W a l k e r for m u c h help a n d g u i d a n c e over the past t w o d e c a d e s , as well as his k i n d n e s s in p r o v i d i n g access to the British M u s e u m tablet collections.

372

N . A . Roughton

References H U N G E R , H e r m a n n . 1999. " N o n - m a t h e m a t i c a l A s t r o n o m i c a l T e x t s a n d T h e i r Relationships". In: N o e l M . S w e r d l o w (ed.), Ancient Astronomy and Celestial Divination: 7 7 - 9 6 . C a m b r i d g e , M A : M I T P r e s s — and PINGREE, D a v i d . 1999. Astral Sciences in Mesopotamia. Leiden: Brill L A N G D O N , Steven; FOTHERINGHAM, J o h n K., a n d S C H O C H , Carl. 1928. The Venus Tablets of Ammizaduga. L o n d o n : Oxford University P r e s s . S A C H S , A b r a h a m . 1948. " A Classification o f the B a b y l o n i a n A s t r o n o m i c a l T a b l e t s of the Seleucid P e r i o d " . Journal of Cuneiform Studies 2: 2 7 1 - 2 9 0 . —

and H U N G E R , H e r m a n n . 1 9 8 8 - . Astronomical diaries and related text from Babylonia, V o l u m e I ( 1 9 8 8 ) , II ( 1 9 8 9 ) , III ( 1 9 9 6 ) a n d V ( 2 0 0 0 ) . W i e n : Osterreichische A k a d e m i e der Wissenschaften.

A Study of Babylonian Normal-Star Almanacs and Observational Texts

F i g u r e l a . B M 3 2 2 4 7 O b v e r s e (Copyright T h e British M u s e u m ) .

373

A Study of Babylonian Normal-Star Almanacs and Observational Texts

375

10' |î^f^*^^4^w*ffa^'4^][i

15' ^

AAA

20'

0

5 cm Figure 2. C o m p u t e r G e n e r a t e d C o p y of B M 3 2 2 4 7 R e v e r s e .

10

-
377

A STUDY OF BABYLONIAN NORMAL-STAR ALMANACS AND OBSERVATIONAL TEXTS

N . A . ROUGHTON

376

"«t

f*^

rn f*^ f i pL)

w w

oo

^ so — o ^ ^

w

O

S:

s o d e s T d o o T T . i O s V O N O O o e s

«3 W3 «3 Kl t » «3 c » . ^ CL,

iri r-i w-i Ti- ..^ o f i ^ — « e s e s o - ^ o s — •^so'^ _ _ TfOsOOt^O'^t^'^ c P o 6 H ' t o o e s i ^ ' n ^ - a ^ a ^ - r Q O ^ o ^ o s t ^ o - H e s 2 v i è s f i s o s o s o o o ^ ^ - H r t ^ r s < s e s t s t - o \ — -^-^•-^'^

s o OS ^ v-1 n - H f fn i ^ s o 0^ 0 0

U

^ -^ s? ^

s; S; p:

s

r> 00 fi m

voooo-^'-;»^g.^^(S|f,5\oos O. ~

(N
»

•o c c c c c

• ^ 3 3 3 3 . ' c/3 CM CO c/3 cn

e s e s fi fS

(N fi >r> 00 s o o s ^ f^

^ CT> OS OS 00 i S 00 <^ O ^ ^ ^

a . r.; ( S r i e s O • o - ^ os O 'T

^

m 00 00 o

7 9

ON 00

t-;

In

„,

o

-H o

o

U d r i d d d d - ' - i ' ^ ^ ' ^ 9' ^ 9- ' ^ ' ^ ' ^ d d - - ^ - - - ' o fi es ^ es •/^oof \osoesw-)es Os >n ^ >0 ON fi ' S . T f p o ^ ' ^ ^ ' ^ o s o s ' î f f i O s e S ' - iOeesS-*ïOt0 0 ' e i. 'o< 5 ; p t ^>r> <SO ^ ^ rs ,osos^fja;Os os ^ >—I Oesfi>ososDoo-H.-i.-i-Hrt{seseses

m SO

Os

00 en «-i es 0*1

so

o , 00 e s os ^^ 00 ^ ^ u e s e s en e s —

o

rooesoonoooso^oo m ^ e s p o o ; et-ooo>/^sosoossot^osso(Sfit~-esooooooot~-sDTt o\• r - e s c s - H O O o ò c K f s r i o ó ^ d d^c i ' * a\ ' ^ o\ --o ^ es 00 es 00 H ^ in K c< ôs r< fi O; O; C ^ e s f ^ v o v o s o o o ^ ^ ' ^ ^ - ^ e s e s e s e s r ^ os - H - H ^ ^ ^

to «
00 e s "O 00 «o os

c

es d

ce:

O fi fi es Os es m

^ o o ^ f i — i r - ^

i s ed

^*omu-,asoooot^

o

l e . (ai;!^2c*i-*'tw-ir^fi"*'n''ssor-ov-HfiTi-w-)SOsor^oo os 2

x>

o

os

o

-H ^

o

S o s ? 5 - ^ 2 ^ ^ ^ 2 2 2 2 « ^ ; : e s ^ ^ ? ^ 2 2 ^ ^ 2 ^

2

00

t'es u

f<-j Ov-Q,, ^ 00

00

?5 2

ai

I

-"t ^ e'i e'i e o fi es es es es es I M

I o

- H .-H

—I

Tl-

S) < s

fi (S n I I ["^1

r^")

e s 04

t

fs I

fi ro es I I I

r^l

2

--ï- -«afi fi fi fi rs» ( S e s e s

oc

•

.s

U

I

B O

••n

3

B S

•o

«8 1» 1) ™

N

f i f i < s e s e s e s < s f i f i f i f i f i f i f i f ! S 3 ; S S S S S S 00 00 00 00 00 00^00^00|00|00^00|00^00|00|00|00^00|00^00^00^00 c lli'ùj'rjj'rjj'ti'ti'w'u U W W t i W t i U W W t i t i t i W W 2^KMKîZlC«;/3«3C«(»5«C«y3y35«C<3C«&0!Zi&0;«C/3a!

a'

C/3 «3 tZ3

es « .C J 3 J 3 è ^ Y S

— W

a

3

ii

^

S

3

g

fc^

a s, I I t i i

S

3

5

3

3

O. U U u Clj

S ^ ^ B ^

Ë a s "

«s

ri

l t l t t l l ^

Cl.

sa

ç? o

o

u -c

l i t : ! I t l l

+

E " 2 ë

es

g. Ë s s

s

«

a

l l i l i l l i

ai

ËE Ë Ë Ë Ë Ë Ë Ë Ë

c ° / o b b ^ é s V V V V V V V V V ó - . ^ ^ e s f A f i ; ^ r ~ o s f i f i f i f i f i e . f i ' ^ ' ^ < / ^ ^ r ~ r ^ o s ^ — ^ — ^

QQ

•Ç ^

P

S a a 2

i:

•§< p s i

^

B

E

E i dû u

5 5

J2

.5

-M

B

o o TfTj-int^fi^^u^kessor^-^-Hfn^v-iso^t^oo

o U

I I I ti bJ

2;

•5'

u

U

-

ti

.B

9

U

M

'S

c

I

I

«J V5 C/3 »3 t/3

E:î

;c

B

o

M

I

u

s

C/3 «J OO V5 {/} c o

<j

I

^

t-

52

2;

s o o o so so

0Q

bû

378

N.A. Roughton

o en eti - o< d es 13 ^
o "-^

O

0 0 >r>

OO ON

«n NO

d NO ON

ON ON

es m Cu

ro

ON ON

ts NO

g

fS

NO

(S

2 g

o

ON

(S

es

«o

o

es

m >o es

oo

TT

d d

00 m

^

ON

O

S

NO

ON

00

TT

TT

ON

00

ON

NO

>A

es o

-H

ON

en es en

>0

ON

ON

O ON

*?

en d

NO

m

TT

es

NO

O

ON

NO

00 00 Tt

ON ON ON

ON »o es es d >o <s ON

00

en e<-j en d en es

>n

t-o o NO â^ 0 0 rcs d es d i-H <—t

00

en •"j

ON en Tf ^o 0 0 o eo eo d 1 d

ON

d es

0 0 <>

ON

00 ,—:

NO

00

TT

ri

TI-

00

o t'T t en '7 d es

o

ON

e n Ti-

d

NO 0 0 0 \ 0 0 0 0 >ri en 0 0 en

NO

Tf

00

00 O

NO

d

NO NO

es

ON

d

>o

NO

fO

fS

d

d d en

en

ON

NO (N

NO

ON

«r> «/->

o

00 00 Os

00

r-i d d d

O

(N .—1

d 9 en •«t o d

NO

d

ON

NO (S

d

NO O

d d cK

d d

9

pi

es r-

NO NO

CO rt

es eo

es

en «o

00

o

00

NO

Ti-

es es d

en

es

en 00

es

^ ON

(ai > si

I 9i

U

0\ g)

o o o ^ es es e n f n T r T | - T } - T f T t m w > < o i o < n « r i v - i «n i n >ri v~> < o o o o o o o o o o o o o o o o o o o o o o o o o o o o o I I I I I I I I I I I I I I I I I I I »D«50C0O(>0û0C«0O(Z)C/ia3aiC/ii/ÎCW
PQ

^

<È <E

+

IT) va »3 V3 C/3 ^ ^ ^ ^ S E S S

j:3 TJÌ

3

3

3

a E Es PQ

E

Ë

E

ë

E

S

S

S

E

E

E

E

S

Egyptian Festival Dating and the Moon Anthony

Spalinger,

Auckland

E v e r since t h e m o o n fascinated L u d w i g B o r c h a r d t , t h e intimate cormection b e t w e e n the lunar a n d civil dates o f ancient E g y p t h a s b e e n e m b r a c e d in scholarly j u d g m e n t . ' Y e t the intricate m a t h e m a t i c a l m a c h i n a t i o n s o f c h r o n o g r a p h e r s often a p p e a r t o o difficult for m o s t o f us.^ F o r a solution t o e n d all t h e c o n t i o v e r s i e s a b o u t a b s o l u t e c h r o n o l o g y , the skeptical r e s e a r c h e r is as p e r p l e x e d a s ever. , In ancient E g y p t political as well as religious events w e r e s c h e d u l e d for auspicious days o f t h e m o o n . B u i l d i n g projects, for e x a m p l e , w e r e t h e m o s t anticipated, followed b y p u b l i c spectacles s u c h a s divine n o m i n a t i o n t o t h e k i n g s h i p or the p r e s e n c e o f a deity during a particular religious c e r e m o n y . B e y o n d the inherent historical i m p o r t a n c e o f these o c c u r r e n c e s , there r e m a i n s a w h o l e series o f related difficulties c o n c e m i n g the timing.^ H o w d i d the E g y p t i a n s k n o w w h e n a specific lunar d a y , especially their first, w a s to o c c u r ? " W e r e t h e p r o g n o s t i c a t i o n s o f psdntyw,

for e x a m p l e , tentative, i m p r e c i s e ; w e r e they artificial?

A r e c o i n c i d e n c e s o f lunar d a y X with the s a m e civil d a y real o r d o t h e y h i d e t h e effective o p e r a t i o n o f the civil c a l e n d a r ? O n a n earlier o c c a s i o n I discussed the simplicity o f t h e N e w K i n g d o m lunar sightings a s d o c u m e n t e d i n the t e m p l e calendars that are extant from that t i m e period.^ H e r e I will e x a m i n e s o m e inscriptions o f D y n a s t y X V I I I , c o m p a r i n g t h e m

' Borchardt's work on calendrics, timekeeping, and absolute chronology spanned more than three decades of research. Let me single out his two main volumes BORCHARDT (1920) and BORCHARDT (1935). The weaknesses in his presentation were later removed by PARKER (1950a). Subsequently, we can mention: KRAUSS (1985); LEITZ (1989); SPALINGER (1992); and DEPUYDT(1997).

^ For example, one key date for the reconstruction of Dynasty XVIIl depends upon a lunarcivil equation in the reign of Thutmose III (see below, Part III). A second lunar date occurs during the same king's reign; namely, the Battle of Megiddo reference. For the XlXth Dynasty we are likewise dependent upon a lunar date from the reign of Ramesses 11. The Middle Kingdom, however, poses a disparate situation because its chronology is organized according to the heliacal rising of Sothis. ' Bradley E. Schaefer half The following list SCHAEFER (2000), with DOGGETT (1994) (most

carefully analyzed the specific human problems over a decade and a of his voluminous articles are worthwhile to cite in this context: SCHAEFER (1993) (the complete theoretical paper); SCHAEFER and important for observations); and SCHAEFER (1996). See as well

FERMOR(1993).

* This is discussed in my chapter "Under the Moon of Earth", in SPALINGER (1992). Schaefer and Doggett have proved that about fifteen per cent of human naked eye sightings of the new crescent moon are anticipatory and false. ^

SPALINGER (1995b). Earlier, see GRIMM (1994); and now EL-SABBAN (2000). The latter

380

A. Spalinger

to later o n e s from the p o i n t of v i e w of the timing o f these events, their location, a n d their intimate links with various festivals. Insofar as m a n y prejudices r e m a i n associated with limar dates, especially their p r e s u m e d inviolability, a n analysis of Egyptian lunar counting and the c o m p l e x lunar-civil p r o b l e m o f ancient E g y p t i a n d o c u m e n t s will reveal h o w fragile our p r e s u m e d "firm" data t m l y are.

I. The Beginning of ttie Lunar Month: Ancient Difficulties and Modern Problems N e w K i n g d o m historical r e c o r d s are a b u n d a n t e n o u g h for diverse aspects of P h a r a o n i c life to b e observable, w h e r e a s the lack of written e v i d e n c e is o n e o f the fatal w e a k n e s s e s of O l d K i n g d o m chronology. Hieroglyphic inscriptions from the N e w K i n g d o m - not j u s t hieratic papyri - p r o v i d e c o r r o b o r a t i o n for lunar dates and their civil equivalencies. B y c o m p a r i s o n b e t w e e n dating m e t h o d s , scholars h a v e b e e n able to set u p a relatively fixed a n d r e a s o n a b l y consistent absolute c h r o n o l o g y for this period. W h i l e questions a n d confroversies continue, the real issues n o w have less to d o with c m n c h i n g a n u m b e r of absolute Julian dates a n d m o r e to d o with calendrics.* It is fortunate that the hieroglyphic r e c o r d s p r e s e r v e sufficient lunar-civil material to s u p p o r t re-evaluation. Often, m o d e m scholars h a v e m a d e a n effort to r e c o n s t m c t an absolute c h r o n o l o g y solely o n the basis o f the few lunar references that are extant in the written record. In 1950 R i c h a r d A. P a r k e r first h y p o t h e s i z e d that a n i n d e p e n d e n t lunar calendar existed side b y side with the official civil c a l e n d a r . ' A s a result, s o m e researchers h a v e suggested that in the N e w K i n g d o m a c o m p l e t e lunar cycle h a d b e e n w o r k e d out to equate (artificially o f course) the civil with the lunar day, w h e n the latter w a s the date o f s o m e religious p e r f o r m a n c e or major secular event.^ B u t there are statistical limitations against reconciling the multiple possibilities within the relatively large time interval of o v e r forty years. O n e m u s t establish a p r o c e d u r e for m a t h e m a t i c a l surety. T h e following criteria h a v e b e e n tested and p r o v e necessary and sufficient for finding any event on a lunarb a s e d calendar, w h a t e v e r that t e r m i m p l i e s : ' (1) A d o u b l e date m u s t b e present. It is n o t e n o u g h for a m o n u m e n t to refer, for e x a m p l e , to lunar day I , psdntyw. B y itself, this d a t u m p r o v i d e s n o w o r k a b l e information. W h e n , however, that date is given in conjunction study wisely avoids the lunar-civil equivalences. * Since the two aspects overlap, let me state that all chronologies for Egypt before 664 B.C. are candidates for revision, none more so than that of the Old Kingdom. It is sufficient to observe that whenever lunar-civil equations can be found, the leeway in absolute dates remains +/- fifty years or so, not a satisfying result. The Sothic dates are also unclear owing to various problems, among which are the location of the sighting and the beginning of the Egyptian day (moming twilight or moming sunrise). ' In SPALINGER (1995a) I felt that Parker's "second lunar calendar" was not necessary to hypothesize. ^ Parker himself pointed out the improbability of this hypothesis in "Once Again the Coregency of Thutmose III and Amenhotep II" (PARKER (1969), p. 79 note 3). ' The following is a summary from my study SPALINGER (1996a), p. 35 note 14. It has been augmented in the light of Schaefer's comments.

Egyptian Festival Dating and the Moon

381

with the civil d a y o f a particular civil m o n t h , a solution c a n b e w o r k e d out. (2) A specific reference to the m o o n has to b e given. If the day in q u e s t i o n is cited b y a n u m b e r (integer), the reference nevertheless m u s t indicate that a m o o n - b a s e d operation, sighted b y the n a k e d eye, t o o k p l a c e . A s a l e m m a , a reference to a d a y of the m o n t h referred to b y name a n d n o t b y a n u m b e r d o e s not automatically indicate that the date is lunar based. (3) D a t a from outside the text m u s t c o r r o b o r a t e the o p e r a t i o n of a lunarb a s e d system. In addition, one m a y b e dealing w i t h a c o s m o l o g i c a l text in w h i c h a lunar set-up c a n b e a s s u m e d . T h e r e are m y r i a d challenges involved in sighting the m o o n . It has b e e n p r o v e n that h u m a n n a k e d eye sighting of the first lunar crescent (in the west) is usually i n a c c u r a t e . " M o r e often t h a n not, p e o p l e will think they see w h a t they really o n l y anticipate. E v e n m o r e so than climatic situations such as cloud cover, dust, light o n the horizon, or forests a n d buildings b l o c k i n g the view, the innate h a n d i c a p is the h u m a n eye (or mind).'^ Extinction effects also raise serious p r o b l e m s b e c a u s e the n e w E g y p t i a n m o n t h b e g a n w h e n the w a n i n g crescent c o u l d n o t b e s e e n (in the east). F u r t h e r m o r e , the m a t c h i n g of any one set of E g y p t i a n lunar dates h a s a l o w statistical significance, as R o n a l d W e l l s has p o i n t o u t . ' ' M u l t i p l e m a t c h e s are possible in e v e r y century. B u t there are other usefiil p a r a m e t e r s w h i c h n e e d to b e c o n s i d e r e d ; n a m e l y , the t w o effects of first and last visibility c o m b i n e d w i t h the o b v i o u s third, the interval b e t w e e n them.''* A c c o r d i n g to W e l l s , the n e t effect of h u m a n i n a c c u r a c y - not uncertainty - " w o u l d b e to s c r a m b l e the s e q u e n c e s of 2 9 - 3 0 d a y intervals in a n y g i v e n set of Egyptian d o c u m e n t s " . ' ^ F o r g u i d a n c e w e m a y consult t w o worthwhile detailed sources. T h e lunar m o n t h s e q u e n c e s a s s e m b l e d b y Otto N e u g e b a u e r from extant lunar a n d p l a n e t a r y e p h e m e r e d e s p r o j e c t e d that the m o o n w o u l d b e visible o n twenty-eight nights in eighty p e r cent o f the m o n t h s tabulated a n d either t w e n t y - s e v e n or twenty-nine nights the rest of the time.'^ Unfortunately, these texts are " m e r e l y calculated p r e d i c t i o n s " a n d there is n o w a y of determining their relationship to reality.'^

'°

See Parker's key reference in PARKER (1950), p. 71 note 30.

''

SCHAEFER and DOGGETT ( 1994).

'^ The problems conceming the inaccuracy of naked eye sighting as being psychologicalphysical are usually ignored. '^ See his comments in the review WELLS (2001 ) in press of ROSE ( 1999). See now Depuydt's study on P. Carlsberg IX (DEPUYDT (1998)). But the appropriate extinction effects need to be taken into consideration: see the review of Wells cited in the previous note. WELLS (2001).

'^ NEUGEBAUER and PARKER (1960), p. 80 note to lines 24-25. The reference is to NEUGEBAUER (1955), pp. 140-150. See note 13 above. There are useful comments by Peter J. Huber on the situation of month-lengths on pages 5-8 of HUBER (1982). He points out that "there are reports of unusually early appearances of the moon, for example Tliompson RM A No. 58 mentions an

382

A. Spalinger

B r a d l e y S c h a e f e r ' s analysis s p e a k s t o the point.'* A l t h o u g h a n a v e r a g e s y n o d i c limar m o n t h is 2 9 . 5 3 d a y s , " t h e actual time p e r i o d b e t w e e n n e w m o o n a n d n e w m o o n varies from ca 2 9 . 2 6 7 9 to 2 9 . 8 3 7 6 d a y s " . " A n d despite the tables from S e l e u c i d B a b y l o n i a m a r s h a l e d b y N e u g e b a u e r , m o d e m sighting u n d e r v e r y r i g o r o u s scientific c o n d i t i o n s h a s c o m e to the c o n c l u s i o n that " t h e length o f t h e o b s e r v e d m o n t h is always 2 9 or 3 0 days long a n d is n e v e r 2 8 or 31 days l o n g " , a p o s i t i o n w i t h w h i c h the A t h e n i a n s w o u l d h a v e concurred. W e c a n dismiss the possibility that E g y p t i a n o b s e r v e r s w a t c h i n g o n cloudless nights or m o m i n g s w o u l d e v e r h a v e arrived at the a b e r r a n t interval o f twenty-eight o r thirty-one. Schaefer assures u s that r e p e a t e d a n d detailed m o d e m sightings r e a c h e d a n " a l m o s t perfect d i v i s i o n b e t w e e n m o n t h s o f 2 9 a n d 3 0 days".^' F r o m 1 8 6 0 t o 1880 Julius S c h m i d t , the director of the o b s e r v a t o r y at A t h e n s , k e p t accurate r e c o r d s o f the visibility o f the lunar crescent for the h i s t o r i a n A u g u s t M o m m s e n w h o w a s studying the b e g i n n i n g o f the d a y in ancient A t h e n s , a city w h i c h a d h e r e d to a n e v e n i n g epoch.^^ In 1910 J. K. F o t h e r i n g h a m a p p l i e d this material to a useful a l g o r i t h m for a r o u g h a p p r o x i m a t i o n o f visibility conditions.^^ F r o m the o b s e r v a t i o n s F o t h e r i n g h a m c o n c l u d e d that w h e n there w a s a clear sky, "the p r o b l e m is a l m o s t p u r e l y astronomical, n o t atmospheric".^'' H i s r e s e a r c h r e p r e s e n t s s e p a r a t e data a n d analyses than Schaefer a n d D o g g e t t ' s results, w h i c h indicated that the h u m a n e y e is not t m s t w o r t h y . " F o t h e r i n g h a m d i d n o t a r g u e for " a c t u a l " visibility as d e t e r m i n e d b y a viewer. Schaefer a n d D o g g e t t e x a m i n e d the latter event, partly p s y c h o l o g i c a l , in detail, a n d their results c a n n o t b e faulted. T h e i r significant c o n c l u s i o n w a s that fifteen in o n e h u n d r e d o b s e r v e r s l o o k i n g for the first crescent o f the m o o n w o u l d see o n e before it w a s p r e s e n t . Surely the ancient data - a n d n o t m e r e l y the E g y p t i a n likewise c o n t a i n a bias o f fifteen p e r cent in w h i c h the dates are t o o early.^* T h e A t h e n i a n s set the first d a y o f the lunar m o n t h u p o n the visibility o f the first lunar crescent.^'' T h u s the " n e w m o o n d a y " for t h e m w a s the first d a y o f t h e m o n t h , n o t the s e c o n d as in ancient Egypt. M o r e o v e r , the twenty-ninth d a y w a s o m i t t e d during a h o l l o w m o n t h " b e c a u s e the lunar o b s e r v a t i o n w h i c h d e t e r m i n e d t h e l e n g t h of file m o n t h w a s m a d e o n the 28th day".^* T h i s p o i n t n e e d s further amplification. appearance on the 28th day". SCHAEFER (1992).

SCHAEFER(1992),p. 32.

The quote is from SCHAEFER (1992), p. 35. For the Athenian evidence, PRITCHETT, (1959). I used his work for the chapter "Under the Moons of Earth" in SPALINGER (1992). SCHAEFER (1992), p. 35.

MOMMSEN (1883), pp. 69-80. FOTHERINGHAM (1910). Schaefer knew this study. FOTHERINGHAM ( 1910), p. 530. "

SCHAEFER and DOGGETT ( 1994).

This statistic derives from over 2500 modem observers making over 1500 observations and is not something based on a few cases. "Under the Moon of Earth", in SPALINGER ( 1992). PRITCHETT (1959), p. 154.

Egyptian Festival Dating and the Moon

383

G i v e n that lunar m o n t h s a r e either twenty-nine o r thirty days long, w h i c h d a y is to b e o m i t t e d w h e n a h o l l o w m o n t h o c c u r s ? W . K e n d r i c k P r i t c h e t t ' s p a i n s t a k i n g analysis,

begun

in c o o p e r a t i o n

with

Otto

Neugebauer,

taught

us to

avoid

a s s u m p t i o n s a n d go t o t h e ancient sources. M a n y Hellenists h a d p r e s u m e d that a reference t o " d a y thirty" automatically implied a fiill m o n t h o f thirty days. S u c h w a s n o t t h e case. Pritchett d e m o n s t r a t e d that if the o l d crescent c o u l d still b e s e e n "just before s u n r i s e " o n d a y twenty-eight, then that e v e n i n g - which was, o f c o u r s e , t h e b e g i n n i n g o f the following d a y - w o u l d b e called " t w e n t y - n i n e " ; if not, b y e v e n i n g " d a y t h i r t y " w o u l d b e recognized. T h e A t h e n i a n s y s t e m w a s partly artificial a n d partly empirical. It d e p e n d e d u p o n n o a s t r o n o m i c a l instrument b u t simple observation. Y e t it also relied u p o n a n a d h o c s y s t e m that c o u l d h a v e o p e r a t e d with o r without a h o l l o w m o n t h . Pritchett eventually d i s c o v e r e d that until t h e e n d o f the fourth century B . C . at A t h e n s t h e final d e c a d e o f a lunar m o n t h w a s n a m e d " a c c o r d i n g to the w a n i n g moon".^^

II. Background to the Egyptian Lunar Days T h i s b a c k w a r d m o v e m e n t w a s in total o p p o s i t i o n t o t h e E g y p t i a n system. T h e t w o calendrical notations contrast as befits their different e p o c h s . T h e E g y p t i a n first d a y , psdntyw, is t h e " n e w o n e " , w h i c h is w h a t t h e n a m e indicates, with t h e s e c o n d , {tp) ibd, d e n o t i n g t h e " b e g i n n i n g o f t h e n e w crescent", a n d t h e third, mspr, b e i n g , finally, t h e "arrivaf.^^ P a r k e r ' s early studies o n t h e b e g i n n i n g o f the E g y p t i a n lunar m o n t h p r o d u c e d conclusive findings. T h e absence o f t h e first crescent w a s n o t e d i n t h e e a s t e m h o r i z o n ; this signified t h e first lunar day. W i t h the p r e s e n c e o f the lunar c r e s c e n t (in the west) t h e s e c o n d d a y w a s reached, a n d t h e m o o n itself w a s said to arrive o n t h e following d a y . A t t h e e n d o f the cycle, t h e twenty-eighth d a y - t h e c m c i a l o n e for the A t h e n i a n s - w a s called "the d a y o f the j u b i l e e o f N u t " , t h e s k y g o d d e s s r e v e r e d in t h e O l d K i n g d o m a s the m o t h e r o f the reigning king a n d the s p o u s e o f the sun g o d Re.^' S h e r e p r e s e n t e d h e a v e n , a n d the d a y designation refers to t h e c o m p l e t i o n o f the c o r e lunar cycle. ( A t this point, w e m u s t r e m e m b e r , t h e situation o f h o l l o w o r c o m p l e t e m o n t h s a r o s e for t h e Athenians.) T h e g o d K h n u m also personified t h e s a m e day. B r u g s c h called it a " d a y o f u n i o n " , b y w h i c h I a s s u m e that h e i m p l i e d t h e r e t u m o f t h e m o o n t o its original starting p o i n t o r else t h e c o m p l e t i o n o f a lunar c y c l e . T h i s d a y is j u s t o n e less than t h e last in a h o l l o w m o n t h , j u s t as t h e d a y o f the crescent {ibd) is o n e r e m o v e d fi-om t h e first, the " n e w " day, psdntyw. Pritchett figured that " i t is p o s s i b l e to h a v e three 2 9 - d a y m o n t h s in a r o w a n d three, four, o r e v e n five 3 0 - d a y m o n t h s . S u c h s e q u e n c e s a r e attested in historical c a l e n d a r s " . " G r a n t e d that h i s k e y reference is actually a table o f calculations b a s e d

PRITCHETT (1959), p. 52.

PARKER(1950a), pp. 11-13. '' Two astronomical studies of Wells may be cited: WELLS (1992) and WELLS (1995). See now Krauss' detailed analysis in KRAUSS (1997). BRUGSCH (1864), p. 57. The study is, of course, very outdated, but the connection to the Egyptian verb hnm ("to unite") is obvious. PRITCHETT ( 1959), pp. 151 - 1 5 2 .

384

A. Spalinger

o n the r e s e a r c h o f F o t h e r i n g h a m a n d others, this statement r e a s o n a b l y expresses the n o t u n c o m m o n situation w h e r e i n a r o w o f h o l l o w m o n t h s c a n occur.^'* F o r o u r p u r p o s e s , h o w e v e r , there is a neat coimection b e t w e e n the E g y p t i a n c o n c e p t for the thirtieth d a y a n d the G r e e k o n e for the last day. E v e n t h o u g h the G r e e k t e r m évr\ KQI véa w a s falsely attributed t o Solon, its m e a n i n g is straightforward: " t h e o l d a n d n e w ( d a y ) " . T h i s t e r m w a s applied t o the final d a y o f either a h o l l o w o r a fiali month. T h e last d a y o f a lunar m o n t h fiilfilled t w o roles, s o m e w h a t like t h e g o d Janus. It referred to the e n d o f the lunar cycle a n d looked forward t o t h e n e x t o n e . C o m p a r e this aspect with the E g y p t i a n prt Mn, lunar d a y thirty. T h e latter designation indicated the m a s c u l i n e procreative capability. M i n will impregnate to a l l o w the next cycle to occur. T h e m o o n , as t h e texts say, "is c o n c e i v e d " (bkS) o n t h e n e w d a y (psdntyw, d a y o n e ) a n d " b o m " (msi) o n the b e g i n n i n g o f t h e n e w crescent (Sbd, d a y two).^^ I n t h e B i r t h h o u s e at E s n a M i n is depicted for b o t h d a y o n e o f P a c h o n s (I imw) a n d d a y fifteen. H i s face, h o w e v e r , is directed inwards a n d o u t w a r d s , a n d B o n n e t r e m a r k s o n t h e link b e t w e e n the sexual deity a n d the general p h a s e s o f the m o o n . B y d a y thirty w e d o n o t w i t a e s s t h e sexual act b u t the g o d r e c o m m e n c e s t h e e t e m a l cycle. H o w this is c o n n e c t e d to the t e r m for the p r e v i o u s d a y , the twenty-ninth, is unclear b e c a u s e t h e translation o f '^h^ siw, " t h e station o f the g u a r d i a n " , r e m a i n s a m b i g u o u s . " B u t w e k n o w from Pritchett that the E g y p t i a n h a d t o figure o u t then a n d there w h e t h e r their m o n t h w a s over o r w h e t h e r there w o u l d b e a d a y thirty. T h e visual o b s e r v a t i o n w a s n o t b a s e d o n calculations, a n d there w a s a great deal o f uncertainty. C o n s i d e r the order o f lunar-determined days in the temple o f R a m e s s e s III at M e d i n e t Habu.^* T h e y d o n o t c o m m e n c e with psdntyw. In a totally u n e x p e c t e d maimer, a list o f "feasts o f h e a v e n " (repetitive festivals t h r o u g h o u t t h e year) b e g i n s w i t h the twenty-ninth (List 7 ) , t h e effective e n d o f a h o l l o w m o n t h , followed b y d a y thirty, d a y o n e , a n d so forth. H e r e , I believe, w e witness a n archaic configuration in which, o w i n g t o t h e irregular length o f the lunar m o n t h , t h e k e y date is t h e twentyninth. A n d in o r d e r to k n o w whether a h o l l o w o r a fiill m o n t h w a s t o h a p p e n , t h e Egyptians h a d t o m a k e their determination o n d a y twenty-eight. H e n c e , M e d i n e t H a b u starts with the following d a y in its list o f "festivals o f h e a v e n " . T h e festival c a l e n d a r o f R a m e s s e s III is a n excellent b u t n o t stiaightforward source o f material about the annual M i n feast, a n d event that w a s d e t e r m i n e d b y the m o o n . It is a c o m p l i c a t e d task t o sort out the inferences: (1) T h e festival o f the " P r o c e s s i o n o f M i n " is a n ancient o n e , k n o w n fi-om the private feast lists o f the O l d K i n g d o m . W o l f g a n g H e l c k correctly

The reference is to PARKER and DUBBERSTEIN (1942).

Along with others citations of PARKER (1950a), this reference was taken from Brugsch's seminal work BRUGSCH (1864). "

BONNET ( 1952), pp. 465-466.

"

Or

38

read'h'iryl

The key reference is in KITCHEN (1969-1990), Vol. V (hereby abbreviated as KRI V) 130.12.

Egyptian Festival Dating and the Moon

385

identified a fragment of the calendar at the sun t e m p l e of N i u s e r r e as the earliest royal reference to this e v e n t . ' ' (2) T h e s a m e religious celebration will b e found associated with the final d a y o f a fiill lunar month. (3) I n the O l d K i n g d o m feast fists this p r o c e s s i o n c o n c l u d e s the m a i n e n u m e r a t i o n o f the important celebrations. T h e r e m a i n i n g o n e s are either s u m m a r i e s or are rare. Prt Mn a p p e a r s to h a v e b e e n celebrated at a close of a cycle. In List 6 6 at M e d i n e t H a b u w e r e a d "I Smw day 1 1 ; the d a y of the feast of M i n ' s g o i n g forth to the terrace while the n e w m o o n ' s d a y {psdntyw)

is in the moming"."^ In 1950, Parker, k n o w i n g

that the calendar at M e d i n e t H a b u is a c o p y of the R a m e s s e u m , d a t e d this lunarc o n n e c t e d o c c a s i o n to the reign of R a m e s s e s II.'*' P a r k e r accurately c o n c l u d e d , " a n y harvest feast in the ninth m o n t h of the civil y e a r w o u l d b e quite proper"."^ Since the c a l e n d a r ' s annual celebrations (the "festivals of earth", E g y p t i a n hbw nw ti)

are

r e c o r d e d chronologically t h r o u g h the E g y p t i a n civil year from I iht 1 to the end, there should b e n o exception for the ninth m o n t h . If w e list the festivals that M e d i n e t H a b u presents for the ninth m o n t h (I Smw),

a logical order in time is to b e

anticipated:"' List 64:

F e a s t of R e n e n u t e t

List 6 5 :

Feast of Clothing A n u b i s

List 6 6 :

Feast of Prt Mn

List 67

Feast of A m u n

ISmwX I^mwlO I Smw 11 ; psdntyw

in m o m i n g

I Smw,

psdntyw

It ought to b e self-evident that the fourth religious event m u s t take p l a c e after the third, the P r o c e s s i o n o f M i n ; and this is the case. List 6 7 tells us that in the reign o f R a m e s s e s II (a p o i n t reiterated b y P a r k e r in his Calendars) an A m u n feast, regulated

'^ See PARKER (1950a), pp. 47-48; HELCK (1977), pp. 53 and 62. N.B.: his reconstruction of the key fragment mentioning the Min feast is unclear in that no date is given. Most certainly, Helck's supposition of I Snm 11 is false because he based it on the lunar date of the Medinet Habu calendar. Note as well my SPALINGER (1996a), passim, especially Chapter VI. Additional information about the Min feast will be found in KRAUSS (1985), pp. 142-144. "° KRIV 182.4. "'

PARKER (1950a), pp. 39-40; with NIMS (1976).

"^

PARKER (1950a), p. 40.

"' The calendar appears to have been one of the first hieroglyphic texts to be inscribed at Medinet Habu. It is located on the exterior southem wall. However, the calendar stops at I Smw psdntyw (List 67). The original hieratic roll from which this copy was made included religious celebrations until the close of the civil year. Why was this not done here? Space must have been the reason. The calendar proceeds from the rear of the temple to Pylon I in the front. It is self-evident that it was overtly designed to avoid the southern portion of the west wall. (There, one will find fictitious scenes of a Nubian war of Ramesses III.) The available space for the text simply ran out although one wonders why the rear of the pylon could not have been decorated with this text. (Perhaps it was already carved and inscribed with scenes of hunting.) See the basic study NELSON (1934), pp. 25-26; HELCK (1965), p. 136; and now HARING (1997), pp. 54-56.

386

A. Spalinger

b y the m o o n , o c c u r r e d o n lunar d a y one in the ninth m o n t h o f the civil year, o n e lunar - or one civil - d a y after the festival o f the p r o c e s s i o n of M i n . T h i s is implied in the g r a m m a r of the circumstantial iw of the k e y p a s s a g e in List 6 6 . In this detail Siegfried Schott rather than Parker w a s correct."'* D a w n (dwit) is w h e n the first limar day, psdntyw, b e g i n s , a n d this is m o s t certainly not the m o m e n t w h e n prt Mn w a s celebrated, regardless o f whether the latter o c c u r r e d in the evening, at night, or during the early m o m i n g before the next day. T h e c a l e n d a r ' s orderly arrangement supports this interpretation. W e m o v e from the first d a y in the ninth m o n t h to the eleventh day. In the reign o f R a m e s s e s II it m u s t h a v e b e e n o n d a y 11 of I Smw, a n d also very early in the k i n g ' s reign, that the d a y before a psdntyw occurred. N o t e that n o explicit lunar m o n t h is stated. T o the Egyptians in the N e w K i n g d o m it w a s sufficient to r e c o g n i z e the b e g i i m i n g o f a lunar m o n t h ; there w a s n o p u r p o s e in worrying about any cycle of a lunar year. O w i n g to this, festival calendars such as that at M e d i n e t H a b u m e r e l y indicate the lunar day, w h i c h o f course was determined b y the n a k e d eye. List 67 w o u l d h a v e to follow List 6 6 in c h r o n o l o g i c a l order. It d i d n o t matter to the compilers o f such religious calendars w h e t h e r the event was lunar or civil. T h e organization of the material simply d e m a n d e d that, in the civil year, if feast X p r e c e d e d feast Y in time, then X should b e written before Y . List 6 6 w a s written before List 67 b e c a u s e the religious event of List 6 6 o c c u r r e d o n the d a y before that of List 6 7 . T h e r e is a reference to the M i n feast c o n c e a l e d in a h e a d i n g to the a c c o m p a n y i n g reliefs: "I ^mw; in it occurs the feast of Min; it is celebrated on the g o i n g forth o f the Protector o f the M o o n (hw i'^h)".'*^ T o Parker this m e a n t the first d a y o f the lunar m o n t h , but there is n o textual support for such a conclusion."* Y e t w h e n w e find the designation of prt Mn within a (lunar-determined) m o n t h cycle it is always the n a m e for day thirty. T h a t is the key. (1) D a y thirty ( w h e n it existed of course) of the lunar m o n t h is prt

Mn.

(2) T h e M e d i n e t H a b u calendar identifies the annual feast o f Prt Mn with the d a y p r e c e d i n g psdntyw

(List 66). T h e date is in I

Mw.

(3) A t M e d i n e t H a b u there is a second reference to prt Mn in a h e a d i n g over the reliefs of this celebration. T h e date is also in I ^mw. T h e last reference a d d s the u n i q u e wording, "the going forth of the P r o t e c t o r o f the M o o n " , a p h r a s e that c a n n o t automatically b e c o n n e c t e d to the first lunar day. "" ScHOTT (1950), p. 104; see PARKER (1950b) for Parker's comments. One crucial point that was overlooked by Parker and Schott is the lack of the expected phrase, "it is psdntyw which fetched it". This statement is always used in the Medinet Habu calendar — but it appears elsewhere as well — when the day under question was to coincide with the Egyptian lunar day one. See our discussion below. "^ KRI V 201.2. Nonetheless, Parker's conclusion is reasonable. Would not Min, as the father of the moon, be his natural "protector", and would not the key time for such support be on lunar days one and two, those of conception and birth? On the other hand, the short passage does not refer to the day of the new moon. What is specified is the "going forth" of the deity Min, and to the latter is attributed the phrase "Protector of the Moon". "*

PARKER (1950), p.

102.

Egyptian Festival Dating and the Moon

387

Rather, in a religious-mythological sense M i n generates t h e n e w lunar cycle a n d M i n n o w p r o c e e d s . T h e d a y feast is therefore identical in spirit, outlook, a n d e v e n dating, t o the annual o n e . T h e s e facts a r e evident in t h e late D e m o t i c p a p y m s P . C a r l s b e r g I in w h i c h t h e visual aspect o f the lunar cycle is present, albeit in a lapidary fashion."' T h e a c c o u n t r e a d s : " . . . T h e y a r e t h e a p p e a r a n c e s o f the m o o n , those i n w h i c h it m a k e s before other T h e y a r e twenty-eight w h e n they fill t h e circuit". T h i s is t h e basic s t m c t u r e o f the lunar m o n t h . T h e first d a y is eliminated b y definition a s it is the o n e w h e n t h e m o o n is n o t visible. Therefore, this p a p y r u s c o m m e n c e s with lunar d a y two: "The moon of the second day is the feast of Horus" (original italics). A s N e u g e b a u e r - P a r k e r indicated, t h e only w a y t o m a k e sense o f this p a s s a g e is t o take the p o s i t i o n o f the ancient: t h e m o o n a p p e a r s o n d a y t w o . T h e n c o u n t t w e n t y - s e v e n m o r e times t o the twenty-ninth day."* T h i s is the k e m e l o f the lunar m o n t h a c c o r d i n g to t h e E g y p t i a n s . N e u g e b a u e r b u t t i e s s e d this interpretation with h i s edition o f the lunar e p h e m e r i d e s from B a b y l o n i a in w h i c h "the m o o n w a s visible 2 8 nights in 8 0 p e r cent o f the m o n t h s tabulated. I n t h e other 2 0 p e r cent it w a s visible 2 7 o r 2 9 nights"."' T h e n h e cross-referenced t h e twenty-nine d a y s t a n d a r d t o other coeval sources. Let u s e x a m i n e a few lunar dates in festivals with five cautious p a r a m e t e r s : (1) It is u n c l e a r if a lunar-civil e q u a t i o n h a s a h i g h d e g r e e o f certainty. Schaefer found about 21.5 p e r cent o f n a k e d eye sightings t o b e false. (2) N e w K i n g d o m Egyptians d i d n o t predict t h e first d a y o f a lunar m o n t h far in a d v a n c e . T h e y waited for it to h a p p e n (3) T h e y m u s t h a v e h a d a r o u g h system, like t h e A t h e n i a n s did, i n o r d e r t o c o n c l u d e w h e t h e r a lunar m o n t h w a s to e n d o n d a y twenty-nine o r o n d a y thirty. (4) B u t the latter determination might b e false, a n d not j u s t statistically in a few cases. (5) If a n error o c c u r r e d , h o w could it b e rectified? T h e fifth p o i n t e x p o s e s t h e susceptibility o f o u r relying u p o n lunar dates for chronological p u r p o s e s . T h e m e a s u r e m e n t desired b y t h e E g y p t i a n s w a s n o t that o f m o d e m scientists with their precise instruments o f observations a n d a d v a n c e d m a t h e m a t i c a l formulations. T h e k e y references from ancient civilizations offer limited m e a n s o f achieving statistical accuracy.

III. Thutmose Ill's Declaration T h e b e s t - k n o w n lunar date from t h e N e w K i n g d o m is t o b e found in a civil d o c u m e n t o f T h u t m o s e III.'° It is a buttress o f the edifice o f absolute c h r o n o l o g y .

"'

NEUGEBAUER and PARKER ( 1 9 6 0 ) , pp. 7 9 - 8 0 . See now the additional commentary of

Joachim F. QUACK ( 2 0 0 0 ) .

"« Ibid. "' See note 1 6 for references. ^° Urk. IV 8 3 3 - 8 and especially 8 3 5 . 1 7 - 8 3 6 . 2 with the standard interpretation and translation of VON BECKERATH ( 1 9 8 1 ).

388

A. Spalinger

a n d in this light t h e 1970 project of W i l l i a m M u m a n e w a s to e x a m i n e w h e t h e r the so-called " l o w e r d a t e " o f T h u t m o s e III w a s accurate.^' His s k e p t i c i s m o f t h e r e l i a n c e u p o n c h r o n o l o g i c a l matters w a s highly salutary, a n d his reliance u p o n t h e t w e n t y five year cycle of P . C a r l s b e r g I X useful as a s t e p p i n g - s t o n e . " L a t e r p h i l o l o g i c a l w o r k o n this accoimt b y E d w a r d W e n t e w a s e v e n m o r e v a l u a b l e in r e c o n s t r u c t i n g the historical b a c k g r o u n d t o t h e event.^^ B u t w h a t P . C a r l s b e r g I X reveals - dealing only with P a r k e r ' s r e c o n s t r u c t i o n at this point - is the following c o i n c i d e n c e s of civil d a y X = lunar d a y X : ' " M o n t h s with 29 or 30 D a y s W h i c h E q u a l L u n a r M o n t h s (1) Y e a r 1

- 1 ( 3 0 days), 2 (?; 2 9 or 3 0 d a y s ) , 3 ( ? ; 2 9 days)

(2) Y e a r 9

- 8 ( ? ; 29 or 3 0 days), 9 ( ? ; 2 9 d a y s ) ;

(3) Y e a r 12 - 3 ( 3 0 days); 4 (?; 2 9 or 3 0 d a y s ) ; 5 (?; 2 9 d a y s ) ; (4) Y e a r 2 0 - 12 ( 3 0 d a y s ) ; intercalary ( 3 0 d a y s ) ; (5) Y e a r 2 3 - 5 ( 3 0 days); 6 (?; 2 9 o r 3 0 d a y s ) ; 7 ( 2 9 d a y s ) ; T h e c o i n c i d e n c e s are rare in the s c h e m a t i z e d lunar-civil t e m p l a t e . M o r e o v e r , as P a r k e r r e c o g n i z e d , the configuration of the entire series of dates into c o l u m n format is n o t entirely satisfying. (A recent discussion of the p a p y r u s b y D e p u y d t is not essential for this study.'^) B u t for the sake of a p r e l i m i n a r y analysis n o t e that w e h a v e at m o s t 13/309 c a s e s w h e r e i n the first lunar d a y = the first civil day; the figure is v e r y l o w . ( T h e n u m b e r 3 0 9 is the total n u m b e r of lunar m o n t h s in the c a l e n d a r . ) T h a t is t o say, there is a p p r o x i m a t e l y a 4.2 p e r cent c h a n c e that psdntyw will b e the s a m e as civil d a y o n e . T h e p e r c e n t a g e will be a p p r o x i m a t e l y the s a m e for lunar d a y s t w o to twenty-nine; i.e. lunar X = civil X. T h u s for P a r k e r ' s r e c o n s t m c t i o n o f P . C a r l s b e r g I X , there is a n e x t r e m e l y small certainty that a civil d a y will b e the s a m e as a lunar day. T h i s statistically l o w frequency carmot b e o v e r l o o k e d since the case of T h u t m o s e III is a direct e q u a t i o n of d a y o n e civil with d a y o n e lunar:

MURNANE( 1970/71).

"

MURNANE(1970/71),p. 7.

" WENTE (1975). The recent study of Jean-Marie KRUCHTEN (1999) has cast doubt upon Wente's interpretation of the initial verbal form as emphatic (i.e., a non-predicative sdm.f). 1 am not convinced that all of Kruchten's examples have been interpreted correctly. On the other hand, whether we understand the opening verb as an emphatic or not is irrelevant. The timing, however, is important: siw indicates only that one was "waiting" for the first lunar day. It does not say when it was. PARKER (1950a), Chapter II and page 25 in particular. The later interpretation of Edouard Samuel is in SAMUEL (1962) Chapter II. He altered the dates of P. Carlsberg IX in order to establish a working artificial calendar for the Macedonians in Egypt, Additional comments will be found in KRAUSS (1985), pp. 27-30; and WELLS (1986). Add now the reconstruction of Depuydt (DEPUYDT (1998)). " See SCHAEFER (1992) and SCHAEFER (2000), with SCHAEFER (1993) (the complete theoretical paper); SCHAEFER and DOGGETT (1994) (most important for observations); SCHAEFER (1996); and FERMOR(1993) cited in note 3 above.

Egyptian Festival Dating and the Moon

389

M y majesty c o m m a n d e d the p r e p a r a t i o n of stretching the c o r d while waiting for the d a y o f the n e w m o o n (psdntyw)

in order to stretch the

c o r d a r o u n d this m o n u m e n t in regnal year 24 II prt 3 0 , the d a y of the festival, (narhely) the tenth d a y of A m u n in K a m a k . T h e p a s s a g e d o e s not m e r e l y indicate a civil-lunar equation; it also sets the event in a ten-day festival cycle for A m u n in T h e b e s : the last d a y of that lengthy rite. Specifically, in his twenty-forth regnal year T h u t m o s e III sat before his courtiers a n d b e g a n to establish his festival hall to the east of the m a i n t e m p l e in Kamak.^^ T h e w a r at M e g i d d o was already over at this time; a n d the king was at T h e b e s as the text states. O n the last d a y o f the s e c o n d m o n t h of the s e c o n d s e a s o n T h u t m o s e III c o m m a n d e d the usual stretching of the cord c e r e m o n y at East K a m a k . T h e timing w a s deliberate: half o f the civil year h a d p a s s e d , a n d auspiciously, civil d a y thirty w o u l d c o r r e s p o n d to day ten of the celebration for A m u n . T h i s w a s not a c o i n c i d e n c e , b e c a u s e the inscription explicitly states that the m a r v e l s w h i c h c a m e to p a s s at that time o c c u r r e d on "the d a y of the festival, (namely) the tenth d a y of A m u n in K a m a k " . " Hrw m-10 is in apposition to hrw hb. F o r the a d h e r e n t s of absolute c h r o n o l o g y this association of festival a n d date is not important. T h e q u o t e d p a s s a g e is what matters, a n d it indicates that the king w a s "waiting for the d a y o f psdntyw" to c o m m e n c e the first p h a s e o f building. I a m therefore in partial a g r e e m e n t with W e n t e ' s analysis.^* H e c l a i m e d that o n d a y thirty of the sixth m o n t h the king was waiting for the first d a y o f the n e x t lunar cycle. B u t w h e n was psdntyw? S o m e h a v e maintained that it m u s t h a v e o c c u r r e d on the following day; i.e., that psdntyw coincided with III prt 1. If this is true, then the civil first d a y c o i n c i d e d with the lunar first day. Helck, o n the other hand, was rather dubious of the entire lunar schema.^' H i s inclination w a s always to w o r k directly with the historical data instead of m a t h e m a t i c a l calculations. In a series o f small yet p r o v o c a t i v e studies in Gottinger Miszellen, H e l c k applied his p h i l o s o p h y to the T h u t m o s e III decree. H e w a s correct to rest his case o n the key w o r d , siw, "to wait (for)", as it is a m b i g u o u s with respect to exact dating. A c c o r d i n g to Helck, the c m x of the matter is that n o indication is given as to w h e n the "blessed e v e n t " is to occur. In the m i d d l e o f the S e c o n d M i l l e n n i u m B . C . there were n o written charts o f lunar-civil equivalences. H e n c e , he t o o k the s k e p t i c ' s w a y out and did not c o m m i t himself with r e g a r d to the a c m a l o c c u r r e n c e of T h u t m o s e I l l ' s psdntyw. M o s t certainly, the c m c i a l w o r d siw m a k e s it clear that P a r k e r ' s earlier reconstmcfion, against w h i c h W e n t e disagreed, is inaccurate.^" B u t p e r h a p s it is All of this has been known since the study of Sir Alan Gardiner (GARDINER (1952)). There is a useful summary in BJÔRKMAN (1971), pp. 84-90. "

Urk. IV 836.3. WENTE (1975).

HELCK (1983), pp. 4 0 - 4 2 . ^° Parker's classic study, in which he argued for 1490 B.C. as the accession of Thutmose III, is PARKER (1957a). But see now Kruchten's consideration of Dynasty XVIII verbal formations (in this case the sdm.f), KRUCHTEN (1999). If he is correct, then Wente's analysis is not.

390

A. Spalinger

n e c e s s a r y to e x a m i n e the n a t m e of E g y p t i a n feasts a n d d e c a d e s . T h i s feast, o n e o f the m a n y associated w i t h A m u n , occurred o n the last d a y in a d e c a d e : the "tenth d a y " of the m a n y associated with A m u n . Therefore, the cycle o f this A m u n celebration m u s t h a v e b e g u n o n day twenty-one o f the sixth m o n t h a n d c o n c l u d e d o n the final d a y of that m o n t h , exactly in the m i d d l e o f the year (ignoring the e p a g o m e n a l s ) . T h e timing was precise as it w a s auspicious. B u t w a s d a y thirty of II prt the e n d of a lunar m o n t h ? This is the q u e s t i o n that I b e l i e v e to b e the m o s t important. In 1994 I b r o u g h t u p the A t h e n i a n system o f determining w h e n a n e w lunar m o n t h b e g a n , a n d in m y c o m p a r i s o n with the E g y p t i a n s y s t e m with its m o m i n g e p o c h , I p r e s e n t e d s o m e possible solutions.*' In the case imder c o n s i d e r a t i o n II prt 3 0 m u s t h a v e b e e n k n o w n to b e the last day o f a lunar m o n t h . T h e r e f o r e , that date h a d to h a v e b e e n either d a y twenty-nine or d a y thirty, a n d in the case of the latter, there is a p p r o x i m a t e l y a three p e r cent change of c o i n c i d e n c e , if w e follow the data fi-om P. C a r l s b e r g IX; i.e., lunar d a y tiiirty = civil d a y thirty. C a n w e b e sure that it w a s lunar d a y thirty? O r should b e leave the question open, especially as the inscription m e r e l y indicates that one w a s " w a i t i n g " for lunar d a y o n e ?

IV. other Amun Feasts and Lunar Events A virtually identical situation involving the s a m e dates a p p e a r s in a late h i e r o g l y p h i c text. In t e m p l e T at K a w a , K i n g Anlamaimi erected a g r a n d i l o q u e n t stela r e c o u n t i n g the early years o f his r e i g n . " W h e n h e tiaveled firom M e r o e to G e m p a a t e n in o r d e r to m e e t w i t h A m u n in that city, A n l a m a n n i d e p a r t e d in the sixth m o n t h o f the civil year. H e arrived at G e m p a a t e n o n II prt 29. A s with the T h u t m o s e III e x a m p l e , these dates w e r e c h o s e n for their religious cormotations, t h e r e b y e n h a n c i n g o n e o f the great feasts of the land, as the king caused the deity to a p p e a r at the first festival of Amun." M a c a d a m ' s interpretation o f the k e y lunar event is n o longer accepted.*" O n c e m o r e the celebration t o o k p l a c e on the first d a y o f the lunar m o n t h . T h e inscription states: " . . . the first feast of A m u n ; it is psdntyw w h i c h fetched it (in psdntyw in swy\ A n l a m a n n i ' s stela d e m o n s t i a t e s that far south in the S u d a n , a n d m a n y centuries later than T h u t m o s e III, Meroitic kings still followed the s a m e millenniaold lunar practices of the Egyptians. A n d with the r e a d i n g in psdntyw in sw securely established, w e c a n lay e m p h a s i s u p o n the timing: w e h a v e o n c e m o r e r e a c h e d the

*'

"Under the Moon of Earth", in SPALINGER (1992).

*^ MACADAM (1949), pp. 4 4 - 5 0 with Plates 15-16. KORMYSCHEVA (1998) follows this position. *^ MACADAM (1949), p. 46 and note 25 page 48. See my SPALINGER (1998), pp. 56-57 for the correction. In that study I preferred the word "fetches" instead of "fetched" for in, even though the Participle is Perfective. I did this because in the festival calendars it is assumed that the event will occur more than once over a period of years. I now feel that the exact civil date (month plus day) indicates "fetched". In a subsequent year psdntyw would have fallen at a different time in the civil calendar, but the entry indicates that it was recorded for the same month. *" MACADAM ( 1949), pp. 4 8 - 4 9 note 26.

Egyptian Festival Dating and the Moon

391

m i d d l e p o i n t o f the civil year. I n this case, h o w e v e r , the festival c o n t i n u e d o n from lunar d a y o n e a n d e n c o m p a s s e d seven days. A t M e d i n e t H a b u , the feast o f "Lifting u p the S k y " {"h pt) o c c u r r e d at the s a m e time.*^ Before p r o c e e d i n g , let u s note that a m i n o r error o f notation h a s slipped into the festival calendar. List 6 0 places the first d a y of the festival o n II prt 9; b u t it is clear that the date m u s t b e e m e n d e d to 2 9 , a n o t i m p r o b a b l e situation with r e s p e c t to the n u m e r o u s slips of n u m b e r s at M e d i n e t Habu.^* ( T o take the m o s t blatant case, the festival e v e oîWigy a n d t h e d a y o f its o c c u r r e n c e are off b y o n e a p i e c e ; Lists 2 4 a n d 2 5 s h o u l d b e c o r r e c t e d from I iht 16 a n d I iht \9 Xo I iht 17 a n d I iht 18 respectively.)^' It is, h o w e v e r , the final t w o notations that clinch the argument: II prt 3 0 refers to the s e c o n d d a y o f the celebration, a n d the g o d " e n t e r s " o n III prt 1.^* A n identical case is p r e s e n t e d immediately after (List 6 1 : feast o f " E n t e r i n g the S k y " ) , w h e r e A m u n " e n t e r s " {''q) o n d a y twenty-nine of III iht. T h e thirtieth d a y o f the m o n t h is the s e c o n d d a y o f the celebration, a n d o n I V prt 1 the g o d " ( r e ) e n t e r s " {s'^q).^^ T h e only r e a s o n a b l e conclusion is that the i m p o r t a n t event o f "Lifting u p the Sky", w h i c h w a s a n A m u n feast, t o o k p l a c e at the e n d o f the first half o f the year. T h u s it is parallel to T h u t i n o s e I l l ' s ten d a y A m u n feast referred t o o n h i s stela of year twenty-four. F r o m the r e c o r d s o f T h u t m o s e III w e l e a m o f three feasts for A m u n established after the king r e t u m e d to E g y p t from his M e g i d d o c a m p a i g n in h i s twenty-third regnal year.'° After arriving in T h e b e s , h e p r o c e e d e d to institute three "feasts of v i c t o r y " w h i c h c a n n o t b e dated with certainty. (1) U n k n o w n date; this is called "the first feast o f A m u n " a n d w a s o f five d a y ' s duration; (2) C o n n e c t e d to the feast o f s^q ntr, w h i c h is called the s e c o n d feast o f A m u n ; this w a s also five days long; (3) C o n n e c t e d to the fifth feast o f A m u n a n d celebrated in the k i n g ' s m o r t u a r y t e m p l e ; this w a s five days long as well. The

first

of this

friad

is the easiest to analyze. In the list o f local feasts at

Elephantine, T h u t m o s e III m e n t i o n e d that it logically o c c u r r e d o n N e w Y e a r ' s D a y (wp rnpt) a n d lasted three d a y s . " E v i d e n c e from a later p e r i o d b r o u g h t forward b y G o y o n indicates that the o p e n i n g days

"

after I iht 1 in w h i c h major

religious

Conveniently, see KRI V 178.14 (List 60).

This fact is too blatant to be discussed here. Suffice it to say, the figures of quadruple hekats and their four-fold multiple (the old "sack") are often confused; other numbers are likewise incorrect. See HELCK (1963) for a relatively complete discussion. SCHOTT (1950), p. 96 was fully aware of the mistake. KRI V 144.16 (List 24) and 145.12 (List 25); see KRI V 144 note 16a: "'16' error for '17'. But is '17' nonetheless possible? I.e., read day 10 + [ 4 ] + 3". KRI V 179.9-20. KRI V 179.12-14. For the phrase s^q ntr, see PARKER, LECLANT, and GOYON (1979),

p. 58 note 41. The expression refers to the retum of a divine procession. '^^ The great study on this subject was BREASTED (1899). It was updated in BREASTED (1906), pp. 176-178. "

Urk. I V 824.8.

392

A. Spalinger

celebrations t o o k p l a c e could a m o u n t to five.'^ A s ancillary information, the B a n i s h m e n t Stela o f M e n k h e p e r r e (Louvre C 2 5 6 ) also indicates a n A m u n celebration on the first d a y o f the year.'^ B r e a s t e d coimected the second event to that o b s e r v e d b y king Piye ( P i a n k h y ) at T h e b e s . ' " O n his lengthy stela the Kushite m o n a r c h refers to the O p e t festival: " A n d he sent m e in p e a c e to see A m u n in his beaufiful feast o f O p e t " . ' ' P i y e gives the date of III iht 2 as the e n d o f the festival, the event o f the (re)entrance. It c a n b e a r g u e d that in the time o f T h u t m o s e III the second A m u n feast coincided with that o f O p e t , w h i c h b e g a n o n d a y fifteen, a fiirther auspicious time from a lunar-oriented viewpoint. Later, the c o m m e n c e m e n t w a s m o v e d to d a y nineteen o f the s e c o n d civil m o n t h , a date that w a s also c o n n e c t e d with the m o o n . ' * This d a y is also k n o w n fi-om the reign o f A m u n h o t e p I I I . " Certainly b y the r e i g n o f R a m e s s e s II, a n d v e r y early at that, the O p e t festival lasted from II iht 19 to III iht 12.'* D a t a from M e d i n e t H a b u link u p with tiie X V I I I t h D y n a s t y sources. O p e t , w h i c h is the o b v i o u s case, c o m m e n c e s with List 2 9 . Lists 6 0 - 6 1 are related t h r e e - d a y feasts c o n n e c t e d to A m u n : the first is the "Lifting u p the S k y " a n d the second, " E n t e r i n g the Sky". T h e y are also parallel to A m u n ' s first feast, as r e c o r d e d in T h u t m o s e I l l ' s Elephantine calendar. H o w e v e r , neither seems to b e associated w i t h his "festivals of v i c t o r y " if only b e c a u s e the latter lasted five days a n d the third t o o k p l a c e o n l y in the k i n g ' s m o r t u a r y t e m p l e . A recent translation o f List 6 7 records: "First m o n t h o f s u m m e r , the n e w m o o n ' s festival o f A m u n - R e , in his first festival of the first m o n t h o f s u m m e r , w h e n this g o d g o e s out o n the fourth occasion of the n e w m o o n ' s festival".'^ T h e p a s s a g e is highly c o n d e n s e d but unfortunately so, b e c a u s e it r e m a i n s a m b i v a l e n t h o w m a n y first lunar d a y s there were. It is easier to surmise that w e h a v e r e a c h e d d a y four o f the A m u n feast a n d that it coincided with the d a y of the n e w m o o n in I Mw. M o r e o v e r , the question is o n c e m o r e r e o p e n e d whether this celebration, as the others for A m u n set u p b y T h u t m o s e III, also lasted five days? A l t h o u g h w e c a n eliminate this event as the third of T h u t m o s e I l l ' s victory feasts b e c a u s e the latter w a s observed o n the west o f T h e b e s , the p r o b l e m o f ranking arises. T h e first celebration at M e d i n e t H a b u set u p o n a psdntyw w a s the V a l l e y Feast. List 3 is clear: "It is psdntyw w h i c h fetched it".*° T h e s e c o n d o n e , in List 6 3 , '^

GOYON ( 1 9 7 2 ) , pp. 4 1 - 4 6 .

'^

VON BECKERATH ( 1 9 6 8 ) , p. 1 2 (line 9 of the text).

'"

BREASTED ( 1 8 9 9 ) , pp. 1 2 5 - 1 2 6 .

'^

See now GRIMAL ( 1 9 8 1 ) , pp. 4 4 - 4 5 , note 1 1 2 .

'* Gardiner's comments are most apropos (GARDINER ( 1 9 5 2 ) , p. 2 2 with note 3 ) . Day nineteen is the reverse of the eleventh day; both indicate the lunar-solar epact of 3 6 5 - 3 5 4 . In modular terms 1 9 = - 1 1 (mod 3 0 ) . " Krauss correctly observes that a reference on one of the wine jugs delivered to the Malqata palace of Amunhotep III associated with the feast refers to "day nineteen" (KRAUSS ( 1 9 8 5 ) , p. 1 5 2 , note 2 ) . Hence, we cannot be sure that this date refers to the first day of the feast or not. The switch from day fifteen to day nineteen may very well suit a post-Amama reorganization. (Of course, the eleven days fit neatly into the end of the month.) '*

The evidence being, of course, the Medinet Habu calendar.

'^

EL-SABBAN(2000),P. 129.

KRI V 1 2 3 . 5 . Note that this celebration is separated from the annual ones.

Egyptian Festival Dating and the Moon

393

is unclear as to n a m e b u t the date is set o n IV prt 1 : "It is psdntyw w h i c h fetched it".*' T h e third w e k n o w to b e c o n n e c t e d to the feast o f M i n (List 6 6 ) a n d it neatly follows that event; namely, the " p r o c e s s i o n a l feast of Amun".*^ T h e translation for List 67 is, h o w e v e r , difficult. A s usual, it w a s P a r k e r w h o found a link to a lunar-based event, a Saite P e r i o d oracle dated to the reign of P s a m m e t i c h u s I . " H e translated: "first day of the e m e r g e n c e (prt) o f this g o d for the four days of the N e w M o o n Feast". P a r k e r further c o n c l u d e d that the total n u m b e r of days w o u l d r e a c h five, the last b e i n g the " d a y of the divine (re-)entrance (s'^q) of A m u n " . ( T h e parallel to Lists 60 and 61 is self-evident.) B u t a p u z z l e still r e m a i n s , the civil d a y one of List 63 w h i c h coincides with the first lunar day.*" T h e r e were m o r e A m u n feasts. O n e w a s celebrated at K a m a k in the e v e n i n g (hiwt nt hb) o n the twenty-sixth day of the first month.*^ Another, r e c o r d e d o n the B a n i s h m e n t Stela, sets u p the civil/feast equivalence in this m a n n e r : regnal year 2 3 , III Smw 2 9 , c o r r e s p o n d i n g (hft) to the feast of A m o n r a s o n t e r in his . . . feast o f "*^ Schott refers to a third occurring four days earlier, a l t h o u g h the context is unclear.*' T h e Deir el M e d i n e h material is not very informative o n this matter.** A p p a r e n t l y , m a n y of the celebrations r e c o r d e d there for A m u n took place o n the east b a n k . O d d l y , though, there w a s one case, dated to III Smw 1, a significant date, w h i c h is c o n n e c t e d to Opet; t w o additional references unfortunately are u n d a t e d . B u t in the T h i r d Intermediate P e r i o d the K a m a k priests k e p t a long series of " a n n a l s " , w h i c h h a v e b e e n reedited in e x e m p l a r y fashion b y K m c h t e n . * ' G r a n t e d that w e are in D y n a s t y X X I I (the reign of S h e s h o n k III); nonetheless, a n y reference to an A m u n feast at K a m a k is significant.'" T h e date is I èmw 2 6 a n d the event t o o k p l a c e in the chief t e m p l e . C o m p a r i n g this a n d other references, K m c h t e n wisely c o n c l u d e d that in was in the first m o n t h of the third a n d final s e a s o n that the s o called " i n t r o d u c t i o n " of n e w p r o p h e t s of K a m a k o c c u r r e d . " T h e k e y date of inscription seven (and others) p r o v e s that once m o r e this is the A m u n feast o f List 67 in M e d i n e t H a b u (set on the n e w m o o n d a y ) . A n o t h e r inscription b e l o n g i n g to the

*' KRI V 1 8 0 . 9 . Kitchen thought it might be a feast for Sokar; note his question mark. There is no evidence, however, of any Sokar feast at this time in the civil year. I suspect that the original day is inaccurate. Day one in IV prt does not fit the order of the calendar. *^ KRIV 182.16. *'

PARKER ( 1 9 6 2 ) , p. 8.

*" See note 81 above. *^ Urk IV 7 7 0 . 3 ; SCHOTT ( 1 9 5 0 ) , p. 8 2 . The eleven days of celebration for Amun referred to in the calendar of Urk. IV 1 7 7 must be Opet: see HELCK ( 1 9 6 5 ) , p. 1 3 9 . VON BECKERATH ( 1 9 6 8 ) , p. 9 (line 1 of the text). Von Beckerath set up the correspondence between the celebration for Amun and the Epiphi feast: see note d page 1 4 . *'

SCHOTT ( 1 9 5 0 ) , p

1 1 0 (O. C G C 2 5 7 9 5 ) .

** VAN WALSEM ( 1 9 8 2 ) . I have seen copies of all of the unpublished ostraca referred to in this study. For Amun feasts, see cases 8 1 (III Smw 1: O. DEM 3 5 4 ) , 9 4 (undated), and 9 5 (undated). Van Walsem observed in note 8 8 page 2 3 9 that the first does not fit the Opet feast; he tentatively linked it with case 7 8 (II Émw 2 5 : O. C G C 2 5 5 3 8 ) . *'

KRUCHTEN ( 1 9 8 9 ) .

'°

KRUCHTEN ( 1 9 8 9 ) , pp. 5 9 - 6 3 .

"

KRUCHTEN ( 1 9 8 9 ) , pp. 2 4 2 - 2 4 4 .

394

A. Spalinger

s a m e series of priestly r e c o r d s is set on d a y s e v e n t e e n of the third civil m o n t h . Again, M e d i n e t H a b u ' s detailed calendar is helpful: List 3 9 indicates the feast i m m e d i a t e l y after Opet. T h a t protracted festival h a d c o n c l u d e d o n d a y twelve o f III iht in D y n a s t y X I X , b u t it w a s lengthened b y three days in the reign o f R a m e s s e s III.'^ It is not k n o w n w h a t additional religious p e r f o r m a n c e s w e n t on until d a y fifteen. A s for the A m u n feasts, T h u t m o s e III explicitly states that five w e r e a l r e a d y celebrated at T h e b e s — t h i s is a k e y point — before h e established three victory celebrations.'^ If w e list all t h o s e celebrations c o v e r e d a b o v e , the following is the result: (1) N e w Y e a r s (I ?ht l): three days in D y n a s t y X V I I I ; (2) U n k n o w n : H/if 2 7 ; (3) Opet: in D y n a s t y X V I I I it c o m m e n c e d o n II iht 15; b y the r e i g n of R a m e s s e s II it r a n from II iht 19 to III iht 12, with three m o r e d a y s a d d e d at the e n d b y R a m e s s e s III; (4) After Opet: set o n III ?ht 17; k n o w n as late as S h e s h o n k III; (5) II prt 21 to II prt 3 0 ; ten days; partly coincides with the "Lifting u p of S k y " at M e d i n e t H a b u (II prt 29 to III prt

1); also c o i n c i d e s w i t h the

Meroific e v i d e n c e o f A n l a m a n n i (II prt 2 9 ) ; (6) " E n t e r i n g the Sky": III prt 29 to IV prt 1 ; (7) P r o c e s s i o n a l feast: 1 Smw, (8) U n k n o w n : III W

psdntyw;

29.

N o w w e see that the victory celebrations were tied to major A m u n feasts w h i c h h a d to last at least five days. I believe that this fits (1). B r e a s t e d m a d e the r e a s o n a b l e a r g u m e n t that (3) m a t c h e s the s e c o n d of T h u t m o s e ' s n e w feasts, w h i c h h e limited to five days each.'" O p e t lasted eleven days in the reign o f T h u t m o s e III, a n d c o m m e n c e d on II ^/j? 15; a second inscription of T h u t m o s e III at K a m a k also attests to the original e l e v e n - d a y duration.'^ A t M e d i n e t H a b u the c o n c l u s i o n to that event, the s'^q ntr, o c c u r r e d o n III iht 12 o w i n g to later extensions of duration; during the reign of Piye the date was III iht 2. H e n c e , during T h u t m o s e I l l ' s reign w e c a n hypothesize that the victory celebration b e g a n at the e n d o f O p e t . Y e t another celebration for A m u n c a n b e found in List 56 at M e d i n e t Habu.'* H e r e the date is I prt 6 and even if the significance of the event e l u d e s us, the p r o x i m i t y to the first d a y of the second season o u g h t not to. In fact, this feast of " W r i t i n g the K i n g ' s N a m e o n the Ished T r e e " conveniently o c c u r r e d a m e r e five d a y s after that o f N e h e b k a u , often called b y Egyptologists, the " S e c o n d N e w Y e a r ' s D a y " . I d o not feel that this event can b e c o n n e c t e d with those victory rites

'^ The source for this is the Great Harris papyrus. Conveniently, see SCHOTT ( 1 9 5 0 ) , p. 8 5 . '^ HELCK ( 1 9 6 5 ) , p. 1 6 2 refers to an Amun feast celebrated on III imw 2 0 in the first regnal year of Ramesses XI. (The Adoption Papyrus: KRI VI 7 3 5 . 1 0 - 1 2 . ) The situation is unclear, however; i.e., is the reference to a local event at Deir el Medineh or can we connected it ( 8 ) below? '"

BREASTED ( 1 8 9 9 ) , p. 1 2 5 .

'^

Urk. IV 177; see note 8 5 above.

'*

KRI V 1 7 6 . 7 - 6 ; see now HARING ( 1 9 9 7 ) , pp. 9 6 - 9 7 .

Egyptian Festival Dating and the Moon

395

established b y T h u t m o s e III; nonetheless, the auspicious nature o f the date c a n n o t b e left unnoticed. T h e late text n o w called t h e B e n t r e s h Stela presents t w o final usefiil references. T h e first involves a n error. I n lines 6-7 O p e t is n a m e d y e t II is t h e m o n t h . O b v i o u s l y , iht is m e a n t . ' ' T h e s e c o n d reference is t o a celebration for A m u n in I ^mw. This c a n b e e q u a t e d with (7) above.'* In s u m , these A m u n feasts a p p e a r t o h a v e h a d lunar coimections. T h e e l e v e n d a y s o f Opet, for e x a m p l e , i m m e d i a t e l y p o i n t t o t h e lunar-solar epact ( 3 5 4 versus 3 6 5 days). T h e later m o v e from d a y fifteen in II iht t o d a y nineteen c a n b e also m e n t i o n e d . Finally, s o m e o f t h e m w e r e related to psdntyw, p e r h a p s a final indication o f a possible original lunar orientation.

V. Hatshepsut and Thutmose III A n equally i m p o r t a n t religious event j u x t a p o s i n g a n existing A m u n feast w i t h a n e w o n e w a s set at the close o f the first half o f the y e a r . " H a t s h e p s u t ' s p r o c l a m a t i o n of h e r eventual k i n g s h i p could n o t h a v e b e e n timed m o r e auspiciously t h a n at t h e e n d o f m o n t h six: R e g n a l y e a r t w o , s e c o n d m o n t h of prt, d a y twenty-nine, third

festival

o f A m u n c o r r e s p o n d i n g to {hft) t h e Litanies o f Sekhmet, t h e s e c o n d day. Everything that h a s b e e n discussed s o far is reflected in this p a s s a g e . First, it is d a y twenty-nine, almost the e n d o f the half-year. Second, n o t e o n c e m o r e the u s e o f hft, " c o r r e s p o n d i n g t o " , a c m c i a l p r e p o s i t i o n that is often related t o c h r o n o l o g i c a l situations.'"" T h i r d , t h e inscription associates festivals, S e k h m e t ' s o v e r l a p p i n g with A m u n ' s . Finally, t h e w o r d i n g rnpt sp 2 II prt 2 9 'i-nw n hb Imn is p a r a l l e l e d b y T h u t t n o s e I l l ' s "fifth" A m u n feast: 5-nw n hb}^^ I f this c a n b e linked witii t h e data c o v e r e d i n t h e last section, then t h e third A m u n feast o f T h u t m o s e III might c o r r e s p o n d t o the e n d o f the first half o f the year. T h e H a t s h e p s u t e x a m p l e is o f p r i m e i m p o r t a n c e for insight into h e r desire for kingship e v e n t h o u g h it is fiill o f lacunae. A s a result, w e d o n o t k n o w if t h e event was linked to a lunar day. T h e p r o c l a m a t i o n {sr) to P h a r a o n i c status w a s m a d e at L u x o r ( " S o u t h e m O p e t " ) , a p l a c e that w e n o w u n d e r s t a n d t o b e at t h e heart o f divine confirmation o f kingship.'"^ N o n e t h e l e s s , this precedent-shattering revelation o f the

" KRI V 2 8 5 . 5 - 6 ; the date is II Smw 11. The writing of Smw is clear (single S sign) although it is not present until Dynasty XIX. '*

KRIV 2 8 5 . 1 4 .

"

The basic study remains YOYOTTE ( 1 9 6 8 ) . A useful discussion is included in DORMAN

(1988),

pp. 2 2 - 2 8 .

The final edition is LACAU and CHEVRIER ( 1 9 7 7 ) 3 1 - 3 4 .

Add the

following important contribution: ROMER ( 1 9 8 7 ) . '"" For its use in the Middle Kingsom, see DELIA ( 1 9 7 9 ) ; and MURNANE ( 1 9 8 1 ) . Add the

important work of Claude OBSOMER ( 1 9 9 3 ) and his book OBSOMER ( 1 9 9 5 ) .

'"' GARDINER ( 1 9 5 7 ) , p. 1 9 5 with note 8: actually Urk. IV 7 4 1 . 4 . The reference is the account of Thutmose III regarding his three new festivals. '"^ For Luxor, one can refer to the standard analysis of Lanny BELL ( 1 9 8 5 ) ; his BELL ( 1 9 9 7 ) .

396

A. Spalinger

g o d to H a t s h e p s u t did n o t take p l a c e during the O p e t festival. T h e date o f the parallel Seventh P y l o n Inscription of T h u t m o s e III is also unclear. I n that composition, the king reports to his court o n his successes a b r o a d , b u t at the b e g i n n i n g v a g u e c o m m e n t s are m a d e c o n c e m i n g his rise to the throne.'"^ A n additional, overtly propagandistic case, the so-called "texte d e la j e u n e s s e " o f T h u t m o s e III, likewise is n o t specifically dated.'^" T h e exact locality w h e r e i n the g o d A m i m revealed the future to the youth is k n o w n from the early p e r s p i c a c i o u s research o f Breasted, w h o covered this narrative in detail: " O n the o c c a s i o n o f a great feast, w h e n the g o d appears in procession, the ftitiue T h u t m o s e III h a s all arranged so that the g o d should stop before h i m as h e stands in his p l a c e a m o n g the ranks of priests in the c o l o n n a d e d hall"."^' T h e location w a s the n o r t h e m area o f the t h e n hypostyle court at K a m a k . V a r i o u s dates m a y b e p r o p o s e d for this event: civil N e w Y e a r ' s D a y , the e n d of the sixth m o n t h ; the feast o f A m i m in I ^mw; etc. ( O p e t a p p e a r s e x c l u d e d a n d I d o u b t if I iht 1 c a n b e argued.) T h e date o f the k i n g ' s accession t o the throne w a s o n I ^mw 4 , b u t the p r o p h e c y m a y n o t h a v e t a k e n p l a c e at the s a m e time. T h e v e r b .yr is u s e d to grant kingship to T h u t m o s e , as in H a t s h e p s u t ' s case. T h e recounting o f a fiiture event, o n c e b y H a t s h e p s u t a n d twice b y T h u t m o s e III, a p p e a r s to negate the possibility that the past e v e n t t o o k p l a c e o n their accession d a y . B o t h future m l e r s were "foretold" o f their g o d l y status as P h a r a o h during specific feasts, a n d the g o d w h o told t h e m w a s A m u n . Is it p o s s i b l e that T h u t m o s e I l l ' s date with destiny occurred at the e n d o f II prt? For the Late N e w K i n g d o m o n w a r d s w e p o s s e s s a great deal o f ancillary information from the private sector. T h e A m u n p r i e s t h o o d at the close o f D y n a s t y X X compiled: p r o m o t i o n s at the E p i p h i feast (the famous o n e in year s e v e n o f the R e n a i s s a n c e , w i t h P i a n k h as H i g h Priest); o n e major oracle stated to h a v e o c c u r r e d during O p e t ( P . B M 10335); a second dated to the reign o f R a m e s s e s V I (stela at K a m a k - N o r t h ) ; a n d t w o m o r e (that of H r i h o r a n d M e n k h e p e r r e ) , p r o b a b l y at a religious c e l e b r a t i o n . D h w t y - m s ^ s Dynasty X X I account, despite m a n y lacunae, m a k e s clear that v a r i o u s preliminary " s e a n c e s " w e r e staged during the C h o i a k feast, a n d to these o n e c a n a d d the m a i n " s e a n c e " a n d the final r e v e l a t i o n . T h e r e w a s a n

Add MuRNANE (1986); and BELL (1986). For the intertwined nature of kingship and power in Pharaonic Egypt, see ASSMANN (1992). For the moment, see Urk. IV 178.91.4.1 have followed Assmann's ideas on the subject in SPALINGER (1996b). Donald B . Redford and William Mumane in O'CONNOR and SILVERMAN (1994) discuss this matter from a political viewpoint. Summarized in SPALINGER (1997). BREASTED (1906), p. 59.

Urk. IV 157.13. Piankh example: NIMS (1948) with KRI VI 702.5-703.3; and P. B M 10335: DAWSON (1925) with KRI VII 416.5-418.9. The last can only refer to Opet: see Dawson's comments on Plate XXXV for recto line 1. Ill iht is considered "probable", 4 iht is "just possibly". However, the latter makes no sense for the auspicious religious event, pace the scholars cited in KRI VII 416.7 note 9. The Ramesses VI example: VERNUS (1975) with KRI VI 283.1-11. Herihor: KRI VI 709-710.11 with EPIGRAPHIC SURVEY (1981), pp. 14-17 and PI. 132. The High Priest Menkheperre: EPIGRAPHIC SURVEY (1981), pp. 17-20 and PI. 133.

KRUCHTEN ( 1986), pp. 310-324.

Egyptian Festival Dating and the Moon

397

additional sixth " s e a n c e " set b e t w e e n the eighth a n d tenth d a y s (inclusive) o f the first m o n t h o f Smw. P e r h a p s this coincided with the A m i m celebration that w e k n o w o c c u r r e d about the s a m e time. T h e oracle text in regnal year fourteen o f P s a m m e t i c h u s I is p r o b a b l y to b e cormected to this festival as w e l l . ' " ' A further oracle is k n o w n w h e n the H i g h priest o f A m u n , O s o r k o n , took c h a r g e o f T h e b e s u n d e r the reign o f his father, T a k e l o t II. T h i s religious event o c c u r r e d at the N e h e b k a u festival, o n I prt 1, j u s t after C h o i a k . " " M o r e revealing, h o w e v e r , is the oracle d a t e d to the third regnal y e a r o f R a m e s s e s II w h i c h c a n b e found o n the rear south face o f the e a s t e m p y l o n at L u x o r . T h i s fragmentary inscription is r e m a r k a b l e for its m a n y idiosyncrasies, n o t the least o f w h i c h are the p r e s e n c e o f semi-cryptographic writings a n d the arcane c o n n o t a t i o n s of the u n k n o w n m a s t e r scribe w h o wrote i t . ' " R a m e s s e s II directs a n investigation into the mysteries o f the religious t h o u g h t o f Luxor, a n d the narration m a y hint at a c o n n e c t i o n to the festival in that area. T h e text says that R a m e s s e s " f o u n d " T h e b e s ; the P h a r a o h d i s c o v e r e d the ancestry o f the w o r l d a n d its m l e r A m u n - R e . T h e t h e m e of the c o m p o s i t i o n covers the overt (re-) invigoration of the cult o f A m u n - R e , a n d the sun, with a n implicitly " m o n o t h e i s t i c " outlook, is also m e n t i o n e d . R a m e s s e s speaks to his n o b l e s " w h o are in his following". H e desires to b e g i n n e w c o n s t m c t i o n at L u x o r , a n d psdntyw w a s o n c e m o r e the timing o f the stretching the cord c e r e m o n y . " ^ T h e w o r k itself w a s c o m p l e t e d in the fourth m o n t h o f iht, regnal year three; b u t a n a r g u m e n t c a n b e m a d e for c o m m e n c e m e n t in the first year o f the k i n g . ' " Unfortunately t o o little r e m a i n s to extract further calendrical information; b u t w e m a y b e d e m a n d i n g t o o m u c h precision o f time as well as o f p l a c e .

VI. Concluding Remarks T h e s e e x a m p l e s o f official reports often reveal the p r e s e n c e o f a l u n a r - b a s e d event within a festival. B u t c a n w e b e certain that the specific lunar event o f psdntyw, the first d a y in a lunar m o n t h , w a s accurately d e t e r m i n e d ? O u r m o n u m e n t a l hieroglyphic inscriptions a s s e m b l e d here m a y , in fact, b e b a s e d o n a p o s t h o c e r g o propter h o c situation. Pritchett analyzed this p r o b l e m from the A t h e n i a n viewpoint, b u t few h a v e c o n s i d e r e d the natural p h e n o m e n o n from the E g y p t o l o g i c a l side. Could, in fact, such cases h a v e b e e n "ftidged"? B r a d l e y Schaefer h a s r e v e a l e d the inherent w e a k n e s s e s o f the h u m a n eye for m o o n a n d star sightings. W e all n e e d to take a skeptical look as such official calendrical sources a n d reanalyze t h e m the w a y the ancient E g y p t i a n s w o u l d h a v e d o n e : festival days could b e lengthened or shortened; their original significance w o u l d b e lost; lunar-civil equations m i g h t fall into d e s u e t u d e or reflect p i o u s h o p e s o f the past or e v e n b e totally false. S h o u l d w e not take into consideration the possibility o f adjusting lunar dates o f p r o g n o s t i c a t i o n

'"' PARKER (1962), pp. 7-8.

"" CAMINOS (1958), p p 3 4 - 5 . "' REOFORO (1971), pp. 110-119. Add the later edition EL-RAZIK (1974) and EL-RAZIK (1975). The text is now in KRI II 345.11-347.4. We owe it to Redford that this discovery was stressed: REOFORO ( 1971 ), p. 115 note g. " ' See MURNANE (1975); and SPALINGER (1979).

398

A. Spalinger

(or p r o s p e c t i v e ones) t o t h e interests o f rulers, priests, o r t e m p l e s ? " " W h a t a r e t h e implications for c h r o n o l o g y if the ancient seer got it w r o n g ? T h e s e questions a r e n o t raised t o p r o v o k e total s k e p t i c i s m o f lunar-dated o c c i u r e n c e s , b u t rather to focus t h e attention o f future scholars u p o n their timing. ( N o n e t h e l e s s , I p l a c e m o r e faith in a n a c c o u n t a n t ' s entry o f a limar d a y t h a n in a lunar date e m b e d d e d i n a royal hieroglyphic inscription.) P l a c i n g t h e k e y event half w a y t h r o u g h the year o r scheduling it during a n A m u n feast m e a n s m u c h m o r e t o m e psychologically ( a n d politically) than calculating a n absolute date p l u s o r m i n u s fifty years. F o r e x a m p l e o n e major text firom early D y n a s t y X V I I I , the A h m o s e - N o f r e t a r i D o n a t i o n s Stela, m a n a g e s t o reveal t h e i m p o r t a n c e o f its date b y reference t o a major feast, Choiak, without recourse to lunar e v e n t s . " ' W e m u s t n o t let ourselves b e o b s e s s e d with astronomical reconstructions a n d forget the h u m a n d i m e n s i o n .

Abbreviations KRI = KITCHEN (1969-1990) U r k IV = SETHE ( 1 9 0 6 - 1 9 5 8 )

References A S S M A N N , J a n . 1992. Politische Théologie zwischen Àgypten und Israel. M u n i c h : Carl Friedrich v o n S i e m e n s Stiftung V O N B E C K E R A T H , Jurgen. 1 9 6 8 . " D i e 'Stele d e r V e r b a n n t e n ' i m M u s e u m d e s L o u v r e " . Revue d'Égyptologie

20: 7-36.

— 1 9 8 1 . " E i n W u n d e r des A m u n b e i der T e m p e l g r i i n d i m g m K a m a k " . Mitteilungen des Deutschen Archàologischen Instituts, Abteilung Kairo 3 7 : 4 1 - 4 9 . B E L L , Lanny. 1 9 8 5 . " L u x o r T e m p l e a n d the Cult o f the R o y a l K a " , Journal

of Near

Eastern Studies 4 4 : 2 5 1 - 2 9 4 . — 1 9 8 6 . " L a reine H a t s h e p s o u t au temple d e L o u q s o r " . Les dossiers. archéologie 101: 25-26.

Historié

et

— 1997. " T h e N e w K i n g d o m ' D i v i n e ' T e m p l e : T h e E x a m p l e o f L u x o r " . In: B y r o n E. S H A F E R (ed.). Temples of Ancient Egypt: 127-184. Ithaca ( N Y ) : C o m e l l University P r e s s . B J O R K M A N , G u n . 1 9 7 1 . Kings at Kamak, U p p s a l a : A l m q v i s t & W i k s e l l . B O N N E T , H a n s . 1 9 5 2 . Reallexikon der agyptischen Religionsgeschichte. Berlin: Walter de Gmyter. B O R C H A R D T , L u d w i g . 1920. Altàgyptische Zeitmessung J O R D A N (ed.). Die Geschichte der Zeitmessung L i e f e m n g B ) . Berlin-Leipzig: W a l t e r d e G m y t e r .

(Emst VON BASSERMANNund der Uhren, B a n d I,

— 1 9 3 5 . Die Mittel zur zeitlichen Festlegung von Punkten Geschichte und ihre Anwendung. Cairo: Selbstverlag.

der

agyptischen

"" To be specific: adjusting lunar dates of prognostication (or prospective ones) to the interests of rulers, priests, temples, and even the civil calendar. " ' HELCK (1975), pp. 100-103. The datum (lines 21-2) was evidence in the famous ParkerGardiner debate: GARDINER (1955), p. 15; and PARKER (1957b), pp. 104-06.

Egyptian Festival Dating and the Moon

399

B R E A S T E D , J a m e s H e m y . 1899. " T h e L e n g t h a n d S e a s o n o f T h u t m o s e I I L ' s First C a m p a i g n " . Zeitschrift fiir Agyptische

Sprache

37: 123-129.

— 1906. Ancient Records of Egypt II. C h i c a g o : University o f C h i c a g o P r e s s B R U G S C H , Heinrich. 1864. Matériaux pour servir à la reconstruction du calendrier des ancients égyptiens. Leipzig: J. C. H e i n r i c h s . C A M I N O S , R i c a r d o A . 1958. The Chronicle of Prince Osorkon. R o m e : Pontificium Institutum B i b l i c u m . D A W S O N , W a r r e n , R. 1925. " A n Oracle P a p y m s . B M 1 0 3 3 5 " . Journal of Egyptian Archaeology \\: 2 4 7 - 2 4 8 . D E L I A , R o b e r t D . 1979. " A N e w L o o k at S o m e Old D a t e s : A R e e x a m i n a t i o n of the Twelfth D y n a s t y D o u b l e D a t e d Inscriptions". Bulletin of the Egyptological Seminar 1: 1 5 - 2 8 D E P U Y D T , L e o . 1997. Civil Leuven: Peeters.

Calendar

and

Lunar

Calendar

in Ancient

Egypt.

— 1 9 9 8 . "The Demotic Mathematical Astronomical Papyms Carlsberg 9 Reinterpreted". In: W i l l y C L A R Y S S E , A n t o o n S C H O O R S , a n d H a r c o W I L L E M S (eds.), Egyptian Religion the Last Thousand Years, Studies Dedicated to the Memory of Jan Quaegebeur, Part II (Orientalia L o v a n i e n s a A n a l e c t a 8 5 ) : 1 2 7 7 - 1 2 9 7 . Leuven: Peeters. D O R M A N , Peter. 1988. The Monuments of Senenmut: Problems in Historical Methodology. L o n d o n : K e g a n Paul Intemational. E L - R A Z I K , M a h m u d A b d . 1974. " T h e D e d i c a t o r y and B u i l d i n g T e x t s of R a m e s s e s II in L u x o r T e m p l e " . Journal of Egyptian Archaeology 60: 1 4 2 - 1 6 0 . — 1 9 7 5 . " T h e D e d i c a t o r y and Building T e x t s o f R a m e s s e s II in L u x o r T e m p l e " . Journal of Egyptian Archaeology 61: 125-136. E L - S A B B A N , S h e r i f 2 0 0 0 . Temple Festival Calendars of Ancient Egypt. L i v e r p o o l : Liverpool University Press. EPIGRAPHIC S U R V E Y . 1 9 8 1 . The Temple of Khonsu, Volume 2: Scenes and Inscriptions in the Court and the First Hypostyle Hall. C h i c a g o : Oriental Institute of the University of C h i c a g o . F E R M O R , John. 1993. " P e r c e i v e d N i g h t L e n g t h Ratios in A n c i e n t E g y p t " . Vistas in Astronomy 3 6 : 3 6 3 - 3 7 3 . FOTHERINGHAM, J o h n K. 1910. " O n the Smallest Visible P h a s e o f the M o o n " . Monthly Notices of the Royal Astronomical Society 7 0 : 5 2 7 - 5 3 1 . G A R D I N E R , Sir A l a n H. 1952. " T h u t m o s i s III R e t u m s T h a n k s to A m u n " . Journal Egyptian

Archaeology

of

38: 6 - 2 3 .

— 1955. " T h e P r o b l e m of the M o n t h - N a m e s " . Revue d'Égyptologie

10: 9 - 3 1

— 1957. Egyptian Grammar (Third Edition). Oxford: Griffith Institute. G O Y O N , J e a n - C l a u d e . 1972. Confirmation du pouvoir royal au nouvel an : Brooklyn Museum Papyrus 47.218.50. Cairo: Institut français d ' a r c h é o l o g i e orientale. G R I M A L , N i c h o l a s . 1 9 8 1 . La stèle triumphale de Pi('ankh)y au Musée du Caire ( M é m o i r e s publiés p a r les m e m b r e s de l'Insfitut français d ' a r c h é o l o g i e orientale 105). Cairo: Institut Français d ' A r c h é o l o g i e Orientale. G R I M M , Alfred. 1994. Die altàgyptischen Festkalendar in den Tempeln der griechisch-romischen Epoche ( À g y p t e n u n d Altes T e s t a m e n t 15). W i e s b a d e n : Otto H a r r a s s o w i t z . H A R I N G , B e n J.J. 1997. Divine Households: Administrative and Economie Aspects of the New Kingdom Royal Memorial Temples in Western Thebes

400

A. Spalinger

(Egyptologische U i t g a v e n 12). Leyden: N e d e r l a n d s Instituut v o o r h e t N a b i j e Oosten. H E L C K , W o l f g a n g . 1 9 6 3 . Materialien zur Wirtschaftsgeschichte des Neuen Reiches III ( A k a d e m i e d e r Wissenschaften u n d der Literatur, A b h a n d l u n g e n d e r Geistesu n d Sozial-wissenschaftlichen Klasse, 1963.2). W i e s b a d e n : F . Steiner. — 1965. "Feiertage u n d Arbeitstage in d e r R a m e s s i d e n z e i t " . Economic and Social History of the Orient 7: 1 3 6 - 1 6 6 .

Journal

of the

— 1975. Historische-biographische Texte der 2. Zwischenzeit und neue Texte der 18. Dynastie (Kleine àgyptische T e x t e ) . W i e s b a d e n : Otto H a r r a s s o w i t z . — 1977. " D i e 'Weihinschrift' aus d e m T a l t e m p e l d e s S o n n e n h e i l i g t u m s d e s K ô n i g s N e u s e r r e b e i A b u G u r o b " . Studien zur Altagyptischen Kultur 5: 41-11. — 1983. " C h r o n o l o g i s c h e Schwachstellen II". Gottinger Miszellen 6 9 : 3 7 - 4 2 . H U B E R , Peter J. 1982. Astronomical Dating of Babylon I and Ur III ( M o n o g r a p h i c J o u m a l s o f the N e a r East. Occasional P a p e r s o n the N e a r East V o l u m e 1, Issue 4). M a l i b u : U n d e n a Publications. K I T C H E N , K e n n e t h A . 1 9 6 9 - 1 9 9 0 . Ramesside Inscriptions. Oxford: B l a c k w e l l . K O R M Y S C H E V A , E l e o n o r a . 1 9 9 8 . " F e s t k a l e n d e r i m K a w a - T e m p e l . V e r s u c h einer R e k o n s t m t i o n " . In: R o l f G U N D L A C H a n d M a t t h i a s R O C H H O L Z (eds.), 4. Agyptologische

Tempeltagung,

Koln 10-12.

Oktober

1996: Feste

im

Tempel

( À g y p t e n u n d Altes T e s t a m e n t 33/2): 7 7 - 8 9 . W i e s b a d e n : Otto H a r r a s s o w i t z . K R A U S S , R o l f 1 9 8 5 . Sothis- und Monddaten. Studien zur astronomischen und technischen Chronologie Altàgyptens (Hildesheimer Agyptologische Beitrage 2 0 ) . H i l d e s h e i m : Gerstenberg. —1997. Astronomische Konzepte und Jenseitsvorstellungen in den Pyramidentexten (Agyptologische Abhandlungen 59). Wiesbaden: Otto Harrassowitz. K R U C H T E N , J e a n - M a r i e . 1986. Le grand texte oraculaire de Djéhoutymose ( M o n o g r a p h i e s reine Elisabeth 5). Brussels F o n d a t i o n é g y p t o l o g i q u e reine Elisabeth. — 1989. Les annales des prêtres de Karnak (XXI-XXIII dynasties) et autres textes contemporains relatifs à l'initiation des prêtres d'Amon (Orientalia Lovaniensia A n a l e c t a 32). Leuven: D é p a r t e m e n t orientalistiek. — 1999. " F r o m M i d d l e to Late Egyptian". Lingua Aegyptia 6: 1 - 9 7 . L A C A U , Pierre a n d H e n r i CHEVRIER. 1977. Une chapelle d'Hatshepsout à Karnak I, Cairo: Institut français d'archéologie orientale. LEITZ, Christian. 1 9 8 9 . Studien zur agyptischen Astronomie (Àgj^Jtologische A b h a n d l u n g e n 4 9 ) . W i e s b a d e n : Otto Harrassowitz. M A C A D A M , M i l e s F . L . 1 9 4 9 . The Temples of Kawa, I. The Inscriptions. London: Griffith Institute. M O M M S E N , A u g u s t . 1 8 8 3 . Chronologic. Untersuchungen Uber das Kalenderwesen der Griechen, insonderheit der Athener. Leipzig: B . G. T e u b n e r . M U R N A N E , W i l l i a m J. 1 9 7 0 - 1 9 7 1 . " O n c e A g a i n the D a t e s for T h u t m o s e III a n d A m e n h o t e p II". Journal of Ancient Near Eastern Studies 3 : 1 - 7 . — 1975. " T h e Early R e i g n o f R a m e s s e s 11 a n d his C o r e g e n c y with Sety I". Journal of Near Eastern Studies 3 4 : 1 5 3 - 9 0 . — 1 9 8 1 . " I n Defense of the M i d d l e K i n g d o m D o u b l e D a t e s " . Bulletin of the Egyptological Seminar 3 : 7 4 - 7 6 .

Egyptian Festival Dating and the Moon

401

— 1986. " L a g r a n d e fête d ' O p e t " . Les dossiers. Historié et archéologie 101: 2 2 - 2 5 . N E L S O N , H a r o l d H . 1934. " T h e C a l e n d a r o f Feasts and Offerings at M e d i n e t H a b u " . Oriental Institute Communications 18: 1 - 6 3 . N E U G E B A U E R , Otto. 1955. Astronomical Cuneiform Texts III. L o n d o n : L u n d Humphries. N E U G E B A U E R , Otto and P A R K E R , R i c h a r d A . 1960. Egyptian Astronomical Texts I. London: Lund Humphries. NiMS, Charles F . 1 9 4 8 . " A n Oracle D a t e d in " T h e R e p e a t i n g of B i r t h s " . Journal of Near Eastern Studies 7: 1 5 7 - 1 6 2 . — 1 9 7 6 . " R a m e s s e u m S o u r c e s of M e d i n e t H a b u Reliefs". In: Studies in Honor of George R. Hughes (Studies in A n c i e n t Oriental Civilization 39): 1 6 9 - 7 5 . C h i c a g o : Oriental Institute of the University of C h i c a g o . O B S O M E R , C l a u d e . 1 9 9 3 . " L a date de N é s o u - M o n t o u ( L o u v r e C 1)", Revue d'Égyptologie 44: 103-40. — 1 9 9 5 . Sésostris I^^: étude chronologique et historique du règne. B m s s e l s : C o n n a i s s a n c e de l'Egypte ancierme O ' C O N N O R , D a v i d a n d S I L V E R M A N , D a v i d (eds.). 1994. Ancient Egyptian Kingship. Leyden: E.J. Brill. P A R K E R , R i c h a r d A . 1950a. The Calendars of Egypt. C h i c a g o : U n i v e r s i t y of C h i c a g o Press. — 1950b. " R e v i e w of Siegfried Schott, Altagyptische F e s t d a t e n " . Bibliotheca Orientalis 9: 1 0 0 - 1 0 2 . — 1957a. " T h e L u n a r D a t e s of T h u t m o s e III a n d R a m e s s e s II". Journal Eastern Studies 16: 3 9 - 4 3 .

of

— 1957b. " T h e P r o b l e m of the M o n t h - N a m e s : A R e p l y " . Revue dÉgyptologie 85-107.

Near 11:

— 1 9 6 2 . A Saite Oracle Papyrus from Thebes in the Brooklyn Museum (Papyrus Brooklyn 47.218.3) ( B r o w n Egyptological Studies 4 ) . P r o v i d e n c e : B r o w n University P r e s s . — 1 9 6 9 . " O n c e A g a i n the C o r e g e n c y of T h u t m o s e III a n d A m e n h o t e p II". In: Studies in Honor of John A. Wilson (Studies in A n c i e n t Oriental Civilization 35): 7 5 - 8 2 . C h i c a g o : University o f C h i c a g o Press. — and D U B B E R S T E I N , W a l d o H . 1942. Babylonian Chronology 626B. C.-A. D. 45 (Studies in A n c i e n t Oriental Civilization 2 4 ) . C h i c a g o : University o f C h i c a g o Press. — , J e a n L E C L A N T , a n d J e a n - C l a u d e G o YON. 1979. The Edifice of Taharqa by the Sacred Lake of Kamak ( B r o w n Egyptological Studies 8), P r o v i d e n c e : B r o w n University P r e s s a n d L o n d o n : L u n d H u m p h r i e s . PRITCHETT, W i l l i a m K. 1959. " T h e Classical L u n a r M o n t h " . Classical Philology 5 4 : 151-157. QUACK, Joachim F . 2000. "Kollationen und Korrekturvorschlage z u m P a p y m s Carlsberg 1". In: P a u l J. F R A N D S E N and K i m Ryholt (eds.), A Miscellany of Demotic Texts and Studies ( T h e Carlsberg Papyri 3): 1 6 5 - 1 7 1 . C o p e n h a g e n : T h e C a r s t e n N i e b u h r Institute of N e a r E a s t e m Studies. R E D F O R D , D o n a l d B . 1 9 7 1 . " T h e Earliest Y e a r s o f R a m e s s e s II, and the B u i l d i n g o f the R a m e s s i d e C o u r t at L u x o r " . Journal of Egyptian Archaeology 57: 1 1 0 - 1 1 9 .

402

A . Spalinger

RÒMER, M a l t e . 1987. "1st der T e x t a u f d e m B l o c k e n 2 2 2 / 3 5 / 1 8 4 d e r C h a p e l l e R o u g e ein Z e u g n i s fiir eine n e u e ' D i m e n s i o n erfahrbarer G o t t e s n a h e '

(Assmaim)".

Gottinger M i s z e l l e n 9 9 : 3 1 - 3 4 . R O S E , Lyim E. 1 9 9 9 . Sun, Moon, and Sothis. A Study of Calendars, Reforms

and

Calendar

in Ancient Egypt. Deerfield B e a c h , F L : K r o n o s Press.

S A M U E L , A l a n E d o u a r d . 1962. Ptolemaic

Chronology.

Munich: Beck.

SCHAEFER, B r a d l e y E . 1992. " T h e Length o f the L u n a r M o n t h " .

Archaeoastronomy

17 ( s u p p l e m e n t to Journal for the History o f A s t r o n o m y 2 3 ) : Zl-Al. — 1993. " A s t r o n o m y a n d the Limits o f Vision". Vistas in Astronomy — 1996. " L u n a r C r e s c e n t Visibility". Quarterly Society

Journal

36: 3 1 1 - 3 6 1 .

of the Royal

Astronomical

37: 7 5 9 - 7 6 8 .

— 2 0 0 0 . " T h e Heliacal Rise o f Sirius and A n c i e n t E g y p t i a n C h r o n o l o g y " . for the History

of Astronomy

— a n d D O G G E T T , Leroy. 1994. " L u n a r Crescent Visibility". Icarus, S C H O T T , Siegfried. 1950. Altàgyptische Mainz,

Abhandlungen

Journal

31:149-155.

der

Festdaten

Geistes-

und

107: 3 8 8 - 4 0 3 .

( A k a d e m i e d e r W i s s e n s c h a f t e n in Sozialwissenschaftlichen

Klasse

1950.10). W i e s b a d e n : V e r l a g der A k a d e m i e d e r Wissenschaften. S E T H E , Kurt. 1 9 0 6 - 1 9 5 8 . Urkunden der 18. Dynastie.

Leipzig: J . C . H i n r i c h s .

SPALINGER, A n t h o n y . 1979. " T r a c e s o f the Early C a r e e r o f R a m e s s e s I I " . Journal Near Eastern Studies

of

38: 271-286.

— (ed.) 1 9 9 2 . Revolutions

in Time. Study

in Ancient

Egyptian

Calendrics.

San

A n t o n i o : V a n Siclen. — 1995a. " M o n t h R e p r e s e n t a t i o n s " . Chronique

d'Egypte

— 1995b. " T h e L u n a r S y s t e m in Festival C a l e n d a r s : O n w a r d s " , Bulletin

de la Société

d'égyptologie

70: 110-122. F r o m the N e w K i n g d o m

de Genève

19: 2 5 - 4 0 .

— 1996a. T h e Private Feast Lists o f Ancient E g y p t (Agyptologische A b h a n d l u n g e n 57). W i e s b a d e n : Otto Harrassowitz. — 1996b. " S o v e r e i g n t y a n d T h e o l o g y in N e w K i n g d o m Egypt: Tradition". Saeculum

— 1997. " D r a m a in History: Altagyptischen —1998.

Some Cases of

47: 217-238. E x e m p l a r s from M i d D y n a s t y X V I I I " . Studien

zur

Kultur 2 4 : 2 6 9 - 3 0 0 .

"Chronological

Remarks".

Bulletin

de

la

Société

d'égyptologie

de

Genève 22: 5 1 - 5 8 . V E R N U S , Pascal. 1975 " U n texte o r a c u l a n e d e R a m s è s V I " . Bulletin Français

d'Archéologie

Orientale

de

l'Institut

75: 103-110.

V A N W A L S E M , R e n é . 1 9 8 2 . " M o n t h - N a m e s a n d Feasts at Deir e l - M e d î n a " . In: R o b J. D E M A R É E a n d J a c o b u s J. J A N S S E N (eds.). Gleanings

from

Deir

el-Medîna

(Egyptologische U i t g a v e n 1): 2 1 5 - 2 4 1 . L e y d e n : N e d e r l a n d s Instituut v o o r het Nabije Oosten. WELLS,

Ronald

Contributions Egyptological

A.

1986.

to

Egyptian

Studies

"Some

Astronomical

Chronology".

in Honor of Richard

In:

Reflections Leonard

A. Parker:

H.

on

Parker's

LESKO

(ed.),

1 6 5 - 1 7 1 . Hanover, N.H.:

University P r e s s o f N e w England. — 1992. " T h e M y t h o l o g y o f N u t a n d the Birth o f R e " . Studien Kultur —1995.

zur

Altagyptischen

19: 3 0 5 - 3 2 1 . "The Goddess

Aegyptiaca

10: 2 0 5 - 1 4 .

Nut, Pharaoh's

Guarantor

of

Immortality",

Varia

Egyptian Festival Dating and the Moon

— 2 0 0 1 . " R e v i e w of L y n n E. R o s e , Sun, M o o n , a n d Sothis. Deerfield ( 1 9 9 9 ) " . Journal

of Near Eastern Studies,

403

Beach

forthcoming.

W E N T E , E d w a r d F . 1 9 7 5 . " T h u t m o s e I l l ' s A c c e s s i o n a n d the B e g i n n i n g of the N e w K i n g d o m " . Journal of Near Eastern Studies 3 4 : 2 6 5 - 2 7 2 . Y O Y O T T E , Jean. 1 9 6 8 . " L a date s u p p o s é e d u corourmement d ' H a t s h e p s o u t " . Kemi

18: 8 5 - 9 1 .

1

A Simple Function for the Length of the Saros in Babylonian Astronomy John M. Steele,

Durham

O n e o f the major u n s o l v e d p r o b l e m s in the study o f B a b y l o n i a n a s t r o n o m y is the relationship b e t w e e n the n o n - m a t h e m a t i c a l astronomical texts a n d the so-called A C T - t y p e texts of m a t h e m a t i c a l astronomy. T h e n o n - m a t h e m a t i c a l a s t r o n o m i c a l texts include the A s t r o n o m i c a l Diaries, the G o a l Y e a r T e x t s and the A l m a n a c s a n d N o r m a l Star A l m a n a c s , w h i c h contain observations a n d predictions of celestial p h e n o m e n a c o v e r i n g a specific p e r i o d (e.g. the Diaries contain o b s e r v a t i o n s a n d predictions over a p e r i o d of generally six m o n t h s ; the A l m a n a c s contain p r e d i c t i o n s for a specific year b a s e d u p o n past observations). T h e A C T - t y p e texts, h o w e v e r , are exemplified b y e p h e m e r i d e s w h i c h give calculated p h e n o m e n a over s o m e time period, a n d p r o c e d u r e texts w h i c h describe - albeit rather obliquely - the m e t h o d s b y w h i c h these calculations are m a d e . T h e y almost all date to after the mid-fourth century B C . A l t h o u g h e x a m p l e s of the n o n - m a t h e m a t i c a l texts contain r e c o r d s dating b a c k to the m i d - e i g h t h century B C , they continued b e i n g written until at least A D 7 5 , a n d thus co-existed with the A C T - t y p e material. In all likelihood, the n o n - m a t h e m a t i c a l texts p r o v i d e d the r a w empirical data from w h i c h the m a t h e m a t i c a l A C T theories were d e v e l o p e d , a n d several attempts to derive the p a r a m e t e r s of these theories from the p r e s e r v e d observations h a v e b e e n m a d e . ' W h a t is only n o w b e i n g d o n e , however, is to consider if a n d h o w the techniques of the m a t h e m a t i c a l a s t r o n o m y d e p e n d u p o n the m e t h o d s u s e d to p r e d i c t celestial p h e n o m e n a in the n o n - m a t h e m a t i c a l texts. T h i s is n o t an easy m a t t e r since these m e t h o d s are frequently not fully understood. In part this is b e c a u s e there are far fewer p r o c e d u r e - t y p e texts for the n o n - m a t h e m a t i c a l material than for the A C T type astronomy, a n d those that there are h a v e b e e n studied m u c h less, but it is also d u e to the fact that these m l e s are less a m e n a b l e to m a t h e m a t i c a l analysis than the A C T material. N e v e r t h e l e s s , in the past few years s o m e of these m l e s h a v e b e e n uncovered,^ w h i c h is causing a reassessment o f h o w w e u n d e r s t a n d the d e v e l o p m e n t of B a b y l o n i a n astronomy. T h e tablet p u b l i s h e d here m a y p r o v i d e another e x a m p l e of a link b e t w e e n the n o n - m a t h e m a t i c a l astronomical texts and the A C T material. If the interpretation p r o p o s e d here is correct - w h i c h seems to m e to b e likely, although it caimot b e finally p r o v e n unless fiirther evidence is discovered - the text contains an empirical fiinction for the excess length of the Saros over 6 5 8 5 days that c o u l d b e u s e d in predicting the times of eclipses. T h e length of the Saros also lies at the heart of the '

E.g. A A B O E ( 1 9 8 0 ) , B R A C K - B E R N S E N ( 1 9 9 7 ) , SWERDLOW ( 1 9 9 8 ) .

^ In particular rules relating to the prediction of the lunar six and month lengths have been uncovered by BRACK-BERNSEN ( 1 9 9 9 ) and elsewhere in this volume.

406

J.M. Steele

so-called S y s t e m A limar theory. W h i l s t there is n o direct link b e t w e e n this empirical fimction for the length of the Saros a n d the c o r r e s p o n d i n g ( t h o u g h not identical in m e a n i n g ) S y s t e m A fimction, g e n e r a l l y k n o w n as 4>, I b e l i e v e that if nothing else, a w a r e n e s s that the Saros could b e m o d e l l e d m a t h e m a t i c a l l y u s i n g a simple function w a s o n e r e a s o n w h y it was u s e d as a foundation stone for the lunar theory.

BM 45861 (81-7-6, 290) B M 4 5 8 6 1 c o m e s from the 8 1 - 7 - 6 collection o f a p p r o x i m a t e l y 7 0 0 tablets p u r c h a s e d b y the British M u s e u m from J o s e p h M . Shemtob.^ A b o u t half of this c o l l e c t i o n are a s t r o n o m i c a l or a s t i o l o g i c a l tablets. T h e tablet is a p p r o x i m a t e l y 7.5 c m w i d e b y 8.5 c m high. N o e d g e s r e m a i n and the tablet is only p r e s e r v e d o n o n e side (see p h o t o at the e n d o f this article). A horizontal ruling indicates that the traces o f t w o lines at the top o f the tablet m u s t b e fi-om a different section. F u r t h e r m o r e , the s p a c i n g o f the text b e l o w this line suggests that there w e r e originally three c o l u m n s , o f w h i c h only final tiaces of the first a n d only the first few signs o f the third c o l u m n r e m a i n . W h e n c o m p l e t e this tablet m u s t h a v e b e e n an a s t i o n o m i c a l p r o c e d u r e text d i v i d e d into several sections w h i c h m a y - or m a y n o t - h a v e dealt with related issues. In the fransliteration g i v e n b e l o w I h a v e attempted to p r e s e r v e the layout o f the tablet, a l t h o u g h the spacing b e t w e e n the c o l u m n s has b e e n e x p a n d e d s o m e w h a t for clarity.'* I' 2' 3' 4' 5' 6' 7' 8' 9' 10'

ir 12' 13' 14' 15' 16' 17' 18'

tiaces only traces only ...] [•• . ] " x " 2,5 tar-'dP[-tum [...] Y 2 tar-di-tum 5 TJâ^ [...] [...] V 1,55 K I M I N 5 U S L A [ L . . . ] [.. .] V 1,50 K I M I N 5 U § T A L ^ [.. .] V 1,45 K I M I N 1,40 K I M I N [.. • ] 1,35 K I M I N [.. • ] [.. • ] 5 U S [•• • [•• • [•• • [.. • [•• • [•• • [.. • [.. • [•• • [.. •

19' 20' 2 1 ' [.. . remainder ^

] ] ]

Y [ .] LAL-[...] BE'-waAN-"'[...]

DIRI.MES

MU w z u i-sal-x [...]

5 US DIRI 1 \ 4 0 Uâ-?M ^r,45 K I M I N 5 U § D I R I

] ]

^r,50KI M I N 5 US DIRI 'r,55 K I M I N 5 U § D I R I KIMIN 5 U S T)IRr

]

T,5

] ]

7\10 KIMIN 5U§[...] ^2\15KI MIN 5 U â [ . . . ] 7 \ 2 0 KI M I N 5 U S [ . . . ]

] ]

'2\\5 ] 7\10 lost.

KIMIN

[...]

5 US LAL 5 US LAL 5 US LAL

- [...;

[...] [...] [...] [...]

5US[...]

V^-tum 5 [ U § . . . ] [5 U S . . . ]

READE(1986), p. XV.

" I wish to thank Christopher Walker for additional collations. He is, of course, not responsible for any errors that remain.

A Simple Function for the Length of the Saros in Babylonian Astronomy

407

O n l y the m i d d l e c o l u m n b e l o w the m l i n g is sufficiently well p r e s e r v e d to b e interpreted.^ E a c h line contains a n u m b e r followed b y the n o i m tardïtu a n d t h e n either 5 U S L A L or 5 US D I R I m e a n i n g "5 U S d e c r e a s e " a n d "5 U S i n c r e a s e " . T h i s refers to the first n u m b e r in the line that h a s always either increased or d e c r e a s e d b y 5 c o m p a r e d t o the p r e c e d i n g line. Indeed, at line 1 0 ' , w h i c h c o m e s after the lowest n u m b e r in the s e q u e n c e , w e read 5 US D I R I . M E S "5 U â are the i n c r e a s e s " . T h i s indicates that w e are to u n d e r s t a n d the n u m b e r s at the b e g i n n i n g of the lines as b e i n g given in U S . In a s t r o n o m i c a l contexts, U § c a n m e a n either a d e g r e e of arc ( g e n e r a l l y celesfial l o n g i t u d e ) , or a d e g r e e of time. F o l l o w i n g the n u m b e r s w e have the n o u n tardïtu written phonefically in lines 3 ' a n d 4 ' , and in lines 1 1 ' and 2 0 ' using the l o g o g r a m U § p l u s the p h o n e t i c c o m p l e m e n t s -tit a n d -tum respectively (in the other lines w e s i m p l y h a v e K I M I N "ditto").^ In general tardïtu m e a n s " a d d i t i o n " or p e r h a p s " a d d i t i o n a l " or "exfra". F o r e x a m p l e , in sections 8-16 o f the O l d B a b y l o n i a n m a t h e m a t i c a l p r o b l e m text Y B C 7 1 6 4 ( = M C T L), w h i c h deal with the c o n s t m c t i o n of canals, terdîtu s e e m s to b e u s e d as a n a m e for the extia water v o l u m e in the c a n a l . ' I k n o w o f only t w o cases w h e r e tardïtu is attested in p u b l i s h e d a s t i o n o m i c a l p r o c e d u r e texts. Unfortunately, in both the context is largely desfroyed. O b v . 7 ' of A C T N o . 2 0 2 r e a d s : [...] zi 4 0 ii tar-di-tum [ . . . ] . T h i s section o f the text apparently gives s o m e c o r r e c t i o n s ( B A R N U N ) that are p o s s i b l y to b e applied to the longitude of the m o o n , b u t t o o little r e m a i n s to m a k e a n y real sense out of it. L B A T 1515 is a p l a n e t a r y p r o c e d u r e text that gives the errors o n s o m e o f the goal-year p e r i o d s o f the planets. R e v . 11-12 includes the p h r a s e s 6 K A S G A L - B U 'tar^-di-tû and 5 K A S G A L - B U tar-di-tii, b u t the text is so b a d l y d a m a g e d that it is n o t at all clear to w h a t these additions o f 5 o r 6 bëru relate to. It is quite p o s s i b l e that this section of the text is n o l o n g e r d e s c r i b i n g p l a n e t a r y p r o c e d u r e s , b u t n o t h i n g m o r e t h a n that c a n b e said. N e i t h e r of t h e s e t w o e x a m p l e s s e e m to relate to the n u m b e r s called tardïtu in B M 4 5 8 6 1 . T h e B M 4 5 8 6 1 n u m b e r s form a linear z i g z a g fiinction. U s i n g the t e r m i n o l o g y of A C T , the fimction c a n s u m m a r i s e d as: M = 2,20 U S m = l,35Uâ d-5US 1,57;30 U § A = 45 U S P e r i o d = 18 lines W h a t is n o t explicitly given, h o w e v e r , is w h a t the p e r i o d relates to. Is it 18 d a y s , 18 m o n t h s , 18 years, or s o m e t h i n g c o m p l e t e l y different? N o r is it a b s o l u t e l y clear w h e t h e r the n u m b e r s are to b e taken as d e g r e e s of arc or d e g r e e s of t i m e . H o w e v e r , since 18 days a n d 18 m o n t h s are n o t p e r i o d s k n o w n e l s e w h e r e from B a b y l o n i a n a s t t o n o m y (nor, t o m y k n o w l e d g e , are there a n y a s t i o n o m i c a l p h e n o m e n a w i t h these * One is tempted to read the few signs remaining in line 9' of the rightmost column as BEma AN[-KU,o...] "If an eclipse but too little of the AN is preserved to be sure that these are not just the first few wedges of another sign. Walker (personal communication) suggests MU NU ZU in line 10' might mean something like "unknown year". ^

I thank Hermann Hunger for a discussion of this issue.

'

P o w E L L ( 1 9 8 8 ) , p 165.

408

J.M. Steele

periods), w e can safely disregard t h e m from further discussion. 18 years, on the other hand, is the a p p r o x i m a t e length of the so-called Saros, w h i c h is well k n o w n in B a b y l o n i a n astronomy.

The Saros and its Length T h e Saros is a p e r i o d of 2 2 3 synodic m o n t h s (the time interval b e t w e e n the m o o n a n d sun reaching successive conjunctions or oppositions). It is also v e r y close to b e i n g a w h o l e n u m b e r of anomalistic m o n t h s (the time interval b e t w e e n successive r e t u m s of lunar velocity) a n d dracontic m o n t h s (the time interval b e t w e e n successive p a s s a g e s of the m o o n b y the lunar n o d e ) : 2 2 3 syn. m . « 2 4 2 drac. m. « 2 3 9 a n o m . m. « 18 years 10 or 11 days^ B e c a u s e o f the relation b e t w e e n synodic and dracontic m o n t h s the Saros acts as an eclipse period. T h u s , one Saros after an eclipse, the m o o n a n d sun are a g a i n at syzygy, a n d are v e r y close to being at the s a m e p l a c e in the m o o n ' s orbit with respect to the lunar n o d e . F u r t h e r m o r e , b e c a u s e the Saros is also very close to a w h o l e n u m b e r of anomalistic m o n t h s , eclipses separated b y one Saros h a v e v e r y similar m a g n i t u d e s a n d durations. It should b e m e n t i o n e d , h o w e v e r , that 2 2 3 synodic m o n t h s actually corresponds to about a third of a d a y m o r e than a w h o l e n u m b e r of days. This m e a n s that eclipses do not occur at the same time o f d a y after one Saros. In fact, b e c a u s e they are shifted b y a p p r o x i m a t e l y 8 h o u r s , a lunar eclipse s e e n t o w a r d s the end of the night will b e followed o n e Saros later b y an eclipse that takes p l a c e during the d a y w h e n the m o o n is b e l o w the h o r i z o n a n d c a n n o t b e seen. T h i s gives rise to the c o n c e p t of eclipse possibilities: syzygies w h e n an eclipse is possible, as o p p o s e d to those w h e n an eclipse defmitely will not occur. A t a b o u t half o f all eclipse possibilities the eclipse will not b e seen. This m a y b e b e c a u s e the eclipse t o o k p l a c e w h e n the luminary was b e l o w the horizon, b e c a u s e the m o o n was t o o far a w a y from the n o d e at that syzygy, or, in the case of solar eclipses, b e c a u s e the eclipse s h a d o w p a s s e d either entirely to the north or to the south of the o b s e r v e r ' s location, although this concept p l a y s n o role in, and I d o not think can e v e n b e c o m p r e h e n d e d in terms of, B a b y l o n i a n m e t h o d s o f eclipse prediction. T h e usefulness of the Saros is not only in identifying eclipse possibilities 2 2 3 synodic m o n t h s after an o b s e r v e d eclipse. B y c o m b i n i n g the Saros with the realization that eclipse possibilities generally occur every six m o n t h s , b u t occasionally after only five m o n t h s , it is possible to predict all p o s s i b l e dates of eclipses. If within a Saros of 2 2 3 m o n t h s there are a eclipses at a six m o n t h interval a n d b eclipses at five m o n t h s , then 6a + 5b = 2 2 3 Since a a n d b m u s t b e w h o l e n u m b e r s , and a m u s t b e m u c h b i g g e r than b, simple m a t h e m a t i c s s h o w s that a = 3 3 and b^ 5. T h u s within a Saros of 2 2 3 m o n t h s , there are 38 eclipse possibilifies, of which 3 3 are at a six-month interval, a n d 5 are at a five-month interval. B y simply spreading out the five-month intervals as e v e n l y as possible w e get five g r o u p s of eclipses, of w h i c h the first eclipse in each g r o u p is

* More precisely: 223 mean synodic months = 6585.322 days, 242 mean dracontic months = 6585.357 days and 239 mean anomalistic months = 6585.538 days.

A Simple Function for the Length of the Saros in Babylonian Astronomy

409

separated from the p r e c e d i n g ecHpse b y five m o n t h s , a n d the r e m a i n i n g e c h p s e s are at a six-month interval. T h e first, thfrd a n d fifth g r o u p s contain 8 eclipses, whilst the s e c o n d a n d fourth g r o u p s contain only 7 eclipses. U s i n g the fact that eclipse possibilities r e p e a t after one Saros, this results in a simple matrix like s c h e m e b y w h i c h all eclipse possibilities c a n b e predicted. T h e only decision the p e r s o n u s i n g this s c h e m e m u s t m a k e is w h i c h m o n t h to take as the first eclipse in the first c o l u m n . ' T h i s s c h e m e for predicting eclipse possibilities is illusfrated in figure 1.'° W e k n o w that this type o f m e t h o d to p r e d i c t lunar eclipses w a s b e i n g u s e d in B a b y l o n b y at least 6 0 0 B C . T h r e e tablets ( L B A T * 1 4 1 4 , 1 4 1 5 + 1 4 1 6 + 1 4 1 7 , * 1 4 1 9 ) that originally formed p a r t o f a large c o m p i l a t i o n of lunar eclipse o b s e r v a t i o n s a n d predictions c o v e r i n g the p e r i o d from (probably) 7 4 7 to 3 1 5 B C , " are laid o u t in the form o f a large m a t r i x with e a c h cell representing one eclipse possibility. S u c c e s s i v e eclipses in the s a m e c o l u m n are separated b y either six or five m o n t h s (in the latter case this is always explicitly indicated b y the w o r d s 5 ITTI " 5 m o n t h s " after the d a t e o f the eclipse possibility), a n d entries in the s a m e r o w are separated b y o n e Saros. O n l y a small p a r t o f the original collection is still p r e s e r v e d , b u t it h a s b e e n p o s s i b l e to r e c o n s t m c t the layout of the w h o l e compilation using eclipses r e p o r t e d o n other tablets.'^ M o s t i m p o r t a n t for this r e c o n s t m c t i o n is the tablet L B A T * 1 4 2 0 w h i c h p r o v i d e s slightly m o r e than o n e S a r o s ' s w o r t h of enfries for the p e r i o d a b o u t 6 0 0 B C . T h i s tablet is a simple list of eclipse possibilities w h i c h w a s p r o b a b l y c o m p i l e d shortly after the dates o f the entiles it contains. It displays precisely t h e a r r a n g e m e n t of five- a n d six-month intervals w e w o u l d expect in the large compilation, indicating that b y 6 0 0 B C at the latest, lunar eclipses were b e i n g p r e d i c t e d in j u s t this fashion. R e c o r d s of solar eclipses are only p r e s e r v e d b a c k to the fifth century B C , a n d a l m o s t all date from after a b o u t 3 5 0 B C . F r o m these r e c o r d s it is evident that solar eclipse possibilities w e r e identified in exactly the s a m e fashion as lunar eclipse possibilities. B e c a u s e far m o r e lunar eclipses are visible at a given location than solar eclipses, I suspect that the b a s i c s c h e m e s for identifying eclipse possibilities w e r e d e v e l o p e d for lunar eclipses a n d then the techniques applied to solar eclipses. B e c a u s e the relation b e t w e e n synodic a n d dracontic m o n t h s is n o t exact, S a r o s series d o n o t last forever. If w e think of e a c h syzygy as h a v i n g a p o s i t i o n o n a stiaight line, a n d only syzygies within a certain r a n g e o n that line (the " n o d a l z o n e " ) b e i n g eclipse possibilities, then the p o s i t i o n of successive syzygies in a S a r o s series gradually m o v e s forward along that line (if the Saros relation were exact, t h e n the position w o u l d r e m a i n fixed). T h e y first enter, then fravel through, before eventually leaving the n o d a l z o n e . T h i s m e a n s that within the Saros s c h e m e d e s c r i b e d a b o v e , there are certain r o w s w h i c h are " a c t i v e " a n d s o m e that are " d e a d " . W h e n the lunar Saros s c h e m e w a s p u t into u s e at B a b y l o n , the first eclipse w a s c h o s e n - p e r h a p s b y accident, b u t m o r e likely b y design - such that the active r o w s w e r e t o w a r d s the t o p of e a c h g r o u p of eclipses, w h e r e a s the d e a d r o w s were t o w a r d s the b o t t o m (i.e., j u s t '

See STEELE (2000b) for further details.

'" It is useful at this point to define some of the terminology I will use in discussing the Saros. By "Saros series" I mean a group of eclipses that are each separated from the preceding one by a Saros. A "Saros scheme", however, refers to the matrix for predicting the 38 eclipse possibilities within a Saros. " The texts are published in HUNGER (2001). For a discussion of their contents and layout, see my appendix to that volume, and WALKER (1997). STEELE (2000b).

410

J.M.Steele

III

II

I

IV

V

1 2 3 4 5 6 7 8

-746 -746 -745 -745 -744 -744 -743 -743

Feb 6 Aug 2 Jan 26 Jul 22 Jan 15 Jul 11 Jan 3 Jun 30

-728 -728 -727 -727 -726 -726 -725 -725

Feb 17 Aug 12 Feb 5 Aug 2 Jan 25 Jul 22 Jan 15 Jul 11

-710 -710 -709 -709 -708 -708 -707 -707

Feb 27 Aug 23 Feb 16 Aug 13 Feb 6 Aug 1 Jan 25 Jul 21

-692 -692 -691 -691 -690 -690 -689 -689

Mar 10 Sep 3 Feb 27 Aug 23 Feb 16 Aug 12 Feb 6 Aug 2

-674 -674 -673 -673 -672 -672 -671 -671

Mar 21 Sep 14 Mar 10 Sep 4 Feb 27 Aug 23 Feb 16 Aug 12

9 10 11 12 13 14 15

-743 -742 -742 -741 -741 -740 -740

Nov 25 May 20 Nov 14 May 10 Nov 3 Apr 28 Oct 22

-725 -724 -724 -723 -723 -722 -722

Dec 6 May 31 Nov 25 May 20 Nov 14 May 10 Nov 3

-707 -706 -706 -705 -705 -704 -704

Dec 16 Jun 11 Dec 6 May 31 Nov 25 May 20 Nov 13

-689 Dec 28 -688Jun 21 -688 Dec 16 -687 Jun 11 -687 Dec 5 -686 May 31 -686 Nov 24

-670 -670 -670 -669 -669 -668 -668

Jan 7 Jul 2 Dec 28 Jun 22 Dec 17 Jun 11 Dec 5

16 17 18 19 20 21 22 23

-739 -739 -738 -738 -737 -737 -736 -736

Mar 20 Sep 19 Mar 9 Sep 1 Feb 26 Aug 22 Feb 15 Aug 11

-721 -721 -720 -720 -719 -719 -718 -718

Mar 31 Sep 23 Mar 19 Sep 12 Mar 9 Sep 1 Feb 26 Aug 22

-703 -703 -702 -702 -701 -701 -700 -700

Apr 10 Oct 3 Mar 31 Sep 23 Mar 20 Sep 13 Mar 8 Sep 1

-685 -685 -684 -684 -683 -683 -682 -682

Apr 22 Oct 15 Apr 10 Oct 3 Mar 30 Sep 23 Mar 19 Sep 13

-667 -667 -666 -666 -665 -665 -664 -664

May 2 Oct 25 Apr 21 Oct 15 Apr 10 Oct 4 Mar 29 Sep 23

24 25 26 27 28 29 30

-735 -735 -735 -734 -734 -733 -733

Jan 5 Jul 1 Dec 25 Jun 21 Dec 15 Jun 10 Dec 5

-717 -717 -716 -716 -716 -715 -715

Jan 16 Jul 13 Jan 6 Jul 1 Dec 25 Jun 20 Dec 15

-699 -699 -698 -698 -697 -697 -697

Jan 27 Jul 23 Jan 16 Jul 12 Jan 6 Jul 1 Dec 26

-681 -681 -680 -680 -679 -679 -678

Feb 7 Aug 3 Jan 28 Jul 22 Jan 16 Juin Jan 6

-663 -663 -662 -662 -661 -661 -660

Feb 17 Aug 14 Feb 7 Aug 3 Jan 28 Jul 23 Jan 17

31 32 33 34 35 36 37 38

-732 -732 -731 -731 -730 -730 -729 -729

Apr 30 Oct 24 Apr 19 Oct 13 Apr 9 Oct 2 Mar 30 Sep 22

-714 -714 -713 -713 -712 -712 -711 -711

May 11 Nov 4 May 1 Oct 24 Apr 19 Oct 13 Apr 9 Oct 2

-696 -696 -695 -695 -694 -694 -693 -693

May 21 Nov 15 May 11 Nov 4 May 1 Oct 24 Apr 20 Oct 13

-678 -678 -677 -677 -676 -676 -675 -675

Jun 2 Nov 26 May 22 Nov 15 May 11 Nov 3 Apr 30 Oct 24

-660 -660 -659 -659 -658 -658 -657 -657

Jun 12 Dec 6 Jun 2 Nov 25 May 22 Nov 15 May 12 Nov 4

Figure 1. A Saros s c h e m e for predicting lunar eclipses. In each c o l u m n successive eclipses fall into five g r o u p s w h i c h I h a v e indicated b y an e m p t y r o w . W i t h i n each g r o u p , the first eclipse is separated fi-om the p r e c e d i n g eclipse possibility b y a fivem o n t h interval, a n d all of the others are at a six-month interval. Eclipses in the same r o w are separated fi^om the one another b y o n e Saros. D a t e s of eclipses w h i c h were theoretically visible in B a b y l o n (at least in part) are given in b o l d . T o save space dates are given in the Julian, rather than the B a b y l o n i a n , calendar.

A Simple Function for the Length of thc Saros in Babylonian Astronomy

411

before the five m o n t h interval). Since the active r o w s migrate d o w n w a r d s t h r o u g h the r o w s over time, this a r r a n g e m e n t m e a n t that the Saros s c h e m e c o u l d b e u s e d for s o m e 2 7 cycles before the date o f the eclipse in the first r o w h a d to b e reset to a c c o m m o d a t e all o f the active rows.'"' It s e e m s that different t e r m i n o l o g y w a s u s e d in the Diaries e t c . for eclipse possibilities o n the active r o w s t o t h o s e o n t h e d e a d rows.''* T h e s e latter eclipse possibilities m a y h a v e simply b e e n n o t e d for r e c o r d keeping purposes. U s i n g the Saros in the w a y outlined a b o v e allows o n e to p r e d i c t w h i c h m o n t h s are eclipse possibilities, b u t the B a b y l o n i a n r e c o r d s also s h o w that the time o f e x p e c t e d eclipse w a s also predicted. This time related to the m o m e n t the eclipse w a s expected t o begin, n o t the time o f syzygy as might at h a v e b e e n expected.'^ T h e r e a s o n for this m a y h a v e b e e n astrological. A late text from U r u k d e s c r i b e s a ritual to b e p e r f o r m e d during a n eclipse o f the m o o n . It s e e m s that the ritual w a s t o b e g i n w h e n the m o o n b e g a n to b e eclipsed a n d e n d e d w h e n the m o o n b e g a n to r e c o v e r its brightness (not at the e n d o f the eclipse as might h a v e b e e n e x p e c t e d ) . T h u s , in m a k i n g p r e p a r a t i o n s for the ritual it w o u l d h a v e b e e n usefiil to k n o w w h e n to expect a forthcoming eclipse to begin. T h i s same p r o p e r t y o f the p r e d i c t e d eclipse times also tells u s s o m e t h i n g about h o w they were predicted. A l m o s t all o f the r e c o r d e d observations of eclipses give a m e a s u r e m e n t o f the time w h e n the eclipse b e g a n , together with the durations o f the various p h a s e s of the eclipse. B e c a u s e eclipses in the same Saros series h a v e r o u g h l y the same m a g n i t u d e a n d duration, a d d i n g o n o n e Saros to the time o f a n eclipse will give the a p p r o x i m a t e time w h e n the next eclipse in that series will b e g i n . ' ' A s m e n t i o n e d a b o v e , the Saros is a p p r o x i m a t e l y one-third o f a d a y longer than a w h o l e n u m b e r o f days, a n d so o n e c a n get a r e a s o n a b l e estimate o f the time o f a n eclipse b y simply a d d i n g o n a third o f a d a y (8 hours) to the time o f the eclipse o n e Saros earlier. M o r e precisely, h o w e v e r , the excess length o f the Saros o v e r 6 5 8 5 days c a n vary b e t w e e n about 6 1/3 a n d 9 h o u r s . This variation is d u e to solar a n d lunar anomaly. I n S y s t e m A o f the A C T lunar theory, the length o f t h e S a r o s a s s u m i n g a constant solar velocity o f 30° p e r m o n t h is g i v e n b y a d d i n g to 6 5 8 5 d a y s the famous - i n d e e d infamous - c o l u m n ^ w h o s e origin a n d exact m e a n i n g h a s caused m u c h d e b a t e over the p a s t fifty years. C o l u m n $ is defined b y a linear z i g z a g fimction with m a x i m u m 2 , 1 7 ; 4 , 4 8 , 5 3 , 2 0 U § , m i n i m u m 1,57;47,7,46,40 U § , a n d m o n t h l y difference 2 ; 4 5 , 5 5 , 3 3 , 2 0 U § , b u t tiimcated at 2 , 1 3 ; 2 0 U S a n d 1,58;31,6,40 U S w h e n actually b e i n g used. A fiinction to correct this value for the solar velocity w o u l d also b e n e e d e d t o obtain the actual length o f a Saros, b u t such a fimction is so far unattested in the p r e s e r v e d texts. In a n y case, the fimction «ï> w a s almost certainly c o n s t m c t e d m a n y years after eclipses times h a d first b e e n predicted. T h u s w e m u s t look for other values o f the length o f the S a r o s that c o u l d h a v e b e e n u s e d in m a k i n g the predictions o f the times o f eclipses. It j u s t so h a p p e n s that the p r e s e r v e d

"

STEELE (2000b).

'"

STEELE (forthcomi ng). STEELE and STEPHENSON ( 1 9 9 7 ) , STEELE ( 1 9 9 7 ) . BROWN and LINSSEN ( 1 9 9 7 ) .

" From a Babylonian perspective the situation is slightly complicated by the fact that times were measured relative to sunrise or sunset, and so any method of predicting the time of an eclipse needed to take into account the change in the length of day throughout the year.

412

J.M.Steele

fragments o f the large c o m p i l a t i o n o f limar eclipses m e n t i o n e d a b o v e m a y p r o v i d e a tantalising g l i m p s e into w h a t these values were. O f the three p r e s e r v e d fragments o f the large c o m p i l a t i o n o n l y o n e , L B A T * 1 4 1 4 , h a s part o f the left e d g e remaining. T h e p r e s e r v e d part c o v e r s o n l y o n e r o w ( r o w 35) o f the m a t r i x o n the o b v e r s e , and t w o ( r o w s 3 4 a n d 3 5 ) o n the reverse. B e c a u s e the tablet t u m s sideways rather than from t o p to b o t t o m as is usual, these entries c o r r e s p o n d to c o l u m n s I a n d X X I V o f the compilation, a n d thus p r o v e its extent. Just before the date o f the eclipse in c o l u n m I o f the o b v e r s e the n u m b e r 1,50 is written. N o similar n u m b e r is found e l s e w h e r e o n the tablets in the c o m p i l a t i o n , a n d so it s e e m s p o s s i b l e that this n u m b e r was intended to refer to all o f the eclipses m that r o w o f the matrix. P r e s u m a b l y sùnilar n u m b e r s w e r e written at the b e g i i m i n g o f every r o w o f the matrix b u t these are n o w lost. H o w e v e r , the n u m b e r s 1,40 a n d 2,10 are found written o n the e d g e o f L B A T 1 4 1 3 . ' ^ T h i s text is a collection o f eclipse r e c o r d s apparently covering the accession a n d first few y e a r s of t h e r e i g n of N a b o n a s s a r (i.e., 7 4 7 B C o n w a r d s ) . T h e n u m b e r s 1,40 a n d 2 , 1 0 are written n e x t to the eclipses of 6 F e b m a r y 7 4 7 B C a n d 2 A u g u s t 7 4 7 B C , w h i c h w o u l d c o r r e s p o n d to the eclipses in r o w s 1 a n d 2 o f c o l u m n I o f the large c o m p i l a t i o n . I a m not suggesting, or at least a m not requiring in the analysis g i v e n b e l o w , that these n u m b e r s are exactiy t h o s e that w o u l d h a v e b e e n found o n the e d g e o f the large c o n ç i l a t i o n h a d it b e e n preserved. It is sufficient to a s s u m e that n u m b e r s o f this type w e r e written there. A s fnst r e c o g n i s e d b y B R O W N ( 2 0 0 0 ) , p . 2 0 5 , a n d t h e n i n d e p e n d e n t i y b y m y s e l f a n d Lis B r a c k - B e m s e n , ^ " all three n u m b e r s ( 1 , 4 0 a n d 2,10 o n L B A T 1413 a n d 1,50 o n L B A T * 1 4 1 4 ) ^ ' are, if they are a s s u m e d to b e in U S , of the right o r d e r o f m a g n i t u d e to represent the excess length of the Saros o v e r 6 5 8 5 d a y s (hereafter j u s t called the length o f the Saros). F u r t h e r m o r e , they actually agree fafrly well w i t h the c o m p u t e d mtervals b e t w e e n the times o f eclipses in their particular S a r o s series. F o r e x a n p l e , as B r o w n has remarked, m o d e m eclipse c a n o n s s h o w that the p r e d i c t e d time of the eclipse o f 9 A p r i l 7 3 1 w a s roughly 1,50 U S later in the d a y t h a n the time of the visible eclipse o n e Saros earlier o n 28 M a r c h 7 4 9 . It is worthwhile mvestigating h o w the length o f the Saros c h a n g e s for e a c h eclipse p r e d i c t i o n in the Saros scheme.^^ Figure 2 s h o w s the length o f the S a r o s for

'* The number 2,10 on LBAT 1413, 3 has only become readable after recent cleaning of the tablet. To my knowledge the other two numbers have only been commented upon twice, first by Huber in his unpublished manuscript of 1973 where he remarks on p. 2 that "the number 1,40 in line 1 (of LBAT 1413) is unexplained, but compare also the next text" (i.e., LBAT *1414 where we have the number 1,40), and then by BROWN (2000), p. 205 (see below). "

On the problems of dating this text, see, for example, STEELE (2000a), pp. 43-45.

^° Reported in my Appendix to HUNGER (2001), to which a reference to Brown's publication should therefore be added to footnote 4. Brown associates these numbers with the actual eclipses next to which they are written, whereas we believe that at least the number on LBAT *1414, and probably also those on LBAT 1413, apply to all eclipses in those rows of the Saros scheme. At this point I wish to express my deepest thanks to Lis Brack-Bemsen who initially suggested investigating how the Saros length varies for the different eclipses, and who generously helped compile all of the relevant data. The data itself was extracted from a printout of the times of syzygies calculated by Peter Huber and kindly made available to Lis Brack-Bemsen. As mentioned, the times of the Babylonian eclipse observations and

A Simple Function for the Length of the Saros in Babylonian Astronomy

413

e a c h ecUpse in the s c h e m e . T h e eclipses are d e n o t e d b y their n u m b e r within this s c h e m e , so the eclipse in r o w 1 c o l u m n I is eclipse n u m b e r 1, the eclipse in r o w 3 8 c o l u m n I is eclipse n u m b e r 3 8 , and the eclipse in r o w 1 c o l u m n II is eclipse n u m b e r 3 9 , etc. T h u s a n eclipse at r o w r in c o l u m n c has eclipse n u m b e r = r + 3 8 c . It s h o u l d b e n o t e d that eclipse n u m b e r d o e s not m e a s u r e a u n i f o r m timescale; generally a n interval of o n e eclipse n u m b e r c o r r e s p o n d s to six m o n t h s , b u t occasionally it is o n l y five m o n t h s . T h e o n l y uniform time unit is 3 8 eclipse niunbers, w h i c h c o r r e s p o n d s to one Saros. In figure 1 only the first 95 eclipse n u m b e r s ( c o r r e s p o n d i n g to 2 1/2 Saroi) are s h o w n . 150

-1

19

38

57

76

Eclipse Number Figure 2. T h e length of the Saros in excess of 6 5 8 5 days b e t w e e n o n e eclipse a n d the next in its Saros series. It is evident from figure 2 that the data splits into two sets b a s e d u p o n w h e t h e r the eclipse n u m b e r is o d d or even. T h i s is to b e expected, of course, since there is in general a six-month interval b e t w e e n o d d a n d e v e n n u m b e r s , a n d so the solar a n o m a l y is at a p p r o x i m a t e l y opposite points in its yearly cycle. If w e think o f e a c h dataset as a separate ftinction, w e see that o n e is a p p r o x i m a t e l y a reflection o f the other in the m e a n value of the fiinction. F u r t h e r m o r e , b o t h fiinctions s e e m to r e p e a t after 1 Saros. T h i s w o u l d b e fairly o b v i o u s from observations m a d e o v e r say three or four Saros cycles, a n d I think it quite plausible to a s s u m e that these t w o features o f the variation in the length of the Saros w e r e k n o w n to the B a b y l o n i a n astionomers. Since the o d d a n d e v e n fimctions are related, it is o n l y n e c e s s a r y to l o o k at o n e set of data to investigate their b e h a v i o u r over longer timescales. In the followmg a r g u m e n t I will o n l y consider the o d d n u m b e r e d eclipses, although the e v e n n u m b e r e d o n e s w o u l d d o j u s t as well. F i g i u e 3 s h o w s the o d d n u m b e r e d eclipses over 11 Saroi. A l t h o u g h the ftinction is still basically periodic w i t h a p e r i o d o f 1 Saros, clearly there are also other factors that c o m e into p l a y o n longer timescales. predictions relate to first contact, not syzygy, but since we are here taking the difference between the times of eclipses separated by one Saros (and so having almost the same duration), the discrepancy is negligible.

414

J.M. Steele

In fact it w o u l d a p p e a r that the S a r o s length is a siunmation o f t w o fimctions: a d o m i n a t m g o n e w i t h a p e r i o d o f 1 Saros, a n d a small p e r t m b a t i o n h a v i n g the effect o f s o m e t i m e s reinforcing the d o m i n a t i n g function, s o m e t i m e s dimiitishing it, w h i c h drifts earlier a n d earlier w i t h respect to the d o m i n a t i n g function. P r e s m n a b l y the drift o f this pertiu-bation is c a u s e d b y the fact that the S a r o s is a b o u t 10 d a y s longer t h a n a w h o l e n u m b e r o f years, a n d so there is a small c h a n g e m solar a n o m a l y after 1 Saros.

0

38

76

114

152

190

228

266

304

342

380

418

Eclipse Number F i g u r e 3 . T h e length o f the S a r o s associated with the o d d n u m b e r e d eclipse n u m b e r s . T h e figures j u s t d i s c u s s e d w e r e all b a s e d u p o n m o d e m determinations o f the times o f syzygy. Before w e c a n go any further w e n e e d to c o n s i d e r h o w m u c h o f the information in these figmes c o u l d h a v e b e e n extracted from the Babyloitian r e c o r d s of eclipses. N o t all o f the eclipses possibilities p r e d i c t e d b y this S a r o s s c h e m e could h a v e b e e n o b s e r v e d . In fact, less than a half o f the d a t a points c o u l d h a v e b e e n obtained from the eclipse record. F u r t h e r m o r e , the distribution o f o b s e r v a b l e data p o m t s is subject to a selection effect; for e x a n p l e , the eclipse possibilities j u s t before a five-month mterval in the Saros s c h e m e almost n e v e r result in an o b s e r v a b l e eclipse. T h i s w o u l d m a s k s o m e o f the perturbations n o t e d a b o v e . M o r e seriously, h o w e v e r , the m e a s m e d S a r o s lengths are subject to timing errors o n the t w o terminal eclipses. Before about 5 7 0 B C , eclipses w e r e a p p a r e n t l y o n l y t u n e d to the nearest 5 U â , a n d although the later eclipses w e r e timed to the nearest U â , their actual a c c u r a c y w a s n o greater than that of the earlier eclipses. M u c h o f the fine detail in the figures, therefore, w o u l d b e u n o b s e r v a b l e . In fact, I believe that all that c o u l d h a v e b e e n d e d u c e d about the function in figure 3 from the o b s e r v e d eclipse r e c o r d w a s that it h a d a m a x i m u m o f about 2,15 U S , a m i n i m u m o f a b o u t 1,35 Ul§, a n d h a d the basic form o f a linear zigzag function with a p e r i o d o f 1 S a r o s . T h e linear z i g z a g fimction w a s one of the fimdamental t e c h n i q u e s for d e a l i n g with varying fimctions in M e s o p o t a m i a n astronomy, stretching b a c k as far as e x p r e s s i o n s for the c h a n g e in d a y a n d night lengths in Enuma Anu Enlil 14, a n d still u s e d extensively in the A C T asfronomy o f the late p e r i o d , so it is not surprising that it w o u l d h a v e b e e n c h o s e n to m o d e l this variation.

415

A Simple Function for the Length of the Saros in Babylonian Astronomy

T h e r e a r e 19 o d d eclipse n u m b e r s in o n e S a r o s , a n d s o this z i g z a g fimction h a s a p e r i o d o f 19 eclipse n u m b e r s . T h e u n d e r l y i n g p e r i o d relation o f a z i g z a g function is given by P = 2A Id. Unfortunately, 19 is not a n i c e n u m b e r a n d so o n e c a n n o t o b t a i n integer n u m b e r s for t h e a n ç l i t u d e a n d difference o f a z i g z a g function w i t h 19 as its period. H o w e v e r , 1 S a r o s is o n l y slightly m o r e than 18 years ( i n d e e d the S a r o s w a s often simply referred t o as 18 M U . M E § " 1 8 y e a r s " in c u n e i f o r m a s t r o n o m i c a l texts), a n d 18 d o e s lead t o n i c e n u m b e r s for the p a r a m e t e r s o f the z i g z a g function. If w e take the a n ç l i t u d e to b e 4 5 U S , w e simply get a difference o f 5 US p e r year. T h e s e are n i c e r o u n d n u m b e r s . F u r t h e r m o r e , t h e y are j u s t the kind o f n u m b e r s that w o u l d c o m e out o f l o o k i n g at eclipses t i m e d t o tìie nearest 5 U § . T h e y are also the s a m e kind o f n u m b e r s that s e e m to h a v e b e e n written o n the e d g e o f t h e large con^ilation. It w o u l d therefore s e e m to b e p o s s i b l e to derive a simple function to give the length o f the S a r o s o v e r 6 5 8 5 d a y s that c o u l d b e u s e d in p r e d i c t i n g the times o f lunar eclipses. A s w i t h the Saros s c h e m e for identifying eclipse possibilities this function, o n c e d e v i s e d , c o u l d also b e applied t o solar eclipses. T h e function h a s a p e r i o d o f 18 y e a r s , a n a n ç l i t u d e of 4 5 U S , a n d a yearly difference o f 5 U S . B u t this is exactly the function found on B M 4 5 8 6 1 . F u r t h e r m o r e , the m a g n i t u d e o f the function o n B M 4 5 8 6 1 is in v e r y g o o d a g r e e m e n t with o b s e r v e d S a r o s lengths. I therefore suggest that this is v e r y likely w h a t the z i g z a g function o n B M 4 5 8 6 1 represents. T h i s interpretation also p r o v i d e s a r e a d y e x p l a n a t i o n for w h y the n m n b e r s o n B M 4 5 8 6 1 are designated the tardïtu "the addition": they are the extra a m o u n t b y w h i c h a S a r o s is in excess o f a w h o l e n u m b e r o f d a y s . 150

0

38

76

114

152

190

228

266

304

342

380

418

Eclipse Number Figure 4 . T h e variation o f the Saros length for the o d d n u m b e r e d eclipses. T h e plain line gives the Saros lengths associated w i t h the o d d n u m b e r e d eclipses a n d the d a s h e d line is the z i g z a g function o n B M 45861. Figure 4 s h o w s the zigzag function from B M 4 5 8 6 1 s u p e r i m p o s e d o n the empirical function for the length of the Saros. A l t h o u g h there is not a perfect m a t c h , the a g r e e m e n t is still v e r y g o o d . A s explained a b o v e , the slightly h i g h v a l u e for the m a x i m u m o f the z i g z a g function is a result o f the desire to use nice n u m b e r s in the

416

J.M.Steele

function, a n d w o u l d in any case b e m a s k e d s o m e w h a t b y the errors in the clocks u s e d to time the eclipses. T h e r e is also not a perfect m a t c h b e t w e e n the p e r i o d o f the function from B M 4 5 8 6 1 and nature. T h i s is simply a refection o f the fact that the S a r o s is not a perfect eclipse period. Nevertheless, the fimction g i v e s a sufficiently g o o d fit to nature that it c o u l d b e u s e d for at least 12 Saros cycles, p e r h a p s longer, without serious error. T o s u m m a r i s e , therefore, I believe that the zigzag fimction o n B M 4 5 8 6 1 represents the variation in the length of the Saros at yearly intervals, a n d that it w a s p r o b a b l y u s e d to p r e d i c t the times of eclipses.

Historical Implications In the p r e c e d i n g section I h a v e s h o w n that the simple zigzag function o n B M 4 5 8 6 1 is j u s t tiie type of function that could b e u s e d in p r e d i c t m g the times o f eclipses, and that a fimction of this type could b e obtained fairly easily from a collection o f t i m e d lunar eclipse observations. F u r t h e r m o r e , I b e l i e v e it is quite p o s s i b l e that the n u m b e r s written o n the e d g e of L B A T 1413 and * 1 4 1 4 m a y h a v e c o m e from a fimction o f this kind, or indeed p o s s i b l y from this v e r y function. All o f this certainly suggests that this interpretation is plausible, b u t I w o u l d like to m a k e a few cautionary r e m a r k s . If this simple function d o e s indeed represent the length of the Saros a n d w a s u s e d to p r e d i c t the times o f eclipses, it is s o m e w h a t sttange that w e a p p a r e n t l y h a v e so little e v i d e n c e for it. T h i s m a y well b e , of course, simply b e c a u s e the a p p r o p r i a t e texts w h i c h w o u l d describe it h a v e not b e e n preserved, or h a v e not b e e n studied in recent times. It s e e m s to m e quite possible that a function of this type c o u l d h a v e b e e n d e v e l o p e d fairly early. O n e w o u l d need, say, only a b o u t 100 years w o r t h of empirical data to constinct it, and it s e e m s that eclipses w e r e o b s e r v e d o n a regular basis fi-om the m i d - e i g h t h century B C o n w a r d s . V e r y few original a s t i o n o m i c a l texts are p r e s e r v e d from this early time. It is generally the case that our imderstanding of the p r o c e d u r e s u s e d b y the B a b y l o n i a n a s t i o n o m e r s on a day-by-day basis for predicting a s t t o n o m i c a l p h e n o m e n a in the n o n - m a t h e m a t i c a l texts is m u c h p o o r e r than o f the m a t h e m a t i c a l a s t t o n o m y . T h u s w e d o n o t k n o w exactly h o w the goal-year p e r i o d s w e r e u s e d to obtain the data in the A l m a n a c s etc. Part of the r e a s o n for this is that they are less a m e n a b l e to mathematical analysis than the A C T - t y p e asfronomy. B u t a n o t h e r is that the p r o c e d u r e texts that give t h e rules for this type of a s t t o n o m y h a v e b e e n studied m u c h less t h o r o u g h l y than the A C T p r o c e d u r e texts. Therefore, unless other evidence for this simple function can b e found, I d o not believe w e c a n b y fully confident that the interpretation I p r o p o s e here is correct, a l t h o u g h I will b e surprised if it is not. O f course, the m o s t likely source of such c o n f m n a t i o n w o u l d b e if a j o i n to this p r e s e n t fragment c o u l d be m a d e , a n d this is s o m e t h i n g I continue to look for. If the interpretation p r e s e n t e d here is indeed correct, then it still leaves several questions u n a n s w e r e d . H o w w o u l d such a function b e u s e d in practice? T h e function has only 18 values, c o r r e s p o n d i n g to the 18 years o f the Saros, b u t within those 18 years there are 19 o d d and 19 even eclipse possibilities. T h e o d d a n d e v e n functions are j u s t m i r r o r images o f each other, a n d so the asfronomers could get from o n e to the other quite easily, b u t still s o m e h o w 18 years h a v e to b e e q u a t e d with 19 eclipse possibilities. W a s a value repeated if there were two o d d eclipses in the same

A Simple Function for the Length of the Saros in Babylonian Astronomy

417

calendar year (say in M o n t h s I a n d XII2)? H o w were the five-month intervals c o p e d with? G i v e n that this fimction c a n b e d e d u c e d directly from empirical data, c a n w e say anything a b o u t w h e n the fiinction w a s c o n s t i n c t e d ? D o e s this type of fimction a n d the empirical data it is b a s e d u p o n tell u s anything a b o u t the lunar a n d solar a n o m a l y ? Lis B r a c k - B e m s e n a n d I will discuss s o m e of these questions e l s e w h e r e . O n e final issue that is w o r t h discussing here is h o w this simple fimction for the length o f the Saros relates to the m o r e c o m p l i c a t e d o n e found in the A C T asfronomy. T h e simple zigzag fimction o n B M 4 5 8 6 1 is, in effect, a n empirical fiinction. It uses a simple mathematical s c h e m e to reflect the o b s e r v e d c h a n g e in the Saros length at different eclipse possibilities. B y contiast, the Saros length in lunar S y s t e m A of the A C T asfronomy is given b y c o l u m n $ w h i c h m o d e l s the c h a n g e in tìie length of the Saros at e a c h syzygy, a s s u m i n g a solar velocity of 3 0 ° p e r m o n t h . T o get the t m e length of the Saros, a correction has to b e applied to $ to t a k e into a c c o u n t the c h a n g e in solar velocity. T h i s solar correction c a n b e d e d u c e d from the tight theoretical s t m c t u r e of S y s t e m A,^^ b u t it h a s not yet b e e n attested in any texts. T h u s , o n the b a s i s of the p r e s e r v e d texts it w o u l d a p p e a r that the t m e length o f the S a r o s c a n n o t b e (or rather w a s not) calculated in S y s t e m A . F u r t h e r m o r e , as Lis B r a c k - B e m s e n has s h o w n , c o l u m n $ c a n n o t b e obtained directiy from, say, eclipse observations b e c a u s e the solar confribution to tìie variation in the length of the Saros is greater than the lunar contiibution.^'* T h u s c o l u m n 4> m u s t either b e interpreted slightly differently, or b e s o m e w h a t r e m o v e d from an empfrical function. I d o n o t think that the simple function for the Saros length could h a v e b e e n u s e d directly in the c o n s t m c t i o n o f c o l u m n $ . H o w e v e r , it c o u l d well h a v e b e e n the first step o n the j o u r n e y . It reflects the realisation that there is such a thing as a variation in the lengtii of the Saros, a n d that this c a n b e m o d e l l e d arithmetically. U s e o f this fimction over a n u m b e r of years m a y h a v e led to the a w a r e n e s s that the variation in the Saros is not a simple periodic function, but actually a s u m m a t i o n o f t w o fimctions with different p e r i o d s . This p e r h a p s points the w a y to attempting to separate out these t w o fimctions, w h i c h t u m out to b e d u e to lunar a n d solar anomaly. In addition, the simple function p r o v i d e d a v e r y g o o d m e a n value for the length o f the Saros o f 6 5 8 5 days plus 1,57;30 U S . O f course, the a c c u r a c y o f this value is in part fortuitous, b e i n g i m p o s e d b y the use o f nice n u m b e r s in the z i g z a g function. B u t it leads to a m e a n length for the synodic m o n t h o f 2 9 ; 3 1 , 5 0 , 1 2 , . . . w h i c h c o m p a r e s v e r y favourably with the well k n o w n S y s t e m B m e a n s y n o d i c m o n t h of 2 9 ; 3 1 , 5 0 , 8 , 2 0 that w a s later a d o p t e d b y H i p p a r c h u s . T h e existence o f this simple Saros function also serves to blur the line b e t w e e n the n o n - m a t h e m a t i c a l a s t i o n o m i c a l texts a n d the m a t h e m a t i c a l a s t i o n o m i c a l texts. T h e division o f B a b y l o n i a n a s t i o n o m i c a l texts into these t w o categories is well established a n d useful - 1 h a v e u s e d the t e r m s frequently t h r o u g h o u t this p a p e r - b u t w e m u s t not forget that it is anachronistic, a n d p o s s i b l y misleading. A s f r o n o m y that w e customarily call n o n - m a t h e m a t i c a l - n a m e l y the p r e d i c t i o n o f eclipses for the Diaries a n d related texts - turns out to m a k e use o f a m a t h e m a t i c a l d e v i c e , the zigzag function. W e should not b e surprised about this, o f c o u r s e , since it w a s the s a m e individuals w h o w e r e responsible for the n o n - m a t h e m a t i c a l a n d the

"

AABOE(1969). BRACK-BERNSEN ( 1 9 8 0 ) .

418

J.M.Steele

m a t h e m a t i c a l a s t r o n o m i c a l texts.^^ T h e r e are i n d e e d differences b e t w e e n w h a t w e classify a s these t w o k i n d s o f text, b u t they a r e p e r h a p s m o r e subtle t h a n h a s generally b e e n a c k n o w l e d g e d . W h e r e a s t h e n o n - m a t h e m a t i c a l texts o n l y u s e m a t h e m a t i c a l tools s u c h a s the zigzag fimction to m o d e l o b s e r v a b l e p h e n o m e n a such a s t h e length o f d a y , the daily retardation o f the m o o n , or, m this case, the length o f t h e S a r o s , c o l i m m s m the m a t h e m a t i c a l a s t r o n o m i c a l texts m a y relate t o p u r e l y constructed fimctions without a physical, o b s e r v a b l e m e a n i n g , c o l u m n s $ a n d G b e m g g o o d e x a m p l e s . I n addition, in t h e n o n - m a t h e m a t i c a l a s t r o n o m i c a l texts zigzag fimctions a l w a y s h a v e a n integer p e r i o d ( e g . 12 m o n t h s , 3 0 d a y s , 18 years), w h e r e a s m the m a t h e m a t i c a l texts, the fi-equently h a v e non-integer p e r i o d s ( e g . t h e m o n t h l y variation in c o l u m n 4> h a s p e r i o d o f 1 3 ; 5 6 , 3 9 , 6 , . . . ) . T h e relationship b e t w e e n t h e g o a l s a n d techitiques o f the n o n - m a t h e m a t i c a l a n d t h e m a t h e m a t i c a l a s t r o n o m i c a l texts still requires m u c h m o r e clarification if w e a r e t o g e t a n y closer to u n d e r s t a n d i n g t h e practice o f a s t i o n o m y in t h e L a t e B a b y l o r u a n p e r i o d .

Acknowledgements I a m greatly i n d e b t e d to L i s B r a c k - B e m s e n for h e r h e l p in u n c o v e r i n g t h e m e a i u n g o f t h e n u m b e r s o n B M 4 5 8 6 1 . M y w o r k o n t h e eclipse texts h a s b e e n greatly mfluenced b y m a n y suggestions m a d e t o m e b y C h r i s t o p h e r W a l k e r a n d H e r m a i m H u n g e r in various discussions over the past few years; without their h e l p it is doubtful if a n y o f this w o r k w o u l d h a v e b e e n p o s s i b l e . I also w i s h t o t h a n k D a v i d B r o w n for c o m m e n t s o n a n earlier draft o f this paper. A l l errors a n d i n p e r f e c t i o n s that r e m a i n a r e , o f c o u r s e , m y responsibility a l o n e . B M 4 5 8 6 1 is p u b l i s h e d b y k i n d p e r m i s s i o n o f the T r u s t e e s o f the British M u s e u m . M y w o r k o n tiie text w a s m a d e possible b y a L e v e r h u l m e T m s t R e s e a r c h F e l l o w s h i p a n d a R o y a l S o c i e t y R e s e a r c h Grant.

Abbreviations A C T = NEUGEBAUER (1955)

L B A T = SACHS (1955) M C T = N E U G E B A U E R and S A C H S (1945)

References A A B O E , A s g e r . 1 9 6 9 . " A C o m p u t e d List o f N e w M o o n s for 3 1 9 B C to 3 1 6 B C fi-om B a b y l o n : B M 4 0 0 9 4 " . Det Kongelige Danske Videnskabemes Selskab Matematisk-fysiske Meddeleser 3 7 : 3 . — 1980. " O b s e r v a t i o n a n d T h e o r y in B a b y l o n i a n A s t r o n o m y " . Centaurus 2 4 : 1 4 35. B R A C K - B E R N S E N , L i s . 1980. " S o m e Investigations o n the E p h e m e r i d e s o f the B a b y l o n i a n M o o n T e x t s , S y s t e m A " . Centaurus 2 4 : 3 6 - 5 0 . — 1 9 9 7 . Zur Entstehung der babylonischen Mondtheorie. Stuttgart: F r a n z Steiner Verlag.

ROCHBERG (1993,2000).

A Simple Function for the Length of the Saros in Babylonian Astronomy

419

— 1999. " G o a l - Y e a r T a b l e t s : L u n a r D a t a a n d P r e d i c t i o n s " . In: N o e l M SWERDLOW (ed.), Ancient Astronomy and Celestial Divination: 149-178. Cambridge, MA: The M I T Press. B R O W N , D a v i d . 2 0 0 0 . Mesopotamian Planetary Astronomy-Astrology. Groningen: Styx. — a n d L I N S S E N , M a r c . 1 9 9 7 . " B M 134701 = 1965-10-14,1 a n d the Hellenistic P e r i o d E c l i p s e Ritual from U r u k " . Revue d'Assyriologie 91: 147-166. H U B E R , Peter J. 1 9 7 3 . Babylonian Eclipse Observations 750 BC - 0. U n p u b l i s h e d manuscript. H U N G E R , H e r m a n n . 2 0 0 1 . Astronomical Diaries and Related Texts from Babylonia Volume 5. W i e n : Osterreichische A k a d e m i e d e r Wissenschaften. N E U G E B A U E R , O t t o . 1955. Astronomical Cuneiform Texts. London: Lund Humphries. — a n d S A C H S , A b r a h a m J. 1 9 4 5 . Mathematical Cuneiform Texts. N e w H a v e n : A m e r i c a n Oriental Society. P O W E L L , M a r v i n A . 1 9 8 8 . " E v i d e n c e for Agriculture a n d W a t e r w o r k s in B a b y l o n i a n M a t h e m a t i c a l T e x t s " . Bulletin on Sumerian Agriculture 4 : 1 6 1 - 1 7 2 . R O C H B E R G , F r a n c e s c a . 1 9 9 3 . " T h e Cultural L o c u s o f A s t r o n o m y in L a t e B a b y l o n i a " . In: H a n n e s D . Gaiter (ed.), Die Rolle der Astronomie in den Kulturen Mesopotamiens ( G r a z e r M o r g e n l a n d i s c h e Studien 3 ) : 3 1 - 4 5 . G r a z : RM-Verlag — 2 0 0 0 . " S c r i b e s a n d Scholars: T h e tupsar Enuma Anu EnliF. In: J o a c h i m M A R Z A H N a n d H a n s N E U M A N N (eds.), Assyriologica et Semitica: Festschrift fiir Joachim Oelsner: 3 5 9 - 3 7 5 . Miinster: U g a r i t V e r l a g . R E A D E , Julian E . 1986. " R a s s a m ' s B a b y l o n i a n Collection: T h e E x c a v a t i o n s a n d the A r c h i v e s " . In: E r i e LEICHTY (ed.), Catalogue of the Babylonian Tablets in the British Museum Volume VI: Tablets from Sippar I: x i i - x x x v i . L o n d o n : British M u s e u m Publications. S A C H S , A b r a h a m J. 1 9 5 5 . Late Babylonian Astronomical and Related Texts Copied by T. G. Pinches and J. N. Strassmaier. P r o v i d e n c e : B r o w n University P r e s s . STEELE, J o h n M . 1997. " S o l a r Eclipse T i m e s P r e d i c t e d b y the B a b y l o n i a n s " , Joumal for the History of Astronomy 2 8 : 1 3 3 - 1 3 9 . — 2 0 0 0 a . Observations and Predictions of Eclipse Times by Early Astronomers. D o r d r e c h t : K l u w e r A c a d e m i c Publishers. — 2 0 0 0 b . " E c l i p s e P r e d i c t i o n in M e s o p o t a m i a " . Archive Science 5 4 : 4 2 1 ^ 5 4 .

for

History

— forthcoming. " T h e M e a n i n g o f B A R D I B in Late B a b y l o n i a n T e x t s " . Archiv fiir Orientforschung, forthcoming. —

of

Exact

Astronomical

and S T E P H E N S O N , F . R i c h a r d . 1997. " L u n a r E c l i p s e T i m e s P r e d i c t e d b y the B a b y l o n i a n s " . Joumal for the History of Astronomy 2 8 : 1 1 9 - 1 3 1 . S W E R D L O W , N o e l M . 1 9 9 8 . The Babylonian Theory of the Planets. Princeton: P r i n c e t o n U n i v e r s i t y Press. W A L K E R , Christopher. 1997. " A c h a e m e n i d C h r o n o l o g y and the B a b y l o n i a n S o u r c e s " . In: J o h n C U R T I S (ed.), Mesopotamia and Iran in the Persian Period: Conquest and Imperialism 539-331 BC: 1 7 - 2 6 . L o n d o n : British M u s e u m Publications.

420

J.M. Steele

B M 4 5 8 6 1 (Copyright T h e British M u s e u m )

The Earliest Datable Observation of the Aurora Borealis F. Richard Stephenson,

Durham and David M. Willis,

Warwick

Introduction T h e Late B a b y l o n i a n astronomical diary from the 37th year o f K i n g N e b u c h a d r e z z a r II ( = 5 6 8 - 5 6 7 B C ) is well k n o w n ; it is the o n l y diary extant from his reign. A m o n g the m a n y astronomical r e c o r d s p r e s e r v e d o n the tablet is at clear reference to the aurora b o r e a l i s o r " n o r t h e m lights". A s far as w e are a w a r e , this is the o n l y k n o w n example o f a n auroral display r e c o r d e d o n the extant Late B a b y l o n i a n a s t r o n o m i c a l texts. F u r t h e r m o r e , although later C h i n e s e reports o f aurorae - from a b o u t 2 0 0 B C o n w a r d s - are n u m e r o u s , the B a b y l o n i a n observation is b y far the earliest datable reference to this p h e n o m e n o n in the history of any civilisation. A l t h o u g h the A s s y r i a n tablet R m 2 1 1 ( R M A 2 7 5 ) m e n t i o n s a r e d g l o w (a-ku-ku-tum), this is the form o f conditional statements a n d there is n o specific m e n t i o n o f t h e p h e n o m e n o n occurring at night.' A u r o r a e are typically seen as m o v i n g b a n d s o f Ught o f various colours, w h i c h are p r o d u c e d in the u p p e r a t m o s p h e r e o f the E a r t h - usually at altimdes o f b e t w e e n a b o u t 8 0 a n d 5 0 0 k m . T h e y are caused b y collisions b e t w e e n i n c o m i n g ionised particles, w h i c h spiral a r o u n d the lines of force o f the E a r t h ' s m a g n e t i c field, and the n i t i o g e n a n d o x y g e n a t o m s o f the terrestrial a t m o s p h e r e . T h e s e collisions result in the emission o f light, similar to the o p e r a t i o n o f a fluorescent tube. Characteristic colours are green, b l u e , white and red. A u r o r a e are m o s t frequent in h i g h latitudes. H o w e v e r , at sites far from the magnetic p o l e s o f the E a r t h - such as B a b y l o n - t h e y are e x t i e m e l y r a r e , occurring o n average p e r h a p s o n c e in a b o u t 5 0 years. A l t h o u g h the B a b y l o n i a n a s t i o n o m i c a l diaries r a n g e o v e r s o m e 7 0 0 years, m o s t o f t h e texts are v e r y fragmentary and less t h a n 10 p e r cent o f the original material a p p e a r s to b e extant. W e are thus fortunate that e v e n the isolated auroral report from t h e time o f K i n g N e b u c h a d r e z z a r h a s survived.

The Babylonian Auroral Record A p h o t o g r a p h o f the r e v e r s e . o f the tablet ( V A T 4 5 9 6 ) is p u b l i s h e d b y S A C H S and H U N G E R ( 1 9 8 8 ) . T h e report describing the aurora is the last d a t e d entry in lunar m o n t h X I . A tiansliteration b y Sachs and Hunger^ of this r e c o r d r e a d s as follows: GEfi 2 9 a-ku6-ku6(-ku6)-tU4 ina § Û K U R 2 D A [ N N A . . . . ]

'

HUNGER ( 1 9 9 2 ) , p

^

SACHS and HUNGER ( 1 9 8 8 ) , p. 5 0 .

19.

422

F.R. Stephenson and D.M. WilHs

A s was r e c o g n i s e d b y N e u g e b a u e r a n d Weidner,^ the kuf, p l a c e d in p a r e n t h e s e s should b e omitted; it represents a scribal error. F o l l o w i n g Sachs and Himger,'* the report m a y b e translated as follows: " N i g h t o f the 2 9 t h (of lunar m o n t h XI), red g l o w flared u p in the west; 2 double-[hours]".

Nature of the phenomenon T h e Chicago Assyrian Dictionary (A, p . 2 8 5 ) in conmienting o n the a b o v e text, mterprets the g l o w in the sky as occurring " w h e n d u s k w a s falling". T h i s w o u l d , of course, i m p l y n o m o r e than a c o n m i o n m e t e o r o l o g i c a l p h e n o m e n o n : a " r e d sky at dusk". H o w e v e r , the u n i q u e n e s s o f the record a m o n g the vast n u m b e r o f Late B a b y l o n i a n astronomical diaries seems to p r e c l u d e this interpretation. Further, w e imderstand from P r o f H e r m a i m H i m g e r (personal c o m m u n i c a t i o n ) that the k e y w o r d K U R should b e franslated as " w e s t " rather t h a n " d u s k " . T h e diaries h a v e a very rigid terminology, and it is clear from m a n y parallel texts that K U R is u s e d to indicate direction only. H a d dusk b e e n mtended, it w o u l d b e e x p e c t e d that the t e r m for sun (samds) should h a v e b e e n mcluded. T h e event c o u l d h a v e o c c u r r e d at any time during the night; for instance there is n o m e n t i o n of the three night watches: U S A N , MURUB4 or Z A L A G . B o t h N e u g e b a u e r a n d W e i d n e r , a n d Sachs a n d H u n g e r r e n d e r 2 D A [ N N A . . . ] as 2 d o u b l e hours.^ T h i s p r e s u m a b l y alludes to the duration o f the event. A l t h o u g h the sign for D A N N A in the text is d a m a g e d . P r o f H u n g e r (personal c o m m u n i c a t i o n ) assures u s that there can b e little d o u b t about its interpretation. A s the p h e n o m e n o n t o o k p l a c e at the end o f a lunar m o n t h , interference b y m o o n l i g h t w o u l d b e negligible. Similar brief reports of a red light seen at night are found o n various dates in East A s i a n history. S o m e t i m e s these reports m e n t i o n constellations near w h i c h the light appeared. T h e r e is sufficient variety in the r e c o r d e d directions to indicate that the twilight or d a w n g l o w is not intended. E x a m p l e s from C h i n e s e a n d K o r e a n history m a y b e franslated as follows: A D 762 M a y 20 (Chma): "At night, at Jiangling, a r e d light w a s seen; it penefrated Beidou (Jiu Tangshu, chapter 3 6 ) .

( = the P l o u g h ) "

A D 1137 F e b 14 ( C h i n a ) : "At night, in the N direction, there was a red v a p o u r lasting until d a w n " (Songshi, chapter 64). A D 1116 O c t 10 ( K o r e a ) : "At night, a red v a p o u r w a s seen at the n o r t h - w e s t " (Koryo-sa, chapter 5 3 ) . F o r fiirther e x a m p l e s , see the catalogue c o m p i l e d b y Y a u et al.^ ^

NEUGEBAUER and WEIDNER ( 1 9 1 5 ) .

"

S A C H S a n d H u N G E R ( 1 9 8 8 ) , p. 5 1 .

'

NEUGEBAUER and WEIDNER ( 1 9 1 5 ) and SACHS and HUNGER ( 1 9 8 8 ) .

*

Y A U , STEPHENSON and WILLIS ( 1 9 9 5 ) .

The Eariiest Datable Observation of the Aurora Borealis

423

Date of the Late Babylonian Diary T h e tablet twice gives the date as the 37th year of N e b u c h a d r e z z a r - b o t h at the b e g i n n i n g of the o b v e r s e and at the e n d of the reverse. A l t h o u g h the text originally c o v e r e d a full year, the surviving section only covers parts of lunar m o n t h s I, II a n d III (obverse) a n d X , X I a n d X I I (reverse). T h e equivalent Julian date of 5 6 8 - 5 6 7 B C , as identified b y N e u g e b a u e r and W e i d n e r , a n d S a c h s and H u n g e r , s e e m s well e s t a b l i s h e d . ' H o w e v e r , it is important to verify this. T h e m a n y lunar o b s e r v a t i o n s w h i c h are p r e s e r v e d o n the tablet enable the date to b e investigated b y a s t r o n o m i c a l computation. T h i s investigation also enables the general reliability o f the astronomical r e c o r d s to b e tested. O b s e r v a t i o n s involving the m o o n are especially valuable for dating the B a b y l o n i a n astronomical diaries since the m o o n m o v e s so rapidly t h r o u g h the s k y o n average 13 d e g daily. T h e lunar observations on the tablet are o f t w o m a i n types: "lunar t h r e e s " (three time-intervals r e c o r d e d n e a r the beginning, m i d d l e a n d end of e a c h m o n t h ) ; a n d conjunctions of the m o o n with " N o r m a l Stars". W e shall c o n s i d e r these t w o sets o f data in t u m . In m a k i n g our lunar c o m p u t a t i o n s , w e h a v e initially a s s u m e d that the year 5 6 8 - 5 6 7 B C is correct and h a v e provisionally a d o p t e d the equivalent Julian dates derived from the tables of Parker a n d Dubberstein.^ W e h a v e c o m p a r e d the various observations with the results of c o m p u t a t i o n . In o u r calculations, w e h a v e m a d e allowance for both: (i) the gradual increase in the length o f the m e a n solar day d u e to tides a n d other causes; and (ii) the m o o n ' s parallax. A t the e p o c h 5 6 8 - 7 B C , the ciunulative c l o c k error A T a m o u n t s to a b o u t 5 h o u r s or about one-fifth o f a d a y . ' In this investigation w e h a v e m a d e full u s e of the transliterations, translations, editorial c o m m e n t s a n d star identifications of Sachs a n d H u n g e r . T h e B a b y l o n i a n d a y b e g a n at sunset, r o u g h l y six h o u r s before the m o d e m civil date. In o r d e r to a v o i d possible confiision, w e h a v e systematically u s e d civil dates in all calculations. Lunar Threes D u r i n g each lunar m o n t h the following three time-intervals w e r e systematically r e c o r d e d : (i) sunset to m o o n s e t on the first of the m o n t h - the first e v e n i n g that the y o u n g crescent m o o n b e c a m e visible after conjunction (na); (ii) s u m i s e to m o o n s e t a r o u n d the m i d d l e of the m o n t h - the first m o m i n g that almost full m o o n set after sunrise (na); (iii) m o o n r i s e to sunrise near the end of the m o n t h - the last m o m i n g that the w a n i n g crescent w a s visible before conjimction ( K U R ) . A t the latitude of B a b y l o n , about 3 2 . 5 ° , e a c h interval increases b y s o m e 12° o n average from o n e d a y to the next. H e n c e it is usually possible to derive the exact date b y c o m p a r i n g the r e c o r d e d interval with computation. In later diaries six time intervals w e r e r e c o r d e d e a c h m o n t h : four at full m o o n and one e a c h at the b e g i n n i n g a n d e n d o f the m o n t h . S e v e n "lunar t h r e e s " are p r e s e r v e d intact o n the tablet. H e r e w e shall give t w o examples.

''

NEUGEBAUER and WEIDNER ( 1 9 1 5 ) and SACHS and HUNGER ( 1 9 8 8 ) , p. 5 2 .

*

PARKER and DUBBERSTEIN ( 1 9 5 6 ) .

'

STEPHENSON and MORRISON ( 1 9 9 5 ) .

'°

SACHS and HUNGER ( 1 9 8 8 ) .

424

F.R. Stephenson and D.M. WilHs

(i) " M o n t h III, (the 1st of w h i c h w a s identical with) the 3 0 t h ( o f the p r e c e d i n g month)... sunset to m o o n s e t : 2 0 ° ..." F r o m the tables o f P a r k e r and D u b b e r s t e i n , " lunar m o n t h III b e g a n o n Jime 21 in 5 6 8 B C . T h e c o r r e s p o n d i n g civil date w a s the e v e n i n g of Jime 2 0 . W e c o m p u t e that o n this evening the interval b e t w e e n sunset a n d m o o n s e t at B a b y l o n w a s actually 2 2 . 7 ° . H o w e v e r , o n the p r e v i o u s a n d following evenings the respective intervals w e r e 6.4° a n d 3 7 . 0 ° . H e n c e the date a c c o r d i n g to P a r k e r a n d D u b b e r s t e i n is quite acceptable. (ii) " M o n t h XII, (the first o f w h i c h w a s identical with) the 3 0 t h ( o f the p r e c e d i n g month)... sunset to m o o n s e t : 2 5 ° , m e a s u r e d " . F r o m P a r k e r a n d D u b b e r s t e m , lunar m o n t h X I I b e g a n o n M a r c h 15 in 5 6 7 B C . T h e equivalent civil d a t e w a s the evening o f M a r c h 14. O n this e v e n i n g , the c o m p u t e d interval b e t w e e n sunset a n d m o o n s e t w a s actually 2 5 . 7 ° - a l m o s t identical to the m e a s i n e d value. O n the p r e v i o u s a n d following e v e n m g s the a p p r o p r i a t e intervals w e r e 10.0 a n d 4 1 . 8 ° . H e n c e o n c e again the tabular date is confirmed. O u r c o m p a r i s o n s b e t w e e n the various r e c o r d e d time intervals a n d their c o m p u t e d equivalents are s u m m a r i s e d in T a b l e 1. In this table, are listed the limar m o n t h , day, equivalent Julian date, n a m e of interval, m e a s u r e d time, c o m p u t e d time a n d difference. Month

Julian D a t e Computed Day Interval M e a s u r e d 4 14 568 M a y 5 3.5 I SR-MS 5 6 8 J u n 17 23 23.2 26 MR-SR II 568 Jun 20 20 22.7 III 1 SS-MS 1 5 6 7 F e b 12 14.5 17.0 XI SS-MS 5 6 7 M a r 14 S S - M S 25 25.7 XII 1 1.5 0.7 12 567 Mar 26 SR-MS XII T a b l e 1. Analysis o f "lunar threes": c o m p a r i s o n b e t w e e n m e a s u r e d values.

Difference 0.5 0.2 2.7 2.5 0.7 0.8 and computed

W e c o n c l u d e that the various lunar threes on the text are quite in k e e p i n g w i t h a date for the tablet o f 5 6 8 - 5 6 7 B . C . In addition, reference to T a b l e 1 reveals that e v e n at this early date, timing errors w e r e typically o f the o r d e r o f 1° - n o m e a n achievement. C o n j u n c t i o n s of t h e M o o n w i t h N o r m a l Stars F r o m an early period, the B a b y l o n i a n s recognised 31 " n o r m a l s t a r s " s p r e a d a l o n g the z o d i a c . W h e n the m o o n or a planet was in conjunction with o n e o f these stars, the separation in " c u b i t s " ( K Ù § ) was estmiated - usually to the nearest 1/2 cubit or a b o u t 1°. F o r a detailed study o f the angular equivalent o f the cubit, as d e t e r m i n e d from Late B a b y l o n i a n observations o f planetary conjunctions, see FATOOHI a n d S T E P H E N S O N ( 1 9 9 7 - 1 9 9 8 ) . T h e s e authors obtained a r e s u h o f a b o u t 2.2°.

''

PARKER and DUBBERSTEIN ( 1 9 5 6 ) .

The Earliest Datable Observation of the Aurora Borealis

425

In ail, s e v e n observations are p r e s e r v e d o n o u r text for w h i c h the star is identified. W e shall discuss three e x a m p l e s in detail b e l o w . F o r " b e g i n n i n g o f the night", or "first part o f the n i g h t " w e h a v e a s s u m e d a solar d e p r e s s i o n o f 10° in the w e s t - so that the s k y w a s dark e n o u g h to see s o m e stars. S u c h an a p p r o x i m a t i o n s e e m s a d e q u a t e for our p u r p o s e since the m o o n m o v e s at only a b o u t 0.5° in a n hour. 2 7 JUNE 5 6 8 B C : 20*' "MONTH III...NIGHT OF THE 8TH, FIRST PART OF THE NIGHT, THE MOON STOOD 2'<2 CUBITS BELOW P LIBRAE"

-+10"

-+9°

-+8"

2V2 CUBITS = 4 . 3 DEG (1 CUBIT = 1.7 DEG)

TU Û

-+6°

+5°

•

€ 196°

195°

194°

Moon +4"

193°

192°

+3" 191°

LONGITUDE Figure 1. T h e conjunction b e t w e e n the M o o n a n d jS L i b o n 2 7 J u n e 5 6 8 B C

(i) " M o n t h III... night o f the 8th, first part o f the night, the m o o n s t o o d 2 1/2 cubits b e l o w /3 L i b r a e " . A s discussed in the p r e v i o u s section, the first d a y o f m o n t h III w o u l d b e g i n o n the e v e n i n g of June 2 0 (civil date) in 5 6 8 B C . H e n c e the 8th d a y w o u l d b e g i n o n the e v e n i n g o f J u n e 2 7 . W e c o m p u t e that at a local time of 2 0 h, w h e n the s u n h a d r e a c h e d 10° b e l o w the horizon, the longitude a n d latitude o f the m o o n w o u l d b e 194.0° a n d + 4 . 5 ° . F o r /3 Lib, tìie c o r r e s p o n d i n g co-ordinates w e r e 193.8° a n d + 8 . 8 ° . H e n c e the m o o n w o u l d b e 4.3° to the south of j3 Lib (see Figure 1). T h e tabular date is thus confirmed. F u r t h e r m o r e , equating 2 1/2 cubits w i t h 4.3° yields the result 1 cubit = 1.7°. (ii) " M o n t h III... night o f the 10th, first part o f the night, the m o o n w a s b a l a n c e d 3 1/2 cubits a b o v e a Scorpii". T h i s conjunction o c c u r r e d o n l y t w o days after the p r e v i o u s one involving j3 Lib. A t 2 0 h on J u n 2 9 , the c o m p u t e d l o n g i m d e a n d latitude o f the m o o n w e r e 2 1 8 . 1 ° a n d + 3 . 4 ° . T h e star a S c o ( = Antares) w o u l d b e in longitude 2 1 4 . 1 ° a n d latitude -4.2°.

426

F.R. STEPHENSON AND D . M . WILHS

2 9 JUNE 5 6 8 B C : 20*' "MONTH ILL...NIGHT OF THE 10TH, FIRST PART OF THE NIGHT, THE MOON WAS BALANCED 3'I CUBITS ABOVE ASCORPII" +4

MOON

o

+3°

+2"

+r 0 3*^ CUBITS = 8.6 DEG ( I CUBIT = 2.5 DEG)

g

\ \

-3°

-4°

a

J

220°

219°

SCO

I

I

\

\

L

218°

217°

216°

215°

214°

-5°

213°

LONGITUDE Figure 2. T h e conjunction b e t w e e n the M o o n a n d a S c o o n 2 9 June 5 6 8 B C H e n c e the m o o n w o u l d b e 7.6° to the north a n d 4.0° to the east o f a S c o (see Figure 2). O n c e again the date is confirmed. If only the difference is latitude is considered, then 3 1/2 cubits w o u l d b e equivalent to 7.6°, yielding 1 cubit = 2.2°. H o w e v e r , if the longitude difference is also taken into account, 3 1/2 cubits = 8.6°, so that 1 cubit = 2.5°. (iii) " M o n t h XII... night of the 2nd, (begiiming of the night), the m o o n w a s b a l a n c e d 4 cubits b e l o w rj T a m i " . F r o m the p r e v i o u s section, the fnst d a y of m o n t h X I I w o u l d b e g i n o n the e v e n i n g of M a r c h 14 (civil date) in 567 B C . H e n c e the 2 n d d a y w o u l d b e g i n o n the e v e n i n g of M a r c h 15. W e c o m p u t e that at a local time of 19 h, w h e n the S u n h a d r e a c h e d 10° b e l o w the horizon, the longitude and latitude of the M o o n w o u l d b e 2 3 . 9 ° a n d -3.8°. F o r rj T a u , the c o r r e s p o n d i n g co-ordinates were 2 4 . 4 ° a n d + 3 . 8 ° . H e n c e the M o o n w o u l d b e 7.6° to the south o f rj T a u (see Figure 3). N o t only is the tabular date confirmed, b u t equating 4 cubits with 7.6° yields the result 1 cubit = 1.9°. O i n investigations of the seven r e c o r d e d conjunctions of the M o o n with stars are s u m m a r i s e d m T a b l e 2. I n this table, w e list the lunar m o n t h , day, equivalent Julian date, star, separation in cubits, actual separation in d e g and ratio of cubits to d e g . For

427

The Earliest Datable Observation of the Aurora Borealis

15 MARCH 5 6 7 B C : 19*' "MONTH XII...NIGHT OF THE 2ND, THE MOON WAS BALANCED 4 CUBITS BELOW R| TAURI"

* TI TAU

4

4 CUBITS = 7.6 DEG (1 CUBIT = 1.9 DEG) 0

Ê

Moon

27°

26°

25°

24°

23°

22°

LONGITUDE Figure 3 . T h e conjunction of the M o o n w i t h 7} T a u o n 15 M a r c h 5 6 7 B C . the first a n d third observations w e h a v e a m e n d e d the r e c o r d e d d a y o f the m o n t h b y 1, o n the basis o f calculation; p r e s u m a b l y scribal errors h a v e o c c u r r e d h e r e . In the case o f the sixth observation, the date is missing a n d w e h a v e inserted the calculated day of the m o n t h . T h e s e e m e n d a t i o n s are indicated b y p a r e n t h e s e s . Month

Day

Julian D a t e

Star

I II III III III XI XII

[8] 1

568 568 568 568 568 567 567

iSVir j3 G e m

[4] 8 10 [11] 2

Apr 29 M a y 22 Jun 24 Jun 27 Jun 29 F e b 22 M a r 15

13

VÎT

/3 L i b a Sco «Leo

Cubits 1 4 1 2.5 3.5 1 4

Degrees 1.9 7.3 3.1 4.3 8.6 2.5 7.6

Ratio 1.9 1.8 3.1 1.7 2.5 2.5 1.9

r| T a u T a b l e 2. Analysis of conjunctions of the m o o n with N o r m a l Stars: c o m p a r i s o n b e t w e e n m e a s u r e d a n d c o m p u t e d values C o m p u t a t i o n shows that four o f the dates in T a b l e 2 are accurately r e c o r d e d . O f the a m e n d e d dates, the a c c o r d b e t w e e n the o b s e r v e d and c o m p u t e d separations b e t w e e n m o o n a n d star is so g o o d that our e m e n d a t i o n s s e e m fiiUy justified. All s e v e n observations are, in fact, well s u p p o r t e d b y calculation a n d are in g o o d a c c o r d with a

428

F.R. Stephenson and D.M. Willis

date for ttie tablet o f 5 6 8 - 5 6 7 B . C . W e d e d u c e that the m e a n v a l u e o f the cubit at this p e r i o d w a s a p p r o x i m a t e l y 2.2°.

Conclusion T h e observations a n a l y s e d here are sufficiently diverse a n d accurate to e n a b l e the accepted date of the tablet - i.e., 5 6 8 - 5 6 7 B C - to b e confidently affirmed. It s h o u l d b e e m p h a s i s e d that although the circumstances o f conjunctions o f the m o o n w i t h stars tend to r e p e a t at 19-year intervals (the M e t o n i c cycle), this is not the case for lunar threes. T h e B a b y l o n i a n auroral observation, w h i c h is the m a i n subject o f this p a p e r , w a s r e c o r d e d o n the night o f the 2 9 t h o f m o n t h X I . T h e m e a s u r e d interval b e t w e e n sunset a n d m o o n s e t o n the 1st d a y o f this lunar m o n t h w a s 14.5°. F o r t h e e v e n i n g o f F e b m a r y 12 (civil) in 5 6 7 B C , c o n f u t a t i o n indicates 17.0°. T h i s a n d other observations in the m o n t h i n p l y that the auroral display o c c u r r e d o n t h e night o f M a r c h 12/13 in 5 6 7 B C . In particular, the visibility o f a n a u r o r a so far from the n o r t h g e o m a g n e t i c p o l e suggests that solar activity w a s at a high level m 5 6 7 B C .

References FATOOHI, L o u a y J. a n d S T E P H E N S O N , F . Richard. 1 9 9 7 - 8 . " A n g u l a r M e a s u r e m e n t s in B a b y l o n i a n A s t r o n o m y " . Archiv. fur Orientforschung 44-45: 210-214. H U N G E R , H e r m a r m . 1992. Astrological Reports to Assyrian Kings. Helsinki: Helsinki U n i v e r s i t y Press. N E U G E B A U E R , P a u l V . a n d W E I D N E R , Ernst F . 1 9 1 5 . " E i n a s t r o n o m i s c h e r B e o b a c h t u n g s t e x t aus d e m 3 7 . J a h r e N e b u k a d n e z a r s II. ( - 5 6 7 / 6 6 ) " . Berichte Uber die Verhandlungen der Koniglich Sdchsischen Gesellschaft der Wissenschaften zu Leipzig, Philosophisch-Historische. Klasse 6 7 / 2 . P A R K E R , R i c h a r d A . a n d D U B B E R S T E I N , W a l d o H . 1 9 5 6 . Babylonian Chronology: 626 B.C.-A.D. 75. P r o v i d e n c e , R.I.: B r o w n U n i v e r s i t y P r e s s . S A C H S , A b r a h a m J. a n d H U N G E R , H e r m a i m . 1 9 8 8 . Astronomical Diaries and Related Texts from Babylonia, Vol. I: 652-262 B.C. V i e n n a : O s t e r r e i c h i s c h e A k a d e m i e der Wissenschaften. S T E P H E N S O N , F . R i c h a r d a n d M O R R I S O N , Leslie V . 1 9 9 5 . " L o n g - t e r m Fluctuations in the E a r t h ' s rotation: 7 0 0 B . C . - A . D . 1 9 9 0 " . Philosophical Transactions of the Royal Society, Series A , 3 5 1 : 1 6 5 - 2 0 2 . Y A U , K e v i n K . C . ; S T E P H E N S O N , F . R i c h a r d , a n d WiLLlS, D a v i d M . 1 9 9 5 . A Catalogue of Auroral Observations from China, Korea and Japan: 193 B.C.1770 A.D. Rutherford A p p l e t o n L a b o r a t o r y T e c h n i c a l R e p o r t R A L - T R - 9 5 - 0 7 3 .

The 'Transit Star Clock' from the Book of Nut Sarah Symons,

Leicester

Introduction T h e Book of Nut is a n E g y p t i a n reUgious text o f w h i c h t h e eariiest surviving e x a m p l e o c c u r s o n t h e ceiling o f t h e S a r c o p h a g u s C h a m b e r o f t h e O s i r e i o n at A b y d o s ' ( b u i h b y Seti I, 1 3 0 6 - 1 2 9 0 B C ) . T h e graphical p o r t i o n o r vignette o f the Book of Nut also occurs in t h e t o m b o f R a m e s s e s I V i n t h e V a l l e y o f the Kings^ ( 1 1 6 3 - 1 1 5 6 B C ) . T w o papyri, Carlsberg 1 a n d la,^ from the s e c o n d c e n t u r y A D a n d apparently written b y d i e same scribe, contain c o m m e n t a r i e s t o t h e Book of Nut. Q u o t a t i o n s from t h e vignette a r e given b u t the d i a g r a m itself is n o t i n c l u d e d i n t h e surviving p o r t i o n s o f the p a p y r i . T h e vignette is a picture o f the s k y - g o d d e s s N u t b e i n g raised b y h e r father, t h e air-god Shu, w h o is p o r t r a y e d cenfrally. T e x t labels o c c u r t h r o u g h o u t t h e vignette b u t the m a i n textual c o m p o n e n t s are t w o lists, the first m i m i n g along the b o d y o f t h e g o d d e s s consisting o f n a m e s o f t i m e k e e p i n g stars o r decans,'' t h e s e c o n d , o r ' d a t e list', occurring i n vertical c o l u m n s in the s p a c e b e t w e e n the b o d y o f the g o d d e s s a n d the ground, w i t h s o m e entries b e l o n g i n g t o the date list also a p p e a r i n g o n the a r m o f the g o d d e s s a n d a b o v e a w a v y line d e n o t i n g s a n d (see F i g u r e 1). T h e date list is cyclical a n d r e a d s , i n the Osfreion version,^ i n a n anfi-clockwise maimer. E a c h entry in the date list c o n p r i s e s a set o f dates in the form: '® date 5(® date date\

|

T h e labels )(® a n d | p r e s e n t n o r e a d i n g difficulties, m e a n i n g respectively

' E n c l o s e d in t h e D u a t ' ^ a n d ' B i r t h ' . T h e label ^ (tpt),

h o w e v e r , is m o r e difficult.

N e u g e b a u e r a n d P a r k e r use the tianslation 'Ffrst H o u r ' (wnwt tpt) a l t h o u g h the w o r d wnwt

'

' h o u r ' d o e s n o t a p p e a r in the b o d y o f the table. Wnwt o n l y o c c u r s i n the text

FRANKFORT ( 1 9 3 3 ) , NEUGEBAUER and PARKER ( 1 9 6 0 ) , p. 3 6 .

^

NEUGEBAUER and PARKER ( 1 9 6 0 ) , p. 3 6 .

^

Transcriptions and translations are given in NEUGEBAUER and PARKER ( 1 9 6 0 ) , Chapter 2 .

Carlsberg 1 was published in LANGE and NEUGEBAUER ( 1 9 4 0 ) .

* The number of decans in this list as well as its composition leads to the conclusion that this list is not associated with the date list, and so will not be pertinent to the following discussion. ' The version in the tomb of Ramesses IV is not only cramped into a rectangle of taller aspect than the Osireion version, but is also completely mirrored (Nut's head is to the left). To simplify description, the orientation of the Osireion version only will be noted in this and subsequent statements. ^

The Egyptian 'Netherworld'.

Ml 430

431

The 'Transit Star Clock' from the Book of Nut

S. Symons

t;

Ik

O M

^ o 1 VO ^ t: o t i ^ a : o a , ca

P

s

•S i2 CO

I

1

CD

-S

1

I

oo

4) VO VO b *o ^ -

c I

ex

I f

di

NO g VO

^ II [§ B

H

e •S

% ^ o

I

«

•g ^ t!

2 § —

S

to

o

(_

fa

a

o

so

t--

C

an

a

1

N-

T3 L^^ t ^ S ^ ci; O 0, ffl'*^ tu B B

Qq (u

û

I I

"S'

•s| S o

H

CM

-s I O

" tì — S

*->

U

O

O

1 JO

a u ^ 3

1/3

.S"

II

NO

—

VO § 2

1

VO

1

432

S. Symons

w h i c h N e u g e b a u e r a n d P a r k e r label text V w h i c h runs horizontally from right to left a b o v e the sand line u n d e r the m a i n b o d y of the table. This text states: ' A s for that w h i c h is b e t w e e n the star w h i c h m a k e s the first h o u r a n d the star w h i c h is enclosed in the D u a t : there are 9 stars. N o w as for that w h i c h is b e t w e e n the star w h i c h is b o m a n d the star w h i c h m a k e s the first hour: 2 0 stars. G i v i n g 2 9 of those living a n d w o r k i n g in h e a v e n . ' ' F o r the m o m e n t , the three w o r d s 'Tpt\ ' E n c l o s u r e ' , a n d ' B i r t h ' will b e u s e d to denote the h e a d i n g s for the three dates in every set. T h e date sets will b e n u m b e r e d starting at the set fijrthest to the left along the sand line a n d continuing w i t h the sets along the sand line, u p N u t ' s arms, a n d across the tabulated list u n d e r N u t ' s b o d y from right to left. T h e first six sets ( o n e o f t h e m e m p t y of dates) a p p e a r along the s a n d line a n d are interspersed b e t w e e n d e c a n n a m e s . T h e order is given in T a b l e 1 ( u p p e r part). A s s o c i a t i o n of the sets with k n o w n d e c a n n a m e s , a n d the contents o f texts like T e x t V a b o v e s h o w that the dated events are relevant to t i m e k e e p i n g stars. T h r e e m o r e date sets (two of t h e m identical) a n d t w o d e c a n n a m e s o c c u r o n the a r m of N u t . O n e date set a p p e a r s u n d e r N u t ' s breast, directly u n d e r a label sbhn. T h e s e are s h o w n in T a b l e 1 (lower part). T h e label sd is thought to b e a g a r b l e d attempt at ipds^ T h e r e m a i n d e r of the list occurs in c o l u m n s w h i c h fill the areas to the right a n d the left o f the g o d Shu, u n d e r the b o d y of N u t . N o d e c a n n a m e s o c c u r n e a r these entiles. T h e first entry repeats the set o f dates u n d e r N u t ' s breast (see T a b l e 2 ) . In the O s i r e i o n version, e a c h set of three dates occupies a single c o l u m n . In contiast, the R a m e s s e s I V version u s u a l l y ' has Tpt a n d E n c l o s u r e in o n e c o l u m n a n d B i r t h in the next. T h e r e are thirty-nine entiles in the c o m p l e t e list including the enfries o n the a r m s a n d a b o v e the sand line. T w o are duplicates ( = 7 a n d = 8 ) a n d o n e (4a) contains n o dates.'° T h i s leaves thirty-six distinct entiles.

T h e Decan List In addition to the d e c a n n a m e s associated directly with the date sets {Stw, hry hpd knmt, hit dit, phwy

dit, tmit hrt hrt, sd (ipds),

bkiti,

a n d sbhn),

two decan names

a p p e a r o n the b o d y o f N u t u n d e r the unrelated d e c a n list in the O s i r e i o n version.

'

Author's own translation.

^

NEUGEBAUER and PARKER ( 1 9 6 0 ) , p. 8 4 .

' Sets 1 4 and 1 8 are omitted entirely and set 3 2 is contained within the second column of set 3 1 . The first occurrence of set 8 appears written horizontally above the first three columns. This empty set occurs in both the Osireion and Ramesses IV versions. In the latter, the label 'Enclosure' is doubled. The execution of decan names and date sets in this portion of the Ramesses IV vignette is markedly different from that of the remainder. The repeated sets 7 and 8 are present in both versions.

The 'Transit Star Clock' from the Book of Nut

=8 9

433

Tpt II Prt 6 E n c l o s u r e I Smw 6 Birth IIII Èmw 16 Tpt II Prt 16 E n c l o s u r e I Èmw 16 Birth IIII Èmw 2 6

10

Tpt II P r r 2 6 E n c l o s u r e I Èmw 2 6 B i r t h IIII Èmw 6

11

Tpt III Prt 6 Enclosure II Èmw 6 Birth IIII ^^mw 2 6

12

7>r III Prt 16 E n c l o s u r e II Èmw 16 Bfrth IIII T^rr 2 6

13

Tpt III P r / 2 6 E n c l o s u r e II Èmw 2 6 Birth I iht 2 6

14

Tpt IIII P r r 6 E n c l o s u r e III Èmw 6 Bfrth II iht 6

15

Tpr IIII Prt 16 E n c l o s u r e III Èmw 16 B i r t h II iht 16

16

Tpr IIII Prr 2 6 E n c l o s u r e III Èmw 2 6 B i r t h II iht 2 6

17

Tpf I Èmw 6 Enclosure IIII >^mw 6 B i r t h II

18

Tpt I Èmw 16 Enclosure IIII Èmw 16 Birth II iht 16

19

Tpt I Èmw 2 6 E n c l o s u r e IIII Èmw 26 B i r t h II itff 2 6

20

T/?r II Èmw 6 E n c l o s u r e ? iht 6 Birth III iht 16

21

7>? II Èmw 16 E n c l o s u r e H ^ M 5 B i r t h III iht 26

22

Tpt II Èmw 26 E n c l o s u r e ? iht 6 Birth III iht 6

23

Tpf III Èmw 6 E n c l o s u r e II iht 6 Birth III iht 16

24

Tpf III >^mw 16 E n c l o s u r e II

25

Tpr III ^ m w 2 6 Enclosure II iht 26 Birth I Prt 6

26

Tpt IIII Èmw 6 E n c l o s u r e III iht 6 Birth I Prt 16

21

Tpt IIII Èmw 16 E n c l o s u r e III ?/if 16 B i r t h I P r r 2 6

28

Tpt III W

29

Tpt I iht 6 E n c l o s u r e IIII iht 6 Birth II Prt 16

30

T/?r H / i n 5 E n c l o s u r e IIII iht 16 B i r t h II Prt 26

31

Tpt I iht 26 E n c l o s u r e IIII iht 26 Bfrth III Prt 6

32

Tpt 11 iht 6 E n c l o s u r e I Prt 6 Birth III Prt 16

33

Tpt II ^Ar 16 E n c l o s u r e I P r M 6 Bfrth III Prt 26

34

7>f II iht 26 E n c l o s u r e I Prt 26 Birth IIII Prt 6

35

Tpt III

6 E n c l o s u r e II Prt 6 Birth IIII Prt 16

36

Tpt III

16 E n c l o s u r e II Prt 16 Birth IIII Prt 26

6

16 B i r t h III

26

2 6 E n c l o s u r e III iht 26 B i r t h II Prt 6

T a b l e 2. D a t e sets without d e c a n n a m e s (from the O s i r e i o n version).

434

S. Symons

T h e s e t w o d e c a n s a r e w'^rt hrt (under rmn hry) a n d sbhn (under hry hpd n knmt a n d hSt dit). I n the R a m e s s e s I V version a n d possibly in the Osireion a s well, the n a m e ts •^rit" a p p e a r s u n d e r '^rt. D e c a n s m e n t i o n e d within text labels a r e spdt, knmt, Stw, a n d "^b ( p r o b a b l y a p o o r writing o f sbhn). F r o m its p o s i t i o n within the vignette a n d from t h e s u r r o u n d i n g texts, t h e date list is pivotal t o , o r i n d e e d m a y b e the m a i n t h e m e of, this p a r t o f the Book of Nut. It is conjectured that in t h e original p r e s e n t a t i o n o f t h e Book, o r its a n t e c e d e n t d o c u m e n t s , the scattered n a m e s o f decans outside the list o n N u t ' s b o d y o n c e formed a c o m p l e t e d e c a n list with e a c h d e c a n related t o a set o f three entries. S u b s e q u e n t c o p y i n g , u p t o a n d including the versions i n t h e O s i r e i o n a n d t h e t o m b of R a m e s s e s I V , altered t h e layout a n d content o f the Book w h i c h h a s resulted in d e c a n n a m e s b e i n g lost o r misplaced. T h e d e c a n s collected from labels a n d the d e c a n n a m e s associated w i t h date sets form a fragmentary d e c a n list. Other d e c a n lists'^ c a n b e u s e d t o r e c o n s t m c t t h e o r d e r in w h i c h t h e d e c a n s should appear. Bkiti a n d sd (ipds) a r e in r e v e r s e o r d e r o n N u t ' s arm. T h e o r d e r o f the fragmentary d e c a n list, w h i c h will b e labelled L , i s : ts "rk, w^rt hrt, spdt, Stw, knmt, hry hpd knmt, hit dit, phwy dit, tmit hrt hrt, bkiti, sd (ipds), sbhn. T h e question o f w h i c h d e c a n applies t o w h i c h set o f dates m u s t n o w b e addressed. A l t h o u g h t h e s e q u e n c e o f date sets is from left t o right a l o n g t h e sand line, the direction o f writing o f each date set a n d d e c a n n a m e is right t o left throughout t h e vignette. A l o n g t h e sand line, d e c a n n a m e s c o u l d either b e written at the h e a d o f their date sets (that is, t o t h e right o f the dates) o r at t h e e n d o f thefr associated date sets (that is, to the left o f the dates). F o r e x a m p l e , s e t 3 c o u l d b e l o n g to either hit dit o r hry hpd knmt. T h e positioning o f hry hpd knmt n e a r t o s e t 2 with S h u ' s legs b e t w e e n t h e d e c a n a n d set 3 m a k e s it clear that the first a l t e m a t i v e is the m o r e likely: the n a m e o f the d e c a n h e a d s its date set.'^ Associating set 2 with hry hpd knmt also fixes t h e d e c a n Stw t o set 3 6 , knmt t o set I, hit dit t o set 3 , a n d phwy dit to set 4 . T h e next label is tmit hrt hrt w h i c h i n the

' ' These are the earliest appearances of this decan, which next occurs as a written name in the time of Ptolemy I on a ceiling designated monument 40 'Hermopolis A' (illustrated in plate 26) in NEUGEBAUER and PARKER (1969).

NEUGEBAUER and PARKER (1960 -and 1969). Decans contained within orderly diagonal star clocks and astronomical ceilings display consistency in their order. Although several different families of decan lists have been identified, many individual decans appear in sources from more than one family. Comparison enables a fairly reliable order of decans to be constructed. Questions about order remain where individual decans only appear in disordered or fragmentary sources, and also within the decans in the region of Orion (NEUGEBAUER and PARKER (1969), pp. 112-114). In papyri Carlsberg 1 and la, the dates of set 5 are associated with phwy dit. This would occur if decans followed their dates. Set 1 would belong to Stw (written to its left), 2 = knmt, 3 = hry hpd knmt, and 4 = hit dit (written to its left). The position of Stw and the reference to phwy dit in the papyri are the only positive results of this arrangement: all the following decans are out of place. However, Neugebauer and Parker believed the scribe of papyri Carlsberg 1 and 1 a to be in error, choosing the wrong date set. This mistake could have been caused by reading from left to right (i.e. in the same sense as the decan list): the decan name phwy dit, the empty headings of 4a, then the dates in set 5.

The 'Transit Star Clock' from the Book of Nut

435

earlier d e c a n lists from d i a g o n a l star clocks''* a p p e a r s as t w o separate d e c a n s : tmit hrt a n d tmit hrt. T h e r e is also the p r o b l e m with the fr)llowmg decan. T h e w o r d bkiti a p p e a r s a l o n e as a label, w h e r e a s in earlier d e c a n lists either the single d e c a n bkM is p r e c e d e d b y the single d e c a n wSiti or the t w o a p p e a r as o n e d e c a n wSiti bkiti. T h e t w o p r o b l e m s t o g e t h e r give fom possible c o m b i n a t i o n s o f d e c a n s : tmit hrt hrt, wSiti bkiti ( t w o d e c a n s ) , tmit hrt hrt, wSiti, bkiti (three decans), tmit hrt, tmit hrt, wSiti bkiti (three d e c a n s ) , or tmit hrt, tmit hrt, w^iti, bkiti (four d e c a n s ) . N e u g e b a u e r a n d P a r k e r favoin the first c o m b i n a t i o n a n d g r o u p o f later lists w h i c h they call 'Seti I B ' . ' ^ T h e y ignore w h i c h a p p e a r s at the correct p o s i t i o n for tmit hrt reinforcing n a m e label refers to t w o d e c a n s . T h e possible d e c a n configinations m c o m b i n a t i o n w i t h n e x t to date sets p r o d u c e the following possible cases: C a s e 1 : (tmit hrt hrt, wJliti

tie this d e c a n list to a the e m p t y date set 4a, the possibility that the the p o s i t i o n o f labels

bkiti)

Sd (ipds) is correctly p l a c e d a n d bkiti is b o t h out of p l a c e a n d b a d l y written. Sbisn is p l a c e d n e a r the first o c c u r r e n c e o f set 8 a n d is related to that set. C a s e l a : (tmit hrt, tmit hrt, wSitl

bkiti)

If the e m p t y date set 4 a w e r e to refer to tmit hrt, set 5 g o e s w i t h tmit

hrt

a n d the following d e c a n s fall as for C a s e 1. C a s e 2: {tmit hrt hrt, wSiti,

bkiti)

T h e u n l a b e l l e d set 6 w o u l d apply to wSiti. Bkiti w o u l d b e correctly written a n d p l a c e d m front o f the s e c o n d o c c u r r e n c e of its date set 7. Sd {ipds)

is

then out o f p l a c e as well as b a d l y written and m u s t b e associated w i t h set 8, w i t h sbSsn h a v i n g set 9. C a s e 2a: {tmit hrt, tmit hrt, wSiti,

bkiti)

If the e m p t y date set 4 a w e r e to refer to tmit hrt, set 5 g o e s with tmit

hrt

a n d the foUowmg d e c a n s fall as for C a s e 2. C a s e 3 : {tmit hrt, tmit hrt, wSiti

bkiti)

Tmit hrt w o u l d b e associated with set 5 and tmit hrt with set 6. T h i s leaves the label 'bkitV

(for wSiti bkiti)

correctly p l a c e d in front o f the s e c o n d

o c c u r r e n c e o f set 7. Sd {ipds) and sbhn

must b e associated with sets 8 a n d

9, as h a p p e n e d with case 2. C a s e 4 : {tmit hrt, tmit hrt, wSiti,

bkiti)

B o t h the writing of bkiti a n d the e m p t y date set are a c c o u n t e d for, b u t the labels bkiti a n d sd {ipds) sbhn

( w h i c h are k n o w n to b e in the w r o n g o r d e r ) a n d

are all out o f p l a c e .

T a b l e 3 illusfrates the six cases. N o t e that 'w^iti' d o e s not a p p e a r at all in either the Osfreion o r R a m e s s e s I V versions o f the vignette. A s afready noted, the partial list p r o d u c e d b y accepting C a s e 1 c o n f o r m s to that o f a family of lists d a t m g m a i n l y from the G r a e c o - R o m a n P e r i o d called the 'Seti I B ' family. T h e lists formed b y the other three cases m a t c h n o other list exactly.

See NEUGEBAUER and PARKER ( 1 9 6 0 ) . NEUGEBAUER and PARKER ( 1 9 6 9 ) , pp.

133-140.

S. SYMONS

436

uate set

Case 1

1

Case l a

Case 2

C a s e 2a

knmt

knmt

knmt

knmt

knmt

knmt

hry hpd

hry hpd

hry hpd

hry hpd

knmt

knmt

hit dit phwy dit

hit dit phwy dit

tmit hrt

tmit hrt

tmit hrt

wSiti

wSiti

tmit brt

tmit hrt

bkiti

bkiti

^

knmt

knmt

knmt

3 4

hit dit phwy dit

hit dit

hit dit

hit dit

5

WÊSÊÊÊêÊÊ tmit hrt hrt

6 7

sd

phwy

dit

tmit hrt wSiti

bkiti

bkiti

{ipds)

8

sbhn

...

phwy

dit

tmit hrt

wm

9

sd

{ipds)

sbSsn

tmit hrt hrt

phwy dit tmit hrt

wSiti bkiti

s_d(ipds) sbhn

tsH

sbhn

w'^rt hrt

35 36

spdt Stw

spdt Stw

[ 1 2 DECANS, 7 CORRECTLY POSITIONED]

[ 1 3 DECANS, 8 CORRECTLY POSITIONED]

w^rt

brt

sdiipds) sbhn

ts ""r^ w*>r brt

w^rf brt

spdt Jltw

spdt

spdt

itw

Stw

spdt Stw

[ 1 3 DECANS, 6 CORRECTLY POSITIONED]

[ 1 4 DECANS, 7 CORRECTLY POSITIONED]

[ 1 3 DECANS, 6 CORRECTLY POSITIONED]

[ 1 4 DECANS, 5 CORRECTLY POSITIONED]

tsH

?

wSiti bkiti

sbhn

10

?

Case 4

hry hpd

hry hpd knmt

4a

Case 3

w'^rt hrt

îs'r^

w*^rt hrt

T a b l e 3 . Possible combinations of decan n a m e s a n d date sets. U n s h a d e d d e c a n s indicate that the decan n a m e appears n e a r to the correct p o s i t i o n in association with its date set in the N u t vignette. A dash indicates that the e m p t y date set 4 a w o u l d not h a v e a n associated d e c a n in this case, that is, its p r e s e n c e is an error. T h e n u m b e r of decans following sbSsn a n d their n a m e s are not k n o w n (indicated b y ellipsis). Ts '^rk a n d w'^rt hrt caimot with certainty b e associated with particular date sets (indicated b y ?).

N e u g e b a u e r a n d P a r k e r p l a c e L as the earliest list in a family of d e c a n lists ('the Seti I B family of d e c a n s ' ) w h i c h persisted until the time of Trajan ( A D 9 8 - 1 1 8 ) . T h e y state that 'the title list is an incomplete one but nonetheless secure in its p l a c e as the earliest in its family.' It is w o r t h reviewing the links b e t w e e n the i n c o m p l e t e list L a n d the later lists within the family. T h e filli 'Seti I B family' list contains not only thirty-six ordinary d e c a n s starting with (1) spdt a n d e n d i n g with (36) tpy-^ spdt, but also twelve extra d e c a n s (one for e a c h g r o u p of three ordinary decans) and eleven deities of the epact ( p e r h a p s instead o f triangle d e c a n s ) . T h e later lists h a v e figures a n d associated m i n e r a l s or w o o d s , but no n a m e d deities a n d no stars. All these features are u n i q u e a m o n g N e u g e b a u e r

The 'Transit Star Clock' from the Book of Nut

437

a n d P a r k e r ' s d e c a n a l families. O n l y t w o m e m b e r s ' ^ o f t h e family include p l a n e t s , b u t n o t in t h e p o s i t i o n o r format e x p e c t e d from other d e c a n lists. T h e r e a r e nine fiirther m e m b e r s ' ' o f the ' S e t i I B family' b e s i d e s the t w o N e w K i n g d o m lists from the Osfreion a n d the t o m b o f R a m e s s e s I V . M o s t family m e m b e r s p r e s e r v e only figines rather than written n a m e s in the part o f the list w h i c h equates t o the p r e s e r v e d p o r t i o n o f L . N a m e s are only forthcoming from 'Edfii' a n d ' D e n d e r a A , D , a n d F ' . T h e a p p e a r a n c e o f individual d e c a n s i n this r e g i o n o f the list m s o m e other m e m b e r s o f the family is a s s u m e d b y conç)aring figines. T a b l e 4 c o m p a r e s the d e c a n n a m e s in the four later lists with t h e early list L . T w o i n p o r t a n t p o i n t s e m e r g e . Ffrst, the d e c a n n a m e d w'^rt hrt in the N e w K i n g d o m lists d o e s n o t a p p e a r m the later lists. T h e d e c a n w'^rt c o u l d b e the s a m e d e c a n a s w'^rt hrt b u t this is n o t explicitly demonsfrated here. S e c o n d , the r e a d i n g o f tmit hrt hrt as a single d e c a n h a s n o confirmation m these later lists. H e r e , t h e d e c a n following phwy dit is n a m e d tmit (written as tmi, dm,^^ a n d dmi). D e s p i t e the fact that tmit hrt hrt is never written as the n a m e o f a single d e c a n outside the t w o N e w K i n g d o m o c c u r r e n c e s o f the Book of Nut, N e u g e b a u e r a n d P a r k e r refer to 'tmit hrt hrt' as a d e c a n n a m e m t w o separate d e c a n families: ' S e t i I B ' a n d 'Tanis'.^*' T h e writing o f tmit hrt hrt o w e s m o r e t o t h e practice (dating asfronomical c e i l m g o f S e n e n m u t ) o f w r i t m g the t w o d e c a n s tmit hrt a n d a single c o l u m n with a n a b b r e v i a t e d form o f tmit a b o v e the w o r d hrt. C a r l s b e r g l a N e u g e b a u e r a n d P a r k e r restored the d e c a n n a m e as tm hr tm comment.^'

from t h e tmit hrt in Indeed, m hr witiiout

E v e n if the d e c a n tmit hrt hrt is a t m e d e c a n b e l o n g i n g to the list associated w i t h the date list in the Book of Nut, the differences a n d uncertainties b e t w e e n t h e N e w K i n g d o m list a n d the later lists are such that N e u g e b a u e r a n d P a r k e r ' s statement o f ' s e c u r i t y ' q u o t e d earlier m u s t surely b e c o n s i d e r e d optimistic in the exfreme. G i v e n tiie u n i q u e n a t u r e o f the Book of Nut source, it is n o t i m p o s s i b l e that the list L s h o u l d b e u i u q u e a m o n g the k n o w n d e c a n lists. T h e recurrence o f the d e c a n ts '^rk in m u c h later lists o n l y indicates that this n a m e w a s still u s e d for a g r o u p o f stars a n d d o e s n o t p r o v e conclusively that the later d e c a n lists d e s c e n d e d directly from the Book of Nut s o i n c e . It is definitely n o t safe, g i v e n the argiunents a b o v e , to state w i t h certainty (as N e u g e b a u e r a n d P a r k e r d o ) that t h e d e c a n s missing from L c a n b e s u p p l i e d from t h e later m e m b e r s o f the 'Seti I B family'.

Numbers 49 'Edfu' and 62 'Esna B ' in the notation of NEUGEBAUER and PARKER (1969) (pis. 30A and 43 respectively). These are numbers 31 'Osorkon I F (pi. 17), 47 'Esna A,' (pi. 29), 49 'Edfu' (pi. 30A), 50a 'Philae B ' (pi. 57), 53 'Dendera A' (pis. 33 and 34), 56 'Nag Hamad A' (pi. 38A), 59 'Dendera D' (pi. 41), 62 'Esna B ' (pi. 43), and 64 'Dendera F ' (pi. 44) (NEUGEBAUER and PARKER (1969), p. 134).

Denoted as numbers 5 'Seti I B ' and 20 'Ramesses I V B ' in NEUGEBAUER and PARKER (1969). Carlsberg 1 has the writing dm in the phrase 'opposite knmt to dm are they, these five stars'. (NEUGEBAUER and PARKER (1960), p. 56). This phrase does not occur in Carlsberg la. ^°

NEUGEBAUER and PARKER ( 1969), pp. 140-149. NEUGEBAUER and PARKER (1960), p. 91.

S. Symons

438

c Q

S \

C (1>

S

I

Q co

S S :2 S

c Ci

60

•S c

Q

S

1 ^ (L>

-a

•4«:

S PQ

I ^ ji,:

-Ê -C.

^

^

CI,

5

u >i

>1

e t ì o O O N O t-.. ^

r<^

s

<^

•J3

a>

e«»/-iVOt-^c « < N ( ^ ' g - -rf

O

m

CI

O

U ^ û5

I I I ;

Ì3.

The ' Transit Star Clock' from the Book of Nut

439

It s e e m s that N e u g e b a u e r and P a r k e r ' s a r g u m e n t in favour o f C a s e 1 is that a family exists into w h i c h the list, in C a s e 1 form, c a n b e inserted, a n d that their dismissal without discussion o f C a s e s l a , 2, 2a, 3 , a n d 4 is similarly b a s e d on the o b s e r v a t i o n that n o o b v i o u s ' f a m i l y ' exists into w h i c h these possibilities will fit. T h i s is clearly not a n acceptable argument, especially in the light of the imique context of the list L in E g y p t i a n a s t r o n o m i c a l timekeeping.

The 'Star Clock' O n e noticeable feature o f the date list is the regularity o f the dates. W i t h e x c e p t i o n o f writing errors, only the 6th, 16th, and 2 6 t h days of the month^^ m e n t i o n e d . T h e r e is a p e r i o d o f exactly three m o n t h s b e t w e e n the dates of 'Tpt' ' E n c l o s u r e ' , t w o m o n t h s and ten days b e t w e e n ' E n c l o s u r e ' and ' B i r t h ' , a n d m o n t h s and twenty days b e t w e e n ' B i r t h ' a n d 'Tpt'. N o account is m a d e of e p a g o m e n a l days.

the are and six the

T h e regularity of the list m a k e s it easy to c o n s t m c t an ideal list o f date sets b y restoring lost signs, correcting writing errors, a n d eliminating duplicate entries. T h i s ideal list is s h o w n in T a b l e 5. T h e p e r i o d of t w o m o n t h s a n d ten days b e t w e e n ' E n c l o s u r e ' a n d ' B i r t h ' is clearly the seventy-day p e r i o d of invisibility w h i c h the Book of Nut implies is a characteristic of a d e c a n a l star and w h i c h is m e n t i o n e d as a p e r i o d of 'purification in the e a r t h ' . T h e r e a p p e a r a n c e or ' B i r t h ' of a star is equally clearly the a s t i o n o m i c a l event called heliacal rise, and the ' E n c l o s u r e ' o f a star is its d i s a p p e a r a n c e at the begiiming of its p e r i o d of invisibility. This leaves the event called 'Tpt' to b e determined. After ' B i r t h ' , a star rises e a c h night a p p r o x i m a t e l y four minutes earlier than the night before, a n d r e a c h e s a higher altitude in the sky before the light o f the sun causes the star to b e c o m e invisible. Eventually, the star will h a v e risen before night fall a n d will a p p e a r after sunset at a n increasing altitude a b o v e the horizon. A t the date w h i c h the text d e n o t e s as 'Tpt', N e u g e b a u e r a n d Parker assert that the associated star transits the meridian at the end of the first hour of the night?'^ N e u g e b a u e r a n d P a r k e r also assert^^ that since the table uses the w o r d 'Tpt' (which in text V is u s e d for 'First H o u r ' ) the table m u s t b e m t e n d e d to m a r k the h o u r s of the night. T h e s e t w o assertions h a v e led to this type of list b e i n g called a 'tiansit star clock'^^ and the list L a hst of 'tiansit decans'.^ F u r t h e r m o r e , the other

The Egyptian civil year comprises three seasons {iht, Prt, and èmw) each containing four months (conventionally labelled by uppercase Roman numerals I, II, III, and IIII or IV) of exactiy thirty days. Therefore, IIII iht 26 means 'the twenty-sixth day of the fourth month of the season iht\ the 116th day of the civil year. After IIII Èmw 30, the 360th day, five extra 'days upon the year' occurred, bringing the length of the civil year to 365 days. This five-day epagomenal period is ignored in the date list. Text Aa and the Dramatic Text from the Book of Nut (NEUGEBAUER and PARKER (1960), pp. 62 and 67-68). PARKER (1974), p. 56 states that 'Since a star spends 80 days in the east before working, it is clear that it [the star] is transiting when it marks an hour'. NEUGEBAUER and PARKER ( 1960), p. 41.

For example in CLAGETT ( 1995), pp. 56-59.

S. Symons

440

Tpt

Enclosure

Birth

1

III iht 2 6

II Prt 26

I Émw 6

2

IIII iht 6

III Prt 6

I Émw 16

3

IIII iht 16

m Prt 16

I Émw 26

4

IIII iht 26

III P r r 2 6

Il Émw 6

\Prt6

IIII Prt 6

II Émw 16

D a t e set

5 6

\Prt\6

IIII Prt 16

II Émw 26

7

I Prt 2 6

IIII PA-? 2 6

III Émw 6

8

II Prt 6

I Êmw 6

m ^ m w 16

9

Il Prt \6

I i^mw 16

III Émw 26

10

II Prt 2 6

I Èmw 26

IIII ^mvv 6

11

III Prt 6

II ^ m w 6

IIII ^ w w 16

12

m

Prt\6

II ^ m w 16

IIII Émw 26

13

III Prt 26

II Émw 26

liht6

14

IIII Prt 6

III .^mw 6

I

15

I l l l P r f 16

m Émw

16

I iht 26

16

IIII Prt 26

III ^WH- 2 6

II iht 6 Il iht 16

16

17

I Èmw 6

IIII Émw 6

18

I Èmw 16

IIII ^ m w 16

II iht 26

19

I Èmw 26

IIII Émw 26

III ^/ir 6

20

II èmw 6

liht6

III

21

I I W 16

l iht 16

III iht 26

22

II Èmw 26

l iht 26

IIII ?/ir 6

23

III èmw 6

II iht 6

IIII iht 16

24

m ^ m w 16

II iht 16

IIII

25

III Èmw 26

II ^/ir 2 6

lPrt6

26

IIII ^mH' 6

III iht 6

l Prt 16

27

IIII ^ m w 16

m iht 16

I Prr 26

28

IIII Èmw 26

III

29

\iht6

IIII iht 6

Il Prt 16

26

16

26

II Prt 6

30

I ^Ar 16

IIII ^/1/16

Il Prt 26

31

I iht 26

IIII iht 26

III Prt 6

32

II iht 6

\Prt6

m Prt 16

33

\\iht\6

IPrt 16

III Prt 26

34

Il iht 26

I Prt 2 6

35

III iht 6

II Prt 6

nu Prt 6 un Prt 16

36

m ^/ir 16

Il Prt 16

IIII Prt 26

T a b l e 5. Idealised version o f date list from the Book of Nut.

NEUGEBAUER and PARKER ( 1960), p. 115.

The 'Transit Star Clock' from the Book of Nut

441

lists w h i c h form the family 'Seti I B ' are subsequently labelled 'transit d e c a n s ' as well. T h e collection of date sets 1 to 3 6 together with a c o m p l e t e d e c a n list w o u l d create a type o f star c l o c k different in nature from the diagonal star clock. T h e a r r a n g e m e n t is different, b e i n g p r e s e n t e d as a list instead o f as a table. T h e lack o f e m p h a s i s o n h o u r s in the date lists is b a l a n c e d b y a greater interest in the yearly cycle of the stars.^^ T h e observational p r o c e d u r e for u s i n g the table is also different. T h e 'fransit star c l o c k ' w o u l d b e u s e d b y o b s e r v i n g the m e r i d i a n to see w h i c h d e c a n is fransiting and, b y k n o w i n g the date, finding the h o u r a n d m a k i n g s o m e calculation o f the h o u r using the date list. T h e diagonal star clock instead uses observations o f stars r i s m g a b o v e the e a s t e m horizon.^^ T h e usual c h r o n o l o g y of t i m e k e e p i n g in E g y p t p l a c e s the 'fransit star c l o c k ' as a d e v e l o p m e n t a n d i m p r o v e m e n t of the diagonal ' r i s i n g ' star clock. T h e p r i m a r y r e a s o n for this o r d e r of events is obvious: surviving diagonal star clocks p r e - d a t e the two surviving 'fransit star c l o c k s ' . T h e 'fransit star c l o c k ' also offers t w o potential areas for increasing the a c c i n a c y of the star c l o c k p r o c e s s . First, the observations w o u l d b e r e m o v e d from the h o r i z o n area, eliminating m a n y observational difficulties, and second, using the m e r i d i a n (a great circle) as d a t u m w o u l d allow equal h o u r s to b e m e a s u r e d with ease. Fio w e ver, b o t h o f these r e a s o n s s t e m from a m o d e m desfre for accurate observing p r o c e d u r e , a n d almost certainly d o not reflect Egyptian timekeeping priorities.^" T h e fragmentary list of d e c a n n a m e s associated with the date list has a l r e a d y b e e n s h o w n to h a v e only circumstantial links with later d e c a n lists. A l s o , n o n e o f these later lists is found in conjimction with a t i m e k e e p i n g m e t h o d . T h e s e p o i n t s , a n d the d o u b t s raised a b o u t the E g y p t i a n ' s desire for ' a c c u r a t e ' o b s e r v a t i o n s , lend n o support to the 'fransit star c l o c k ' theory. Far m o r e important, a n d cenfral to the c o n c e p t o f a 'fransit star c l o c k ' is the identification of the event 'Tpf with a concrete, observable asfronomical event: the fransit of a star across the m e r i d i a n at the e n d of the first h o u r of the night. This b e h a v i o u r , m e a n i n g that the fransit of a star m a r k s the first h o u r of the night 2 0 0 days after the s t a r ' s ' B i r t h ' , could b e t m e ^ ' of Sirius w h i c h h a s a visual m a g n i t u d e o f - 1 . 6 0 b u t d o e s not hold for fainter stars. A l t h o u g h the time o f fransit in relation to the time of sunset (the condition for 'first h o u r ' ) is n o t d e p e n d e n t o n m a g n i t u d e , the date of ' B i r t h ' certainly is. Therefore, if Sfrius w e r e in exactly the s a m e position, b u t h a d m a g n i t u d e 1.0, the date of ' B i r t h ' w o u l d b e five d a y s later. Similarly, if Sirius h a d a m a g n i t u d e o f only 3.0, the date o f ' B i r t h ' w o u l d b e 17 d a y s later. It is possible that s o m e decans are e v e n fainter than the third m a g n i t u d e . A difference of 10 days at ' B i r t h ' m e a n s that fransit at the date given b y 'Tpf o c c u r s one E g y p t i a n ' h o u r ' later than required, a n d a difference of 2 0 days at ' B i r t h ' m e a n s that fransit at the date g i v e n b y '7/?f' occurs t w o E g y p t i a n ' h o u r s ' later than required.

However, the diagonal star clock takes into account the five epagomenal days whereas the 'transit star clock' does not. LEITZ ( 1 9 9 5 ) contrary to most authors, believes that settings, not risings, were used. SYMONS ( 2 0 0 0 ) , pp.

112-113.

Assuming a reasonable condition for the time of the first hour of the night.

442

S. Symons

T h e r e is, therefore, a n inherent and insurmountable p r o b l e m w i t h the date list if it is t o b e u s e d a s a star clock in the m a i m e r w h i c h N e u g e b a u e r a n d P a r k e r suggest. T h e observer w o u l d h a v e his m e r i d i a n set u p using a n external reference for south a n d sight line, b u t t h e stars w o u l d n o t b e h a v e as e x p e c t e d w i t h r e g a r d s t o t h e position Tpt. Specifically, the decans m a y not transit in the correct order. ' B i r t h ' a n d ' E n c l o s u r e ' w o u l d b e correct, but the information o n w h i c h the identification o f the date list a s a c l o c k rests d o e s not hold. M a j o r e v i d e n c e w h i c h h a s b e e n u s e d t o s u p p o r t t h e theory o f the 'transit star c l o c k ' c o m e s from t h e t w o papyri Carlsberg 1 a n d Carlsberg l a . T h e surviving portions o f the p a p y r i include a n interpretation o f the text labelled ' U 4 ' w h i c h u s e s the w o r d ' w o r k i n g ' for t h e p e r i o d 120 days before 'Tpt' for e a c h star.^^ T h e nature of this ' w o r k i n g ' is n o t frilly explained within the text, b u t it is implied that the star m a r k s the first h o u r o f the night for ten days before the date called 'Tpt\ t h e s e c o n d h o u r o f the night for the ten days before that, a n d s o o n u p t o the twelfth h o u r for the first ten d a y s o f the 120-day ' w o r k i n g p e r i o d ' . T h e scribe b r e a k s the year o f a typical star into the following s e q u e n c e o f events: 110 days w o r k IIII iht 26 begins t o w o r k 10 days w o r k T p r ' I Prt 6 stops w o r k 9 0 days in the west ' E n c l o s u r e ' IIII Prt 6 7 0 days in the D u a t ' B i r t h ' II Émw 16 8 0 days in the east In contrast, text V s i m p l y states that there a r e '29 [decans] living and working in heaven' a n d gives n o fiirther information about ' w o r k i n g ' . N e u g e b a u e r a n d P a r k e r infer^^ that the 120-day w o r k i n g p e r i o d m e n t i o n e d in the papyri is the ' w o r k i n g ' (bik) referred to in the Osireion version o f the text labelled V a n d that the date list was regarded, from the N e w K i n g d o m o n w a r d s , a s a star clock: a device for finding the h o u r o f the night. A l t h o u g h the c o m m e n t a r y b y the scribe o f papyri Carlsberg 1 a n d l a implies that the stars in the Book of Nut m a r k e d twelve h o u r s o f the night o v e r a p e r i o d o f 120 d a y s , t h e assertion that the Book of Nut itself is a star clock is n o t m a d e n o r s u p p o r t e d strongly b y the s c r i b e ' s choice o f w o r d s . T h e r e are t w o p o s s i b l e explanations for the s c r i b e ' s c o m m e n t s w h i c h infroduce a p a s s a g e o f 120 d a y s w o r k i n g : h e either formed his conclusions from h i s e x p e r i e n c e of other t i m e k e e p i n g m e t h o d s , o r h e h a d access to a m o r e c o m p l e t e s o u r c e . T h a t the scribe h a d difficulty in understanding the Book of Nut is clear from the w a y h e deals with t h e texts from t h e diagram. F o r e x a m p l e , h e r e a d s t h e entire d i a g r a m in a n illogical order (sunrise to sunset instead o f sunset t o sunrise) a n d h e c h o o s e s t h e w r o n g dates for his e x a m p l e d e c a n phwy dit. It m u s t b e r e m e m b e r e d that the scribe

NEUGEBAUER and PARKER ( 1 9 6 0 ) , pp. 5 5 - 5 7 . The star is also said to 'begin to do work' only 1 0 days before it 'stops working' at 'Tpt\ "

NEUGEBAUER and PARKER ( 1 9 6 0 ) , p. 5 6 , commentary to line 44.

The ' Transit Star Clock' from the Book of Nut

443

w a s about fifteen centuries r e m o v e d from the O s i r e i o n text ( a n d p e r h a p s u p to t w o t h o u s a n d years r e m o v e d from the original text) a n d so caimot b e a s s u m e d to b e a n expert o n the original m e a n i n g of the text. T h i s m a k e s it possible that h e w a s i n d e e d forming his o w n u n d e r s t a n d i n g o f the text using outside information. C o n v e r s e l y , it has already b e e n n o t e d that the Book of Nut as p r e s e r v e d in the O s i r e i o n a n d the t o m b o f R a m e s s e s I V is a n incomplete d o c u m e n t and it is therefore p o s s i b l e that the scribe h a d a fuller version o f the text. H o w e v e r , it is n o t a b l e that if this is the case n o m e n t i o n is m a d e o f a n y d e c a n s w h i c h are n o t p r e s e r v e d in the N e w K i n g d o m version, a n d also that the scribe d o e s n o t allude t o any other h o r n s o f the n i g h t in t h e context o f t i m e k e e p i n g , w h i c h m a y h a v e h e l p e d h i m clarify his description. T h i s m a y indicate that the former case is m o r e likely, but it c a n n o t b e d e t e r m i n e d w h i c h of the t w o possibilities, if either, is t m e . T h e 'transit star c l o c k ' theory d o e s n o t answer the questions that the existing N e w K i n g d o m sources raise about the intended u s e o f the Book of Nut date list. T h e lack o f e m p h a s i s o n ' h o u r s ' suggests a very different function for the list. T o infroduce this function, the origin o f the d e c a n s must first b e discussed.

O r i g i n of t h e D a t e L i s t It is fairly certain that the b e h a v i o u r of Sirius gives a m o d e l for all other d e c a n s . I n particular, the Book of Nut states that decanal stars h a v e a p e r i o d o f invisibility lasting seventy days.^"* Sirius w o u l d theoretically h a v e h a d this p e r i o d of invisibility at the latitude o f E g y p t aroimd 3 5 0 0 BC,^^ if v i e w i n g w a s perfect a n d o b s e r v a t i o n s were carried o u t a n d a v e r a g e d over several years. T h e observational factors p r e v i o u s l y m e n t i o n e d w o u l d tend to lengthen the p e r i o d o f invisibility, altering the hypothetical date o f first calculation o f the invisibility o f Sirius to u p to o n e t h o u s a n d years later,^^ d e p e n d i n g u p o n the quality o f observational data u s e d a n d the n u m b e r o f observations over w h i c h the p e r i o d w a s averaged. S e v e n t y days is n o t o n l y the length o f the invisibility o f Sirius, b u t is also the time set aside for funerary p r e p a r a t i o n s , w h e r e status permitted, after a death. It is therefore possible that the u s e o f the d e c a n s to m a r k the h o u r s w a s a b y - p r o d u c t o f

The earliest occurrence of the text is in the New Kingdom, far later (by at least one thousand years) than the origin of Egyptian interest in the motions of stars, therefore the use of the exact period of seventy days may be a later concept. The difficulty of dating the origin of the text means that the seventy-day funerary period and the seventy-day ideal invisibility of the decans cannot be considered absolute; however, it is still certain that the invisibility of Sirius was significant. Using the heliacal rise and set prediction algorithm presented by SCHAEFER ( 1 9 8 5 ) and converted into a computer program listed in SCHAEFER ( 1 9 8 7 ) . This period ( 3 5 0 0 BC to 2 5 0 0 BC) ties in with the most likely of three possible dates for the establishment of the civil calendar based on an analysis of Sothic periods: 2 7 8 1 - 2 7 7 8 BC, given by CLAGETT ( 1 9 9 5 ) , p, 3 1 . NEUGEBAUER'S ( 1 9 4 2 ) rejection of an astronomical basis for

the foundation of the civil year which leads Clagett to date the introduction of the calendar to c. 3 0 0 0 BC still lies within the estimate of the time period during which the disappearance of Sirius lasted ' 7 0 days', as does PARKER'S ( 1 9 5 0 ) estimate (based on an average lunar year) of c. 2 9 3 7 - 2 8 2 1 BC. The date of the introduction of the civil calendar is significant to the present argument because it would only be after the introduction of the civil calendar that counting days in packets of ten would become natural.

444

S. Symons

another, m o r e i m m e d i a t e l y practical, activity: m a r k i n g a p e r i o d o f d a y s . If the d e a t h a n d burial o f a p e r s o n c a n b e likened to the d i s a p p e a r a n c e a n d heliacal rise o f a star, then the p e r i o d b e t w e e n death a n d burial c a n similarly b e linked to the p e r i o d b e t w e e n the d i s a p p e a r a n c e a n d r e a p p e a r a n c e o f a star. This idea leads to a hypothetical course o f d e v e l o p m e n t o f the rising star c l o c k w h i c h is outlined b e l o w . T h e link b e t w e e n the n u m b e r days o f flmerary p r e p a r a t i o n s a n d the d i s a p p e a r a n c e o f a star p r o v i d e s the m o t i v a t i o n to o b s e r v e stars a n d t h e dates o f their d i s a p p e a r a n c e a n d r e a p p e a r a n c e , with the objective o f m a k i n g a list o f stars w i t h a p e r i o d o f invisibility o f about seventy days. If observations were b e g u n o n this basis, the e n o r m i t y o f t h e task w o u l d very s o o n b e c o m e apparent. S o m e short cuts w o u l d b e n e e d e d to simplify the w o r k a n d the e n d p r o d u c t . Firstly, t h e brightest stars w o u l d b e the easiest a n d m o s t o b v i o u s to o b s e r v e . U s i n g o n e star p e r d a y w o u l d n e e d a list o f 365 stars. Bright stars are not frequent so a list this long w o u l d contain very faint stars a n d also g a p s where n o star w a s suitable. O n e star p e r m o n t h is t o o long in c o m p a r i s o n with the seventy-day p e r i o d w h i c h w a s to b e m e a s u r e d . S o m e frequency b e t w e e n o n e d a y a n d thirty days w o u l d b e chosen. E x a c t l y h o w e a c h star w a s selected is n o t k n o w n , but a c o m b i n a t i o n o f m a g n i t u d e , familiarity, p e r i o d o f d i s a p p e a r a n c e , a n d a certain a m o u n t o f h u m a n error a n d l e e w a y w o u l d a l l o w a list to b e m a d e with a r e a s o n a b l e a m o u n t o f ease. In order to u s e the list o f stars for the original p u r p o s e it w a s intended, o n e w o u l d find o n the list the star which d i s a p p e a r e d o n or a r o u n d the date o f death. T h e r e a p p e a r a n c e of that star w o u l d m a r k the time o f burial, or o f the d e c e a s e d ' s entrance to the D u a t , or s o m e other important m o m e n t in the existence o f the d e a d . T h e transformation o f this list o f star n a m e s (the p r o t o t y p e o f a d e c a n list) a n d dates into a star c l o c k requires the addition o f a n e w axis m a r k e d in h o u r s a n d s o m e observation of the m o v e m e n t s o f the listed stars after their r e a p p e a r a n c e . V e r y few observations w o u l d b e n e e d e d to p r o d u c e the diagonal p a t t e m w h i c h is characteristic of diagonal star clocks. F r o m the a r g u m e n t a b o v e for the evolution o f the diagonal star c l o c k from a list of stars created t o m e a s u r e the p a s s a g e of days, it c a n b e seen that the date list contained within the Book of Nut resembles m o r e closely a n e m b e l l i s h e d list o f this type (with the event 'Tpf b e i n g a d d e d t o the n e c e s s a r y ' E n c l o s u r e ' a n d ' B i r t h ' ) than an i m p r o v e d diagonal star clock. S o m e s u p p o r t for the importance o f ' E n c l o s u r e ' a n d ' B i r t h ' o v e r T p f ' in the date list c a n e v e n b e g l e a n e d from p a p y m s Carlsberg l a , w h i c h contains a p a s s a g e w h i c h lists dates o f ' E n c l o s u r e ' with dates o f ' B i r t h ' citing ten d a y ' m a r g i n s o f e r r o r ' . Five entries h a v e survived at the beginning o f C o l u m n III o f the p a p y m s . T h e s e contain (confused) dates o f ' E n c l o s u r e ' a n d ' B i r t h ' for four date sets ( 2 2 , 2 5 , 2 8 , a n d 3 1 ) from the Book ofNut?^ 'Tpt' is not mentioned. T h e survival o f a d e s c e n d a n t o f a n early form o f date a n d star list t o the N e w K i n g d o m is certainly n o t a n impossible concept, especially w h e n it o c c u r s in the Osireion c o m p l e x w h e r e a type o f diagonal star c l o c k also w a s u s e d as d e c o r a t i o n for another ceiling.^^ T h e unique nature o f the Osireion m a k e s it likely that decoration for the building w a s chosen to represent the oldest available k n o w l e d g e . " Neugebauer and Parker suggest (NEUGEBAUER and PARKER (1960), p. 89) that it is not impossible for a column to have existed between the surviving Columns II and III in which 'Enclosure' and 'Birth* from other date sets may have been listed. NEUGEBAUER and PARKER (1960), pp. 32-35.

The 'Transit Star Clock' from the Book of Nut

445

A date for the star calendar c a n b e o b t a i n e d from the heliacal rising o f Sirius. B y a s s u m i n g that date set 35 relates to spdt, N e u g e b a u e r a n d P a r k e r ^ ' s a w that ' B i r t h ' o c c u r r e d o n IIII Prt 16, w h i c h places the date list in the M i d d l e K i n g d o m , aroimd 1850 B C , a n d implies that the date list, a n d h e n c e any t i m e k e e p i n g m e t h o d it represents, c o u l d h a v e b e e n m o r e than five h u n d r e d years old at its i n c o r p o r a t i o n into the d e c o r a t i o n of the Osireion. Leitz''*^ argues for a m u c h earlier date. H e c o m b i n e s an earlier year for prt spdt o n IIII Prt 16 ( = 16th July), a 7 0 - d a y p e r i o d of invisibility of Sirius, a n d culmination of Sirius at m i d n i g h t o n I iht Ì, to arrive at a date of a r o u n d 3 3 2 3 B C for the compilation o f the date list. T h i s date caimot b e confirmed: the u s e s of ' c u l m i n a t i o n ' a n d ' m i d n i g h t ' are p r o b l e m a t i c , a n d there are difficuhies associated with determining a n era for the b a s i s of the 7 0 - d a y p e r i o d of invisibility of Sirius. W h a t is indisputable is the r e m a r k a b l e survival o f the Book of Nut as a k n o w n a n d studied text into the present era in the form o f p a p y r i C a r l s b e r g 1 a n d l a ( w h i c h date from the s e c o n d century A D ) . If a text c a n survive from the N e w K i n g d o m to R o m a n times, it is certainly possible that the basis for, or e v e n the original of, that text p r e - d a t e s its earliest surviving occurrence, a n d that the m e t h o d the text describes m a y b e e v e n older.

Conclusion T h e star c a l e n d a r m the Book of Nut is a n e m b e l l i s h m e n t of a list of stars w h i c h displayed a c o m m o n p e r i o d of invisibility, n o t a n i m p r o v e d diagonal star c l o c k n o r a 'transit star c l o c k ' . F r o m the N e w K i n g d o m sources only, the date list in itself is not sufficient to find the time during the night. Firstly, the d e c a n list included with the date list in the Book of Nut is too fragmentary to p e r m i t the use o f the date list in s u c h a way. Secondly, the information contained within the date list a n d its a r r a n g e m e n t d o e s not lend itself to this use. It s e e m s p r u d e n t w h e n trying to determine the nature of the date list first p r e s e n t e d in the N e w K i n g d o m to allow N e w K i n g d o m sources to h a v e the p r i m a r y focus. T h e label 'Tpt' c a n surely b e taken to m e a n wnwt tpt 'Ffrst H o u r ' w i t h the m e a n i n g ' b e g i n n i n g of the n i g h t ' , without implying that the entire table as it stands should b e interpreted as a clock. It has also b e e n suggested'*' that the 120 d a y s related to 'Tpt' simply represent the p e r i o d of time during w h i c h a star culminates during the h o u r s of darkness. If ' c u l m i n a t i o n ' is t a k e n in a g e n e r a l sense o f the highest altitude a star attains, rather than in the p r e c i s e definition o f ttansiting the meridian, this p r o v i d e s a m e a n i n g for the third event in the s t a r ' s year: d i s a p p e a r a n c e , r e a p p e a r a n c e , a n d attaining greatest height at the b e g i n n i n g o f the night (being ' t o p d e c a n ' after simset). S u c h a r o u g h definition o f 'Tpt' is m o r e in k e e p i n g with the nature of the list as a calculated, rather than o b s e r v e d , description of annual events. T h e existence o f a m e t h o d for determining the h o u r s of the night using the fransits o f d e c a n s across the m e r i d i a n has b e e n disputed a n d it has b e e n s h o w n that NEUGEBAUER and PARKER ( 1 9 6 0 ) , p. 5 4 . L E I T Z ( 1 9 8 9 ) , p. 5 2 . DEPUYDT ( 1 9 9 8 ) , p. 9 .

446

S. SYMONS

the Book of Nut star calendar is not an ' i m p r o v e m e n t ' of the d i a g o n a l star c l o c k b u t is closely related to the p r e c u r s o r of the diagonal star clock: a list of stars w h i c h d i s a p p e a r e d for s e v e n t y days.

References CLAGETT, M a r s h a l l . 1 9 9 5 . Ancient Egyptian Science Vol. 2: Calendars, Clocks, and Astronomy. Philadelphia: A m e r i c a n Philosophical Society. D E P U Y D T , L e o . 1998. " A n c i e n t Egyptian star c l o c k s a n d their theory". Bibliotheca Orientalis 5 5 : 5 - 4 4 . F R A N K F O R T , H e n r i . 1933. The Cenotaph of Seti I at Abydos. London: Egypt E x p l o r a t i o n Society. L A N G E , H a n s O . a n d N E U G E B A U E R , Otto. 1940. Papyrus Carlsberg No. 1. Ein hieratisch-demotischer kosmologischer Text (Det Kongelige Danske Videnskabemes Selskab. Historisk-filologiske Skrifter, B d I, N r . 2). C o p e n h a g e n : E. M u n k s g a a r d . LEITZ, Christian. 1989. Studien zur agyptischen Astronomie (Àgyptologische A b h a n d l u n g e n 4 9 ) . W i e s b a d e n : Otto Harrasowitz. — 1 9 9 5 . Altagyptische Sternuhren (Orientalia L o v a n i e n s i a analecta 6 2 ) . L e u v e n : Uitgeverij P e e t e r s e n D é p a r t e m e n t Orientalistiek. N E U G E B A U E R , O t t o . 1942 " T h e Origin of the E g y p t i a n C a l e n d a r " . Journal Eastern Studies

of

Near

1: 3 9 7 - 4 0 3 .

— a n d P A R K E R , R i c h a r d A . 1960. Egyptian

Astronomical

Texts Vol. 1. P r o v i d e n c e :

Astronomical

Texts Vol. 3. P r o v i d e n c e :

B r o w n University Press. — a n d P A R K E R , R i c h a r d A . 1969. Egyptian B r o w n U n i v e r s i t y Press. PARKER,

Richard

A.

1950.

The

Calendars

of

Ancient

Egypt

(Studies

in

A n c i e n t Oriental Civilization 26). C h i c a g o : U n i v e r s i t y of C h i c a g o P r e s s . — 1974. " A n c i e n t Egyptian A s t r o n o m y " . In: F r a n k R. H O D S O N (ed.), The place astronomy

in the ancient

of

world: 5 1 - 6 5 . L o n d o n : O x f o r d University P r e s s for the

British A c a d e m y . SCHAEFER, B r a d l e y E. 1985. "Predicting Heliacal Risings and Settings". Sky Telescope

and

70: 2 6 1 - 2 6 3 .

— 1 9 8 7 . " H e l i a c a l Rise P h e n o m e n a " . Journal

for

the History

of Astronomy

11:

19-33. S Y M O N S , Sarah. 2 0 0 0 . " A c c u r a c y Issues in A n c i e n t Egyptian T i m e k e e p i n g " In: A n g e l a M C D O N A L D a n d Christina RIGGS (eds.). Current Research

in

Egyptology

2000 ( = B A R I n t e m a t i o n a l Series 9 0 9 ) : 1 1 1 - 1 1 4 . Oxford: A r c h a e o p r e s s .

Enûma Anu Enlil Tablets 1-13* Lorenzo

Verderame,

Rome

T h e first thirteen tablets o f the astrological series EnUma Anu Enlil ( E A E ) , ' together with T a b l e t XIV,^ a sort o f astronomical c o m p e n d i u m are a h o m o g e n e o u s a n d a u t o n o m o u s g r o u p inside the series itself, a n d are c o n s i d e r e d to b e a sub-series with a n a m e o f its o w n , I G I . D U g . A . M E sd 3 0 "visibilities of the m o o n " . T h i s sub-series deals w i t h p h e n o m e n a associated with the a p p e a r a n c e s or visibilities o f the m o o n , as suggested b y the title. In particular, m o s t of the o m e n s deal with the first a p p e a r a n c e o f the m o o n a n d contain in the protasis the expression ina IGI.LÀ-fw {ina tâmartisu) "at its a p p e a r a n c e " . T h e B a b y l o n i a n s c o n s i d e r e d t w o days to b e particularly n o t e w o r t h y in the month:^ the first day, n e w m o o n , a n d the 14/15th day, ideal fitll m o o n ; since the latter also c o n c e m e d lunar eclipse p h e n o m e n a , extensively treated in the T a b l e t s X V - X X I I o f E A E , the 14/15th d a y is rarely m e n t i o n e d in T a b l e t s I-XIII.'* E v e n the d a r k e n i n g o f the m o o n , generally d e n o t e d b y the v e r b adâru, that usually indicates eclipses, that are the m a i n topic o f E A E X V - X X I I , are rarely attested in tablets I-XIII.^ T h e u s e o f the v e r b ekëlu " t o b e d a r k " in the incipit o f This paper is part of a more general study about the Tablets EAE I-XIII, that will be published soon; see VERDERAME (2002). 1 would like to thank John Steele and Christopher Walker for the invitation to participate in UOS meeting. 1 would also like to thank Christopher Walker, together with the Authorities of the British Museum and the staff of the Student Room, for the assistance given to me in the Student Room in this last three years; David Brown, Salvo de Meis, Sveva Mazzini, who read this text in its early phases and who made many helpful suggestions conceming its content. Any errors found herein are my own. ' For a general reconstruction of the series see WEIDNER (1941-1944; 1954-1956; 1968-1969). For editions of celestial omina from EAE see GEHLKEN (2000); LARGEMENT (1957); REINER and PINGREE (1975, 1981, 1998); AL-RAWI and GEORGE (1991-1992); VAN S o L D T (1995); ROCHBERG-HALTON (1985).

^

WEIDNER (1941-1944), pp. 317-318; AL-RAWI and GEORGE (1991-1992), pp. 52-72;

HUNGER and PINGREE (1999), pp. 4 4 - 5 0 .

^

LANDSBERGER (1915), pp. 99ff; STOL (1992), pp. 245-250; COHEN (1993), pp. 3ff.

Notice that omina conceming the visibility of the moon and sun on day 14, 15, etc. (Die 30 M 20 im-iah-ru-ma, sit-qu-lu, su-ta-su-û, su-ta-tu-ù, KI a-ha-mes IGI.MES, etc.), well known from the Reports and commentaries, are very rarely from Tablets I-XIII, except for an omen conceming the moon and sunlight, Die 30 u 20 \}D.Dk-su-nu du-^-it-mat (Tablet I: 823-23, 16 + Rm 151/2Sp2 + Sp8: r.34, and Sm 955: r.ll'; Tablet VI: K 1348 + Rm 591/2Sp3+SplO: 11'). ^ For adâru (KAxMI) in EAE I-XIII, see e.g. K 2311 + 3624 (Spi): 1-8; K 6570: 1-4, both belonging to Tablet II; VAT 7813 (WEIDNER (1941-1944), Pl.XIII): 1 (incipit of Tablet IV, see Uruk Cat.: 4). The verb da'âmu (DARA4) refers principally to luminous phenomena, as

448

L. Verderame

T a b l e t 11,*^ suggests that t h e m a i n topic o f this T a b l e t w a s t h e d a r k e n i n g o f the m o o n ; nevertheless ekëlu is e m p l o y e d chiefly in c o n n e c t i o n with l u m i n o u s p h e n o m e n a ' a n d d o e s n o t indicate t h e d a r k e n i n g resulting from eclipse a s is the case with the v e r b adâru^ T h e first a p p e a r a n c e o f the crescent, that is the b e g i n n i n g of a n e w lunar m o n t h , is p r e c e d e d b y a p e r i o d of lunar absence {Um bubbulif\}^^k.kM)^ a time of danger and death. T h e transitional character o f the m o n t h ' s last d a y is also e m p h a s i z e d in the h e m e r o l o g i e s a n d in m a n y p a s s a g e s from literary texts: in this d a y S i n a n d S a m a s are i n council a n d take decisions. A n anticipated o r d e l a y e d visibility o f the crescent, that is t o say, a n irregularity in the standard length o f the m o n t h , 3 0 days,"^ is c o n s i d e r e d inauspicious. ' ' T a b l e t I b e g i n s , after the S u m e r i a n a n d A k k a d i a n introduction,'^ with p h e n o m e n a that h a p p e n i n t h e days b e t w e e n t h e m o o n ' s d i s a p p e a r a n c e a n d t h e n e w m o o n , i.e. from t h e 27th o f a m o n t h to the 2 n d o f the next month.'^ T h e B a b y l o n i a n observer w a t c h e d a t t h e early o r irregular {ina là simânisu, ina lâ adanni-su, ina là minâtisu/SÏDME^-suy^ a p p e a r a n c e s o f the m o o n , at the shape a n d n u m b e r o f the

the 5ë/M/UD.DA or the agûlKGK; rarely to the moon (Tablet VI, K 1348 + Rm 591/2Sp3+SplO: r.44, DIS 30 ina È-sû DARA4) or its homs (Tablet VIII, K 2267+/2Spla: iii 3 // TU 17: 14, DiS SI.MEè-:?!/ DARA4.MES-/Mfl WJL.Is le-e ana \G\-su GUB). ^

Uruk Cat.: 2, 30 ina IGI.LÂ-iw e-kd "the Moon is dark at its appearance".

^ Referring to the light of the moon (^e/w/UD.DA), e.g. DIS UD.DA 30 {ma-gal) ek-let, in Tablet I (82-3-23, 16 + Rm 151/2Sp2 + Sp8: r.29'; Sm 955: r.7'; LBAT 1527: r.l5'), VI (K 1348 + Rm 591/2Sp3+SplO: 13') and VIII (TU 17: 29; LBAT 1529: 11'; K 2267+/2Spla: iii23'; K 3741b: 4'); to the appearance of the day, e.g. U4-mi ek-le-tu (Sm 751/SpI5: r.3), or of a planet/constellation, e.g. REINER and PINGREE (1998), pp. 218-219 1. 51 (MUL.SU.PA e-kil) and SAA8 502: 14 (DIS MUL.GÎR.TAB i-da-tu-sâ uk-ku-la). See CAD E sv ekëlu mng. a 1 and b 1, p. 64. *

VERDERAME (1999).

^ See note 3; especially LANDSBERGER (1915), pp. 9 9 f and 141ff.; HOROWITZ (1998), pp. 162 and 256; CAD B sv. bubbulu mng. 2, pp. 298-300; AHw sv. bubbulu, p. 135. '° It is useful here to remember that 30 is also the number with which the moon and moongod were coinmonly designated. "

BEAULIEU (1993), p p 66-69.

See my forthcoming paper "The bilingual introduction to the astrological series Enuma Anu Enlil." The period of lunar absence is rarely as long as three or more days and only under certain metereological conditions: UD.27.KÂM 30 it-ta-[bal\ \ UD.28.KÀM UD.29.KÂM ina AN-e bu-û-ut \ Ù UD.30.KAM it-tan-mar \ im-ma-tim-ma li-in-na-mir \ ba-ab-ti 4 U4-mi-i ina AN-e li-bit \ im-ma-tim-ma 4 U4-mu ul i-bit "The moon disappeared on the 27th; the 28th and the 29th it stayed inside the sky, and was seen on the 30th: when (else) should it have been seen? it should stay inside the sky less than 4 days, it never stayed 4 days" (SAA8 346: r.2-7). See BEAULIEU (1993), p. 66 and notes 2 and 3. Tablet I: K 4027 (Sin 1): 26, 32; 82-3-23, 16 + Rm 151 (2Sp2 + Sp8): 14-20; Sm 955: 12, 24-30; LBAT 1527: 3 - 9 K 8145 (2Sp2): 9, 18-24; K 6505: 11-16; BM 36828 + 80-6-17, 618: 8-9. For adannu see CAD A] sv a. mng. 2a-2', p. 100; M2 sv minîtu, pp. 86-87; S sv simânu, p p 268-269. WEIDNER (1911), p. 72; KUGLER (1913), pp. 251 and 2 6 4 - 2 6 5 ; G A D D (1966), pp. 3 1 - 3 2 ; PARPOLA (1983), p. 72; ROCHBERG-HALTON (1985), pp. 4 2 - 4 3 ; BROWN (2000),

p p 148 and 154n.367.

Enûma Anu Enlil Tablets 1-13

crescents {uskârufU4.SA¥iJiK),^^ (5êm/UD.DA).'^

and

at

the

appearance

449

of

the

lunar

light

M o s t of the p h e n o m e n a treated in the T a b l e t s I-XIII are, h o w e v e r , a t m o s p h e r i c . H a l o s , " c r o w n s " , s h a p e s a n d c o l o i n s o f the m o o n are p h e n o m e n a c a u s e d b y ice crystals, w a t e r particles, a n d dust p r e s e n t in the a t m o s p h e r e . M o s t of the o m e n s c o n c e m i n g a t m o s p h e r i c p h e n o m e n a , d e s c r i b e in the p r o t a s i s the visibility o f the m o o n , w h e t h e r or n o t it is o b s c u r e d b y c l o u d s ( I M . D I R I ) or b y a sabThuJ^ or the aspect o f the m o o n ' s light (sêtu/VD.DA), w h e t h e r it is faint (unnutu) or b r i g h t {ba alulZALAG), dark or m a d e u p of different colours.'^ A n o t h e r a t m o s p h e r i c p h e n o m e n o n is that o f the lunar halos {tarbâsuITlJ^: one or m o r e liuninous circles that encircle (/amw/NIGIN) the m o o n c a u s e d b y refraction a n d reflection o f light p a s s i n g through ice crystals in the a t m o s p h e r e . ' ^ T h e limar h a l o s are the m a i n topic of T a b l e t s X - X I I , b u t are also freated in other tablets (VIII a n d IX).^° T a b l e t V I I I deals in general with h a l o s that c o u l d h a p p e n from m o n t h V I to m o n t h X I I ; particular attention is p a i d to the p r e s e n c e o f o n e or m o r e halos^' a n d to the different c o l o u r s they m i g h t h a v e . T h e p r e s e n c e inside the h a l o of a n o p e n i n g ("gate", bâbufKÂf^ or of a h e a v e n l y body^^ are the topic of T a b l e t X . U n d o u b t e d l y ,

DIS 30 NU ìGl-ma 2(/3) U4.SAKAR.MES IGI, Tablet I: K 4027 (Sinl); r.4'-5'; 82-3-23, 16 + Rm 151 (2Sp2 + Sp8): r.36'-37'; Sm 955 (2Sp2): r.l3'-14'; BM 37377: 2'-3'. For U4.SAKAR see AHw sv usk/qâru, p. 1438; WEIDNER (1911), pp. 2 9 - 3 2 ; CIVIL (1966), p. 92; STOL (1992), pp. 245-247. See note 18. CAD S sv sêtu, pp. 150-151. " K 11094 + 11252 (Sinl+2): ii6'-7'; cf K 6021: 11, REINER and PINGREE (1998), pp. 46-49,11. 8 0 - 8 4 and parallels. For sabThu, some kind of meteorological phenomenon, see CAD S, sv s., p. 11, and A2 sv aramu mng. Id, p. 229; WEIDNER (1936-1937), p. 362, n.20; VON SODEN (1955), p. 141, n.l, "Wolken oder Nebelablagerung"; REINER and PINGREE (1998), p. 14. The term employed are (UGU sà gi-na-a) da'âmu and ekëlu "to be dark" (cf note 7), SIG7 {araqu) "to be green", {ma-gat) ZALAG{/namaru) "to be bright", K O R . K O R {nakâru Gt/Dt) "to change" (cf REINER and PINGREE (1998), pp. 2 3 2 - 2 3 3 , 1 . 17 and parallels), kasû "to become cold", galàtu "to quiver". Tablet I: 82-3-23, 16 + Rm 151 (2Sp2 + Sp8): r.26'34'; Sm 955 (2Sp2): r l ' - l l ' ; LBAT 1527: r.9'-17'. Tablet VI: K 1348 (2Sp3 + SplO): 9', 10', 12'. STT339: 25', 29', 31', r.4. "

KUGLER (1909-1924), p. 103; REINER and PINGREE (1981): 2.2.7, p. 20, and (1998),

pp. 11-12; D E MEIS and HUNGER (1998), pp. 20, 35f, passim. For solar halos see KUGLER (1911), pp. 107-130; VANS0LDT(1995),pp. 6f, 23 I Va 1-3, passim. See also Iqqur Ipus §79 and 80. Tablet X reports cases of two, libbânul^k-nu "inner" ( K 2887/2Spl5: 3', 5', 7'; K 200+/SinlO + 2Spl4: r.22', 30'; 81-7-27, 96: 4'-6'; CT51 143: 18', r.5') and kidû "outer" ( K 2887/2Spl5: 9'; K 200+/SinlO + 2Spl4: r.22', 30', 31', 40'; 81-7-27,96: 7';CT51 143: 18', r.5', 6', 14'), or three halos ( K 2887/2Spl5: 3', 5', 7', 10', 13', 16', 17'; K 200+/SinlO + 2Spl4: r.23'; 81-7-27, 96: 5'; CT51 143: 19'), 1 "first" (22°), sanûll-û "second" (46°) and 3SÛ "third" (90°). 2^ DIS 30 TUR ... NÌGIN-/na KÀ ana IM.x BAD {petû) I KUD (paràsu, nakàsu); also the absence of this opening is recorded in the omina of Tablet X : DI§ 30 TÙR ... NIGlN-ma KÀ NU TUK, quoted in Iqqur Ipus §79. K 2887 (2Spl5): 3', 10', 13', 16', 17'; K 2398 (Sp21): 2'-20'; K 200+ (SinlO + 2Spl4): 8'-5r, r.1-5', 33'-40'; 81-7-27, 96: 8', 10', 12'; K 2140: 2'8;K 10731: l'-6';K 13786: r-6';CT51 143: 8'-14'; LTTN 4: r. Ì3-23; Sin24: 59-69.

450

L. Verderame

halos were dealt with in Tablet IX and, perhaps, in Tablet XI, judging b y their incipits^'* and their closeness to Tablet X. These two Tablets are difficult to restore at present, but the supûm/AMAS or "great halo","^^ together with the usurtu/GlSMUR^^ could be the topic o f the fragmentary Tablets IX or XI. The agli/AGA, lunar "tiara" or crown, is the main topic o f Tablet III. The term agû refers to at least two, maybe, three different phenomena:^' 1) earthshine; 2) cloud-banks that encfrcle the m o o n as headgear; 3) the hypothetical meaning o f "fiill moon". The o m e n s ' protases note the agûs number and colours, besides the interaction with atmospheric phenomena (clouds, storm, etc.) and heavenly bodies. The agii, conceived as headgear (tiara or crown) o f the moon, is imagined to be made up b y different metals and materials.^^ It is also compared with objects^' and, in a way less understandable to us, with abstiact concepts.^'' a. MUL.SIPA.ZI.AN.NA (Orion; K 2134+/2Spl6: 22, 25); b. M U L . B A N (part of Canis Maior; K 2134+/2Spl6: 31); c. MUL.BIR (part of Puppis; K 2134+/2Spl6: r.l; K 9574: 2'); d. Zibânïtu (Scales; K 2134+/2Spl6: r.3; K 10442: 6'; 81-7-27, 134 : 9'); e. MUL.SA5 (Mars; K 2134+/2Spl6: r.3; K 9574: 4'); f MVL.C^)Anim (K 2134+/2Spl6: r.5; K 9574: 6'); g. MUL.MU.BU.KÉè.DA (Bootes; K 200+/SinlO+2Spl4: 5); h. (MUL.'')UDU.IDIM (a planet generally. Mars, Mercury; K 200+/SinlO+2Spl4: 6); i. IM.DIRI SA5 (red cloud; K 200+/SinlO+2Spl4: r.7-11; CT51 143: 3-7); j . MUL.NIN.URTA (Sirius; K 200+/SinlO+2Spl4: r.l8; CT51 143: 14); k C*)âUL.PA.È.A (Jupiter; K 200+/SinlO+2Spl4: r.l9, 20; CT51 143: 15, 16); 1. (MUL.'')TIR.AN.NA (Andromeda?; K 200+/SinlO+2Spl4: r.21; CT51 143: 17); m. MUL ("a star;" K 200+/SinlO+2Spl4: 4, r.24); n. MUL.MES ("stars;" K 200+/SinlO+2Spl4: r.25, 26; CT51 143: r.l, 2). Tablet IX: 30 ina IGI.LÀ-5M TUR NIGIN-wa ka-bar u su-par-ru-ur "The Moon at its appearance is surrounded by a halo and it is large and extended" (Uruk Cat.: 9). DIS 30 AMAS NIGIN-wû id-lip na-da-nu pa-le-e ana LUGAL sà T Ù R ra-bu-û NIGINma "If the moon is surrounded by a sheepfold and lingers on: giving of the reign to the king. I.e., it is surrounded by a large halo" (SAA8 494: 7-9). Any of the fragments belonging to EAE I-XIII treats the supûm/AMAS , well known from other sources (BM 38295: r.l; BM 38160: r.2; BM 26185: r.5; Sin 16: Iff., 3 : 131-134, 24 : 10; LTTN 1: 3-6; Iqqur Ipus §80; SAA8 494: 1-2, 7-9); see CAD S sv supûru mng. 2, p. 398. Cf the epithet of Sin "the Lord of the horn (and) the halo" (EN qar-ni su-pu-ri) in a babylonian hymn, LANGDON (1915), p. 191, 1.7; STOL(1992), p. 255. The usurtu, lit. "design", is a light phenomenon that encircle the moon, perhaps an halo; Cf Iqqur Ipu§ §77, 77', 78, 80. For the cruciform shape of the usurtu on the basis of K 4336: i l 8 (WEIDNER (1941-1944), pi. 7: il8), [is\-pal-lu-ur-tû : ù-sur-tû, see LARGEMENT (1957), pp. 262f ; REINER (1974), pp. 258-259, n.l4; ROCHBERG-HALTON (1985), pp. I56f and note 4. " CAD A, sv agû, p p 153-4, i.e. mng. 2a r-3'. WEIDNER (1911), p p 2 4 - 5 2 ; STOL(1992), pp. 249-250; BROWN (2000), p. 89, n.226. KÙ.BABBAR "silver"; KÙ.GI "gold";, ZABAR "bronze"; URUDU "copper";, AN.NA "lead"; etc. See Tablet III: K 6984+ (Sp2): r.3-5, 8-10; K 6276 (Sp3): 4-9; K 4768 (2Spl7): 3, 7-12; K 6291 (2Spl7): 8, 12-18; K 11244: 5-7; K 3855 (Sin20): 4. SAGâU "turban"; GIS.GIDRI "scepter". See Tablet III: K 6984+ (Sp2): r.9; K 10608 (Sp4): 2; K 2890 (Sp5): r.5-7; BM 32373: 6; LTTN I: 2. N I G . H A . L A M . M A "destruction"; saggastu "slaughter"; sagaltu "homicide"; saqummatu "mourning"; salummatu "terrifying splendor"; da'ummatu "darkness"; HÉ.NUN "abundance"; ME "battle"; mêsaru ( / N Î G . S I . S À ) "justice"; uzzu "anger"; ummatu "heat". See Tablet III: K 2311 + 3 6 2 4 (Spi): 28, 31-35; K 6984+ (Sp2): r.13-14; K 6276 (Sp3): 3, 13, 15, 17; K 10608 (Sp4): 1, 6-7; K 2890 (Sp5): r.8-9; K 4768 (2Spl7): 6; K 6291 (2Spl7): 11, 20,

EnQma Anu Enlil Tablets 1-13

451

M a n y tablets deal with aspects o f t h e m o o n ' s h o m s (qarnu/Sl), b u t this is the central topic o f T a b l e t V a n d m o s t o f T a b l e t X I I . T h e h o m s p h e n o m e n a c o n c e m m o s t l y their s h a p e : they could b e , principally, p o i n t e d (edédu^^) o r b l u n t e d (kepû^^), stretched (tarâsu^^), t i n n e d t o w a r d t h e g r o u n d o r the s k y (lit. " l o o k / p i e r c e t h e ground/sky", K I / A N IGl^^/terû^^), otherwise they r e a c h (kasadu^^) o r t o u c h (nenmudu^^) e a c h other. M o r e difficult t o interpret a r e p h e n o m e n a that a p p e a r t o b e impossible, a n d s o t o b e imagined. W e h a v e t w o solutions for these eventualities: either w e d o n o t u n d e r s t a n d yet w h a t t h e scribe really m e a n t w h e n h e u s e d these expressions, or w e refer t o the M e s o p o t a m i a n t e n d e n c y t o w a r d s c o m p l e t i o n , the result o f w h i c h is a list o f p h e n o m e n a that n e v e r could h a p p e n in nature.^^ Other kinds o f p h e n o m e n a a r e astronomical: the vicinity a n d conjimction (occultation) o f h e a v e n l y b o d i e s (planets, stars a n d constellations) b y t h e moon.^^ E x a m p l e s o f this kind o f p h e n o m e n a a r e scattered t h r o u g h o u t E A E , they a r e systematically treated in T a b l e t V I a n d in s o m e w a y also in T a b l e t s III a n d X I I . T a b l e t V I is d i v i d e d into sections that c o r r e s p o n d t o a h e a v e n l y body'*" particularly in different positions next to the moon."*'

24-26; K 3625: 4-7; 1902-5-10, 23: 3, 6-8, 11; 1902-5-10, 31:4; 80-7-19, 158: 5, 9,10; 81-24, 333: 6-8, 12; DT 74: 5-6; K 8493: 2; BM 32373: 2, 5, 10; LTTN 1: 6-7. Tablet V: DT 104(Spl8): 1', 2', 5', 6', 9', 10'; 89-4-26, 25: 3'; Rm2 302: 23'; K 3559 + 3777 (2Spllb): 6; K 6833 + 7948: 4-6; K 10964: 2', 7'; Sm 1931 ( S p l l ) : 3'; K 11094+ (Sinl+2): 13', 14'; BM 38289 + 38762: r. 2, 3, 6, 7, 10, 11, 14, 15, 26; BM 99072: 6. Tablet VII: Rm2 39 (Spl3): 1 '-9'; K 1348 (2Sp3 + SplO): r.55; K 2267+ (2Spla): i I9'-25'; K 3103: 7'; K 6396: 5'-8'; K 10102: V-T; K 12429: 8'-10'; Sm 182: 8'. Tablet VIII: TU17: 12-13; K 12282: 4; LTTN 13: ii I', 4'. K 9505: 8'; SAA8 105: 3-8, 106: 4-7, 252: 1-3, 257: 5-7, 389: r.1-4, 362: 1-3,505: 5-6. Tablet V: DT 104 (Spi8): 1', 2', 5', 6', 9', 10'; 89-4-26, 25: 4'; K 2223: 3, 10; K 3114 + 7155 (2Spl2): r.4; K 7192 (2Sp6): 1'; K 7661 + 11109 (2Sp8): 3 ' , 11'; K 10964: 2', 7'; BM 38289 + 38762: r. 2, 3, 6, 7, 10, 11, 14, 15; LTTN 2: iil7', 18'. Tablet VII: K 2133 + 2247 (2Splb): 31', 43'; K 6396: 9'-ir. Tablet VIII: K 2267+ (2SpIa): iii 10, 21 ; K 3741b: 2'; LBAT 1529: 9'; TU17: 21. SAA8 265: 5-r.l. Tablet V: K 2223: 6-8; K 3559 + 3777 (2Spl lb): 2-4; K 6833 + 7948: 7', 8'; K 7661 + 11109 (2Sp8): 7'-9'; BM 38289 + 38762: 8'-10'. K 9505: 10'. Tablet V: K 11094+ (Sinl+2): 6'-8'; K 2223: 11; K 6833 + K 7661 + 11109 (2Sp8): 2', 5', 12'; 7948: 1', 17', 19';Rm2 302: 18'; LTTN 2: i34'-36'. Tablet VII: K 3135: 6; K 2133 + 2247 (2Splb): 31'. K 75 (Sp 17): 1; SAA8 391: r. 1-3,498: 8-9. " Tablet V: K 2223: 1-2, 4, 5; K 7661 + lI109(2Sp8): 1', 2', 4', 5'; K 11094+(Sinl+2): r, 3'; Rm2 302: 19', 21'; LTTN 2: Ì30', 32'. VII: K 2133 + 2247 (2Splb): 43'; K 1348 (2Sp3 + SplO): r.45'. K 9505: 15'; STT339: 16'; SAA8 57: 5-6 and r. 3-5. T a b l e t V : D T 1 0 4 ( S p l 8 ) : 3 ' , 4 ' , 7 ' , 8', 11'; BM 38289 + 38762: r. 4, 5, 8, 9, 12, 13. "

Tablet V: K 2223: 9; K 7661 + 11109 (2Sp8): 10'.

BOTTÉRO (1974), pp. 144-168; TROLLE-LARSEN (1987), pp. 203-225; BROWN (2000), Chap. 2 and 3. PARPOLA (1983): pp. 385ff".; D E MEIS and MEEUS (1991), pp. 1-3.

0. U D . A L . T A R (Jupiter, probably this section belongs to Tablet V; K 7661 + 11109/2Sp8: r-4';K 15458: r-4'; K 7062: 6'-12'); 1. MUL.MUL (Pleiades; K 5281/Sin21: r-4'; K 7661 + I1109/2Sp8: 5'-8'; K 3773/2Sp9a: r-6'; K 3556/2Spl la: r-6'; K 3559 + 3777/2Spllb: il-9; Sm 1148: 1-6; K 9966: r ; B M 45894: r-2'); 2. SIPA.ZI.AN.NA (Orion; K 5281/Sin21: 5'-6'; K 3773/2Sp9a: 7'-8'; K 3556/2Spl la: 7'-8'; Sm 1148: 7-8; K 9966: 2'-

452

L. Verderame

Little is p r e s e r v e d o f s o m e Tablets o f t h e I G L D U g . A . M E sub-series. W e therefore d o not k n o w w h i c h other p h e n o m e n a w e r e treated. In particular, r e g a r d i n g T a b l e t XIII, the last o n e o f the astrological type o f the I G L D U g . A . M E sub-series, not e v e n incipit is p r e s e r v e d in t h e U r u k C a t a l o g u e . Significantly the sub-series I G L D U g . A . M E ends with T a b l e t X I V , w h i c h contain "tables giving s c h e m e s for, a m o n g other things, t h e lunar cycle";''^ a Tablet that, situated at the e n d o f this s u b series, constitutes a kind o f useful appendix.

Abbreviations C T = C u n e i f o r m T e x t s from B a b y l o n i a n Tablets in t h e British M u s e u m . L o n d o n : British M u s e u m Publications. 1 8 9 6 Iqqur Ipus = L A B A T ( 1 9 6 5 ) L B A T = SACHS (1955) L T T N = WISEMAN A N D BLACK (1996)

SAA8 = H U N G E R (1992) Sin = VIROLLEAUD ( 1 9 0 8 ) Sp = VlROLLEAUD(1910) 2 S p = VIROLLEAUD ( 1 9 1 2 ) S T T = GURNEY a n d H U L I N ( 1 9 6 4 ) T U = THUREAU-DANGIN (1922)

U r u k Cat. = U r u k C a t a l o g u e ( V A T 7 8 1 4 + A O 6 4 7 0 ) , s e e W E I D N E R ( 1 9 4 1 - 1 9 4 4 ) , pi. I

References B E A U L I E U , Paul-Alain. 1 9 9 3 . " T h e I m p a c t o f M o n t h - l e n g t h s o n the N e o - B a b y l o n i a n Cultic C a l e n d a r " . Zeitschrift fiir Assyriologie 83: 66-87. B O T T E R O , Jean. 1974. " S y m p t ô m e s , signes, écritures e n M é s o p o t a m i e a n c i e n n e " . In: Jean-Pierre V E R N A N T et al. (eds.), Divination et rationalité: 7 0 - 1 9 7 . Paris: Éditions d u Seuil. B R O W N , D a v i d . 2 0 0 0 . Mesopotamian Planetary Astronomy-Astrology (Cuneiform M o n o g r a p h s 18). G r o n i n g e n : Styx. CIVIL, M i g u e l . 1966. " a / u s k a m < U4-sakar2(SAR)". Revue d'Assyriologie 60: 9 2 . 3'; B M 45894: 3'-4'); 3. IM.DIRI (a cloud of different colours; K 3773/2Sp9a: 9'-13'; K 12654: 1'; K 9966: 4'-8; K 13436: 4'; BM 45894: 5'-7'); 4. BAN (part of Canis Maior; K 3773/2Sp9a: 14'; K 12654: 2'; K 9966: 9'; K 13436: 5'); 5. Sukudu (Sirius; K 6982/Sinl8 r-6';K 3773/2Sp9a: 15'-2r; K 12654: 3'-8'; LTFN 14: r - 1 2 ' ; K 9 9 6 6 : 10'-15'; K 13436 6'-8'); 6. GÌR.TAB (Scorpio; K 6982/Sinl8: 7'-12'; K 3773/2Sp9a: 22'-32'; K 7995/2Sp9b ii2'-12'; K 12654: 9'-12'; LTTN 14: I4'-27'); 7. UDU.IDIM (a planet generally. Mars, Mercury; K 6982/Sinl8: 13'-14'; K 3773/2Sp9a: 33'-34'; K 7995/2Sp9b: iil3'; LTTN 14:

28'-3r). a. ina k-sû "at one side"; b. ina ^k-sû "inside"; c. ina SI.MES-JM "inside the moon's homs"; d. ina §À SI ZkG-sû "inside (or near) the right horn"; e. ina SI GÙB-5M "near the left hom"; f ina bi-rit (/DAL.BA.NA) SI.MES-5M "between the moon's homs"; g. ana \G\-su "in front"; h. ina UGU-^u "above"; i. ina MAâ.QA-5« "near the moon's shoulder"; j . GIM TÙR NfGIN-5w "encircling the moon like a halo". See LARGEMENT (1957), p. 259. *^

AL-RAWiand GEORGE (1991-1992), p 52.

EnQma Anu Enlil Tablets 1-13

453

C O H E N , M a r k E. 1 9 9 3 . The Cultic Calendars of the Ancient Near East. B e t h e s d a : C D L Press. D E M E I S , Salvo a n d H U N G E R , Hermarm. 1998. Astronomical Dating of Assyrian and Babylonian Reports (Serie Orientale R o m a 81). R o m a : Istituto Italiano p e r l'Africa e l ' O r i e n t e . D E M E I S , Salvo a n d M E E U S , Jean. 1 9 9 1 . " A p r o p o s d'occultation d e planètes p a r la Lime d a n s des textes b a b y l o n i e n s " . L'Astronomie Juin 1 9 9 1 : 1-3 G A D D , Cyril J. 1966. " S o m e B a b y l o n i a n Divinatory M e t h o d s a n d their Interrelations". In F . W E N D E L et al. (eds.), La divination en Mésopotamie ancienne et dans les régions voisines : XlVe Rencontre Assyriologique Internationale: 2 1 - 3 4 . Paris: P r e s s e s Universitaires de F r a n c e . G E H L K E N , Erlend. 2 0 0 0 . " S o n n e n a u f g a n g in Sippar: Tafel 2 7 der Serie E n û m a A n u EnHl". In: Simonetta ORAZIANI (ed.). Studi sul Vicino Oriente Antico dedicati alla memoria di Luigi Cagni, I (Series M i n o r 61): 3 4 5 - 3 5 3 . N a p o l i : Istituto Universitario Orientale. G U R N E Y , Oliver R. a n d H U L I N , P. 1964. The Sultantepe Tablets IL L o n d o n : British Institute of A r c h a e o l o g y in A n k a r a . H O R O W I T Z , W a y n e . 1998. Mesopotamian Cosmic Geography. Winona Lake: Eisenbrauns. H U N G E R , H e r m a i m . 1992. Astrological Reports to Assyrian kings (State A r c h i v e s o f Assyria V I I I ) . Helsinki: University Press. —

and PINGREE, D a v i d . 1999. Astral Sciences Orientalistik 1). L e i d e n - B o s t o n - K o l n : Brill.

in Mesopotamia

( H a n d b u c h der

KUGLER, Franz X. 1 9 0 9 - 1 9 2 4 . Sternkunde und Sterndienst in Babel: assyriologische, astronomische und astralmythologische Untersuchungen, II. M u n s t e r : Aschendorff. — 1 9 1 1 . "Confribution a la m é t é o r o l o g i e b a b y l o n i e n n e " . Revue

d'Assyriologie

8:

107-130. — 1913. Sternkunde und Sterndienst in Babel: assyriologische, astronomische und astralmythologische Untersuchungen, Erganzungen I. Miinster: A s c h e n d o r f f L A B A T , R e n é . 1 9 6 5 , Un calendrier babylonien des travaux, des signes et des mois (Séries iqqur îpus). Paris: H o n o r é C h a m p i o n . L A N D S B E R G E R , B e n n o . 1 9 1 5 . Kultische Kalender der Babylonier und Assyrer (Leipziger semitistische Studien 6.1/2). Leipzig: J.C. Hinrichs'sche Buchhandlung. L A N G D O N , Stephen. 1915. " A F r a g m e n t of a Series of RituaUstic P r a y e r s to Asfral Deities in the C e r e m o n i e s of Divination". Revue d'Assyriologie 12: 1 8 9 - 1 9 2 . L A R G E M E N T , R e n é . 1957. "Confribution à l'Etude des A s t t e s errants d a n s l'Asfrologie c h a l d é e n n e ( 1 ) " . Zeitschrift fiir Assyriologie 52: 2 3 5 - 2 6 4 . P A R P O L A , S i m o . 1 9 8 3 . Letters from Assyrian scholars to the kings Esarhaddon and Assurbanipal, II (Alter Orient u n d A h e s T e s t a m e n t 5/2). N e u k i r c h e n - V l u y n : B u t z o n & B e r c k e r Kevelaer. A L - R A W I , F a r o u k N . H . and G E O R G E , A n d r e w R. 1 9 9 1 - 1 9 9 2 . " E n û m a A n u Enlil X I V a n d other E a r l y Asfronomical T a b l e s " . Archiv fiir Orientforschung 38-39: 52-72. REINER, Erica. 1974. " N e w Light o n S o m e Historical O m e n s " . In: K u r t BiTTEL, et al. (eds.), Anatolian Studies Presented to H.G. Giiterbock on the Occasion of his

454

L. Verderame

65th Birthday: Institut.

257-261.

Istanbul:

Nederlands

Historisch-Archaeologisch

— a n d PINGREE, D a v i d . 1 9 7 5 . Babylonian Planetary Omens, I. The Venus Tablet Ammisaduqa (Bibliotheca M e s o p o t a m i c a I I / l ) . M a l i b u : U n d e n a Publications.

of

— a n d PINGREE, D a v i d . 1 9 8 1 . Babylonian Planetary Omens, II. Enuma Anu Enlil, Tablets 50-51 (Bibliotheca M e s o p o t a m i c a II/2). M a l i b u : U n d e n a Publications. — a n d PINGREE, D a v i d . 1998. Babylonian Planetary Omens, III ( C u n e i f o r m M o n o g r a p h s 11). G r o n i n g e n : Styx. R O C H B E R G - H A L T O N , Francesca. 1985. Aspects of Babylonian Celestial Divination: The Lunar Eclipse Tablets of Enuma Anu Enlil (Archiv fiir Orientforschung Beiheft 2 2 ) . H o m : B e r g e r & Sôhne. S A C H S , A b r a h a m J. 1955. Late Babylonian Astronomical and Related Texts ( B r o w n University Studies 18). P r o v i d e n c e : B r o w n University Press. V O N S O D E N , W o l f r a m . 1955. " Z u m a k k a d i s c h e n W o r t e r b u c h . 6 1 - 6 6 " . Orientalia NS 24: 1 3 6 - 1 4 5 . V A N S O L D T , Wilfred H . 1995. Solar Omens of Enuma Anu Enlil: Tablets 23 (24) 29 (30). Leiden: N e d e r l a n d s H i s t o r i s c h - A r c h a e o l o g i s c h Instituut te Istanbul. S T O L , M a r t e n . 1992. " T h e M o o n as seen b y the B a b y l o n i a n s " . In: D i e d e r i k J.W. MEIJER, Natural Phenomena. Their Meaning, Depiction and Description in the Ancient Near East: 2 4 5 - 2 7 7 . A m s t e r d a m - O x f o r d - N e w Y o r k - T o k y o : R o y a l N e t h e r l a n d s A c a d e m y of A r t s and Sciences. T H U R E A U - D A N G I N , F r a n ç o i s . 1922. Tablettes d'Uruk (Textes C u n é i f o r m e s V I ) . Paris: P a u l Geuthner. TROLLE-LARSEN, Morgens. 1987. " T h e Mesopotamian Lukewarm Mind. Reflections o n Science, Divination and Literacy". In: F r a n c e s c a R O C H B E R G H A L T O N (ed.). Language Literature and History: Philological and Historical Studies Presented to Erica Reiner ( A m e r i c a n Oriental Series 6 7 ) : 2 0 3 - 2 2 5 . N e w H a v e n : A m e r i c a n Oriental Society. V E R D E R A M E , L o r e n z o . 1999. "I colori n e l l ' a s t r o l o g i a m e s o p o t a m i c a " . In: H a r t m u t W A E T Z O L D T (ed.). Von Sumer nach Ebla und zuriick. Festschrift Giovanni Pettinato zum 27 September 1999 gewidmet von Freunden, Kollegen und SchUlern ( H e i d e l b e r g Studien z u m A l t e n Orient 9 ) : i - x i . H e i d e l b e r g : H e i d e l b e r g e r Orientverlag (IN P R I N T ) . — 2 0 0 2 . Le Tavole I-VI della serie astrologica Enuma Anu Enlil ( N i s a b a 2). M e s s i n a : Università di M e s s i n a . V I R O L L E A U D , Charles. 1908. L'astrologie chaldéenne. Texte cunéiforme, Sin. Paris: Paul Geuthner. — 1 9 1 0 . l ' a s t r o l o g i e chaldéenne.

Supplément,

texte. Paris: Paul G e u t h n e r .

— 1912. L'astrologie chaldéenne. Second supplément, texte cunéiforme. Paris: Paul Geuthner. W E I D N E R , E m s t F . 1 9 1 1 . "Beitrâge zur b a b y l o n i s c h e n A s t r o n o m i e " . Beitràge zur babylonischen Astronomie 8/4: 1 - 1 0 1 . — 1 9 3 6 - 1 9 3 7 . "Anfrage an S a m a s u n d A d a d w e g e n einer M o n d f m s t e m i s " . fur Orientforschung —1941-1944. Orientforschung — 1954-1956. Orientforschung

Archiv

11: 358-369.

"Die

Astrologische

Serie

Enûma

Anu

Enlil".

Archiv

fiir

Enûma

Anu

Enlil".

Archiv

fur

14: 1 7 2 - 1 9 5 and 3 0 8 - 3 1 8 . "Die

Astrologische

17: 7 1 - 8 9 .

Serie

EnQma Anu Enlil Tablets 1-13

—1968-1969. Orientforschung

" D i e Astrologische

Serie

Enûma

455

A n u Enlil".

Archiv

fìir

22: 6 5 - 7 5 .

W I S E M A N , D o n a l d J. a n d B L A C K , J e r e m y A . 1 9 9 6 . Literary Texts from the Temple of Nabu. ( C u n e i f o r m T e x t s from N i m r u d 4 ) . L o n d o n : British S c h o o l o f A r c h a e o l o g y i n Iraq.

456

L. Verderame

Appendix Incipit, a c c o r d i n g to the U r u k Catalogue, and p h e n o m e n a o r d e r e d b y Tablet. I U D A N E N . L Î L . L [ À ] " W h e n A n , Enlil", U m k Cat.: 1. P h e n o m e n a related to the disappearing period o f the m o o n and the n e w m o o n , i.e. from day 2 7 of the p r e c e d i n g m o n t h and d a y 2 of the next m o n t h (unusual or irregular a p p e a r a n c e s , etc.). A s p e c t of the m o o n ' s light (sëtu/UDDA). Similitude ( D l S 3 0 G I M "if the m o o n is like ...") of the m o o n , i.e. c o l o u r a n d light, to object. II 3 0 ina IGI.LÀ-5M e-kil "the M o o n is dark at its a p p e a r a n c e " , U r u k Cat.: 2 . D a r k e n i n g (ekëlu) o f the m o o n . Fragmentary... Ill 3 0 ina I G I . L À - ì m A G A a-pir " T h e M o o n at its a p p e a r a n c e w e a r s a c r o w n " , U m k Cat.: 3 . L u n a r crown/tiara (agû/AGA). H e a v e n l y b o d y next the m o o n . IV 3 0 ina IGI.LÀ-5M a-dir " T h e M o o n is dark at its a p p e a r a n c e " , U m k Cat.: 4 . Variation o f m o o n ' s p o s i t i o n a n d light/appearance. Fragmentary...

3 0 ina IGI.LA-5M a-dir-ma SI 15-5M ki-pat SI 2,30-5w ed-de-e[d] "If at the a p p e a r a n c e o f the M o o n its right h o m is blunt and its left h o m p o i n t e d " , U m k Cat.: 5.

M o o n ' s h o m s (see notes 3 1 - 3 7 ) . VI 3 0 ina IGI.LÀ-i^M M U L . M U L ina k-sû G U B - / [ z ] "If at the a p p e a r a n c e o f the M o o n the Pleiades stand at its side", U m k Cat.: 6 . H e a v e n l y b o d y next the m o o n (see note 4 0 ) . VII 3 0 ina I T I . B À R U D . l . K A M ina IGI.LÀ-iw SI 1 5 - 5 « A N - e te-rat "In N i s a n on the first d a y at the a p p e a r a n c e of the M o o n its right h o m p i e r c e s the sky", U m k Cat.: 7 . M o o n and h o m s aspect (form a n d colour), m l e d b y m o n t h (I-VI).

ENQMA ANU ENLIL TABLETS 1 - 1 3

457

VIII 3 0 ina ITI.DUé U D . l . K À M ina IGI.LÀ-5M T Ù R N I G I N "In Tishri o n the first d a y the M o o n at its a p p e a r a n c e is s u r r o u n d e d b y a h a l o " , U r u k Cat.: 8. Different p h e n o m e n a ruled b y m o n t h ( V I I - X I I / X I I " ) . P r e s e n c e o f a h e a v e n l y b o d y in the lunar halo.

IX 3 0 ina IGI.LÀ-5W T Ù R N I G I N - w a ka-bar u su-par-ru-ur " T h e M o o n at its a p p e a r a n c e is s u r r o u n d e d b y a halo a n d is large and e x t e n d e d " , U r u k Cat.: 9. L u n a r halos. Fragmentary...

X ina I T I . B À R 3 0 T Ù R N I G I N - m a K À N U T U K U - i / " I n N i s a n the M o o n s u r r o u n d e d b y a h a l o that has n o ' g a t e ' " , U r u k Cat.: 10. P r e s e n c e of an o p e n i n g ("gate", K Â ) in the halo (see note 2 2 ) . H a l o s of different colours.

is

P r e s e n c e of a h e a v e n l y b o d y or a cloud in the h a l o .

XI ana § À 30 T U enters the M o o n " , U r u k Cat.: 1 1 . P r e s e n c e o f a " d e s i g n " ( G I S . H U R ) or h e a v e n l y b o d i e s a r o u n d / n e x t the m o o n ? Fragmentary...

XII M U L ina S À SI 15 3 0 G U B " A star stand inside the right h o m of the M o o n " , U m k Cat.: 12. P r e s e n c e o f a h e a v e n l y b o d y next the m o o n .

XIII 3 0 X x[... " T h e M o o n ...", U m k Cat.: 12. 9

The Role of Astronomical Techniques in Ancient Egyptian Chronology: The Use of Lunar Month Lengths in Absolute Dating Ronald A. Wells,

Berkeley

Introduction Egyptological literatine has p r o m u l g a t e d the c o n c e p t that exceptionally precise absolute dates c a n b e c o m p u t e d for A n c i e n t E g y p t using astronomical t e c h n i q u e s . H o w e v e r , these publications often exhibit little u n d e r s t a n d i n g of the m e a n i n g of the fundamental a s t r o n o m i c a l equations involved; faulty statistics; n o a t t e m p t s to duplicate the o b s e r v i n g conditions that might h a v e existed; a n d n o p a r a m e t r i c error analysis w h a t s o e v e r o f either the equations or the o b s e r v i n g conditions a n d w h a t these errors m i g h t m e a n for the final result. T h i s report describes s o m e of the basic pitfalls that can b e e n c o u n t e r e d . T h e description includes the m e a n i n g a n d significance of the original observation, a n d treats the various physical aspects required to duplicate that o b s e r v a t i o n as closely as possible. T h i s p r e s e n t a t i o n deals primarily w i t h E g y p t i a n p a p y r i h a v i n g b o t h lunar a n d civil dates r e c o r d e d on them while a related article d e m o n s t r a t e s the necessity of carrying t h r o u g h a detailed error analysis to assess the reliability o f specific results.

Lunar Month Length Sequences A persistent attempt to determine absolute c h r o n o l o g y in A n c i e n t E g y p t u s e s various d o u b l e - d a t e d d o c u m e n t s , that is, r e c o r d s containing b o t h a n E g y p t i a n date in the civil calendar and, simultaneously, one in the religious limar calendar. Since all Egyptian civil m o n t h s contain 3 0 days each, a d o u b l e - d a t e d r e c o r d p e r m i t s the determination o f the civil date for d a y 1 of the given lunar m o n t h , called psdntyw. L u n a r m o n t h s t h e m s e l v e s m a y have either 2 9 or 3 0 d a y s (because the lunar synodic p e r i o d is 2 9 . 5 3 0 5 8 9 . . . days, i.e., the interval b e t w e e n r e p e a t e d p h a s e s , ca. 2 9 . 5 o n a v e r a g e ; h e n c e , the alternating n u m b e r o f days p e r lunar c a l e n d a r m o n t h , t w o o f w h i c h give the s a m e a v e r a g e ) . G i v e n a series o f similarly dated d o c u m e n t s , it is thus possible to d e r i v e a s e q u e n c e of lunar calendar m o n t h lengths consisting o f repeating 2 9 or 3 0 days. B e c a u s e the p e r i o d is slightly greater than the a v e r a g e , such a s e q u e n c e d o e s not a l t e m a t e exactly, but m a y h a v e t w o or m o r e 3 0 - d a y m o n t h s in a row, or s o m e t i m e s t w o or m o r e 2 9 - d a y m o n t h s in a r o w . Egyptologists h a v e then a t t e m p t e d to p l a c e such a s e q u e n c e within a given absolute time frame b y c o m p u t i n g the o c c u r r e n c e of the begiiming of lunar m o n t h s to find a m a t c h i n g s e q u e n c e . B y

WELLS ( 2 0 0 3 ) .

460

R.A. Wells

c h a n c e , a g r o u p o f M i d d l e K i n g d o m p a p y r i from l l l a h u n form s u c h a series. O n e of t h e m contains the frequently q u o t e d 12th D y n a s t y Sothic rising in the 7th regnal year of Sesostris III,^ a n d the r e m a i n d e r (including several m o n t h entries o n o n e p a p y m s ) allow the d e d u c t i o n of a lunar s e q u e n c e , the correct n u m b e r i n g o f w h i c h has b e e n discussed at length b y Borchardt,^ Parker,"* Krauss,^ Luft,^ a n d m o r e recently R o s e . ' T h e s e d o c u m e n t s illustrate the p r o b l e m s c o n n e c t e d with absolute dating, b u t the r e m a r k s g i v e n b e l o w apply to a n y g i v e n s e q u e n c e of lunar m o n t h lengths. T h e typical a p p r o a c h is to determine the s e q u e n c e o f 2 9 - 3 0 - 3 0 - 3 0 - 2 9 - 2 9 - e t c . days for all the d o c u m e n t s in the series as the o b s e r v a t i o n a l d a t a b a s e . L u n a r m o n t h lengths are then c o m p u t e d for a similar series o f c o n s e c u t i v e m o n t h s from a s t r o n o m i c a l tables such as those b y P . V . Neugebauer,^ C. Schoch,^ or m o r e recently H . H. G o l d s t i n e , ' " until a m a t c h is found for the o b s e r v e d s e q u e n c e (see also C h a p r o n t - T o u z é a n d Chapront,*' and E s p e n a k ' ^ ) . Since the Julian year is p a r t of the c o m p u t a t i o n , the m a t c h s u p p o s e d l y then gives a " p r e c i s e " asfronomical date for

^

EDGERTON(1942).

^

BORCHARDT (1935).

"

PARKER (1950), Excursus C , 63-69.

^

KRAUSS (1984).

^

LuFT(1992).

'

ROSE (1999). See WELLS (2002), for a review of this book

* NEUGEBAUER (1912, 1914, 1925, 1929, 1937). These tables are out of date and difficult to use. They do not take into account the tidal deceleration of the Earth (length-of-day error) (HUBER (2000a)). In the derivation of crescent visibility parameters with topocentric coordinates, the algorithm on which the tabular computations are based does not adequately take into account the larger variations of refraction and extinction near the horizon as discussed in text. Recent trends using commercial or other planetarium programs suffer the problem that the particular author never cites error limits or other qualifications built into the program such as refraction and extinction effects near the horizon on the visibility of rising objects which can have drastic effects on the obtained results (see footnotes 15 and 19). ' SCHOCH (1928), p. 94. A dated discussion by Fotheringham on lunar crescent visibility is also included in this work. See also SCHOCH (1927). These both do not take into account the same effects mentioned in the previous foomote. '° GOLDSTINE (1973). The length-of-day error (HUBER (2000a)) in the geocentric computations is also not taken into account in these tables. There is also no algorithm for estimating first crescent visibility. Attempts to use a rule-of-thumb predictor, e.g., age of the Moon, are notably unreliable. " CHAPRONT-TOUZÉ and CHAPRONT (1991). A good source for computing lunar phases is given by this outstanding work for professional and amateur astronomers and programmers. The provided computer programs, however, require modifications to yield the desired phases. Also, the secular acceleration parameter used in these tables is much too low and needs updating (see HUBER (2000b)). There is, of course, no estimator with which to predict first or last lunar crescents. Fred Espenak, http://sunearth.gsfc.nasa.gov/eclipse/phase/phasecat.html provides the simplest geocentric catalogue of lunar phases from 2000 B C to 3000 AD with the length-ofday (AT) error included. The tables do not, of course, provide the means of predicting the first or last visible crescents (see text for further discussion). This website is a subset of the more extensive NASA Goddard site: http://sunearth.gsfc.nasa.gov/ eclipse/eclipse.html.

The Role of Astronomical Techniques in Ancient Egyptian Chronology

461

the d o c u m e n t s . All investigators w o r k i n g with these p a p y r i h a v e u s e d this a p p r o a c h . B u t they h a v e all arrived at different results. T h e principal r e a s o n for these differences is that n o n e of the investigators agree o n the order of the 2 9 - 3 0 - 2 9 - 2 9 30-30-etc. lengths in the sequence d e d u c e d from the E g y p t i a n r e c o r d s b e c a u s e o f a lacuna o b s c u r i n g part o f a date, a n e m e n d e d date d u e to p o o r scribal transcription, etc. Pinpointing the precise differences a m o n g investigators is not the p u r p o s e o f this discussion. All b u t one, h o w e v e r , at least a g r e e d that the p a p y r i date to s o m e t i m e in the early 1 2 * Dynasty, that is to say, s o m e t i m e in the early 17*** century BC. R o s e , o n the other hand, argues that the s e q u e n c e better fits a n era a r o i m d 3 9 5 BC.'^

O b s e r v a t i o n a l D a t a Set Statistics T h i s p r o c e d u r e ignores a basic mathematical point, n a m e l y , that the m a t c h i n g of a n y set of E g y p t i a n m o n t h lengths in any century is of l o w statistical significance b e c a u s e there will always b e multiple m a t c h e s t h r o u g h o u t the r a n g e . S o e v e n a s s u m i n g perfect ancient r e c o r d i n g o f lunar dates, m o n t h s e q u e n c e s from multiple d o c u m e n t s written in the s a m e era like those firom Illahun caimot b e assigned the appropriate fixed yearly interval solely o n the basis of c o m p u t e d s e q u e n c e s . M o r e o v e r , a statistically valid determination of an absolute date b y this m e t h o d , a s s u m i n g perfectly o b s e r v e d , error-free data, w o u l d require b e t w e e n 2 5 0 a n d 3 0 0 precisely s p a c e d m o n t h lengths a c c o r d i n g to Huber''* w h o has e x a m i n e d this p r o c e d u r e for B a b y l o n i a n cimeiform texts! Since a g i v e n s e q u e n c e c a n h a v e multiple m a t c h e s in a n y g i v e n century, w h a t w o u l d b e the effect o f a n error for a n y o n e o f those m o n t h lengths, that is if the o r d e r of the 2 9 a n d 3 0 d a y s p e r lunar m o n t h is c h a n g e d ? Investigators usually m e n t i o n the possibility that a c o m p u t a t i o n o f a lunar date fi-om a table m a y b e in error b y ± 1 day, t h o u g h this admitted possibility is n e v e r evaluated in terms o f the multiple m a t c h e s j u s t m e n t i o n e d . In fact, it d o u b l e s the c o m p u t a t i o n a l effort to take o n e such error into a c c o u n t for any g i v e n s e q u e n c e , w h i c h caimot h a v e less t h a n 2 9 n o r m o r e than 3 0 d a y s . B e c a u s e one d o e s not k n o w a priori w h i c h m o n t h length in the s e q u e n c e m i g h t b e in error, to take into a c c o u n t all possibilities, the error m u s t b e p r o p a g a t e d t h r o u g h the entire s e q u e n c e , requiring {n + 2) c o m p u t a t i o n s instead, w h e r e n is the n u m b e r o f m o n t h lengths in the s e q u e n c e , a n d the constant 2 represents the original s e q u e n c e plus the case for a n error at e a c h interval. S o m e t i m e s the investigator m a y also m e n t i o n that c l o u d y skies m a y h a v e c a u s e d a priestly error in m a k i n g the observation of lunar crescent visibility r e q u k e d to d e t e r m i n e the start of a lunar month. B u t this possible error has also n o t b e e n t a k e n into a c c o u n t c o m p u t a t i o n a l l y n o r evaluated in terms o f its likelihood. T h e observational situation is actually m u c h w o r s e than has generally b e e n a s s u m e d in the p a s t as will b e c o m e evident b e l o w . A series of publications pertain to the p r o b l e m o f n e a r h o r i z o n celestial observations, b o t h theoretically a n d observationally. T h e principal p a p e r s are: Schaefer a n d Liller;'^ Schaefer;'* D o g g e t t a n d Schaefer;'^ a n d Schaefer.'^"'^ ROSE (1999).

HUBER (2000b). The author thanks Dr. Huber for discussions conceming the lunar month length statistics freated in this paper. SCHAEFER and LILLER (1990): A compilation of refraction observations very close to and

462

R.A. Wells

Until this past d e c a d e , there were n o statistical data o n errors c a u s e d b y g r o u p s of o b s e r v e r s in trying to detect n e w crescents observationally. S o the ancient scribal r e c o r d i n g s of various lunar dates h a v e b e e n blindly a c c e p t e d as is, e x c e p t for o b v i o u s lacunae. B u t D o g g e t t and Schaefer^° w e r e able to test o b s e r v a t i o n a l capabilities b y a c c u m u l a t i n g over 1500 i n d e p e n d e n t observations m a d e b y m o r e than 2 5 0 0 a m a t e u r astronomers in 5 special M o o n w a t c h o b s e r v i n g p r o g r a m s w h i c h they designed. T h e y s h o w e d that in a n y g r o u p of 100 observers, 15 o f t h e m will honestly report false positive observations of the n e w lunar crescent. B u t e v e n m o r e interesting results were found w h e n s o m e o f the data w e r e g r o u p e d a c c o r d i n g to eyesight, e x p e r i e n c e , a n d age groupings. T a b l e 1 indicates that in the g r o u p s b e t w e e n n o r m a l a n d g o o d e x p e r i e n c e d o b s e r v e r s , the p e r c e n t a g e o f sightings increased from 6 8 % to 8 2 % , respectively; a n d b e t w e e n n o r m a l a n d g o o d eyesight, 7 1 % to 8 2 % , respectively. T h e age g r o u p 3 0 - 3 9 years w e r e only able to detect the crescent 7 7 % o f the time, while the 4 0 - 4 9 g r o u p sighted it 6 9 % . B o t h a g e g r o u p s together a v e r a g e d 7 3 % ( 1 3 5 / 1 8 4 ) sightings. If w e take g o o d eyesight a n d e x p e r i e n c e a n d b o t h these a g e g r o u p s together as typical o f E g y p t i a n 'Imy-r wnwt priests, these results r e m a r k a b l y suggest that the latter w o u l d h a v e m i s s e d the first crescent sighting s o m e w h e r e b e t w e e n 1 8 % and 2 7 % of the time!

below the horizon which indicate that the range is far greater than had been previously suspected. The amount of refraction determined by these authors is considerably greater than theoretical equations, which also allow for temperature and pressure effects, themselves predict at or below the horizon. Modem planetarium programs generally allow only for standard atmospheric temperature and pressure (STP = 0°C, 760 mm Hg). SCHAEFER (1993): A detailed theorefical treatment describing the equations and factors involved in observing near-horizon celestial events. DOGGETT and SCHAEFER (1994): The data obtained from 5 separate Moonwatch observing programs are listed, processed, and discussed. This article represents the only major statistically valid study describing the capabilities of groups of observers in detecting first crescents. SCHAEFER (1996): An update of the Moonwatch analysis presented in Icants containing more data and more discussions of the accuracy and reliability of computing first crescent visibilities using 12 ancient, medieval, and modem algorithms compared with the observations. SCHAEFER (2000): A discussion of computing Sothic risings with his improved algorithm as well as the efficacy of using lunar month length sequences for ancient Egyptian chronological purposes. He also provides an estimate of extinction valid for ancient Egypt determined by 3 different methods. The value, 0.35 ± 0.09 (1 a) magnitude per airmass at the horizon in summer, is significantly larger than estimates for non-desert climates, key factors in the determination being the dust content of the atmosphere near the horizon and humidity close to the Nile. This value together with a horizontal airmass factor of 40 implies an extinction of 14 magnitudes. Since Sirius, the brightest star in the sky, has a magnitude of -1.46, this extinction indicates, for example, that it was impossible to see the star rising within the first degree above the horizon. This large near horizon extinction also makes the rising last crescent of the Moon difficult to observe even for a cloudless sky. ^°

DOGGETT and SCHAEFER ( 1994).

The Role of Astronomical Techniques in Ancient Egyptian Chronology

Total N o . O b s e r v e r s Experience Normal Good EyesigJit Normal Good Age 04-09 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89

T o t a l Sightings

463

Percentage

299 90

204 74

68 82

373 22

264 18

71 82

8 16 31 100 84 51 30 14 1

5 8 23 77 58 37 17 7 0

63 50 74 77 69 73 57 50 0

T a b l e 1. M o o n w a t c h 2 Sightings Distributed b y Visual Acuity, E x p e r i e n c e , a n d A g e G r o u p s (derived from T a b l e V I data o f D O G G E T T and

SCHAEFFER ( 1 9 9 4 ) ) .

T h e ancient E g y p t i a n lunar calendar is, of course, not like the Islamic or B a b y l o n i a n o n e w h i c h d e p e n d s o n n e w (first) crescent visibility to start the limar m o n t h . In fact, the observational situation for the E g y p t i a n lunar calendar m o n t h w h i c h starts o n the d a y of d i s a p p e a r a n c e o f the old (last) lunar crescent {psdntyw) is, if anything, m u c h w o r s e b e c a u s e three effects are involved: last and first crescent visibilities a n d the interval b e t w e e n them. D e t e r m i n a t i o n of last crescent visibility is c o n s i d e r a b l y m o r e difficult than first crescent visibility b e c a u s e the former involves detection o f a rising object w h i l e for the latter, the M o o n is generally 1 0 ° o r m o r e a b o v e the h o r i z o n w h e n the sky b e c o m e s d a r k e n o u g h after sunset for the crescent to b e apparent. B e c a u s e o f the large extinction factor on the order o f 1 4 m a g n i t u d e s at the horizon,^' it is easier to m i s s the thin rising crescent w h i c h will s o o n d i s a p p e a r b e c a u s e o f i m p e n d i n g sunrise. T h e interval o f n o visible M o o n b e t w e e n last/first crescent a p p e a r a n c e s is variable in length, d e p e n d i n g o n s e a s o n a n d the a p p a r e n t tilt o f the lunar orbital plane.^"^ Should the w a n i n g last crescent rise b e m i s s e d b e c a u s e o f weather or b y accident, then determination of the d a y o n w h i c h psdntyw had o c c u r r e d prior to the a p p e a r a n c e of the n e w crescent will b e uncertain. W i t h regards to lunar dated d o c u m e n t s , the M o o n w a t c h statistics o f first crescent visibility j u s t q u o t e d i m p l y that for a n y g i v e n year o f 1 2 o b s e r v e d m o n t h s 2 to 3 o f t h e m were p r o b a b l y incorrectly recorded. T h i s conclusion a s s u m e s , o f c o u r s e , that the M o o n w a t c h samplings reliably represent the E g y p t i a n Umy-r wnwt p r i e s t s ' observational capabilities. B u t b e c a u s e the observational condition for d e t e r m i n i n g the start of the Egyptian calendar is certainly m o r e p r o n e to error than the

See footnote 19, especially last sentence. PARKER (1950), §§12-17,45-46, pp. 5-6, 13-14, long ago recognized this variability.

464

R.A. Wells

B a b y l o n i a n calendar as j u s t explained, there m a y likely b e e v e n m o r e than 2 - 3 incorrectly o b s e r v e d dates p e r lunar year. D O G G E T T a n d S C H A E F E R ( 1 9 9 4 ) (their T a b l e V ) also found that 6 8 % ( 1 1 0 / 1 6 2 ) of observers were able to detect the thin crescent M o o n w i t h t e l e s c o p e s or b i n o c u l a r s , but that this n u m b e r d r o p p e d to only 4 3 % ( 2 1 1 / 4 9 4 ) w i t h the u n a i d e d eye — and these observations apply to crescents of ca. 1 0 ° elevation. T h i s additional finding c o m b i n e d with the p r e v i o u s statistics d o e s not bolster confidence that the ancient Egyptian a s t r o n o m e r s were a n y better in performing their o b s e r v a t i o n a l duties. It is therefore quite likely, o n the basis o f this evidence, that a significant fraction of all ancient E g y p t i a n papyri carrying lunar dates contains errors o f o n e or m o r e days simply b e c a u s e the start of the lunar m o n t h was incorrectly o b s e r v e d . W h a t is the effect of these errors o n the l l l a h u n s e q u e n c e ? T h e net effect w o u l d b e to scramble the s e q u e n c e s of 2 9 - 3 0 day intervals in any g i v e n set o f E g y p t i a n d o c u m e n t s , c o m p l e t e l y obliterating a n y internal consistencies that m i g h t otherwise b e u s e d to validate use of the d o c u m e n t s for c h r o n o l o g i c a l p u r p o s e s . T h i s is a realistic a s s e s s m e n t of j u s t the so-called " o b s e r v e d " data set with w h i c h o n e has to work. W h a t about c o m p u t i n g a date of first or last crescent visibility?

G e o c e n t r i c and T o p o c e n t r i c C o o r d i n a t e s : Old T a b l e s , C o m p u t e r P r o g r a m s , R u l e s - o f - T h u m b , a n d t h e T h r e s h o l d of Visibility T h e c o m p u t a t i o n o f the o c c u r r e n c e o f the first d a y of a lunar m o n t h in antiquity is not the simple p r o c e s s that s o m e investigators h a v e a s s u m e d in the past a n d in m o r e recent publications. T h e r e are t w o distinctive parts to the p r o c e d u r e . O n e is theoretically p r e c i s e , a n d practically accurate, although o n e aspect o f it n e e d s to b e clarified here. T h e other is highly subjective; a n d until the M o o n w a t c h p r o g r a m described in the p r e v i o u s section, essentially n o data existed to indicate its statistical limitations. T h e first part p r o v i d e s only the geocentric position, i.e., with r e s p e c t to the E a r t h ' s center, of the M o o n in its orbit which, in relation to b o t h b o d i e s with respect to the Sun, enables the time to b e d e t e r m i n e d w h e n any desired lunar p h a s e occurs or has o c c u r r e d in the past. F o r e x a m p l e , the timing of n e w or fiill M o o n s , first or last quarters, or any p e r c e n t a g e crescent for that matter, with r e s p e c t to the G r e e n w i c h m e r i d i a n or any other reference meridian. W h a t this first p a r t d o e s not indicate is w h e n the first or last visible lunar crescent occurs. T h e latter is a determination m a d e with respect to the local horizon, i.e., a t o p o c e n t r i c n o t a geocentric relationship. T h e c o n v e r s i o n of the g e o c e n t i i c p o s i t i o n into a topocentric one then is the c m x of the p r o b l e m for the d e t e r m i n a t i o n o f first or last crescent visibilities, and is the part m o s t p r o n e to error. In the past, data from the theoretical geocentric equations h a v e b e e n p r e s e n t e d in tabular form from w h i c h the user generally d e t e r m i n e s the o c c u r r e n c e o f n e w m o o n s for given dates b y looking u p the values in c o l u m n s , or m a k i n g s o m e simple adjustments to c h a n g e the longitude from the catalog reference m e r i d i a n to Egypt. T h e s e results c a n b e categorized as generally accurate, apart from o n e distinction noted b e l o w , n o matter w h i c h of the previously cited^^ tables are used. S o m e i m p r o v e m e n t s over time w e r e m a d e to t h e m b y later investigators. T h o s e p u b l i s h e d b y P. V . Neugebauer,^'* however, p r o v e d difficuh to use b e c a u s e errors in the "

See footnotes 8-12. NEUGEBAUER (1912, 1914, 1925, 1929, 1937).

The Role of Astronomical Techniques in Ancient Egyptian Chronology

465

Tidal Deceleration Effects 16 SlBphensonSHoulden(1986)

14 -

SlBphen9on(1997)

|io^

8

>» nj Q O O) c (1)

20 -

-3000

-2000

-1000

0

1000

2000

Y e a r (astronomical)

Figure 1. T h e ciunulative length of d a y error caused b y tidal d e c e l e r a t i o n o f the E a r t h ' s rotation. A l t h o u g h observational data d o not exist for the p h a r a o n i c p e r i o d o f Egypt, the observations from ancient r e c o r d s in B a b y l o n a n d C h i n a c a n b e mathematically exfrapolated t h r o u g h that p e r i o d (with s o m e error p r o g r e s s i v e l y increasing t o w a r d s earlier dates^^). T h e solid line represents the c o m p u t a t i o n s p u b l i s h e d b y S T E P H E N S O N and H O U L D E N ( 1 9 8 6 ) . T h e d a s h e d line b e g i n n i n g at - 7 0 0 represents a revision b y S T E P H E N S O N ( 1 9 9 7 ) for later times. For 2 0 0 0 B C , AT = E T - U T = ca. 12 to 15 h o u r s (including error r a n g e ) , w h e r e E T is uniform E p h e m e r i s T i m e , a n d U T is U n i v e r s a l T i m e b a s e d o n the E a r t h ' s rotation (roughly G M T ) . original tables w e r e corrected in later publications b y modifications to the laborious data extraction t e c h n i q u e s . T h e m o s t critical aspect of lunar orbital theory not u s e d effectively in s o m e of these tables is the lack of accounting for the tidal deceleration of the E a r t h ' s rotation w h i c h contributes to the receding of the M o o n from the E a r t h a n d its s p e e d i n g u p in orbit. T h i s secular acceleration of the M o o n w a s k n o w n prior to L a p l a c e , w h o g a v e an explanation o f its origin as due to the secular reduction o f the orbital eccentricity o f the Earth. T h i s p h e n o m e n o n , h o w e v e r , only a c c o u n t e d for part of the o b s e r v e d acceleration. It w a s n o t until m u c h later that it was realized that tidal deceleration of the E a r t h ' s rotation a c c o u n t e d for that remaining. T h e latter t e r m is usually called the length-of-day error.^* Figure 1 shows the variation of this error from 2 0 0 0 B C to 1600 A D . A n y table p u b l i s h e d prior to the late 70s will not h a v e included a large

For a description of the error, see HUBER (2000b), Figure 8.1. The error is cumulative. It does not mean that the day length during the pharaonic era was significantly longer than it is today, but that the very small differences added over 4000 years, for example, can be very significant.

466

R.A. WELLS

e n o u g h secular acceleration factor t o account for a significant length-of-day error. T h e latter c a n a m o u n t t o about a half d a y a r o u n d 2 0 0 0 B C . ^ ' T h i s factor alone w o u l d p r o d u c e a n error o f ± 1 d a y in the determination o f a lunar m o n t h length for that era. Consequently, all p r e v i o u s attempts t o establish a n absolute c h r o n o l o g y in E g y p t from lunar m o n t h lengths are incorrect solely o n this basis. T h e C h a p r o n t tables, w h i c h a r e really c o m p u t e r p r o g r a m s for t h e e x p e r i e n c e d user t o set u p , a r e t h e best o f these; b u t a c c o r d i n g t o H u b e r , e v e n they h a v e u s e d a secular acceleration t e r m w h i c h is m u c h t o o small.^^ A s a result, t h e s u g g e s t i o n o f F i m e i s ^ ' t o u s e t h e C h a p r o n t tables t o calculate "relevant lunar d a t e s " will b e in error unless this correction is included. T h e tables p u b l i s h e d o n t h e w e b b y Espenak^° are the simplest t o use and d o include the length-of-day error. O t h e r m o d e m c o m p u t e r p l a n e t a r i u m p r o g r a m s a r e c o m m e r c i a l l y available. Unfortunately, these rarely specify the nature o f the equations u s e d in t h e sky plots a n d give n o information about corrections for t h e length-of-day error o r for extinction a n d refraction (see next section). T h e p r o g r a m s were generally n o t written for the historian to c o m p u t e dates in antiquity, b u t rather for a s t i o n o m e r s a n d a m a t e u r a s t i o n o m e r s to p l a n observing p r o g r a m s o r to contiol h o m e t e l e s c o p e s , i.e. for t h e m o d e m e r a . W h i l e it is possible to c o m p u t e s k y a p p e a r a n c e s for dates in antiquity with t h e m , without t h e extia information there is n o w a y t o assess t h e statistical reliability o f the results for a given date. T h e s e c o n d p a r t o f the p r o c e d u r e for c o m p u t i n g a lunar m o n t h length in antiquity requires t h e t o p o c e n t i i c (i.e., altitude a n d azimuth coordinates) d e t e r m i n a t i o n o f the a p p e a r a n c e o f the last a n d first visible lunar crescents, respectively, events that o c c u r on either side o f the occurrence o f n e w M o o n , o r conjunction. T h e r e is n o exact m e t h o d o f c o m p u t i n g these events; a n d a s the p r e v i o u s section s h o w e d , o b s e r v a t i o n s are highly subjective in nature. Rules-of-thumb h a v e d e v e l o p e d i n t h e past, attempting to p r e d i c t w h e n they occur. F o r e x a m p l e , s o m e ancient o b s e r v e r s b e l i e v e d t h e first crescent w o u l d always b e visible if t h e M o o n set at least 4 8 minutes after sunset ( M o o n s e t lag). Others h a v e u s e d t h e a g e o f t h e M o o n a s a predictor, e.g., if it is 15.4 h o u r s o l d t h e crescent should b e visible. T h i s value w a s for m a n y years t h e earliest detection o f a first crescent; b u t w a s s u p e r s e d e d b y several observations in t h e M o o n w a t c h p r o g r a m . T h e earliest d e t e c t i o n o f t h e crescent is n o t a general predictor, however, a n d is essentially useless for c o m p u t i n g lunar m o n t h lengths^' b e c a u s e it m a k e s n o allowance for location, season, a n d observer. A m l e - o f - t h u m b predicting last crescent visibility is also unreliable b e c a u s e o f the variable interval b e t w e e n last a n d first crescents. T h e m o s t accurate m e t h o d used i n the p a s t d e p e n d s o n t h e arcus visionis, or altitude a n d a z i m u t h relations b e t w e e n the M o o n a n d Sun after sunset. T h e s e will b e characterized in m o r e detail in t h e following section. Historically, they h a v e b e e n b a s e d o n t h e crescent observations o f Julius S c h m i d t at t h e A t h e n s O b s e r v a t o r y in the m i d - 1 8 0 0 s to w h i c h several investigators later a t t e m p t e d t o fit a c u r v e that defined the threshold for visibility. T h e tables o f N e u g e b a u e r , for e x a m p l e , included

HUBER (2000a) AND STEPHENSON (1997). SEE ALSO STEPHENSON AND HOULDEN (1986). CHAPRONT-TOUZÉ AND CHAPRONT (1991); HUBER (2000b).

FIRNEIS (2000). SEE FOOTNOTE 12. FIRNEIS (2000) INDICATES THAT 15.5 OR 17.5 HOURS GUARANTEESFIRSTCRESCENT VISIBILITY!

The Role of Astronomical Techniques in Ancient Egyptian Chronology

467

an arcus visionis determination for crescent visibility.^^ A s will b e c o m e clear, his algorithms d o not always p r o d u c e reliable results.

Effects of E x t i n c t i o n / R e f r a c t i o n o n C o m p u t a t i o n s of C r e s c e n t Visibility N e i t h e r the table algorithms n o r the p l a n e t a r i u m p r o g r a m s , h o w e v e r , realistically accoimt for the t w o principal effects influencing near h o r i z o n observations, n a m e l y the large nearly horizontal airmass extinction, a n d the refraction of the light ray d u e to this same airmass. H o r i z o n t a l airmass extinction is p r o d u c e d b y v a r i o u s factors such as aerosol (particulate) scattering, o z o n e content, relative humidity a n d the like w h i c h h a v e very significant effects o n visibility b e c a u s e o f the large p a t h lengths near the h o r i z o n t h r o u g h w h i c h light m u s t p a s s . T h e w o r k o f Schaefer,^^ h o w e v e r , has s h o w n that extinction a n d refraction effects h a v e b e e n seriously u n d e r e s t i m a t e d in the past. F i g u r e 2 shows that the o b s e r v e d range in refraction, for e x a m p l e , is m u c h greater than the standard theoretical equations a l l o w near a n d b e l o w the horizon.

Observed vs. Theoretical Refraction Effects

Theoretical; 760 mm-Hg, 10 C Hawaii Observations La Serena, Ctiile Cerro Tololo, Chile Vina del Mar

80 -

60 -

•S. «

40 -\

•o <

20 -

-20 -10

0

10

20

30

40

50

60

70

80

90

100

110

Refraction (arc min)

Figure 2. Variations in refiraction near and b e l o w the h o r i z o n o b s e r v e d b y S C H A E F E R a n d LILLER ( 1 9 9 0 ) from four different regions in the n o r t h e m a n d s o u t h e m h e m i s p h e r e s as indicated b y the plotted symbols. N o t e that the observations r a n g e from sea level to m o u n t a i n t o p observatories. T h e theoretical c u r v e is taken from C. W . Allen^" for standard p r e s s u r e a n d an a t m o s p h e r i c t e m p e r a t u r e o f 10 °C. T h e m a x i m u m p r e d i c t e d value is ca. 35 arc m i n ( d e p e n d i n g on pressure, temperature). A s c a n b e seen, the o b s e r v e d range c a n b e almost 3 times that amount.

" "

Footnote 8. Footnotes 15, 16, and 19. ALLEN (1973).

R.A. Wells

468

Alt/Azimuth Fits vs. Moonwatch Data

Fotheringham (1910)

20

10

0

-10

-20

Lunar Azimuth (degrees)

Figure 3 . A c o m p a r i s o n o f four separate altitude/azimuth {arcus visionis) algorithmic fits to the same 1 9 * century observational data set o b t a i n e d at A t h e n s with the m o d e m M o o n w a t c h observational p r o g r a m o f D O G G E T T a n d S C H A E F E R ( 1 9 9 4 ) . T h e vertical axis denotes the altitude of the M o o n a b o v e the Sun, w h e r e a s the horizontal axis shows the relative a z i m u t h of the M o o n with respect to the Sun. T h e thin, fitted curves represent the t h r e s h o l d v a l u e s of the arcus visionis for limar visibility (i.e.. M o o n visible a b o v e , invisible b e l o w curve). T h e thicker curves denote actual values d e r i v e d from 4 sessions of the M o o n w a t c h p r o g r a m . T h e large dispersion o f the fits to the A t h e n s data indicates little a g r e e m e n t a m o n g the researchers in the theoretical representation of the observing conditions. T h e large variations in the M o o n w a t c h data, w h i c h represent observations from different areas o f the globe, s h o w the real effects of climatic conditions, i.e., variations in h a z i n e s s and relative h u m i d i t y as a fimction of season, latitude, a n d elevation. ( A d a p t e d from S C H A E F E R ( 2 0 0 0 ) , Fig. 4). T h e effects o f extinction/refraction variations w e r e particularly evident w h e n Schaefer^^ c o m p a r e d the M o o n w a t c h data with 4 different alt/azimuth (i.e., the arcus visionis) algorithmic fits to the same set of data, the series of o b s e r v a t i o n s m a d e b y Julius S c h m i d t in A t h e n s alluded to in the p r e v i o u s section. F i g u r e 3 s h o w s the altitude o f the M o o n a b o v e the sun as a fiinction of the relative a z i m u t h (or distance) of the M o o n from the Sun. T h e long, thin curves w e r e attempts b y Fotheringham,^^

"

SCHAEFER (1996).

FOTHERINGHAM (1910). An attempt to fit a crescent visibility algorithm to the lunar observations of Julius Schmidt at Athens observatory.

The Role of Astronomical Techniques in Ancient Egyptian Chronology

469

Maunder,^^ a n d Ilyas^^ to fit the G r e e k observations. T h a t they are s o w i d e l y dispersed is a n indication that n o n e o f the investigators h a v e r e p r o d u c e d conditions at the time o f the original observations. T h e M o o n w a t c h data are m a r k e d b y the thicker, solid lines. T h e inability to duplicate the M o o n w a t c h observations is a result of not taking into a c c o u n t the appropriate airmass extinctions n e a r the h o r i z o n as a function o f season, latitude a n d elevation, w h i c h the data s h o w to b e highly variable.

C o m p a r i s o n of V i s i b i l i t y A l g o r i t h m s D o g g e t t a n d Schaefer a n d Schaefer also c o m p a r e d 12 ancient, m e d i e v a l , a n d ' m o d e m ' m e t h o d s o f calculating lunar crescent first visibilities with their m o r e than 1500 M o o n w a t c h observations. T h e 4 8 minute M o o n s e t lag m l e - o f - t h u m b in fact p r o v i d e d very p o o r predictions, a m o n g the worst reliability in the series o f algorithms tested. E v e n the altitude/azimuth (arcus visionis) algorithms u s e d b y S c h o c h , F o t h e r i n g h a m , P . V . N e u g e b a u e r a n d others (i.e., the t h r e s h o l d arcus visionis values in F i g u r e 3 , for e x a m p l e ) , o n w h i c h m a n y m o d e m E g y p t o l o g i s t s rely for c o m p u t i n g crescent visibility, yielded only o n e correct p r e d i c t i o n in four! If this m u c h - u s e d c o m p u t a t i o n a l o r tabular technique reliably predicts o n l y 2 5 % o f the o c c u r r e n c e s in a k n o w n database, consider h o w m u c h p o o r e r it predicts in a d a t a b a s e in w h i c h at least 1 5 - 2 7 % o f the occurrences themselves are incorrect! C o n s e q u e n t l y , all p r e v i o u s investigations related to attempts to d e t e r m i n e absolute c h r o n o l o g i e s for ancient E g y p t b y this m e t h o d o l o g y are imreliable, including R o s e ' s m i s p l a c i n g the M i d d l e K i n g d o m b y 1500 years as well as D e p u y d t ' s recent attempt to date the lunar table in P . Carlsberg 9."*° W h e n Schaefer'*' c o m p a r e d his o w n m o d e m a l g o r i t h m v ^ t h the M o o n w a t c h observations, h e found that its predictability w a s 9 times better than the worst ancient m e t h o d a n d b e t w e e n t w o a n d four times m o r e reliable t h a n the best o f the ' m o d e m ' alt/azimuth m e t h o d s . T h e r e a s o n for the difference w a s that his a l g o r i t h m m o r e accurately t o o k into account the extinction a n d refi-action. Schaefer'*^ v e r y recently c o n c l u d e d " . . . that the current large uncertainties in predicting lunar visibility a n d in ancient Egyptian p r o c e d u r e s d o n o t a l l o w for a n y p o s s i b l e astronomical solution o f Egyptian absolute c h r o n o l o g y with lunar d a t e s " .

"

MAUNDER ( 1 9 1 1 ) . Another attempt, following Fotheringham's (FOTHERINGHAM ( 1 9 1 0 ) ) ,

to provide an algorithm for Julius Schmidt's observations. ILYAS ( 1 9 8 4 ) . Contains a more modem attempt to fit the Athens data of Julius Schmidt. And ILYAS ( 1 9 8 8 ) . A revision of his 1 9 8 4 published algorithm. DOGGETT and SCHAEFER ( 1 9 9 4 ) and SCHAEFER ( 1 9 9 6 ) .

ROSE ( 1 9 9 9 ) and DEPUYDT ( 1 9 9 8 ) . Depuydt's calculations from the Goldstine tables (see footnote 1 0 ) to find new moons do not take into account the appropriate length-of-day error. No explanation of how crescent visibilities were determined was given. One suspects rule-ofthumb. Consequently, his attempt to fix the era to which his revised table from P. Carlsberg 9 might belong is unreliable. SCHAEFER ( 1 9 % ) . SCHAEFER ( 2 0 0 0 ) .

470

R . A . Wells

Conclusions T h i s b e i n g the case, is it w o r t h asking whether there are a n y c o n d i t i o n s in w h i c h the t e c h n i q u e applied t o E g y p t might b e useful for a b s o l u t e c h r o n o l o g y ? T h a t is, w o u l d it b e worthwhile re-investigating, for e x a m p l e , the l l l a h u n s e q u e n c e u s i n g the m o r e rigorous a l g o r i t h m p r o v i d e d b y Schaefer. T h e a n s w e r h a s t o b e n o , that n o n e o f t h e E g y p t i a n lunar dates offer a n y p r o m i s e o f yielding a n absolute date for t w o r e a s o n s : (1) t h e likelihood that a n y o b s e r v e d s e q u e n c e o f multiple m o n t h lengths c o n t a i n s at least o n e error, b u t m a y b e m o r e , invalidates its u s e ; a n d ( 2 ) the large n u m b e r o f c o n s e c u t i v e m o n t h lengths, given perfect o b s e r v a t i o n s a n d perfect c o m p u t a t i o n s o f p a s t events, r e q u i r e d for statistical validity far e x c e e d s the available E g y p t i a n record. B a r r i n g the d i s c o v e r y o f a cache o f d o u b l e - d a t e d E g y p t i a n p a p y r i statistically c o m p a r a b l e to t h e B a b y l o n i a n cuneiform tablets, then, w h a t r e m a i n s to aid Egyptologists in d e t e r m i n i n g absolute dates? T w o m e t h o d s h a v e p r o v e d very important in the p a s t a n d m u s t always b e c o n s i d e r e d in a n y a b s o l u t e date analysis. O n e is a study o f the s y n c h r o n i s m s with n e i g h b o u r i n g countries; t h e other, a r e v i e w of t h e internal consistencies o f the p r o p o s e d d a t e s , s u c h a s that initially d r a w n u p , for e x a m p l e , b y Kitchen,'*^ Homung,'*'* o r v o n Beckerath'*^ in w h i c h lengths o f reign, g e n e a l o g i e s , events, climatological c h a n g e s , s e a s o n s , a n d t h e like, form a c o h e r e n t data set relatable t o similarly c o h e r e n t data sets from other countries. Kitchen'*^ h a s g i v e n the m o s t r e c e n t analysis o f this type, w h i c h m u s t b e c o n s i d e r e d a fiindamental c o m p a r i s o n s t a n d a r d for E g y p t i a n c h r o n o l o g y . It is t m e that this k i n d o f appraisal has limitations, i n t h e b e s t o f cases o n the o r d e r o f ± a d e c a d e , o r s o m e t i m e s ± 2 5 o r ± 5 0 years, i.e., a g e n e r a t i o n o r t w o . B u t other dating t e c h n i q u e s ' " h a v e similarly imprecise limitations, w h i c h m u s t a l w a y s b e clearly outlined in a n y p r e s e n t a t i o n in o r d e r to d e m o n s t r a t e c o n v i n c i n g l y that a n o b t a i n e d result h a s statistical validity.''^

References A L L E N , C l a b o n W . 1 9 7 3 . Astrophysical Quantities (3'** ed.). L o n d o n : A t h l o n e . V O N B E C K E R A T H , Jiirgen. 1 9 9 7 . Chronologie des Pharaonischen Àgypten: Zeitbestimmung

der Agyptischen

Geschichte

von der Vorzeit

( M û n c h n e r À g y p t o l o g i s c h e n Studien 4 6 ) . M a i n z : P h i l i p p v o n Z a b e m . B O R C H A R D T , L u d w i g . 1 9 3 5 . Die Mittel zur zeitlichen Festlegung von Punkten agyptischen

Geschichte

und ihre Anwendung.

Die

bis 332 v. Chr. der

C a i r o : Selbstverlag.

KITCHEN (1989). HORNUNG(1989). VON BECKERATH ( 1997). KITCHEN (2000).

*^ For example, even though radiocarbon (C''') dating can be calibrated against tree rings, the error limits on a determined age are no better than synchronistically determined dates (see http://www.C14dating.com for a detailed discussion of the methodology and error limits). Methods involving pyramid orientations, or planetary motions of Venus and Mercury are discussed in WELLS (2003). The author thanks Dr. Anthony J. Spalinger for bringing a number of references to his attention. He also thanks the editors and reviewers of this volume for several helpful suggestions which improved the manuscript.

The Role of Astronomical Techniques in Ancient Egyptian Chronology

471

C H A P R O N T - T O U Z É , M i c h e l l e a n d C H A P R O N T , Jean. 1 9 9 1 . Lunar Tables and Programs from 4000 BC to A.D. 8000. R i c h m o n d , V A : W i l l m a n - B e l l . D E P U Y D T , L e o . 1998. " T h e D e m o t i c M a t h e m a t i c a l A s t r o n o m i c a l P a p y r u s C a r l s b e r g 9 Reinterpreted". In: W i l l y C L A R Y S S E , A n t o o n S C H O O R S , a n d H a r c o WlLLEMS (eds.), Egyptian Religion the Last Thousand Years, Studies Dedicated to the Memory of Jan Quaegebeur, Part II (Orientalia L o v a n i e n s a A n a l e c t a 85): 1 2 7 7 - 1 2 9 7 . L e u v e n : Peeters. D O G G E T T , L e r o y E. a n d SCHAEFER, B r a d l e y E.. 1994. " L u n a r C r e s c e n t Visibility," Icarus, 107: 3 8 8 - 4 0 3 . E D G E R T O N , W i l l i a m . 1942. " C h r o n o l o g y of the Twelfth D y n a s t y , " Journal Eastern Studies 1: 3 0 7 - 3 1 4 .

of Near

FiRNEis, M a r i a G. 2 0 0 0 . " H e h a c a l Sirius-dates a n d first lunar crescent dates d e p e n d i n g o n geographical latitude for the use in absolute c h r o n o l o g y " . In: M a n f r e d BlETAK (ed.), The Synchronisation of Civilisations in the Eastern Mediterranean in the Second Millennium B.C. ( P r o c e e d i n g s o f an i n t e m a t i o n a l s y m p o s i u m at Schloss Haindorf, 15th-17th o f N o v e m b e r 1996 a n d at the A u s t r i a n A c a d e m y , V i e n n a , 11th-12th of M a y 1998): 5 8 - 5 9 . V i e n n a : V e r l a g der Ôsterreichischen Alcademie der Wissenschaften. FOTHERINGHAM, J o h n K. 1910. " O n the Smallest Visible P h a s e o f the M o o n " . Monthly Notices of the Royal Astronomical Society 7 0 : 5 2 7 - 5 3 1 . G O L D S T I N E , H e r m a n H . 1 9 7 3 . New and Full Moons lOOI BC to A.D. 1651 ( M e m o i r s of the A m e r i c a n Philosophical Society 9 4 ) . Philadelphia: A m e r i c a n Philosophical Society. H O R N U N G , Erik. 1989. " C h r o n o l o g y of die N e w K i n g d o m (with d i s c u s s i o n ) " . In: Paul Â S T R O M (ed.). High, Middle or Low?: Acts of an International Colloquium on Absolute Chronology held at the University of Gothenburg, 20th-22nd August, 1987, V o l . 3 : 3 4 - 3 6 . G o t h e n b i n g : Paul Â s t r ô m s Forlag. H U B E R , Peter J. 2 0 0 0 a . " M o d e l i n g the L e n g t h o f D a y a n d Extrapolating the R o t a t i o n of the Earth". In: Fabrizio BÒNOLl, Salvo D E M E I S , a n d A n t o n i o P A N A I N O (eds.), Astronomical Amusements, Papers in Honor of Jean Meeus: 9 1 - 1 0 4 . M i l a n o : Mimesis. — 2000b. "Astronomy and Ancient Chronology" ^ ^ W / c a 1 1 9 - 1 2 0 : 1 5 9 - 1 7 6 . I L Y A S , M o h a m m e d . 1984. Islamic Calendar, Times & Qibla. K u a l a L u m p u r : Berita Publishing C o . — 1988. "Limiting Altitude Separation in the N e w Criterion". Astronomy & Astrophysics 206: 133-135.

Moon's

First

Visibility

K I T C H E N , Keimeth A . 1989. " S u p p l e m e n t a r y N o t e s o n the B a s i c s o f E g y p t i a n C h r o n o l o g y " . In: Paul A S T R Ò M (ed.). High, Middle or Low?: Acts of an International Colloquium on Absolute Chronology held at the University of Gothenburg, 20th-22nd August, 1987, V o l . 3 : 1 5 2 - 1 6 0 . G o t h e n b u r g : Paul Â s t i ô m s Forlag. — 2 0 0 0 . " R e g n a l a n d genealogical data of ancient E g y p t (absolute c h r o n o l o g y I) T h e historical c h r o n o l o g y of ancient Egypt, a current a s s e s s m e n t , " In: M a n f r e d Bietak (ed.), The Synchronisation of Civilisations in the Eastern Mediterranean in the Second Millennium B.C. ( P r o c e e d i n g s of an i n t e m a t i o n a l s j o n p o s i u m at Schloss Haindorf, 15th-17th of N o v e m b e r 1996 and at the A u s t r i a n A c a d e m y , V i e n n a , l l t h - 1 2 t i i of M a y 1998): 3 9 - 5 2 . Vienna: V e r l a g der Osterreichischen A k a d e m i e der Wissenschaften.

472

R.A. Wells

K R A U S S , Rolf. 1984. Sothisund Monddaten (Hildesheimer Àgyptologische Beitràge 2 0 ) . H i l d e s h e i m : G e r s t e n b e r g V e r l a g . L U F T , Ulrich. 1992. Die chronologische Fixierung des agyptischen Mittleren Reiches nach dem Tempelarchiv von lllahun (Sitzungsberichte der Osterreichischen A k a d e m i e der Wissenschaften, Philosophisch-Historische Klasse, 598). V i e n n a : V e r l a g der Osterreichische A k a d e m i e der Wissenschaften. M A U N D E R , E d w a r d W . 1 9 1 1 . " O n the Smallest Visible P h a s e of the M o o n " . Journal of the British Astronomical Association 21: 355-362. N E U G E B A U E R , P a u l V . 1912. Tafeln zur astronomische Chronologie I. Sterntafeln von 4000 V. Chr. bis zur Gegenwart. Leipzig: J.C. Hinrichs — 1914. Tafeln zur astronomische Chronologie II. Tafeln fiir Sonne, Planeten und Mond. Leipzig: J.C. Him-ichs. — 1 9 2 5 . Tafeln zur astronomische Chronologie IH (2. Auflage). Leipzig: J.C. Hinrichs. — 1929. Astronomische Chronologie I-II. Berlin: W . d e G m y t e r . — 1937. Hilfstafeln zur technischen Chronologie (Astronomische Nachrichten 2 6 1 , N r . 6 2 5 0 - 6 2 5 4 ) . Kiel: V e r l a g der A s t r o n o m i s c h e n N a c h r i c h t e n . P A R K E R , R i c h a r d A . 1950. The Calendars of Ancient Egypt (Studies in A n c i e n t Oriental Civilization, N o . 2 6 ) . C h i c a g o : U n i v e r s i t y of C h i c a g o P r e s s . R O S E , L y n n E. 1999. Sun, Moon, and Sothis. A Study of Calendars, and Calendar Reforms in Ancient Egypt. Deerfield B e a c h , F L : K r o n o s Press. SCHAEFER, B r a d l e y E. 1 9 9 3 . " A s t r o n o m y a n d the Limits of V i s i o n " . Vistas in Astronomy 36: 3 1 1 - 3 6 1 . — 1996. " L u n a r C r e s c e n t Visibility". Quarterly Journal of the Royal Astronomical Society 3 7 : 7 5 9 - 7 6 8 . — 2 0 0 0 . " T h e Heliacal Rise of Sirius and A n c i e n t Egypfian C h r o n o l o g y " . for the History of Astronomy 3 1 : 1 4 9 - 1 5 5 .

Journal

— a n d LILLER, W i l l i a m . 1990. "Refracfion N e a r the H o r i z o n , " Publications Astronomical Society of the Pacific 102: 7 9 6 - 8 0 5 .

of the

S C H O C H , Carl. 1927. Planetentafeln fiir Jedermann zur Berechnung der geozentrischen Orter der grofien Planeten (und des Mondes) fiir den Zeitraum von 3400 v.Chr. bis 2600 n.Chr. ohne Anwendung der Logarithmen und trigonometrischen Funktionen bis auf ein Zehntel Grad unter besonderer Beriicksichtigung der Babylonischen Astronomic. Berlin: L i n s e r - V e r l a g . — 1 9 2 8 . " T a b l e s for Computafion". In S t e p h e n L A N G D O N a n d J o h n K. FOTHERINGHAM, The Venus Tablets of Ammizaduga. L o n d o n : O x f o r d University Press. S T E P H E N S O N , F . Richard. 1997. Historical Eclipses And Earth's Rotation. C a m b r i d g e ( U K ) a n d N e w Y o r k : C a m b r i d g e University Press. — a n d M i c h a e l A . H o u l d e n . 1986. Atlas of Historical Eclipse Maps: East Asia 1500 BC - AD 1900. C a m b r i d g e ( U K ) a n d N e w Y o r k : C a m b r i d g e U n i v e r s i t y .Press. W E L L S , R o n a l d A . 2 0 0 2 . " R e v i e w o f R O S E ( 1 9 9 9 ) " . Journal 61:311-315. W E L L S , R o n a l d A . 2 0 0 3 . The Role of Astronomical Chronology: The Use of Statistics and Error preparation.

of Near Eastern

Studies

Techniques in Ancient Egyptian Analysis in Absolute Dating. In

Signs from the Sky, Signs from the Earth: The Diviner's Manual Revisited Clemency

Williams,

Providence

In the closing w o r d s o f his article, " A B a b y l o n i a n D i v i n e r ' s M a n u a l " ( 1 9 7 4 ) , O p p e n h e i m a p p e a l e d to "historians of s c i e n c e " to h e l p "identify t h e specific [astronomical] m e t h o d s alluded to in the text." H i s r e q u e s t w a s n o trivial o n e , for the p a s s a g e , a n d n o less the w h o l e text is b y n o m e a n s straightforward. H i s a p p e a l w a s w h a t first d r e w m e t o w a r d s studying this astronomical p o r t i o n in m o r e detail.' T h i s section, fifteen lines in length (lines 5 7 - 7 1 ) , occupies a p o s i t i o n b e t w e e n t w o sections o f the text, w h i c h contrast with it in b o t h style a n d content. T h e p r e c e d i n g section (lines 1 - 3 5 ) gives a list o f the protases o f 14 "signs o f the e a r t h " [i-da-at K I tim] a n d 11 "signs o f the s k y " [i-da-at A N - e ] a c c o m p a n i e d b y s o m e c o m m e n t a r y (lines 3 6 - 5 6 ) . T h e section following (lines 7 2 - 8 4 ) c o m p r i s e s t w o tables: the first, a m e n o l o g i c a l table, sets out a listing o f the twelve m o n t h s o f the year b e g i n n i n g with B A R (Nisaimu), a n d the second lists the three watches. B o t h tables give s e e m i n g l y r a n d o m designations o f favourable [ S E / S I G S ] or not (favourable) [ N U ( S E ) ] for e a c h entry. T h e a s t i o n o m i c a l section contains references to m e t h o d s c o n c e m i n g the observations o f various astronomical p h e n o m e n a . O n a m o r e detailed e x a m i n a t i o n , these m e t h o d s c a n b e g r o u p e d into t w o distinct types: m e t h o d s c o n c e m e d with intercalation a n d m e t h o d s w h i c h c o n c e m the c o m p o s i t i o n o f the year (the n u m b e r o f m o n t h s in the year, the n u m b e r o f days in e a c h month, etc.). T h e first g r o u p , those dealing w i t h intercalation m e t h o d s , outline techniques w h i c h serve to check the c o m p o s i t i o n o f the year; that is, to d e t e r m i n e w h e t h e r the c a l e n d a r date is in sufficient a g r e e m e n t with respect to the astronomical p h e n o m e n a , a n d m a k e the n e c e s s a r y intercalations to correct it if required. T h e s e c o n d g r o u p urges the diviner to establish the c o m p o s i t i o n o f the year; that is, h a v i n g established the p r o p e r intercalations, to set out the y e a r ' s format, with its m o n t h s , the n u m b e r o f d a y s e a c h m o n t h contains, a n d also to establish exactly w h i c h is the present d a y in the current year. M a n y o f the m e t h o d s alluded to resemble those techniques e x p o u n d e d in w o r k s such as M U L . A P I N a n d T a b l e t 14 o f the astrological c o m p e n d i u m Enûma Anu Enlil (henceforth E A E ) . B e l o w is a n attempt to identify the m e t h o d s o n e b y one, a n d then to p u t e a c h o f the m e t h o d s into a larger context. First I give a ttansliteration a n d a franslation of this section o f the text:

' For the full details of the Diviner's Manual, see OPPENHEIM ( 1 9 7 4 ) . This passage has also been (partially) translated by BROWN ( 2 0 0 0 ) , p. 1 2 1 [lines 5 7 - 6 2 and half of line 7 1 ] and HOROWITZ ( 1 9 9 8 ) , p. 1 5 1 [lines 5 7 - 6 3 ] .

^

The transliteration is that of OPPENHEIM ( 1 9 7 4 ) , lines 5 7 - 7 1 , p. 2 0 0 .

474

C. WILLIAMS

57. 58.

12 I T U . M E S sà M U 1 . K A M 6 U § \JD-me-sâ mi-na-at ina SU-ka DlB-ma bi-ib-li UD.DA.ZAL.LÀ-e MUL.MES

59. 60. 61. 62. 63.

a-dan-na-ti-sû-nu mit-hur-ti S A G . M U sà M U L A â . G Â N ta-mar-ti "ES u ' ' U T U sà I T U § E u I T U K I N ni-ip-ha u I G I . D U g . A . M E â ' ' E S ar-hi-sam IGl-ru K I N . K I N - w « sit-qul-ta sà M U L . M U L u ''ES S E â - m a li-pu-ul-ka-ma sà M U ITU.ME§-5Ó I T U . M E S U D . M E â - f w kin -ma mim-mu-û te-pusu su-ul-lim e-nu-ma ina IGI.DUg.A ' ' E S UD-mw er-pw GÀL-A:âf li-ti-ik-sû D[UG mas-qu-u] e-nu-ma ina bi-ib-lu UD-mw er-pw GAL-/:a li-ti-ik-sû D U G mas-qu-u a n a /a-raA: èi-iò-// M na-an-mur-ti G U R U N E N [ I T U ] M sa-at-tum ^""^' 12 I T U . M E S ina S U - ^ a rw-^a/ a - n a /a-tóA; U D - w e ki-nu-tim sit-qul-ti M U L . M U L u ''ES m a âU^^-)ta às-ri sip-ki K I N . K I N - w a U D . M E S D I R I . M E S lu-û ti-di-ma M U . A N . N A ki-in-ma di-ri-sa su-ul-lim it-i-id la te-eg-gi

64. 65. 66. 67. 68. 69. 70. 71.

zag-muk sà ta-mar-ti

T w e l v e are the m o n t h s of the year. T h r e e h m i d r e d a n d sixty are its d a y s . T a k e into yom- h a n d the e x p e c t e d time of the zagmukku. Investigate t h o r o u g h l y the d a y o f d i s a p p e a r a n c e o f the m o o n , the dates of the first a p p e a r a n c e s of the stars, their time p e r i o d s , the c o r r e s p o n d e n c e of the b e g i n n i n g o f the year a n d the Field, the first a p p e a r a n c e s of the M o o n a n d the S u n in the m o n t h A d d a m (XII) a n d in the m o n t h U l u l u (VI), the risings a n d first a p p e a r a n c e s o f the M o o n as o b s e r v e d e a c h m o n t h . K e e p w a t c h for the " b a l a n c i n g " o f the P l e i a d e s ( M U L . M U L ) a n d the M o o n . M a y this a n s w e r you. Establish the m o n t h s of the year a n d the days of the m o n t h a n d carry out perfectly w h a t e v e r you d o . W h e n , at the first a p p e a r a n c e of the m o o n , the d a y is c l o u d y for you, its checking device^ is a masqu vessel. W h e n , at the d i s a p p e a r a n c e o f the M o o n , the day is c l o u d y for you, its c h e c k i n g d e v i c e is a masqu vessel. F o r the text of d i s a p p e a r a n c e a n d r e a p p e a r a n c e , take into y o u r h a n d Inbu Bel Arhi a n d Y e a r broken twelve m o n t h s . F o r the test o f the correct days take into your h a n d the " b a l a n c i n g " of the P l e i a d e s a n d the M o o n . S e a r c h t h o r o u g h l y the p l a c e of the sipku. M a y y o u l e a m the extra days. Establish the year arid carry out perfectly its intercalation. P a y attention! D o n o t b e careless! Let us consider the astronomical content of this section statement b y statement: T w e l v e are the m o n t h s of the y e a r . T h r e e h u n d r e d and sixty are its d a y s . (12 I T U . M E § M M U l . K A M 6 U § VX)-me-sà) This e m p h a t i c statement immediately evokes the notion of the so-called " s c h e m a t i c " (ideal) calendar, a systematic a r r a n g e m e n t of certain a s t r o n o m i c a l p h e n o m e n a (solstices a n d e q u i n o x e s , risings of the stars & c ) established within a c o n s t m c t e d time frame. In this schematic calendar, thirty days c o m p r i s e one m o n t h , a n d twelve

FOLLOWING BROWN, FERMOR, AND WALKER ( 1 9 9 9 ) , P. 1 3 9 .

Signs from the Sky. Signs from the Earth: The Diviner's Manual Revisited

475

m o n t h s , that is, three himdred a n d sixty d a y s , c o m p r i s e o n e year. T h e schematic calendar has m a n y uses in a n astronomical context. O n e o f these is to function as a " b l u e p r i n f w h i c h c a n b e u s e d to c o m p a r e the actual date o n w h i c h a p h e n o m e n a occurs against the c o r r e s p o n d i n g schematic date. S u c h a c o m p a r i s o n c a n b e useful to indicate w h e t h e r or n o t o n e n e e d s to intercalate. T h e schematic calendar is first attested in a n astronomical context in the O l d B a b y l o n i a n text B M 17175 +17284 a n d is subsequently foimd in E A E 14 a n d MUL.APIN.'* T h i s reference is also a r e m i n d e r o f the j u x t a p o s i t i o n o f the schematic c a l e n d a r ' s twelve 3 0 - d a y m o n t h s , o n w h i c h all o m e n literature is based, a g a m s t the " a c t u a l " calendar w h e r e all the m o n t h s are either 29 o r 3 0 days in length. A large part o f the d i v i n e r ' s j o b , then, is to interpret t h e data given according to the schematic calendar and convert it so it m a k e s sense v ^ t h respect to the actual calendar. T a k e into y o u r h a n d t h e e x p e c t e d time of t h e z a g m u k k u . (mi-na-at SV-ka D I B - m o )

zag-muk

ina

Zagmukku, from the S u m e r o g r a m Z A G . M U (literally: t h e threshold o f the year) is a t e r m often associated with the n e w - y e a r festivals, b u t in this context it refers to the b e g i n n i n g o f the n e w year. T h i s t e r m r e a p p e a r s t o w a r d s the v e r y e n d o f the text (line 72) w h e r e it is contrasted v ^ t h qi-ti-sù (literally: its e n d ) , confirming its interpretation o f the b e g i n n i n g o f the year. It s e e m s likely that the m o t i v a t i o n for having this figure at h a n d could simply b e a n instruction to establish t h e starting date from w h i c h to determine the dates o f the various p h e n o m e n a in the year. In o r d e r to date p h e n o m e n a so as to c h e c k t h e m against the schematic calendar, o n e m u s t k n o w w h e r e to b e g i n the year from. O f particular interest here is the u s e o f the t e r m minïtu, a t e r m which, a m o n g other usages, c o n v e y s the n o t i o n that the item it qualifies h a s b e e n established t h r o u g h calculation!' A p a r a m e t e r w h i c h is 'minïtu', b e it a m o m e n t in time, a m e a s u r e o r a length, is o n e w h i c h h a s b e e n calculated. P e r h a p s the n u a n c e o f this t e r m t h e n suggests a n element o f flexibility in the selection o f this d a t e . In a c c o r d a n c e with this s e c t i o n ' s emphasis o n c h e c k i n g a n d r e c h e c k i n g the c o m p o s i t i o n o f the year a n d whether it h a s b e e n intercalated p r o p e r l y , it is fitting that the m o s t logical p l a c e to start o n e ' s checks is with the begiiming o f the year, especially given that the b e g i n n i n g h a s b e e n calculated, a p r o c e d u r e that m a y h a v e b e e n carried out less than perfectly. (Investigate t h o r o u g h l y ) (...KIN.KIN-i,ifl))

t h e d a y of d i s a p p e a r a n c e

of the moon

{bi-ib-li

T h e a b o v e reference t o the investigation o f the d a y o f the d i s a p p e a r a n c e o f the m o o n suggests techniques that establish the lengths o f the m o n t h s . T h e q u e s t i o n o f w h e t h e r a m o n t h w a s h o l l o w (29 days in length) or full ( 3 0 days in length) w a s a p r o b l e m w h i c h greatly o c c u p i e d a s t t o n o m e r s . T o ascertain this, they w e r e interested

^ MUL.APIN's schemafic calendar can be derived from the second section of the work (1 ii 36-iii 12: sequential calendar dates of the heliacal risings of certain constellations) and has the following arrangement, namely that 1 month = 30 days and 1 year = 12 months = 360 days. See the discussion in HUNGER and PINGREE (1989) pp. 139-40. *

See CAD "M" Vol. 10 part 11, minitu, 1 d), p. 87.

476

C. Williams

in determining h o w m a n y days the interval w a s b e t w e e n the last visibility o f the m o o n at the e n d o f a m o n t h a n d its first visibility that m a r k e d the b e g i n n i n g . M a n y sophisticated m e t h o d s were d e v e l o p e d for predicting this, o n e o f w h i c h included the technique pertaining to the K U R value, o n e o f the so-called lunar six that c o n c e m time p e r i o d s b e t w e e n p h e n o m e n a o f m o o n a n d sunrise a n d set. A l t h o u g h these techniques w e r e not d e v e l o p e d to a suitable level o f sophistication until a r o u n d the sixth century B . C . , it is likely that general, ' m l e - o f - t h u m b ' m e t h o d s that h a d b e e n noticed existed prior to this.^ T h e K U R value is the interval o f time from m o o n r i s e to sunrise o n the d a y w h e n the m o o n is visible for the last time before conjunction. In v e r y general terms, a large K U R value i m p l i e d that conjunction ( a n d h e n c e the beginning o f the next m o n t h ) w a s farther off t h a n w h e n the K U R value w a s small, so that accordingly in the former case the m o n t h will b e longer than in the latter case. Therefore this type o f m e a s u r e m e n t o f the K U R value allows o n e , at least in a very c m d e sense, to predict the length of the current m o n t h . T h e application o f this principle could b e what is alluded to here in this text. T h e diviner c a n u s e this information w h e n h e is trying to establish h o w long the m o n t h h e is in is g o i n g to last, before the first sighting o f the n e w crescent. It lets h i m determine, in a d v a n c e , h o w m a n y days h e m u s t allow before h e starts the b e g i n n i n g o f the n e x t m o n t h . t h e d a t e s of t h e first a p p e a r a n c e s of ( U D . D A . Z A L . L À - ^ sa ta-mar-ti M U L . M E S

the stars, their a-dan-na-ti-sû-nu)

time

periods

T h i s p r o c e d u r e refers to a s c h e m e similar t o (if n o t identical to) that set out in M U L . A P I N I ii 36-iii 12 in w h i c h the heliacal risings ( d e n o t e d b y the w o r d I G I . L À = a k k a d i a n tamartu) o f 3 5 constellations are laid d o w n in order o f their o c c u r r e n c e in the schematic calendar. T h i s is usefiil as an intercalation check as o n e c a n o b s e r v e the p h e n o m e n a they describe a n d c o m p a r e the date o f o c c u r r e n c e against the o n e p r o v i d e d in M U L . A P I N a n d shift the date as necessary. T h i s intercalation is also practical, as it involves p h e n o m e n a which are h a p p e n i n g t h r o u g h o u t the s p a n o f the schematic year, rather than p h e n o m e n a that o c c u r only o n c e or twice in the year. A s the diviner wishes to c h e c k the composition o f the year right at the m o m e n t h e is working, a n d m a y n o t h a v e the liberty to wait six m o n t h s or m o r e for a certain p h e n o m e n a to o c c u r with w h i c h h e c a n check the intercalation, the ideal situation is to h a v e intercalation checks that are distiibuted in such a w a y that there is o n e close at h a n d from a n y given m o m e n t in the year. S u p p o s i n g the p r o c e d u r e for risings given in M U L . A P I N w a s envisioned. as w e h a v e seen, listing the thirty-five constellations spread out t h r o u g h the of the schematic year, o n e has at h a n d a specified rising time at least at a five, a fifteen, or a twenty d a y interval (with t w o e x c e p t i o n s b e i n g as long as thirty so that there will b e a intercalation check b y this m e a n s close at hand.

Then, length a ten, days),

^ See, for example, the observations and predictions set out in the Letters and Reports, as discussed by HUNGER and PINGREE ( 1 9 9 9 ) , pp. 1 1 6 - 1 1 7 , in which similar practices are used, involving the observation of the Sun and the Moon in opposition to each other in the middle of the month for the predictions both of the day of opposition and of the day of the New Moon.

Signs from the Sky. Signs from the Earth: The Diviner's Manual Revisited

T h e correspondence of the beginning of the year and the Field

477

{mit-fiur-ti

SAG.MU M M U L AS.GAN)

T h e m e a n i n g o f this is s o m e w h a t o b s c u r e , as the b e g i i m i n g o f t h e year follows the rising o f the Field. T h e r e f o r e it is difficult t o imagine t h e p r e c i s e n u a n c e o f the w o r d ' c o r r e s p o n d e n c e ' (mithurti)

w h i c h is u s e d h e r e . A c c o r d i n g t o M U L . A P I N , t h e Field

rises o n m o n t h X I ( S a b a t u ) d a y 5: D I § ina '"ZIZ U D 5 K A M " " " ' G U . L A " " " ' A S . I K U U """"^lu-lim l O r ' ^ " O n t h e 5 * o f Sabatu, the G r e a t O n e , t h e Field a n d the S t a g b e c o m e v i s i b l e " ( I iii 10).^ M U L . A P I N r e p e a t s this information later ( I iii 42ff), stating that the F i e l d rises 7 0 days before the V e r n a l E q u i n o x , o r o n d a y 3 0 5 o f the ideal year. I n 1000 B C P P e g ( t h e first star o f t h e Field) rises a p p r o x i m a t e l y 6 0 d a y s b e f o r e t h e V e m a l E q u i n o x . W i t h i n M U L . A P I N ' s limits, if the V e r n a l E q u i n o x falls o n I 1 t h e n P P e g rises o n X 3 0 / X I 1. Similarly if the V e m a l E q u i n o x falls o n I 3 0 t h e n P P e g rises o n X I 3 0 / X I I

1. I n a c c o r d a n c e with the a s t r o n o m i c a l reality, a p l a u s i b l e

intercalation m i e c o u l d b e s o m e t h i n g like t h e following: If p Peg rises in month XII (i.e. has not risen before XII) this year

is intercalary

(i.e. add an intercalary

XII^.

In this w a y there is s o m e c o n n e c t i o n b e t w e e n the b e g i n n i n g o f the year a n d t h e F i e l d pertaining t o intercalation. T h e r e is also a potentially meaningful association b e t w e e n t h e Field a n d N i s a i m u ( M o n t h I ) in E A E . I n the assiuned T a b l e t 5 1 , * a n interesting astrological s c h e m e , b a s e d o n a tradition closely associated with A s t r o l a b e B , is set u p in w h i c h v a r i o u s constellations a r e c o r r e l a t e d t o the first n i n e m o n t h s o f the year.^ A l t h o u g h this s c h e m e is astrologically b a s e d , a n d thus is n o t necessarily c o n s t r a i n e d b y t h e n e e d t o reflect a s t r o n o m i c a l reality, t h e s e q u e n c e set out is n o t grossly d i v e r g e n t from t h e a s t r o n o m i c a l p h e n o m e n a . ' " O f particular n o t e is t h e c o i m e c t i o n o f m o n t h I w i t h the Field: M U L . A S . G À N ina I T I . B À R [ I G I - m a r B E - m a M U L B I N I M - m a I G I . . . ] B E ma M U L B I ZAL-ma

I[TI-5M D I B - m a I G I . . . ] " T h e Field rises helically i n N i s a i m u ;

if this star rises early : [ . . . ] if this star is late a n d p a s s e s b y its m o n t h a n d rises [ . . . ] " " ( I X 1), a n d similarly in I X 16, X 1 a n d X I I 1. T o further sustain the c o n n e c t i o n b e t w e e n t h e b e g i n n i n g o f t h e y e a r a n d the Field, in a p r e v i o u s section o f the D i v i n e r ' s m a n u a l itself, in the list o f the " S i g n s o f the S k y " , t h e following o m e n is cited: D I S M U L A S . G A N ina I T U B A R IGI-i> " I f the Field is s e e n in t h e m o n t h N i s a n n u " (line 3 4 ) . H o w e v e r , in t h e O l d B a b y l o n i a n S c h e m a t i c calendar, P P e g rises o n e m o n t h earlier t h a n m o n t h X I , in m o n t h X ( a s the e q u i n o x dates are p l a c e d o n e m o n t h before), so it is difficult t o s e e w h y it w o u l d b e a s s o c i a t e d w i t h the b e g i n n i n g o f the year. T h e r e f o r e , it s e e m s m o r e p l a u s i b l e that the intercalary m i e is linked with a later N e o A s s y r i a n c o n v e n t i o n . T h u s it s e e m s that there w a s s o m e special relationship b e t w e e n the b e g i n n i n g o f the year a n d the Field for t h e c o m p i l e r o f this text, w h e t h e r it b e that h e identified t h e h e l i a c a l rising of the Field with m o n t h I, o r s o m e other c o n n e c t i o n b e t w e e n t h e t w o , w h i c h h e s a w ^

HUNGER and PINGREE ( 1989), p. 46.

*

See REINER and PINGREE ( 1981 ), p. 52 and following.

' I =AS.GÀN, II-MUL.MUL, I1I=S1PA.ZI.AN.NA, 1V=KAK.S1.SÀ, V=BAN, V I = B I R , VU=zibanitu, VIII=GIR.TAB, IX=UD.KA.DUg.A. '° REINER and PINGREE (1981), p. 53 rightly point out that "this is not a bad sequence astronomically; MUL.APIN has the above nine constellafions rise respectively on X 1 5; 11 1; 111 10; IV 15; V 15 VI 10 VII 15; VHl 5; and IX 15." ''

REINER and PINGREE ( 1981 ), p. 56.

478

C. Williams

fit to include a m o n g s t material dealing with intercalation m e t h o d s . Its g e n e r a l fijrmat is the same as other intercalation m e t h o d s included in this text w h i c h involve a constellation rising in a n e x p e c t e d m o n t h . T h e first a p p e a r a n c e s of t h e M o o n a n d the S u n in t h e m o n t h A d d a r u (XII) a n d in t h e m o n t h U l u l u ( V I ) {ta-mar-ti u "UTU M I T U § E u I T U K I N ) T h e precise significance o f this is unclear. H o w e v e r , t h e m o n t h s o f A d d a m ( X I I ) and U l u l u ( V I ) a r e significant as these a r e the t w o m o n t h s w h i c h a r e d e s i g n a t e d as intercalary m o n t h s . If a n extra m o n t h is to b e a d d e d , either o f these m o n t h s will b e the ones w h i c h are repeated. T h e s e m o n t h s are also significant b e c a u s e they a r e t h e m o n t h s in w h i c h t h e e q u i n o x e s are p l a c e d in the O l d B a b y l o n i a n v e r s i o n o f the ideal calendar,'^ n o t a b l y o n e m o n t h before their dates in t h e schematic calendar as set o u t in M U L . A P I N . ' ^ T h e risings a n d first a p p e a r a n c e s of t h e M o o n a s o b s e r v e d e a c h m o n t h ba u IGI.DU8.A.MES M ''ES ar-bi-sam IGl-ru K I N . K I N - m a )

(ni-ip-

T h e first a p p e a r a n c e s o f the M o o n determine t h e b e g i n n i n g o f the m o n t h . O n e m u s t take account o f these to c h e c k t h e m against the lengths o f the m o n t h s in days that h a v e already b e e n established. N o precise m e t h o d is specified; h o w e v e r , it is likely that m e t h o d s which u s e d a c o m p a r i s o n b e t w e e n t h e N A value with its calculated value from E A E 14 w e r e meant.''* T h e length o f the observed N A c o m p a r e d w i t h t h e calculated N A indicates whether the m o n t h b e g a n o n t h e correct d a y . T h e e m p h a s i s o f the i n ^ o r t a n c e o f t h e first d a y o f the m o n t h , w h e n t h e n e w m o o n appears, h a s also b e e n suggested b y Livingstone, w h o finds that it is frequently favourable in hemerologies.'^ K e e p w a t c h for t h e " b a l a n c i n g " of t h e Pleiades ( M U L . M U L ) a n d t h e M o o n . {sit-qul-ta M M U L . M U L u " E S SES-WIA) M e t h o d s involving t h e Pleiades a n d the M o o n a r e a c o m m o n intercalation c h e c k a n d are d o c u m e n t e d in several other sources. Such a s c h e m e , the so-called " P l e i a d e n Schaltregal", first r e c o g n i z e d b y Schaumberger,'^ is described in M U L . A P I N II G a p A 8 - 9 " as: See, for example, BM 1 7 1 7 5 + 1 7 - 2 8 4 , the Old Babylonian tablet published in the Appendix of HUNGER and PINGREE ( 1 9 8 9 ) , p. 1 6 3 .

MUL.APIN has an intercalation scheme (discussed by HUNGER and PINGREE ( 1 9 8 9 ) , pp. 1 5 0 - 1 5 1 ) which is based on observations of the Sun and the Moon in equinoctial months (also includes observations of only the sun at solstices). Of course, the scheme is a month out, but the general idea is similar. Table A in AL-RAWI and GEORGE ( 1 9 9 1 - 1 9 9 2 ) , p. 5 5 gives the NA value as a function of the day since the conjunction in an equinoctial month based on a linear zigzag scheme (from day 5 - 2 5 ) . LIVINGSTONE ( 1 9 9 3 ) , p 1 1 0 . SCHAUMBERGER ( 1 9 0 9 / 1 0 ) .

HUNGER and PINGREE ( 1 9 8 9 ) , pp. 8 9 - 9 0 . See also II Gap A ii 1 - 2 , an intercalation rule

which also includes the Pleiades and the Moon being in conjunction on different months (Arahsamnu and Kislimu). BROWN ( 2 0 0 0 ) p. 1 2 1 mentions " a scheme very similar to that in

Signs from the Sky. Signs from the Earth: The Diviner's Manual Revisited

479

8. [ D I § ina I T I . B À R U D 1 K ] A M M U L . M U L u ''Sin sit-qu-lu M U B I GI.NA-to 9. [Dia m a , I T I . B À R U ] D 3 K A M M U L . M U L u '^Sin L À L M U B I D I R I - ó r 8. [If on the r' of Nisannu] the Stars and the Moon are in conjunction, this year is normal. 9. [If] on the 3'''^ [of Nisannu] the Stars and the Moon are in conjunction, this year is a leap year T h e P l e i a d e s a r e at longitude 18° a n d latitude 3° in 1000 B C . O n I 1 t h e S u n is at 345° (given that the V e m a l E q u i n o x is o n I 15) a n d the M o o n is a r o u n d 0° ( b e c a u s e it is the night o f First Visibility). Therefore at b e g i n n i n g o f night o f I 2 , the M o o n is at a p p r o x . 13°, a n d b y the e n d o f the night, 18.5°-19°, w h i c h is in the P l e i a d e s . S o in order t o m a k e s o m e sense o f the m i e , (understanding that sitqultu = b a l a n c i n g (i.e. at the s a m e longitude b u t not necessarily s a m e latitude)), s o m e t h i n g like t h e following could b e u n d e r s t o o d : If on the second night the Moon passes into the Pleiades this year is normal. If this occurs a night later (because the Sun and Moon are farther back) intercalation is needed. H o w e v e r , this interpretation n o t a b l y d o e s n o t u s e d a y 1, a s t h e m i e m e n t i o n s . Pingree h a s ingeniously suggested a solution w h i c h c o n c e m s t h e rising o f the M o o n before a n d after t h e Pleiades a s a c h e c k . W h i l e this solution for the m o s t part w o u l d w o r k a s an intercalation m i e , it fits a w k w a r d l y with the attested m e a n i n g s o f sitqultu. A m o r e elaborate form o f this m e t h o d - o n e that c a n b e a p p l i e d i n all twelve m o n t h s o f t h e year - is t h e s c h e m e explained b y H u n g e r a n d R e i n e r . ' ' I n this s c h e m e , e a c h m o n t h a particular d a y is assigned for checking t h e longitude o f the M o o n a n d t h e Pleiades. If they a r e t h e " s a m e " , then intercalatibn is n o t necessary; h o w e v e r , if they a r e different, then intercalation is necessary. T h e general s c h e m e H u n g e r a n d R e i n e r extract from t h e text is the following: " I f in m o n t h n o n t h e ( 2 7 - 2 n ) t h d a y y o u observe t h e Pleiades a n d t h e M o o n a n d they h a v e t h e same longitude, then this year is n o r m a l ; if they fall down, it is left behind."^" W h e n , at t h e first a p p e a r a n c e o f t h e m o o n , t h e d a y is c l o u d y f o r y o u , its c h e c k i n g d e v i c e is a maSqu vessel, {e-nu-ma ina IGI.DUg.A U D - m u er-pu GkL-ka li-ti-ik-M D [ U G mas-qu-u]) W h e n , at the d i s a p p e a r a n c e o f the M o o n the d a y is c l o u d y f o r y o u , its c h e c k i n g d e v i c e is a m a s q u vessel, {e-nu-ma ina bi-ib-lu UD-/w« er-pu G A L - ^ a li-ti-ik-M D U G mas-qu-u) T h e situation referred to here is o n e in w h i c h the first a p p e a r a n c e o f the n e w m o o n that m a r k s t h e begiiming o f the m o n t h , o r the final a p p e a r a n c e at almost the e n d o f the m o n t h c o u l d not b e v i e w e d d u e t o inclement weather conditions. T h e p u r p o s e o f

MUL.APIN GapA10-ii6 for determining when a year should be intercalated is described", an identification which misses out the obvious Gap A 8-9 and includes methods which, although similar in format, do not include the Moon and may deal with constellafions other than the Pleiades. HUNGER and PINGREE (1989), p. 152. HUNGER and REINER (1975). HUNGER and REINER ( 1975), p. 24.

480

C. Williams

sighting the first s c e n a r i o is o f course to establish the begiiming o f t h e m o n t h . a b o v e reference gives the diviner a n altemative option to establishing observation should the weather p r e v e n t it, b y m e a n s o f a later o b s e r v a t i o n m e a s u r e m e n t o f a time interval b y using a specific vessel called a masqu (from to fill (with water)),^' p r o b a b l y s o m e sort o f water clock.

The this and saqû

T h i s is a clear reference to the e m p l o y m e n t o f the techniques a n d time p e r i o d s similar to those in T a b l e t 14 o f EAE.^^ T h e s e tables set o u t differing time p e r i o d s o f lunar visibilities a n d invisibilities, calculated b y m e a n s o f linear zigzag fimctions, with values tabulated, in t w o o f the tables, for e a c h d a y o f the m o n t h , a n d in the other t w o , o n the 1 5 * a n d 3 0 * / P * d a y o f each m o n t h . T h e desired values c a n b e d e t e r m i n e d b y using a c o m b i n a t i o n of tables, T a b l e A a n d T a b l e D . A s m e n t i o n e d previously. T a b l e A gives the N A value as a fimction o f the d a y since conjunction for a n equinoctial m o n t h a n d T a b l e D gives the deviation o f this i n c r e m e n t b y m o n t h . Therefore, g i v e n a n y m o n t h a n d day, o n e c a n take the v a l u e o f calculated N A for an equinoctial m o n t h from T a b l e A c o r r e s p o n d i n g to the d a y in questions, and the increment for that m o n t h in T a b l e D , divide b y twelve a n d m u l t i p l y it b y the first value to obtain a calculated N A which c o r r e s p o n d s to the d a y a n d m o n t h in question. S o that, if the d a y is c l o u d y at the time o f the d i s a p p e a r a n c e o r r e a p p e a r a n c e o f the m o o n , o n e simply waits the n u m b e r o f days until the w e a t h e r is clear, a n d m a k e s a m e a s u r e o f the appropriate p h e n o m e n a . U s i n g this value, o n e selects the appropriate tables a n d calculates the theoretical value. C o m p a r i n g these t w o values, the o b s e r v e d a n d the calculated, o n e c a n then " c o u n t b a c k " to establish w h i c h d a y is the first d a y o f the m o n t h , that is, the d a y o n w h i c h the first visibility o f the crescent should h a v e b e e n sighted w e r e it n o t cloudy. T h i s m e t h o d o f c h e c k i n g the b e g i n n i n g o f the m o n t h is n o t a b l e for its practicality. A g a i n it is w o r t h mentioning, that n o t only does this take note o f the difficulty o f practicing a s t i o n o m y with certain i n t m s i v e weather elements, b u t also, this is an intercalation c h e c k that is n o t exclusive to o n e particular instance in the year. F o r t h e test of d i s a p p e a r a n c e a n d r e a p p e a r a n c e , t a k e into y o u r h a n d I n b u B e l A r h i a n d Y e a r broken t w e l v e m o n t h s , {ana la-tak bi-ib-li u na-an-mur-ti G U R U N E N [ITUI « sa-at-tum 12 I T U . M E â ina ^V-ka tu-kal) A n astiological text b y the n a m e Inbu Bel Arhi ("Emit, L o r d o f the M o n t h " ) is extant. Livingstone h a s called it " p n e o f the m o s t enigmatic m a g i c a l texts"^^ a n d described the content o f the text as follows: "individual tablets for e a c h d a y o f a n ideal m o n t h a n d e x t i a tablets for intercalary N i s a n a n d Elul. E a c h tablet specifies m l e s o f cultic hygiene, behavior, dietary prohibitions d a y b y d a y t h r o u g h the m o n t h . C o m b i n e d with this is a n e x t i e m e l y detailed d a y b y d a y account o f w h a t type o f offerings the king is to m a k e to w h i c h particular deities t h r o u g h o u t the d a y and/or

As observed by BROWN, FERMOR and WALKER ( 1 9 9 9 ) , p. 1 3 9 n. 3 5 , such an instrument is

also attested in a commentary text in the Sin section of EAE: [li]-ti-iq-su

mas-qu-u, ACh 2

Supp. Sin 1 9 = K 3 1 2 3 (8).

AL-RAWI and GEORGE ( 1 9 9 1 - 1 9 9 2 ) , pp. 5 2 - 7 3 . 1 thank John Britton who helped me work through this. " See LIVINGSTONE ( 1 9 9 9 ) and also LIVINGSTONE ( 1 9 9 3 ) . For other references to this series, see CAD inbu mng. 1 d).

Signs from the Sky. Signs from the Earth: The Diviner's Manual Revisited

481

night. T h i s is then followed for e a c h m o n t h b y sections w h i c h ring t h e c h a n g e s o n o m e n s o f the Iqqur Ipus type, b u t with the king as subject o f the protasis. T h i s o f c o i u s e a m o u n t s to a prescription o f royal b e h a v i o u r through the seasons o f the year."^'* A s Livingstone explains, Inbu Bel Arhi is a series o f h e m e r o l o g i c a l tables. Therefore, given the inclusion o f the m e n o l o g i c a l material at the e n d o f the D i v i n e r ' s M a n u a l , a reference to it is n o t out o f p l a c e at all. B u t while its format does fit in with the character o f the contents o f the D i v i n e r ' s M a n u a l in general, its content d o e s n o t s e e m to b e in a n y sort o f c o r r e s p o n d e n c e with the function the D i v i n e r ' s M a n u a l suggests, w h e n it says that o n e should u s e it to " c h e c k the d i s a p p e a r a n c e a n d reappearance."^^ It could b e ù a t the p h r a s e ana la-tak bi-ib-li u na-an-mur-ti is to b e u n d e r s t o o d as pertaining to the p r o c e d u r e s o n the p r e v i o u s t w o lines w h i c h are closer in content. F o r t h e test of t h e correct d a y s t a k e into y o u r h a n d the " b a l a n c i n g " o f t h e P l e i a d e s a n d t h e M o o n {a-na la-tak VD-me ki-nu-tim Mt-qul-ti M U L . M U L u ' ' E S ina §U"-/fca tu-kat) This, o f course, is repeating the statement m e n t i o n e d a b o v e w h i c h h a s a l r e a d y b e e n discussed, this time h o w e v e r with a v i e w to establishing the c o m p o s i t i o n o f the year, as it m e n t i o n s a coimection with days. I n v e s t i g a t e t h o r o u g h l y t h e p l a c e of t h e sipku ( ds-ri Sip-ki K I N . K I N - w a j T h i s is a suggestion to the diviner to k e e p observing, r e m i n d i n g h i m o f the m i p o r t a n c e o f the astral p h e n o m e n a in his work. W h a t is w o r t h y o f m e n t i o n h e r e is the p h r a s e asri sipki. T h e r e h a s b e e n s o m e uncertainty as to h o w to translate this phrase.^^ Sipku, in a n a s t i o n o m i c a l setting, is c o m m o n l y c o m b i n e d in the p h r a s e sipku same to m e a n horizon^' (literally "the base o f the sky"). Asm is a g e n e r a l w o r d for a " p l a c e " , b u t w h e n c o m b i n e d with w o r d s denoting the n e t h e r w o r l d (KUR.NI.GI4.A, dannïnaf^ it refers to the h e a v e n s . Therefore, given t h e asfronomical context a n d given that m a n y o f the observations referred to in the p r e c e d i n g lines are ones that are m a d e at the horizon, it m a y not b e w r o n g to a s s u m e asri sipki m e a n s " t h e h o r i z o n . " T a k e n also in its p r o x i m i t y to the next p h r a s e " m a y you l e a m the exfra d a y s " , it s e e m s to further suggest that h o r i z o n is meant. U n d e r s t a n d i n g U D . M E S D I R I . M E S as b e i n g the exfra days, or the epact, that is the days in excess o f 12 m o n t h s c o r r e s p o n d i n g to a year, these days are d e t e r m i n e d b y using observational data (rising a n d settings) o b s e r v e d o n the horizon. H o w e v e r , the use o f this p h r a s e for "horizon", w h e n the phrase sipku same a l r e a d y exists, is p u z z l i n g a n d so this franslation should b e taken with caution. LIVINGSTONE (1999), p. 137.

Although LIVINGSTONE ( 1993) does suggest that there is a relationship between the phases of the moon and the favourability and unfavourability of the days and the months, with the days of disappearance and reappearance being particularly important (p. 102, p. 108, and p. 110). OPPENHEIM (1974) renders the phrase as "Look up the ..." and in a footnote (n. 41) he comments "the sign group às-ri sip/me-ki remains a mystery". See respectively the enfries in the CAD for sipku vol. 17 p. 70-71 mng. Id) and also supku vol. 17, p. 323-324, mng. c). See CAD vol. 1 part 11 asm p. 456-460, mng. 2e).

482

C. Williams

M a y y o u learn the extra days. Establish the y e a r a n d carry o u t perfectly its intercalation. P a y attention! D o not be careless! ( U D . M E S D I R I . M E § lu-ii ti-dima M U . A N . N A ki-in-ma di-ri-Sa su-ul-lim it-i-id la te-eg-gi) T h u s w e see that t h e astronomical section of the D i v i n e r ' s m a n u a l sets forth a n u m b e r of different intercalations and m e t h o d s to establish the correct n u m b e r of days and their a r r a n g e m e n t in each year. T h e phrase used to describe this list o f astronomical intercalation techniques is especially pertinent. In the second half o f line 5 6 , t h e astronomical section is introduced by the phrase: an-nu-û NAM.BÙR.BI-5M-rtM, translated literally as "this is their Moosing'",^^ that is, t h e m e a n s t o dispel the unfavourable consequences of the o m e n s . This establishes t h e expectation that the list of what is to follow are m e t h o d s to subvert the unfavourable results predicted by t h e o m e n s . It s e e m s then that the purpose of including these intercalations are, as their introduction implies, to r e m o v e the evil c o n s e q u e n c e s of an o m e n which h a s been found to be unfavourable.''^ This section presents o n e option the diviner h a s for r e m o v i n g these evil c o n s e q u e n c e s , that is, by checking the validity of the date on which t h e o m e n occurred, and if he is able, to c h a n g e this date, either by a n u m b e r of days or even by a whole month, using o n e of the intercalary t e c h n i q u e s mentioned as a justification for his date-alteration. T h e s e astronomical techniques are not so m u c h for the diviner to necessarily establish the "exact d a t e " of his omen, but m o r e to provide him with options for " c o r r e c t i n g " the date should he need a way to avert an ill-boding o m e n . T h e s e techniques are not for strict astronomical calculations but rather for divinatory purposes. T o support this o n e must consider the nature of these techniques described. N o n e of the schemes mentioned a b o v e are a reflection of any sort of astronomical precision. T h e y seem not so much to be an attempt at a highly accurate representation of the heavenly p h e n o m e n a they deal with, but are rather s i m p l e .

For discussion of the Sumerian word NAM.BUR.BI, and the Akkadian loan word namburbu, see REINER (1995), p. 81-82. See also BROWN (2000), pp. 121-122 (section 3.1.2), who comes to the same conclusion. However, this conclusion fits in uneasily with his overall thesis set out in this section of his book, namely that "...when a celestial body was behaving according to the ideal and when it was not...the evidence is that the former was considered propitious and the latter not" (p. 113. repeated again with reference to the astrolabes, p. 115, with reference to MUL.APIN, p. 117, and with reference to the Pleiades rule, p. 119). He asserts that this thesis applies to all the texts he discusses in the previous section (3.1.1) of which the Diviner's Manual is one. However, the emphasis of the Diviner's Manual is quite distinct from this. It is focused not on deriving significance from the date of when the omen occurred and its aberration from the ideal, but in fact, it uses the astronomical phenomena in question to provide "alternative options" for establishing the date, and thus the prediction of the omen. That is, if these phenomena are distinct from their "ideal" predicted times, the diviner has a justifiable reason to recalculate the date so that the phenomena are synchronized with their ideal times, which will in turn alter the forecast of the omen, hopefully, for the diviner, for the better. In this text, their aberration from the ideal is not significant in itself but only as far as it allows the Diviner to be able to justify an intercalation to change the date.

Signs from the Sky. Signs from the Earth: The Diviner's Manual Revisited

483

ruie-of-thumb type schemes,"" which seek to highlight general astronomical trends. For e x a m p l e , the P l e i a d e s / M o o n intercalation s c h e m e (line 7) as discussed by H u n g e r and Reiner^^ is a major simplification o f the astronomical reality. T h e E A E 14 type s c h e m e s are based on mathematical linear a p p r o x i m a t i o n s of t i m e p e r i o d s and the t i m e periods seen in M U L . A P I N for the risings o f the stars are also approximations. A l s o noticeable is the inclusion of m a n y parallel m e t h o d s for intercalation, s o m e o f w h i c h h a v e conflicting sets of underlying astronomical p a r a m e t e r s (such as different dates for solstices etc as seen in the references to both M U L . A P F N a n d also Old B a b y l o n i a n m a t e r i a l ) . " This is further support for the idea that these s c h e m e s are primarily for divinatory uses rather than strictly astronomical o n e s . T h e inclusion of m a n y different techniques is an advantage of the diviner for a n u m b e r o f r e a s o n s . Firstly their variety will allow him to select o n e appropriate for w h e r e he is in the year. Secondly, the diviner will have several intercalafion p r o c e d u r e s at his disposal, s o m e of which m a y give different results with respect to the need to intercalate or not, and the a m o u n t he needs to intercalate. Such flexibility is crucial in e m p h a s i z i n g that it is not the astronomical techniques in t h e m s e l v e s that are important, but rather their ability in allowing the diviner to readjust the date should he need t o . T h e r e is m u c h e v i d e n c e to suggest that the forecasts of o m e n s cast with respect to similar p h e n o m e n a changed (sometimes substantially) w h e n one readjusts the date. T h i s is especially true with astral phenomena.^'* T h e connection between the time o f o c c u r r e n c e and forecast of o m e n is readily apparent in E A E . For e x a m p l e , in lunar eclipse p h e n o m e n a , the date (the variables b e i n g a month, a day, one of the three w a t c h e s etc) is the d o m i n a n t parameter for organizing and c a t a l o g u i n g the o m e n s and directly impacts the forecast of the result. If o n e selects identical astronomical o c c u r r e n c e s on t w o c o r r e s p o n d i n g dates, there is (almost always) a large variation in the a p o d o s e s . For e x a m p l e , c o m p a r i n g lunar eclipse p h e n o m e n a in E A E 22,^^ one finds that if an eclipse occurs on the 14"' day of the first m o n t h

^'

See also BROWN ( 2 0 0 0 ) , pp. 1 2 1 - 1 2 2 .

" HUNGER and REINER ( 1 9 7 5 ) , p, 2 7 observe the discrepancies which occur when they try and derive the beginning of the year from the rule and get many different values. They comment: "These discrepancies are caused by the nature of the text, which tries to devise a simple scheme for the observation of the Moon and Pleiades throughout the year, and ignores some of the more complicated factors." " This point is pertinent for previous techniques that have been employed to estimate the date of this text. For example, BROWN ( 2 0 0 0 ) , pp. 1 2 0 - 1 2 1 , uses evidence from this astronomical section with the rising of Mul.AS.GAN coinciding with month 1 as an OB feature and also the possible connection between the OB equinoxes and months Xll and VI to date the text as "...substantially older than Oppenheim suggests". However, he ignores the material that is clearly from later sources, and as we see, because of this misses one of the distinctive features of this text, namely that the inclusion of methods based on seemingly contradictory astronomical parameters (such as the month difference between the OB and MUL.APIN ideal calendar) is done quite purposefully, for it offers the diviner more scope and flexibility in changing the date with methods that are backed up by the literature. For two examples of date-changing that are not astronomical see OPPENHEIM ( 1 9 7 4 ) n. 4 6 , who provides one with respect to gall bladders and the other with respect to snake encounters. "

ROCHBERG-HALTON ( 1 9 8 8 ) , p. 2 5 3 , 1 1 I, p. 2 5 4 III

1,1 III 1.

484

C . Williams

" . . . t h e r e will b e lamentation in t h e land o f the e n e m y and t h e land will d w i n d l e ; t h e king will d i e " , on 1 4 * d a y o f the second month, " . . . t h e r e will b e destruction a n d there will b e perpetual famine a n d t h e king will d i e " , on t h e 1 4 * d a y o f the third m o n t h , " . . . r a i n in the sky, flood in the source will c o m e ; A d a d will t r a m p l e ; t h e vast army o f A k k a d will d i s a p p e a r " and so o n . T h e c o n c e p t behind t h e m e n o l o g i e s a n d h e m e r o l o g i e s , that certain m o n t h s or days can b e favourable o r unfavourable, also attests the i m p o r t a n c e o f the date for divinatory o u t c o m e s . T h e practice o f diviners subverting t h e unfavourable c o n s e q u e n c e s o f an event by m a n i p u l a t i n g the date o f a planned activity is also d o c u m e n t e d . For e x a m p l e , in a report sent b y B a i a s i t o t h e Assyrian King, there is a plea for intercalation, lest t h e m o n t h pass unfavourably: " L e t them intercalate a m o n t h ; all t h e stars o f t h e s k y have fallen behind. A d a r must n o t pass unfavourably; let them intercalate!"^^ M o r e generally, t h e actions o f the diviners to surpass ill effects by ingenious m e t h o d s a r e well k n o w n . For e x a m p l e , in a letter from lssar-sumu-ere§ t o the king,^^ in w h i c h h e , recognizing from t h e h e m e r o l o g i e s that t h e 1 5 * d a y o f the month for a planned activity w a s unfavourable, b y p a s s e d this by r e c o m m e n d i n g that t h e activity b e carried o u t u n d e r t h e influence o f astral p h e n o m e n a that would effectively cancel out t h e ill effects: " . . . h o w e v e r , it is written as follows in t h e h e m e r o l o g i e s o f the m o n t h N i s a n : ' H e should n o t swear on t h e 1 5 * day, ( o r else) t h e G o d s will seize h i m . ' ( H e n c e ) they should en[ter] t h e treaty o n t h e 1 5 * d a y , at d [ a w n ] , ( b u t ) c o n c l u d e it only in the night o f the 16[th] day before the stars." In conclusion then, lines 5 7 - 7 1 o f the D i v i n e r ' s M a n u a l form a set o f t e c h n i q u e s which involve intercalary checks and p r o c e d u r e s . N o n e o f the m e t h o d s listed w e r e technically sophisticated e n o u g h t o represent accurately the behavior o f the celestial p h e n o m e n a they pertained t o , but their purpose w a s n o t t o c o o r d i n a t e o r rigorously systematize t h e calendar, b u t rather to provide diviners with m e t h o d s t o intercalate, should they, having exhausted all other m e t h o d s , need t o dispel t h e evil c o n s e q u e n c e s o f an ill-boding o m e n by altering the date on which the o m e n fell.

Acknowledgements M a n y thanks must g o t o John Britton, David Pingree a n d John Steele for their patient and helpful c o m m e n t s . T o them 1 a m very grateful. References B R O W N , David. 2 0 0 0 . Mesopotamian Styx Publications. —,

FERMOR,

John

and WALKER,

Planetary

Astronomy-Astrology.

Christopher.

1999. " T h e Water

Groningen: Clock

in

M e s o p o t a m i a " . Archiv fiir Orientforschung Al : 1 3 0 - 1 4 8 . H U N G E R , H e r m a n n . 1 9 9 2 . Astrological Reports to Assyrian Kings. Helsinki: Helsinki University Press. — and PINGREE, David. 1 9 8 9 . MUL.APIN, An Astronomical Compendium in Cuneiform ( A r c h i v flir Orientforschung Beiheft 2 4 ) . Horn: Ferdinand Bergern & Sôhne.

HUNGER ( 1 9 9 2 ) , p. 5 7 , lines 8 - 1 0 . "

P A R P O L A ( 1 9 9 3 ) , number 6 , p. 7 - 8 esp. lines r. 1 1 - 1 9 .

Signs from the Sicy. Signs from the Earth: The Diviner's Manual Revisited

— and PINGREE, David. 1999. Astral Sciences

in Mesopotamia.

485

Leiden: Brill.

— and REINER, Erica. 1975. " A S c h e m e for Intercalary M o n t h s from B a b y l o n i a " . Wiener Zeitschrift fiir die kunde des Morgenlandes 67: 2 1 - 2 8 . HOROWITZ, W a y n e . 1998. Mesopotamian Cosmic Geography. Indiana: E i s e n b r a u n s . LIVINGSTONE, Alasdair. 1993. " T h e Case of the H e m e r o l o g i e s : Official Cult, Learned F o r m u l a t i o n and P o p u l a r Practice". In: Eiko M A T S U S H I M A (ed.). Official Cult and Popular Religion in the Ancient Near East: 9 7 - 1 1 3 . Heidelberg: C. Winter. — 1999. " T h e M a g i c of T i m e " . In: Tzvi A B U S C H and Karel V A N DER T O O R N (eds.), Mesopotamian Magic Textual, Historical, and Interpretative Perspectives: 1 3 1 - 1 3 7 . G r o n i n g e n : Styx Publications. O P P E N H E I M , A. Leo. 1974. " A Babylonian D i v i n e r ' s M a n u a l " . Journal of Near Eastern Studies 33: 1 9 7 - 2 2 0 . P A R P O L A , S i m o . 1993. Letters from Assyrian and Babylonian Scholars (State A r c h i v e s o f Assyria 10). Helsinki: Helsinki University Press. A L - R A W I , F a r o u k N . H . and G E O R G E , A n d r e w R . 1 9 9 1 - 1 9 9 1 . " E n i i m a A n u Enlil 14 and O t h e r Early Astronomical T a b l e s " . Archiv fiir Orientforschung 38/39: 52-73. REINER, Erica. 1995. Astral Magic in Babylonia. Philadelphia: T h e A m e r i c a n Philosophical Society. — a n d PINGREE, David. 1 9 8 1 . Babylonian Planetary Omens 2: Enûma Anu Enlil Tablet 50-51. M a l i b u : U n d e n a Publications. R O C H B E R G - H A L T O N , Francesca. 1988. Aspects of Babylonian Celestial Divination: The Lunar Eclipse Tablets of Eniima Anu Enlil (Archiv fur Orientforschung Beiheft 2 2 ) . Horn: Ferdinand Berger & Sohne. K U G L E R , F r a n z X. and S C H A U M B E R G E R , Johann. 1935. Sternkunde und Sterndienst in Babel. E r g a n z u n g e n 3. M u n s t e r in Westfalen: Verlag der Aschendorffschen Verlagbuchhandlung. V I R O L L E A U D , Charles. 1 9 0 5 - 1 9 1 2 , L'astrologie chaldéenne, le livre intitulé «Enuma (Anu) ''"Bel». Paris: Librairie Paul Geuthner.

Indices

Subject Index algorithm Almanacs anomaly lunar solar archons arcus visionis Aristarchos Astrolabes astrology aurora calendar Assyrian Babylonian Callipic civil

149ff 367, 405,416 1,36-37, 172,411,417 1 , 3 , 2 1 , 3 7 , 39, 4 3 , 5 3 , 411,414,417 223-226, 228 177, 182-183, 190, 469 295-296 23 173,206, 209ff,223ff, 477 42 Iff

24 51, 177-178 94, 295-296 28, 380-381, 386, 388390, 392, 397, 424, 459 Egyptian 195,206,379ff,459ff Hebrew 81-82, 84-86, 96-98, 100-104 Hindu 81 Julian 24, 30, 33, 177-178, 424 lunar 66. 79, 82-83, 95, 173, 176,214, 379ff,459ff Islamic 83, 463 schematic 23-23, 33, 52, 70, 474475, 477-478 chronology 79-81,86, 95-98, 101, 3 8 0 , 3 8 9 , 3 9 8 , 441, 459ff conjunction 1,3, 5, 16, 82-84,91, 95, 466 Copernicus 46, 93, 295 crescent visibility 83-84,381-383,448449, 461,463-464, 469, 479 cubits 110-111, 114, 117-118, 283ff, 299, 307, 309.

curriculum decans Dendera Diaries digits divination dodekaoros eclipse general lunar magnitudes omens possibility solar ecliptic education epact

ephemeris general lunar planetary equinox

exercises extinction Eye of Horus Eye of Ra false position festivals

311,424-428 330, 336-337, 344, 346ff, 356, 360, 361 224, 228, 429ff 109-110, 113, 115, 118, 287, 437 7, 178, 367, 3 6 9 , 3 7 1 , 405,411,421 110-112, 117-118 209ff, 447ff, 473ff 224 16.37. 40. 52-53,68. 94, 173. 295 7, 18,37-38, 46, 82, 9 1 , 2 0 6 , 3 5 8 , 408-409 36,38 214-215, 447-448, 483 36,215-216, 408ff 7,36, 8 2 , 9 1 , 3 5 8 , 408409 6, 17, 82, 175-176, 186-187, 2 0 3 , 2 2 6 325ff 22, 24, 28, 36, 43, 48, 53, 75, 78, 392, 395, 481 62, 405 36,381,387 381 21,23-24, 2 9 , 3 3 , 4 3 , 62, 66-69, 78, 95-96, 170,358, 474, 477 238, 246, 253, 356, 361 177, 180-184,381, 463, 467 193ff; 297ff 194.301 155,158 379ff, 475

488

Indices

findspots 237, 248, 326-329 finger 283 ff fractions 148, 297ff Goal Year method 8, 11-12, 14, 16-18 Goal Year periods 61, 177, 190, 407, 416 Goal Year Texts 177, 190, 367,370371,405 halo 449-450 height 109ff Hipparchus 2, 47, 49,51-53,91-96, 100-103,417 Horus eye fractions297ff House B 358,361 House F 326ff hypsoma 226, 231,287 intercalation 23-26,30-31,33-36, 52, 6 9 , 7 1 , 101, 176, 473,475-476,478-480, 482-484 Jupiter 2 1 , 2 6 , 6 1 , 170, 176, 199, 225-226, 228, 230-231 Kapo Kepler latitude lunar leitourgoi length lenticular tablets Letters line of sight longitude general lunar solar lunar six lunar three lunar theory lunar velocity Mars Mercury

metalanguage Metonic cycle

metrology

month

121,138-141 83 42, 53, 425-427, 478 224 114 253,330 10, 473 113 36-37, 39-40, 47, 407 1, 11,38, 168, 171, 425-427, 478 1-4, 2 6 , 3 3 , 167, 171 6ff, 358-359, 476 423,428 5, 79 1-3, 14,37, 4 5 , 4 9 61, 170, 176, 225,367, 370 61, 170, 194-195, 197199, 202-205, 225-226. 231,367,370 124, 130ff 26, 3 0 - 3 1 , 3 3 . 3 6 , 3 9 40, 43, 45, 48, 50, 5254, 74, 85, 295, 428 238-240, 244, 246, 249, 260, 297, 303304, 325

anomalistic dracontic "full" "hollow"

2, 8, 45, 82, 92, 408 8, 14, 82, 408-409 5ff, 384, 475 5ff, 3 1 , 7 4 , 382-384, 475 length 5ff, 24, 28-29, 36, 5152, 54, 80, 382-384, 459ff schematic 23-24 sidereal 26, 28-29, 37, 40, 4850, 82 synodic 1,8, 1 4 , 2 8 - 2 9 , 3 7 , 4 7 , 49-50, 54, 79, 8Iff, 167, 173,295-296, 408-409,417, 459 nodes 38, 408-409 Normal Stars 368, 423 NS Almanacs 367-369, 371,405 Observational texts367, 370 omens 206, 209ff, 447ff, 477. 482 opposition 1, 10, 16-18, 82 precession 52, 103 problem texts I21ff, 147ff problems 241,250, 253,259ff, 353,357 planetary periods 225, 231 planetary theory 42, 53, 168, 176-177 Ptolemy 2, 47, 49,51-54, 75, 91-101, 167, 177, 182, 228-230, 295 rectangular tablets 253 Reports 10, 476 Saros cycle 7-8, 11, 14, 16, 18, 36, 295, 405ff Saturn 175ff, 199, 225 schools 325ff seasons 21,29, 4 3 , 5 3 , 6 6 , 68, 85 sexagesimal system 7, 91, 240, 243-244, 248, 295, 358 Sirius 21,23-24, 26,28-30, 43, 62-63, 69-70, 189, 194, 202, 206, 286291, 368, 441, 443, 445 solar velocity 1, 3, 411. 417 solstice 21-24, 29-30, 36. 43, 51-52, 62, 66-71,7475, 78, 94-96, 170, 358, 474, 483 Sothic dates 380, 460 square tablets 253ff, 358 star clock 429ff

Indices statistics statues step function synodic arc System A (lunar)

461-462 I09ff 2 3 7 , 3 9 , 47, 49, 172 i-3, 36-43, 45-47, 4953, 90, 176, 405,411, 417 column O 2, 36-38, 40, 42, 45, 50, 52-53, 405,411, 417-418 column B I column F 37 column G 1-2, 42, 45, 53,418 column J 1-2 column A 37 System A (planets) 2 System B (lunar) 1-4, 36-37, 42, 45-51, 53,90-91, 167-173, 176, 295-295,417 column T 173 column A 2-3,47, 168-169, 171173 column B 168, 172-173 column AH' 45 column AT' 45 column F 2, 46, 52, 172 column G 1-4, 45-48, 90-91, 167, 170, 172-173 column H 1-4, 171-172-173 column J 1-4, 45, 167, 172-173 column K 173

489

System B' (lunar) 49-51,53 column A 49 syzygy 1-2, 1 6 , 3 6 , 3 7 , 8 2 , 9 1 , 168, 173,408-409,411 table texts 148, 325 tithi 21-22, 28, 3 0 , 3 6 , 48, 62-63, 66, 74, 81 Uruk Scheme 22,30, 43-44,51,53, 69-70, 74-75, 78 Venus 61, 176, 189, 193ff, 225-226, 228, 230-231 vocabulary 124ff year anomalistic 1,47 length 21,26, 2 8 , 3 1 , 4 1 - 4 2 , 49,51-54, 74, 80, 85, 94-96 sidereal 1,30, 47-49, 52-54, 171 tropical 1,30, 48,51-52, 54, 295-296 ziqpu stars 24 zig-zag function 2-4, 7, 45, 90, 92, 171172, 4 0 6 , 4 1 1 , 4 1 4 416,418, 480 zodiac 36, 39, 52, 82, 224226, 228, 358

Text Index A 30211 A 30279 A 33446 ACT 119 ACT 122a ACT 123 ACT 123a ACT 123aa ACT 199 ACT 202 ACT 210 Almagest AO 6578 AO 6770 AO 8862 AO 8865 ARAK 455 Astrolabe B

see3N-T316 see3N-T611 see 3 N-T 914.x 173 49 49 49 49 see U107+124 407 50-53 4 6 , 5 1 , 9 1 - 9 5 , 97, 101, 177, 182, 295 24 130, 136 130-131, 140 330 see SAA8 455 477

Atyp. K see BM36722(+)40082 BCM 1845-1982.2 48 BE 13918 24 BM 13901 133 BM 15285 133 BM 17175+17284 475, 478 BM 26185 450 BM 32247 367-369, 373-376 BM 32373 450-451 BM 35339+... 367, 370-371, 377-378 BM 36311 37 BM 36599 36 BM 36712 28-31 BM 36722(+)40082 11, 14 BM 36731 21-22,29-30, 49, 6269, 74-75 BM 36737 36 BM 36747+37018 13 BM 36782 13

490

BM 36822 36, 39-42 BM 36828+... 448 BM 37377 449 BM 38160 450 BM 38289+38762 451 450 BM 38295 BM 38369+... 24 BM 40094 40 BM 41004 176 BM 42282+42294 12, 16 BM 45659+45685 269 21,26-29, 59-61 BM 45728 BM 45861 406ff BM 45894 451-452 see BM 35339+... BM 46235+... BM 76738+76813 178-179, 185 BM 85196 264 BM 85200+... 125 129 BM 96957+... BM 99072 451 BM 15285 357 Book of Nut 429ff CBM 12648 137 CBS 154+921 131 355 CBS 3551 CBS 19761 133 CG 25367 148 148 CG 25368 Codex Askewkanus 223 CT 44, 49 61 CT51, 143 449-450 De divinatione 210-211 Diviner's Manual 473 ff D T 74 451 D T 104 451 Enûma Anu Enlil 13,23, 176,212,215, 414, 447ff, 473, 475, 477-478, 480, 483 264 Erm 15073 HS232 355 lllahun fragments 148, 155-156, 298,319 IM 31210 265 IM44134 351 IM 52301 128 IM 52685+52304 132 132 IM 52916 IM 54472 263 IM 55357 136-137 IM 58436 see 3 N - T 342 IM 58446 + 58447 see 3 N-T 362+366 see 3 N - T 594 IM 58573 IM 121613 135 I.NAM.GIS.HUR.AN.KI.A 23 Inbu bel Arhi 480-481

Indices

Ivory Prism K.75 K.200+ K.1348+Rm591 K.2133+2247 K.2134+ K.2140 K.2223 K.2267+ K.2311+3624 K.2398 K.2887 K.2890 K.3103 K.3114+7155 K.3123 K.3135 K.3556 K.3559+3777 K.3625 K.3741b K.3773 K.4027 K.4336 K.4768 K.5281 K.6021 K.6276 K.6291 K.6296 K.6396 K.6505 K.6833+7948 K.6982 K.6984+ K.7062 K.7I92 K.7661+11109 K.7995 K.8145 K.8493 K.9505 K.9574 K.9794 K.9966 K.10102 K. 10608 K . 10731 K. 10964 K.I 1094+11252 K. 12282 K . 12429 K. 12654 K . 13436

24 451 449-400 447-449, 45 451 450 449 451 448, 451 447, 450 449 449 450 451 451 480 451 451 451 451 448, 451 451-452 448-449 450 450 451 449 450 450 451 451 448 451 452 450 451 451 451 452 448 451 451 450 24 451-452 451 450 449 451 449, 451 451 451 452 452

Indices

K. 13786 K. 15458 KRI II 345 KRI V 144 KRI V 145 KRI V 176 KRI V 178 KRI V 179 KRI V 180 KRI V 182 KRI V 201 KRI V 285 KRI VI 283 KRI VI 702 KRI VI 703 KRI VI 709 KRI VI 735 KRI VII 416 LABS 160 LBAT 1413 LBAT 1414 LBAT 1415+... LBAT 1419 LBAT 1420 LBAT 1515 LBAT 1527 LBAT 1529 LTTN 1 LTTN 2 LTTN 4 LTTN 13 Ll IN 14 MDP27, 61 MLC 651 MUL.APIN

449 451 397 391 391 394 391 391 393 393 386 395 396 396 396 396 394 396 21,26,61 39,412,416 409,412,416 409 409 409 407 448-449 451 450-451 451 449 451 452 330 259-260, 273 5, 7, 12-13,23-26,28, 52, 69, 176, 213, 473ff. 483 N 3971 355 NBC 8082 260-261,274 NCBT 1913 260 Ost. Semnut 153 148 Ost. Turin 57010 148 pBerlin 3038 298,319 pBerlin 6619 148-149, 155, 166 pBerlin 10500 299, 319 pBerlin 23252+... 310 pBoulaq 18 298 pBM 274 285 pBM 10250 148 pBM 10335 296 pCarIsberg 1 387, 429, 434, 442, 445 pCarlsberg la 429, 434, 442, 444-445 pCarlsberg 9 381,388, 390, 469

pColker pEbers pHarris pLeiden I 384 pLouvre E 3226 pMoscow E 4676

49

167, 169, 173 298 298 285 298,319 148-149, 155-157, 165, 301 pOxy ined. 23 3B 1/0 (l-4)c 168-169 pRhind (pBM 10057-58) 148ff,298, 300,303,319 pRoUin 298,319 pTebtunis Tait 8 285 pUC32134 156-157, 166 Pistis Sophia 223ff PSI 1491 169 Pyramid Texts 193-194, 205 Rm211 421 Rm2 39 451 Rm2 302 451 SAA8 57 451 SAA8 105 451 SAA8 106 451 SAA8 252 451 SAA8 257 451 SAA8 362 451 SAA8 265 451 SAA8 389 451 SAA8 391 451 SAA8 346 448 SAA8 455 61 SAA8 494 450 SAA8 498 451 SAA8 502 448 SAA8 505 451 SBTU IV, 171 178-179, 187-188 Sm 182 451 Sm 751 448 Sm955 447-449 Sm 1148 451 Sm 1931 451 Str 362 265 Str 366 129, 137 Str 367 129 Str 368 129 STT 339 449, 451 STT 340 24 TMS V 131 TMS VI 131 TMSX 265 TSS 50 238-239, 241-246, 250 TSS 81 238, 241,246, 248 TS§ 554 238 TSS 671 238, 241,243-246,250 TU 11 5ff

492

Indices

137, 2 6 5

T U 17

451

YBC 4673

U107+124

2 1 - 2 2 , 4 3 , 4 8 , 50, 52-

YBC 6504

129,133

53, 76-78

YBC 7164

264, 407

Urk. Urk. Urk. Urk. Urk. Urk. Urk. Urk.

IV 157

396

YBC 7234

253-254, 267

IV 1 7 7

394

YBC 7235

256-257, 269

IV 178

396

Y B C 7273

263, 277-278

IV 741

395

YBC 7284

260

IV 770

393

YBC 7289

357

IV 8 2 4

391

YBC 7290

261

IV 8 3 6

389

Y B C 7291

259, 272

IV

883-8

387

YBC 7294

262-263, 277

121

136, 140, 2 6 5

YBC 7326

263-264, 278-279

UETV,

U E T V, 8 5 8

140

YBC 7345

358

U E T V, 8 5 9

136, 140

Y B C 7353

257, 270

U E T V, 864

133, 136, 140

YBC 7354

254, 267

U E T VI/2 211

356

YBC 7355

255, 268

U E T VI/2 2 2 2

356

YBC 7356

258, 271

U E T VI/2 2 3 6

357

YBC 7358

256, 269

U E T VI/2 2 3 7

357

YBC 7359

261-262, 275-276

U E T VI/2 295

356

YBC 7997

130

U E T VI/2 321

356

YBC 9856

130, 2 6 5 , 2 8 0 - 2 8 1

U E T VI/2 3 8 7

355

YBC 9874

130, 2 6 4 , 2 7 9 - 2 8 0

UM

29-15-192

260

Y B C 10722

261,275

U M 55-21-289

262,276

UM 55-21-360

see see see see

UMB

260

2 N-T 30

VAT 6 7 2

137

2 N-T 115

358

VAT 4596

421

2 N-T 116

260

V A T 6505

353

2 N-T 472

260

VAT 6597

265

2 N-T 496

358-359

VAT 7528

263

2 N-T 500

358-359

V A T 7531

136-137

3 N - T 261

338

VAT 7532

129

3 N-T 316

335-336

VAT 7535

129

3 N-T 342

337

VAT 7813

447

3 N-T 362+366

353,359

VAT 8522

263

3 N-T 594

336

V A T 12431

see W F 7 3 see W F 9 8 see W F 9 9

3 N-T 604

359

3 N-T 605

355, 360

3 N-T 608

338

3 N-T 611

354,358

U M 55-21-356 U M 55-21-357

VAT

16/2

12455

V A T 12537 Vat. gr. 3 8 1 fol

3 N-T 261

Y B C 10801

3 N-T 604

Y B C 10802

260, 274

3 N-T 605

Y B C 11125

257-258, 270-271

YBC

255,268

3 N-T 360

163^296

11127

260, 357

2 1 - 2 2 , 3 0 , 3 6 , 62, 70-

3 N - T 914.x

337

75

8 0 - 7 - 1 9 , 158

451

W 23291

136

81-2-4,333

451

WF73

247

81-7-27, 96

449

WF 98

238, 247-249

8 2 - 3 - 2 3 , 1 6 + ...

447-449

WF 99

247-248

89-4-26, 25

451

YBC 4662

136-137, 140

1902-5-10, 23

451

W 22801+22805

YBC 4663

136, 140

YBC 4669

137, 2 6 3 , 2 6 5

Indices

493

W o r d Index Akkadian

and

a.na a.na.àm a.râ a.§à AB àd adannu adâru

AGA agû a-ku-ku-tum alaku alâkum

-àm amârum

AMAS AN.NA ana

AFIN arâqu arum

asas asri sipki asm assum awâtum

BA BA.A ba.se.e ba.si ba.sig ba^âlu bâbu bâmtum

bân banûm

BAR BÀR BAR-NUM barig bariga bâssu basûm

BE beru beriim

dah dakâsum dannino

DARA4

Sumerian 136 136, 138, 140 124-126, 133, 135,338 124, 132, 135, 138 22 39 448 447-448 448, 450 448, 450 421 63 126 140 133 450 450 212 22 449 125-126 134 481 481 131, 136, 138 219 129 129 135 128 124, 128, 134-135, 138 449 449 129-130 240, 333 126 22 473 407 333 240 129 128, 135 219 24 125 124, 135, 139 139 481 447

da^âmu da^ummatu dikistum

DIR DIS

du DUé

duy.du? dug en.dè edëdu ekëlu eli... elûm

watârum

en.nam eqlum

e§e3 esëpum

gâ.gâ gaba gâl gâl.bi galàtu

GAN GÀN gar gar.gar GAR gaz gen GEfe GENNA ge§ gesù gi.na GIN gin7(.nam) GÎR-TAB gi§-as4-lum gis-§urumx-ma GIS-GIDRI Glâ.HUR GU4 GUB gU7 GUR gur-mah harâsum hasâbum

HÉ.NUN

447, 449 450 139 22, 24, 30,71 219, 239 125 22 127, 135 128, 135 137 451 447-449 135 126, 133, 137 136 132 333 126, 135, 139 134-135 127 127, 129 127,339 449 22 134 133, 135-136, 138 124, 135, 139 333 129, 139 125 15 61 239 239 133 6, 63 131, 136 185 332 332 450 450 22 71 126 6 239-240 125, 135 125 450

494

Indices

hepûm

129, 1 3 8

KÙS

126, 188, 4 2 4

i.gU7.gU7

126-127, 135, 1 3 9 140

LAGAB lai

127, 132, 135, 1 3 8 125

135

lamû

449

124, 127-129, 132,

lapâtum

133, 1 3 6

134-135, 137-138 140

laqûm Ubbànu

132 449

'gi

124, 128-129, 135, 137, 3 3 9

263

igi.bi

135

lillidu lui -ma

igi.dù

137

igi.dug

139

igi.gub IGI

124, 135

mahârum

127

23, 29, 43, 62, 6 9

mahïru

261

ì.ib.tag4 ib.si ib.sig ib.tag4

133 131, 133, 136, 138, 140 449

ma-gal

IGI.LÀ

476

mala

igibûm

135

igigubbûm igûm

135 129, 1 3 5

mala mano

il

126, 135

illiakkum IKU

133 134

131, 136, 1 3 9 masi

marû masqu matûm ME

136, 1 3 9 7, 3 3 3 134 480 125 15,43 450

IM.DIRI

449

MÈ

in.ne.sè

136

mehrum

127

inanna

131, 136, 139 131, 136, 1 3 9

mësaru mïnâm

450

minîtu mïnûm

475

IZI

265 22

izzuzu KÀ

71 449

mislum mithartum

kalakku kamàrum

187, 1 8 9

mithurti NA"

inuma iskarum

kamurrûm kasû

124-125, 135, 138 124

131 131-132, 136, 138, 140 129, 1 3 5 127-128, 132,1 3 5 477 15

nadûm

136

449

nadânum

kasâdu

451

nakâru

129, 133, 135-137, 1 4 0 449

kepû

451

namaru

ki mas

namâru

63 449

NAM.BÙR.BI namburbu

481 481 138

Kl M I N

131, 136, 139, 1 4 0 407

kidû

449

kïma

131, 136, 139

nam-lù.ulù

kimrâtum kiyâ KIN

125, 135

nakasum

125

131-132, 136, 1 3 9 22

nasâhum nasûm

125, 134-135, 1 3 8 126, 137 451

KIN-A

22

nenmudu

KIN-DIR KIN-2-KAM kïnum

71 22

KÙ.BABBAR

450 450

NIG.HA.LAM.MA 450 NÎG.SI.SÀ 450 nigin 136 nigin(.na) 131, 136 NIGIN 125, 127, 1 3 5 NIGIN 449 nim 137 NIM 29, 6 3 . 126, 135 ninda 132,333

KÙ.Gl kud kullum kumurrûm KUR KUR.NI.GI4.A KÛR.KÙR

133

125 126 135 15,422 450 449

nitkupum

127

NU

473

Indices pa-ni

pad

pat patânim puhrum pûtum qaqqari qarnu qi-ti-sù

RÀ rabu re7

rësica lilâl rësum

sag sag an.na sag ki.ta SAGSU SAG-US sahârum sarrum

SI sig SIG SIG5 SIG7 sila sila4-gub SUg sum supûru sënu sêtu SA-nu sanû sabïhu saga I tu saggastu salummatu saJiânum saqu saqû saqûm saqummatu

sar §aru SE siddum sitqultu sipku sipku same

SU SÛ

129 138, 140 133 1.28, 135 219 132 24 451 475 125 63 125 133 132,135 124, 132-135, 138 134-135 134-135 450 61 131, 136, 139 133 451 134 22 473 449 239-240 263 125 129, 133, 135, 140 450 263 448-449 449 449 449 450 450 450 129, 133-136 63 480 126 450 239 239 22, 473 132-133, 135 479 481 481 22 15, 23,29, 43,62-63, 69

495 âu.ri.a §u-si summa sutakûlum sutâkulum sutakûlum sutamhurum sutbûm

tab tabâlum takïltum tamartu tammar tarâsu tarbàsu tardïtu târum

TÙR u

ù.ub U4 U4.da U4.NÀ.ÀM U4.SAKAR Ug-udu-hi-a UD UD.DA ugu ... dirig UL.GAR UL.UL ûm bubbuli umattu unnutu

UR.UR URUDU uskâru usurtu

us us 2.kam US uzzu wasâbum

ZABAR ZAG.MU zagmukku

ZALÂG zi zi.zi zi.zi.i zibânîtu ziqpu

ziz

129-130, 135 333 130, 135. 138, 212, 219 126 127 126-127, 135 127, 135 125 126, 135, 138 125 126 476 133, 137-140 451 449 407,415 131, 136, 138 449 124-125. 239 140 22 135, 136 448 449 263 22, 24 448-449 135 124,135 127 448 28, 450 449 127, 135 450 449 450 124, 132, 134-135, 138 134 7, 22-23, 92, 187, 407 450 124, 135, 138 450 457 475 62, 449 125, 128, 134-135, 138 135 134-135 185 24 22

496

Indices

Egyptian ibd ipt ipds irt iti-w

^h^siw 'hpt wigy w'^rt w^'rt hit wSiti wSUi bkiti wdit bik hki bkiti prt Mn phwj dit psdntyw

mst mspr ri rmn

hry

rdj

383, 384 301,308 432, 434-435 308 290 434 310 434 154 384 391 391 437 434, 437 435 435 297, 300, 308 442 384 434-435 386 432, 434, 437, 442 379, 380, 383-386, 388-390, 392, 395, 397, 459, 463 429 383 302 434 308

Hebrew heleq mol ad rega'

79ff 85-86 81,85

hnw hrw hb hit dit hbw nw ti hqit hw i^h hnm hry hpd n knmt siw s^q ntr sbSsn spdt sd (ipds) sdm.hr=f Stw knmt

tpt tpy-^ spdt tmit tmit hrt hrt tmit hrt ts ''rq dpit) dwit d^mw mi''

302 389 432, 434 385 297, 301-302 386 383 432, 434 388 391 432, 434-435 434, 436 432, 434-435 151 432, 434 113-115 434 310 308 429, 432, 439, 441442, 444-445 436 437 435,437 432, 434-435 434 305 386 308

Ugarit-Verlag

Munster

Ricarda-Huch-StraBe 6, D-48161 Miinster (www.ugarit-verlag.de) Lieferbare Bande der Serien AOAT, AVO, ALASP(M), FARG. Eikon und ELO:

Alter Orient und Altes Testament

(AOAT)

HERAUSGEBER: Manfried DIETRICH - Oswald LORETZ 43

NILS P. HEEBEL, Babylonisch-assyrische

Diagnosti/c. 2 0 0 0 ( I S B N 3 - 9 2 7 1 2 0 - 8 6 - 3 ) , XII + 471 S . + 2

ABB., E 98,17. 44

RYKLE BORGER, Zeichenlexilcon. 2 0 0 3 ( I S B N 3 - 9 2 7 1 2 0 - 8 2 - 0 ) (I.V.)

245

FRANCESCO POMPONIO - PAOLO XELLA, Les dieux d'Ebla. Étude analytique l'époque

des arcliives

royales du Ille millénaire.

des divinités éblaïtes

à

1997 ( I S B N 3 - 9 2 7 1 2 0 - 4 6 - 4 ) , VII + 5 5 1 S., E

59,31. 246

ANNETTE ZGOLL, Der Recittsfall der En-hedu-Ana im Lied nin-me-sara,

1997 ( I S B N 3 - 9 2 7 1 2 0 - 5 0 - 2 ) ,

X I I + 6 3 2 S., G 6 8 , 5 1 . 248

Religion und Gesellschaft.

Studien zu ihrer Wechselbeziehung

Orients. Veròffentlichungen

in den Kulturen des Antiiœn

des Arbeitsicreises zur Erforschung der Religions- und

des Antilcen Vorderen Orients (AZERKAVO),

Vorderen

Kulturgeschichîe

Band l. 1997 ( I S B N 3 - 9 2 7 1 2 0 - 5 4 - 5 ) , VIII + 2 2 0 S.,

G 43,97. 249

KARIN REITER, Die Metalle

im Alten Orient unter besonderer

Beriicksichtigung

altbabylonischer

Quellen. 1997 ( I S B N 3 - 9 2 7 1 2 0 - 4 9 - 9 ) , XLVII + 4 7 1 + 160 S . + 1 TAF., G 7 2 , 6 0 . 250

MANFRIED DIETRICH - INGO KOTTSIEPER, HRSG., "Und Mose schrieb dieses Lied auf. Alten Testament und zum Alten Orient. Festschrift

Oswald Loretz.

Studien zum

1998 ( I S B N 3 - 9 2 7 1 2 0 - 6 0 - X ) ,

XVIII + 9 5 5 S., G 112,48. 251

THOMAS R. KÂMMERER, Èimâ millca. Indulction und Reception der mittelbabylonischen Ugarit, Emâr und Tell el-'Amârna.

252

Dichtung von

1998 ( I S B N 3 - 9 2 7 1 2 0 - 4 7 - 2 ) , X X I + 3 6 0 S., G 6 0 , 3 3 .

JOACHIM MARZAHN - HANS NEUMANN, HRSG., Assyriologica OELSNER anlafilich seines 65. Geburtstages

et Semitica.

Festschrift fiir

Joachim

am 18. Februar 1997. 2 0 0 0 ( I S B N 3 - 9 2 7 1 2 0 - 6 2 - 6 ) , XII

+ 6 3 5 S. + ABB., G 107,88. 253

MANFRIED DIETRICH - OSWALD LORETZ, HRSG., dubsar anta-men. Studien zur Altorientalistilc.

Fest-

schrift fur W.H.Ph. Romer. 1998 ( I S B N 3 - 9 2 7 1 2 0 - 6 3 - 4 ) , XVIII + 5 1 2 S., G 72,60. 254

MICHAEL JURSA, Der Tempelzehnt

in Babylonien vom siebenten bis zum dritten Jahrhundert

v.Chr.

1998 ( I S B N 3 - 9 2 7 1 2 0 - 5 9 - 6 ) , VIII + 146 S., G 4 1 , 9 3 . 255

THOMAS R . KAMMERER - DIRK SCHWIDERSKI, Deutsch-Aklaidisches

Worterbuch.

1998 ( I S B N 3 -

9 2 7 1 2 0 - 6 6 - 9 ) , XVIII + 5 8 9 S., G 79.76. 256

HANSPETER SCHAUDIG, Die Inschriften Nabonids von Babylon und Kyros' des Groften. 2 0 0 1 ( I S B N 3-927120-75-8), XLII + 7 6 6 S.. G 1 0 3 , - .

257

THOMAS RICHTER, Untersuchungen zu den lolcalen Panthea Siid- und Mittelbabyloniens

in altbabylo-

nischer Zeit. 1999 ( I S B N 3 - 9 2 7 1 2 0 - 6 4 - 2 ) , XXII + 5 1 8 S., G 85,39. 258

SALLY A.L. BUTLER, Mesopotamian

Conceptions

of Dreams and Dream Rituals.

1998 ( I S B N 3 -

9 2 7 1 2 0 - 6 5 - 0 ) , X X X I X + 4 7 4 S. + 2 0 PL., G 75,67. 259

RALF ROTHENBUSCH, Die kasuistische Rechtssammlung im Licht altorientalischer

260

TAMAR ZEWI, A Syntactical

Parallelen.

im Bundesbuch und ihr literarischer

Kontext

2 0 0 0 ( I S B N 3 - 9 2 7 1 2 0 - 6 7 - 7 ) , IV + 6 8 1 S., G 65,10.

Study of Verbal Forms Affixed by -n(n) Endings in Classical

Biblical Hebrew, El-Amarna Aldcadian and Ugaritic.

Arabic.

1999 ( I S B N 3 - 9 2 7 1 2 0 - 7 1 - 5 ) , VI + 2 1 1 S., G

48,06. 261

HANS-GUNTER BUCHHOLZ, Ugarit, Zypem

und Agais - Kulturbeziehungen

im zweiten

Jahrtausend

v.Chr. 1999 ( I S B N 3 - 9 2 7 1 2 0 - 3 8 - 3 ) , XIII + 8 1 2 S., 116 TAFELN, G 109,42. 262

WILLEM H.PH. RÓMER, Die Sumerologie. EinfUhrung in die Forschung und Bibliographie

in Auswahl

(ZWEITE, ENVEITERTE AUFLAGE). 1999 ( I S B N 3 - 9 2 7 1 2 0 - 7 2 - 3 ) , XII + 2 5 0 S., G 61,36. 263

ROBERT ROLLINGER, Friihformen historischen Denkens. Geschichtsdenken, Ideologie und Propaganda im alten Mesopotamien am Ubergang von der Ur-lll zur Isin-Larsa Zeit (ISBN 3-927120-76-6)(I.V.)

Recommend Documents

Flight and Freedom in the Ancient Near East (Culture and History of the Ancient Near East)

CULTURE AND HISTORY OF THE ANCIENT NEAR EAST FLIGHT AND FREEDOM IN THE ANCIENT NEAR EAS1 EDITED BY B. HALPERN, M. H. ...

Figurative Language in the Ancient Near East

SCHOOL OF ORIENTAL AND AFRICAN STUDIES FIGURATIVE LANGUAGE IN THE ANCIENT NEAR EAST The Burney Relief Figurative La...

Slavery in the Ancient Near East

SLAVERY IN THE ANCIENT NEAR EAST The earliest representation of war captives. A drawing made from the impression of a...

A Companion to the Ancient Near East

A COMPANION TO THE ANCIENT NEAR EAST BLACKWELL COMPANIONS TO THE ANCIENT WORLD This series provides sophisticated and...

The Ancient Near East: A History

THEANCIENTNEAR EAST A Histoy Second Edition William W. Hall0 William Kelly Simpson Yale University I I Harcourt Brace...

A Companion to the Ancient Near East

A COMPANION TO THE ANCIENT NEAR EAST BLACKWELL COMPANIONS TO THE ANCIENT WORLD This series provides sophisticated and...

The Ancient Near East: A History

Prophets and Prophecy in the Ancient Near East

Writings from the Ancient World Theodore J. Lewis, General Editor Associate Editors Billie Jean Collins Jerrold S. Coop...

Prophets and Prophecy in the Ancient Near East

Writings from the Ancient World Theodore J. Lewis, General Editor Associate Editors Billie Jean Collins Jerrold S. Coop...

Ritual, Performance, and Politics in the Ancient Near East

RITUAL, PERFORMANCE, AND POLITICS IN THE ANCIENT NEAR EAST In this book, Lauren Ristvet rethinks narratives of state fo...