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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 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*^ 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-
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;34,35n
oo (N
>o v-i >r> > >
> >
>
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
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r*-
g s
il ilia
O SI G
III
HP
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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 > 5:
U
5 " r-I r-' ? >= £3 > > > 5 > >M
60 N
m — ON
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>
> = >
\0 *n "«t m > a
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=
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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
=
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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
+
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— <s — X X
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>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
> >>
— > ^ ?
^
u
3 OD s
^
2
— (N > 3
— Q 00 00 00 00 00
VO
f- 00 0\ o —
ON
a
— U-I
>< R
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=
^
00
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01) N C
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5
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> >>
>
= > = ;=
ON
00 r-
T)- m IN — o
(Nm-5tuivot^ooov lN 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
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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
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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
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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.
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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.
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( 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.
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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.
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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.
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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).
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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.
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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.
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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.
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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
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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 ) .
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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.
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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
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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
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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
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— 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.
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— ; 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.
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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.
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— 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.
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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:
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— 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
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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 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.
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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'
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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