Astronomical Cuneiform Texts: Babylonian Ephemerides of the Seleucid Period for the Motion of the Sun, the Moon, and the Planets (Sources in the History of Mathematics and Physical Sciences, 5) [1955 ed.] 9781461255093, 9781461255079, 1461255090

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Sources in the History of Mathematics and Physical Sciences

5

Editor

G.J. Toomer

Advisory Board

R.P. Boas P.J. Davis T. Hawkins M.J. Klein A.E. Shapiro D. Whiteside

ASTRONOMICAL CUNEIFORM TEXTS BABYLONIAN EPHEMERIDES OF THE SELEUCID PERIOD FOR THE MOTION OF THE SUN, THE MOON, AND THE PLANETS

I Edited by

0. NEUGEBAUER

Published with the Assistance of the

INSTITUTE FOR ADVANCED STUDY PRINCETON , NEW JERSEY

Springer Science+Business Media, LLC

0. Neugebauer Institute for Advanced Study Princeton, NJ o8540 U.S.A.

AMS Subject Classification: OIA17 Library of Congress Cataloging in Publication Data Main entry under title: Astronomical cuneiform texts. Reprint. Originally published: London: Published for the Institute for Advanced Study, Princeton, N.J., by Lund Humphries, 1955. "Published with the assistance of the Institute for Advanced Study, Princeton, New Jersey." Bibliography: v. 2, p. Includes index. Contents: v. 1. Introduction, the moon-v. 2. The planets, indices-v. 3· Plates. 1. Astronomy, Assyro-Babylonian. 2. Assyro-Babylonian language- Texts. I. Neugebauer, 0. (Otto), 1899. II. Institute for Advanced Study (Princeton, N.J.) QB19.A87 1983 528'.835 83-622 With 225 Illustrations.

© Springer Science+Business Media New York 1955 This book was originally published in 1955 by Lund Humphries, London, England, for the Institute for Advanced Study.

ISBN 978-1-4612-5509-3 ISBN 978-1-4612-5507-9 (eBook) DOI 10.1007/978-1-4612-5507-9

DEDICATED TO THE MEMORY OF FATHERS J. N. STRASSMAIER, S.J. (I 846-I 920) J. EPPING, S.J. (I835-I894)

F. X. KUGLER, S.J. (I862-I929)

PIONEERS IN THE INVESTIGATION OF BABYLONIAN ASTRONOMY

VOLUME I INTRODUCTI ON THE MOON

IX

PREFACE TO THE SPRINGER EDITION

When this collection of Babylonian astronomical texts was published in 1955 (a date omitted by mistake from the title page), it contained all texts of this type that I could lay my hands on. As was to be expected, the past 25 years provided more fragments, identified by A. Sachs and A. Aaboe in the British Museum and listed below. Also, some new joins could be made and some errors of mine corrected. Nevertheless, I think one still can consider the material of 1955 to be representative of what has been preserved of the mathematical astronomy of the Seleucid period. In the meantime, far more progress has been made in our understanding of Babylonian astronomy, mainly by the publications of Aaboe, Hamilton, Maeyama, Sachs, van der Waerden, and others. As an example, I mention here only the elucidation of the

purpose of column of the lunar ephemerides (by Aaboe) and the explanation of the method of computing the eclipse text ACT No. 6o (by Hamilton and Aaboe). Some of these advances I have tried to incorporate into my History of Ancient Mathematical Astronomy (1975), which should be used as a guide to the more recent literature. My sincerest thanks go to Springer-Verlag for making this work again available to students of ancient astronomy. The Institute for Advanced Study, which together with Brown University has supported my work for more than four decades, has graciously given its permission for this reprint.

Princeton November 1982

0.

NEUGEBAUER

BIBLIOGRAPHY Aaboe A Seleucid Table of Daily Solar(?) Positions. JC S'*' 18 (1964), 31-34. On a Babylonian Scheme for Solar Motion of the System A Variety. Centaurus I I (1966), 302f. Some Lunar Auxiliary Tables and Related Texts from the Late Babylonian Period. Danske Vid. Selsk., Mat.-fys. Med. 36 (12) (1968). A computed List of New Moons for 319 B.C. to 316 B.C. from Babylon: B.M. 40094. Danske Vid. Selsk., Mat.-fys. Med. 37 (3) (1969). Remarks on the Theoretical Treatment of Eclipses in Antiquity. JHAt 3 (1972), 105-II8. Lunar and Solar Velocities and the Length of Lunation Intervals in Babylonian Astronomy. Danske Vid. Selsk., Mat.-fys. Med. 38 (6) (1971). Scientific Astronomy in Antiquity. Phil. Trans. R. Soc. London, A 276 (I 974), 21-42. Aaboe and Norman T. Hamilton Contributions to the Study of Babylonian Lunar Theory. Danske Vid. Selsk., Mat.-fys. Med. 40 (6) (1979). • JCS: journal of Cuneiform Studies.

t JHA:

Aaboe and Janice A. Henderson The Babylonian Theory of Lunar Latitude and Eclipses According to System A. Arclzi·ves lnternationales d' Histoire des Sciences 25 ( 197 5), 181-222. Aaboe and Peter J. Huber A Text concerning Subdivision of the Synodic Motion of Venus from Babylon: BM 37151. Connecticut Acad. Arts and Sciences, Memoir 19 (1977), 1-4. Aaboe and A. Sachs Some Dateless Computed Lists of Longitudes of Characteristic Planetary Phenomena from the Late Babylonian Period. JCS 20 (1966), 1-33. Two Lunar Texts of the Achaemenid Period from Babylon. Centaurus 14 (1969), 1-22.

0. Neugebauer and A. Sachs Some Atypical Astronomical Cuneiform Texts. I. JCS 21 (1967), 183-218; II. JCS 22 (1969), 92-113.

journal for the History of Astronomy.

XI

PREFACE

This edition of Astronomical Cuneiform Texts is intended to furnish the basis for a chapter on Babylonian Mathematical Astronomy in a larger History of Ancient Astronomy. In the present work, however, no attempt has been made to arrive at general historical conclusions, though the introductions to volumes I and II provide the reader with the necessary background of Babylonian lunar and planetary theory. The publication of this work has been made possible by the generosity of the Institute for Advanced Study in Princeton, New Jersey. The underlying research was begun at the Mathematical Institute of the University of Copenhagen and continued at Brown University and during repeated stays in Princeton. It is only through the support and understanding which I met in these institutions that I have been able to carry out a program of so large a scale. It is with a feeling of sincere gratitude and indebtedness that I conclude these volumes.

It was my aim to reach completeness so far as the special texts under consideration are concerned. Accordingly, I have republished about fifty texts which were previously published by Kugler (1900, 1907), Thureau-Dangin (1922), and Schnabel (1924, 1927). About thirty of these texts have been substantially enlarged by joining new fragments to the already published parts or by adding unpublished columns or sections. The material presented here in its entirety amounts to about 300 tablets and fragments; one may estimate that the present edition contains about four or five times as much material as was known previously. About 170 texts concern the moon; the rest have to do with the five planets, Jupiter being better represented than all the other planets combined. About one-third of all texts come from Uruk; two-thirds, in all probability, from Babylon. In 1881 the key to the understanding of Babylonian mathematical astronomy was found by Father Epping, S.J., in British Museum tablets which had been identified as astronomical by Father Strassmaier, S.J. Around this time Strassmaier was copying many thousands of texts - tablet by tablet, fragment by fragment- which had been sent to the British Museum in the tens of

thousands. Whenever he ran across an astronomical text of a worthwhile size he recopied it for study by Epping and, after Epping's death, by Father Kugler, S.J. It was not until the 1920's that more astronomical texts from Paris and Berlin became available. When I began to work on the present edition in 1935, it was again Strassmaier's material that formed the basis. Strassmaier's notebooks, by that time, were in the custody of the Pontificio Istituto Biblico in Rome; from these notebooks, astronomical texts were extracted by Father Schaumberger, C.Ss.R., for the .continuation of Kugler's work. Father Schaumberger not only sent me the copies of relevant texts, but also drew my attention to unpublished astronomical texts from Uruk which were in Chicago at the Oriental Institute. Finally, with the kind help of the late H. Ehelolf, I obtained access to the texts in Berlin. The work on this material was practically completed in 1945. At that time, contact with Istanbul was reestablished. Dr. F. R. Kraus kindly sent me, from his excellent catalog of about 60,000 texts, a list of more than a hundred astronomical fragments from Uruk, and, subsequently, a microfilm of the texts themselves. Many of these fragments could be joined with one another or with tablets in Paris, Berlin and Chicago. The result was that the U ruk texts became about as uniform a group as the Babylon texts in the British Museum. This entailed almost a complete rewriting of my manuscript, a task which took about three years. In the meantime, it had become clear that Strassmaier's notebooks contained additional material which I had not yet seen. In 1949, on the recommendation of Father A. Deimel, S.J., all of Strassmaier's relevant notebooks were placed at my disposal through the courtesy of the Pontificio Istituto Biblico. Dr. A. Sachs went through some thousands of such copies and identified those which belong to my class of texts. The yield was about 100 new fragments, which, when reduced in number by joins, became 83 more or less complete texts. The photographing of the originals in the British Museum and the working out of the details required more than two years, again resulting in a rewriting of about half of the manuscript.

XII

Strassmaier's notebooks cover only texts with the inventory numbers between BM 32,000 and BM 36,000. He did, however, make notes about similar texts, numbered between BM 45,000 and 47,000, which had been quoted to him by Pinches. Thus it was clear that the astronomical archive had a much greater extent than the part explored by Strassmaier. This conclusion was confirmed in 1952. A travel grant by the Rockefeller Foundation enabled Dr. Sachs to work during the summer at the British Museum. There he was given access to about 1800 sheets of copies of astronomical texts, made by Pinches in the years preceding 1900. Many of these masterly copies duplicated texts which we knew through Strassmaier. But there were also many that were new and that substantially increased our knowledge. For the present edition, about 60 new fragments had to be incorporated, about half of which joined previously known texts. This process of successive approximation has left its traces on the present edition. Quite a few texts were slowly pieced together from many fragments scattered not only over the different collections of the same museum, but sometimes over two or three museums on different continents. Each new join required the recomputing of hundreds of numbers or changes in the numbering of lines, columns, sections, and texts. There are many texts which went through this process five or six times. In spite of all attempts to keep track of these continuous changes which went on, year after year, it is only too evident that many mistakes must have been made which I have been unable to eliminate. A serious student of these texts must not only be indulgent toward small inconveniences, e.g., in the counting of texts and plates, or with inconsistencies in transcnption or translation, but he also must be aware of the necessity of continually checking all possible ramifications of whatever statement he may doubt. Furthermore, the reader should have no illusions with respect to the completeness of the material. We know the Uruk archive only insofar as it has reached Istanbul and the collections of Berlin, Paris, and the United States. The Babylon archive is now available as far as it was explored by Strassmaier and Pinches, or, on the basis of the original inventory numbers, the material that came to the British Museum between 1876 and 1882. But we have no estimate, e.g., about the contents of the collections of the Iraq Museum and others, while the British Museum promises to produce still more texts as the recently begun process of systematic cataloging proceeds. Indeed it was this prospect

which induced me to publish this edition now, at a moment when we have reached the end of Strassmaier's and Pinches' material. Since the present edition has already occupied the main part of my time for research for a period of twenty years, it is clear that the possibility of doubling the source material would jeopardize the publication even of the limited section which is accessible now. A great debt of gratitude I owe to Mr. D. A. Jonah, Librarian of the Brown University Library, for many years of patience and helpfulness in all my bibliographical requests. And the final task of putting my manuscript into print has been performed by Lund Humphries in London with great skill and with understanding for my exacting requirements. I wish to express my thanks to the curators and keepers of the following collections for their cooperation and helpfulness; Berlin, Staatliche Museen; Chicago, The Oriental Institute of the University of Chicago; London, British Museum; New Haven, Yale Babylonian Collection of Yale University and Morgan Library Collection; New York, Columbia University Library and The Metropolitan Museum of Art; Paris, Musee du Louvre; Philadelphia, The University Museum of the University of Pennsylvania. British Museum tablets are published by courtesy of the Trustees of the British Museum. How much I owe to my friend and colleague, Dr. A. Sachs, for his help in all phases of the preparation of this work cannot be explained in a few sentences. For ten years he has read and reread the manuscript in all its stages. There is scarcely a page where his suggestions did not contribute to the clarity of formulation and correctness of detail. During the summer of 1952 and again since September 1953 there was scarcely a day when I did not ask him for collations of texts in the British Museum, for help with photographs and copies or readings. Without him I would never have been able to complete this work. And finally I should like to express my respect to the shades of the scribes of Eniima-Anu-Enlil, descendants of Ekur-zakir or of Sin-leqe-unninni, and of all the other scribes who computed and wrote the texts which are published here. By their untiring efforts they built the foundations for the understanding of the laws of nature which our generation is applying so successfully to the destruction of civilization. Yet they also provided hours of peace for those who attempted to decode their lines of thought two thousand years later. O.N.

X111

TABLE OF CONTENTS

PART I. INTRODUCTION

§1. The Texts. Description and Notation A Description of the Texts B The Present Edition § 2. Provenance and Dates A The Provenance of the Texts B The Dates of the Texts c The Seleucid Kings D Conclusions § 3. The Colophons A General Remarks B The Scribal Families c Text of the Colophons D Index of Personal Names and Place Names E Concordance § 4. Errors in the Texts

1 1 1 4 4 6 7 7 11 11

13 16 24 26 27

§ 5. The Mathematical Methods for the Computation of Ephemerides A General Concepts B Step Functions . c Linear Zigzag Functions § 6. Time Reckoning and Dating A Eras B The Continuous Calendar c Dating § 7. Astronomical Ideograms . A Months and Zodiacal Symbols B General Astronomical Ideograms . § 8. Metrological Units . A Angular Distances B Tithis

28 28 29 30 32 32 33 35 38 38 38 39 39 40

PART I I. EPHEMERIDES oF THE MooN INTRODUCTION

§1. The Lunar Theory in General . § 2. System A A Column T B Column I Monthly Variation II Daily Variation * c Column B I General Structure II Rules for Checking III Diophant IV Daily Motion B* . D Column C E Column E I ~E II Column E outside the Nodal Zone III Column E inside the Nodal Zone IV Dating by Diophant v Continuation of E over N Years VI Rules for Checking

41

44 44 44 44 45 45 45 46 46 47 47 47 47 48 49

so 52 54

VII 18-Year Cycle VIII Column E* IX Approximate Column E F Column 'P I Definition of 'P II Column 'P' I II Significance of 'P G Column F and related Columns I Column F . 1. Unabbreviated Parameters 2. Unabbreviated Parameters. Variant . 3. Abbreviated Parameters II Column F* H Column G I Definition of G II Definition of G III Computation of G from F IV Diophant for G v Checking Rules for G . A

A

54 54 55 55 55 55 57 58 58 58 58 58 58 59 59 60 61 61 61

XIV

Column J K Column C' L Column K M Columns M and P I The Hours of the Syzygies II The Dates of the Syzygies III The Computation of Column P IV Visibility Conditions v Summary N Eclipses I Introduction II The Columns T to C III The Columns E and 1Jf IV The Columns from F onwards B System 3. § Introduction . A Column T B The Columns A and B I Column A . 1. Unabbreviated Parameters 2. Abbreviated Parameters II Column B . III Column B* c The Columns C, D, and related Columns I Column C . II Column D' III Column D . D Column P' and related Columns I Column P" 1. Parameters expressed in degrees 2. Parameters expressed Ill eclipse magnitudes II Column L:llJf' 1. Unabbreviated Parameters 2. Abbreviated Parameters III Column P' E The Columns F, G, and related Columns I Column F . 1. Unabbreviated Parameters 2. Abbreviated Parameters II Column F' . III Column F* IV Column J: F* v Column G . F The Columns H and J I Column H . II Column J 1. Unabbreviated Parameters 2. Abbreviated Parameters 3. Diophant for J G The Columns K and L I Column K .

J

61 62 63 63 64 64 65 66 67 68 68 69 69 69 69 69 70 70 70 70 71 71 72 72 72 73 73 73 73 73 74 74 74 75 75 76 76 76 76 76 76

77 78 78 78 78 78 79 79 79 79

II Column L . H Column M J The Columns from N to P I Column N II Column 0 III Column Q IV Column R v Column P. Visibility Conditions K Eclipses

79 80 81 81 82 82 83 83 85

CHAPTER I. SYSTEM A

§1. Ephemerides . Full moons. S.E. 124 and 125 No.1. New moons. S.E. 124 to 126 No.2. New and full moons. S.E. 141 No.3. No. 3aa. New moons. At least S.E. 141 No. 3a. New and full moons. S.E. 142 No. 3b. New and full moons. S.E. 142 New moons. At least S.E. 145 to No.4. 149 No. 4a. New moons. At least S.E. 146 to 149 New moons. S.E. 146 to 148 No.5. No. Sa. New moons. At least S.E. 146 Full moons. At least S.E. 149 No.6. No. 6aa. New moons. At least S.E. 150 to 155 No. 6ab. New moons. At least S.E. 154 No. 6a. New and full moons. S.E. 155 No.6b. New moons. At least S.E. 172 New and full moons. S.E. 176 No.7. No. 7a. New moons. S.E. 180 and 181 New moons. At least S.E. 182 No.8. and 183 No. Sa. Full moons. At least S.E. 183 No. 8b. New moons. At least S.E. 183, 184 New moons and full moons. S.E. No.9. 185 No.10. New and full moons. S.E. 186 No. 11. New moons. S.E. 188 and 189 No. lla. New moons. At least S.E. 189 No.12. New moons. S.E. 190 and 191 No.13. New and full moons. S.E. 19+ and 195 No. 13a. New moons. At least S.E. 201 and 202 No.14. New and full moons. S.E. 202 No.15. New and full moons. S.E. 209 and 210 No. 16. New and full moons. S.E. 219 No. 16a.

New moons. At least S.E. 229

No. 16b. Full moons. At least S.E. 248, 249

86 86 86 87 87 87 88 88 88 89 90 91 91 91 91

92 92 95 95 95 95 95 96 96 97 97 97 98 98 98 99 99

99

XV

§ 2.

§ 3.

§ 4.

§ 5.

No. 17. Full moons. At least S.E. 253 No. 18. New and full moons. S.E. 263 No. 18a. Full moons. At least S.E. 266 to 269 or S.E. 41 to 44. No. 19. New moons for at least two years No. 20. New moons and full moons for at least one year No. 21. New moons and full moons for at least one year No. 22. New moons and full moons for at least one year No. 23. New moons for at least one year . No.24. Full moons for at least one year No. 25. New moons for at least one year No. 26. New moons for two years . Eclipses. No. 60. Lunar eclipses. S.E. 137 to 160 No. SO. Solar eclipses. At least S.E. 141 to 147. No. 61. Eclipses(?). At least S.E. 177 to 199(?) No. 61a. Solar eclipses. At least S.E. 191 to 194 No. 51. Solar eclipses. At least S.E. 199 to206 No. 5 1a. Lunar eclipses(?). S.E. 206 to 220 No. 52. Solar eclipses. At least S.E. 244 to 248 No. 53. Solar and lunar(?) eclipses. At least S.E. 298 to 253(?) No. 54. Eclipses(?) for at least seven years No. 55. Eclipses or excerpts for several years Auxiliary Texts A Latitudes No. 70. Full moons. At least S.E. 49 to 60 B Excerpts No. 75. New moons. At least S.E. 181 to 185 No. 76. New moons. S.E. 204 to 221 Daily Motion No. 80. Moon. S.E. 178 I No. 81. Moon. S.E. 178 VII . Ephemerides of Undetermined System from Babylon No. 90. Fragment. Columns B, E, C No. 91. Fragment. Columns E, T No. 92. Full moons for at least two years No. 92a. Longitudes and latitudes of the moon in four separate years No. 93. Latitudes(?) and eclipses magnitudes for at least seven years

100 100 101 102 103 103 103 104 104 105 106 106 106 109 109 112 112 113 114 115 116 116 117 117 117 117 117 118 118 118 118 120 120 120 121 122 122

CHAPTER II. SYSTEM B

Introduction Arrangement of the Texts § 1. Ephemerides A Ephemerides from Uruk No. 100. New moons. S.E. 106 to 108 No.101. Newmoons.S.E.118and119 No. 102. New moons, last visibility, and full moons. S.E. 121 No. 103. New moons. S.E. 123 No. 104. New moons and full moons. S.E. 124 No. 105. Full moons. S.E. 135 to 137 . No. 106. New moons. At least S.E. 136 and 137 No. 107. Full moons for at least one year No. 108. New moons for several years . No. 109. New moons for at least one year No. 110. Full moons for at least one year B Ephemerides from Babylon . No. 119. New moons. At least S.E. 176(?) No. 120. New moons and full moons. S.E. 179 No. 121. New moons and full moons. S.E. 181 No. 121a. New moons. At least S.E. 185 to 188 No. 122. New moons. S.E. 208 to 210 . No. 122a. New moons. At least S.E. 221 No. 123. New moons and full moons. S.E. 235 No. 123aa. New moons and full moons. S.E. 236 No. 123a. New moons and full moons for two years No. 124. Full moons for at least one year No. 125. Fragment of ephemeris for at least one year No. 125a. Full moons for at least one year No. 125b. Fragment of ephemeris for at least three years No. 125c. Fragment of ephemeris for at least one year No. 125d. Fragment of ephemeris for at least two years

124 124 126 126 126 129 132 136 136 138 139 139 140 140 140 140 140 141 143 144 144 146 146 150 150 152 153 154 154 154 155

XVI

No. 12Sf. Eclipse magnitudes for at least two years. Ephemeris(?) No. 126. Full moons for four years No. 126a. Full moons for at least one year No. 127. Fragment of ephemeris for at least one year No. 128. Last visibility for at least two years . No. 129. Full moons for at least two years . § 2. Eclipses. A Solar Eclipses No. 130. Solar eclipses for at least ten years, incl. S.E. 126 to 130 B Lunar Eclipses . No. 135. Lunar eclipses for S.E. 113 to 130 No. 136. Lunar eclipses for at least seven years, incl. S.E. 121 to 124 § 3. Auxiliary Tables A Longitudes No. 140. S.E. 115 to at least 130 No. 141. At least S.E. 121 to 124 No. 142. S.E. 123 to 142 No. 143. At least S.E. 146. No. 144. S.E. 148 to 161 No. 145. At least S.E. 126 to 139 No. 146. Several years B Eclipse Magnitudes No. 149. At least S.E. 54 to 67 No. 150. At least S.E. 115 to 138 No. 151. At least two years No. 152. Fragment c Lunar Velocity . No. 155. S.E. 104 to 124 No. 156. At least S.E. 122 to 131 D Columns H and J No. 160. S.E. 123 to at least 154. No. 161. S.E. 124 to 156 No. 162. At least S.E. 133 to 151 No. 163. At least S.E. 117 . No. 164. At least S.E. 127 to 132 No. 165. At least S.E. 137 to 156 No. 166. Several years No. 167. Fragment E Syzygies No. 170. S.E. 104to 112 No. 171. S.E. 115 to 124 No. 172. At least S.E. 117 . No. 173. S.E. 123 to 130

155 155 157 157 158 159 160 160 160 161 161

163 164 164 164 165 166 166 166 167 167 167 167 168 168 168 169 169 169 169 170 170 170 170 170 171 171 171 172 172 173 175 175

No. 174. S.E. 124 to 131 No. 175. Fragment F Visibility No. 180. New moons. S.E. 120 to at least 125 No. 181. New moons. Several years No. 182. New moons. Several years § 4. Daily Motion A Solar Motion No. 185. S.E. 124 No. 186. Several months No. 187. Fragment B Lunar Motion No. 190. For 248 days No. 191. S.E. 117 No. 192. S.E. 118 No. 193. S.E. 119 No. 194. S.E. 130 No. 194a. S.E. 243 No. 194b. Several months No. 195. Fragment No. 196. Several months Appendix. Solstices and Equinoxes . No. 198. S.E. 116 to 131 No. 199. A least S.E. 143 to 157

175 176 176 177 177 177 178 178 178 178 179 179 179 180 180 181 181 182 183 183 183 184 184 185

CHAPTER III. PROCEDURE TEXTS

Introduction

§ 1. Procedure Texts from Babylon No. 200. Introduction Section 1 Section 2. Column C1 Section 3. Column B2 Section 4. Column P' Section 5. Columns (/) and F Section 6. Column E Section 7. Solar velocity Section 8. Variation of daily solar velocity Section 9. Monthly solar velocity; eclipse magnitudes Excursus: The term hab-rat Section 10. Determination of the extremal velocities in general Section 11. The seasons of the year Section 12. Column J Section 13. Columns K and M Section 14. Columns(/) and G Section 15. Column P 1 , day numbers in column M Section 16. Column P3

186 186 186 186 187 187 188 188 189 190 193 194 194 197 198 199 200 201 202 204 208

XVll

No. 200a Introduction Section 1. Column If> Section 2. Column B1 Section 3. Column C1 No. 200aa Sections 1 to 5. Column If> Section 6. Column B1 Section 7. Column P1 No. 200b Section 1. Column If> Section 2. Column C1 Section 3. ~C 1 Section 4. ~E No. 200c. ~E No. 200d Section 1. Solar velocity; eclipse magnitudes main differences aE; 2. Section Section 3. aE; intermediate differences No. 200e. aE No. 200f. Eclipses; ~E . No. 200g. Eclipses No. 200h. Eclipses; ephemeris for S.E. 60 and 61 No. 200i Section 1. Column E Section 2. Column F Section 3. Column P 2, 3 Section 4 No. 201 Introduction Sections 1 to 4. First and last visibility Sections 5 and 6. Coefficients for ecliptic and latitude. No. 201a Section 1. First visibility Section 2. Coefficients for ecliptic and latitude Section 3 No. 201aa. First and last visibility No. 202 Transcription Section 1. Coefficients Section 2. Coefficients Section 3. Day numbers of syzygies Section 4. Change of epoch, System B(?)

210 210 210 211 211 211 211 213 213 213 213 214 214 215 216 217 217 218 218 219 219 220 221 222 222 225 225 226 226 226 227 239 240 240 241 241 241 242 242 243 244 244 244

No. 203. Column If> No. 204 Section 1. Columns E and lJI Section 2. Columns lJI and E Section 3. Columns E and lJI Section 4. Columns lJI and E Section 5. Columns If> and F Section 6. Column t1> Section 7. Longitudes(?) Section 8. Latitudes . No. 204a. Columns f!J, G, and F No. 205. Columns f[J and G No. 206. Columns If> and G No. 207. Columns t1> and G No. 207a. Columns t1> and G No. 207b. Columns t1> and G No· 207c. Columns f[J and G No. 207ca. Columns f[J and G No. 207cb. Columns t1> and G No. 207cc. Columns f[J and G No. 207cd. Columns f[J and G No. 207d. Columns f[J and A No. 207e. Columns f[J and A No. 207f. Columns t1> and A(?) No. 208. Columns F and G No. 210 Section 1. System A, Column B Section 2 Section 3. System B, Sidereal and Synodic Periods. Section 4. Section 5. System B, Column A Section 6. Eclipse cycle Sections 7 and 8 No. 211 Section 1. Eclipse cycle Section 2. System B, Column ~lJ''(?) Section 3. System B, Columns F and G Sections 4 and 5 Section 6. Eclipse Magnitude Section 7. System B, Column ~lJ''(?) Section 8. Eclipse cycle

§ 2. Procedure Texts from Colophons Introduction . No. 220. Eclipses No. 221. Daily Motion of the Moon .

244 245 245 246 247 247 248 249 249 250 251 252 253 254 255 256 257 258 261 262 263 263 268 269 269 271 271 271 271 273 273 273 273 274 274 274 274 275 275 275 276 276 276 276 277

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Sources in the History of Mathematics and Physical Sciences

5

Editor

G.J. Toomer

Advisory Board

R.P. Boas P.J. Davis T. Hawkins M.J. Klein A.E. Shapiro D. Whiteside

ASTRONOMICAL CUNEIFORM TEXTS BABYLONIAN EPHEMERIDES OF THE SELEUCID PERIOD FOR THE MOTION OF THE SUN, THE MOON, AND THE PLANETS

I Edited by

0. NEUGEBAUER

Published with the Assistance of the

INSTITUTE FOR ADVANCED STUDY PRINCETON , NEW JERSEY

Springer Science+Business Media, LLC

0. Neugebauer Institute for Advanced Study Princeton, NJ o8540 U.S.A.

AMS Subject Classification: OIA17 Library of Congress Cataloging in Publication Data Main entry under title: Astronomical cuneiform texts. Reprint. Originally published: London: Published for the Institute for Advanced Study, Princeton, N.J., by Lund Humphries, 1955. "Published with the assistance of the Institute for Advanced Study, Princeton, New Jersey." Bibliography: v. 2, p. Includes index. Contents: v. 1. Introduction, the moon-v. 2. The planets, indices-v. 3· Plates. 1. Astronomy, Assyro-Babylonian. 2. Assyro-Babylonian language- Texts. I. Neugebauer, 0. (Otto), 1899. II. Institute for Advanced Study (Princeton, N.J.) QB19.A87 1983 528'.835 83-622 With 225 Illustrations.

© Springer Science+Business Media New York 1955 This book was originally published in 1955 by Lund Humphries, London, England, for the Institute for Advanced Study.

ISBN 978-1-4612-5509-3 ISBN 978-1-4612-5507-9 (eBook) DOI 10.1007/978-1-4612-5507-9

VOLUME II THE PLANETS INDICES

Vll

TABLE OF CONTENTS

PART Ill. EPHEMERIDES OF THE PLANETS

INTRODUCTION

§ 1. The Planetary Theory in General A. Introduction B. The Leading Ideas of the Planetary Theory C. Periods D. Mean and True Motion E. Dates F. Concluding Remarks § 2. Theory of Mercury A. Introduction B. System A1 . I. Heliacal Rising (T). Positions II. First Appearance in the Evening (E). Positions III. Continuation of B(T) and B(E) IV. Dates of rand E V. Continuation of T(T) and T(E) VI. Last Visibilities (.E and Q). Positions . VII. Last Visibilities (.E and Q). Dates . C. System A2 I. Last Visibility in the Morning (.E). Positions II. Reappearances as Evening Star (E). Positions III. Last Visibility in the Evening ( Q). Positions IV. Heliacal Rising ( T). Positions V. Dates D. Daily Motion § 3. Theory of Venus A. The Main Parameters B. System A0 • C. Systems A1 and A2 § 4. Theory of Mars A. General Properties B. Subdivision of Synodic Motion C. System A. I. Positions for (/>, r and Q II. Positions for e and lJf .

279 279 279 281 284 285 286 287 287 288 288 290 291 292 293 293 294 295 295

III. Dates for T, (/>, and Q IV. Dates for 8 and lJf § 5. Theory of Jupiter A. System A. I. Positions II. Dates B. System A' I. Positions II. Dates c. Modifications of the Systems A and A' D. System B I. Positions II. Dates E. System B' F. Subdivision of Synodic Motion. System A' G. Daily Motion § 6. Theory of Saturn A. System A. B. System B . I. Positions II. Dates c. Subdivision of Synodic Motion. System A

311 313 313 313 314 314 314 314

MERCURY

296

CHAPTER

296 297 298 299 300 300 300 301 302 302 303 303 303 305

Introduction §1. System A2 No. 300a. At least S.E. 4 to 22 . No. 300b. At least S.E. 10 to 18 § 2. System A1 No. 300. S.E. 118 to 143 No. 301. S.E. 133 to 153 No. 302. S.E. 166 to 189 No: 303. At least S.E. 183 to 186 No. 303a. At least S.E. 214 to 220 No. 303b. At least S.E. 216 to 229 No. 304. At least S.E. 224 to 226 No. 305. For at least five years § 3. Daily Motion No. 310. For at least seven months

I.

306 306 307 307 307 308 308 308 309 310 310 311 311 311

316 316 316 317 317 317 318 321 324 324 324 325 325 326 326

Vlll CHAPTER II. VENUS

§ 1. System A0 No. 400. S.E. 111 to 135 No. 401. S.E. 175 to 303 § 2. System A1 No. 410. S.E. 236 to at least 259 No. 411. S.E. 246 to at least 254 No. 412. At least S.E. 265 to 281 § 3. System A2 No. 420. S.E. 180 to 242 No. 421a. At least S.E. 183 to 242 No. 421. At least S.E. 187 to 204 No. 430. S.E. 96 to at least 111

329 329 329 330 330 330 331 332 332 332 333 333

CHAPTER III. MARS

§ 1. System A At least S.E. 89 to 131 S.E. 123 to 202 At least S.E. 170 to 187 At least S.E. 172 to 187 First and last visibility for about 80(?) years No. 503. Stationary points for several years No. 504. Longitudes for at least 102 years . §2. System X No. 510. Last visibility for at least 18 years No. No. No. No. No.

500. 501. SOla. 501b. 502.

335 335 335 336 336 336 337 337 338 338

CHAPTER IV. JUPITER

§ 1. System A No. 600. S.E. 113 to 173 No. 606. S.E. 113 to at least 161 No. 601. S.E. [115] to 181 No. 602. At least S.E. 130 to 205 No. 603. S.E. 147 to 218 No. 604. At least S.E. 157 to 191 No. 604a. At least S.E. 185 to 197 No. 605. At least S.E. 188 to 222 No. 607. At least S.E. 209 to 218 No. 608. At least S.E. 217 to 237 § 2. System A' No. 609. At least S.E. 134 to 146 No. 610. At least S.E. 142 to 195 No. 611. S.E. 180 to 252 No. 612. At least S.E. 187 to 230 No. 613. At least S.E. 197 to 206 No. 613aa. At least S.E. 202 to 210 No. 613ab. At least S.E. 202 to 273 No. 613a. S.E. 203 to at least 274 No. 614. At least S.E. 239 to 247 § 3. System B No. 620. At least S.E. 127 to 194

339 339 339 340 340 341 341 342 342 342 342 343 343 343 344 344 345 345 346 346 346 347 347

No. No. No. No. No. No. No. No. No.

624. 620a. 620b. 621. 621a. 622. 622a. 623. 625.

At least S.E. 161 to 170 347 S.E. 171 to 243 347 S.E. 171 to at least 180 348 At least S.E. 182 to 205 348 At least S.E. 185 to 221 348 S.E. 190 to 231 349 At least S.E. 200 to 232 349 S.E. 202 to 267 350 First visibility for at least 17 years 351 No. 625a. Fragment for at least 14 years 351 No. 626. Fragment for at least six years 351 No. 627. First stationary points for at least 12 years . 351 No. 628. Fragment for at least seven years 352 No. 629. Fragment for at least nine years 352 § 4. System B' 352 No. 640. At least S.E. 131 to 161 352 § 5. Daily Motion 353 No. 650. Fragment 353 No. 651. Fragment 353 No. 652. Fragment 354 No. 653. Fragment 354 No. 654. At least S.E. 147 IX to 148 V 354 No. 655. At least one month 355 CHAPTER V. SATURN

No. No. No. No. No. No. No. No. No.

700. 701. 702. 703. 704. 704a. 705. 705a. 706.

At least S.E. 86 to 134 357 At least S.E. 108 to 118 357 At least S.E. 123 to 182 357 Last visibility for at least 19 years 358 S.E. 155 to 243 358 At least S.E. 201 to 224 359 At least S.E. 203 to 225 359 At least S.E. 229 to 252 360 Second stationary points for at least six years . 360 No. 707. Fragment for at least 25 years 360 No. 708. Fragment for at least 47 years 361 No. 709. Fragment for at least 11 years 361 CHAPTER VI. PROCEDURE TEXTS

Introduction § 1. Procedure Texts from Uruk No. 800. Mercury. Mean synodic arc No. 800a. Mercury. Table for rand I: No. 800b. Mercury. Table for rand I: No. 800c. Mercury. Table forE and Q No. 800d. Mercury. Table forE and Q No. 800e. Mercury. Fragment of table No. 801. Mercury and Saturn Introduction

362 362 362 364 364 365 365 365 366 366

lX

Section 1. Mercury, rand .E Section 2. Mercury, E and Q Sections 3, 4, and 5. Saturn, System A Sections 6, 7, and 8. Saturn, Periods; System B No. 802. Saturn Introduction Sections 1, 2, and 3. System A Sections 4, 5, and 6. Periods; System B No. 803. Mars. Retrogradations No. 804. Mars; cf. Pis. 211 and 212 . No. 805. Jupiter Section 1. System B Section 2. System A' § 2. Procedure Texts from Babylon No. 810. Jupiter Sections 1 and 2. System A', arcs . Section 3. Daily motion, slow arc Section 4. Daily motion, medium arc Section 5. Daily motion, fast arc Section 6. Daily motion, medium arc No. 811. Jupiter, Saturn, and Mars. Section 1. Jupiter, modified System A', arcs; approximate periods Section 2. Saturn, approximate periods Section 3. Mars, approximate periods No. 811a. Mars Sections 1 and 2. Coefficients Section 3. Derivation of AT from A,\ Sections 4 and 4a. Dates of r, (/), Q (?) . Section 5. (/) and Q Section 6. Components of synodic time Sections 7 to 9. Derivation for the three components Section 10. Velocities Section 11. Periods; mean synodic arc No. 811b. Mars No. 812. Jupiter and Venus Section 1. Jupiter, System B, dates Section 2. Jupiter, System B, derivation of AT from A,\ . Section 3. Jupiter, System A', arcs Section 4. Jupiter, System A', daily motion, slow arc Sections 5 to 9. Jupiter, System A'; fragments . Section 10. Jupiter, approximate periods Sections 11 to 16. Venus, longitudes Sections 17 to 24. Venus, dates Sections 25 to 26. Venus, summary of motion Section 27. Venus, longitudes

366 367 368 370 371 371 371 372 373 374 375 375 375 376 376 376 377 378 378 379 379 379 380 380 381 382 382 383 384 384 386 388 390 391 392 392 393 393 394 394 395 396 397 399 400

Sections 28 to 31. Venus, synodic period, fragments . No. 813. Jupiter Section 1. System A; approximate periods Section 2. System A, arcs; motion Sections 3 and 4. Fragments, coefficients Section 5. Trapezoid Section 6. Velocities Section 7. System A", arcs Section 8. System A"', arcs Section 9. System A', daily motion Section 10. System A, arcs, motion Section 11. System A modified, arcs, motion Section 12. System B, dates Section 13. System B, derivation of AT from A,\ Sections 14 to 16. System A', derivation of AT from A,\ Sections 17 and 18. System A', arcs, motion Sections 19 to 22. Systems B and B' Section 23. System A or A', motion Section 24. System A or A', fast arc Section 25. System A', arcs Sections 26 and 27. Systems B and B', motion Section 28. Motion; coefficients . Section 29. Coefficients for motion Section 30. Motion, fast arc, times Section 31. Motion, fast arc, longitudes Section 32 No. 813a. Jupiter Sections 1 to 3. System A, arcs, motion Sections 4 and 5 No. 813b. Jupiter Section 1 Section 2. System A, arcs Section 3. System A"', arcs No. 814. Jupiter Section 1. System A; approximate periods Section 2. System A, motion, coefficients; System A', arcs Sections 3 and 4. System A', motion No. 815. Venus. Approximate periods No. 816. Mercury, System A3 Section 1. Yearly motion, Q and r Section 2. Yearly motion, E Section 3. (/)and 'l', synodic arcs Section 4. Q and r, 20-year period

402 403 403 404 405 405 405 406 406 406 408 408 411 411 411 413 413 415 416 416 417 417 418 418 419 420 420 420 420 421 421 422 422 422 422 423 424 425 425 425 426 427 428

X

Section 5. E, 20-year period No. 817. Jupiter. Mathematical Problem Section 1. Jupiter, coefficients for motion; invisibility Section 2. Jupiter, NA(?) Section 3. Jupiter, velocity (?) Section 4. Trapezoid No. 818. Jupiter Sections 1 and 2. System A or A', motion Section 3. Phenomena for S.E. 61 and 62 No. 819a. Jupiter Sections 1 to 5 Sections 6 to 8 . No. 819b. Jupiter

-4-28 429 429 429 430 430 431 431 432 432 433 433 434

No. 819c. Mercury(?) and Saturn, with data for S.E. 61 to 64 § 3. Procedure Texts from Colophons of Ephemerides No. 820. Jupiter, System B' No. 820aa. Jupiter, System A No. 820a. Mercury, System A10 rand 1:, E and Q No. 821. Jupiter, System A No. 821aa. Mars, arcs No. 821a. Jupiter, System B No. 821b. Venus, Systems A1 and A2 • No. 822. Jupiter, System A' No. 823. Jupiter, System B No. 823a. Jupiter, System A' No. 824. Planet(?) .

435 436 436 437 437 439 439 439 440 443 444 444 444

PART IV. FRAGMENTARY AND UNIDENTIFIED TEXTS 445 445 445 445 446 446 446 446 446 447 447 448 448 448 448

Introduction § 1. Sun and Moon No. 1000 No. 1001 No. 1002 No. 1003 No. 1004 No. 1005 No. 1006 No. 1007 No. 1008 No. 1009 No. 1010 No. 1011 § 2. Planetary Texts from Uruk

A. Nos. 1013 to 1021 No. 1013 No. 1014 No. 1015 No. 1016 No. 1017 No. 1020 No. 1021 B. Nos. 1030 to 1032 No. 1030 No. 1031 No. 1032 § 3. Planetary Texts from Babylon No. 1050 No. 1051

448 448 449 449 449 449 449 450 450 450 450 450 451 451 452

PART V. INDICES AND BIBLIOGRAPHY

§ 1. Concordance of Texts § 2. Bibliography and Abbreviations § 3. Glossary

453 461 467

A. General Glossary B. Technical Terminology § 4. Subject Index

467 498 504

xii

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150

II -I 00

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11

'I

11

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I

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150

1

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2.50

300

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Sources in the History of Mathematics and Physical Sciences

5

Editor

G.J. Toomer

Advisory Board

R.P. Boas P.J. Davis T. Hawkins M.J. Klein A.E. Shapiro D. Whiteside

ASTRONOMICAL CUNEIFORM TEXTS BABYLONIAN EPHEMERIDES

OF THE SE LEU C ID PERIOD FOR THE MOT IO N OF THE SUN , TH E MOON, AND THE PLANETS

I Edited by

O. NEUGEBAUER

Published wit" tlle Assistance of the I NS TIT U TE FOR A DV ANC ED STU DY PRINCETON, NEW JERSF.\'

Springer Science+ Business M e di a, LLC

O. Neugebauer Institute for Advanced Study Princeton, NJ 08540 U.S.A.

AMS Subject Classification: olAI7 Library of Congress Cataloging in Publication Data Main entry under title: Astronomical cuneiform texts. Reprint. Originally published: London: Published for the Institute for Advanced Study, Princeton, N.J., by Lund Humphries, 1955. "Published \\'ith the assistance of the Institute for Advanced Study, Princeton, Ne\\' Jersey." Bibliography: v. 2, p. Includes index. Contents: v. I. Introduction, the moon-v. 2. The planets, indices-v. 3. Plates. I. Astronomy, Assyro-Babylonian. 2. Assyro-Babylonian language- Texts. I. Neugebauer, O. (Ouo), 1899. 11. Institute for Advanced Study (Princeton, N.J.) QB19.A87 1983 528'.835 83-622 With 225 IIIustrations.

© Springer Science+Business Media New Vork 1955 This book was originally published in 1955 by Lund Humphries, London, England, for the Institute for Advanced Study.

ISBN 978-1-4612-5509-3 ISBN 978-1-4612-5507-9 (eB ook) DOI 10.1007/978-1-4612-5507-9

VOLUME 111 PLATES

TABLE OF CONTENTS The numbers reler to the plates

I

11

Moon

1 to 150

A System A

1 to 53

B System B

54 to 136

C Procedure Texts

136 to 139

D Syzygies, First and Last Visibility

140 to 150

Planets

151 to 212

A Mercury

151 to 169a

B Venus

170 to 172

C Mars

173 to 175

D Jupiter

176 to 205

E Saturn

206 to 210

F Procedure Texts

210a to 212

III Fragments and Unidentified Texts

213 to 216

IV Copies

217 to 226

V

.

Photographs

228 to 255

The arrangement of the plates follows, in the main, the arrangement of the texts in Vois. land II. Arrows indicate that the table continues on another plate in the direction of the arrow.

1

Part I.

Introduction

§ 1. THE TEXTS. DESCRIPTION AND NOTATION

A. Description of the Texts Lists of positions of the sun, the moon, and the planets computed for regular time intervals (e.g., from day to day) are called "ephemerides". The majority of the texts published here are of this character;1 they give, for instance, the position of the moon from month to month for a single year or the positions of Jupiter at consecutive heliacal risings. The purpose of these texts, however, is not restricted to the finding of the place of a heavenly body at regular intervals in order to give a picture of its movement. The final goal is to predict phenomena like new moons, last visibility, eclipses, etc. The whole collection corresponds very much in tendency to modern tables like the "American Ephemeris and Nautical Almanac". In order to compute a lunar ephemeris, many separate columns are needed, referring to the velocity of the sun, of the moon, certain corrections, etc. Such specific functions were frequently computed for many years in advance and collected on separate tablets. We shall say that such texts contain "auxiliary functions''. Closely related to all these texts are tablets which indicate the rules for computing ephemerides. These "procedure texts" ("Lehrtexte" in Kugler's terminology) give the rules for computing an ephemeris step by step. Unfortunately, these texts do not contain much of what we would call the "theory" behind the method. Such a theory must have existed because it is impossible to devise computational schemes of high complication without a very elaborate plan. Texts like No. 811a, a procedure text for Mars, give us only some glimpses of the formulation of a more general theory. Most of the tablets are written very neatly in carefully arranged columns and lines though there are also tablets which were copied with less care. 2 There can be very little doubt that the majority of these tablets are "final copies" made from preliminary "manuscripts" which contained the necessary auxiliary calculations. Certain columns of our texts frequently give only

rounded-off values, which presuppose the existence of columns of unabbreviated numbers. The tablets with auxiliary functions certainly served this purpose. Many tablets begin with an invocation and end with a colophon, giving the owner, scribe, and date of the tablet. 3 The invocation is usually written on the upper edge. The colophon either runs over the whole width of the tablet 4 or is written in the last column. 5 A few texts also have a brief label of the contents of the tablet; e.g., the lower edge of the obverse of No. 60 bears the notice, "for the 14th day" (because the text refers to lunar eclipses), and the upper edge of No. 155 says, "velocity of the moon on the 28th". Occasionally 6 short rules are added for the computation of the text; these rules are apparently extracts from procedure texts. The invocations, colophons and other additional remarks are usually written in a more cursive ductus than the numbers of the main tablet. The script shows everywhere the characteristic features of texts written in the Seleucid period. In comparison with nonastronomical texts of this period, the shape of most of the tablets is peculiar, namely, very long as compared with the width. This shape obviously conforms to the need, especially in the lunar ephemerides, for many parallel columns. B. The Present Edition

General Arrangement For the sake of convenience and for technical reasons, all transcriptions of the ephemerides are collected in volume III. Critical apparatus and commentaries, however, are to be found in the text volumes I and II. \Ve abandon here the use of the word "ephemeris" introduced by Kugler (SSB II p. 464 ff.), who applied the term to astronomical texts of a different character. 2 Examples of very clearly written texts are Nos. 100 and 101 (Pl. 229 and Pl. 230); of less orderly writing, Nos. 51 (Pl. 220) or 170 (Pl. 233). 3 Cj. § 3, p. 11 ff. 4 E.g. No. 702 (Pl. 249). 5 E.g. No. 161 (Pl. 232). 6 E.g. Nos. 135 + 220,603 + 821, 611 + 822, 622 + 823; cf. p. 276 and p. 436 ff. 1

2

§ 1, B.

ARRANGEMENT

Part I contains all information which pertains to the material as a whole. Sections 1, 2, and 3 deal with general features and historical problems, such as dates, provenance and scribes. Sections 5 and 6 explain the general mathematical methods and chronological concepts. The specific methods for the computation of the ephemerides, however, are described in the introductions which precede the corresponding sections in Parts II (moon) and III (planets). Concordances, abbreviations, glossary, etc. are given at the end, in Part V.

The texts are arranged according to their contents in larger groups: Part II contains lunar texts; Part Ill, planetary texts in the modern order: Mercury, Venus, Mars, Jupiter, and Saturn. Both lunar and planetary texts can be subdivided into "Systems" according to the method of computation employed. In each of these groups the texts are arranged chronologically as far as possible. The procedure texts form the last chapter in each part. In Part IV are collected fragments and unidentified texts. Most of them are very small fragments, but the largest of these texts, No. 1050 (p. 451 ), is well preserved and was recognized as an astronomical text as early as 1881 but is still unexplained. Single Texts The commentary to each ephemeris is preceded by information about date, provenance, previous publication, etc. Omission of a reference to another place of publication or an asterisk indicate that the tablet was not published previously. The reason for giving a certain provenance is indicated in [ ]; thus Uruk followed by [U] means that the tablet belongs to the U ruk collection in Istanbul and that this fact is the main basis for our statement. Similarly [BM] denotes a group of texts which, in all probability, came from Babylon and are now in the British Museum. The details for these criteria are explained in § 2, A (p. 4). The symbols 0/R and 0-R

respectively, listed under "arrangement", indicate how the text must be turned in order to proceed from obverse to reverse (cf. Fig. 1). It must be remembered that in ephemerides the first column is always on the left-hand side, in contrast to the arrangement in ordinary cuneiform writing, where the first column of the reverse is the right-hand column (cf. Fig. 2). In the present material this ordinary arrangement is found only (but not always) in procedure texts, excepting one ephemeris (No. 5la).

In our transcriptions the left-hand margin of the first column of a tablet is marked by a double line, and the same holds for the right-hand margin of the last column. A single line at the beginning or end of a O~v.

I

lr

:nr

lY

mll •••~ 6-

o.m 1J[ , .'•

~

r-m tJ_JJJ

. The numbers in the ephemerides are consistently expressed in the sexagesimal notation, except in the case of the year numbers in some tablets, which use 1-me for "one hundred". Sexagesimal numbers are always transcribed by using commas to separate the places. 7 In commentaries, integers are separated from fractions by a semicolon. Thus we write 1,40 for 100

but

1;40 for 1

2

+3.

Zero, indicated in the text by the separation symbol, 8 is transcribed by a dot, in commentaries by the ordinary zero. Thus 1,.,40 = 3640 = 1,0,40. In translating, sexagesimal writings are kept unchanged, but decimal writings are replaced by numbers in our ordinary notation, e.g., 1-me 5 by 105. I have considered it unnecessary to give "translations" of ephemerides because they contain, besides numbers, only a few ideograms for months, zodiacal signs and some astronomical concepts. A list of all these terms is given in § 7 (p. 38). For each transcription there is a critical apparatus for the readings or restorations. Usually I refrain from mentioning the deviations of my transcriptions or restorations from previous editions or older copies. All deviations from previous copies have been checked with photographs or the original tablets. Errors of the texts are, of course, mentioned only either in the critical apparatus or in the commentary.

3

In spite of numerous controls I have no doubt that many errors which I have committed remained undiscovered. No numbers should be used without consulting apparatus and commentary and without checking their correctness by means of preceding and following values. The commentaries are intended merely to explain problems of the individual tablet and are not meant to develop consequences for our understanding of Babylonian astronomy in general. The ordinary colophons of all tablets are collected in chronological order in§ 3 C (p. 16 ff). Concordances for colophons and the numbers of texts are given on p. 26. For procedure-text colophons cf. p. 276 and p. 436. A few remarks for the Assyriologist must be added to explain the principle followed in my transcriptions. The basis is, of course, Thureau-Dangin's system, modified only by a few convenient additions as follows: samas for utu, samds for MAN; or and for DIS and absin 0 for KI when used for absin, gam 0 for a GAM with three corner wedges. By mistake I have failed to distinguish between -kam and -kam in the writing of ordinal numerals. In the use of capitals, however, I deviate in part from the usual custom of distinguishing between certain and uncertain values. Our material contains three different types of texts which require different treatment: (a) the ephemerides proper with their auxiliary tables, (b) the colophons, (c) the procedure texts. The ephemerides contain only a very limited number of symbols, consisting for the most part of the well-known list of symbols for the months and the zodiacal signs in common use in the Seleucid period. It seems to me of no interest to complicate the transcription of an ephemeris with the distinction between known or unknown sign values (gun or LAL), or between real Sumerograms (se) and mere abbreviations (dr), or graphic variants (SIG for sig 4). I therefore transcribe all symbols in the ephemerides by lower-case letters; it goes without saying that I have chosen, when possible, the most plausible value. There are, however, many cases where I have no real reason for the choice of a transcription; Ia!, e.g., is used in such different meanings as "subtraction" and "positive latitude" that it would not be at all surprising if the equivalents would be similarly different. The colophons offer no special difficulties and are consequently transcribed as is usual with N eo- Babylonian texts. Serious problems arise, however, in the transcription of the procedure texts. Here we meet a great number of technical terms of wholly unknown reading, if not 7 8

Cj. Neugebauer [3] for this principle of transcription. Deimel, SL 378.

§ 2, A.

PROVENANCE

unknown meaning. We are far from being able to give the Akkadian correspondences for many words, not to mention details such as determining the special verbal forms, etc. to be used. I have therefore adopted the principle of not transcribing Sumerograms except when the text explicitly indicates the Akkadian reading. Thus I write matu(1a1-u), but merely lal even when there is no doubt that both forms were read matu. Here, however, I have followed the principle of distinguishing between fairly certain values (like ge 6 for night) and totally unknown values (like BE for elongation). The reader must therefore keep in mind, e.g., that "be" in the ephemerides and "BE" in a procedure text might represent the same concept but that, on the

§ 2.

other hand, BE might also be read summa. In the glossary p. 4-67 ff. cross-references can be found between these various possibilities.

Plates The main body of volume III consists of transcriptions of the ephemerides. Copies and photographs appear towards the end of the volume. Wherever possible, I have given a scale with the reproduction of a text; wherever it is missing, the scale is unknown to me. Some joins of photographs of fragments which belong to different museums have not been made because the rephotographing in equal scale and proper position could not be effected (e.g., Pl. 228: Berlin-New York).

PROVENANCE AND DATES

All available information, explicit dates as well as palaeographical evidence, etc., concurs in proving that all the texts published here were written in the Seleucid period, i.e., during the last three centuries B.C. This does not imply, however, that they form a uniform mass of material. We shall indeed demonstrate the possibility of a geographical classification which is in turn reflected in the chronology of the texts.

A. The Provenance of the Texts About one hundred of our texts come from an archive in Uruk. The rest, about 200 tablets, are more difficult to localize, but we will show presently that there are good reasons to believe that the non-Uruk tablets came from Babylon.

Uruk Several of the best preserved tablets, now in the Berlin, Chicago, Istanbul, and Paris collections indicate their provenance from U ruk in the colophon.l During the German excavations at U ruk before 1914-, several fragments of tablets were found and photographed. From these photographs I succeeded in identifying many fragments of ephemerides, several of which join bigger pieces in the above-mentioned museums. 2 This certified stock of Uruk tablets furnished the basis for the development of criteria which are more or less characteristic for this archive and led to the identification of many more U ruk tablets whose colophons were broken away. Long after this classification was completed, I obtained the photographs of the Uruk collection in Istanbul. All but two minute fragments 3 known to me from the German photographs appeared to have finally reached Istanbul. Many new fragments permitted joins with already identified texts, thus leading

to a never hoped-for completeness of the astronomical archive in U ruk. It is rarely necessary to rely on secondary criteria for the provenance from U ruk now that most of the texts are identifiable either from "colophon" or from the German photos (quoted as "Warka photo") and from the Istanbul collection (quoted "U"). Yet it is of interest to know that the Uruk texts also form groups in the collections in Berlin (VAT 7800 ff.) and Chicago (A 34-00 ff. and 64-00 ff.). It was furthermore possible to establish definite scribal habits which sharply contrast the scribal schools of Uruk and Babylon: 1.

Arrangement: Uruk: preferably 0-R Babylon: 0 /R

very rarely 0-R

2.

Writing of tens followed by units (like 10,1 ): Uruk: always 10,.1 never 10,1. Babylon: 10,1 very rarely 10,.1.

3.

Writing of year numbers in the text of an ephemeris 4 (column T): Uruk: frequently using 1-me for 100. Babylon: never 1-me, always sexagesimally 1,40.

There are no ephemerides among the Uruk tablets in the Morgan collection (at present deposited in the Yale Babylonian collection). 2 Only one fragment of an ephemeris from this excavation seems to have reached Berlin (VAT 9154, now part of No. 171 ). Dr. Sachs identified two economic texts (VAT 8562 and 8569) in Schroeder KSW (Nos. 16 and 9). 3 Called here Warka X 40 ( = No. 800e) and Warka X 45 (= No. 1017). Cf. also PI. 248 No. 805 for an example of a \Varka fragment (X 56) and its present state of preservation U 180(10). The photograph which contains the fragments from "\Varka X 1" to "X 51" is reproduced on PI. 6b of Neugebauer, Ex.Sci. 4 This does not include dates in colophons, in which the decimal notation is common for all archives. 1

§ 2, A.

PROVENANCE

4.

Writing of zodiacal symbols: Uruk: never gir (for Tit), never zib-me (for)() Babylon: very rarely lu (for cy:> ).

5.

Notation for leap years: Uruk: never dir-Se. Babylon: never kin or kin-a (for VI 2) or a (for XII 2).

6.

Colophons: U ruk: preferably colophon added. Babylon: colophons rare. Babylon

The situation is much more complicated for the nonUruk tablets. A large part of this second group belongs to the Spartali collections ("Sp.") of the British Museum. Unfortunately, only contradictory statements about this collection are available. Bezold says 5 that Babylon is the place of origin of the Spartali tablets. The text Sp.171, however, is a copy of a Borsippa original and written in Borsippa, 6 as is stated in the colophon. 7 Accordingly, Strassmaier assumed that "ein grosser Theil der sogenannten SpartoliSammlung im Britischen Museum von Birs Nimrud, 8 speciell vom Tempel des Nebo daselbst, stammt". 9 Finally, Sippar has been proposed as the location of the astronomical school where our tablets were written. Strassmaier himself assumed this latter origin, 10 probably influenced by the famous passage in Pliny where he speaks about the three Babylonian astronomical schools at Uruk, Babylon and Sippar.U This assumption seemed to be confirmed when Kugler read in the colophon of a large lunar ephemeris12 the name of Sippar. Since this text, composed of several fragments of the Shemtob and Spartali collections, belongs to System B of the lunar theory, it has become customary to localize this whole group of texts at SipparP Fortunately, Kugler's reading now turns out to be a mistake, 14 and we are therefore left with no direct evidence for astronomical texts from Sippar.l 5 The existence of astronomical texts at Babylon is directly attested in the case of our texts. The first, No. 207e, a procedure text for the moon, was found by the German Babylon expedition in 1902. Unfortunately the text was found not in a building but in debris near the city wall in Amran. The second text, the lunar ephemeris No. 155, written in Uruk, is a copy of a Babylon original, according to a short line written on the upper edge of this tablet (colophon A p. 16). And indeed, more arguments can be found for the assumption that most of the non- U ruk tablets came from

5

Babylon (or Babylon and Borsippa). In 1896, G. A. Reisner published a collection of religious texts belonging to the Berlin museum16 which certainly came from one archive in Babylon, as is proved by many colophons,l7 All these texts, copied in the Seleucid period, either show low inventory numbers (below VAT 600) or numbers between VAT 1700 and VAT 1900.18 To the very same group belong also three astronomical texts: the lunar ephemerides VAT 209 + MM 86-11405 (No. 18) and VAT 1770 (No. 125) and the Jupiter ephemeris VAT 1753 + 1755 (No. 611). The colophon of VAT 209 is badly preserved but the scribal names Etir-Marduk and Nanna-utu also speak for a Babylon origin. 19 VAT 1753 + 1755 is a fragment to be joined to five fragments now in the Spartali collection. This confirms Bezold's above-quoted statement concerning the Babylon origin of the Spartali collection. Additional information about this group of texts in the Berlin museum can be obtained from Ungnad's publication of the economic documents. From Ungnad's list 20 it follows that the above-mentioned three ephemerides were purchased in the same lot, containing tablets from Babylon, Sippar21 and Dilbat. None of the Sippar tablets is from the Seleucid period, which speaks very strongly against Sippar as the provenance of our ephemerides. The same is true of the Dilbat tablets; in addition, the absence of religious material from Dilbat could scarcely be explained if a temple literary archive had been found at this place. Thus Bezold, Lit. p. 149 (ad 18). In the year S.E. 111 ( = 201 B.C.). No. 603 = Sp.II,43 mentions in its colophon (Zl, p. 21) as the owner or scribe a "citizen of Borsippa", but this does not imply that the tablet was found in Borsippa, as Schnabel assumed (Schnabel, Ber. p. 213, misquoting Kugler, SSB I p. 125). 7 Epping-Strassmaier [1] p. 244 and p. 228. 8 I.e. Borsippa. 9 Epping-Strassmaier [3] p. 280. 10 Epping AB p. 6. 11 Pliny NH VI, 121-123 (ed. Ian-Mayhoff I p. 481, trans!. Rackham II p. 431). Cf. also Pliny NH VII, 193 (ed.lanMayhoff II p. 68, trans!. Rackham II p. 637). 12 No. 122 (colophon Zo on p. 22). 13 E.g. Schnabel, Ber. p. 213 or Langdon according to Fotheringham [1) p. 720. ~< Misreading of enuma(as-u,)-An-na for Sip-par-an-na, as Dr. A. J. Sachs recognized. 15 It is perhaps significant that Sippar does not appear among the schools which are responsible for the arrangement of the great astrological series "Enuma-Anu-Enlil". Weidner [3] p. 181 and p. 193 knows of five schools: Uruk and BabylonBorsippa in the South, Assur, Kalgu ( = Nimrud) and Nineveh in the North. 16 Reisner SBH. 17 Reisner SBH, introduction p. XI. 18 A small group between VAT 2170 and 2190 can be added. 19 For the scribal family of Nanna-utu see Reisner SBH, introduction p. XIII (Reisner reads Sin-ibni for Nanna-utu). 20 Ungnad VS 6 p. XII. 21 This might be the reason for Schnabel's statement that VAT 290 + 1836 (an almanac) came from Sippar (Schnabel [3] p. 66, repeated Schaumberger [1] p. 279). 5

6

§ 2, B.

6

Babylon remains as a center whence came not only Reisner's religious texts but also a large number of business documents and astronomical texts. 22 The above-mentioned facts make it at least very plausible that the ephemerides of the Spartali collection and the SH 81-7-6 collection, 23 the three texts VAT 209, 1770, 1755, and all texts joined with these tablets form a uniform group written by the scribes of a temple in Babylon. The continued existence of temples in this city in spite of the removal of inhabitants to Seleucia is proved by Reisner's texts. This is confirmed by Pausanias 24 who reports that the "Chaldeans" were left in their quarters around the temple of Bel. Joins were made between the Spartali collection and almost all other groups of astronomical texts in the British Museum known to me, thus making a common provenance from Babylon very likely. Only small groups of texts have not been directly connected with the rest. On the other hand, both dates and contents are in perfect agreement with the Babylon texts and I therefore have very little doubt that also these tablets have the same origin. And it is also worth noticing that not a single join could be established between the U ruk texts and any of the texts from the British Museum. Thus it seems to be a fair conclusion that all the texts published here originated from only two major archives, one in Uruk, the other in Babylon.

B. The Dates of the Texts If our texts were all well preserved we would have no trouble with their dates because it is a characteristic feature of ephemerides that at least one column gives the dates to which the positions which are listed in the following columns refer. Unfortunately the date column is usually the first column and therefore particularly exposed to damage. Accordingly, the dating of texts frequently requires special investigation. In principle, there are two possibilities: comparison of the positions given in the text with astronomically computed positions; or the use of the ancient methods of computation, starting from a dated text and extending the calculation until values which are contained in the text under consideration are reached. The second method has the decided advantage of avoiding all questions about the significance of an agreement between ancient and modern calculations and has therefore been followed here wherever possible. It will be shown (in § 6, p. 35) how the continuation of the ancient computations can be abbreviated very considerably so that one can dispense with the finding of intermediate positions which are not needed. Thus a great number of texts can be restored completely and

DATES

their dates established without an appeal to modern elements. Only a few smaller fragments remain undated, chiefly in cases where the results are only given in rounded-off numbers, as is frequently the case in certain columns of our ephemerides. The result of the dating of these texts can best be illustrated by the diagrams given as Frontispiece of Vol. I and of Vol. II respectively. Each stroke in these diagrams corresponds to a single tablet and indicates by its length the time covered by the ephemeris inscribed on the tablet. 25 The date of the actual writing of a tablet can, of course, be estimated only approximately from the dates for which an ephemeris is computed. It is, on the other hand, very plausible to assume that the ephemerides were computed in advance for a time interval not too far ahead. This guess is largely confirmed by those instances where a colophon gives us the date of the actual writing of a tablet. The complete list of such cases, given below, shows that the majority of the texts were written close to the beginning of the time interval covered by the ephemeris. Colophon J K L M

Q

s

T

u

v w y

z

Zb Zc

Text No.

Written 26

101 192 600 601 640 180 102 135 161 174 501 702 620 194

[117]2 7 X 7 118 III 13 118 VII 12 118 [.. ..] 119 [.. ..] before 121 120 XII 12 12[1] J2 8 ) 124 IV 5 124 v 124 IX 4 124 [.. ..] before 127 29 13[0] VI 28

Ephemeris for 118 and 119 118 113 to 173 [115] to 181 ... 131 to 161(?) 120 to 125 ... 121 113 to 130 124 to 156 124to 131 123 to 202 [123] to 182 [125] to 194 130 [I] 30 to [XII]

22 I am indebted to Dr. Sachs for the investigation of this whole material. 23 Joins between this group of SH-texts and Sp.-texts are, e.g., Nos. 122 and 420. "Pausanias I, 16, 3 (ed. Schubart, p. 34, Loeb Cl.L. p. 80/81). Cf. also Bikerman IS p. 176 and CAH VII p. 187 f. 25 The lack of a horizontal line at the end of a stroke indicates that the text is a fragment which does not permit us to establish its accurate extent. 26 The first number refers to the year of the Seleucid Era (cf. below p. 32); the second gives the Babylonian month; the third, the day. 27 Also [118] or [119] would be possible; cf. the discussion of this colophon on p. 17. 28 Cf. the discussion on p. 19. 29 Cf. the discussion of this colophon on p. 20. 30 Cf. p. 181.

§ 2, Colophon Zmab and Zo Zkc Zq

Text No. 430 SOla 122 194a 18

Written

C. THE SELEUCID KINGS.

Epheme ris for

186 (?)

97 to (?) 187

209 IX 18 243 [.. ..] 263 [.. ]

208 to 210 243 263

..

C. The Seleuci d Kings

Uruk Date (S.E.)

(104-112 ... ) (104-124) (106-108) ( ... 108-118 ... ) (111-135) (113-161 ... ) (115-124) (115-130 ... ) ([115]-[138]) ( ... 117 ... ) 117 [117] X 7 ( ... 117 ... ) 118 III 13 118 VII 12 118 [ ... .. . ] (118-143) 119 [ ... . .. ] (119)

Date of the Colophon Text Colophon (S.E.) No. No. 100

J

101

K

192

L

600 601

Q

640 102

T y

X Zb Zc

501 185 620 194

Text No. 700 500

A B

c

D E F G

H

J K L M p Q

170 155 100 701 400 606 171 140 150 163 191 101 172 192 600 601 300 640 193

Parker-D ubberstei n BCh. p. 20. Schroede r KSW, No. 32 Rev. 32. (Reference kindly called to my attention by Dr. W. Dubberst ein.) 33 MLC 2182 (unpublis hed), mentione d by Clay BR 2 p. 14 and Goetze [1] p. 46. Parker-D ubberstei n BCh. p.20 quotes as earliest date 188 B.C. July 17 (or 7) = S.E. 124 III 28 (or 18). 34 Cj. Parker-D ubberstei n BCh. p. 20. 35 Antiochu s III, here and in all the following lines. 36 h.s. = his son. 37 Antiochu s, the son of Antiochu s III, died in 120 S.E. (Bouche- Leclercq HS II p. 625). Our text, dated at the end of this year, would therefore hardly have mentione d the younger Antiochu s. Cf. also Parker-D ubberstei n BCh. p. 20. 38 Seleucus IV Philopato r, here and in all the following lines. 39 A square bracket means that the correspon ding date is restored but practically certain, whereas dots indicate an unknown number of preceding or following years. 31

32

M

Colophon

( ... 86-134 ... ) ( ... 89-131 ... )

The Arsacid Era is attested only once in our whole material. Colophon Zq of No. 18, an ephemeris from Babylon, equates the year S.E. 263 with A.E. 199 (= 49/48 B.C.), which agrees with the well known relationship between the two eras. King Arsaces and Queen Piriustanii are mentioned in No. 194a for S.E. 243 (= 69/68 B.C.).

B

7

We are now able to combine our previous results and to arrange the texts from Uruk and Babylon in their chronological order (cf. the lists below). In the case of texts the date of whose actual writing is unknown, the beginning of the time interval covered by the ephemeris can be substitu ted as a fairly good indication of the date. These cases are indicated in our lists by dates enclosed in parentheses. 39

Antiochus III and Seleucus IV h.s. Earliest date: S.E. 123 VII 1433

[Antiochus 35 and Antiochu]s h.s. 36 [106] Antiochus and Antiochus h.s. [117] X 7 Antiochus and Antiochus h.s. 118 III 13 Antiochus and Antiochus h.s. 118 VII 12 Antiochus and Antiochus h.s. 118 Antiochus and Antiochus h.s. 119 Antiochus ...... .. 37 120 XII 12 Antiochus and Seleucus 38 h.s. 124 IX 4 Antioch us and Seleucus h.s. 124 Antiochus ...... .. before 126 [Seleucus] 13[0] VI 28

CoNCLUSIONS

D. Conclusions

The Seleucid rulers mentioned are Antiochus III (the Great) and Seleucus IV (Philopator), and Antiochus (the oldest son of Antiochus III) who died before his father but had been associated with the throne during the last years of his life. The colophons supply us with a series of dates listed in the table given below. From the other documents of this period the following dates are know: Antiochus III and Antiochus h.s. 36 Earliest date: S.E. 10431 Latest date: S.E. 119 X 21 32

Ruler

D.

§ 2, D.

8

CONCLUSIONS

Babylon-continued

Uruk-continued Text No.

Colophon

Date (S.E.) ( ... 120 ... ) (120-125 ... ) 120 XII 12 12[1] I ( ... 121-124 ... ( ... 121-124 ... ( ... 122-133 ... ( ... 123 ... ) ((123]-[130]) (123) (123-142) (123-155) 124 IV 5 124 v 124 124 IX 4 124 [ ... .. ] (124) (124, 125) (124-126) ( ... 125-130 ... ([125]-194) ( ... 127-132 ...

s T

u

) ) )

v w X

y

.

z Za

I

) Zb )

13[0] VI 28 ( ... 133-151 ... ) ([135]-[137]) ( ... 136, 137 ... ) ( ... 137-156 ... )

I

Zc

Ze

1032 180 102 135 136 141 156 1031 173 103 142 160 161 174 185 501 702 104 1 2 130 620 164 194 162 105 106 165

( ... 143-157 ... ) ( ... 146 ... ) (148-161) 149

199 143 144 6

( ... 157-191 ... )

604

Babylon Text ColoNo. phon ------------------300a ( ... 4-22 ... ) Date (S.E.)

( ... 10-18 ... )

300b

Date (S.E.)

Colophon

Text No.

( ... 49-60 ... )

70

( ... 60, 61 ... ) ( ... 61, 62 ... ) ( ... 61-64 ... )

200h 818 819c

( ... 130-205) (133-153) ( ... 134-146) (137-160) (141) (141 ... ) ( ... 141-147 ... ) ( ... 142) ( ... 142-195 ... ) (142) ([145]-[149]) ( ... 146 ... ) ( ... 146-149 ... ) (146-148) (147' 148) (147-218) (149) (150-155) (154) (155) (155-243)

820aa 820a Zj

821, Zl

Zla

602 301 609 60 3 3aa 50 3a 610 3b 4 Sa 4a 5 654 603 6 6aa 6ab 6a 704

( ... 161-170 ... ) (166-189)

624 302

(171-180 ... ) (171-243) ( ... 172 ... ) ( ... 172-187 ... ) ( ... 176(?)) (176) (177-199(?) ) (178 I) (178 VII) (179)

620b 620a 6b 501b 119 7 61 80 81 120

Zlb

Zlc

§ 2, D.

9

CONCLUSIONS

Babylon-con tinued

( ... 180-186 ... ) (180, 181) (180- 242) (180- 252) (181) ( . . . 182, 183 ... ) ( ... 183 . .. ) ( ... 183, 184 ... ) ( .. . 183-1 86 .. . ) ( . .. 183- 205 . .. ) ( ... 183-219 ... ) ( .. . 185-188 ... ) ( ... 185-197 ... ) ( .. . 185- 222 .. . ) (185) (186) ( ... 187- 204 .. . ) ( ... 187-230 . .. ) (188, 189) ( . .. 189) ( ... 189- 222 ... ) -

--

Text No.

Colophon

Date (S.E.)

-

Babylon-continued

- - --

-

82lb, Zld 822, Zm

-

- -- - - - - - - -

(190, 191) (190-231) (191-1 94 ... ) (194, 195) ( . . . 197- 206 . . . ) ( . .. 199- 206 . .. ) -

( ... 200- 202 ... ( . . . 200- 232 . .. ( ... 201 - 210) ( .. . 201 - 224 . . . (202) (202-267) ( . .. 202- 273) ( . . . 203- 225 ... (203-276 ... ) (204-221) (206- 220) ( ... 209-218 . . . 209 IX 18 (209, 210)

-

) ) )

75 7a 420 611 121 8 Sa 8b 303 621 421a 121a 604a 621a 9 10 421 612 11 11 a 605

12 622 61a 13 613 51

823 Zn

-

~---- --

I 823a, Zna I I

Zp )

) Zo

13a 622a 613aa 704a 14 623 613ab 705 613a 76 51a 607 122 15

Colophon

Date (S.E.)

Text No.

( ... 214-220 ... ) ( ... 216-230 ... ) ( . .. 217-237 ... ) (219)

303a 303b 608 16

( ... 229 ... )

16a

- - - - 1 - - - -- - - - - - -

(235) ( . . . 236-258 .. . ) (236) ( ... 239- 247) ( ... 243-249 ... ) (243) (246-254 . . . ) ( ... 248, 249 ... ) - ------ - - - -(253) --;63 [~~~-.-..

Zk

123 410 123aa 614

Zkb

Zkc

- --

J______

52 194a 411 16b - - - -17

Zq--~-~8---

I

( .. . 265- 281 .. . ) (266- 269 ... ) - - ( -. . . 298- 353 . .. )

412 18a

I

----~--~3--

The chronological distribution of the texts, as listed in the above tables, is once more represented in the graph of Fig. 2a, where I have plotted the number of

25

, ,r

-··

.,'•'

20 I I

15

•' '

/

''I

'

Uruk

Bab,lon

,'

10

''

5

o.

50

100

1.50

200

25 0

300

lflri>er ot OpM>II>1'1do• vhlch bepn 1n t.h• .IUD d6C'ade of ;.a

Seleucid tr,a

Fig. 2a

texts per decade. The fact that often the exact beginning or end of a tablet is unknown has very little effect on the lunar ephemerides , which rarely concern more

S.E.

10

§ 2, D.

CONCLUSIONS

than two or three years; for the planetary texts, a possible displacement of more than one decade is rather unlikely. Thus our graph can be assumed to give a fair representation of the texts whose dates can be ascertained. The distribution of the texts from Babylon is very peculiar. On the whole, these texts begin when the material from U ruk ends. But the earliest texts of all are three texts from Babylon, Nos. 300a and b and No. 70, which precede the remaining texts from Babylon by more than a century. No. 70 is exceptional in two other respects: it is the only auxiliary table 41 known from Babylon, and it is the only auxiliary table of System A-all other auxiliary tables (30 texts) come from U ruk. There are four texts from Babylon in our material which are not represented in the graph of Fig. 2a because the dates do not seem to be significant. These are first the three procedure texts No. 200h (moon), No. 818, and No. 819c (planets), all of which contain numerical samples ranging over the years S.E. 60 to 64. Unfortunately, none of these numerical sections is computed with the regular methods, and their association with procedure texts leaves the possibility open that we are dealing with excerpts from earlier sources which belong to a level different than the rest of our ephemerides. Finally No. 430 + SOla (+ 821aa) is a very unusual combination of ephemerides for Venus and Mars extending from S.E. 97 to 111 . . . and from ... 170 to 187, probably written in S.E. 186. Once the texts have been arranged in two major archives, the question naturally arises whether or not these archives have anything to do with the various "Systems" of computation. Looking at the distribution of the planetary systems, one realizes, however, that no clear correlation exists between computing system and locality. 4 2 This result is not very surprising because the planetary ephemerides are computed according to many more than two systems, and the various types are not too sharply separated so far as the basic principles are concerned. 43 Quite different, however, is the situation in the case of the lunar ephemerides. Here we have two clearly distinct methods, the in some respects more primitive "System A" and the refined "System B". From the above-given lists of Uruk and Babylon texts, it is evident that Uruk represents lunar System B (text numbers beginning with 100) because only two lunar texts (Nos. 1 and 2: S.E. 124 to 126) from Uruk are computed according to System A against 52 of System B. Babylon, on the contrary, favors System A (text numbers below 100). If we add undated fragments, we obtain the following total:

Lunar Ephemerides

Uruk

Babylon

System A System B

2 52

61 27

This result has some unexpected consequences. The above lists show that the ephemerides from U ruk cover the short period from about 80 S.E. to 160, whereas the Babylon texts begin later, about 130 S.E., but extend to the very latest period of cuneiform writing (S.E. 360 = A.D. 49). 44 In other words the older archive at U ruk represents the more highly developed System B, but the later archive at Babylon favors the supposedly older System A. The obvious explanation of this fact is that the chronology of the archives has nothing to do with the chronology of the systems. The early end of the Uruk group is probably related to the occupation of Babylonia by the Parthians in 171 S.E. In particular it seems as if the dates of the U ruk texts were determined by the history of the Res sanctuary. 45 We know from building inscriptions that this sanctuary was rebuilt in the years S.E. 68 and 110 46 whereas the latest mention of it is found in a text written in S.E. 173.47 We do not know the exact locality where the Uruk fragments were found, but the German excavations of 1912/13 centered around this region, 48 and the colophons show a close relationship of the scribes to the Res sanctuary. Thus we come to the result that the density of our material reflects only the fact that we possess a fairly complete selection of texts from two archives whose 41 I refer here to auxiliary tables in the strict sense, that is, extensive lists of single columns only, in the present case B 2 and E 2 • 42 Example: Jupiter, System A: Nos. 600, 601 from Uruk; Nos. 602, 603 from Babylon. Jupiter, System B: No. 620 from Uruk; Nos. 621, 622 from Babylon. Mercury, System A,: No. 300 from Uruk; No. 301 from Babylon. Saturn, System B: No. 701 from Uruk; No. 704 from Babylon. 43 Schnabel's distinction of three schools, following Pliny, finds no support in the actually preserved texts (Schnabel, Ber. p. 212 f.). 44 The latest text published here is No. 53 (S.E. 353 =A.D. 42). Only recently Dr. Sachs discovered a solar eclipse table (B:\!136599 [= 80-6-17,328 + 444] + BM 36941 [= 80-6-17, 682] plus duplicate BM 36737 [ = 80-6-17,470] and BM 47912 l = 80-11-3,619]) which concerns the years S.E. 342 to 360 ( = -30 to A.D. 49) and which contains columns F and (/> computed according to System A. The remaining columns are only approximate, as far as can be seen in the present state of preservation.-For Uruk the same upper limit which we found here can be recognized also in the economic texts (cf. Schroeder [1] p. 20).--The latest date known from cuneiform texts is an "almanac" for 75 A.D. (from Dropsie College in Philadelphia, U.S.A., to be published by Sachs and Schaumberger). 45 Traditionally called bit-res. For the correct reading cf. Falkenstein TvU p. 4. 46 Cf. Falkenstein TvU p. 4 ff. 47

Falkenstein TvU p. 9.

Called Wuswas by the Arabs; cf. MDOG 47 (1911) p. 47, 51 (1913) p. 47 ff., 53 (1914) p. 9 ff. 48

§ 3.

THE CoLOPHONs

local history determines the beginning and end of our information. Consequently no conclusions can be drawn from the dates of our texts as to the real time of origin of the underlying astronomical theory. Unfortunately, Schnabel's attempts 49 to establish the date of origin of the methods used in the ephemerides have enjoyed widespread acceptance in spite of serious objections raised by Kugler. 50 Schnabel's procedure is based on entirely unfounded suppositions

§ 3.

11

as to the accuracy of numbers which he used and regarding the initial observations which he assumed as the point of departure. 51 All that can be said with safety at present is that the methods for computing lunar and planetary ephemerides were in existence around 250 B.C. Their previous history is unknown to me. 49 50

51

Schnabel, Ber. p. 219 ff. and [1] p. 15 ff. SSB II p. 604 ff. Cf. Neugebauer [18] for details.

THE COLOPHONS

A. General Remarks In the preceding chapter we made frequent use of the information furnished by the colophons which are found on several tablets, especially those from the U ruk archive. Transcriptions and translations of all these colophons are collected in chronological order in section C of this chapter.1 We will now analyze these colophons in detail, one aim being to extract information about the scribes of our tablets. A colophon in the usual sense of the word is a note at the end of a tablet indicating owner, scribe and date of the tablet. In addition, we frequently find on the upper edge 2 of a tablet an invocation and sometimes also a brief remark ("title") concerning the contents of the text, or even rules how to compute certain parts of the ephemeris. 3 The most complete version of the invocation known to me is given in Clay BR 4 No. 8, a text written in Uruk in the year 61 S.E.: "According to the command of Anu and Antu may whatever I do go well in my hands; 4 may I be satisfied with its abundance". 5 This prayer is shortened in all astronomical U ruk texts to ina amat Anu u Antu liSlim 6 "According to the command of Anu and Antu, may it go well". In the texts from Babylon, Anu and Antu are replaced by Bel and Belti "The Lord and my Lady". Some of the Uruk tablets contain at the end a sentence which is a kind of counterpart to this initial prayer, begging for the preservation of the tablet. sa The shortest version is given by No. 171 (F): piilib Anu u Antu lii itabba!Su

"He who worships Anu and Antu shall not remove it (the tablet)". This is elaborated by MLC 1873 7 and Nos. 600 (L) and 180 (S): prilib Anu u Antu ina surqu 8 la itabbalsu

"He who worships Anu and Antu shall not remove it by thievery." A still more elaborate form is given by No. 194 (Zc):

piilib Anu u Antu ina surqi lii ittabalsu sa itabbalusu Adad u Sala litbaliinisu

"He who worships Anu and Antu shall not remove it by thievery. Whoever does carry it off, 9 may Adad and Sala carry him off." In No. 135 (U) one finds this formula in a cryptographic writing: piilib 21 50 10 40 la [ ..... ] which makes it highly probable that we have to read 21 as Anu, 50 as u, 10 40 as An-tu. A similar cryptogram is used in AO 645810 where we read tuppi 121 35 35 26 44 mar 121 11 20 42. In No. 600 (L) we find the name 121-aba-ut-ter-ri. This shows that the interpretation of 21 as Anu fits excellently with these names, but I have not succeeded in deciphering more.U Perhaps 35 35 represents a a = abu. Related to these formul~

1,S7

*"

J,Zl.

3, 2S

I 3"1*"'

l 31 ** 328~

4:,-

I, 36 1, 3~ I, 42.

z, 36

47

1,4-1

31 ~2.

so 53

1,47 I, SO

s6** '· ~-3*"'

z, 33

l,

31

z, 4 1 2,44

z,47 2.,

!J{)

4,~6 ~. 19 "t,S2

40 3, '3 ' -43 .3,6 3. ~ "'* -4 6 -4, g 3, II 4, II 3,14 3, 17 4, tit .3, ,, 16

3,0

ll. *1
10° (1) Longest day M = 3;36H Shortest day m = 2;24H From the application made, it follows that the same values in inverse order were considered valid for the longest and shortest night. The computation of column C presupposes column B, i.e., the longitudes of the moon. In the following tabie Bl

c

B2

-"'

Daylight in 10° of large-hours

T ~

)(

9 6\ !JP

.;1' ()

=

3

3; zo J.

32_

3.3,

==l;lZ

)(

J,zo

~

'J>

III

t1

3

""'

()

=

)(

){ 9 6'l 1Jl?

Z;40 l-Z8

Z._l.4

z,ze

z,4o

Interpolation per degreg beyond 10

+ 0; o, 4o + o, 0. 24 + 0.

0

- o, o,

-

g

I

0; 0, Z4

0.

o,

o 4o

!';0.~ 0.

0

+

0;

o,

¥

0,

l~

-

+ +

q

r

0. 0, 40

the values of C can be obtained directly for every 1Oth degree of the zodiacal signs. For every position beyond

C, AND

E

47

10°, linear interpolation by means of the last column must be carried out. Examples: No. 1 obv. III,10: B2 = 61. 6;36 = Q15 10 + 26;36. Thus, multiply 26;36 by +0;0,8 and add the result +0;3,32,48 to the value 2;24 given for C in Q15 10. The sum 2;27,32,48 is given in obv. IV,10. No. 16, obv. III,6: B1 = n:l! 20;16 = n:l! 10 + 10;16. Thus, multiply 10;16 by -0;0,40 and add the result -0;6,50,40 to the value 3;20 given for C in n:l! 10. One finds 3;13,9,20 as given in obv. IV,6, Rules for checking are found in procedure text No. 200b Section 3 (cf. p. 215) in the form that C(n + 12) can be found from C(n) if both B(n + 12) and B(n) belong to the same zodiacal sign and to the same type of arc (slow or fast).

E. Column E This column describes the latitude of the moon's center. The details, however, of the computation of E arc so involved that it seems convenient to describe them by means of several steps even though they were in practice condensed into a single process. The general remark must be made in advance that all units of the first two places are to be interpreted as "barleycorn" (se). For example, the maximum 7,12,0,0 of E as given by (5) p. 48 is to be interpreted as M = 7,129• Because 1" = 0;0,50° (cf. General Introduction p. 39), we have finally M = 7,128 = 6° which is an adequate value for the extremal latitude of the moon, augmented by the parallax. All other numbers are to be interpreted accordingly; e.g. in (2): d = 1,58;45,428 = 1;38,58,5°; D = 2,6;15,428 = 1;45,13,SO etc. Column E is not a simple zigzag function but depends upon variable differences LlE. Consequently, we can no longer speak of the periods of E in the same sense as with linear zigzag functions. Nevertheless, we can define "mean" periods which play the same role for the general theory as the ordinary periods in the simple case. 6 One finds the following values for these mean periods: 3,5,32 Po = 3 21 20·24 22 11 = 0;55,17,22,18, •• • (1)

0, 40

-

B*,

Po

=

, ' , , '

3,5,32 15,48;24,22,11

=

11 ;44 •15 •18• ..•

I. LlE Our first goal is the computation of the differences LlE of E, though none of the texts of System A gives these differences as a separate column. 5 6

Cf. above p. 33. Cf. Neugebauer [7] p. 237 f.

48

SYSTEM

A,

The function ~E which we shall now describe is closely related to column B. It is a step function with jumping points at exactly the same places as column B, i.e.:

(1)

t=lll'13

~ = )( 27

El:

The values in between are d = 1,58,45,42

(2)

t=)(13 ~ = 1ll' 27 D = 2,6,15,42

respectively. For an interval which contains a jumping point, the resulting value is given by

d' = 1,58,45,42 + 15,0 (s1 - s) D' = 2,6,15,42- 16,0 {u1 -a)

(3)

t- jump ~-jump

where s1 - s and u1 - a represent, as in column B, the arcs beyond the jumping points, expressed in degrees (Fig. 18 p. 46). Ifxample: No. 9 rev. III,13 gives B2 = =::= 20. Thus, we have u 1 - a= =::= 20 - 1lJ' 27 = 23° and therefore D' = 2,6,15,42- 16,0 · 23

= 2,6,15,42- 6,8,0 = 2,0,7,42 . The essential point in the computation of ~E lies, of course, in the determination of the values d' and D' because (2) is sufficient elsewhere. It is therefore convenient to possess formulae which permit us to compute d' and D' in a row without using the formulae (3) more than once. This can be done by means of the following rules: d' decreases from t to t to by 2,46,0 but not below d = 1,58,45,42. When subtraction of 2,46,0 would result in a value less than d, the amount 4,44,0 must be added to the last value above d. D' increases from ~ to ~ by 2,46,0 but not beyond D = 2,6,15,42. When addition of 2,46,0 would result in a value greater than D, the amount 4,44,0 must be subtracted from the last value below D. Example: D' = 2,4,51,42 + 2,46,0 = 2,7,37,42 No.9 rev. 0: >D 4,44,0 2,0,7,42 2,46,0 + No. 10 rev. 13: D' = 2,2,53,42

No.9 rev. 13:

II.

D' =

CoLUMN

E

OuTSIDE THE NODAL ZoNE

We assume ~E to be computed according to the rules described in the preceding section. If we would start to compute E by adding the values from ~ E step by step and reflecting at values ± M, we would obtain a function composed of two linear zigzag functions, one with difference d, the other with difference D, and joining each other exactly at the jumping points of B.

COLUMN ~E

Actually, this is the rule for computing E only for such values y of E which do not belong to the strip (4)

K

= 2,24,0,0 > J > -

K

= - 2,24,0,0.

We call this strip the "nodal zone" and shall postpone the rules for the computation of E inside the nodal zone to the next section. Outside the nodal zone, we proceed as usual with zigzag functions, using either d or D, or the values d' or D' as given by ~E. If we approach a maximum or minimum determined by M = -m = 7,12,0,0

(5)

we change the direction according to the parameters 2M- d = -(2m + d) = 12,25,14,18 2M- D = -(2m+ D)= 12,17,44,18

(6)

in all cases where the extrema belong to an interval where ~E = d or ~E = D . For intervals which contain jumping points, (6) must be replaced by the specific values 2M - d' = -(2m + d') 2M- D' =-(2m+ D')

(7)

required by d' and D'.

Examples: (a)

No. 1 obv. 111,7: B 2 = H 6;36 thus ~E 2 = D and therefore 6,0,46,18 E2 obv. V,6 6,16,58,0 obv. V,7 2M- D = 12,17,44,18 [cf. (6)].

(b)

No.1 rev. [-II],1: B2 = Tll4;7,30 thus ~E 2 = d and therefore -6,35,56,30 E 2 obv. V,12 - 5,49, 17,48 rev. [0], 1 (2m+ d)= -12,25,14,18 [cf. (6)].

(c)

No.1 rev. [ -II],13: K and the differences given by L\E all equal D for the subsequent six lines. Thus, we have Y2 = Yo - (2D + K) = 2,42,11,54 - 6,36,31,24 = -3,54,19,30 < - K as the point on the other side of the nodal zone. Therefore, we can find y 1 either from

YI = 2(yo - (D

(9b)

=

as follows from (Sa). In the second case, (Sb ), two values will fall inside the nodal zone. We define them as follows:

(lOa)

49

Fig. 19

(Sa)

(Sb)

COLUMN

Y1

= 2(Yo - (dl + z) K

)

+ Z) ) K

2(2,42,11,54 - 3,1S,15,42) 2. 36,3,48 = - 1,12,7,36

= or from

+ 2) 3,54,19,30 + 3,18,15,42)

Y1 = 2(y2 + D

= 2 (= - 2. 36,3,48

K

-1,12,7,36

=

Our result is therefore:

(lOb)

Yo =

or, because of (Sb ), by the equivalent rule (lOc)

All these rules are the consequence of the following corstruction of the function E: outside the nodal zone of width 2K, the slope of E is determined by the differences given in L\E. Inside the nodal zone, however, the slope is twice the difference given in L\E (cf. Fig. 19). Before we give examples, it is convenient to list the numerical values needed in (S), (9) and (1 0) in the special case where the differences are either d or D.

2,42,11,54

-------------------

Yl

= - 1,12,7,36

y2

=

- 3,54,19,30

as given in the text rev. V,6 to 8. In all cases where (a) the difference in L\E is constant and (b) two steps are sufficient to pass from one side of the nodal zone to the other, the actual differences furnish a valuable check because we have from (8

L\E

K

Y1

Y2 < -

K

Yo - Yr = -y2 Y1 - Y2 = Yo

50

SYSTEM

In our example above, we find E Yo= 2,42,11,54 ---------------------

Y1 = -1,12,7,36 y 2 = -3,54,19,30

A,

3,54,19,30 = -y2 2,42,11,54 =Yo

Y1 =Yo+ Y2 where y 2 is given by

+ K)

or Y2 =Yo - (2D

+ K).

Example 2. Jumping Point of L\E Inside the Nodal Zone No.9 rev. V,ll gives the valuey 0 = -4,10,53,24 forE in an increasing section. The subsequent differences in L\E are d1 = D = 2,6,15,42 d2 = D' = 2,0,7,42 d3 = d = 1,58,45,42 Thus

Yo

+ (d1 + d + K) 2

= - 4,10,53,24

+ 6,30,23,24 =

2,19,30,0
M obtained by (Sb) a reflection on M or m = - M according to the usual procedure with zigzag functions. Comparison of (5a) and (3) shows that in the case of small values of E (i.e. E = e) (6)

if sign K = +1 if sign K = -1

'P' = 'P 'P' = -'P

This gives the precise meaning of the above expression that o/' is "essentially" equivalent with lJ' for the nodal zone.

Example: We compute from a given section of E2 the corresponding values which lead to o/' 2 (cf. table below):

E2

----·-----·... '· zo. 18, " ' < I
1< + .s: 4! 40,48 > I( + 6, 3.s; 33, 30 >

I
~i--~'--~1~--· \ --+1~• ---+------+--night

d~ytime

1....: ---6-H

N3.----=C:.!...,~.:....:l~-J+l:l' M,--J {,M,-C, ·I· I C,-1

--~-----~------4-----·~------4-----~r---

t-ol•.---- 6

H

---~--11

M,-----1

Fig. 30

Similarly it follows from Fig. 30 33 that N 3 1s gtven either by (Sa)

the crescent and with the delay of moon set. It might be added that similar criteria are found in Arabic astronomy and especially in Maimonides. 36 '.

(Sb) In each individual case it has to be decided which of these values of N 1 or N 3 is to be used for the computation of the elongation. The majority of cases will cause no difficulties and (7b) will be the proper interval. The main importance of the computation of P 1 lies, however, just in the treatment of doubtful cases where (7a) or (7c) might be possible. 34 We know from the ephemerides that in critical cases two solutions were computed, one under the assumption of a full moon, one for the possibility of a hollow month (expressed by the phrase ina 30-su). The precise criteria for accepting or rejecting alternative possibilities are not known to us. The study of the few preserved cases of alternative solutions as well as similar experience with ephemerides of System B seems to show that the values of P alone cannot have been sufficient for a decision. Tentatively a criterion of the form

might be offered as explanation of the selection made by the texts. 35 A good reason can be given for considering the sum of elongation and time of setting. The elongation measures not only the angular distance between sun and moon but also the width of the visible crescent. In Fig. 31 the lunar diameter LL indicates the

·.:

,t\

•

- · · E

~

0

---------• 0

Fig. 31

For P 3 the alternative solutions indicated in No. 5 seem to suggest a condition

(10) for last visibility, where the value of c3 seems to lie between 18 and 22, these limits included.

v.

SUMMARY

We are now in a position to describe the whole procedure of the computation of M in its relationship to K, P 1 and P 3 • We illustrate it by an example taken from Fig. 30 illustrates two cases which lead to (Sa). The smallest values of M, where (7a) is applied are 4;12H in No.5 rev.11, 4;26H inNo.18 obv. 11 and4;26Hin No. 12 line 9. Kugler (SSB II p. 434) considers 15.5h = 3 ;52,30H as smallest possible value. According to Schoch (Langdon-FotheringhamSchoch VT p. 97), the age of the moon seems to vary between 16.5b ( = 4;7,30H) and 42.0b ( = 10;30H) in order to be seen at Babylon. In the ephemerides, however, we find many instances where more than 48h ( = 12H) were considered necessary (e.g., No. 10 month VI: conjunction about 1 ;40H before sunset on the 28th of a 30-day month; thus N , = 13;40H). 30 Cj. the commentaries to Nos. 5, 7, 12 and 18. 36 Cf. Neugebauer [16] p. 356 fl. 33

(9)

.

5 1:-

or by

34

68

SYSTEM

A,

VISIBILITY CONDITIONS. ECLIPSES

No. 5 (cf. also the corresponding graphical representation in Pl. 14-0ff., as well as the commentary to No. 200 Section 15, p. 206f.).

K,

No.5

X

obv. II. 12. 13. 14.

3,3,,27 3,19,35 J,29.23

:xr

][

r.

28 29

2.9 2B

"'o.·

9, 19

.so

2,32.,52 .56 s,l3.17 .so 1.43.S~

so

P, X[ I

I

(b) We will now use K 1 in order to compute the date of the next conjunction according to (1) p. 64-. We thus obtain

- K1 hence

=

X 29 6;9, 19H before sunset -3 ;36,27H X 29 2;32,52.

Now we add the 29 days, which are always understood to be added in K, and take into account that month X has 29 days. Thus the new conjunction will be XI 29 2;32,52H before sunset. (c) The first visibility after the conjunction on the 29th will certainly give to the month XI a length of 30 days. P 1 is found 38 to be zoo. The new crescent will therefore be visible at XII 1

zoo

duration of visibility.

(d) Before going to the next month, we determine the last visibility which precedes the conjunction just found. The sunrise of the 28th seems a plausible guess and one obtains 39 for P3 the value 13°, which shows that XI 28 13° before sunrise (kur) describes the last visibility before the conjunction. (e) We now proceed in the same fashion to the next conjunction XI 29 2;32,52H

- K1 thus conjunction:

=

14,10

XI 20

(a) We start with the conjunction (M 1) at the end of month X at the 28th 0;9, 19H before sunset (su ). The new crescent can be expected for the evening of the next day. We therefore compute P1 for this evening and find 37 for the duration of the visibility 14-; 10° which is sufficiently large to guarantee the reappearance of the moon. The month XI therefore begins on the 30th of month X (or month X has 29 days).

X 28 0;9, 19H

(f) The next visibility is either to be expected at I 1 or at XII 30. In the first case, we obtain P1 = 25°, which is a rather high value, or P1 = 10;30° in the

XI 30 8;32,52H before sunset - 3;19,35H XI 30 5;13,17 29d (XI has 30 days)

+

XII 29 5;13,17H before sunset.

25 ina 30- .Su 10.30

13,2.0

P:s XI:

28

:r:

2.7

11: 2.7

13 kvr 15,10 kur 9.30 kur

26 .•.

second case; and for the elongations 21 o and 9° respectively. The totals 0 1 P1 of 4-6° and 19;30° seem to exclude the second solution. Month I is therefore assumed to be a month of 30 days.

+

N. Eclipses I.

INTRODUCTION

The source material for the investigation of the Babylonian theory of eclipses is very restricted: we possess ten texts from System A and only three fragments from System B. 40 Only one of the texts of System A, the lunar eclipse table No. 60, is well enough preserved to show us at least the major part of the procedure followed, but also this text is incomplete. The remaining texts furnish only checking material, except for No. 61, which seems to follow a different principle but is so badly preserved and written in so disorderly a manner that no results can be obtained from this fragment. Our discussion will therefore be based almost exclusively on No. 60. The lunar ephemerides, discussed in the preceding sections, are so arranged that one can directly compose from them the eclipse tables. The consecutive lines in the ephemerides correspond to complete series of conjunctions or oppositions respectively, i.e., exactly those moments when eclipses are possible. The second condition for the occurrence of an eclipse, the nearness of the moon to the ecliptic, can immediately be checked from column E, the moon's latitude. Accordingly, eclipse tables can be obtained as excerpts from the lunar ephemerides. One has simply to select those lines in which the smallest absolute values of E occur. The characterization of the eclipse tables as mere excerpts from the lunar ephemerides implies a severe restriction of the usefulness of these tables for the prediction of solar eclipses. The ephemerides do not take into account the size and distance of the celestial bodies involved. Consequently, neither geographical 37 38

39 40

The elongation would be 12°. Elongation: 17°. We obtain for the elongation 12;30°. Nos. 130, 135 and 136.

A,

SYSTEM

coordinates nor parallaxes are included in the attempt to predict solar eclipses. It will therefore be easy to exclude the possibility of solar eclipses in certain cases, but hardly possible to predict the actual visibility of an eclipse of the sun. Lunar eclipses can be predicted, of course, with a much higher chance of success.

II.

THE CoLUMNS T

TO C

The arrangement of the columns of eclipse tables corresponds, in general, exactly to thP. arrangement in ordinary ephemerides. The first column i3, consequently, the column of dates which gives the year {in the Seleucid Era) and the month. In addition, the cases of a five-month interval, instead of the usual six months, are marked by 5 ab "five months". The next column is (/1, following the arrangement of the ephemerides. From the isolated values, however, it would not be clear whether they belong to an increasing or decreasing section. The signs tab and lal are therefore added for t and ,), respectively in Nos. 60 and 52. Column (/1 is omitted in No. 50. The next column is B, followed in No. 50 by a column of unknown significance. Nos. 60 and 52 give the expected column C.

III.

THE COLUMNS

E

AND

lJ'

The values, given in column E, are the values of the latitude of the moon at the syzygies nearest the ecliptic. Two signs are added to these numbers. The first sign indicates whether the latitude of the moon is northerly (lal, according to our modern terminology "positive" latitude) or southerly (u, modern "negative" latitude); the second sign refers to the direction of change in the moon's latitude. If the latitude changes from north to south of the ecliptic u is added (denoted here by ,), ) whereas the opposite movement is indicated by lal (our t ). This notation, used in Nos. 60 and 51, also occurs in the ephemerides, e.g., Nos. 1, 4, 9, etc. The next column, lJ', measures the magnitude of an expected eclipse. If lJ' :2:; 34,48,0 no eclipse is possible. The same holds for lJ' ~ 0; this latter case is characterized by an added be in No. 60 (and perhaps in No. 61). No. 51, however, has no be in the only two cases where it should be expected in order to §3.

69

EcLIPSES

distinguish between lJ' < 0 and 1JI > 0. In the ephemerides, be occurs in Nos. 9 and 10. In No. 61 (rev. 11,10 and 12) we find the words "north" and "south" apparently related to 1JI but the details escape us.

IV.

THE COLUMNS FROM

F

ONWARDS

The remaining part of the eclipse tables must serve the purpose to decide about the actual visibility of an eclipse. Some cases are already eliminated because of Column lJ'. Additional cases, however, must be discarded because the conjunction takes place at night or the opposition falls in the daytime. The exact moment of the syzygies mu.. t therefore be determined. For thi.., the remaining columns are needed, some of which are preserved in our texts: F in Nos. 51 and 60 and G in No. 60 only. In No. 61a we find the totals of G, J, and C' for the 6- or 5-month intervals which separate eclipses. We must, of course, assume that also K and M were contained in complete eclipse texts. From M the moment of the middle of an eclipse would be known. It would be very interesting to know how far the theory went beyond this point. The main point would be to compute the duration of an eclipse of given magnitude. Our scanty text material does not permit us to answer this question. The values given for F are again characterized as increasing by an added tab and as decreasing by lal. The values of G cannot be directly used as given in the ephemerides, where G indicates the excess beyond 29 days of the time which elapsed between two consecutive conjunctions or oppositions. Between eclipses, however, not one month but five or six must be breached. This total can be obtained by adding all values of G for the intermediate months, disregarding, of course, an integer number of days. A column with these values, which we call .EG, is incompletely preserved in No. 60 and still more fragmentary in No. 51 (cf. p. 113). It must be said, however, that the comparison of the expected values with the values given in No. 60 shows many deviations which I am not able to explain satisfactorily. For the details, see the commentary to No. 60. Much more serious discrepancies appear in the values of .EG in No. 61a. The reason may be the use of different interpolation schemes, the existence of which is also suggested by procedure texts (No. 207ca, p. 259 f.).

SYSTEM B

Introduction The classification of the ephemerides into two "Systems", A and B, is based on the different way in

which the solar anomaly is accounted for: step function versus linear zigzag function. Parallel with this fundamental distinction go many other features, e.g., different

70

SYSTEM

B,

COLUMNS

values of parameters for periods, amplitudes, etc. This does not exclude, however, that essential parameters are identical for both systems, e.g., the relation

(1)

1 year= 12;22,8 synodic months

The differences in parameters and in methods permits us to classify not only ephemerides but also procedure texts. The majority of the latter follow strictly System A. In some cases, however, we find sections following System B even in texts which otherwise belong to System A. Two procedure texts (Nos. 210 and 211) are essentially of System B type and it is from one of these (No. 210) that we obtain informutiuu about such ua:sil: parameters as the length of months and years. The value of 29;31,50,8,20d for "the month of the moon" is identical with the value which is characteristic for the mean value of Column G of System B (cf. below, p. 78) and thus assigns the whole section to this system. The following parameters are listed: (2)

12 months of the moon returning to its place = 5,27;51,20d

(3)

18 years of the moon returning to (its) place = 1,49,44;31,20d

(4)

18 years of the moon = 1,49,45;19,20d

The relation (2) gives the lengths of 12 sidereal months, relation (3) the corresponding length of 18 years which contain 241 sidereal months. In (4) we have the length of 18 years or 223 synodic months. Then follows

(5)

18 years of the sun returning to its place in 18 rotations= 1,49,34;25,27,18d

which gives a length of 6,5;14,44,51d for one sidereal year, or simply for one "year" since all available evidence points to the sidereal period of the sun as the only astronomical definition of the solar year beside the calendaric year of 12 or 13 lunar months. The same texts give also two "epacts," one of 10;53,52,42d for the excess of 18 lunar years (4) over 18 solar years, and the other as 11;4 for the excess of a solar year over 12 lunar months, a parameter which plays a fundamental role in the planetary theory (measured in tit his). Both parameters seem to be called "gaba-ri mu-an-na of the sun." These seem to be the basic parameters upon which System B was built. For the practical computation of ephemerides many variants were introduced, as will become evident from the following discussion of the single columns. From now on we follow the arrange-

T

AND

A

ment given on p. 43. The actual grouping on the tablets often deviates very considerably from this list. Moreover, no ephemeris contains all columns, and rounding off is very common. Consequently, the investigation of texts of System B faces, in: general, many more difficulties than System A.

A. Column T For the column T of ephemerides, the remarks which we made for System A (p. 44) hold. In auxiliary tables, column T frequently covers many years, thus giving us a better chance to find undamaged dates. The year numbers are written either in sexagesimal or in decimal notation.

B. The Columns A and B I.

COLUMN

A

According to our definition, a lunar tablet is said to belong to "System B" if the solar velocity is assumed to be a linear zigzag function. This is more than a purely formal definition. The assumption of a continuously variable solar velocity, in contrast to the simplified model of only two velocities adopted in System A, has far-reaching implications which influence the whole structure of the subsequent procedure. Consequently, texts can be assigned to System B even if the column A for the solar velocity is not preserved or is not given. Two types of column A are attested, of four and three sexagesimal places respectively, the latter being obviously an abbreviated form of the former. The difference is common to both of them (18 units of the second place), but the periods deviate already in the third place (12;22,8,53,20 and 12;22,13,20 respectively). It must be assumed that the abbreviated form was not used without corrections over too long a period. I.

Unabbreviated Parameters

The parameters are as follows:

d = 18,0,0 M = 30,1,59,0 m = 28,10,39,40

(1a) and consequently

(1b)

2M- d = 59,45,58,0 2m d = 56,39,19,20

+

Ll

=

11- =

1,51,19,20 29,6, 19,20

and (lc)

p

11

2,46,59

= -i3 30 = 12;22,8,53,20 =

' 13,30 p

=

2,46,59

SYSTEM

B,

CoLUMNS

The value P = 12;22,8,53,20 gives the number of months after which the solar velocity returns to the same value or, using modern terminology, the length of the anomalistic year expressed in mean synodic months. If we ask for the return to the same longitude, 6,0 the sidereal year, we obtain - = 12;22,7,51,53, ... fL

The distinction between anomalistic and sidereal year is, of course, not to be taken as a historically adequate description. There is no reason to assume that a theoretical distinction between different types of years was made. For the Diophant of column A, we obtain as necessary condition: if y 0 and Yn both belong to the same type of section (either both on an increasing or both on a decreasing section), then

Yn - y 0

(2)

=0

must be satisfied. In this case, the distance, expressed by the number n of lines, is given by

(3)

mod. 2,46,59,

Yo and y n both belonging to increasing sections. 2.

Abbreviated Parameters

The following parameters are attested: d = 18,0

M = 30,2,0 m = 28,10,40

(4a)

2M - d 2m +d

= =

59,46,0 56,39,20

~ =

fL =

1,51,20 29,6,20

p =

5,34 27

=

12;22,13,20

n=

27 P

=

5,34 .

For the Diophant, it is necessary that

(5)

II.

CoLUMN

B

The longitudes of the moon at the syzygies are found by summation of column A. If B1(N) indicates the longitude of sun and moon for the conjunction in month N, or B2(N) the longitude of the moon at opposition in month N, then we find the longitude of the moon at the syzygies in month N + 1

B1(N + 1) B2(N + 1)

+ A1(N + 1) + A (N + 1) where A1(N + 1) and A2(N + 1) denote the increase

(2)

B1(N) B2(N)

=

=

2

of the longitude of the sun between the consecutive conjunctions and oppositions respectively.

Tl S.E. 132 I II III

AI

Bt 1;38,42,38 (; 30;10,23,16 II 28;24,3,54

28,31,40,38 28,13,40,38

26;49,42,36

28,25,38,42

(;

IV

Q15

Column A1 is here based on the unabbreviated parameters. Frequently column B is given in rounded-off values without column A, e.g., in Nos. 100, 101, 102. It is impossible in these cases to decide whether A was originally computed with abbreviated or unabbreviated parameters.

Both for abbreviated and unabbreviated parameters

[P]

=

12 even

holds. For the derived zigzag functions A' we obtain:

and

(4c)

the solar velocity by giving the increase in longitude since the preceding syzygy.

Diophant for B

with

(4b)

71

Example: No. 142 Obv. II/IV, 40 ff.

mod. 1,20

1,8,8 n = 4Q(Yn- y 0 )

A AND B

Yn - Yo

=0

mod. 40,

Yo and Yn both belonging to the same type of section. We then find for values from increasing branches mod. 5,34 . The units of the first place of M, m, ~ and fL are degrees per month. In other words, column A indicates

Unabbreviated parameters

= 28,10,39,40

m'

M' = 28,13,49,0

d' = 1,56,0

2m'

+ d' =

56,23,15,20

2 M' - d' = 56,26,2,0 38 ll' = 4,59 P' = 3 + 1,27

with the contribution of s

=

5,49;35,48° = - 10;24,12°

mod. 6,0°

for each normal intervaJ.l 1 The computation of A' can be reduced to much smaller numbers if one operates withy - m instead of withy. Then one obtains:

m'

M'

= 0,0,0 =

3,19,20

d' 2M' - d'

= =

1,56,0 4,42,40.

SYSTEM B, COLUMNS B* AND

72

If this condition is satisfied, we obtain for

Abbreviated parameters

= 28,10,40 M' = 28,14,0

2m'

m'

d' = 2,0

P

I

=

+ d' =

56,23,20

2M'- d' = 56,26,0 1 ll' = 10 3 3

+

each normal interval contributing the amount

s = 5,49;36° = -10;24°

mod. 6,0° .

The values of s are, of course·, also useful for checking of step-by-step computations in intervals of 12 lines. Example: No. 142, unabbreviated parameters. A line preceding a minimum of column A is obv. I,5. Thus we have 27;0,43,56 - 10;24,12 II 16;36,31,56

II

(2)

COLUMN B*

Column B1 indicates the longitude of the sun only for the consecutive conjunctions, while B2 only indirectly furnishes solar positions by means of the diametrically opposite positions of the full moon. In contrast thereto, column B* gives the longitudes of the sun from day to day without relation to the moon, except in the column which we call T*, where the months and days are listed and where the variable length of the months, 29 or 30 days, must be known in advance from the ·ephemerides. There are texts preserved (Nos. 185 ff.) which give the longitudes of the sun from day to day, assuming a ·constant velocity of 0;59,9. If one assumes a year of ·6,5; 14,45d, one would obtain a mean velocity of the sun ·of 6 /;~ 45 = 0;59,8, 17, .. old. In the procedure text ,, , No. 200 of System A, we find 0;59,go/d as mean velocity. The available material does not seem to furnish an explanation of the value 0;59,9.

Diophant for B* Let B* be tabulated in three-place numbers, i.e., consider seconds of arcs as the unit. Each zodiacal sign then corresponds to 30,0,0 units. Let Yv and Yo be two values of B where 11 indicates thaty. is 11 days later than y 0 • The necessary condition for a number Yv to be obtained by addition of 11d mod. 30,0,0 is then (1)

Yv -Yo

=

0

mod. 3.

1,54,7 3 -(y. - y 0)

11

mod. 10,0,0

II= -

Example: It should be investigated whether there is an essential error in No. 186. There will be no essential error if the last line can be obtained by continuous computation from the first line in the number of steps indicated by the total of lines in the text. We have Yv = 26,36,35 Yo= 26,12,8 0 mod. 3 . 24,27

last line (rev. VII,2g); first line (obv. 1,20): thus Yv -Yo

=

Consequently 11

= 1,54,7 · 8,9

=

= 3,3

mod. 10,0,0 .

+ +

87 = 3,3. 90 The number of lines is given by 6 essential excludes The identity with the value of 11 errors.

as in the next line (obv. I,l7) preceding a minimum of A. III.

c

C. The Columns C, D, and Related Columns I. COLUMN C In contrast to System A, the vernal equinox is assumed to fall at cy> go instead of cy> 10°. The extremal values of the length of daylight, however, are the same in both systems. Thus we have (1)

Equinox: Longest day: Shortest day:

cy> go

M = 3;36H m = 2;24H

The scheme of computing the length of daylight (Column C) for any given longitude of the sun also deviates from the scheme in System A. Kugler restored the following scheme2 Bl

B2

8° or

'T'

~

!)

"t

)(

9

61 IJf ""= "l

.1' '!;

:.:= )(

~

lS

= )(

'P 1:1

l.

9 6l '!JP

c

Daylight in le.rge-hours

3

Interpolation per degree beyond 8°

...

o; o,3,

l; II .3. 30

+ 0; ·O,Z4 + o. 0 IZ.

3;30 3; It

-- o; a,z.:

7.'37 ==-==3 Z; 4Z z. 30 l.Z4

Z;lO l;1l

-

0; O,IZ.

o. 0 Jt 0; o, 3'

-

o, o, z~

+

0; 0, IZ

-

o. o 1a

J'

0, O,l-4 + 0, 0,

+

2 Kugler BMR p. 99. His method was based purely on trial. A systematic deduction of this scheme can be found in Neugebauer [1). Another scheme which Kugler, SSB II p. 587, derived from column C in No. 101 is to be discarded; cf. the commentary to No. 101.

SYSTEM

B,

No text is preserved which uses exactly this scheme, but the rounded-off values given agree sufficiently well with the expected results to guaranty its correctness. Example: No. 123, rev. 12

thus

D'

COLUMNS

B2 = 11l' 9;33,30 = 11l' 8 + 1;33,30 c2 = 2;42 + 1;33,30 . 0;0,36 = 2;42 + 0;0,56,6 = 2;42,56,6

AND

lJI"

73

velocity. Column lJI", however, is essentially a linear zigzag function operating with constant difference. Because Column lJI" is of simpler structure than lJI', we begin our discussion with column lJI" though it is clear that lJI'' is only a simplified form of lJI'.

I.

CoLUMN

lJI"

The shape of the true function is indicated by Fig. 32.

The text gives 2;42,56.

II.

CoLUMN

D'

The length of the daylight and the length of night together add up to 6H. Consequently, we obtain the column D' for the length of the night by subtracting C from 6H. Because of the symmetry of the scheme for column C, one can find D' directly by using B2 in order to find D' 1 and B1 for D' 2 • Example: No. 104 obv. 11

B1 = }( 20;40 = X 8

+

12;40

t

Fig. 32

Except for the discontinuity after passing the line lJI" = 0, this function is constructed like an ordinary linear zigzag function. The parameters are either expressed in degrees or in eclipse magnitudes, assuming the value 1 for the greatest possible eclipse.

hence, using column B2 in the scheme for C, we obtain D' 1 = 3;18- 12;40 · 0;0,36

= 3;18 - 0;7,36 = 3;10,24

CoLUMN

t D' = H6 -

The linear zigzag function is determined by (1)

D

Column D gives half of the length of the night. We therefore can write D =

Parameters Expressed in Degrees

d

abbreviated in the text to 3; 10.

III.

I.

C)

D. Column lJI' and Related Columns In our material no text of System B is preserved which contains a column for the latitude of the moon comparable to column E of System A. Also column 1JI is only preserved twice, namely in the eclipse tables No. 130 and No. 135. We know, however, from System A that the eclipse magnitude 1JI was extended to a function lJI' which coincides with 1JI for ascending nodes, with - 1JI for descending nodes, and which is also defined for syzygies where eclipses are excluded. Column lJI' of System B has the same qualities, though the method of connecting subsequent nodes is slightly different from the procedure of System A. 3 Moreover, we must distinguish between two different types of columns which we shall denote by lJI' and lJI" respectively. Column lJI' is the exact analogy of column lJI' of System A in so far as its differences depend on the solar

M

= 3,52,30 = -m = 9,52,15

2M - d = - (2m + d) = 15,52,0 ~ = 19,44,30 p. = 0

The first sexagesimal place in these numbers represents degrees. If we measure periods, e.g., by the distance of consecutive maxima, we find

p (2)

=

p =

1,30,58 1,38 43 , 1,30,58

=

-7,45 =

0;55,17,22,16, ... 11;44,15,29,1, ...

II = 7,45 · P = 1,30,58

.

The astronomical significance of the value of p lies in the fact that it indicates that the nodical month is assumed to be 0;55, 17, ... mean synodic months. The discontinuity after passing the zero line has the total value of

(3)

2c = 3,0,0 .

Eclipses are possible if lJI" lies between 0 and 2c for an ascending node or between 0 and -2c for a descending node. The largest eclipse has the value c = 1,30,0. 3

For a comparative study see Neugebauer [11] and [7].

74

SYSTEM

B,

COLUMNS

The procedure for taking the discontinuity (3) into account is as follows. In approaching the zero line, one uses the difference d as usual in computing a linear zigzag function. When the sign of lJ'" changes, one continues with d until a value would be reached which has the new sign but whose absolute value would exceed 2c. In this case, d has to be replaced by d - 2c = 52,30

(4)

Example: No. 122 obv. V,2 ff.

+ 1,51 + 2,43,30 + 6,36

d = 3,52,30 d = 3,52,30 d = 3,52,30

d- 2c = 52,30 d = 3,52,30

tp"

Difference

+ 4,22,30 + 0,30

d = 3,52,30

- 0,22,30 - 4,15

d- 2c = 52,30 d = 3,52,30

It should be noted that, in the first example, the reduced difference appears in the line following the change of signs because the first positive value is still smaller than 2c = 3,0,0. In the second example, however, change of sign and reduced difference coincide because otherwise the first negative value would be -3,22,30, thus exceeding the value -2c = -3,0,0. Diophant The discontinuity near the zero line makes it necessary that we assume that two values y 0 and Yn not only lie on branches of the same type but also that Yo and Yn either both preceed or both follow the discontinuity of their respective branch. If this is the case, we obtain as necessary condition Yn - y 0 == 0

mod. 30

2

·10 (Yn- y 0 ) ,

=

Yn- 1,1,0

disregarding the signs of y n and y n + 12 2. Parameters Expressed in Eclipse Magnitudes This type is represented by the ephemerides No. 123 and probably No. 125. All parameters are two-thirds of the preceding ones. Hence we have d = 2,35,0

M

= -m =

6,34,50 2M - d = -(2m + d) = 10,34,40 !::. = 13,9,40 fL = 0 .

(7)

The periods are, of course, again given by (2) p. 73. The discontinuity is determined by

2c

mod. 1,30,58

=

2,0,0

and consequently we obtain for the greatest possible eclipse the value c = 1,0,0. The necessary condition for a Diophant is given by

(9)

Yn -Yo

=

0

mod. 20

and then (10)

n

3

== 51,27 · 10 (Yn- y 0 ) mod. 1,30,58 '

if Yo andy n satisfy the same conditions as assumed in (5) and (6).

II.

COLUMN

!11J''

The differences of lJ'' form a linear zigzag function !11J''. Two types of !11J'' are attested, the second being an abbreviated form of the first. 1. Unabbreviated Parameters The parameters are

.

Assuming (5) satisfied and that Yo and Yn both belong to an increasing branch, we obtain for n n == 51,27

Yn+12

(8)

Similarly No. 122 rev. V,1 ff.

(6)

If Yn and Yn+ 12 are two values of lJ'", both of which belong to the same type of branch, then the following numerical relation holds:

Difference

- 9,46,30 -5,54 - 2, 1,30

(5)

Checking

.

After that, d has to be used again until a new change of signs occurs.

tp"

tp" AND !11J''

(11)

d = 33,20,0,0 M = 48,13,4,26,40 m = 44,46,55,33,20 2M - d = 1,35,52,48,53,20 2m+ d = 1,30,7,11,6,40 !::. = 3,26,8,53,20 fL = 46,30,0,0

SYSTEM B, COLUMNS /11J'' AND lJ'' and therefore 46,23 P.= 3 45

'

(12)

III.

=

Diophant. Necessary condition Yn- Yo= 0

=

=

.6.~' ~;,.~z.S'~ZO

I.

4S: 1. 331 lO 44, S7, 37, ~~ 40 'f:.~ JO, ~-7, 'tl, ~0

4.

4~

4,17,'41, 40

4t, 37. 3~ 41. 4o ~ 7, 10, S7, 4(. 4o

7.

-

Example: No. 121 rev. I/II,1 ff. (cf. table below).

'f'

I, -4, S7, J( 4o .7••• £1,:tt..Mg __ 'I' z~: 1.~ l( + Ji, ~0, 3l, I J, lO + tzo 44tJ"O + 1,~1,31, 3l,13,ZO + 1, Z, l~ J'f. l~ 40

2. Abbreviated Parameters In several ephemerides (e.g., in No. 123) the following abbreviated parameters occur:

(15)

d = 33,20 M = 48,13,4;30 m = 44,46,55;30 /1 = 3,26,9

= 1,35,52,49 = 1,30,7,11 !"' = 46,30,0 .

2M- d 2m d

+

Note that the values of M and m require one more sexagesimal place than actually appears in this threeplace column. The periods are p (16)

=

3,26,9

Diophant. Necessary condition Yn- Yo= 0

mod. 2

Yo and y n both belonging to the same type of branch. For values on increasing branches, one finds

(18)

n

=

"'o

'1' 1(1) ,3i. ve n lfl'(z) = 4''(1) + L\ o/'(a) lj)'{J) = lt'! J(J)- H (.4)

.s:

s; 30,

12,/7,30

7.

.!1£..2.

, ,

2,43 . 1,4,56,0 - 8,0

,7

In using .EH for the computation of J one has to watch whether J is computed with abbreviated or unabbreviated parameters. For an example see the General Introduction p. 37.

G. The Columns K and L

H

~

,

which shows the change of J (for unabbreviated and abbreviated parameters) corresponding to a full period of H.

Example: No. 122 obv. VIII/IX 1 ff. (abbreviated parameters)

1.

.EH = 2 56 24 0 O = { 2,43 · 1,4,56,12- 40,36 ,

and mean period

(8)

We may finally note that

Abbreviated Parameters M = -m = 32,28,0 /). = 1,4,56,0 p. = 0

(7)

79

46,23 3,45 = 12;22,8

p =

2.

J, K, AND L

J(l)

=

J'(s)={2.m + H£s))-J(4) 1(6) = J(s) + H (~) 1(7) = J (6)+ H£7)

::-_t_.tM.M + 3, 30,30

I.

~-

7.

I.

(1)

29d

=

contribution of a normal interval:

s = 1,4,52,30; parameters of the derived zigzag function H': 2m' + d' = 32,30 ll' = 30 2M' - d' = 42,30 30 30 p' = 13 = 6 + 2,43

m' = 0 M' = 37,30

d' = 32,30

Because the number period of H is comparatively small (ll = 16,48) it is useful to have the total of all values of H in a complete number period. One finds II

E H = 2,56,24,0,0 1

K = 29d

+

G

+J

Example: No. 171 rev. IV/III/V, 54 ff. G2 J2 K2 Line 54. 2;58,24,10H- 0;9,10,36H = 2;49,13,34H 55. 2;35,54,10 + 0;11,11,54 = 2;47,6,4 56. 2;13,24,10 + 0;24,46,54 = 2;38,11,4

J

6 even;

+

the largest unit in K being large-hours.

For the summation of H over a gtven interval the following _parameters are needed:

[P]

K

The time interval between consecutive syzygies is given by

For an interesting example of correct and incorrect relation between the extrema of H and J, see the commentary to No. 126 (p. 156).

]. Diophant for

COLUMN

II.

COLUMN

L

Column L gives the moment of the syzygies in terms of midnight epoch. The use of midnight epoch instead of the evening epoch of the civil calendar eliminates the need for a special column C', introduced in System A to take into consideration the variation of the moment of sunset during the seasons. Another advantage of the procedure of System B consists in the counting of time "after midnight" in contrast to System A, where time is counted in negative direction "before sunset". In System B, the moment of the syzygies can therefore be expressed simply by (2)

L(N

+

1)

=

L(N)

+

K(N

+

1)

The numbers obtained by this formula must be reduced mod. 6H if necessary, that is, if they would be greater than 6H = 1d • 7

42,51,15 periodically repeated.

80

SYSTEM

B,

Example: No. 171 rev. IV/V, 4 ff. (cf. the preceding example) Line 53 54 55 56

L2

2;49,13,34" 2;47,6,4 2;38,11,4

IV 5;10,43,4" v 1;59,56,38 VI 4;47,2,42 VII 1;25,13,46

Here we have, e.g.,

=

= 1;59,56,38

after

before}

illlfter

.sunset

used in

No.

kur

kur

~ur

ntm

11Lm me 11im, 111e

.i~

d.u 101,/0l., 107, Ill,

lu s•' roo

UN 5H, we may use it for N 1 ; otherwise a whole day, i.e., 6H, or even two days, must be added. Thus (1b)

N1 = C

+D -

L

+{

~=

12H

where (la) furnishes (in general) the decision between the various possibilities. Similarly, one finds the time difference between the morning of last visibility and the moment of conjunction by

(2a) The limits seem to be (2b)

In No. 102 a column N 2 is given which is discussed in detail in the commentary to this text. From this discussion it follows that the dates given are the dates of sunrise following opposition. The large-hours, however, are either the time interval from opposition to the following sunrise, or from opposition to the preceding sunrise. The first type is to be used if the opposition occurs during daylight, the second if the opposition falls in the night. Consequently, we have (cf. Fig. 33)

-----+--~~--~-0----~--o.---~----

N,.

This procedure is essentially equivalent to the method followed by System A though one can observe deviations in detail, which will be mentioned in the discussion of the single columns.

I.

CoLUMN

N

Column N 1 gives the time interval from the moment of conjunction (column L 1 ) to the evening for which the first visibility of the new crescent might be expected.

(3a)

D'} < C

Fig. 33

N2

;;-;;:

6H

oppos. at{daytime . h mgt

10 Already Epping succeeded in explaining correctly the meaning of columns N, 0 and P (Epping, AB p. 93 ff) whereas the explanation of Q and R is due to Kugler SSB II p. 592 ff. and to Schaum berger, Erg. p. 380 ff. (cf. the review Neugebauer [4]). Cf. also Sidersky [1]. 11 The greatest value attested is 15;52H in No. 100 obv. VIII,6. 12 Or 6H ! D'.

82

SYSTEM

B,

CoLUMNS

and (3b)

N2 = D - L

+ 6H

oppos. at daytime

but (3c)

N2 = L - D

+ {~: . ht . h t {before m1.dmg f oppos. at mg a ter

The reason for this procedure is not really clear. The interesting case is, of course, the case of opposition at night because only then would a lunar eclipse be visible. This might be the reason for computing the circumstances for the preceding morning in order to be able to check the accuracy of the prediction for the following night. It is not clear, however, why the date of the following morning is given nor are the elements for the morning after a daytime opposition of interest. The ideogram for all columns N is ktir, written at the beginning in Nos. 100, 101, 102, 105, 108, and 122, but at the end in No. 120 and perhaps in No. 107. The date of N 1 is 29 or 30 and refers to the civil day which ends at the sunset which is followed by the expected first visibility of the new crescent. The dates of N 3 and N 2 offer no ambiguity because the date of sunrise is always uniquely determined.

II.

CoLUMN

0

The next step consists in finding the elongation, expressed in degrees, of the moon from the sun for the evening or morning in question. To this end, the time difference N, expressed in large-hours, must be multiplied by the relative velocity between sun and moon, expressed in degrees per large-hours. This relative velocity is approximately given by F' -0; 1()/H, assuming a solar velocity of about 1"1d. Consequently, we have (4a)

Elongation= N(F' - 0;10)

The values given in column 0 can be represented by (4b)

0 = N(F'- 0;10)-

€

where E is a small positive quantity. The values of E 1 , (case of first visibility), show very small variations around the value 1;30 in the texts No. 100 and No. 101 whereas rather irregular oscillations (between 1 and 3;40) occur in Nos. 120 and 122. Similar oscillations appear for € 3 (last visibility) in No. 122 (between 2 and -0;50) with a mean value of 0;30. No. 120 was investigated by Sidersky ([1]) who interpreted column

N, 0,

AND

Q

0 1 as elongation corrected for parallax. The numerical agreement is not good, however, nor does this theory explain the different order of magnitude of E1 and E3 , the latter being about t of the first. Also the expected seasonal variation is nowhere visible; the texts Nos. 100 and 101 show practically constant values of € whereas the deviations in the other texts do not show seasonal dependence. Schaumberger (Erg. p. 388 f.) considers the correction € as a correction for twilight, thus explaining satisfactorily at least the difference between E1 and € 3 ; also parallax may have influenced the empirical determination of these values. In No. 102 columns 0 2 and 0 3 are abbreviated to N 2F' 2 and N 3 F' 3 respectively, thus even the solar movement is ignored. Cf. Schaumberger, Erg. p. 388, 389. III.

CoLUMN

Q

Column 0 indicates the main part of the circumstances which influence first or last visibility, the difference in longitude between sun and moon. Column Q gives the (positive or negative) correction which must be added to 0 in order to replace a difference in longitude by the corresponding oblique ascension. In the ephemerides No. 101, No. 102 and No. 120 this correction is called "sd lu-mas" which means "because of the zodiac"; no title is given in No. 100. Column Q 1 (first visibility) is completely computed in No. 120 whereas Nos. 100 and 101 contain negative values of Q 1 only. The reason is evidently that negative values of Q1 might diminish the values of 0 1 so much that the visibility of the crescent is excluded although it seemed possible from 0 1 alone. If, however, Q1 is positive, then 0 1 alone is sufficient to guaFantee visibility. No column Q1 is found in No. 122, but it can be shown from the final values in P 1 that Q1 must have been computed completely. A column Q2 (full moons), computed for positive and negative values, is found in No. 102. The same ephemeris also contains a column Q3 (last visibility) but omitting negative values. The reason for this is similar to that for the omission of positive values in Q1 • If 0 3 is so small that visibility at a given sunrise before the opposition seems to be excluded, then a positive correction Q 3 might improve the conditions so much that visibility should nevertheless be expected. Negative Q 3 , however, can only confirm the exclusion based on 0 3 alone. A complete column Q:1 must have been used in computing P 3 in No. 122, but it was unfortunately not included in the final copy. The only way to reach at least qualitative understanding of the computation of Q lies in the analogy to

SYSTEM

B,

CoLUMNS

System A. The procedure texts of System A, especially No. 200 Section 15, show that the oblique ascension Q is obtained from the elongation 0 by means of coefficients q which form a linear zigzag function dependQ ing on the longitude. In System B the sum 0 A. corresponds to the oblique ascension Q of System Thus one might expect that Q can be obtained by means of given coefficients q in the form

Q, R,

AND

=

q· 0

where the values of q should be not too different from the values of q - 1 in System A. In order to test this hypothesis one can compute the ratios Q/0 in all preserved cases and plot the result as function of the corresponding longitude. It turns out that the coefficients q obtained in this way belong, in the majority of cases, to a linear zigzag function for the ephemerides Nos. 100, 101 and 102, whereas they must result from a more complicated scheme in the case of No. 120. But even the first three ephemerides do not lead to exactly the same result. No. 100. Only negative values of q1 are available. The best approximation is obtained if we plot q1 as ~0 1 which corresponds roughly to function of B1 the midpoint of the arc between the sun and the new crescent. The corresponding zigzag function has the minimum -0;20 at=::.:: 8 and the value of 0 at QD 8 and }') 8. 13 Thus the difference per zodiacal sign is 0;6,40. No. 101. Only negative values of q1 available. Minimum --0;24 at=::.:: 8 if considered as function of B ]> i.e., for the conjunction itself. The value 0 corresponds then to QD 8 and }') 8. The difference per zodiacal sign is 0;8 as in System A. No. 102. For opposition we find as function of B2 the values of q2 forming a linear zigzag function with m = - 0;24 in=::.:: 8, and M = +0;24 in "Y' 8; thus the difference is 0;8 per zodiacal sign. For last visibility, only positive values of q3 are available; they agree with the values of q2 by forming a linear zigzag function with the value 0 in QD 8 and }') 8, and the maximum +0;24 in =::.:: 8. The difference is again 0;8 per zodiacal sign. No. 120. The values of q1 as function of B1 -i-- 1:! 0 1 lie on a curve with a flat minimum of about -0; 18 at =::.:: 8, and 0 near QD 8 and l') 8. The maximum cannot be more than +0;9, on a very flat curve. It is evident that we have here a much more accurate scheme than in the preceding ephemerides from U ruk. While the coefficients q of System A can be related to the scheme of the length of daylight (column C), this cannot be done with the q's of No. 120 because one would obtain no asymmetry for the q's from the symmetric scheme for column C. Greater similarity, however, exists with

+

I

83

the coefficients given by Maimonides14 for the transformation of the "third length" into the "fourth", which vary between -0;20 and +0;12.

+

Q

P

IV.

CoLUMN

R

The last correction needed concerns the latitude of the moon. Positive latitude increases the difference of right ascension between moon and horizon and consequently increases also the duration of visibility whereas negative latitude has the opposite effect. The magnitude of this effect depends, however, on the seasons, being smallest at the vernal equinox, largest at the autumnal equinox. This variation is reflected in the linear zigzag function given in the procedure texts for System A No. 200 Section 1515 (new moons) and No. 201 Step dl6 (full moons). The values given in these procedure texts vary for r 1 between 0 and 1 in No. 200 and for r 2 between 0;18 and 0;42 in No. 201. If f3 denotes the latitude of the moon for the moment under consideration then R = rf3 is the corresponding correction. A :imilar scheme can be assumed to have existed for System B, but the scantiness of our material makes its restoration very difficult. A column R 1 is attested in Nos. 100, 101, 102 and 120, R 2 and R 3 in No. 102, and R 1 and R3 must have been computed for No. 122 without being incorporated in the final copy. The investigation of these texts shows general agreement with the expected trend, though the numerical values frequently show considerable deviation. For details, see the commentaries to the abovequoted texts.

V.

CoLUMN

P.

VISIBILITY CoNDITIONs

Column P gives the final result for the visibility of the moon. It can be described as (5) which means that P is obtained from the elongation 0 by applying the correction Q for oblique ascension and the correction R for the latitude. The values for P are everywhere limited to two places and rounded off to full tens in the last place. The smallest unit considered is therefore 0;0,10H (the equivalent of 40 seconds). 13 It must be understood that there is no possibility to deterrome these positions with great accuracy from our material. The value 8' is chosen only because of its significance for System B. About 2 degrees more would approximate equally well the given values. 14 Daneth [1] p. 166; Neugebauer [16] p. 354. 15 Cj. p. 206. 16 Cj. p. 233.

84

SYSTEM

B,

VISIBILITY CONDITIONS

The texts Nos. 100, 101 and 102 give in P 1 year and months, Nos. 110, 120 and 122 months only, followed by 30 or 1, thus indicating whether the preceding month had 29 or 30 days respectively. For P 3 we find in No. 100 and in No. 102 year, months and days (usually 28 or 27), in Nos. 108 and 122 the date only. No. 102 gives months and dates for P 2 • If L(N) denotes the value of column L for month N we can say that P1 (N + 1) is the value of P1 for the beginning of month N 1 which follows the conjunction L 1(N). Consequently we always find L 1(N) and P1 (N 1) in the same line. The values of P3(N) for the last visibility in month N, preceding the conjunction L 1(N), are given in the same line with L 1(N) in Nos. 100 and 102. In No. 122, however, L 1(N) and P3 (N + 1) are coordinated. Of great interest is the problem of visibility conditions. A careful investigation of our material shows that the values of P alone cannot have been the criterion used. As in System A a criterion of the form

+

+

(6)

leads also here to a satisfactory explanation though it is difficult to determine the exact value of c. It also must be kept in mind that c might depend on the longitude beside additional criteria for very small or very large values of 0 or P alone, considered sufficient to decide the case. Alternative solutions given for P1 in Nos. 100 and 101 suggest a value of c1 of about 23° whereas No. 120 accepts values :;;;: 20" and No. 122 goes as low as 0 1 + P 1 :2;; 17°. For last visibility a value c3 = 23° would satisfy both No. 102 and No. 122. The expectation that in extremal cases P alone might have decided the visibility question is supported by lists of P 1 alone, apparently collected for several years in succession (Nos. 180, 181 and 182). The lowest values which occur in these texts are 11,20 11,40 and 11 ,50, followed by a phrase ina pi of unknown meaning. The highest value preserved is 25,10 (No. 180 obv. I, 11) whereas 20,30 and 10,30 are given as alternative solutions in No. 181 obv. I,8. The same month for which No. 180 (obv. I,11) gives the value 25,10 is also preserved in the ephemeris No. 102 (obv. XIII,14) but there we find the corresponding low alternative value 12 used. We now can return to the description of the procedure of the computation of column L, illustrating it in the following example, taken from No. 120. No. 120 obv. III,4 L 1(III): conjunction: Month III, day 28 3;43,42,10H after midnight

assuming that the new crescent appears already on the 30th, we obtain obv. IX,4 P1 (IV): month IV, beginning on (III) 30th; duration of visibility 0;14,20H = 14;20°. For the elongation we have obv. VI,4 0 1 (III): 15;0° thus

This total is sufficient for the visibility of the crescent and month III will be hollow. Thus we can proceed to the next conjunction. We have: obv. II,S K 1 (IV)

=

1;41,13H

therefore we obtain for month IV 29d

III 28d 29d

L 1 (III) 1 (IV)

+K

3;43,42,10H

+ 1;41,13H

thus 5;24,55, 1oH 5;24,55, 10H as given in obv. III,S. Similarly the next step: obv. III,S obv. VI,S obv. IX,S

L 1(IV): conjunction: IV 28 5;24,55,10 0 1(IV): 22;30° v 1 18;40° P 1(V):

and therefore

guaranteeing visibility for the evening following sunset at the end of the 30th. Hence month IV is full and we find IV 28d + 29d IV S8d v 28d

5;24,55, 10H 2;21,30,30H (cf. obv. II,6) 1;46,25,40H 1;46,25,40H

as given in obv. III,6. Consequently we find m L 2 rev. VIII,4 rev. VIII,S

IV V

30 i.e., month III is hollow 1 i.e., month IV is full.

If we know that the opposition in month IV fell according to

rev. VIII,4

IV

14

2; ... H

SYSTEM

and that according to rev. VII,5 we know that the opposition of month V will be given by L 2(V)

=

IV

Hd 2; ... H

+ 29d 4;1,4,40H

= IV+ 44d 0; ...H = V Hd 0; ...H as indicated in rev. VIII, 5.

B,

85

EcLIPSES

K. Eclipses Three eclipse tables are preserved in System B, two for lunar eclipses, one for solar eclipses. The principle of selection is, of course, the same as in System A. Two of these texts are unfortunately too fragmentary for the investigation of details. In No. 135, however, the better part of a complete 18-year cycle of lunar eclipses is preserved. The discussion of this material is given in the commentary (cf. below p. 161 ff.).

86

CHAPTER I. SYSTEM A Texts Nos. 1 to 26:

Ephemerides

Nos. 50 to 61a: Eclipses Nos. 70 to 76:

Auxiliary Texts

Nos. 80 and 81: Daily Motion of the Moon Nos. 90 to 93:

Ephemerides of Undetermined System

§ 1.

EPHEMERIDES

No.1 3420 +A 3435 +A A 3412 Contents: Full moons for S.E. 124 and 125( = -187 j 186) Arrangement: 0-R Provenance: Uruk (??) [in favor of Uruk: 0-R and A 3400; against Uruk: dates in T written sexagesimally and no colophon] Transcription: Pis. 1 and 2; Photo of rev.: Neugebauer, Ex. Sci., Pl. 7b. Critical Apparatus

Obv. II,8 [2, 1]5, 12,36,33,[20]: The reading 36 is certain; it should, however, be 35. Isolated error. Rev. [0],13 5,19,39,24: read 5,15,39,24. IV,5 4,5,20: sic, instead of 4,5,30. The error is irrelevant for V,5 if 59 units of the fourth place were disregarded; with the error the fourth place would be 49. IV,7 10: sic, without us. Commentary

This text is the oldest ephemeris of System A, referring to full moons of two consecutive years. For the earlier year, S.E. 124, we have also an ephemeris of System B in No. 104 rev. The distribution of columns is as follows: III IV V VI VII II I B2 C2 E2 F2 G2 J2 C'2 K2 M2 @2 T2 Rev. [-IV] [-III] [-II] [-1] [0] I II III IV V VI Obv.

Column J 2 is of interest because of the use of an approximate value - 57,4,0 for g = - 57,3,45. The valuesg(t) andg({-) in rev. 111,6 and 13 are, however, computed without rounding off. For the distribution of full and hollow months cf. Fig. 34.

No.2 127

u

Contents: New moons, S.E. 124 to 126 ( = -187/186 to -185/184) Arrangement: 0 jR Provenance: Uruk [U] Colophon: only the heads of a few wedges visible Transcription: Pl. 2 Rev. 1,1

Critical Apparatus

[abs ]in : the traces fit absin better than ki; cf. next line. 1,2 absin : apparently written absin 0 and not absin, as in the preceding line. I,8 29,3,45 : sic, instead of 29, 18,45. Cf. next line. 1,9 27,11,15 :sic, instead of 27,26,15. Consequence of the error in the preceding line. II, 1 3,2[3, ... : traces look more like 2,25, ... or 2,28, .. . II,8 and 9 : the values restored are computed from the erroneous values in 1,8 and I, 9. Commentary

The size of the preserved fragment makes it practically certain that the obverse contained two years, the reverse

Nos. 3 one year only, followed by one blank line and a colophon. Preserved are only small sections of B1 and C1 for S.E. 124 to 126. The corresponding values for full moons for S.E. 124 and 125 are contained in the ephemeris No. 1.

TO

with the ideograms u u, according to the custom in all similar texts. The obverse ends with 2,22 I and not with the last month of a year as is customary for new-moon ephemerides.

No. 3aa

No.3 Rm.721

+

87

3a

BM 42152 ( = 81--6-25,775)

Rm.810

Contents: New moons and full moons for S.E. 141 (= -170/ 169)

Contents: New moons for (at least) S.E. 141 ( = -170/ 169)

Arrangement: 0 j R

Arrangement: ? (only one side preserved)

Provenance: Babylon [BM]

Provenance: Babylon [81-6-25]

Transcription: Pis. 3 and 4

Transcription: Pl. 5; Copy: Pinches No. 1

2..s r .II ]I[

riZ'

y l![

'i[ J!l(

ll'

z

:xr m 2,6 I

,-13

-.....,_

Fig. 34

Commentary Distribution of columns: Obv. [-V] [-IV] [-III] [- II] [-1] [0] I II III EI o/'1 f1 GI J1 CI BI T1 cpl E2 o/'2 f2 G2 J2 C'2 C2 B2 T2 ct>2 Rev. [-IV] [-III] [-II] [-I] [0] I II III IV V Though a complete restoration of the text would be very easy, only those columns are given which are needed to check the preserved columns. The columns G and J suffice to restore ([> and B respectively, each of which can be used for dating by a Diophant.1 Of the column rev. I, only one number, 6, is preserved in line 11. Because column II contains F 2, one has to restore in column I either E 2 or 2 lfl' 2. Continuation from other texts which contain these columns shows that accidentally both columns end in line 11 with a number 6. Nevertheless, the restoration of lfl' 2 seems to be certain in view of the fact that column E 2 would not end with a number alone but

Commentary This is a fragment from a duplicate of No. 3 (join excluded). Only a small section of columns }1 and C' 1 is preserved.

No. 3a

BM 41467 + BM 41865 + BM 41937 + BM 41968 ( = 81- 6-25,78 + 81-6-25,485 + 81--6-25,558 + 81- 6-25,590) Contents: New moons and full moons for S.E. 142 (= - 169/ 168) Arrangement: 0 j R Provenance: Babylon [81-6-25] Colophon: Invocation Zja (p. 21) Transcription: Pis. 4/ 5; Copy: Pinches Nos. 2 to 4 1 The date obtained is explicitly confirmed by text No. 50 (solar eclipses), where the values in column B1 coincide with values found in the present ephemeris. The same holds for No. 60 (lunar eclipses) for all columns from T 2 to 'l'' 2 whereas F 2 shows different values. Cj. also No. 5, note 1. 2 Column 'l' can be excluded because line 11 would be empty.

Nos. 3a

88

Commentary Columns: Obv. I Tl T2 Rev. I

II III IV cpl Bl Cl cp2 B2 C2 II III IV

[V] El E2 V

[VI] 1J''1 1J''1 [VI]

[VII] VIII IX X Fl Gl J1 C\ F2 G2 J2 C'2 K2 [VII] VIII IX X XI

Dr. Sachs, who collated this text in London, remarks that the obverse is the convex side of the tablet, contradictory to the accepted rules. A fragment of a duplicate is No. 3b which contains parts of the columns C'l Kl Ml K2 M2 The two texts Nos. 3a and 3b are certainly not pieces of the same tablet as can be seen from a clear difference in ductus and in width of lines.

No. 3b

TO

4a

Critical Apparatus Obv. II,26 3,32,5,[30]: perhaps 22 instead of 32. Rev. II,8 [2,17,5]3,51: sic, instead of [2,11,4]7,42. The value of the text ignores the fact that it still belongs to the critical region near the nodal zone. This error has no consequences for the subsequent lines. IV,19 2,40,13, ... : sic, instead of 2,40,3, .... No.5 rev. II,18 has another mistake in the same place. In violation of the accepted rules of cuneiform writing, the "obverse" text is written on the curved side, the "reverse" on the flat side.

Commentary The distribution of the columns is as follows:

BM 37186 ( = 80-6-17,939)

Obv.

[-1]

Contents: New moons and full moons for S.E. 142 (= -169/168)

Rev.

Tt [-II]

Arrangement: 0 jR Provenance: Babylon [BM]

[0] I cpl Bt [-1] [0]

II cl I

III Et II

IV Ft III

v Gl IV

VI Jl

v

The years 2,26 to 2,28 are also covered by No. 5 (cf. p. 90 especially with regard to column F1).

Transcription: Pl. 5

Critical Apparatus Obv. III,4 3,58,54: sic, instead of 3,58,55. III,S 1,59,42: influenced by the preceding error. Rev. II,3 15: lower part destroyed. II,4 2,1 ]0: traces.

Commentary This text is a duplicate to No. 3a. The present column I of the obverse corresponds to col. X (C' 1) of No. 3a, followed by K1 and M1. The reverse shows the beginning of the columns P 2,1 and P2 ,2 probably followed by P 2,3 and P2,4•

No.4

BM 34575 + BM 34687 ( = Sp.II,47 + Sp.II,174 + 82-7-4,164 + 82-7-4,189)

Contents: New moons for (at least) S.E. 145 to 149 (= -166/165 to -162/161) Arrangement: 0 jR Provenance: Babylon [Sp.; writing 50,3 in rev. IV,14] Previously published: Sp. II,47: Kugler BMR Pl. 10 Sp. 11,174 + ... : * Transcription: Pis. 6 and 7; Copy: Pinches Nos. 5 and 6

+

No. 4a

BM 32499 + BM 32704 + BM 32773 + BM 32785 ( = st 76-11-17,2148 st 76-11-17,2238 + st 76-11-17,2473 + st 76-11-17,2545 + st 76-11-17,2557)

BM 32414

+

Contents: New moons for S.E. 146 to (at least) 149 (= -165/164 to -162/161) Arrangement: ? (no fragment with two sides preserved) Provenance: Babylon [BM] Transcription: Pis. 8 and Sa

Critical Apparatus Obv. II,14 : wedges which could be read bar-bar or su-su or the like; one might expect gab or bar. VII,2 3,40,19 : sic, instead of 3,40, 9. This error influences all subsequent values in column VIII; cf. note to VIII,l. VIII,1 5,29,[42: traces of 40visible. This proves that in IX,2 the incorrect value 3,40,19 (instead of 3,40,9) has been used because otherwise we would have here 5,29,32. VIII,3

4,31 ,3]3 : traces.

Nos. 4a Commentary

This text is restored from four disconnected fragments which, however, belong in all probability to the same tablet. One of the fragments contains a short section of the upper edge, sufficient to determine the date of the first line as the conjunction at the end of S.E. 145 XII 2 • The obverse must have covered at least two years (S.E. 146 and 147). One of the fragments concerns S.E. 148 and the beginning of 149 and it is therefore plausible to assign it to the reverse, thus assuming a coverage by the whole text for S.E. 146 to 149. The division between obverse and reverse as shown on Pis. 8 and Sa is based on this assumption. The distribution of columns is as follows: Obv. [-III] [-II] [-I] (/}1 T1 B1 Rev. [-VII] [-VI] [-V] Obv. IV G1 Rev. [0]

v

VI C'1 II

J1 I

I II III [0] lf'' 1 F1 c1 E1 [-IV] [-III] [-II] [-1] VII K1 III

VIII M1

The text is a partial duplicate of No. 4 which begins at least one year earlier. The only difference between the two texts consists i.n the addition of a column lf''1 in No. 4a. Note that column F 1 is identical in Nos. 4 and 4a whereas the third text for this period, No. 5, gives somewhat different values.

No.5

BM 34041 + BM 34094 + BM 34253 + BM 34354 BM 34420 + BM 34734 + BM 34778 ( = Sp.l37 Sp.l93 + Sp.360 + Sp.469 + Sp.540 + Sp.II,82 + Sp.II,224 Sp.II,270)

+ +

+

Contents: New moons for S.E. 146 to 148 ( = -165/164 to -163/162) Arrangement: 0 /R Provenance: Babylon [Sp.; writing 50,3 in rev. II, 13] Previously published: Sp.137: Kugler BMR Pl. 10 (obv. and rev. interchanged). Other parts: * Transcription: Pis. 9 to 11; Photo: Pl. 255; Copy: Pinches Nos. 7 to 13. Critical Apparatus

Obv. II,8, 9 : reading of g[fr and p[a doubtful. VI, 19 3,25,41, ... : sic, instead of 3,25,40, ... This error has no influence on the next line (i.e., rev. II,1). Cf. also note to obv. IX,19. IX,2 : cf. No. 4a, obv. VII,2. IX,5 2,33,13 : 2,33,12 not excluded. IX,6 2,20,46 : 2,20,45 not excluded.

AND

5

89

IX,7 2,51,22: 2,51,23 not excluded. IX,8 and 9 ... 5[5] : traces of 5. IX,19 2,23,5[5] :this restoration is based on the values found in the continuation (rev. V,l). This shows that in obv. VI,19 the correct value 3,25,40 was used (cf. note to obv. VI,19) because with this value alone one obtains here 3,23,55,55, ... or abbreviated 3,23,55. X,1 : cf. No. 4a, obv. VIII,l. X, 16 . ,5,51 : initial zero written over erased 5. X,17 2,53,44: sic, instead of 2,43,44. Because the value in rev. VI, 1 is correct one may restore correct values in the two intermediate lines. XI,16 [..... ] : Strassmaier's copy shows the upper ends of three vertical wedges. Rev. [0],7 2,11,47,42: I restored here the correct value which is replaced by [2, 17 ,5]3 ,51 in No.4 rev. II,8. II, 18 2,40,31,51 ,6,40 : sic, instead of 2,40,3,31 ,6,40. The following line is again correct; cf., however, the note to rev. V,18.- It is a peculiar accident that also No.4 (rev. IV,19) contains an error at this place, namely 2,40,13,[31,6,40]. V,4 2,28,23 : sic, instead of 2,38,23. Cf. the note to rev. VI,4ff. V,18 2,32,13: assuming the correct value 2,40,3, ... in rev. II, 18, we should obtain here 2,31,55 whereas the uncorrected value would result in 2,32,23. Hence neither assumption explains the new error. VI,4ff. : the restored numbers are computed under the assumption of the correct value 2,38,23 in rev. V,4. This is required by the numbers in rev. VI,18 and 19. VI,10/11 28 : restored according to the values in VII,l0/11. VII,9 15,20 ... :Pinches 16,20; the 20 is followed by two small horizontal wedges in high position (tab ?); meaning unknown. VII,10/11 1 and 30 : should probably be 30 and 1. VII,12/13 30 and 1 : restored according to the values of P 1 (cf. VI,12/13). Edge Written below obv. VI and IX on the edge between obverse and reverse in a direction perpendicular to the rest of the tablet (cf. Pl. 255). 13 2]3[ .. ] : probably nothing missing after 23. 14/15 [kur] : rest of the line destroyed. 15 .. ]2,[4]0: the heads of two vertical wedges are visible in front of corner wedges which are probably 40 or 50. 23 15,[ .. ] : or 16, followed by tens. 24 11[ .. ] :or 12.

Nos. 5

90 Commentary

Out of eight fragments it was possible to rebuild what is now the most complete sample of ephemerides of System A (cf. Fig. 34a and Pls. 140 and 255). The righthand part actually reaches the end of the tablet; the left-hand part is destroyed but can be completely restored. This gives us the following distribution of columns: Obv. [0] Tt Rev. [-IV] Obv. VII

Jl

Rev. III

I 1

[-III] VIII C' 1 IV

III

II Bt [-II]

cl

[-I]

IX

X

Kx

Mx

v

VI

[IV] Et [0]

v

XI PI VII

Edge Pa

Ft I

AND

Sa

For last visibility four cases with alternative solutions are sufficiently well preserved; they are given in the following list: Edge

Pa Text

Oa computed

6 (9,20) 27 10 (12,20) 19,10 9,30 ([2]3, ... ) 13 19 8,10 ( ...... )

VI Gt II

7,20 5,0 13,30 8,0

03

+ P3

21,40 (16,40) 48,40 17,40 (17,20) 35,50 24,0 23,0 (47, ... ) 21,50 16,10 ( .. . ... )

The values enclosed in parentheses are the ones given as alternatives in the text.

,,- --------------,.----------:1

No. S Obv. Fig. 34a

One interesting feature is the parallelism between this text and Nos. 4 and 4a, which cover the same years. All three texts contain a column F 1 but the numbers of Nos. 4 and 4a deviate from the numbers in No. 5 by 3 units in the last place 3 (i.e., 0;0,3ofd). This explicitly confirms the result of theoretical considerations that F is only a column of approximate values, accumulating relatively rapidly an appreciable error. It is for this reason that F does not run unaltered through all the texts of System A, as do the other columns. The main interest of No. 5 lies, however, in the fact that it contains two columns beyond M 1 (moments of conjunctions) namely P1 and P3 (first and last visibility, cf. Pl. 140). Of special importance are the cases where alternative possibilities for visibility are investigated. 4 In the case of first visibility only one case is preserved (obv. XI,13) with 25° and 10;30° as alternative values for P1 . For the corresponding elongations 0 1 , one finds 21;10° and 9;50° respectively and therefore for the total 0 1 + P1 the values: 46;10° or 20;20°, the first of which seems to have been given preference.

No. Sa BM 42248 ( = 81-6-25,871) Contents: New moons for (at least) S.E. 146 ( = - 165/

164) Arrangement: ? (only one side preserved) Provenance: Babylon [81-6-25] Transcription: Pl. 11; Copy: Pinches No. 14 Commentary

This is a fragmentary duplicate of Nos. 4, 4a and 5, columns B1 and C1 • In spite of the very close similarity of the break no join with No. 5, obv. II, III is possible.

3 Values from a column F 2 for the same years are contained in the eclipse table No. 60. The maxima ofF, in No. 5 precede

the minima of F 2 in No. 60 .

1

by~ + 4-~2.; for Nos. 4 1

ever, one obtams only 2 + 42 . 4 Cj. Introduction p. 66 f.

and 4a, how-

91

Nos. 6 ro 6a

No.6 BM 46076 ( = SH 81-7-6,524) Contents: Full moons for (at least) S. E. 14-9 ( = -162 j 161) Arrangement: ? (only one side preserved)

Commentary

This is an ephemeris, probably for six years (S.E. 150 to 155) of which the following columns are preserved: Obv.

I

II

(III]

IV

Provenance: Babylon [BM] Transcription: Pl. 12 Critical Apparatus

I, 7

1,11 ,]4-8,26 : lower parts of 4-8,26 destroyed.

The two fragments, BM 32302 and 3274-2, of which this text is composed, have no direct contact but there can be little doubt that they are pieces of the same tablet.

Commentary

Distribution of columns: [-IV]

T2

[-III]

cp2

[-II] B2

Line 6 of the present text is identical with line obv. 27 in the eclipse table No. 60.

No. 6ab BM 42081 ( = 81-6-25,703)

Contents: New moons for (at least) S.E. 154- ( = -157/ 156) Arrangement:? (only one side preserved) Provenance: Babylon (81-6-25] Transcription: Pl. 12b; Copy: Pinches No. 15

No. 6aa BM 32302 BM 32742 ( = st 76-11-17,2031 + st 76-11-17,2512)

+

Contents: New moons for (at least) S.E. 150 to 155 (= -161/160 to -156/155) Arrangement: 0 jR Provenance: Babylon [BM]; cf. writing 10,7 obv. II,S, 30,3 rev. 11,9, and 10,9 rev. III,H Transcription: Pis. 12a and 12b Critical Apparatus

Obv. 1,6 [se] : traces of a sign near the edge, but rather (3] than (se]. V,18 4-,21,24-,27: sic, instead of 4-,31, ... ; isolated error. Rev. 11,24- 2,8,25,27: sic, instead of 1,52,50,54-. The scribe overlooked the fact that two values belong to the nodal zone. An error of this type has no influence on subsequent values. III,6 16 : traces look more like 17. 111,9 4-6,1 : traces look more like 4-6,5; perhaps dittography from line 8. IV : The restoration of this badly preserved column depends essentially on the reading in line 4- where the last 15 seems to be a plausible rendering of the traces. IV,3 14-,[57] :traces of tens (atleast 4-0) and of units (between 5 and 8) support the restoration [57].

Commentary

This is a small fragment of a column B1 which has its direct continuation in No. 6a obv. [-V]. Note that the last line concerns the month XII, not (as usual) the first month of the following year. A partial duplicate is No. 6aa rev.

No. 6a BM 41029 + BM 41075 + BM 41153 ( = 81-4-28,576 + 81-4-28,622 + 81-4-28,700) Contents: New moons and full moons for S.E. 155 (= -156/155) Arrangement: 0 jR Provenance: Babylon [BM] Transcription: Pis. 13 and 14Critical Apparatus

Obv. 1,7 (9,4-4-,55] :traces which would fit 9,4-0,4-5 better. 1,9 6,11,34-: sic, instead of 6,11,4-4-. 1-IV,H: this line is uninscribed as far as the preserved fragment is concerned, except for IV,H. Edge 13, 14- : left blank. Rev. II,7 10 us: written so far to the left that the final vertical wedge of the us is in line with the 7 above and the 8 below.

Nos. 6a

92

III,3 [1,49,47]: derived from column IV; expected: 1,49,48. III,4 [2,18,53]: derived from column IV; expected: 2,18,54. III,5 [2,48,12]: derived from column IV; expected: 2,48,13. IV,4 1)3 :the reading 13 is very uncertain; 12 equally possible.

TO

7

No.7

BM 34582 ( = Sp.II,54)

Contents: New moons and full moons for S.E. 176 (= -135/134) Arrangement: 0 jR Provenance: Babylon [Sp.] Previously published: Kugler BMR Pl. 12 Transcription: Pis. 14 and 15; Copy: Pinches No. 18

Commentary

Columns: Obv. [-VII) [-VI] l/)1 T1 l/)2 T2 Rev. [-VI] [-V] Obv.

Rev.

I C'1 C'2 II

II K1 K2 III

[-V] B1 B2 [-IV]

[-IV] c1 c2 [-III]

[-I] G1 G2 [OJ

III M1 M2 IV

IV p1

Edge Pa

[0] J1 J2 I

For the final columns cf. the graph on Pl. 141.

No.6b

BM 34966

+ BM 35455 ( =

Sp.II,488

+ Sp.II,1044)

Contents: New moons for (at least) S.E. 172 (= -239/ 238) Arrangement: ? (only one side preserved)

Critical Apparatus

Obv. IV,ll 3,1[1,1]3 :only two of the last 3 wedges visible. Rev. I,5-9 : the blank space after these short lines IS preserved. II,7-11 :the blank space is preserved. IV,1 3,12,50: sic, instead of 3,12,51. IV,2 2,50,41 :sic, instead of 2,50,42. IV,3 2,28,23 : the next place would be 59, which is, however, disregarded. IV,4 2,6,4 : sic, instead of 2,6,5. IV,5 1,49,16: sic, instead of 1,49,17. IV,6 2,26,6 : sic, instead of 2,26,5. The next place would be 53. IV,12 4,18,15: sic, instead of 4,18,1. The next place would be 25. V,9 2,1,[ .. : also 2,2,[ .. or 2,3,[ .. possible.

Provenance: Babylon [Sp.) Transcription: Pl. 12; Copy: Pinches Nos. 16 and 17 Critical Apparatus

I, 9 bah : not certain, resembles bar 1. II,7 53,45,21 : the first number looks more like 52 but 53 is required.

Commentary

Contents: Obv.

Rev. Obv.

Commentary

Distribution of columns:

[-III] [-II) [-I] B1 lP1 T1

[OJ Cl

Rev.

I E1

As usual column F is the only column which cannot be connected with all other texts of System A. The writing 40,.7 in II,8 is one of the rare cases of this form in a text from Babylon. Also the terminology for the node in column E (I,9) is unusual. We find here bah(?) added to the value of E in the nodal zone; the .corresponding eclipse would be a small partial one.

[-V] T1 T2 [-V]

[-IV) [-III] [-II] [-I] l/)1 El cl Bl l/)2 B2 E2 c2 [-IV] [-III] [-II] [-I]

II Jl J2 II

III C' 1 C'2 III

IV Kl K2 IV

v Ml M2

[OJ Fl F2 [OJ

I Gl G2 I

[VI] pl

v

It is of interest to notice that the dates in rev. V show that four consecutive months (IV to VII) were each computed to be 30 days long (cf. Fig. 35). This is in agreement with the results of Parker-Dubberstein computed with Schoch's tables. 5 The months IX and X, however, are found to be 30 and 29 days long, • Parker-Dubberstein BCh. p. 41 (136 B.C.). Schoch himself thought that such a sequence of four 30-day months is excluded (cj. Langdon-Fotheringham-Schoch, VT p. 98).

No.7

93

cl ,-,D t:.!L .c' -e.!.. v6~/ ~

E m J!l

~

c::.!L c:!L

~o-'',,~

c:!L

J![

:w

~-~P",e.__

D Jx:

,0

X~

~oL'

if-i3

,0'

c:L

c:!!....,_

~

:xr t::.!L r:r"'"' c:,!!__

No.7 Rev. Fig. 35

respectively, whereas Parker-Dubberstein gtves the inverse order. The case of four consecutive months of 30 days is interesting enough to justify tl1e investigation of the special circumstances which led to this result. Unfortunately, the dates and hours in M 1 are not preserved but it is possible to restore them within narrow limits. To this end we utilize the lengths of all months from I to X, as established in Fig. 35, to give us the background for the dates of the new moons (Fig. 36). Then we use column K 1 to establish the moments of all preceding and subsequent conjunctions by starting from one initial moment which can be chosen with a high degree of probability. It is evident that the conjunctions in months I and III must fall as near as possible to the evening of first visibility in order to make a sequence of four consecutive late appearances possible. We assume that the smallest possible interval is about SH (cf. Introduction p. 67, note 34) and thus start with I 29 SH before sunset. This leads to the

following reconstruction of column M 1 and to the corresponding graph in Fig. 36. 6

0.

s.

:m2

c:£_

][

~

.m nr

c£...

JZ: 1![

1ZJ[

'lliii

IX X

xr

d!,_

c:!!...

c£...

c:.£_

~

c:!L c!r...... I::!L.

c2s

~

··-

[;!..29 -· •• .•

c:!.?... c1!. t::1!...

~

10,

d!... . . . .. ---~ d!_ d?,../

t:.!!...

,..._.

c:.3!...

c::.!!.-

.... --~

--

t

y

4, 2, 10 s, 4, Jl 10, Sl

3, 41, 2

llf

'K

l li

!![ £i![

iK !:

E

~­ l8

21 '1

.z, Z1

l~

21 lll l.6

21

u

21

z, J4, "'

sJ

o, sr, f,~-

sJ

Z, 31, I,] .S: 3~ IS Z, 7, 24 4. $, 14

su

0, 4Z

su

s

s;

0.

.s.;

o, zq 3

l, 1, IZ. 'i,U,IO I, lt,S]

~. 3o~; '

vI

v' su

VI

su

S,

vI

su vI

. .su' .su'

• I su su'

v

10,

1

SLI

6 In the transcription, Pl. 15, obv. V, only such day numbers are given which remain unchanged even if all conjunctions fall somewhat earlier. This would mean in Fig. 36 a translation of the curve, parallel to itself, to the left. Towards the right, only a very small motion would be permissible.

... c:.:L·· 1(..!-

~r'

c::.!L

-~

_...---- -~ 1(.' ...__

1(...!.__

.•'~

~ ~

t::;30

~.!.,._

. • . -· c:!L

1, s~ss

3, "· 13 2,-IIJI

c 29 .. 29

~ --

r:::..~?····

r; w·· ,......._

!!!

Xifz

2, l/l, 13 2,S7, 3l J, Z(,SI

-1, Ito. 31

~

t::.!!..

~ I, JS, Itt

4,

'X[

2,.56I

M,

K,

I(.!-

I

~...!..,_

~ 2 and B2 •

No. 8b BM 34881 ( = Sp.II,388) Contents: New moons for (at least) S.E. 183 and 184 (= -128/127 and -127 /126) Arrangement: ? (only one side preserved) Provenance: Babylon [Sp.] Transcription: Pl. 17; Copy: Pinches No. 21 Commentary Ephemeris which probably contained the following columns:

[-V] Critical Apparatus II,2 [2,4]1,1[3] : one corner wedge of 40 and one vertical wedge of 13 visible. II,9 3,4,6: the 6 is damaged but 6 is the best reading. Accurate computation results m the value 3,4,5,35,56. 11,10 2,57,2[3] : two wedges of the final 3 are visible. Accurate computation results in the value 2,57,23,7. II,ll 2,26,3[4] :one wedge of the final 4 is still visible. II,12 3,13,[54] :sic, instead of 3,53,[54]. Of the final 54 three corner wedges are still visible.

1\

[-IV] cpl

[-III] Bl

[-II] c1

[-I] El

etc. This fragment might be a part of No. 8 which contains the beginning of the year 183. The direct continuation is found in No.9.

No.9

BM 34088 + BM 34493 + BM 35282 + BM 35356 (= Sp.187 + Sp.617 + Sp.II,105 + Sp.II,851 + Sp.II,932) Contents: New moons and full moons for S.E. 185 (= -126/125)

Nos. 9 TO 11

96 Arrangement: 0 /R

Provenance: Babylon [Sp.; writing 20,6 in obv. VI,13] Previously published: Sp.II,105 and Sp.187: Kugler BMR Pl. 8 and 9 (join not recognized) Sp.617 + Sp.II,851 + Sp.II,932: * Transcription: Pis. 18 and 19; Copy (except for part of Sp. 187): Pinches Nos. 22 to 25 Commentary

Contents: Obv. I Tl T2 Rev. I

II 2 , the values of which we had to assume to have been influenced by an essential error. Computation of E2 for the years 277 to 280 shows indeed that E2 assumes nodal values two months ahead of the dates obtained for C/>2, and a similar statement holds for E1 • For the first part, however, we should expect perfect agreement between C/> 2 and E2. The nodes of E 2 fall into S.E. 177 XII, 178 VI and XI, whereas if>2 requires S.E. 177 XI, 178 V and XI. The latter dates, however, correspond exactly to the nodal values of E 1 • We thus face the strange fact that the values of (/) 2 were selected from dates determined by El" We shall see that also the columns B and C belong to new moons (B 1 and C1) and not to full moons. It therefore cannot be doubted that our text is the result of an erroneous combination of elements from both a solar and a lunar eclipse table.

l:J

and for the second part

[2, 4, 8,31, 6,40] 2,13,2[5,33,20] 1,5[8,45,5] 5,33,[20] (2, 12,35,33,20) 2, 4,58,31, 6,40 2, 7,12,57,46,40 [2,]10,21, 6,40 [2, 1,5]0,22,13,20

Rev. II, 1 3 5 7 9 11 13 15

Text

I

I; 'tl, /:;"'-';~"'G

lJ; /11 li>1~; ~-z

!!

ll.;'tl, 4f

"l

4; '~

.:e..

zM~

11

..

l.l(

~~-

)(

zs;~.,

~

ll; -41,11;ll;40

}(

B2

0

r t.z.o

g

"1

z.l.~o

"1.

g IJ

["'J

[b' "!. !1

=b )(

.!!!' X

I

I

l[..J;Jo Mib'O

b

If

"l

g

l,.

1;'t0

zz:

l'

30

11 30 l]; -40 1.3

"t

I'; 7, 30 II; Jz. I; ~S"

:::!b

zr; zz, Jo

)(

11;~

[~J"o

9

IZ; 3, 4S

8 1;41,/S

~~

u,~-6

tJ

l 3; II, 1>-

[ ...]30

"1

ts. ;-z

1:1

IZ;

I

117

(3. 44

~

'!f

)l

11S'

z:Z~tS

== )\

11..;

If(

·HII

)(

S.E. 113 ill.

411,45

"l lJ

ll: i,J, I

ll;

Text

Tl

ZJ;"O 23

This comparison rules out V) 8;8 (e.g. because of lT\. 27 and TT\.16 in the second and third lines) and strongly favors V) 8. We are now in a position to determine the date of B1 • Using V) 7;52, we obtain S.E. 90 IX ( = 222 B.C.) and S.E. 315 IX(= 4 A.D.) as possible dates. These dates should correspond to the dates of minimal latitude. If, however, we compute E 1 for these years we find that E 1 crosses the nodal zone in 90 VI and 315 VI. Thus there is no reason to consider these dates any further. The value V) 8, however, leads to the date S.E. 193 IX, which is only 15 years later than the date obtained for (/) 2 , Unfortunately, this date is not confirmed by E; according to E1, the months 193 VII and 194 I should have been chosen, not 193 IX. Though the spacing of B1 corresponds exactly to the spacing of nodal values of E 1 I could not find any date where longitudes and nodes agree. We must therefore assume a systematical error in the coordination of the lines of B1 and E 1 .

.

ll; 40

)(

Text

z. ~"0 ~~-o .ito ' ' l;41, 1. w

23; ..;4 l3;41

,!!,

..;tl IS

12/tO l; lf,.t.S

'; 3&

'1'

(3]

Jg_

JJ[

:0ii I

.n I

4 [S)

!E

6

li XII

7

:r

:r N

111 y M:

8

tn

The agreement is not too good, but could be made perfect if 196 I and 197 I could be replaced by the preceding months 195 XII and 196 XII. The years could be the regnal years of Antiochus VIII because his first year 121 B.C. corresponds to S.E. 191 6 ; but he seems not to have ruled in Babylon. Column lJf

In several instances, B is followed by numbers which are in all probability to be interpreted as eclipse magnitudes; e.g., in obv. II,4 we have 8 V) 8,10 be. The 6

Cf. Parker-Dubberstein, B.Ch., p. 21.

112

Nos. 61, 61a,

Contents: Solar eclipses for S.E. 191 to (at least) 194 (= -120/119 to -117/116) Arrangement: ? (only one side preserved)

~ g

K,

10

zo

12[...

lit

l6;SO f.I 30

:!!:: )(

23,40 23

s. 20

'?f

IZ; 30

)(

IZ; 30

~

2; 10

I;

1:!

No. 61a BM 77238 ( = 83-6-30,18)

Provenance: Babylon [BM]

B 9

10, 40 7,10

II 1,30 13,20

Transcription: Pl. 40

be

...]

Commentary This small fragment contains part of the columns

.s[ 2 in obv. II,3 and 5 and E leads only to a descrepancy in lJf (cj. table below).

s.

51

tion was to compose a table of lunar eclipses. The dates of the first part of 1>2 , S.E. 177 and 178, would then be the starting point for obv. 11/rev. II, leading to about 185 at the end of rev. II. The dates of B1 range from 193 to 199, i.e., well within the possible extension of an eclipse table covering also the period from 177 to 185 (24 years are covered, e.g., by No. 60). No agreement can be established, however, with E or lJf for these later years.

ideogram be is known from the eclipse table No. 60 to indicate negative values of lJf (eclipse excluded). In obv. 11,8 and rev. II,12, the ideogram which is added can be read si or gab, the latter being known to indicate lJf > 0 (Nos. 9, 10 and 18). The reading si "north", however, is perhaps preferable because it matches the ideogram gal "south" in rev. II, 10, though neither "north" nor "south" hitherto occur in connection with eclipses. Another strange feature of these numbers is the distribution of their values. One should expect, of course, that the values of lJf oscillate around the value 17,24 of the greatest total eclipse (cf. e.g., the values in No. 60, column VI). The values which we find here are much closer to 10 than one should expect, as the following list shows:

0

AND

qy

E1

'P.

determined

178 1I ~

E2

lj)2

T2

l78l. + 1,14,27 E[ + zo, 30, ~-,

qy

+ 21,

-

a,~

31,46

for this fact than the use of a different form of interpolation for the non-linear parts of G.

No. 51 BM 34608 ( = Sp.II,87) Contents: Solar eclipses from (at least) S.E. 199 to 206 (= -112/111 to -105/104)

Nos. 51

AND

113

Sla

[ -]'[]

Arrangement: ? (one side only preserved) Provenance: Babylon [Sp.]

[5, 38:~0. SJ. Sf, ( ~0] [ZA9. ro, s.s; 3~ zo] z..[3B;-IZ. 34, 48,S3, zo]

1K

I.

Previously published: Kugler BMR Pl. 12

JY

3,20 1[

ll!l.

Transcription: Pl. 42; Photo: Pl. 229; Copy: Pinches No. 46

J3.z.o] z, 4~~~zs.ss. ,53,31, 6. '10]

3,2.1][

liii

5.

3,Zl I.

Critical Apparatus [-III],6 3,22 I : see Commentary. II,6 and 7 : ljl is here negative but no "be" is added. II,14 and 15 : damage to the text makes it difficult to decide whether the first place was empty or a symbol for zero was used. II,14 and 15 : ljl is here negative but no "be" is added, IV, 1 [5 : traces. IV,4 3[7: traces of 30. IV,S 4[9 : sic, without initial zero. IV,l2 5[9: only two wedges of 50 preserved. IV, 13 [5 : traces.

[OJ Cl

Tl

I E1

II lJ'l

III F1

6.

VII

xuz. 3, 23 TI 10. Eli

s. [..

3,24}1

XI

3,2SXI

[4, I, 12,Zq44,z6,418

Xi

No. Sla BM 37062 ( = 80-6-17,807) Contents: Perhaps lunar eclipses for S.E. 206 to 220 (= -105/104 to -91/90) Provenance: Babylon [BM] Transcription: Pl. 43

IV .EGI

Fragments of complete ephemerides corresponding to lines 4 to 8 are preserved in Nos. 13a and 14. In line 6 the wrong month was selected because the moon is nearer the node in 3,22 II than in 3,22 I (cf. Pl. 42). The error has no practical importance because in neither case is an eclipse to be expected. In column II four instances occur where o/1 is negative (lines 6, 7, 14, and 15). One would expect that this would be indicated by "be" as in other eclipse tables, but this is not the case here. Column F 1 is symmetrical with respect to a point between lines 5 and 6 because we have 5.

s.

2, [1,.37,13,20] 2. t3.ts"z.to. 22. 13, zo] •, -41, 2~ II, [6, 40] 14, 6, 40] s, 'I, 2~,18, 12. 3S. 33, zo] 10. 1. [7, 7, z~. zt. ~o] I, sf9. I, Z.S?,-4( 4o] [S. II, l, 58, Jl, 6. 4o]

Arrangement: 0/R (cf. Commentary)

Commentary Solar eclipses for eight years. The columns are

[-III]

I.

14,12 14,54 15,36 15,36 14,54 14,12

Th;s symmetry is visible from line 0 to 11. Column IV contains the totals of column G between the ecliptic months. The table given herewith is computed on the basis of the relations between (jJ and G as stated in the Introduction (p. 60). The preserved initial numbers agree with this reconstruction (cf. especially line 8).

Critical Apparatus Obv. 1,6 8,29,20 : or 7,29,20. I, 7 ] 1,46 : ]2 or ]3 equally possible. 1,8 ]6 be : ]4 or ]5 possible; instead of be one might read 20. II,8 2]5,5[6 : three wedges of 5 and three of SO preserved. III,3 4,24,46 : or 45. III,4 5,5,[ : traces of following 20. III,S 5[ : or any other integer between 5 and 9.

Rev. III,2 5,[45 : traces of 45. III,S : uninscribed. Commentary This small tablet contained four columns of eight lines each on the obverse and similarly on the reverse, written in an arrangement which follows the ordinary scheme of cuneiform writing but which is otherwise never attested for astronomical tables:

Obv. Rev.

I

II

ljl

B

III 1J'

[IV] B

IV

III

[II

I]

A single vertical line is visible between columns I and II on the obverse and between III and IV on the

114

Nos. Sla

reverse; a double line separates column II from column III. Thus it is clear that the values which are written beside values of B always belong to the neighbor on the right. No dates appear in the text. The date of the text can be easily established from the values of column B, which concerns full moons and which can be connected with any other text of System A. In this way one finds for obv. II:

7;30 TTl_ 29;8 'Y' 27;7,30 18;4 'Y' 16;45

7

'Y'

6;22,30 [=::=)

S.E. 3,26 I VII 3,27 I VII 3,28 I VI 2 XII

and similarly for rev. III:

5;[45 )-(] 5;16 [11J7] 25;22,30=

3,39 VI XII 3,40 v

These dates are only determined modulo the period of column B, that is, within multiples of 3,45 years. Thus an interval from S.E. -19 to -5 would in principle be equally possible. The above list might suggest a list of intervals consistently 6 months apart. This is excluded, however, by the fact that 177 months lie between S.E. 3,40 V and 3,26 I. Thus an eclipse table remains the only alternative. This is supported by the remaining columns whose numbers are of a type to be expected for eclipse magnitudes lJf. Also the occurrence of the ideogram be for excluded eclipses points in the same direction. Nevertheless none of the dates in the above list corresponds to a possible eclipse, because the moon is close to its nodes two months (or occasionally) one month earlier than these dates. The same holds for the interval S.E. -19 to -5. Furthermore, if one computes the values of the continued function lf'' of eclipse magnitudes corresponding to the given dates, one obtains totally different values. Thus there is no good reason for the selection of the dates which are listed nor is it possible to explain the values in column "lf'". One can go a step further. We can completely disregard the association of column B with a definite date and assume that the corresponding lJ' values are correctly associated with the values in B. Then we are able to compute to a given value lJ' the corresponding latitude e from

e = sign K · 6 (lf' - c)

c = 17,24,0.

The differences ~E of the latitude being known via column B, it is possible to reconstruct the whole

AND

52

column E to which the values of lJ' should belong. The sign of the node remains undetermined but the numerical values of E remain otherwise unaltered. The values of lJ' are not influenced by this possibility of interchange of + and -. In this way one can, e.g., use the value 1Jf3 = 14,15,20 of obv. 1,3 to compute the corresponding e and one finds e3 = ± 18,52,0. Six months later one obtains from this e4 = =f 16,11,36 and again six months later e5 = =f 48,24,48. The corresponding values of lJf are lf'4 = 14,42,4 and 1Jf5 = 25,28,8.

The text, however, has

16,55,20 and 19,30,10 respectively. Similarly one may start with the correct value lf'4 = 16,55,20 and compute six months forward and backward: lf'3 = 16,28,36 and lf'5 = 27,41,24

as compared with

14,15,20 and 19,30,10. This suffices to prove that none of the three values of lJ' in 1,3 to 5 can be combined with column B so as to

give a latitude column E. Thus the origin of the values in lJ' remains unexplained. It seems possible that the present text is related to the eclipse table No. 61 (p. 112). Both tablets in all probability come from the same archive (Babylon) and both have in common the feature that the values of lJ' are in the average very low instead of showing a mean value of about 17,24,0.

No. 52

BM 33748 ( = Rm.IV,306)

Contents: Solar eclipses from (at least) S.E. 244 to 248 (= -67/66 to -63/62) Arrangement: ? (only one side preserved) Provenance: Babylon [BM) Transcription: Pl. 43 Commentary Solar eclipses. Only the beginning is preserved.

I ([Jl All lines correspond correctly to the values of E which are nearest the nodes (cf. Pl. 43).

No. 53

No. 53 BM 34083 ( = Sp.181) Contents: Solar and lunar eclipses for (at least) S.E. 298 to S.E. 353 (= -13/12 to + 42/43) Arrangement: 0-R (!) Provenance: Babylon [Sp.) Transcription: Pis. 44 and 45; Copy: Pinches No. 49 Critical Apparatus Obv. [-I) : the month names are the correct ones; cf. Commentary. V,3 5,36 : the 6 is damaged but only 5 or 6 is possible. Rev. I : the surface is badly damaged; the readings are thus very uncertain. I,1 5 :Pinches read 2 (damaged). I,2 57 : Strassmaier read 3, 17, the 3 being damaged; Pinches gives ]8,33, the 8 being almost completely destroyed, the 30 damaged. I,5 ]7 ,35 : or )4,35. I,10 48: Pinches: 15. II,8 sig : following Pinches, but su does not seem completely excluded. II, 9 g[ an] : following Pinches. III,3 13 : or 27? III,4 5 : or 8? III,5 15 : or 18 etc.; Pinches: 8 alone. III,6 26 : or 25. III,7 25 :or 24; Pinches: 15 (damaged). III,8 13 :or 14 etc.; Pinches: 13. Commentary This table of eclipses is the latest text in all the material published here, reaching well into the first half of the first century A.D. Groups of three columns, T, B, and 'P (or 'P') form a unit covering 18 years each, a fact obviously related to the eclipse cycle of that length and clearly visible in the similarity of the numbers of the eclipse magnitudes in columns I and IV of the obverse (solar eclipses). Regardless of the possibility of eclipses the sign l}ab is used throughout, but only positive values of 'P are chosen in those cases where two consecutive syzygies lead to latitudes inside the nodal zone. In columns II to IV of the obverse the values of B and 'P are computed correctly but the names of the months are consistently one month too early. In column V the months are given correctly. Also in my restoration of columns [-I] to I, I have used the correct

115

months. For the complete functions B, E, and 'P cf. Pl. 45. The reverse causes great difficulties, partly because of its very poor state of preservation but also because of contradictions which are inherent in the figures which can be read. We begin our discussion with the dates given in column II. The year numbers are either the numbers between 5,36 and 5,43 or between 5,46 and 5,53. To be on the safe side we call these year numbers n + 6, n + 7, ... where n may be any multiple of 10. What one can read with fair security is the following sequence of lines: line 1. XII 2 2. n + 7 [VI] 3. XII 4. n + 8 VI 5. XI 5 months 6. n+9[ . .. ] 7. X Thus we know for sure that the yearn + 6 is a *-year. Consequently n + 7 is an ordinary year and the restoration VI in line 2 is certain. We now must distinguish two cases: (a) the year n + 8 is an ordinary year, and (b) it is a *-year (** being excluded because there are two ordinary years preceding a **-year). In case (a), n + 8 XI will be followed by n + 9* (or **) V and XI (or X). The text shows X in line 7; thus case (a) requires that the year n + 9 is a **-year. Within the Seleucid calendar this is only the case for n = 3,0, n = 6,10 etc., whereas we have only n = 5,30 or 5,40 at our disposal. In case (b) n + 8* XI will be followed by n + 9 IV and X in good agreement with traces in line 6 and with line 7. Thus case (b) leads to n + 6* and n + 8* as a pair of intercalary years with only one ordinary year between them. This happens only once in each 19-year cycle and the endings 6 and 8 for *-years are only possible in the Seleucid calendar for n = 20, n = 3,30, n = 6,40 etc. Again 11 = 5,30 or 5,40 is excluded. Thus there must be some error either in the year numbers or in the month names or both. The subsequent months, m both cases, would be IV X IV X III IX whereas we find in the text III IX [III] IX II VIII This suggests the assumption of an erroneous 5-month interval somewhere after n + 8 XI. The correct sequence, beginning with the year n + 9, would be V XI IV X IV X III IX which may have been changed to IV X III IX III IX II VIII . In this case n = 5,40 is the only possibility.

Nos. 53

116

So much being established it is not surprising to find complete disagreement between whatever can be read in the text and the computation for the years 5,36 ff. as well as 5,46 ff. One example may suffice to illustrate this situation. In line 4 we can read 5,38 or 5,48 ki[n] (= VI) followed by 5 (or 8?) in the column of longitudes. Computation, however, requires either 5,38 IV 12,22,30 ~ or 5,48 IV 21,7,30 ~. Even if one would disregard the requirement of small latitudes and would compare only the longitudes with the dates given by the text complete disagreement results. Hence no accurate date can be assigned to the reverse except for the plausible conclusion that roughly the same period will be covered as on the obverse, that is, years in the neighborhood of S.E. 300 to 360. No. 54

BM 35231

+ BM 35355 ( = + Sp.II,931)

Sp.II,797

Contents: eclipses for (at least) 7 years Arrangement: ? (only one side preserved) Provenance: Babylon [Sp.] Transcription: Pl. 46; Copy: Pinches Nos. 47 and 48 Critical Apparatus

II,1 ]1,47 : or ]4,47. II,3 25,27,43 : the first 20 is damaged and could be read 30 or 40. Instead of 27 Strassmaier read 28. II,8 3[0, . ,2]9 : traces only. III,8 4,42,22 : 42 and 22 damaged. III,9 [4], . ,22 : traces of 4. Column III is followed by a strong double ruling. IV,12 3,5,7 : sic, instead of 3,4,7. Commentary

The interpretation of this fragment as an eclipse table is based on the following considerations. The numbers in column III are obviously large-hours and can therefore be considered as values of }; K. The numbers in column II could be }; C' under the assumption that the nodes practically coincide with the points where the signs of C' change. Adding up whole groups of 6 or 7 consecutive lines of C' such that all numbers have the same sign leads to totals between 34 and 35, as shown in column II of our text. And obviously these totals must show alternating signs, as is the case in column II. For the preceding column one should expect 1: J, that is, numbers which are either negative or zero. The corner wedges which are still visible at the end of column I allow an interpretation as parts of the sign lal which we would expect (cf. Pl. 40, No. 61a, col. II).

TO

55 No. 55 BM 46015 ( = SH 81-7-6,461)

Contents: eclipses or excerpts for several years Arrangement: 0 jR Provenance: Babylon [SH] Transcription: Pl. 46 Critical Apparatus

Obv. (?) 1,4 ]24: or 44; Strassmaier read 24. 1,5 [1]1,12: Strassmaier read ]1,2 which is indeed the best reading. But 11,2 < 11,4 = m. Thus one would have to read [1 ]2,2 which is not impossible but is no better than the reading 12 instead of 2. 1,6 ]8 : Strassmaier: traces of 4. 1,8 14,14: only faint traces of 10 or 20. A better reading would perhaps be 11,11 or 11,21 but a "one" cannot appear as a last figure in column F. Thus only 4 remains as a possible reading of the last figure, and from analogy also the first unit should be 4 and not 1. II,1 ]4,6,5[0 : 4 and 50 very uncertain. II,2 [5]3,20 : or [3]3,20. II,4 tab : Strassmaier read a damaged lal, but tab is certain. II,6 ]9,2,5 : or ].,2,5? II,6 lal : error for tab; cf. commentary. II,7 [4, .. ,4]5 : traces of 4 and 40 or SO. II,8 3,[10, : or any number from 10 to SO. II,9 3,[4]5, ... : traces of 40 or 50. 11,10 ]4: or 5. II,ll ]8 : or 7, or 6. III,3 [30 : traces, perhaps 40. III,4 [5]7 : 50 partly preserved. 111,6 [5]9 : traces of 40 or 50. 111,7 to 10 : traces of the restored numbers. Rev.(?) 1,1 2,[15,47: traces. 1,5 [27] : traces. 1,6 [20,3]6 : traces of 20 and 30. 1,7 32 : or 33. Commentary

A glance at the preserved numbers suffices to show that we are not dealing with an ordinary ephemeris but with a list of excerpts, thus either a table of eclipses or equidistant excerpts, presumably in 12-month intervals. Unfortunately, neither of the two hypotheses suffices to explain the text. Obv. I is obviously a column F. In order to explain

Nos. 55 TO 75 the preserved numbers as excerpts for eclipses, thus 6 (or occasionally 5) lines apart, one should have 12,58 15,24 11,12 15, 8 13,6 or 12,24 13,14

t

t t

t t t

text: [ ... ]2 [ .. ]24 [1]1,12 [ ... ]8 [ ..... 14,14

t

(!)

t

(!)

[ .] [ .] [ .] .]

117

t,

of eclipses or excerpts as well. If we keep the sign then a Diophant gives the dates S.E. -3 or A.D. 191, both being historically excluded. And again, no other value in our text appears as the continuation of 4,24,16, ... Thus the possibility of excerpts seems definitely disproved though the positive proof for eclipses is equally wanting. Obv. 111,4 shows the number -57,3,45 which is characteristic for column J and would be the expected value for a list of equidistant excerpts. The next six lines, however, show +21,2,59. This is the difference between two values of g( or g( in column J, twelve lines apart (cf. Introduction p. 62). This number could be useful for the computation of excerpts, but should not occur in excerpts themselves or in eclipse tables. The number 2,39,23,[30] in obv. III,3 would be a value g(,j,) or (t) if 23 could be emended to 36. Obv. IV and rev. I are probably columns of the same type and the preserved numbers do not contradict the possibility of excerpts from a column C'. Rev. II could then be related to a column K.

t.

Similar discrepancies appear in the case of steps of 12 lines. In order to obtain in the third place the value 11,12 the first line should be 13,56 t and the last one 15,2 Thus neither hypothesis is supported by column F. The next column seems clearly to favor eclipses because the minimum of column G is 2;40H whereas in obv. 11,2 and 3 occur 2;32, .. Hand 1;0, .. H respectively. This excludes excerpts of single values of G but is perfectly possible for a column }; G as is needed for eclipses, the totals being reduced mod. 6H (cf. e.g., No. 60 column VIII). On the other hand, the signs tab and lal make sense only for single values which can lie on increasing or decreasing branches respectively. Furthermore, there occurs in line 4 the value 4,24,16, 32,35,33,20 which, although with the sign,),, is found as the value of G 1 in No. 8 for S.E. 3,2 III. Direct computation shows, however, that none of the neighboring values in our text can be explained for the cases

t.

t)

t)

No. 60 Seep. 106. Nos. 61 and 61a See p. 109 ff.

§ 3. AUXILIARY TEXTS A. LATITUDES No. 70 Sp.II,453

+

BM 34934 ( = Sp.II,604) Contents: Auxiliary table for full moons for (at least) S.E. 49 to 60 (= -262/261 to -251/250) Arrangement: 0 jR

connectible with the corresponding columns in all the remaining texts. Probably the text contained a total of 4 columns, each of which covered T 2 + B2 + E2 for 4 years. The obverse would then concern the years from 45 to 52, the reverse from 53 to 60. This text is one of the earliest texts of our whole material; cf. p. 10.

Provenance: Babylon [Sp.] Previously published: * (Sp.II,453 was mentioned by Kugler BMR p. 55 and p. 58)

B. EXCERPTS

Transcription: Pl. 47; Copy: Pinches No. 51 Critical Apparatus

Obv. 11,7 3,20,.6 : the 6 is written below 42, the zero below 51. Commentary This table contains the values of T 2 , B2 , and E2 for at least 10 years. Diophants show that both functions are

No. 75 BM 45976 ( = SH 81-7-6,419) Contents: Excerpts from ephemerides for at least S.E. 181 to 185 (= -130/129 to -126/125) Arrangement: ? (only one side preserved) Provenance: Babylon [SH] Transcription: Pl. 48

Nos. 75 TO 81

118 Commentary

This is a table of values of E1, lfJ~, and F1 for moments exactly 12 months apart. Whereas eclipse tables give excerpts from an ordinary ephemeris for those months for which the latitude is closest to zero, we have here excerpts without any astronomical significance. The only plausible explanation for this type of excerpts seems to be their usefulness for checking purposes. Indeed, equidistantly spaced values follow a pattern which can be independently computed and which is closely related to the Diophantine methods which I have used for checking purposes. A hint in the same direction can be found in the marks which appear in No. 190 (cf. p. 179). The present text is only a small fragment of a larger tablet. Probably No. 76 formed a direct continuation for the years S.E. 204 to 221. Column F 1 is computed with abbreviated parameters. It is plausible to assume some indication as to whether a value belonged to an increasing or to a decreasing branch, indicated here by and respectively. All values of tp' 1 are positive in the present section but preceding values would be

t

t

§4.

Provenance: Babylon [Sp.] Colophon: Zlc (p. 21) Transcription: Pl. 49; Copy: Pinches No. 54; Photo of rev.: Pl. 230 Critical Apparatus

Rev. I,4 [2,]6,11, .. : sic, instead of [2,]4,11, ... Commentary

Rev.

I

q,• I

II B* II

+ BM 34749 ( =

+ Sp.II,240)

- is

Sp.II,63

Contents: Excerpts from ephemerides for S.E. 204 to 221 (= -107/106 to -90/89) Arrangement: 0 /R Provenance: Babylon [Sp.] Transcription: Pl. 48; Copy: Pinches Nos. 52 and 53 Commentary

Excerpts for 12-month intervals of B1 , C1 , and E1 • An exception to this pattern occurs in lines 5 and 6 of the obverse. From 3,24 VIII in line 1 to 3,27 VII in line 4 twelve-month intervals are used. Then, without any visible motivation, the text jumps to 3,28 I and II (that is, 6 months and 1 month respectively). From 3,28 II until the end (3,41 IX in rev. 7), twelvemonth intervals are chosen as before.

S.E. 178 VI 29 to VII 30 is found in No. 81, which latter text is to be consulted for the dating of both fragments.

No. 81 BM 34803 + BM 34815 ( = Sp.II,296 + Sp.II,314) Contents: Daily motion of the moon for S.E. 178 VII (= -133 Sept./Oct.) Arrangement: 0 /R Provenance: Babylon [Sp.] Transcription: Pl. SO; Copy: Pinches Nos. 55 and 56 Critical Apparatus

Rev. V,2 lallal : sic, instead of lal u.

The text contains the following columns:

[0] T* [0]

No. 76

BM 34590

+ or

DAILY MOTION

No. 80 BM 34606 ( = Sp.II,84) Contents: Daily motion of the moon for S.E. 178 I (= -133 March/April) Arrangement: 0 /R

Obv.

negative. The omission of an ideogram for surprising.

III

c•

III

The dates are expressed in tithis (cf. p. 40) and the subsequent columns are normed accordingly. The mean conjunctions are associated with the date 28. The present fragment covers the lunar days from S.E. 177 XII 28 to 178 I 29. Its continuation for

Commentary

The following columns are preserved: Obv. Rev.

I T* I

II

q,• II

III B* III

IV

c•

IV

v

E*

v

VI F* VI

The first four columns can be obtained by continuation from the corresponding columns in No. 80, a text

119

No. 81 which concerns the first month of the same year of which the present text concerns month VII. The date of both fragments can be established as follows. In No. 81 just enough is preserved at the beginning of the obverse and at the end of the reverse to establish all day numbers (tithis) in column I. Column C/J 1 of an ordinary ephemeris contains only six places whereas our text gives seven places. Consequently, as elements of C/J 1 are possible only those values of C/J* which have zero in the last digit. This excludes all dates near the end of a month except the 25th, the 28th, and the 1st. A Diophant shows that the values of C/J* for the 25th and the 1st cannot be connected with C/J1 whereas this is possible for the value which corresponds to the 28th. Indeed 2,3,43,53,20 U,) in rev. 13 is a value of C/J 1 of the ordinary ephemerides, corresponding to the date S.E. 178 VII. From column III it follows that the corresponding solar longitude is === 24 and this is again in agreement with the data obtainable for column B1 for 178 VII. The connectibility of Nos. 80 and 81 can be established directly and leads for the 28th day to the correct values known from the ordinary ephemerides. The relevant section of T 1, C/J 1, and B1 is Xii

Z, II, 4, 1.>: 33, lO

z,

11:

u. s~. 11,

l

3~

6 40

l, J.4,.., ~

JS', JJ,

R

l, ll, I, 40

z,ss* r

1if

:X: li

!.!I

lb

l, 1, l, '.

6, 4o

zo

t:.: "~. l '· 40 c1,ftK, S'J, lO

l, 3,4~.S-~ lO

-~)~·~:--- y,· z, ~~- 1.:>

a

lj, ~'. IS lK, J,4S

~

o, ~~ ,,:,- .li:

No. 80, obv. No.80, rev.

I. II.

)(

£U!..Ui__ §1. l4

.24

'.!!'

~

No. Sl, rev. 13.

Finally we can compute column E for 2,58 VI and VII by means of continuation from another ephemeris of System A. One obtains for E 1 : 2,58 VI VII

·- 2,50,2,3 - 4,56,17,45

The first value would appear in column V in the line which precedes obv. 1 (for the 28th of month VI) whereas the second value is identical with the value in rev. V,13 (for the 28th of month VII). This confirms the previously obtained date. Obviously VI 28 and VII 28 are the dates of the conjunctions whereas the opposition falls on VII 13 (obv. 15). The values of C/J*, B*, C*, and certainly also ofF*, are obtained by linear interpolation for thirtieths of a synodic month, i.e. for "tithis". The same holds in principle forE*. Because we are dealing with a fast arc of the ecliptic, E* outside the nodal zone increases in

30 steps by D = 2,6,15,42,0. This leads for a single step in E* to the difference D* = 52,12,31,24 (cf. Introduction p. 54). The crossing of the nodal zone between K = 2,24,0,0,0, and -- K = - 2,24,0,0,0 was handled rather carelessly by the scribe. Both in obv. V,9 and in rev. V,8 only the first value y 1 within the nodal zone is correctly computed from the preceding value y0 according to

Y1

=

+ ~)) 2

2 (Yo =f (D*

(cf. Introduction p. 49). The subsequent values still lie within the nodal zone such that

Y2

=

Y1 =f 2D*

Y3

=

Y2 =f 2D*

should have been used and only y 4 would be given by

Y4 =Yo =f (4D*

+ K)

whereas the text finds already y 2 from y 0 =f (2D* For example, one finds for obv. V,10 ff.:

+

Yo= 2,51,52,43,0 Y1 = + 1,35,20,23,12 9,4,39,36 y2 = Y3 = - 1,53,29,42,24 Y 4 = - 3,0,57,22,36

Text:

+

K ).

+ 2,51,52,43,0 + 1,35,20,23,12 - 1,16,32,19,48(!) - 2,8,44,51,12(!) - 3,0,57,22,36

and similarly for rev. V, 7 ff. Of course errors of this type do not influence the values of E* outside the nodal zone. The scribe obviously overlooked the fact that E* can have three or four points inside the nodal zone and operated as in the case of a single value for E. Column VI is unfortunately only partly preserved. This much, however, is clear: it gives the lunar velocity F* for single tithis, using values close to

8 = 20,56

m

= 11,4,0

M = 15,57,0

(cf. Introduction p. 59). It is obvious that the complete column gave more than two places. As is to be expected, the difference in· the second digit is ordinarily 21, with an occasional 20. In obv. VI, 9 to 11, however, two differences 22 seem to occur, compensated by a 20 in line 8 and in line 14. The preserved numbers suffice to show that no essential error can have occurred in column F* in the part which is given in the text. On the other hand, it can be shown that F* in our text cannot be obtained by using exactly the same position for the minimum of F* and of C/J*, though the deviation is probably quite small.

Nos. 81 To 91

120

§ 5. EPHEMERIDES OF UNDETERMINED SYSTEM FROM BABYLON The following texts (Nos. 90 to 93) form a group by themselves because of a strange combination of elements of both Systems A and B. The occurrence of a column(/> and the use of 7,12,0 as maximum of the latitude relates them to System A. A simple zigzag function for the latitude, however, and a column G of the same type assimilates them with System B. The parameters for column F, the longitudes B and the length of daylight or night agree with neither one of the two major systems in detail. The present fragments do not suffice to reconstruct an independent system of the lunar theory. The irregularities in detail and the great number of obvious errors make it not very plausible to see in this group of texts more than a somewhat abortive attempt of one computer to reproduce (or possibly modify) ephemerides of the ordinary type. Whether the last text (No. 93) is at all related to the preceding group cannot be made out at present. No. 90 BM 36636 ( = 80--6--17,367) Contents: New moons (?) for (at least) three years Arrangement: 0 fR Provenance: Babylon [BM] Transcription: Pl. 51 Critical Apparatus

Obv.fRev. : This arrangement is certain because several lines are left empty at the end of the reverse. Obv. II,11 3,16: sic, instead of 3,13. Isolated error. III,19 3[] :perhaps nothing missing. Rev. I,8 ] 1,18 : reading very doubtful. I,9 ]10,50 : reading of 50 doubtful; the 10 is perhaps preceded by units, perhaps 5 or 8. II,3 5,51,45 : sic, instead of 5,50,45; isolated error. II,4 3,23,35 : sic, instead of 3,23,30; isolated error. Commentary

The three preserved columns are B, E, and C. Column E is based on the parameters M

=

-m

=

7,12;08

d

=

2,27;15 8

hence (cf. p. 55) p

= 1,55,12 = 11;44,6, .... 9,49

as compared with the ordinary value 11 ;44, 15, . . . . Column B is too badly preserved to check the method

of computation but the few numbers which are preserved in rev. I,8 to 11 do certainly disagree with System A. The last column is obviously an abbreviated column C or D. Because the maxima fall near Q15, the minima near ~, we are dealing in the first case with new moons, in the second case with full moons. Whatever the case may be, the values in C (or D) disagree in detail with all known schemes. The repetition of 61 in obv. I,16 and 17, and of Q15 in rev. I,lO and 11, shows that the corresponding longitudes must be close to the first and last degree of the respective signs. This leads to the following comparison:

61 0 61 30 Q15 Q7i

0 30

System A 3,33;20 3,24 3,34;40 3,33;20

System B 3,31;36 3,21;12 3,34;24 3,31;36

Text 3,35 3,28 3,35 3,23

The intermediate values seem to be computed with a rather crude scheme of almost constant difference 12 though such impossible pairs as 2,56 and 3,0 occur (obv. III,12 and 13). A final difficulty lies in the problem of determining the extent of the tablet. A Diophant shows that obverse and reverse are connectible but only at a minimum distance of about 110 years. This result being obviously excluded for a single tablet, we have to assume some error since both sides, which concern either full moons or new moons, should be connectible. The obverse of No. 90 would be about 90 years earlier than No. 91 rev., and 217 years earlier than No. 93. The reverse, however, would be about 18 years later than No. 91 rev. and 109 years earlier than No. 93. None of these relations seems very plausible. No. 91 BM 32351 (= st 76-11-17,2083) Contents: Auxiliary table for the latitude of the moon for (at least) four years Arrangement: 0 /R Provenance: Babylon [BM] Transcription: Pl. 51 Critical Apparatus

Obv. and Rev. may be interchanged. Obv. II,2 z[fz ] : traces of and? Rev.

II, 1 [and : traces.

Nos. 91

Commentary Column I is the same type of column E which we know from No. 90. The terminology, however, is different in so far as u is now replaced by sig. Column II gives consecutive months with the addition of a note for values near the nodes, introduced by and and at least once followed by RfN which is an otherwise unknown term in connection with latitude; otherwise rin = Libra or zalag = daylight would be possible interpretations. No. 92

BM 34581

+ BM 34610 ( = + Sp.II,89)

Transcription: Pl. 52; Copy: Pinches Nos. 58 and 59

Critical Apparatus The greater part of the surface is badly worn, and thus many signs are only visible in traces. II, 12 : the final zero looks like a big 10, different from the zero, e.g., in the next line and elsewhere in the text where zero is written as usual by two small corner wedges above each other. III,6 ff. : because of a break in the tablet the 7 is nowhere completely preserved but it seems a better reading than 8, which is the only other possibility. IV,6 25 :Pinches 35,[1]6 (?). V,15 3,29: Pinches 3,27 (?). V,16 3,16: Pinches 3,25 (?). V, 17 2,57 : Pinches 2,58 (damaged). VI,12 5,54,45 :sic, instead of 5,44,45; isolated error. VII, : on Pl. 52 the last column should be called VII, not V. VII,11 1,5[0: Pinches: 1,55 (?). VII,15 3,29,23,10 : Pinches copied 2,.9,13,1[0 but traces allow also my reading. VII,16 3,52,53[: Pinches 3,52,34. VII,17 4,15,2[0: Pinches 4,15,40. VII,18 4,25[: Pinches 4,15 (?). Commentary This text concerns full moons as is evident from the combination of months in I and longitudes in IV. The arrangement of the columns is quite unusual:

III F

of System A but not connectible with 2 (or 1) of any other text. Column F. The difference is d = 42,0,0,0. For the maximum we obtain from lines 10 and 11:

2M- d = 31,3,26,15,0 thus M

IV

v

B

D

VI E

=

15,52,43,7,30.

The minimum cannot be determined directly from the text. It is, however, a natural assumption to suppose that as usual F and have identical periods. This leads to

fL

Provenance: Babylon [Sp.]

II

121

+d=

22,41,46,52,30

and finally to agreement with the preserved numbers in lines 3 and 18. The mean value of this new variant of function F in System A is

Arrangement: ? (only one side preserved)

92

m = 10,59,53,26,15 and 2m

Sp.II,53

Contents: Full moons for (at least) two years

I T

AND

VII G

also in details many irregularities can be observed. Column . Computed with the ordinary parameters

= 13,26,18,16,52,30

thus somewhat smaller than the ordinary mean value of F in this system fL = 13,30,29,31,52,30.

The period is, of course, the same in both cases. Contrary to expectation, the maximum of F does not exactly coincide with the maximum of but is located about one half interval later. Column B. Between Y) and II (that is for the sun between ill5 and 1 ) a monthly velocity of 29° is applied without revealing a clear scheme for the rest of the zodiac. This looks almost as if a constant mean motion of the sun (29;6, ... ) had been intended to be used. Column D. Because the highest numbers occur in the months VIII and IX we cannot interpret this column as length of daylight (C) but only as length of night (D). The values agree 5 times with System A, 5 times with System B and a few times they seem unexplicable by either system. Column E. Here we have the same function for the latitude which we know from Nos. 90 and 91. All these columns are connectible at distances which were listed in the commentary to No. 90. The terminology for the latitude itself, Ia! : u, in our present text is the same as in No. 90. But there is added a second sign for the increasing (Ia!) and decreasing (u) branch, as is common in texts of System A. Column G. This last column is only partially preserved but its character cannot be doubted: it is a linear zigzag function similar to column G in System B. Nevertheless the difference seems to be not 22,30 but perhaps 23,30. Also the maxima and minima do not agree. Unfortunately the text is so badly preserved that it is impossible to determine the values accurately.

122

Nos. 92a AND 93 No. 92a

BM 36323

+ BM 36700 ( = 80-6-17,49 + 80-6-17,432

+ 80-6-17,715)

Contents: Longitudes and latitudes of the moon in four separate years Arrangement: 0 jR Provenance: Babylon [BM] Transcription: Pl. 52a

Critical Apparatus Obv. I,6 Rev. I,12 11,2 11,3 IV,2

[30,32 : expected 30,35; cj. Commentary. [5],1[7]: 6 units of 7 are visible. 1,27 : or 1,28. 28 : written over erasure of 1,28. ff : the differences suggest an error.

Commentary The present text is written in four columns each on the two sides of a tablet of which I call "obverse" the side which contains the longitudes, "reverse" the side with the latitudes, since this is the usual sequence of columns. Neither side is fully inscribed; the fourth column as well as the last line are followed by blank space of about the width of a narrow column and five or six lines respectively. The purpose of this text is unintelligible to me. The omission of date columns would suggest a procedure text, but nothing is really typical in the two types of columns. I am not even sure whether the four columns of the obverse are supposed to give the longitudes corresponding to the four columns of latitudes of the reverse. Each column of the obverse can be obtained from its predecessor by adding 10;20° in each line. Since 10;20 is ! of 31, it would require 1080 columns before this scheme repeats itself. It is also obvious that the consecutive columns are not continuations of each other. Thus we have four columns of similar structure but of entirely unknown mutual relation. The differences (column ~ on Pl. 52a) suggest the assumption that the longitudes in line 6 should be 30;35 lEi, 10;55 61., 21;15 61., 1;35 ~. respectively. We then obtain a much more symmetric distribution for the differences which must represent monthly motions of the moon. This latter interpretation finds a confirmation in the fact that the arithmetical mean of the differences in each column is 29;6 and this value is known to us as mean monthly motion of the moon from the procedure text No. 200, Section 9, since 29;6 is the

mean value of the extremal monthly solar motions of

27;36 in n and 30;36 in f. Unfortunately, it is only the mean value for which we obtain agreement. The minimum in our text is 29; 1 and lies in the middle between n and f and similarly for the maximum. In other words, the solar apogee is a whole quadrant behind its correct position. For the latitudes we have to assume extrema of ±5,46 and two essential differences of 1,55 or 2,0 respectively for the upper and lower half of each column. If we consider the longitudes of the obverse as belonging to the latitudes of the reverse, we can say that the difference 1,55 belongs to the part from 'Y' to 61., and 2,0 to the part from ~ to )( of the ecliptic. This is reminiscent of the latitudes in System A where d = 1,58,45,42 holds from )( 27 to ~ 13 and D = 2,6,15,42 from~ 13 to )( 27. The smaller differences can be explained as a change of units. If we reduce the values of d and D in the ratio of 5,46 : 6,0 of the extrema, we obtain 2,1 and 1,54 respectively, instead of 2,0 and 1,55 in the text. In using ±6,0 as the extrema of the ordinary theory we have disregarded the nodal zone. This is only partially justified by our text which shows no influence of a nodal zone for increasing branches. The decreasing branches, however, are slightly modified near zero, though not by a constant amount as expected. I see, of course, no motivation for a differentiation between the two branches. The values of the latitudes in the four columns of the reverse differ only very little from column to column; the values in the nodal zone, e.g., decrease only by 0;4 (line 3) or 0;1,20 (line 9). If the obverse is, as we assumed, associated with the reverse, then we would have to suppose that the distance between consecutive columns corresponds to a period which nearly restores longitudes and latitudes. This would relate our text to the theory of eclipses.

BM 34705

No. 93 BM 34960 ( = Sp.II,193 + Sp.II,482)

+

Contents: Latitudes (?) and eclipse magnitudes for (at least) 7 years Arrangement: 0 jR Provenance: Babylon [Sp.; cf. however, writing 30,.4 in obv. 11,17] Transcription: Pl. 53; Copy: Pinches No. 57

Critical Apparatus Obv. 1,13 ]8,46,40 : or ]7,46,40. 11,20 21,16: sic, without preceding u.

123

No. 93 III,3 10,13,15,26,[40: 26 or 23. III,9 7,48,17,46,40: 48 or 58; if necessary 17 could be read 27. Rev. I,17 ]8,46,40: or ]7,46,40. 11,2 35,[56] : upper part of SO visible. x. III,9 15,31,1[0: or 1[0 III,15 9,2[0: or .,2[0+x; Strassmaier read "zero", which I restore to 9, the lower part being broken. Pinches read 20,2[0.

+

In this way it can be shown that the reverse is the continuation of the obverse such that rev. 0 = obv. 58. Thus the complete text may have contained about 90 lines. For checking purposes it is useful to know that the values which directly follow a discontinuity form again a linear saw function whose parameters are:

d' (4)

This text is unfortunately too badly damaged to allow more than the statement that it contains an otherwise unknown method for the computation of a function of the lunar latitude, presumably eclipse magnitudes. The text is very carefully written and shows a wide empty space between column I and II, whereas the numbers in III follow II closely. No column ruling is visible. Of column I only two (or sometimes three) final digits are preserved, mostly multiples of 6,40 and thus much too common to be identifiable. It seems to me possible, however, that the endings in III plus the endings of the corresponding numbers in I add up to zero. The numbers of column II are closely preceded by the ideograms Ia! and u at distances of 6 or 5 lines, thus probably indicating increasing and decreasing branches respectively. In these lines, which obviously correspond to eclipses, we find the numbers of III, preceded by sig and Ia! respectively, probably indicating negative and positive latitude. These numbers in III could well be eclipse magnitudes measured in fingers (given by the first digits) though with an accuracy of 5 places as compared with the ordinary 3. Also the addition of GAR is not to be explained from our ordinary texts. The numbers in II form a simple linear saw function increasing with a differenced= 3,4 from m = 21,0 to M = 39,0. If y 0 and y 1 are values before and after a discontinuity, we have

Yo - y 1 =

~

- d = 14,56

The other parameters are: d= 3,4

(2)

fl.= 30,0

m = 21,0

M = 39,0

= 18,0 2,15 p = 23 = 5;52,10,26, ...

~

If Yn - Yo is divisible by 8, then the number n of lines between Yn and Yo is given by

(3)

n

= g47 (Yn- Yo)

Yo' - Y1' = 2,40 23 P' = - = 7·40 ' 3

24

m' = m = 21,0

M' = 24,4

Commentary

(1)

=

mod. 2,15

The period P = 5;52,10, ... in (2) gives the distance between two nodes of opposite sign. Thus the period of the latitude itself between nodes of the same sign is determined by

(Sa)

2P =

4,30

23

= 11;44,20,52, ...

from which a true period

(Sb)

4,30

p = 4 53 = 0;55,17,24, ... '

is derived (cf. Introduction p. 31 ). This is obviously an approximation of a more accurate period, e.g., known from System B: (5c)

1,30,58

p = 1 38 43 = 0;55,17,22, ...

' ' A (cf. p. 47) or of System for and similar (5d)

p=

1,55,12

Zsl ' '

=

0;55,17,19, ...

in No. 90 and No. 91. (cf. p. 55) The main question I cannot answer: what significance did the numbers tabulated in column II have? If we call these numbers y we know that eclipses are characterized by the condition

(6)

21 < y

:s:: 24,4

where the exclusion of 21 and the inclusion of 24,4 is taken from rev. 4. If I am right in assuming that the numbers in III are the eclipse magnitudes lf', then I would furthermore conjecture that lJI tends towards c = 17,24 when y tends towards 21. It would be natural to think that y represents the latitude at the syzygies but this would require a mean value 0 and not 30. The fragmentary character of our text makes it impossible to associate it with either one of the two main systems.

124

CHAPTER II. SYSTEM B Texts Nos. 100 to 129:

Ephemerides

Nos. 130 to 136:

Eclipses

Nos. 140 to 182:

Auxiliary Functions

Nos. 185 to 187:

Daily Motion of the Sun

Nos. 190 to 196:

Daily Motion of the Moon

Nos. 198 and 199: Solstices and Equinoxes

Introduction Texts of the lunar System B show nothing like the uniformity of System A. This might be due in part to the fact that System B is represented in two archives, the earlier texts belonging to the Uruk archive, the younger ones coming from Babylon. But also within each of these two main groups we find a marked lack of uniformity. This holds not only so far as external appearance is concerned (e.g., arrangement of columns, rounding-off of numbers, terminology, etc.), but even similar texts are frequently not connectible because of changes in parameters. Furthermore, while the texts of System A are complete ephemerides1 of almost identical type, the majority of Uruk texts of System B are "auxiliary tables", containing only selected groups of columns, e.g., A and B, or H, or Hand J, etc. The arrangement of the texts in this chapter is determined by the above facts. We give first the complete ephemerides in chronological order, thus separating automatically the Uruk archive from the Babylon texts. The second group consists of eclipse tables, all from Uruk. Then follow auxiliary tables, again all but one from U ruk, arranged first according to contents and then chronologically. The last section concerns the daily motion of sun and moon, including three Babylon texts and one of doubtful origin (No. 190 = MLC 1880). As an appendix I added tables of solstices.

Arrangement of the Texts

§ 1 A. Ephemerides from Uruk No. 100 S.E. 106-108

T 1 B1 F1' D/ 11\" L 1 M1 Nl 01 Ql Rl Pl Na Pa

101

118,119

102

121

103

123

... Bl Fl' ...

104

124

T B F' D' IJ'" L ... (for 1 and 2)

105

135-137

T 2 B2 F 2 ' D 2 ' M2 [N2 ...

106

.. 136,137..

[T1 B1] ... D 1' IJ'1" L 1...

T 1 B1 F 1' }1 D 1' "~P/ L 1 Ml N 1 01 Ql Rl P1 T B F' D' J IJ'" L M N 0 Q R P (for 1, 3, and 2)

•••

L2

107

... L 2 M 2 N 2

•••

108

... B1 D 1 '

M 1 N 3 P3

109

... L 1 M 1 ...

110

... R 2 P 2

.••

•••

§ 1 B. Ephemerides from Babylon No. 119 S.E ... 176(?) ... [~IJ'1 '] IJ'/ H 1 }1 G 1 Kl ... 120 179 ... ~IJ'' IJ'' F' F G H J K L M (for 1 and 2) N1 01 Q1 R1 Pl 121 181 ... ~IJ'' IJ'' F' F G H ... (for 1 and 2) 121a .. 185-188 ..... F 1 G 1 H 1 }1 K 1 L 1 pl .. . 122

208-210

[Td A1 B1 C1 D1 IJ'1" F1 Gl H1 h K1 Ll M1 N1 01 p1 Pa Oa

1 I disregard here for the moment the eclipse tables and the excerpts from complete ephemerides.

125

SUMMARY OF CONTENTS

122a 123 123aa

.. 221 ..

T 1 A 1 B1

235

T A B C lJf" fj.lJf' lJf' F G H ... (for 1 and 2)

236

§ 3 C. Auxiliary Tables. Lunar Velocity

•••

TAB C ... (for 1 and 2)

123a

T A B C lJf" F G H J K L M (for 1 and 2) P 1

124

... A2 B2 D2' ...

125

... B C lJf" ..•

125a

... B2 c2 .. .

125b

... CD .. .

125c

••. lJf" fj.lJf' lJf'

125d

•.. fj.lJf' lJf'

125f

... fj.lJf' lJf' .••

126

· · · 'P2" f2 G2 H2 J2 K2 L2 M2 ...

126a

... [G2J L2 ...

127

... H

128

... Kl Ml Pa

129

· · · G2(?) 12 K2 L2(?) · · ·

F G ...

F G ...

[H2J

12

K2

1 ...

S.E ... 126-130 .. T 1 B1 D 1 '

156

S.E. 104-124

T 1 F 1'

.. 122-131 .. T 2 F 2'

§3D. Auxiliary Tables. Columns Hand J No. 160

S.E. 123-154. .

T 2 H2 T 2 H2

161

124-156

162

.. 133-151 .. T 1 H 1

163

.. 117..

164

.. 127-132 .. T 2 H 2

165

.. 137-156 .. T 2 H 2 }2

T 2 H 2 12

166

T H J

167

T H J

}2

§ 3 E. Auxiliary Tables. Syzygies No. 170

S.E. 104-112..

T 1 B1

}1

G1 L1

171

115-124

T 2 B2 }2 G 2 K 2 L 2

172

.. 117 . .

. .. G 2 K 2 L 2

173

123-130

... Br 1r Gr Kr Lr

174

124-130

T2 B2

175

12

G2 K2 L2

... G2 ...

§ 3 F. Auxiliary Tables. Visibility

§ 2 A. Solar Eclipses No. 130

No. 155

.•.

L 1 M 1 'P1

No. 180

S.E. 120-125..

T 1 P1

181 182

§ 2 B. Lunar Eclipses No. 135 136

S.E. 113-130

T 2 B2 D 2' L 2 'P2 M 2 G 2

.. 121-124 .. T 2 B2

•••

M 2 'P2

§ 3 A. Auxiliary Tables. Longitudes No. 140 141 142

S.E. 115-130 .. Tr Ar Br .. 121-124 .. T2 A2 B2 123-142 Tr Ar Br

143

.. 146 ..

T 1 [A 1] B1

144

148-161

Tr Ar Br

145

.. 126-139 .. T2 A2 B2

146

... [A] B

§ 3 B. Auxiliary Tables. Eclipse Magnitudes No. 149 150

S.E ... 54-67 . .

B1 /j.lJfr' 'P1 '

§ 4 A. Daily Solar Motion No. 185

.•. fj.lJf' lJf'

152

... [/j.lJf'] lJf'

T* B*

186

T* B*

187

T* B*

§ 4 B. Daily Lunar Motion No. 190

S.E.?

T* F*

191

117

T* .EF*

192

118

T* .EF*

193

119

T* .EF*

194

130

T* .EF*

194a

243

F* T* L'F*

194b

F* T* .EF*

195

T* .EF*

196

T* .EF*

Appendix. Solstices

.. 115-138 .. T 2 B2 'P2"

151

S.E. 124

No. 198 S.E. 116-131 199

Equinoxes and Solstices

.. 143-157 .. Summer Solstices

No. 100

126

§ 1. EPHEMERIDES A. Ephemerides from Uruk It is a common feature of these ephemerides to give in several columns only rounded-off values (e.g., column B) and to omit auxiliary columns (like A and H) in the first part, which ends with the moment of the syzygies (columns L and M). The second part, however, ending with the magnitude of visibility (column P), gives all the auxiliary columns needed. This is in marked contrast to System A where column M is immediately followed by column P. No. 100 AO 6475 U 126

+

Contents: New moons for S.E. 106to 108( = -205/204 to -203/202) Arrangement: 0 /R Provenance: Uruk [TU and U, writing 20,.1 in obv. 111,11 and 10,.3 in obv. VIII,14] Previously published: AO 6475: TU No. 22, Pis. 43 f. u 126: * Colophon: Invocation and colophon B (p. 16) Transcription: Pis. 54 to 56; Photo of obv.: Pl. 229 Critical Apparatus

Obv. I and II: no separation line between these columns in the text. II, 1 gun: this seems the best reading though the text otherwise uses lu for Aries. 11,2 18,20: sic, instead of 19,20. The error affects all subsequent lines. Cf. the commentary. 11,15 6,20: sic, instead of 6,10. C.f. the commentary. V,15 1,28: the copy in TU Pl. 43 gives 2,28, but the photo shows that the text has the correct value 1,28. The impression of a 2 is due to interference from the number above and a crack. VI: separated in the text into two columns. VI,1 31: sic, instead of 21; cf. VII,l. VI,12 2,[7]: the restoration 2,[2] suggested by Thureau-Dangin's copy is excluded because of the next column. VII: separated in the text into two columns. VII,1 55 kur: based on the erroneous value m VI,l. One should have 1,5 kur. VII,12 29 nim: 8,15 in VIII,12 and 3,17 in IV,12 would require 28 nim. VIII: separated in the text into two columns.

VIII,20 29: sic, instead of 30. Isolated error. IX,6 24,30: sic, instead of 25,30. IX,18 23,30: sic, instead of 25,30. This error influences the value in XII, 18. X,18 7,40: sic, instead of 4,40 (or 5,40 ?). This error influences the value in XII, 18. XI, 1 2: only lower half preserved. XI, 11 and 17 50: probably error for 40, assuming the values in IX, X and XII to be correct. XII and XIII: the text takes into one column our whole column XII and the first three signs of column XIII, leaving one narrow column for the last numbers of column XIII. XII,18 14: sic, instead of 19, because of the errors in IX,18 and X,18. XIII, 1 28: sic, instead of 29. XIII,7 9,29: the first 9 written over 2. XIII, 11 9,[7]: written over erasure; [7] agrees best with traces. XIII, 13 9, 9: written over erasure (2, 10 ?); one would expect 9,10. XIII, 14 28: sic, instead of 29. XIII,18 7,7: second 7 written over erasure (10). XIII,20 [8,15]: traces of 8 preserved. XIV,11 20: reading certain on photo, omitted in copy. XIV,12 12: or 13,[ ... ]. One expects 14,[ ... ]. XIV, 14 [11 ]: traces visible on photo; one expects 13. XIV, 19 27: sic, instead of 26. Rev. I and II: as in obv., columns not separated. V,4 7,40: the copy in TU gives 5,40 (partly damaged). Photo shows clear traces of 7 as expected by computation. VI: separated as in obv. XII and XIII: separation as in obv. There is, however, no space left for alternative solutions in column XII. XIII,2, 5, 7 28: sic, instead of 29. XIII,4 to 8: traces of all restored numbers visible. XIII,11 [6,4]6: restored from computation using the value in XIV,11; traces visible. XIV: all names of the months should be lowered by one line; cf. commentary. XIV,8 27: 20 very cramped. XIV,lO and 11 28: sic, instead of 27. XIV,16 16,40: reading 26,40 not excluded.

No. 100 Commentary This text is the oldest complete lunar ephemeris of the whole material published here, covering the years S.E. 106 to 108, though the auxiliary tables No. 155 and 170 began with S.E. 104. The present text is comparatively well preserved, permitting us to give the complete list of its 14 columns.

III F' 1 VIII

IX

v lf'" 1

XIII

XIV

It should be remarked, however, that the separation lines between columns as given on the tablet do not always agree with the actual contents_! Column B The solar positions are given by numbers which are rounded off to full tens of minutes; the corresponding column A is omitted. It is not difficult, however, to restore, at least approximatively, a column A and a corresponding column B which accounts for the numbers found in the text. This procedure reveals two essential errors. The first was committed at the very beginning of column B. The two positions 'Y' 20;40 and t) 18;20 are only 27;40° apart, which is less than the minimum of the monthly solar velocity. From the subsequent values it follows that about 28;40° would be the value to be expected here, thus leading to t) 19;20 instead of t) 18;20. Consequently, all subsequent values in B should be raised by 1o. This is confirmed by the corresponding position of the perigee which would be about f 18;30 according to the text instead of between f 19 and f 20. A second essential error was committed in obv. 17 by repeating the preceding line of about 28; 11,50 in column A. This error decreases the values in B by only 0;18° but it pushed the whole column A one line down. Schnabel ([1] p. 23) realized that exactly the same displacement of column A appears in obv. 41 of N o.170, an auxiliary table covering also the years of No. 100. Also the first error of No. 100 is reflected in No. 170, thus suggesting that the values in column B of No. 100 were actually taken from No. 170. 2 Though the part of No. 170 which contained the passage in question is destroyed, we can use the preserved later part to restore exactly columns A and B, whereas No. 100 shows at what points the errors were committed which distorted all following lines inN o. 170. Thus we obtain the table, given on Pl. 57, for the complete and for the abbreviated column B. The rounding off in line 15 is certainly a mistake.

127

It is interesting to notice that the erroneous introduction of an extra line in column A is still felt in the ephemerides No. 101 (S.E. 118 and 119) and No. 102 (S.E. 121), but no longer in No. 103 (S.E. 123) and No. 104 (S.E. 124). Column F' The years S.E. 106 ff. are also covered by the text No. 155 which gives the auxiliary function F 1' only. The values do not, however, agree with the present ones, as can be seen from the following comparison:

No. 100 obv. 1

2,21,40 XII 2,28,41,40 2,15,40 106 I 2,22,41,40 2, 9,40 II 2,16,41,40 2, 3,40 III 2,10,41,40

No. 155 obv.25

It looks as if the values in No. 100 were one line too high and 0,1,1,40 too small in a decreasing section. The opposite deviation occurs in No. 101. Multiplication by 6 changes F' into F. This function F, however, cannot be continued from No. 100 into No. 122. Even the continuation ofF' itself from No. 100 to No. 101 is impossible. The same holds for F' and No. 120. Column lf'" We have here a column lf'" with the following parameters

M = -m = 9,52,15

d = 3,52,30

c = 3,0,0 .

The same parameters for If\" occur also in No. 104 and in No. 122. A Diophant shows that continuation of lf'1 " from No. 100 into No. 104 (S.E. 124) or into No. 122 (S.E. 208) is not possible. In the case of No. 104, however, a continuation becomes possible if one raises the values in decreasing sections by 1,0,0 and correspondingly lowers increasing sections. Column L Only day and hour, in midnight epoch, are given (cf. the graph on Pl. 145). The hour of the conjunction is abbreviated to two sexagesimal places. The unabbreviated values (four places) are found in No. 170 obv. III though erroneously increased by 0,10,0,0. The following comparison reveals that this error did not affect No. 100: 1 Cf. the critical apparatus for columns I and II, VI to VIII, XII and XIII. 2 Cj., however, the discussion of column L.

No. 100

128

No. 100 3 obv. VI, 2

No. 170

2, 1

obv. III, 26

3,59 21 3,14

2,11,29,30 4, 9,31, 0 31,16, 0 3,24,46, 0

etc., showing that at least from the second line onwards the values in No. 100 are correct. This fact is of interest in so far as it weakens very much the assumption that No. 100 was actually based on the tablet No. 170, as suggested by the parallelism between the errors in column A and B in No. 100 and No. 170. Column L cannot be computed directly without having previously computed G, H, J and K. Consequently, the values given in No. 100 cannot have been found by direct computation but must have been copied (and abbreviated) from a more complete text. No. 170 contains L completely and it is hard to understand why the scribe should not have used No. 170 also for column L if he used it for B. It seems therefore that we must assume (at least!) the following relations: auxiliary text for A and B with error in A and B

~~

No. 170 error in A and B; error in L

auxiliary text for A, B, ... , and L; error in A and B; L correct No. 100

Columns N 1 to P 1 Column N 1 (ideogram kur) gives the time difference between the moment of conjunction (column L 1 or M 1) and the moment of sunset after which the new crescent is expected to be visible. This time difference is given to two places, i.e., large-hours and minutes. The preceding date, 29 or 30, is the date of the evening in question and is therefore indicative of the expected length of the month. Column 0 1 (ideogram bi) gives the elongation in degrees and tens of minutes. The majority of the values can be obtained by means of the formula

01

=

N 1 (F/ - 0;10) - 1;30 .

Column Q1 is only given for negative values, abbreviated to full tens of minutes. With two exceptions, these values can be explained as Q1 = q1 0 1 with coefficients q1 depending on the longitude of the mid-

point of the arc between the sun and the new crescent.

These q's form a linear zigzag function which is zero for illi 8 and for V) 8, and which reaches its minimum -0;20 for =::= 8. One exception to this rule is obv. 18, most likely caused by a copyist's error replacing Q1 = -4;40 by -7;40. The second exception is obv. 8, where Thureau-Dangin's copy requires a restoration of the damaged signs as 1,50 instead of the expected 3,50. Column R1 • Correction for latitude in degrees and tens of minutes. No agreement with the method of System A. Column P 1 . Name of the new month (= T 1 + 1) and date of its first day, followed by ripeness of the new crescent measured in degrees and minutes of right ascension, as obtained from

In two cases, obv. 11 and 18, alternative solutions for hollow months are given. In the first case we have for a full month:

01

for a hollow month: 0 1

+ pl = +P

= 1

= =

23;10 47°

+ 23;50

(10;10) 20;40°

+ 10;30

in the second case 4 for a full month: 5

01

for a hollow month: 0 1

+ P1 =

+P

=

1

=

=

23;30 37;30°

+

(12;10) 22;0° .

14;0

+

9;50

The elongations for the alternative cases are not given in the text but computed from the elongations for the full month by subtracting F - 1. No alternative solution is investigated for obv. 3 where

01

+ pl =

12;0

+ 12;20 =

This suggests a value of about c1 lower limit for certain visibility.

=

24;20°. 23° or 24° as the

Columns N 3 and P 3 Column N 3 (ideogram kur) gives the time difference between the morning of last visibility and conjunction, measured in large-hours and minutes. The names of 3 In general, the last two places are simply omitted. Deviations from this rule are found in obv. 15, 16, 19 and 20. • We use here the numbers of the text (though both 0 1 and P 1 should be 2° larger) because these numbers must have been the basis for the scribe. • The numbers given here are the numbers of the text, influenced by errors. Actually one should have 0 1 + P 1 = 25;30 + 19;0 = 44;30.

No. 101 the months agree, of course, with the names in T 1 • The dates, strangely enough, are not the dates of the mornings in question (which would be, in the majority of cases, the 27th) but agree with the dates in L 1, with the exception of obv. 1, 14 and rev. 2, 5 and 7 where N 3 has 28 instead of 29 in L 1 • Column P3 again refers to the same month as T 1 , 6 followed by the date of the morning of last visibility and the number of degrees and tens of minutes which indicate the duration of the visibility of the waning moon before sunrise (kur). These numbers can be obtained (with a few minor exceptions in the obverse) from P3

=

N 3{F3'

-

0;10)

which shows that the elongation is considered to be a sufficiently accurate estimate of the amount in question. The smallest value of P3 is 10;10° (rev. XIV,2) whereas obv. XIV,4 shows that 9;40° is already considered too low. Instead of this value we find in the text P3 = 21° which is the value resulting from adding 6H to N 3 •

129 VII, 1 30: sic, instead of 1. (The conjunction 28 5;2SH corresponds to shortly before midnight of the 29th of the civil calendar, which excludes a month of only 29 days; this is confirmed by the date 28 2;41H of the next conjunction.) IX,6 10,51: sic, instead of 10,52. XII,2 3[0: also 40 and 50 are possible. XIII,9 13,40: sic, instead of 13,10.

Rev. X,7 21,10: according to computation, one should expect 22,10. The result in XIII,7 is based, however, on the value 21,10. XI,5 6,30: only lower part preserved, but reading 3,30 is excluded by computation. XIII,7 16,20: the correct value in X,7 would give here 17,20. Commentary text contains the following preserved well This columns:

No. 101 VAT 7809 Contents: New moons for S.E. 118 and 119 ( = -193/ 192 and -192/191) Written: S.E. [117] X 7 Also [118] and [119] are not excluded (cf. p. 17).

Arrangement: 0-R Provenance: Uruk [colophon; writing 10,.9 m rev. IX,9] Previously published: Schnabel Ber. p. 242/243 and Schnabel [1] p. 28/29 (only transcription with many omissions) Colophon: J (p. 17) Transcription: Pis. 58 and 59; Photo: Pl. 230 Critical Apparatus Obv. II,3 [4],10: [4],20 not quite excluded. II,7 28: sic, instead of 28,10. Cf. commentary. II, 9 26: sic, instead of 26,20. Cf. commentary. II,10 25: sic, instead of 26. Cf. commentary. VI,9 sig: sic, not bar; cf. the corresponding entry in XII,9. 6 All month names in the reverse should be one line lower. The scribe apparently copied the names erroneously from column XII instead of using column XIII, probably misled by the senseless position of the vertical dividing line which combines column XII with the first half of column XIII.

Only part of the lower section of the reverse is occupied by the colophon. The remaining free space shows numerous traces of half erased numbers which indicate that the scribe actually carried out at least some computations when writing this tablet. Both Schnabel and Kugler discussed this ephemeris on different occasions, 1 unfortunately with so much vigor and temperament that their results require essential revisions. Column B As in No. 100, only abbreviated values of B are given and A is omitted. It is evident that errors occurred in copying these numbers because, e.g., f 25 (line 10) and V') 25;50 (line 11) would require more than 30° solar movement. It is possible, however, to restore column A and B with a very small margin of arbitrariness by using the values at the beginning and at the end of the well-preserved parts of column B. The accuracy of this restoration can be increased by extending it for two more years, thus including also S.E. 121, for which we have the ephemeris No. 102. It turns out that it is possible to restore a column A which explains satisfactorily the values in column B of No. 101 and No. 102 1 Schnabel Ber. p. 215 ff., p. 242 f., Schnabel [1] p. 28 ff. and in P. Mich. III p. 314 and p. 317; Kugler SSB II p. 584 ff.

130

No. 101

as well. If one computes the corresponding position of the apsidalline of the solar orbit, one finds it almost one sign too far advanced. This shows that the error, committed in No. 100, of duplicating one line of A still influences the values of A in No. 101 and No. 102. It is therefore only logical to continue the exact values of A and B from No. 100 which are known to us from No. 170. The result obtained is shown in the following list, 2 which starts with the line where the duplication in A occurs (cf. table below).

:m:

The daily velocity of the moon for the first five months of S.E. 118 can also be derived from the text No. 192 but the results show no agreement with either No. 101 or No. 155. Cf. the commentary to No. 192. Column J This column is based on the value

M

'07

JY:

ll1. "· 45, 22. 28, II, 4s; Zl

8

4, 26, 17, 32

No. 170 r-ev.33.

Ill J:t.

Z8,li,IO, 42.

6'1

17, 33, 5.1)4

l9,Z2,l9.J8

')'>

from

No. 170

Xi[

118 I

][. J.[

l9, ...:. 39, /8

28,443~.18

28,2.8,32, 18

I

X

XL

:xi[

6)

2., 38, 2, 54

6, 29,21,58

g

6'1

[r

[4,]30

No. /00 obv: lb.

No. 101 ob~. I.

2, 40

11

s, 34, 4, lb Itr

W~] ~)

6

a

2,50

)(

4, 2£!43,34

:n: [-4.]10

2.

01

Z,49,lZ,SZ 1, o, z.s~

Z!,Sl,/7, 58

11

l3, 43,14, S8

~

z~ 30

IS.

1!:.~ II, ss; 44 14, 47, sz. Zl. I{ S: 49, 0

=

14,40

rev. 8. 9.

1I 2& 10,40,

119

32,28,6

for its maximum. The same value appears also in the

No. 170 ob~. 4o.

continued

=

l9,SJ,S~38

Z.9, JS; s~ 38 2.9,17, st, 38

The agreement is satisfactory only until obv. 6. The later numbers are all smaller than expected by an amount varying between 0;30 and 0; 10. Some smaller error must have occurred in the computation of A or B (or both) which cannot be exactly determined because of the rounding off in B. The main fact, however, remains certain that No. 101 is a continuation of the erroneous second half of Nos. 100 and 170. Column F' Comparison with the contemporary auxiliary table for F 1 ' No. 155 shows disagreement

No. 155 No. 101 obv. 1 1,59,31,40 XII 2, 4,20 rev. 24 2, 5,31,40 118 I 2,10,20 2,11,31,40 II 2,16,20 2,17,31,40 III 2,22,20 In contrast to No. 100 the values of No. 101 are one line lower than the similar values in No. 155 and 0,1, 11,40 too small in a decreasing section. Therefore no continuation of F' is possible between No. 100 and No. 101. On the other hand the same deviation is also found in F 3 ' = F1 ' of No. 102 (S.E. 121) which therefore can be obtained by direct continuation from No. 101.

-

)(

1'

01

I

)(

14,30

'Y'

13,SO

s.

10.

ephemeris No. 102 and in the auxiliary tables Nos. 171 172, and 174. The corresponding column H is omitted but easily reconstructed (cf. Pl. 59). The values of H are in principle obtainable by continuing column H from No. 170, but 31 lines earlier than they should be. Column J cannot be continued from No. 170 because it uses the maximum 32,28,0. Both columns H 1 and }1 of No. 101 can be continued into the columns H 1 = H 3 and J1 = J3 of No. 102. Column D/ It is easy to see that the numbers of this column do not agree with the values to be expected according to the ordinary rule. Consequently Kugler proposed (in SSB II p. 586) a new scheme for the computation of the length of daylight, using 'Y' 3° as vernal point and with differences which also deviate from those in System B. Yet the results obtained by this new scheme again show many deviations from the numbers in the text, as Schnabel did not fail to point out (Schnabel [1] p. 32). Indeed, there can be only little doubt that 2 Schnabel [1] p. 32 gave a continuation of No. 170 which is distorted by the following mistakes in column B: the initial value in 117 XII is given as 6,30,57,58 instead of 6,29,24,58; 118 VII should be 27,21,36 instead of 27,21; 119 XI should be 14,48,49,22 instead of 14,48,49,28. All these errors affect the following lines.

No. 101 Kugler's scheme is erroneous. By far the best agreement can be reached, as has been observed by Dr. Olaf Schmidt, if one keeps the usual scheme of System B with the only exception that 'Y' 0° is made the vernal point. Then one obtains in 19 out of 26 cases exact agreement or a deviation ± 0; 1H, three times the deviation ± 0;2H, and four times the deviation ± 0;3H. This occurrence of 'Y' 0° as vernal point is hardly more than an error of the present text. Such an error can easily originate from using lists of coefficients similar to those found in the procedure texts No. 200 Sect. 15 and No. 201 Sects. 5 and 6 where only 'Y', t5, etc. are mentioned, though the actual limits always lie at the 1Oth degree. A similar error is also recognizable in No. 102. Column 'P" We have here a case of an abbreviated column 'f'", all numbers being rounded off to full tens in the second place. 3 The original parameters were certainly the same, which we find, e.g., also in No. 100 or No. 104:

M= -m d c

9,52,15 3,52,30 3,0,0 .

Column L The length of the first five months according to No. 101 can be compared with the length of the same months in No. 192. We find I II III IV

v

No. 101 29 30 29 29 30

character of the visibility condition shown in column P. Cf. the graph on Pl. 146. Columns N to P Column N 1 gives the date of the expected last evening of the month, 29 or 30, and the time from conjunction to the sunset in question. The values in column 0 1 are closely represented by

rounded off to full tens of minutes. Column Q1 is computed for negative corrections only. The value -0; 10 in obv. 4 cannot be explained by means of the scheme deducable from all the rest; one Continued

fr0111

Tl

No. 100

~ 118 I

r.

No. 192 29 29 30 29 30

The interchange between the lengths of II and III in the two texts is not surprising in view of the doubtful

-

11-

il .E E iii

+

1! E

.:&!'

u

FE2

119 I I

.![

ii y

:[

iii: if

Il E EI

&'

s, lb,30

_':, ),,3j,_ Q

i[

~

This is clearly shown by continuing 'f'" from No. 100 until S.E. 118/119. The result reveals a systematic deviation from the numbers in No. 101 which can be minimized by subtracting 0, 18,0 from the values in decreasing branches. This is shown in the table given on this page. There can be no doubt of the agreement of the general trend in spite of the fluctuations in the last place. Kugler's attempt 4 to explain these deviations as a result of taking the variability of the solar velocity into account is wrong because we know now 5 that this correction was effectuated by means of a function 'f'' obtained from a function ,::1 'f''.

S.E. 118

131

+ i-

Z, ZB, 30 .J,Z.t, 0 /3 30 I, 38,30 4,41, 0

z

_-+;,-- 2"U9 2,S1, 0

-

Modified continuation No. 101 No. 101 !rom No.100

-...- I, '·

4,sr, 3o

4j 0

-

~.I~>~ 30

----- -~3,_Q + 21,30 1-

4,z.z,

0

+ g 14 30 + 7,37,30 ..+ __J,_-iS 0-

- ....

-

7,30 I, o, 0 "t,Sl,JO

[- 4, :. II, So

II, 4Z, JJ, U> II, ll, S1, l o 1o,1~: z~; zo t, 't1_.rt, lD

10.

13,[~0

.ll:

8

6l !lj' .e..

No continuation is possible from F 1 ' = F 3 ' and F 2 ' of No. 102 to F 1' or F 2' of No. 104. Column J Column J1 is completely destroyed. If one, however, continues J1 (and, of course, H 1 ) from No. 101, one reaches for the proper dates the values preserved in }3 • It is therefore evident that J1 = Ja (and H 1 = Ha)· The values of J2 are identical with the values found in the auxiliary table No. 171 rev. 12 to 24 .

1:r.

II,Jo 10, ~-o

1, S'Q _.._ I

Column D' This column gives the length of the night and is therefore marked by the ideogram ge 6 "night". From the identity of B1 and Ba it follows that also D 1' = Da', a relation which helps to some extent to restore broken passages which are needed in order to check later columns. Unfortunately, D 1 ' cannot be computed directly from B1 because the preserved parts of Da' suffice to show deviations from the expected numbers between ± 0;3H. Slightly better agreement is obtainable if one uses 'Y' 0° for the vernal point, an error which also occurred in D 1 ' of No. 101. a It would be plausible to assume the same error for D 2 '; the only preserved number, however, does not agree with this assumption. We have in rev. 6 for B2 the longitude =:=: 20;20 to which D 2 ' = 2,52 (vernal point = 'Y' 8°) or D 2 ' = 2,48 (vernal point = 'Y' 0°) would be associated, whereas the text has 2,50. Also the values in column N 2 do not permit a clear decision, though computation with 'Y' 0° leads perhaps to slightly better results than the ordinary scheme. In my restoration I nevertheless use the ordinary rules because the final deviations are rather small in both cases and it is difficult to distinguish between real errors and rounding-oft's. Column F' Column F 1 ' is completely destroyed. It is plausible, however, to assume the identity of F 1 ' and Fa'. The correctness of this assumption is proved by the fact that exactly the same disagreement can be observed between F 3 ' of No. 102 and F 1 ' of the auxiliary table No. 155 as we observed between F 1 ' of No. 101 and F 1 ' of No. 155. The gap between No. 101 and No. 102 is only one year, and it is clear that we are entitled to restore F/ in No. 102 as the direct continuation of F 1 ' in No. 101, which in turn results in the identity of F 1 ' and Fa' in No. 102.

Column lJf" The poor state of preservation makes it very difficult to come to a clear understanding of this column. The few preserved numbers on the obverse seem to confirm our expectation that o/1 " = Pa"· Only traces of three numbers of o/2 " are preserved. In contrast to o/1 " and Pa" these numbers seem to be rounded off to tens of minutes, as in No. 101. The values agree with the corresponding values found in No. 150 as is shown by the following comparison:

No. 150

No. 102 rev. VI,6 7 8

-20 -4,10 -8,10

-22,50 -4,15,20 -8,7,50

obv. VIII, -2 -1 0

The subsequent lines of o/2 " are destroyed in No. 102 but we can restore at least the positions of the extrema and nodes from column R 2 (rev. XII,6 ff.). The result is again in agreement with No. 150. For o/1 " and Pa", however, serious difficulties arise. In spite of the fact that many of the numbers given in our transcription must be considered very doubtful, it is certain that discrepancies exist between the values of the text and a column lJf" computed according to the ordinary rules, based on the parameters M = -m

= 9,52,15

d

= 3,52,30

c

= 3,0,0

On the other hand, the deviations are small enough to leave little doubt that a column lJf" with the above parameters was intended. This is shown by the comparison in the table p. 135 where the values found in P/ and Pa" are set parallel with a correctly computed column lJf" starting with the same value. If one computes o/1 " for the year S.E. 121 by using the values found in No. 104 for S.E. 124, one again finds disagreement. 4 3 The agreement could be further improved by using the vernal point 'Y' 2" but it is hard to see any motivation for this choice. 4 The values given in the table below are based on the uncorrected values of '1' 1 " in No. 104. No agreement can be obtained with the corrected values either.

No. 102

'¥.". 't','"

f ll.l

lo. 102

:r

! 1i 1r

r

E" !if

!'!!!

iK E &l

!1[

][~

+

J, IO,JO

+ 2s +

I,S3,SO

... $, &,.>o _of:- j,_l.J,JP..

-

l.,l.f,ZO 3,13,/0

- [7, .....lo

- [....... J

- [...1. 30

;-__ J .• J.~Q +

2~,30

+ 4,1$, 30

'f'" expected

+

),10, 30

+ ~ ,1 + t,..;, + ~. S',,30

12

~.·

cont. :t'r0111 No.lo4

+

+

L2

3,31, 30

~.

J~.J~ 8,1K,30

7,~J30

.-:-- }}.1Q

+

.,.

'· 30

3,~1

~.z&

·~

~.II, Jo

+

2, Zl

I,Z,

G,S

zo

I, 11 I, IJ, I, /3

Z,4S

30

~()

4. 3l, 30

Column L Column L gives the name of the month in all three cases, the number 30 or 1 for hollow or full months respectively, and the date of the syzygy in midnight epoch. The ideogram ge 6 "night" must refer to this choice of epoch. The moment of the conjunction is given in large-hours, identical in L 1 and L 3 • The values for L 2 had to be restored by means of the auxiliary table No. 171. 5 The graph on Pl. 147 shows the relation to the dates. Column M The dates are given in evening epoch, i.e., according to the civil calendar. In the restoration of M 2 the hours before and after sunrise and sunset are determined from L 2 and D 2'; passing from D 2' to D 2 = ! D 2 ' causes an incertitude of ± 1 in the last unit. Column N The dates given in N1 and N 3 are the dates of the evening or morning respectively, for which first or last visibility is expected. Of greatest interest is column N 2 (rev. IX,6 ff.) because No. 102 is the only text in our material where such a column is sufficiently well preserved. The dates given (confirmed by the dates in P 2 ) are the dates of the sunrise following opposition. It is very surprising, however, that the time intervals only correspond to the time between opposition and following sunrise in case the opposition falls in the daytime, whereas for oppositions at night the time interval to the preceding sunrise is computed. Using the terminology of the procedure text No. 201 (cf. p. 229) we can say that NA is investigated in case of an opposition in daytime, SO for oppositions at night. That this is the rule followed by the text is clearly shown by the following table. 8 The reason for this procedure is not clear to me.

'·

,,

z

l,lZ

J, IZ.

I, 31

l,~o S'K s, 11.. l.~f l,ft7 3,So 1,~4 Z,43 I, 3 I.JK 4,31 l,l, :.....-.;_.,_.

_-_ __ JUQ +

~L

1,43 J,J3

~.3S,JO

-"=--~-L- _+_-- {3___

-- 2,"1( 30 -- z -- z. - 7, S7, - ..;, s

D2 Day~ht

-4,3-4

J....l~

+ E,lK

+

135

S,[~

z,4[{1

s, '

s, '

~: 30

l,•B

4,11

3,Z7

l,:n

.s:

Tct

Night

L-D L-D+6

S, JO ~J"r3

1r,4o

f4J3c!

S,/8

[.fj/7

-4, ll

3,zr

J,S{

I

B.~j1

s; zs

SJ.,

3,/0

J, II

A different rule seems to have been followed in No. 107 (cf. below p. 140). Column 0 Only four values of 0 1 are preserved but they seem to follow the rule known from Nos. 100 and 101 according to which

01

= N1 (F1 '

0;10) - 1;30 .

-

The values of 0 2 disregard the solar movement and give the elongation of the moon at sunrise before opposition (if opposition occurs at night) or after opposition (if opposition occurs during the daytime), following the distinction made in column N 2• The numerical values of the text are obtainable from 02 = N2F2' rounded off to full tens of minutes. This is shown by the following comparison. F' 2

N2

2, IZ. 2, /g

~s

l, l-'1

.s~ ~

~,-~.g

2,Z.1 l, Zl

Z,4-4

,!,: 30 l,H 4,U

Z,J]

4, Zl

l, II

t>~ 17

z, s I, Si I, S3

I, :>4 ~. 0

J,ll

3,S, S, :,·1 (!)

s, z.(

3, II

F;•N2

accurate rounded 11,11 ~. 17

''· 14

13, 4f 1. 13 11,

z

1,S6

II, JZ 7,/l 7, 48

II, It

.

10,1~

6ll.

ott II, to

Tct

zo

''· 10

/l, IO

ll, /0

13,40

13

6,

1. 10

II, 0 10, 0

li, 30

7,

10

?,So ll,lO

10,

Z()

~ l.O

~ lO

(!)

1,10

II /0

"· 30 7,10

7, ~o(!)

"· zo zo

10

ho

Rounded off by addin·~ 1 in the second place if the rest is ::;;: 45. The deviations in the last units (not exceeding ±2) are caused by a corresponding incertitude in the values of L 2 and D 2 • Slightly smaller deviations would be obtained by using a,.I;Z, {, Zl; ZJ,S~. "1 20;31, z, l..

1S

Z.I;S3, 4S

= )(

J: 3r}J

li;U IS

zo~

The deviations are caused, of course, by the fact that No. 142 is computed for variable solar velocity whereas No. 185 is based on the mean velocity only. The error in obv. XII,7 (3,24,15 instead of 3,27,15) was considered by Schnabel 3 as a proof for an empirical correction of the solar velocity with far-reaching consequences for the history of the discovery of the pre-

cession of the equinoxes. The new fragments joined with the only part available to Schnabel show additional mistakes which completely disprove Schnabel's conclusions.4

No. 186 A 3406 + U 147 + U 160 Contents: Daily motion of the sun for several months Arrangement: 0-R Provenance: C ruk [U and A 3400-number; writing 30,.8 in rev. 111,24] Transcription: Pls. 129 and 130

Commentary

Auxiliary table for the longitude of the sun from day to day, assuming a constant daily velocity of 0;59,9o/d. The date, S.E. 124, is furnished by the title, written on

The extrema entered here are the extrema of A1 • \Ve assume here that No. 104 gives the correct length of month X. 3 Schnabel [1] p. 39 f. 4 Cf. Neugebauer [18]. 1

2

Nos. 186 Commentary No. 186 is an auxiliary table for the daily position of the sun, assuming the constant velocity of 0;59,9°/d, The same assumptions are made in No. 185 which, moreover, also belongs to the Warka archive. It is therefore natural to investigate the possibility of connecting these two texts. Unfortunately, No. 185 contains several essential errors which might or might not influence subsequent texts. 1 One must, therefore, investigate the relation of No. 186 both to the beginning and to the end of No. 185. The result is in both cases negative. Assuming No. 186 to be the older text, one will not reach the beginning of No. 185 by continued computation; nor will the last numbers, which are preserved in No. 185, lead eventually to the beginning of No. 186. These facts deprive us of the possibility of establishing the date of No. 186.

No. 187 u 165 Contents: Daily motion of the sun Arrangement: ? (only one side preserved) Provenance: U ruk [U] Transcription: Pl. 126

2

Critical Apparatus [18,56,50]: lower part of 18 visible. 19,55,5[9]: last 50 only partially visible.

Commentary This fragment of a table for daily positions of the sun can be restored with different numbers in the last place. The restoration given here accounts not only for spacing and traces at the end of lines 3 and 4 but also admits of a continuation of our fragment into No. 186. The last column of the obverse of No. 186 would then be 30 months later than our fragment, 17 months being full, 13 hollow.

B. Lunar Motion The association with System B of the following texts concerning daily lunar motion is purely arbitrary and does not imply that the masters of System A were untamiliar with, or perhaps not the inventors of this procedure. The first text of this group (No. 190) gives a complete number period of F*, the daily velocity of the moon. Unfortunately neither date nor provenance of this text is known. The subsequent four tablets, however, all from Uruk, contain the day-by-day positions

TO

190

179

of the moon for the years S.E. 117, 118, 119 and 130 based exactly on the values of F* found in No. 190. The Babylon archive is represented by texts Nos. 194a ff. which cannot be connected with the U ruk texts though the general structure is the same. The last text (No. 196) stands by itself as is shown, e.g., by the use of rounded-off numbers only.

No. 190 MLC 1880 Contents: Daily velocity of the moon for 248 days Arrangement: 0- R Provenance: ? (perhaps acquired between 1910 and 1920) Transcription: Pl. 131 Critical Apparatus Obv. VI,7 11,(?)4,10: sic, instead of 11,46. Isolated error. VI1,14 11,55: sic, instead of 11,56. Isolated error. Rev. I,lS 14,22: sic, instead of 14,12. Isolated error. Commentary A dating of our text is unfortunately impossible because F* has a number period of 248 days only, covered exactly once in our tablet. None of the characteristic values ft, M or m, occurs. Checkmarks, small corner wedges in raised position, are an interesting feature. I divide these marks into two groups. The first group I interpret as line-counters because they occur in the lOth and 20th line of the first column (in our present counting, obv. I,8 and 18; rev. I,9 and 19). The second group of marks is irregularly distributed over the tablet. Restoring, however, a checkmark in the first line of the tablet, we obtain the following list of corresponding values of F* which clearly indicates that we have here marks for checking the computation by using the fact that constant differences occur between equidistant points:

[11, 7,10] 11,33,10 (!) 11,23,10 11,31,10 11,39,10 11,29,10 11,37,10 11,45,10 11,53,10 1

No error occurs in the preserved part of No. 186.

180

Nos. 190

We have here two groups of values, separated by the dotted line, each of which shows the constant difference 0,8,0 from line to line, except the secon,d "(!)" where the preceding value 11,15,10 should have been taken. Beginning with the dotted line, all subsequent values are taken from one line earlier, as would be the case if we had continued the preceding group. The reason for this change might be that 11,29,10 is the last line of the obverse which should be checked in order to make sure that all values of this part of the tablet are correct. The difference between the dates of the checked values in each group is 28 days, but 27 days at the dotted line. No.191 U96 Contents: Daily positions of the moon for S.E. 117 (= -194/193) Arrangement: ? (only one side preserved) Provenance: U ruk [U] Transcription: Pl. 132 Critical Apparatus

Rev. 1,9 2,9,40: only one wedge of the 2 is preserved. 1,10 [1]6,51,40: one could also read [1]5, but 16 is required by computing back from column III. Also the next line requires this reading because 15 in line 10 would give [30,]15,40 in line 11, and md-md would have been written one line too early. A similar consideration holds for line 7. Commentary

This is the first of three tablets describing the daily motion of the moon for the three consecutive years S.E. 117, 118 and 119 (cf. Nos. 192 and 193). These tables must have been kept up without correction for at least 11 more years because we find the continuation of the present table again in a similar table for S.E. 130 (No. 194). The date of the present fragment can be determined by means of the method described in the Introduction (p. 77). We start from two passages in our text and in No. 192 which belong to the same values of F•. No. 192 X

~ obv.

16 17 18

EF•

F•

6l10,24,20 23,36,20 1lR 6,30,20

13,12,0

EF• QI!

12,54,0

12,39,40

25,51,40 6l 8,45,40

No. 191 Rev. I 14 15 16

A Diophant shows that these two corresponding values

TO

192

of EF• are exactly one number period of F• apart. No. 192 being the later text. Thus No. 191 Rev. 1,15 precedes No. 192 obv. X,17 by k = 4,8d. The later date is known to be S.E. 118 V 28 whereas the earlier date shows the day number 15. Equivalent dates (e.g., both 15th) are therefore separated by 4,8 - 13 = 3,55d = 3 · 3Qd + 5 · 29d or 8 lunar months. Column I of our fragment is therefore dated as S.E. 117 IX. We know from the present text that S.E. 117 IX contains 30 days. The same holds for S.E. 117 XII according to the ephemeris No. 101 (ob,y. VI1,2) and for S.E. 118 III from No. 192 (obv. V/VI). This exhausts all possibilities for full months and shows that S.E. 117 . X and XI were hollow. The result for S.E. 117 XI is confirmed by No. 101 obv. VII,l.

No. 192 A3408 Contents: Daily positions of ~he moon for S.E. 118 (= - 193/ 192) Written: S.E. 118 III 13 Arrangement: 0-R Provenance: Uruk [colophon] Colophon: K (p. 11 and p. 17) Transcription: Pl. 133 Critical Apparatus

Obv. II,17 mul-mul: sic, and not mul as it is written elsewhere in the text. IV,11 2,.,40: reading of. uncertain; it looks like an erasure. X,14 1[3,6,20: beginning of 3 visible. Rev. The last column ended in mul; before that, traces of zib and lu are visible. Everything else is"destroyed. The above-mentioned traces are sufficient, however, to show that the text covered a whole year. Commentary

This text gives EF• from day to day, continuing No. 191. According to 'the colophon, No. 192 was written in the month III of S.E. 118. The same year is also represented· by the ephemeris No. 101, where the months II and III are given 30 and 29 days respectively in contrast to 29 and 30 days here. The lengths of the months I, IV and V are in both texts the same. The values ofF which result from F' given in No. 101 and No. 155 disagree with the values of F• for the dates of the conjunctions given in No. 101:

Nos. 192 To 194

No.IOI ~ S.E. lllf I

No.155

F1

~

12,53,10 13, ~. 10 /3, ..;s, 10

][

1I[

ll

13,

F*

IS, H -48 14.22

14, '"

l't,Sl,IO

conjunction receded by about 5h with respect to the first conjunction. Column III (F') of No. 101 leads to the following comparison with F• in our text

No.l9Z

1~.

14,50 IS, 6, 10

'"· Z.l, 10

r

z.

13,38

13,18 13, 12.

S.E. 119 IV 28 v 29

Cf. for similar discrepancies No. 193.

u Arrangement: 0-R1

Provenance: Uruk [U]

No. 193 obv. III

F• 5,43,50 20,41

17

t: 5,55

Contents: Daily positions of the moon for S.E. 130 (= -181/180)

Provenance: U ruk [colophon]

The date of this fragment can be established by the same method as for No. 191. We notice that the same values of F* correspond to different longitudes in the present text and in No. 192:

16

No. 194 AO 6492

Arrangement: 0 jR

Commentary

0('

13,47 14,23

Written: S.E. 13[0] VI 28

Transcription: Pl. 132

15

14,10 13,26

A similar discrepancy is observable in No. 192.

No. 193 122 + u 142

Contents: Daily positions of the moon for S.E. 119 (= -192/191)

No. 192 obv. II

181

II

14,57,10 15,14

QD

1,13,10

12

16,10,20

13

1,24,20

14

If we call Yo= 0(' 20,41,0 and Yk = II 16,10,20 then we find for k the value k = 8,16d. The value Yo in No. 192 corresponds to the date S.E. 118 I 27. For Yk we know only the day number 22. The distance between corresponding day numbers (e.g., 22 in both texts) is therefore k + 5 = 8,21d = 8 · 30d + 9 · 29d. Thus corresponding dates are separated by 8 full and 9 hollow months, or a total of 17 months. This gives for columns II/III of our text the date S.E. 119 V. It is interesting to see that the ephemeris No. 101 shows in column VII exactly 8 full and 9 hollow months between S.E. 118 I and S.E. 119 V. It also shows that S.E. 119 IV has 29 days only, as required by our text. The conjunction at the end of S.E. 119 IV was found in No. 101 to happen on the 28th (before sunset) in 61. 18;50, and for the next month on the 29th (after midnight) in l1Jl 17;30. The longitudes of the moon for these dates in our text are 61. 20; 1,10 and l1Jl 26;30,20. The discrepancy in the second case corresponds to the fact that the positions in our text must refer to the same time of the day, whereas the second

Previously published: TU No. 25, Pl. 46 Colophon: Zc (p. 12 and p. 20) Transcription: Pl. 134 Critical Apparatus

Obv. [0]: The restoration of the colophon on the reverse shows that little of the beginning of the lines is missing. Thus, the colophon on the reverse requires about the same space as the months VI, VII, and VIII on the obverse. If we assume that the reverse contained the months IX to XII (S.E. 130 is an ordinary year), we would have space for at least II to V on the obverse. A slight unevenness in spacing would permit one to restore the whole year 130 as originally contained in this text. III,26 absi]n: the traces seem to indicate a writing ki for absin, though the complete absin sign occurs in obv. V,23 and VII,22. V,22 19,35,50: szc, instead of 19,38,50; isolated error. V,26 19,15,20: szc, instead of 19,18,20; isolated error. VII,26 13,27,40: sic, instead of 13,28,40; isolated error. Commentary

As in the preceding group of texts (Nos. 191, 192, 193), we have here the function .EF•. Again F* can be computed and is identical with F• given in No. 190. 1 The reverse is completely destroyed with the exception of one number (9) which is sufficient to determine the arrangement.

182

Nos. 194

It was shown in the Introduction p. 77 that the distance between No. 192 obv. IV,15 and No. 194 obv. I,1 amounts to 18 number periods of F* or 1,14,24d. In order to establish the date of No. 194 in ordinary calendaric terms, we note that the distance between the two first days of a month, No. 192 obv. IV,1 and No. 194 obv. I,1 is consequently 1,14,49d. This total must be close to an integer multiple of the mean value for the length of the synodic month, known from column G to be 29;31,50,8,20d. Indeed, using this value, we find that 1,44,49d = 2,32 months+ 0;21d. Thus we know that the distance between the two lines in question amounts to 2,32 months. This total must be composed by a full and f3 hollow months, a and f3 being integers of not greatly different value. Thus we must have a· 30 +

f3 · 29 =

1,14,49

and

a+ f3 =

2,32

The common solution is

f3=

a= 1,21

1,11.

This shows that all calendaric conditions are satisfied and leads directly to the establishment of the month names given in our restorations. No. 194a BM 45818 + BM 45838 + BM 46192 (= SH 81-7-6,239 + SH 81-7-6,262 + SH 81-7-6,653) Contents: Daily positions of the moon for S.E. 243 (= -68/67) Arrangement: 0 /R; cf. No. 194b Provenance: Babylon [SH] Colophon: Zkc (p. 23) and fragment of procedure text: No. 221 (p. 277) Transcription: Pl. 135; Photo of rev.: Pl. 239 Commentary

The first three columns are F*, T*, and .EF* respectively. Thereafter follows a new triple of similar columns, and we shall see presently how this scheme continued over the rest of the tablet. The names of the months are followed by ge 6 "night". In substance this text is of the same type as Nos. 191 to 195, with the sole exception that here not only .EF* but also F* itself is tabulated. Also the general parameters are the same but no continuation is possible ·~ n the previous group and the present text. This seen from the fact the three-place numbers of F*,

AND

194a

ending in 10, form the increasing branches in Nos. 191 to 194 whereas they belong to the decreasing branches in our present text. Numerically, however, all values of F* in No. 194a occur in No. 190, though, of course, in inverse arrangement (No. 190 rev. VI,2 = No. 194a 1,0 to No. 190 rev. IV,14 =No. 194a 1,27). It is not difficult to reconstruct the size of the tablet and the arrangement of its columns. A Diophant shows that there are 1,23 lines between obv. 1,5 and obv. IV,13. The date of the former is 1,11, the date of the latter IV,S or 1,24 days after the first date in case all months were 30 days long. Because 1,23 = 1,24- 1 we see that one month must have been hollow, and, because the text has a date I,30 and a date III,30, we can conclude that II was hollow. If the first column began with I,1, then the second column began with III,17 and 75 lines are needed for column I. Thus our tablet had about 75 lines in each column. Using once more a Diophant one finds that there are either 28 or 4,36 lines between obv. IV,13 and rev. 1,9. The first possibility is excluded because of the length of the columns. Thus 4,36 days elapsed between the date IV,S of obv. IV,13 and the date N,16 of rev. 1,9. If all the months were 30 days long we would haven· 30 + 11 days between these two dates. Because 4,36 = 4,41 - 5 = 9 · 30 + 11 - 5 we see that 9 months fall between IV and N, five of which were hollow. Thus the last month is a month XII 2 (or XII if the year had a month Vl 2), and, of the 13 months of this year, 6 were hollow, 7 full. Thus the obverse must have contained in three parallel sections the first 8 months, having two sections for the months from IX to XII 2 for the reverse, the last column of which contained the colophon, perhaps preceded by a few lines of numbers from the last month. The date of this tablet can be established as follows. As Strassmaier noted in his copy, the same combination of a "king Arshaka" and a queen "Pi-ri-us-ta-na-a" also occurs in the tablet SH 81-7-6,21 which is dated in the year 180 Arsacid Era, corresponding to the year 244 of the Seleucid Era or -67/66 (cf. Kugler SSB II p. 447 and Debevoise, Parthia p. 72 note 7). The king ruling at this time was Phraates III ca. -70 to -56. These are the historical limits for the date of No. 194a. The traces of the year number in the colophon suggest 200 + 13 or+ 43, which would correspond to -98/97 or -68/67 respectively. The first year is an ordinary year, the second contains a months XII 2 • Thus on all grounds 213 is excluded and only 243 remains. This is confirmed by the number 4,[3] on the edge. Also modern computation shows no contradiction in the longitudes of the moon for this year as given in the text.

183

Nos. 194b, 195, 196

No. 194b

BM 34623 ( = Sp.II,103) Contents: Daily positions of the moon for (at least) 5 months Arrangement: ? (only one side preserved) Provenance: Babylon [Sp.] Transcription: Pl. 135a; Copy: Pinches No. 83 Critical Apparatus The pos1t10n of the upper edge could be anywhere between three lines lower or 18 lines higher, and the same holds for the lower edge, maintaining a total of about 43 lines per column. Rev. IV c, 12 rf)n: only a double vertical final wedge is visible, thus either rfn or mas or a. From the continuation of column Illc rin would be best. Perhaps colophon. 1Vc,15 .. ] .: the traces, one little corner wedge, do not support the expected reading p]a (nor zib or zib-me nor rfn). Perhaps colophon. Commentary Three columns (a-b-c) always belong together, giving F*, T*, .EF* respectively. These triple columns proceed from right to left, as is customary for the reverse of literary texts but very rare for ephemerides. We find the same feature in No. 194a, a text of exactly the same structure. Also the mention of ge 6 "night" after the names of the months is common to both texts. It is therefore very plausible to assume that both texts are also close in time. This seems easy to check because the same values ofF* occur in both texts, e.g., 15,0,0(t) in Illa,21, and in No. 194a obv. 1,6. Thus the interval can be any multiple of 4,8 days. The longitudes, however, could only be obtained if the interval were about 4000 years. This points to a scribal error committed in the computation of the longitudes, sorncwhere between No. 194a and No. 194b. If column IV continues the scheme of the preceding columns, then the year in question again contained a month XII 2 as is the case inN o. 194a. Otherwise we have to assume that the traces in IV c belong to a colophon. A direct join between Nos. 194a and 194b is excluded by the fact that the first contains about 75 lines per column against only 42 in the latter.

Provenance: U ruk [U] Transcription: Pl. 135 Critical Apparatus I, 1 ]33,40: or ]36,40. Commentary Fragment of table for L'F* similar to Nos. 191 to 194. From the fact that the moon is in at the end of the month one can conclude that we are dealing here with the last months of a year. Comparison with the correspondingvalues in any of the other texts of this type shows that our fragment belongs to none of them. The preserved numbers do not suffice for an independent restoration.

=

No. 196 Rm. 777 Contents: Daily positions of the moon for several months Arrangement: 0 /R Provenance: Babylon [BM] Transcription: Pl. 136 Critical Apparatus Obv. 13 zib: sic, according to Strassmaier's copy. The recently cleaned text shows 1 written over zib. 14 1[2]: Strassmaier' s copy gives traces similar to 17, which is, however, excluded. Rev. 8-10: The cleaned text shows all restored numbers plus 13 gu in line 11. Commentary This text gives the daily positions of the Moon (.EF*) for several months, rounded-off to integer degrees. That the function represents L'F* becomes evident by comparison with texts like Nos. 191 to 194:

No.196 oov.

2.

24

'

1

u 131 Contents: Daily positions of the moon for unknown year Arrangement: ? (only one side preserved)

l1.

li g,

r

ll

6 lO II,

4

17

No. 195

nr,zc.

Z4 S.

H.

Y,

30.

rev.

z4,J7, J,o 1, ll, 30

.z4,

z'

1.~

l4, z~

f, ~-,

I.

Zl, /0

....

7. ' l.0,44 4, 4

l.

s.

... 11

,,

No. 192. ohv.

][, ~~~

...

lt.

4 E., /.

Ill

S, -41, :.-o

lO, -41 $, ff

z.

.zo, ~.-, :.: l1

11, 41

17, ' Zf,SO

30

l[Z]

No.196

No.IIJJ, ohv.

7.

ll,

ll

The positions in the zodiac are, of course, different because of the rotation of the apsidalline of the moon. This might be utilized for an approximate dating of this fragment.

No. 198

184

APPENDIX.

SOLSTICES AND EQUINOXES

No.198 A 3456 Contents: Planetary positions, observed and computed, mostly for Mercury. Solstices and equinoxes. From S.E. 116 to 131 (= -195/194 to -180/179) Arrangement: 0 /R Provenance: U ruk [A 3400]

Commentary Though this text does not belong to the class of texts published here, it contains interesting information about the schematic computation of solstices and equinoxes. The following dates are preserved: 1 Obv. 1,43 I, 46 II,8 II,20 11,38 11,52 111,10 III,24 III,32 III,51

(1 me 18)

Rev. 1,48 11,22 III, 11

(1 me 26 (1 me 27 (1 me 29

[a]b 8 d§amds gub-za (se-dirig) 11 lal-dt [k]in 16lal-tu se 22 .... u4-bi lal-tu bar 3 lal-tu ab 11 dsamds gub-za sig) 1[6] dfamdsgub (gan) 22 samds gub-za se 25 lal-dt dirig 6lal-tu

(1 me 19) 1 me 21 (1 me 22

(1

me 23)

ab) 6 d§amds gub se) 20 [lal-tu] dirig) 12 lal-tu

Vernal Equinox [1,57 I, Sl*

I,S1

Zl"

•

z, 2. 3* [2,4 l,

£zJ

!!!

30]

i{i1'

u

EFz. I

II

3

[K!"z. 14] X[

Mz. &]I

M

zs (,

17]

l8]

[z, 6" lY!z. il 2,7 X! zo 1] 2, 9"' [I

X!:z.

12.

winter solstice, and between winter solstice and vernal equinox. Thus the simplest possible distribution of intervals is assumed with no consideration of the actual inequality of the seasons. It is interesting to note that exactly the same scheme fits also the following instances from similar texts published by Kugler, SSB I p. 90 ff. and SSB II p. 481 ff. 2,0

[I,S1

[z, o [z,t* z.z.

[Z, 3* [2,..; [l,S

[2,6*

[2, 7

Ii

jjf

:m:

R

lf 11f Iff Iif

][

1![

Z] 13] Z-4]

.s]

rfn

Z7] 8] I~]

lol

1r

ll]

[Z; 10

][

14]

R

I 1

Rm. IV, 435 obv. 1

2,9*

IV 3

Rm. IV, 435 obv. 7

Fall Equinox [I,SB*

I,S1

[z, o [z. r" [z, z [2, 3'* [2,4 [z, s [z,6*

II]

[2, 8

[2,1*

2,9*

Later texts show that this scheme cannot be continued too far. This is not surprising in view of the fact that the difference of 11 r is too small, as is shown, e.g., by comparison with the more accurate scheme of No. 199. Nevertheless, an investigation of all dates of solstices and equinoxes published by Kugler in SSB I and II shows that the same pattern holds for larger groups of texts, e.g., from S.E. 2,58 to 3,8, then again for S.E. 3,54/3,56 and 5,1/5,3. Consequently all attempts to use such dates as if they were based on observations are doomed to failure. On the other hand, it is now very easy to determine the dates of texts which contain references to solstices and equinoxes. It is easy to show, e.g., that the dates for the solstices, given in

Sumner Solstice [I,SII*

SH 81-6-25, 214 obv.

III 24

Fii if

:n

f![

if :E!: 1'!

XI

E

5] 16 Z7] 8] 11] Jo] II]

u] 3]

Winter Solstice I,SII* X

[1, ~-, [z, 0 2, 1* 2, 2.

[2,3* [2, 4 [z,s

jK

1i. X

1K X

IE I!

z,6* .!

I 11] JO] II lZ.

J] Ji]

zs] 6

3]

The structure of this scheme is very simple. The time interval from line to line in each column is 12m+ 11r. This corresponds to a length of 6,5;11,42d for the solar year. From vernal equinox to summer solstice 3m + 2r are assumed, whereas 3m + 3T elapse between summer solstice and fall equinox, between fall equinox and

No. 199, are still in agreement with the dates of solstices found in Almanacs and Diaries from the latest period {first century A.D.) as well as with very early dates, preceding the beginning of the Seleucid era. I follow a transcription of the whole text made by Dr. A. J, Sachs.

1

No. 199

No.199 u 107 u 124

+

Contents: Summer solstices for (at least) 30 years, including S.E. 143 to 157 (= -168/167 to -154/ 153) Arrangement: 0-R Provenance: Uruk [U; writing 10,.9 in obv. 1,11] Previously published: Neugebauer [13] Transcription: Pl. 136 Critical Apparatus Obv. 1,8 to 10 [2]: traces visible. 1,12 21,13]: followed by erasure of 10.

185

II,1 to 13 ... : either 2 or 3. Rev. 1,2 and 3 ... : traces of either sig or su. Commentary

This text gives the summer solstices for many years. Assuming our arrangement as correct, the complete text must have covered about S.E. 138 to 229. The dates are, of course, expressed in tithis and the difference between solar and lunar year is assumed to be 11;3,10r = 10;52,47,23, .. d. The corresponding length of the tropical year is 6,5;14,49, ... d. For further details cf. Neugebauer [13]. See also the procedure text No. 200 Section 11 (p. 199) and No. 813 Sections 14 and 15 (p.411f.).

186

CHAPTER III. PROCEDURE TEXTS Texts Nos. 200 to 211:

from Babylon

Nos. 220 and 221: Colophons

§ 1. PROCEDURE TEXTS FROM BABYLON Introduction The purpose of the "Procedure Texts" is to give rules for computing the ephemerides. The procedure texts normally do not explain why the operations are to be carried out; in fact, they rarely indicate the significance of their rules. It is evident that texts of this type were of use only to the computer, who was already fully familiar with the theoretical background of the ephemerides. The present chapter contains mainly procedure texts for lunar ephemerides of System A. The most important text in this group is No. 200, which is a collection of rules for a variety of columns of System A, frequently used by Kugler. No. 201 is a very systematically arranged procedure text and is exclusively devoted to the computation of the visibility of the moon at opposition. The remaining texts supplement and amplify the material contained in Nos. 200 and 201. No parallel is found, however, for column A (which occurs in Nos. 207d and e), either in procedure texts or in ephemerides. System B is represented by Nos. 210 and 211 and perhaps by No. 202 Section 4, as well as by occasional passages in texts which otherwise belong to System A. No. 200 BM 32651 ( = st 76-11-17, 2418)

Contents: Procedure text for the moon, System A Arrangement: 0 jR Provenance: Babylon [BM] Previously published: Sections 1 to 6 and 12 to 14: Kugler BMR Pis. 5 and 6; Sections 7 to 11 and 15, 16: * Photo: Pl. 234; Copy: Pis. 223 and 224

Introduction This text is by far the most complete procedure text which we possess. It contains the rules for the computation of several columns of the lunar ephemerides of System A, but also deals with additional topics not contained in the ephemerides. The text is subdivided into separate sections without much of a clear principle of arrangement, as can be seen from the list which follows: Section 1: destroyed ........................ p.187 2: Column C1 • • • . • • • • • • • • • • . • • • • . . • . • 187 3: Column B 2 •••••••••••••••••••••••• 188 4: Column lf'' ........................ 188

Section 5: Columns F and(/) ................. p.189 6: Column E ......................... 190 7: solar velocity ....................... 193 8: daily solar motion ................... 194

N 0. 200, SECTIONS 1 AND 2

Section 9: monthly solar motion; eclipse magnitudes ................... . p. 194 10: extremal velocities .................. 198 11: the seasons of the year .............. 199 12: Column J .................. ........ 200

187

Section 13: Columns K and M ................ p. 201 14: Columns ~ and G .................. 202 15: Column P1 . . . . • . . . . . . • . . . . . . . . • • • • • 204 16: Column P3 •••••.••••.•...••.• •••••• 208

The upper part of the obverse of the text is broken, desuoying at least parts of sections 1, 7, 14 and 16. At the end of Section 14, at least 10 lines must be restored and additional material is missing from the beginning of Section 15. This leads to the conclusion that our fragment is only slightly more than one half of the original tablet. Because of the mutual independence of the sections, we treat them here as separate units but follow the arrangement of the text.

Section 1 From the restoration of Section 14 in rev. I, it follows that at least 10 preceding lines are missing. Only a small fraction of the last lines of this section is preserved. All that one can read is ki 1,6 1 nim(?) 1,4[0(?) ....

Section 2 Transcription

Obv. I a 2 [10 gun 3 Sd altO g]un dirig a-ni 40 DU ki 3 tab 3 [10 mul 3,20] sa altO mul dirig a-ni 24 DU ki 3,20 tab 4[t0 mas 3,3]2 sa altO mas dirig a-ni 8 DU ki 3,32 tab 5 [10 kusu] 3,36 sa altO kusu dirig a-ni 8 DU ta 3,36 lal GtO a 3,32 sa al10 a dirig a-ni 24 DU ta 3,32 lal 7t0 absin 0 3,20 sa al10 absin 0 dirig a-ni 40 DU ta 3,20 lal s10 rin 3 sa altO rin dirig a-ni 40 DU ta 3 lal 9 10 gfr-tab 2,40 sa altO gfr-tab dirig a-d. 24 [DU] ta 2,40 lal 10 t0 pa 2,28 sa altO pa dirig a-ni 8 DU ta 2,28 lal 11 t 0 mas 2,24 sa al 10 mas dirig a-ra 8 D U ki 2,24 tab 12 10 gu 2,27 sa al10 gu dirig a-ra 24 DU ki 2,28 tab 13 10 zib 2,40 sa altO zib dirig a-ni 40 DU ki 2,40 tab Critical Apparatus

t2

2,27: should be 2,28 but 7 seems the better reading of the slightly damaged sign. Translation and Commentary

This section contains the rules for the computation of the length of daylight for arbitrary positions of the sun in the zodiac. Each of the twelve lines of this group follow the scheme (For the) 10(th degree of the zodiacal sign) S, (the length of daylight is) a; everything beyond (the) lO(th degree of) S, multiply by b; tfo rom

lj

a {adbd

su tract

.

The lOth degree is chosen because of the position of the vernal equinox at 'Y' 10°. The numbers a give the length of daylight for the lOth degree of each sign; the numbers b are the interpolation coefficients for other longitudes. The result corresponds to the table for C1 on p. 47. The badly preserved section 11 (below p. t99) seems to be related to the present section. 1 Kugler BMR PI. 5 gives 9, written in three rows, which is not the form for 9 used elsewhere in the text. A reading 7, however, is not excluded.

No. 200,

188

SECTIONS

3

AND

4

Section 3 Transcription

Obv. I b 2lu-ma[s .......... ta 13 zib] 3 en 27 absin 0 • • [ • • • ] • • 4 28,7,30 sd al13 z[ib] 5 dirig a-ni 1,4 DU ki 6 13 zib tab ta 27 absin 7 en 13 zib 30 tab sd al 8 27 absin 0 dirig a-r:i 56,15 9 DU ki 27 absin tab

Critical Apparatus

2 ma[s: Kugler's reading ina (BMR p. 67) is certainly wrong. 3 .. [ ... ] .. :probably ab and ab. 5 1,4: the 4 is erroneously omitted in Kugler BMR ' p. 67.

the transcription on p. 67 gives the correct value 27. According to the photograph, 27 seems to be the better reading. absin: sic and not absin 0 as the copy in Kugler BMR Pl. 5 shows.

6 27: the copy in Kugler BMR Pl. 5 gives 28 whereas

9 absin: sic, not absin 0 •

Translation and Commentary

This section deals with the positions of the full moons in the zodiac (column B 2). The first line must have contained the title. If the second sign is actually BAR one might read lu-ma[s-mes and translate "zodiac" or "longitudes". What follows is easy to interpret. We have two parallel rules, the first of which refers to the fast arc of the ecliptic, the second to the slow arc. (a) From 13 )( to 27 ~month by month (you shall add) 28,7,30; anything beyond 13 )( multiply by 1,4 (and) add it to 13 )(. (b) From 27 ~ to 13 )( you shall add 30; anything beyond 27 ~multiply by 56,15 (and) add it to 27

~-

The fast arc begins at )( 13 and ends at ~ 27. Suppose that a full moon falls shortly before this arc, e.g., )( 7 (cf. ephemeris No. 1 obv. III,S). According to rule (a), the next position will be found as follows: we add to )( 7 the arc 28;7,30 which brings us to 'Y' 5;7,30. This point lies already inside the fast arc, namely 22;7,30° beyond )( 13. If we multiply this amount by 1;4 we obtain 23;36. This is the arc we must add to )( 13 in order to obtain the next position 'Y' 6;36 (No. 1 obv. III,6). Using the terminology of the Introduction (p. 46) we computed the arc s1 -s in Fig. 18 (p. 46) by first con-

t

tinuing the slow movement beyond = )( 13 and then multiplying the part inside the fast arc by 1;4 = ~~ = £' . This is exactly formula (Sb) on p. 46. v The same idea is used in the second rule in crossing from the fast arc into the slow arc. Here we have to reduce the arc which falls beyond

t = ~ 27 in the ratio 0;56,15 = ~! = ~

.

Example: last position

~ 6;36 (No. 1 obv.

III,ll); adding 30 we obtain=::::: 6;36, i.e., a point 9;36° inside the slow arc. We therefore multiply 9;36° by 0;56,15 and find 9° as the arc beyond = ~ 2T. The next position will therefore be=::::: 6 (No. 1 obv. III,12).

t

Section 4 Obv. I b 10nim U sig sa bab an-kulO lla-ni 20 DU ina nim 12

ki 17,24- tab ina sig

13zi

Transcription

No. 200,

SECTIONS

4

AND

5

189

Critical Apparatus 10

bab: Kugler BMR Pl. 5 and p. 213 gives KU; the photograph shows a sign which can only be bab or si.

Translation and Commentary The title of this section contains the words "bab of an eclipse" which probably means "eclipse magnitude'' (cf. below p. 197). According to (3) p. 57 the magnitude lJf of an eclipse is given by lJf=17,24,0+signK·

~ (lEI

;S;;K).

We saw, furthermore, that a continuous function IJf' was defined which is given by (5b) p. 56

.

lJf'=stgnK·17,24,0+

2E =f 6

K

(lEI

:2:K)

for values of E outside the nodal zone. Our text obviously deals with one of these relations. We therefore translate tentatively Positive and negative latitude of the eclipse magnitude. You shall multiply by 0;20. In (the case of) positive latitude, you shall add to 17,24,0; in (the case of) negative latitude, you shall subtract (from 17,24,0). The factor 0;20 =

~ points towards the second formula, which can also be written as tp' =sign K. 3

17,24,0 =f

~+ ~ 6

3

where E is either positive (:;;;;; K) or negative ( :s;; - K). The only disturbing element is the omission of the value

of~

6

Section 5

Transcription Obv. I

ipus(du-uS) SQ zi sin ab ana ab 42 tabU 1a1/ib-bu-u sa 15,56,54,22,30 takassad(kur-ad) sd al15,56,54,22,30 dirig ta 15,56,54,22,30 lallib-bu-u sa 11,4,4,41,15 16 takassad(ku[r-a1d) sa a/11,4,4,41,15 matu(lal-u) ki 11,4,4,41,15 tab . and tar-# 2,17,4,48,53,20 17 15,56,54,22,30 zi tasakkan(gar-an) and tar-# 1,57,47,57,46,40 11,4,4,41,15 gar 18 ana tar-~a 2,13,20 15 zi ana tar-~a 1,58,31,6,40 11,15 zi SQ al2,13,20 19 [ga]l tur a-ni 15,11,15 DU and zi tur u gal tabu lal

14

15

Critical Apparatus 15

15,56, ... (twice): Kugler BMR Pl. 5 shows 14,56, ... but the correct reading 15, ... is certain in the second case and possible in the first.

16

takassad(ku[r-ald): Kugler BMR p. 160 and Pl. 5 interpreted the traces as sig gir-tab.

17

gar: omitted in Kugler's transcription.

Translation and Commentary In this section the relations between C/J and F are established. The parameters of F are the unabbreviated parameters. The title (line 14) is Procedure for the velocity of the moon.

No. 200,

190

SECTIONS

5

AND

6

Then follows (lines 14 to 16) the statement that the difference of this linear zigzag function has the value d 42,0,0,0 and the extrema M = 15,56,54,22,30 and m = 11,4,4,41,15:

=

Month by month, 42 add and subtract, until you reach 15,56,54,22,30; anything beyond 15,56,54,22,30 subtract from 15,56,54,22,30 until you reach 11,4,4,41,15; anything below 11,4,4,41,15 add to 11,4,4,41,15. The remainder of the section determines the relative position of F and as follows (lines 16 and 17): Opposite 2, 17,4,48,53,20 you shall put 15,56,54,22,30 as velocity. Opposite 1,57,47,57,46,40 you shall put 11,4,4,41,15. This means MF = 15,56,54,22,30 corresponds toM= 2,17,4,48,53,20

and mF = 11,4,4,41,15 corresponds to m = 1,58,31,6,40 (-}) corresponds to the endpoint of the linearly increasing section of G. The last sentence (lines 18 and 19) is: Everything greater or smaller than 2,13,20 multiply by 15,11, 15, add to the velocity or subtract, smaller or greater. This rule is explained by the fact that the difference dF dF where dq,

=

=

=

42,0,0,0 ofF is given by

0;15,11,15 dq,

2,45,55,33,20 is the difference of.

Section 6 Transcription

Obv. I 2°episu(di1-su) Sd nim u sig Sd sin ab and ab 12 dagal ma-lak dsin 2,24 qabalti(murub 4 -ti) qaq-qar ki-~a-ri 21 ta 27 zib en 13 absin ab and ab 1,58,45,42 tab u Iailib-bu-u Sd qabaltu(murub 4 -tu) takassad(kur-dd) 22 [s]um-ma 3,52,11,39 nim 1,58,45,42 ina lib DUL-DU-ma 1,53,25,57 tag 4 23in(?)-nu-u ki-i al-la 2,24 lal 30,34,3 Ia! 30,34,3 ina 1,52,25,57 2 4lal-ma 1,22,21,54 nim dr 3,52,11,39 nim tasakkan(gar-an) 30,34,3 a-na 1,58,45,42 25tab-ma 2,29,19,45 sig tasakkan(gar-an) lib-bu-u Sd 13 absino takassad(kur-dd) and 1 us ki sa a/13 26 absin 0 dirig GAM 15 DU ki 1,58,45,42 tab and nim lu and sig gar sum-ma lu-mas 13 rin

[t]a 13 absin 0 en 13 rin 30 1 danna 30 GAM 15 DU 7,30 7,30 ki {ki} 1,58,45,42 [tab-m]a 2,6,15,42 [e]n(?) 7,12 nim nim [e]n(?) 7,12 sig sap-lu sa a/7,12 dirig 29 [ina 7,1]2 DUL-DU-ma [and nim lu and sig] gar ta zib en absin ina murub 4 Iu-masgab-bi a-sar 30 [qabalti(murub 4 -t]i) 2 [ ........... ] ... 30 20 ta zib en absin 0 tasakkan(gar-an) an-ta murubt u ki-ta 31 [murub 4 3,52,11,39 nim u 2,29,19,45 sig] ki a-ba-mestab-ma 6,21,31,24 1,58,45,42 32 (ki 1,58,45,42 tab-ma 3,57,3]1,24 2,24 itti(ki)-su tab-ma 6,21,31,24 takassad(kur-dd) ta 13 absin 0 33 [en 27 zib ................................ sa ki-#]r murub 4 KUR gim igi Ia! 27 28

No. 200,

SECTION

191

6

Critical Apparatus sign like TAR, for the second en a damaged KI.

absin: probably written absin 0 •

21

22 [S]um-ma: Kugler BMR Pl. 5 gives DU-ma but this reading is not confirmed by the photograph. 23 23

al-la: Kugler BMR Pl. 5 incorrectly DUL-DUma.

30

ina 1,52, ... : (a) the wedge for "one" is crossed by the horizontal wedge for ina, which gave rise to Kugler's transcription (BMR p. 142) lal 1. The copy on Pl. 5 shows AN. (b) the value 1,52, ... , given by the text, is an error for 1,53, . . . .

24

[qabalti(murub 4-t]i): Kugler restored ku]su, which makes no sense. 2: or any similar sign. ... 30 20: Kugler BMR Pl. 5 gives mul 30 20. Instead of mul I would read ina followed by a square sign (mas?).

31

1,22,21,54: sic, instead of ... ,51,54.

26

1,58,45,42: Kugler BMR Pl.S and p. 142 erroneously 1,55, . . . .

28

[tab-m]a: Kugler BMR Pl. 5 and p. 142 erroneously ultu(ta). en(?) 7,12 nim nim en(?) 7,12 sig: the copy in Kugler BMR Pl. 5 gives for the first en traces of a

[ina: hardly space enough to restore [ina lib. 7,1]2: Kugler BMR Pl. 5 gives KI instead of the damaged 12.

29

2,29,19,45 sig]: traces of 9,45 sig still seem to be visible.

32 33

3,57,3]1,24: Kugler BMR Pl. 5 still shows 58 (sic!, damaged) before 31,24.

sa ki-,l'i]r: this reading according to Kugler BMR Pl. 5. The greater part of these signs is now broken away.

Translation and Commentary This section deals with the latitude of the moon (column E). The first line (line 20) contains the title and the values for the extremal latitudes of ±6° and for the latitude K = ±2,24§ which determines the width of the nodal zone. Procedure for the positive and negative latitude of the moon, month by month. 12 (degrees is) the width of the road of the moon. 2,24 (from the) middle (is) the area of the change (of differences). The "road of the moon" is obviously the zone of ±6° latitude 2 • The "nodal zone" bounded by ± K = ±2,24,0,0 is called (line 20) "2,24 qabalti qaq-qar ki-~a-ri'', later abbreviated to qabaltu (the construct chain ki-#r qabalti seems to have occurred in the now destroyed part of line 33). The translation "2,24 (from the) middle (is) the area of the change (of differences)" is an attempt to find a translation for ki~ru which covers not only our present passage, but also the use of the same word for the extrema of a linear zigzag function in Section 10 (cf. p. 199). We reach safe ground with line 21. The first sentence gives the difference d of E outside the nodal zone and for the slow arc of the solar movement: From )( 27 tony 13 month by month 1,58,45,42 (you shall) add or subtract until you reach the nodal zone. With line 22 the treatment of a special case begins (lines 22 to 24): If 3,52,11,39 ( +) (is given), subtract 1,58,45,42 from it, and 1,53,25,57 remains. Now(?), since (the result) is less than 2,24,0,0, (namely,) it is less by 30,34,3, subtract 30,34,3 from 1,53,25,57 3 and 1,22,51,543 (+)after 3,52,11,39 (+)you shall put down.

The meaning of these rules can be described as follows. Suppose we start with a value y 0 and wish to find the subsequent value y 1 • We form

= + 3,52,11,39 (.})

Yo-- d =o 3,52,11,39- 1,58,45,42 = 1,53,25,57 = y 1 '. 2 The reading ma-lak is not certain since ma-AL seems to fit better the slightly damaged signs.

3

The text has 1,52,25,57 and 1,22,21,54 respectively.

No. 200,

192

SECTION

6

Now Yl'

=

1,53,25,57
between the minimum (lines 3, 4, 6 and 9) and the maximum (lines 2, 5 and 8), though I do not know why two sections are needed. Nor can I offer any explanation of the titles "procedure for 2,13,20 of the 15th day" (line 1), "procedure of the 15th day" (line 7), "procedure for 2,[13,20 ... " (line 11). The value 2,13,20 of(/>(.}) corresponds to the point where G starts to increase from its minimum value 2;40H. It is, however, extremely doubtful whether this relation is significant in the present context. Not only here but also in No. 204 rev. 1 and 9 and in No. 211 rev. 3 the same number 2,13,20 is used where a noun is expected. The following comparison of passages almost suggests a reading si-man "time, duration" though this does not help the understanding. 200aa

obv. 1: obv. 11:

epesu sd 2,13,20 sd u4-15-k[am .... epesu sd 2,[13,20 ....

204

rev. 1: rev. 9:

[2, 13] ,20 and su-~u-u 2,13,20 sd u4-1-kam and u4-14-kam

211

rev. 3:

sd ta 2,13,20 GIS-u

cf. and

rev. 1:

sd ta si-man GIS-u

rev. 5:

sd gab-rat ta si(?)-man GIS-u Sections 4 to 6

BM 32241 is a fragment of the same tablet of which beginning and end are preserved in BM 32172. No exact position can be established but it seems to me plausible that not much is missing between the first three sections and the second fragment since both concern column (/>,

Transcription Section 4. BM 32241 obv.(?). Beginning destroyed. 1 • • • • • • • • • • • • • • • • • • • • • ] 20 DUL-[DU-ma ........... . 2 • • • • • • • • • • • • • • • • • • • 4]0 DUL-DU-ma sd a[l . .... . 3 • • • • • • • • • • . • • • • • • • • ] 2,13,20 sd u4-1-kam tab-ma tasakkan(gar-an) .[.... 4 • • • • • • • • • • • • • • • • • ]40 lal sd a/1,57,47,57,46,40 [ ........ . 5• • • • • • • • • • 4,16,15,4]4,26,40 tepu(tab-u) 4,13,29,48,53,20 [ ....... . Section 5 t]ab(?) sd a/2,17,4,48,53,20 dirig ta [ ... . 1,57,47,5]7,46,40 ma!u(1a1-u) ki 1,57,47,57,4[6,40 ... . 4,17,34,4,26,]40 tab 4,12,11,28,53,20 lal [ ....

8, • • • • • • • • • • • • • • • . 7•••••••••• 8 •••

Section 6

28,7,30] tab sd al13 absin 0 dirig a-r:i 1,[4 DU .... . 1o•••••.••••••••••• sa a]l27 zib dirig a-ni 56,[15 DU .... . 11 • • • • • • • • • • • • • • and 12 ab 1]0,22,30 tab ta 1[3 absino en 27 zib .... . 12 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ] 15(?) tab(?) t[a(?) ..... . Rest destroyed.

9•••••••••••••

Critical Apparatus 3: perhaps first line of a new section. 3 tasakkan(gar-an): or sd a[l . ... 6 t]ab(?): or la]l.

Commentary The number in line 4 is the minimum of(/>, In line 5 we have useful parameters for checking the values of (/>1 (new moons) and (/> 2 (full moons) for moments one-half synodic month apart. If Nand N + 1 denote two consecutive months, then we have

No. 200aa, SECTIONS 4 TO 7; No. 200b, SECTION 1

, G, and F Arrangement: 0 /R Provenance: Babylon [low VAT-number] Photo: Pl. 237 Transcription

Obv.(?) Beginning destroyed. Traces of numbers 6(?) and 18(?) at the end of line 1. Below it probably a section line. 2• • • • • • • • • • • • 2S,48,]38,31,6,40 ta1-pil-t[u ..... s. . . . . . . . . . . . . . . . 2,]4S,SS,33,20 a-ni 9,20[ DU ..... . 4 • • • • • • • • • • • • • • • • ]2,30 sd ina lib-bi ... -... 7,20 gar .. [ .... . 5 • • • • • • • • • • • • • • • 2]7 a-ni 30 DU-ma 9,38,2[0 ..... . 6 • • • • • • • • • • • • • • • 1]0 sa 22,8,S9,13,46,4[0 ...... . 7 • • • • • • • • • • • • • • • • • • • 1],SO,SO,SO ...... [ ...... . s. . . . . . . . . • . . . . . . . . • 1,]S7,47,S7,4[6,40 .... . Probably end of section; rest destroyed. Rev.[?] Beginning destroyed. 1 • . • • • • . . • ]23 tab(?) k[i(?) ......... . 2 • • • • • • • • • ] DU-ma ni[m(?) ......... . a. . . . . . . . • ] (uninscribed) [ ......... . 4• • • • • • • • • • • • •

5• • • • • • • • • • • •

]9,22,S7,46,[40 ..... .

-]ma a-ni 30 D[U ... .

6 • • • • • • • • • • • • • 3]6,S2,2[0,44,26,40 ... . Traces of one more line; rest destroyed.

Critical Apparatus

Obv. 4 ... -... : the second sign is dan or possibly e; it is preceded by a damaged sign which might be la. 4

gar .. [ .. : perhaps gar-a[n = tasakkan.

S 2]7: traces of 20 which also could be 30, 40, or SO. S 9,38,2[0: the last number also could be 30 but hardly 40 or SO. 6 6

sa

22,8, ... : it would also be possible to read

sa

7 .... l],SO, ... : traces of 1. 7 ....... : signs which I cannot read; perhaps 10 nu(?) u u su or ki. 8 .... 1,]S7, ... : traces of 1. Rev. 2 ni[m(?): or u pap ... [ ...

samds 2,8, ... or sd 20,2,8, ... ' or sa-nis 2,8, ...

S 30: 9,20 cannot be excluded.

... ,13,46,4[0: 3 and 40 damaged, thus reading uncertain.

6 3]6: instead of 6 any number between S and 8 possible.

Commentary

Obv. 2 mentions the difference of G, line 3 the difference of (j> and the factor 9;20 by which the latter must be multiplied in order to obtain the former. I cannot reconstruct what was contained in the lines 4, S, and 6. In line 7 there appears a number [1,]SO,S0,50 which is known as the minimum of column F' in System B, F' being the lunar velocity per large hour. The appearance of a parameter otherwise only known from System B makes the present restoration rather suspect. In line 8 we have the minimum of (j>, The reverse contained at least two sections. The number in line 6 (if correctly restored) is the factor by which dF must be multiplied in order to give d = 2,13,20(t) lf>

G

=

4,46,42,57,46,40C(,)

G

=

2,51,29,22,57,46,40(-})

interpolation: -9;20 This makes the section where G = G(,j,) somewhat longer (41;40 steps) than in the ordinary scheme (41 steps) though the initial points lf> = 2,0,59, ... (t) coincide. The remaining lines of the obverse are too badly preserved for establishing more than the fact that we have to decrease G still more as lf> increases-probably again by means of a somewhat cruder interpolation (line 8 contained the coefficient -[ .. ];20) than before. For the reverse one might expect the rules for lf>(,}) and G(t). In fact, however, we find again mentioned (lines [1] and 2)

lf>

= 2,.,59, 15,33,20

followed (in lines 3 and 4) by

lf>

= 2,13,20

G

=

2,51,29,22,57,46,40

or, in other words, the end points of the interval of lf>(t) for which G = G(,}). The fact that we are dealingagainst expectation-with the increasing branch of lf> is underlined by the next line (5) which quotes the maximum of lf> M

=

2,17,4,48,53,20

(though distorting 2,17,4 into 17,4, 17,4). Unfortunately the corresponding value of G is destroyed but the lacuna is so short that one can hardly restore anything but 2,40, i.e., the same minimum of G as in the ordinary theory. This provides an additional argument against the value 2,25,32, .... found in obv. 2 and 3 because it would be less than 2,40. In line 6 the minimum of lf> m = 1,57,47,57,46,40

is associated with a number 5, ...... ,42,13,20 which is probably somewhat less than the maximum 5,4,57,2,13,20 of C but greater than the maximum 4,56,35,33,20 of G in the ordinary scheme. With line 7 we seem to turn once more to the increasing branch of lf>. We find again, as in obv. 4,

lf>

=

1,59,48,8,53,20

followed by a coefficient of interpolation -6;50 (cf. obv. 5), but erroneously applied to lf> = 1,59,48,8,53,20 instead of to G = 4,54,48,53,20 (line 8). Also the end of line 8 seems to consider lf> as the depending variable because a remainder is probably added to (or subtracted from) lf> = 2,0,59,15,33,20. And again in line 9: a remainder is combined with 1,59,4,46,40, a value which can only be a value of lf>. Probably the corresponding value is the maximum of G though the traces in line 10 seem to require a value 4,2,18, ... or 4,1,18, ... where one should expect a value of G close to the maximum 4,56,35,33,20 in the ordinary scheme. Line 11 can be restored completely to "[opposite 2,45,55,33,20] ( = difference of lf>) you put down 25,48,38,31,6,40, the difference of the duration (G)". After a ruling, there follows one more line of text (line 12), namely [ .......... ] from )( to ey 3,24 [ . . . . The meaning of this statement is unknown to me. It is obviously related to similar passages in No. 204 Section 7 (p. 250).

No. 207cb

261

The text ends, after another ruling with a catch line which mentions Jupiter, with a colophon (Zrb cf. p. 24). In the Seleucid period the enumeration of the planets always begins with Jupiter, but it is new that a lunar procedure text refers to a planetary text as its sequel. Summarizing, one may say that our text shows a new variant for the computation of G from r/J though many details remain dark and scribal errors may play a role. Nevertheless the deviations in the values of .EG from the expected values in the eclipse tables No. 60 and No. 61 a (cf. p. 108 f. and p. 112) may ultimately be explained by this or a similar variant. No. 207cb

BM 41990 ( = 81-6-25,612) Contents: Procedure text for the moon, System A, Columns r/J and G Arrangements: 0 jR Provenance: Babylon [81-6-25] Photo: Pl. 236; Copy: Pinches No. 101 Transcription

Obv. Beginning destroyed 1( ........................ ] 1 [ .............. . 2[ ................. 2,13,8,]8,53,2[0 ........... . 3 [and tar-~a] 2,13,8,8,53,20 tab 2,5[3,20 ... . 4[ .............. 4,]46,42,57,46,40 [ ..... . 5[ ................ s]i-man tab (empty) [ .. . &[ ................. ] •. anti tar-~a 1,59,48,8,5[3,20 tab 4,54,48,53,20 .. .

7[ ................................ ] ......... 8[ .... . Rest destroyed Rev. Beginning destroyed 1• • . . • • • • • • • • ] • • • • • • • [ • • • • •

2,]13,20 Ia! dirig a-ni 3,[22,30 DU .... . a••...... ] si-man and tar-~a 1,57,55,[33,20 ... . 4 • • • • • • • • ] a-ni 3,22,30 DU ... [ ......... . 2• • • • •

Critical Apparatus

Obv.

1 ... ] 1 [ ... : or 4 (?). 6 ... ] .. : DU not excluded. 7 .. ] .... 8[: the traces which precede the 8 (or 7) may belong to words rather than numbers.

Rev. 1 .. ] ....... [ ..... : perhaps not followed by 17,46,40.

]su

but certainly

4 ..• [ ... : could be read 10( or, perhaps, k[i.

Commentary

On the obverse we have fragments for the following pairs of (/) and G

rp

=

2,13,8,8,53,20(t)

rp

=

1,59,48,8,53,20 [(t)

G = 2,5[3,20q,) G = 4,46,42,57,46,40 G = 4,54,48,53,20

On the reverse we find

rp = 2,]13,20(-}) and

(/) =

1,57,55,[33,20(t)

[G = 2,40 = m G = 4,51,21,28,53,20{t)

and the coefficient 3,22,30 which is the reciprocal of 17,46,40 (cf., e.g., No. 207a).

lower limit for G = C. upper limit for G = C. cf. No. 207ca obv. 4 f.

No. 207cc

262

No. 207cc BM 36438 ( = 80-6-17,165) Contents: Procedure text for the moon, System A, Columns

c

G.

7• • • • • • • • • • • • • • ]

lal(?)

=

2,13,20 ({.)corresponds the minimum value 2;40H of G. d

v' . . . . . . . . . . . . . ] kusu lal u•••••••••••••• ]14 u 4 -me du 10 • • • • • • • • • • • • • • ]14 u 4 -me du u ........... ] and 14 u 4-me du

8.

The reading 14 is not certain; perhaps one should read in all three lines lal sd instead of and 14.

264

No. 207d, SECTIONS 2 TO 6

® (!) /

-~

®

0!/,

®

®

~-

Rev.

®

Fig. 53a

Sections 2 to 6

Critical Apparatus Obv.

II,26 2,8: sic, without tab. III,15 a-ni 8: sic, instead of 9. III,25 ff. tab]: sic, instead of lal; cf. commentary.

Rev.

III,12 ff.: all numbers wrong but tab correct.

Commentary These sections consist of numerical tables which are transcribed on Pl. 138. A partial duplicate is No. 207e obv. Sections 2 and 3 on the one hand and Sections 4, 5, and 6 on the other hand, forming two major units, the first of 27 lines, the second of 20 lines. Section 2 gives values of tl>, decreasing with constant difference 17,46,40,0 beginning with 2,4,29,15,33,20(.}) in obv. II line -2 down to the minimum m

tabulated at intervals of E ~ 1/ 9;20 mean synodic months. 93 Obviously associated, line by line, with Section 2 is Section 3 which contains a difference sequence of second order which I call A. This function A is strictly symmetric; it begins and ends with the value +23,35,33,20,0 and has in the middle a minimum of -20,13,20,0,0. The first differences increase from -5,22,35,33,20 to +5,22,35,33,20. The second differences are 0 on both ends, then four times 17,46,40,0, once 20,22,13,20, and 35,33,20,0 in the whole middle part (cf. Pl. 139). The factors of interpolation which are listed in column III are proportional to the first differences such that an increase of A by 17,46,40,0 corresponds to an increase of these coefficients by 1. Consequently ~A= 35,33,20,0 corresponds to an increase of 2 and 20,22,13,20 to an increase of A diophant shows that the values in column II are connectible with the values of 2 which are used in the ephemerides of System A (and thus not connectible with 1 ). The resulting 93

date, however, falls outside the three or four centuries which are historically possible. This is not surprising for a scheme found in a procedure text.

No. 207d, SECTIONS 2 TO 6

265

1;8,45 for the coefficients of interpolation. Example: A decreases from line 3 to line 4 by 4,11,28,53,20 (cf. Pl.139). Division of this number by 17,46,40,0 gives 14;8,45 as listed in line 3. 94 While Section 3 gives the values of A near its minimum we find in Section 6 the values near the maximum of A. These values are tabulated only once but are to be used twice: in ascending order in connection with Section 4; in descending order in connection with Section 5. The numbers in Section 6 form a difference sequence of second order, which increases from 3,4,53,20,0,0 to 3,55,15,33,20,0 (cf. right half of Pl. 139). The first differences decrease from 5,20,0,0,0 to 0, the second differences are constant and equal 17,46,40,0. In Sections 4 and 5 we have the values of tP to which the corresponding values of A in Section 6 belong. In both Sections 4 and 5 the constant difference 17,46,40,0 is used. The numbers in Section 4 increase linearly from tP = 2,7,42,13,20,0(t) (obv. 11,25) to tP = 2,13,20,0,0,0(t) (rev. 11,17). In Section 5 the scribe committed an error. He began with tP = 2,13,4,48,53,20 but wrote tab ( = t) instead of lal( = .}). He now correctly added step by step 17,46,40,0 not realizing, however, that he should not go beyond rev. III,ll because the next step would lead him to values greater than MlP. Thus he wrote 2,17,13,42,13,20 in rev. III,12 and continued in this fashion instead of writing again 2,16,55,55,33,20 and going down to 2,15,27,2,13,20 (rev. I11,17 =rev. III,6). On the other hand, from rev. III,12 on, the sign tab( = t) is now correct because the values which should have been listed there belong to the increasing branch of c!J, though arranged in the inverse order. Finally the whole set of cp values in Section 5 should be moved one line up, i.e., one should begin with tP = 2,13,22,35,33,20(.}) (cf. obv. 26) and end with tP = 2,15,9,15,33,20(t ). The reason for this correction is the following. When A increases linearly, tP increases from 1,58,48,53,20,0 to 2,7,42,13,20,0, thus by 8,53,20,0,0 = 30 · 17,46,40,0. Obviously the same relation must hold for the decreasing branch. The lowest value of this section, tP = 2,4,29,15,33,20(.}), cannot be changed without disturbing the symmetry of the arrangement near the minimum of A. Consequently the linear part must begin with tP = 2,4,29,15,33,20 + 8,53,20,0,0, = 2,13,22,35,33,20. The whole scheme can now be explained very simply (cf. Fig. 54). While tP increases from 1,58,48,53,20,0(t)

Fig. 54

to 2,7,42,13,20,0(t) the function A increases with the constant difference 5,22,35,33,20 from 23,35,33,20,0 to 3,4,53,20,0,0. For tP increasing from 2,7,42,13,20(t) to 2,13,20,0,0(t ) the corresponding values of A are listed in Section 6. In this way A has reached its maximum 3,55,33,20,0,0. This value is held constant until cp = 94 In Pl. 139 these coefficients are listed with the corresponding first differences, thus one line lower than in the text. In obv. III, 15 the text writes erroneously 8 instead of 9.

No. 207d, SECTIONS 2 TO 6, SECTIONS 7 AND 8

266

2,15,27,2,13,20(t) or better 2,15,9,15,33,20(t) which should be the entry in rev. 111,17. From here on, Section 6 should be taken once more but in opposite direction, decreasing to 3,4,53,20,0,0 while (]J passes its maximum and goes down to 2,13,4,48,53,20(-}) or better to 2,13,22,35,33,20(-}). From here on, both (]JandA decrease linearly, (]J with the difference 17,46,40,0, A with the difference 5,22,35,33,20, until (]J = 2,4,29,15,33,20(-}) and A= 23,35,33,20,0 is reached. Then the scheme of Section 2 and 3 becomes effective, bringing A to its minimum -20,13,20,0,0 and back again to +23,35,33,20,0 after (]J has passed its minimum. This leads us back to (]J = 1,58,48,53,20,0(t), with which we started. The previous discussion has completely determined the arithmetical relations which lead from (]J to A and vice versa. The main problem, however, the astronomical significance of these functions, remains completely in the dark. Perhaps it is significant that the difference 5,22,35,33,20 of A in its linear parts, is known as the difference of (]J(n) and (]J(n + 12) if both belong to the same branch. It is also evident that the use of intervals of length E = 1/9;20 must have the same reason which gives this parameter its importance for the transformation from (]J to G. But no other feature of A is explicable to me.

Section 7 Though only a small fraction of this section is preserved, it suffices to show that here was described in words the contents of the numerical tables of Section 2 and 3, following the pattern known from No. 200 Section 14 and similar texts. Because of the symmetry of the scheme, the text begins just before the minimum value in col. III,11 and covers the whole remainder of the table. At the right-hand side of this column one line was written perpendicularly to the rest of the text. The traces seem to indicate numbers, perhaps beginning with 25,40 (but hardly 25,48,38,31,6,40) followed towards the middle by sd 5 or 4-4-,5. Section 7 contained 18 lines; the ends of the first 13 can be read fairly securely as follows:

D)U ki 19,55,33,20 tab tasakkan(g[ar]-an) lal a-di 2,.,20,22,13,20 3 • • • • • and tar-~a 2,.,20,22,]13,20 lal19,55,33,20 gar 4 • • • • • • • • • • • • • • • • • • • • • • • t]a 19,55,33,20 nim gar 5 • • • • • • • • • • • • • and tar-~a 1,)59,4-4-,48,53,20 Ia! s..................... a-ra17 DU ta 17,33,20 nim gar 7 • • • • • • • • • • • • • • • • • • • • t]a 1,59,9,15,33,20 Ia! s.... and tar-~a 1,59,9,1)5,33,20 lal 12,48,53,20 gar 9 • • • • • • • • • • • • • • • • • • • • t]a 12,48,5[3,)20 nim-ma gar 10• • • • • • • • • • • • • • • • • • • • l]al en 1,58,33,[42,1)3,20 Ia! 11 • • • • • • and tar-~a 1,58,33,42, 13,2]0 lal 5,42, 13,20 gar 12• • • • • • • • • • • • • • • • • • • • • • 14,8,45 D]U ta 5,42,13,20 n[im] 13 • • • • • • • • • • • • • • l]al sd mu tasakkan(gar-an) 1• • • • • • • • • • • • • • • •

2 • • • • • • • • • • • • • • • • • ) ••

14 • • • • • • • • • • • • • • • • • • • • • ] ••• [ • • • • • • ] • • • • • 15 • • • • • • • • • • • • • • • • • • • • • • • ] •••• [ • • • • • • • • • • ]

1s. . • • • . • • . • . • • • • • • . . . • .

8,]45 DU-ma ]8,45(?)

17 • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

18 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ]

A translation is not needed because the text until line 12 follows the general scheme "opposite a put b; that which exceeds a multiply by c and add it to b" with values of a, b, and c which are all given in the numerical table. In line 13 a "year" seems to be mentioned and at the end appear the coefficients of interpolation given in Section 3.

Section 8 As far as the tablet is preserved one line runs across the reverse, concluding in -]mes sd mu a-na mu, which would mean "the ...... of year by year". This is perhaps a title of what is contained either in sections 4 to 6 above it or section 8 underneath it, and may refer to the methods of checking 12- (or 14-)line intervals.

No. 207d, SEcTION 8

267

The preserved part of Section 8 proper consists only of the ends of 12 lines, too short to permit a consistent interpretation.

D[U]-ma ana tar-~a 2,13,2,13,20 tab 3,55,33,20 tasakkan(gar-a[n]) 2o......... ] .... 3,55,33,20 gar sd al2,15,27,2,13,20 tab dirig GAM 21••• sa al . .... ] ... t[ab] dirig a-ni 3,22,20 D[U] 22 ..................... ] .. u 4-mes 1,5[0 ... ] .... -me8 ... [ ..... ] 23, • • • • • • • • • • • • • • ] •• Ijl SI u ... [ .... . 24.......... -]ma ta 3,55,33,20 n[im] .... [ ...... ] 25, • • • • • • • ] ana tar 2,17,4,48,53,20 tab 26 • • • • • • • • • • 3,]55,33,20 tasakkan(gar-an) 27 • • • • • • • • • ta 3,5]5,33,20 nim sa tag4 2s......... a]na tar 2,13,4,48,[53,20 tab] 29........ S]a tag 4 2,3,53,42,13,20 ao .......... ]5 2,3,53,42, 1[3,]20 [ ....... ]

19 •• ] ••

Rest destroyed

Critical Apparatus 20 ] .... 3,55,33,20: the 3 is preceded by some damaged corner wedges; one could read ]30,13 etc. but no such number occurs in the scheme. 20

GAM: written like "zero".

22

.. u 4-mes: the "u 4 " is perhaps the end of a larger sign like T A which also precedes the second mes, which is followed perhaps by [g]ar-a[n].

23

] .. Ijl: perhaps a number between ]2,40 and ]6,40.

23

... : the last sign is perfectly clear but unknown to me; perhaps DA?

24 n[im] ... : reading of nim uncertain. There follows what may be a 2 or 3, perhaps with 2 written below, or even 8. 30 ]5: or 8?

Commentary We obviously have here the text belonging to the tables in Sections 4, 5, and 6. In line 19 the correspondence of the values in rev. 11,16 and rev. IV,16 is stated. Because this is the next to the last line of the whole scheme it is plausible to assume that Section 8 began with the equivalence of the values in the last line. Line 20, however, repeats the number from rev. IV, 16 or 17 and then jumps to rev. Ill,6. Thus we find a shift from column II to column III and a step of 10 lines up. At the end of line 21 some excess is multiplied by 3,22,20 which is equivalent to a division by 17,46,40. Because 17,46,40 is the constant difference in the columns II and III and because this difference corresponds in a column if> to intervals of length E = 9;20 we may say that the quotient counts the number of £-intervals beyond some limiting value. Lines 22 and 23 are almost completely destroyed and unintelligible. Lines 24, 26, and 27 mention the maximum of A. Line 25 gives Mq, though this value does not appear in the tables of col. II and III but would fall exactly in the midpoint between rev. III,11 and 12. Line 28 mentions the value with which column III begins (obv. 25). Thus we seem to have reached the end of the discussion of the tables in Sections 5 and 6. The text continues by giving in lines 29 and 30 a value which is found in the third line of the table in Section 2 (obv. 11,0). This is the endpoint of the decreasing branch of if> for which also A decreases linearly. Hence we have completed the scheme.

268

No. 207e

No. 207e

BE 15557 Contents: Procedure text for the moon, System A, Column C/J and related columns Arrangement: 0 jR Provenance: Babylon, German excavations; found 1902 in Amran, near the city wall towards the Euphrates 95 Transcription of obv.: Pl. 137; of rev.: cf. below Excavation photo: Pl. 243

Critical Apparatus to obv. I and II II,4

17,33,30: sic, instead of 17,33,20 (isolated error).

I/II,15: The displacement of 2,13, ... towards the right suggests some words (title?) preceding the number from which nothing is missing. I,16

... 13,20: sic, instead of ... 33,20 (isolated error).

Commentary The obverse contains in col. I and II numerical tables which duplicate in part the tables of No. 207d. Beginnings of a few lines of a third column are visible, roughly beside lines 8 to 18 of col. II but without direct relation to them as is seen from the different spacing of these lines. All I can read in these lines is 2[ .. , 20[ .. , 2,10[ .. , DU[.. , 30[ .. , en[ .. (?), 1[ .. (?), 20 .. (?) . The reverse (or what I have arbitrarily called the reverse) is again inscribed in (at least) two columns, the first of which is almost completely destroyed except for a few signs at the beginning of lines. All I can identify is igi[ ... , a[ ... , ki[ ... , sa 5 [ ... or a similar group. A dividing line indicates a new section followed by 1,9[(?) ... , 1,1[0(?) ... , mim-ma[ ... , DU[ ... ; the rest is destroyed. The second column has parts of three sections which remain, however, a complete mystery to me. Not a single one of the subsequently transcribed numbers occurs elsewhere in known context. Of the first section only fragments of the last three lines are preserved: 1]1,50 and 6 [ ...... . ]7,28,46,4[0 ...... . itu izi 5 e(?) .. [ ..... .

1. • . . . • . . . • • • • • • 2• • • • • • • • • • • • • 3• • • • • • ] • • • • • •

The only intelligible words are "month V" in line 3. The and 6 in the first line has perhaps its parallel in the phrase and 6 itu in the lines 5 and 7 to 9 of the next section and again in lines 12 and 14. In the following transcription of the two better preserved sections, it must be underlined that the division of numbers is often a matter of mere guess, in addition to the insecurity of readings.

]-ka 2,7,19 5,23,20 tab x [ ....... ] 5,10 12,18,53,20 tab 2,28 x and 6 [ ...... ] 6 • • • • • ]-ka 2,7 ,26,23,20 matu(lal-u) x y [ ........ ] 7 • • • • • • • ] •• 4,40,27,46,40 matu(lal-u) 11,44 x and 6 [it]u 8 • • • • • • • • • • • ]8,27,46,40 tab and 6 itu 7,6 tab 9 • • • • • • • • • ]49,10,55,33,20 tab and 6 itu 7,6 tab 10 • • • • • • • • • • • • • ] ge 6 babbar sa 5 sig 7 11 • • • • • . • • • • • ] 1 2 3 4 4•••••••

5••••• ]

The sign "x" in lines 4 to 7 looks somewhat like ina su. The group x y in line 6 one could read en 10 or x 1,10 but also the "10" is doubtful because of a horizontal wedge beginning right at the upper end of the corner wedge. •• I owe the knowled!!e of this text to the kindness of Mrs. L. Ehelolf.

Nos. 207e

AND

207f

269

The reading and interpretation of the lines 10 and 11 I owe to Dr. Sachs, who recognized that the four colours black

white

red (or brown)

yellow (or green)

are associated with the numbers 1, 2, 3, and 4 respectively. What this has to do with the rest of the text is unknown. 4]1,26,4,40 1,15 tab 7,6 tab and 6 itu 2,30 29,31,52,30 tab 2,28 tab 14 • • . • • . . • • • ]4, 12 29,31,52,30 tab 2,28 tab 15, • • • • • . . . • • • • ]45,54,22,30 7,6 tab and 6 i[tu] 16 . 29,31,52,30 11,44 [ 17, 2]9,31,52,30 11,[44 Rest destroyed. 12• • • • •

13 • • . • • • • ] •

0

•

0

0

•

0

0

•

0

0

0

••

0

••••••

0

•••

0

0

•

0

•

]

0

•

0.

0

•

0.

0

0

0

•

]

0]

Four times in this section (lines 13, 14, 16, 17) there occurs the number 29,31,52,30 -or should one read 29,31,50,2,30?-which may or may not represent a value for the length of the synodic month. The numbers 2,28 and 7,6, and 11,44 occur as minimum, mean value, and maximum, respectively, in a Louvre text (MNB 1856) which I know through the courtesy of Mr. J. Nougayrol from a copy by Dr. Sachs. This text contains, after certain planetary data, some numerical lists, one of which is a difference sequence of second order (period 14 lines) with the above-mentioned extrema 2,28 and 11,44. The last line is "11,25 tab 7,6 sd 6 itu sa 14" and obviously corresponds somehow to the 7,6 tab and 6 itu of the present text (lines 12, 15 and 8, 9, respectively). Since MNB 1856 quotes "full moon" in this connection, one might assume a similar purpose for the present text. All the rest remains obscure. MNB 1856 is dated by a colophon as written in the year Philip 5 (= -318/317). This early date might explain why the parameters of the present text show no connection with the material known from the ephemerides.

No. 207£ BM 42685 ( = 81-7-1,449) Contents: Procedure text for the moon, System A, Column into F as indicated in Section 5 of the Procedure Text No. 200 (p. 190). The relevant section of the relations between F and G is given in the table below. The column for tf> is added in order to facilitate the comparison with the scheme given on p. 60 of the Introduction. The only deviation consists in adding explicitly the minimum value of F. 96

F

t1> I,S8,13. zo 1

szs:; 33to

I Sl47:.7.4,40 I, ~7,

~

:;¥, f. S1, ZO t

Discovered by Dr. A. J. Sachs.

' 0 ••• could be chosen arbitrarily, or taken from observation. 9 In fact, however, we have rules which determine the motion from r to ([>, from ([> to e, etc., depending on the location in the zodiac. This obviously means that one single element, e.g., r 0 , would suffice to determine not only ri, r 2> • • • but also ([> 0 and thus ([>I, ([> 2> ••• and 8 0 , ••• etc. and thus the whole scheme. On the other hand, one should demonstrate that the rules which connect different phenomena in their natural order (that is, in lines) are compatible with the results which One obtains, e.g., for TI, ([>I, E)I, ••• as computed in vertical columns. It is quite obvious, however, that such a proof has not been given in full generality and

§2.

287

MERCURY

that the rules which connect different phenomena are probably often not much more than auxiliary schemes which are useful, but not binding, for the choice of the initial values and for interpolations. In general, it can be said that the procedure texts contain many more such auxiliary schemes and variants of the main computing processes than are actually used in the preserved ephemerides. Whether this is accidental or not is difficult to say. This much, however, is certain: we do not find anywhere in the planetary texts the strict consistency, extending for almost three centuries, that holds for the lunar ephemerides of System A. To this may be added the fact that one rarely meets accurate numerical agreement between our preserved ephemerides and Almanacs and similar texts. The Babylonian planetary texts certainly represent a much greater variety of possibilities than is usually assumed. " It may be mentioned that none of our ephemerides begins with a set of simple round numbers, as should be the case if they were taken from observation. As everywhere else, the actual texts must be the result of some compromise between observations and the requirements of computation.

THEORY OF MERCURY

A. Introduction We know of three different methods of computing ephemerides of Mercury. System AI is best known because it is represented by the majority of preserved ephemerides and procedure texts. System A 2 is attested only in two early ephemerides (Nos. 300a and 300b, from the first two decades of the Seleucid Era) and therefore not known in all details. Much worse, however, is the situation for System A 3 which appears only in one procedure text (No. 816), leaving many questions open. The following remarks concern exclusively the Systems AI and A 2• For System A 3 the commentary to No. 816 (p. 425 ff.) must be consulted; for the determination of daily positions cf. below Section D (p. 299). In Systems AI and A 2 four phenomena are considered: first and last visibility as morning star (T and I: respectively) and first and last visibility as evening star (8 and Q). In both syste:ns, only two phenomena are computed independently, the two remaining ones being a direct function of the first two. In System AI the risings r and 8 are computed first and .E is then obtained from T, Q from 8. In System A 2 the opposite arrangement holds: the disappearances I: and Q are found first, and from them 8 and r respectively. This can also be expressed as follows. In System AI the

appearances are found independently; to them the stretches of visibility are added in order to find the disappearances. In System A 2 the disappearances are computed first and the given stretches of invisibility determine the reappearances. Both Systems have in common that the longitudes of the primary pair of phenomena are computed by means of synodic arcs which are constant on given segments (three or four in number) of the ecliptic (cf. Fig. SSe on following page, in which I: and Q concern System A 2 , rand E System A1 ). Thus both Systems are of Type "A". From the longitudes the times are derived by small corrections which will be discussed later. Given amounts of "pushes", depending only on the longitude of the initial phenomenon, then lead to the data of the related secondary phenomenon. Probably at the basis of all these methods lies the 46-year period, which is explicitly mentioned in procedure text No. 800, to wit (1)

145 phenomena of the same kind

= 46 years

which yields a mean synodic arc of approximately 1,54;12° or a mean period of 3;9,7, .... Neither of the two systems uses (1) exactly, but, because of the requirements of the zonal arrangement of the synodic

288

MERCURY, SYSTEM

Al

:-

"J

~4o i: _yo

I=

I 'l'l !1

l(

8 1 61

!!f'l:==117ll./lo 1=1

)(

2:~0

2,20 2,0

_140 1.20

2,20

~0 .!:_40 .!J.20

2,20

2,0

I cp I

l:j

)(I

I @16l

wl=:!=l m.l vi' I 01=1 >< I

_j

L_

.n

'l'l t1

)(

®16l17Jt'l::lm. lvl'lo 1=1

)(

)(

1 8 I & 1 w 1:!:1 m.l vi' I 0 1=1

)(

r

1,40 1,20

I 'T' I l:l

Fig. 55c

arcs, each phenomenon shows a slightly different period, though it would take many centuries before these deviations would make themselves felt at all. These periods are

.E: in 388 years

.... ~-

in 480 years

(2)

!J: in 217 years

r:

in 848 years

1223 disappearances as morning star 1513 appearances as evening star 684 disappearances as evening star

the case, as we know from procedure text No. 801 as well as from ephemerides, when r would fall anywhere between 'Y' 10 and ~ 20 or E between =:::= 0 and Til 5. Obviously when the appearances r and E do not take place, the subsequent disappearances .E and Q are equally omitted. The corresponding intervals are not explicitly mentioned but must be close to 'Y' 24 -+ IT 5 for .E and =:::= 18 -+ Tit 30 for Q. In the Almagest (XIII,8) the "beginning of Taurus" and "the beginning of Scorpio" are mentioned for the omission of a period of visibility of Mercury.

2673 appearances as morning star

In this connection the terms "appearance" and "disappearance" are used in an abstract sense in order to give the correct period relations or, expressing the same thing differently, the correct mean value for the synodic arc. In fact, however, some of the stretches from r to .E or from E to Q are "passed by". This is

B. System A1

I.

HELIACAL RISING ( T). PosiTIONS

Three zones of constant synodic arc are used: (1): (2): (3); 10

from 61,1 to V'S 1610 from V'S 16 to ~ 30 from IT 0 to 61,1

The position

w1 = 1,46° w 2 = 2,21;20 w 3 = 1,34;13,20

V'S 16 actually occurs in No. 302 rev. VI,10.

MERcuRY, SYsTEM

A1, LoNanuoEs oF

The corresponding arcs are:

r

289

first preliminary position: 'Y' 2,43;40 = II 1,43;40 0 =II 1,43;40° beyond II 0

t

respectively. Hence we obtain for the period: p

= 2,45 + ~ +

1,1 1,34;13,20

2,21;20

1,46

44,33

= - - = 3;9,7,38, ... 14,8

with

z=

II= 44,33

14,8.

hence first correction:

1,43;40 . - 0;20 = - 34;33,20 first prelimin. pos.: II 1,43;40 second prelimin. pos.: 1,9;6,40 II

t

Thus 2673 heliacal risings occur during 848 years.

= 61.. 9;6,40 =

61.. 1 8;6,40° beyond 61.. 1

Because

w2

w1

-

= 35;20 = 0; 20 1,46

w1

w3

w2

-

_

w2 w1

w3

-

=

w3

47;6,40 2,21;20

11;46,40 1,34;13,20

8;6,40 . 0;7,30

= _ 0; 20

+ 0;7,30 w3

etc. This leads to the rule that every degree beyond & 1, obtained by using w 3 , should be multiplied by 0;7,30 in order to obtain the actual position, every degree beyond l"S 16 by 0;20, every degree beyond II 0 by -0;20.

This method yields all subsequent positions if one initial position is known, computing from line to line. Two modifications are possible: (a) computation from line to line but the final position is obtained directly from the initial position, (b) computing from the value in line n the value in line n + 3. (a) Let us denote the initial position by .\ 0 , considered as given; then the subsequent position .\1 can be found by the following procedure. 11

Examples:

Section

No. 300 obv. II,33 initial position: II 26;6,40 w 3 = 1,34;13,20

t

61.,1,0;20

&1 59;20° beyond 61.. 1

hence correction: 59;20 · 0;7,30

= 7;25

preliminary position:

~

0;20

obv. II,34: final position:

~

7;45

No. 300 obv. II,23 initial position: 'Y' 22;20 w 2 = 2,21;20

I

)(O~>-o~611

][

6l I ~ ).o ~ rtf 30

][

preliminary position: II 2,0;20 = ~ 0;20

= 1;0,50

61.. 9;6,40 61.. 10;7,30

obv. II, 24: final position:

= 0; 7,30

we have w1 = w 3

hence second correction:

::::!!::

0 ~ A0 ~ 15 10;30

oc=O

f3

~=0

)(O

t~X + 8;2.2,30

~0

c9o ==o

0\

+ 16

; ~ + 16

JrtO 150

X 0 10;30:i ).0~ 15 16

-oo

t

y 1516 ~ A0 ~ 'P 10;10

cso

~ 3

-5.~640 ) •

)l_o

!f 'T' 10;10 ~

'PO

~ ~- 6;-,37,30

61o

).0 ~ 'l130

0(0(

9;20

)(O

1. Determine from the table given here the "section" to which ,\ 0 belongs. 2. Find the angle a which is determined by .\ 0 as endpoint and by the point given in the column called a = 0 for the section found in the first step. 11 The description given by Kugler SSB I p. 190 f. is incorrect because he distinguishes 5 sections only.

290

r

MERcuRY, SYsTEM A 1 , LoNGITuDEs oF

AND

3. Compute for the value of a, found in step 2, the value of f1 according to the column called {3.

and add

4. Add the angle found for f1 to the point given in the column called f1 = 0. The result is the new position ,\1 •

we obtain the point

Example:

s

dl = 6,0 - 16;7,30

24

QD

which still lies in zone 3 and

No. 300.

A0

Obv. II,27

=

QD

9

8;22,30

=

in section I before the beginning of zone 1. Hence

54;46,40

a=

S a+

24;46,40

1,10

,\ =

{1

=

3

Q1'j

24

+ -79

=

QD

24·46 40 ' '

as given in obv. 11,27.

thus as given in obv. II,28.

II.

(b) Because P = 3;9, ... , three lines almost restore the initial value. It is therefore possible to find simple rules for the computation of the position ,\ 3 in line n 3 if the position ,\ 0 in line n is given. These rules are: If both ,\ 0 and ,\ 3 belong to one of the zones of the table given below, then the increase in longitude is given by the corresponding number d1 , d 2 or d 3 respectively.

+

Zone (1): from

61.

1 to V'S 16

dl = 6,0 - 16;7,30

~

(3): from

61. 1

II

PosiTIONS

The three zones of constant synodic arc are:

0 to

30

d2 = 6,0 - 21;30 d 3 = 6,0 - 14;20 .

If .\ 0 + d; does not belong to the same zone as ,\ 0 and if 8; _ 1 indicates the length of the missing arc, then we have respectively 1

A3

=

A0

+ d2 + 4

A3

=

A0

+d

A3

=

A0

+ d1 + 9

1

3 -

Z

from

(2):

from=:::= 26 to )( 10

w 2 = 1,46;40

(3):

from )( 10 to

6

w3

= 2,14

a

Q1'i

Q1'i

=

W1

2,40°

= 1,36

Length of the arcs: a1

= 1,50

a

2

3

= 1,56

hence period: p

= 1,50 2 40 ,

1,56 = 25,13 = 3·9 7 30 80 , , ,

2,14,0

+ 1,46 '40 + 1' 36

'

and II= 25,13

Z= 8,0 .

In other words, 1513 appearances occur during 480 years. The period of E is not exactly the same as the period of r:

81 82

1

P(E) = 3.9 7 30 = 44,32;58 ~ ~4,33

83

, , ,

14,8

14,8

= 3;9,7,38, ... = P(T).

Examples:

No. 300 obv. 11,25

A0

=

fll26;7,30

obv. II,28

.\ 3

=

fll10 .

A0

=

61. 10;7,30 in

For the computation of positions we have

1•4 1,36

Both points belong to zone ( 1) and therefore their distance is given by dl' If we, however, start with obv. 11,24

6 to =:::= 26

(1 ):

Increase of longitude

(2): from V'S 16 to

FIRST APPEARANCE IN THE EVENING ( S).

zone (1)

=

0·40 ,

53;20 2,40

- - - - 0;20 10;40

- - - - -0;6 .

1,46;40

MERcuRY, SYsTEM

A1,

This is the basis for the rule given in procedure text No. 801 Section 2: For every degree of the final position beyond l:m 6, multiply by 0;40; beyond ~ 26, by -0;20; beyond )( 10, by -0;6.

24;36 w 3 = 1,36 2,0;36 = 1Tl' 0;36 =

1,0;36 6 l:Il5 54;36 beyond l:m 6 l:m

t=

0\

6 ~ ~0 ~

:::!!:

26

==o

E

"l 23;20

"10

JY

)( 10 ~ ::\0 ~

)(

y

'T'

:n

:\0 ~ )( 10

No. 301 rev. II,8 7

(2):

+=

2,21·- 0;20

hence correction: prelim. position: rev. II,9

beyond

final position:

~

(i)

r

-

(I)

(3) (1) (2.)

(3)

Fi.rst

Point

-47 ~ 2,47 30

w

go

)( 6 ~ A0 ~ g 6

)(o

1,4&

2, 4:>

)( 0

2, l.l; l.O

l.,/1,

4,ZS; 30 J, 4~; 30

)( 0

6l I

1,~;13,l.O

I, I

3,l7;4o

~l6

2,40

)( 10

I,"'; I,

)( 10

9 6

Q15

~

Jt

4o

lf'

01.

+ 11;ZO

::::!!:::0

Increase of longitude

6 to

~

26

dl

d2

6

Q15

= 6,0 - 24;20

= ds =

26 to )( 10

CONTINUATION OF

l'

l

() lb

g 6 l6

Jl 0

~"'+b J

6ll 0 16

=!!:

..ioe-11 10

6,0 - 16;13,20 6,0 - 14;36

B(T)

AND

B(E)

Assume that a value of y 0 of B(T) or B( E) is given. The following rules will give the corresponding value Yn n lines (roughly n/3 years) later. Obviously we may assume 0 < n < II because this condition can always be satisfied by adding or subtracting multiples of the number period by which the value of Yn is not modified. The following table has to be used:

=

Ent.l

=o

'T'O

III

As in the case of r, this process can be modified in two directions: either the final position can be directly

Zone

0\ - 13;2.0

J( 6

A0

0 ~

from~

26

~ 2,0=~

f~tO

)( 0

(3): from )( 10 to

(= )( 17)

2,47 ~ 26 2,21

~

~ 0( +25;2.0

ot-t-6

(1 ): from

· w 1 = 2,40

prelim. position:

{3=0

)( 0

30

Zone ~

@0

:::!:26 ~ \~ ll'l25;2C ~

~

+

1Tl' 37

initial position:

=0

(b) If A0 and As are positions corresponding to line n 3 respectively and if both Ao and As and line n belong to the same zone in the table given below, then their difference in longitude is given by the corresponding value d;.

54;36 . 0;40 = 36;24 hence correction: 1Tl' 0;36 preliminary position: rev. II,8 final position:

291

I

tj

tj

9

I

No. 301 rev. II,7

preliminary pos.:

s

ANn

Section

Examples: initial position:

r

LoNciTunEs oF

w••• wi.

"f·w S, J4; 7, 50

I. lO

7, z~; 3o 4,S7

0.J 40 I; 7,30

1,!)0

S,ll

8,l4;l.O

2,11,

4,ll;S3,l.O 3, 2.

S, Jt; IJ,l.O

I, :,-(,

I

0;40

s. l., 3(;

OJI,

I 1,0

'

computed from line to line, knowing to which of the six sectors of the ecliptic the initial position belongs; or in groups, progressing by steps of three lines.

(a)

Determine the zone (i) to whichy 0 belongs and replace y 0 by the corresponding distance z 0 from the beginning of the zone (i).

(a) The procedure to find from a given position Ao the subsequent position A1 follows exactly the pattern described on p. 289. The numerical values, however, have now to be taken from the following table. 12

(b)

Form ~z

12

=

n · W;

+ z0

The rules given by Kugler SSB I p. 198 are incorrect.

292

MERcuRY, SYsTEM

(c)

A1,

LoNGITUDEs AND DATEs oF rAND s

and determine by means of a division the greatest integer a such that

This has to be added to the first point of the zone (2) + 2 = (1) and we obtain

Three cases are possible:

as found in the text in rev. II,34.

0

(I)

~

zn < l;

l; ~ zn < l;'

(II)

1/ ~ zn < Pw

(III)

Then form respectively

+ zn case (II): Yn = first point of zone (i + 1) + zn' case (III): Yn = first point of zone (i + 2) + z/ case (I): Yn

=

first point of zone (i)

where

IV.

DATEs oF

r

AND

s

For both phenomena the following rule for finding the dates is used in the ephemerides. Suppose the nth synodic arc is ~,,A. degrees, then the corresponding synodic interval is given by

(1) measured in tithis; ~nA has to be taken in its numerical value as equally many tithis. This rule is, as a matter of fact, only a simplified form of (1a)

and

given in the procedure text No. 801 obv. 8 (cf. p. 366). Comparison with (13b) p. 286 shows that we must have

Example:

E

No. 302 obv. II,O gives for E

Using either P(T) (p. 289) or P(E) (p. 290), we obtain

Yo== 11;46,40. Find Yn for n = 1,11. We have i = 2

z0 = nw 2

=

=11;46,40-

~ 26 = 1,45;46,40 1,46;40 · 1,11 = 2,6,13;20

thus ~z

= 2,7,59;6,40 .

Division of this value by Pw 2 = 5,36;13,20 gives the quotient a = 22 and the remainder

zn = 4,42;13,20 . Because

E

= 11;3,59, .. .r

Thus it is obvious that (1a) is based on the value 11;4r of the epact.

Examples for the computation of dates m ephemerides: No. 300. obv. II,33 initial position: obv. II,34 final position:

zn' = (4,42;13,20 - 2,14) · 0;54 = 2,13;24 and (2,13;24 - 1,56) . 1;40 = 29 .

n 26;6,40 date: 2,8 III 26; 18,33 ~

7;45 = n 2,7;45 1,41;38,20

synodic arc: hence time difference:

we are dealing with case III. Thus we have to compute

z n"

= 3;30,39,4,20 • p .

initial date: obv. II,34 final date:

1,41;38,20r

+ 3;30,39T

1,45;8,59T =3m III 2,8 VII

+ 15;8,59T 26;18,33T 11;27,32T

Calculation in groups proceeding by three lines is here possible in the same way as with positions (cf. p. 290 f.). This leads to the following rule: if both A0 and

MERcuRY, SYsTEM

A1,

DATEs oF

,\ 3 belong to the same zone, then the corresponding time difference can be obtained from the table given here.

Increase of Time

Zone

Oll l:o

(I): [rom (2): from

r

-

.n: 0

to

from @6

to

==26

(3):

fr-om

(I): (Z):

frolTI ==26 to

(J);

I'· .

V.

I~

0

to

t1 30 6l I

0 16

H ID

;:-,-~rq

12m - ~~8:3'"

13;.. ~ '"I

A

(T).

S=o

(3

"'=:Q

:m:

~

HELIACAL RISING

We now assume that the longitudes of the acronychal settings ( Q) are known. Then the longitudes of r are obtained by means of "pushes" which depend only on the longitude of Q. For the greater part of the zodiac these pushes are negative, indicating that Mercury is retrograde between Q and T. For the short section

Secti.on

-

11

17.

0; 15

27;7,30 2,15;37,30

27;30 _,..._

14.

12;3,20 = 2,0;33,20

lll

14

line 11.

No. 300a obv. VI:

= 12;3,20 = 0.6 40 1,48;30

w1 w3

r

AND

Example:

Furthermore w2

fJ

LoNGITuDEs oF

.±. s

0

'Y'O

0 0

9

0

0\

+

0;30

@O

~

+

0.30 J

~0

from "('0 to rr5, however, the pushes are positive and the motion becomes direct. The following list shows the values of these pushes

o=

B(T) - B(Q):

"(' 0: from "(' 0 to "(' 15: increasing 0;24 per degree 0= 6 "(' 15: from "(' 15 to ti 15: from

l) Ql5

15 to

l)

15:

decreasing 0;4 per degree

0=4 Ql5

15: decreasing 0; 12 per degree

o=

dl

= 6,0 - 16;30

from

d2

= 6,0 - 18;20

from=::= 15 to ll1_15: decreasing 0;8 per degree

d3

= 6,0 - 16;30

from ll1_15 to )( 15:

d4 = 6,0 - 20;37,30 .

15 to=::= 15:

o=

-8 -12

from )( 15 to "(' 0: increasing 0;48 per degree

298

MERcuRY, SYsTEM

A 2 , LoNGITUDEs oF

r;

DATEs

B-B(il.)

0

-10

Fig. 57a

A graphical representation is given in Fig. 57a. For the actual computation of B(T) the following table can be used in which A = B(.Q).

). =B (.0.) "'" 0 ;:! ~

B(r)

a 'T' IS

'I" 0 + I;Z."i ~ "Y' l l + O;S' (:\- "'"IS) 0 19 + 0; 48 ( ~- 0 IS)

'T' IS ;:l A ~ 0 IS

0 8

IS ;:i IS ~

A a{}

A

IS

A-

~ .e:: IS -Is~ A~ "t 1s "l IS :< :\ ~ )( IS )( IS ~ A ~ )( 30

~7

g

+ O;SlP-"""" IS)

A-

IZ.

)( J 1-

'• ~so- >c 1s)

Example:

~T and ~A was assumed whereas in 61., ~'~and in )( the synodic times ~ T exceed ~A consistently by about 6'" to 9'".

Reappearances Exactly as the longitudes of E and rare obtained from the longitudes A of .E and .Q respectively by means of pushes which depend on A alone, in the same way the dates T(E) and T(r) were obtained from T(E) and T(Q) by the addition of amounts which are determined by A. The omission of fractions in all preserved dates again causes difficulties for the restoration of the exact rules. Nevertheless, in the case of T(T) the following table

No. 300a obv. VI,9: B( .Q)

::\=B(!l)

= A = )( 25

..,.. IS :ii

- )( 15 10 . 1;48

=

18

+ )(

obv. VIII, 9:

3 B(T) = )( 21

t11s

A

DATES

Disappearances The longitudes A of E and .Q being known, one might expect to be able to find the time differences ~T between two consecutive phenomena of the same kind by means of (1)

~T = ~A

+ 3;30,39'"

where ~A is the corresponding synodic arc. Though the preserved examples for dates in System A 2 are all without fractions it is nevertheless clear that ( 1) has not been used since there occur deviations of ± 3'" for T(.E) and between about - 3'" and + 6'" for T(.Q). The preserved material is not large enough to detect the principle of computation. Only in the case of T( Q) does it seem as if in some signs ( ~, t, ~)equality of

i:l

IS

9 1s

9 IS ~ ::>t ~ -'= IS *'-IS~ i\ a "'iS 1s ~ A ~ o 3o

nz

"""0 ~

A

>( IS~ ~

v.

~

~A ~

T(r) -T(.Q)

~

)(

IS

::0 'Y' IS

38 -r - 0; 6 (A

-

'Y' IS )

Js'"-

0;13(A- lJ IS)

zz t"

0; lb

ll,.

-

(A -

:£b.

t-4 1" - o; 3.12... ( -1-

IS)

"l u,-)

=

lOr+ 0; 12. ( i\ 0) l'i r- + 0; U (A- )( IS)

(cf. also Fig. 57b, on p. 299) reproduces the data found in the text so closely that one can be sure that it is essentially correct.

Example: No. 300a obv. VI,S:

A = t 17

rrp5 - 32 . 0;3,12 = -

1;42,24

14 12;17,36 T(.Q)

=

VIII

T(T) = IX

26

- --

8;17,36

text: IX 8

For T(E) we can only tentatively restore the following rules:

MERCURY, SYSTEM

A 2,

DATEs; DAILY MonoN

299

T(r}-T(fl)

40""

30

20

10

Fig. 57b

if n 10 ;;;;::

=

>. ;;;;:: 61. 30 2QT

+

then T(E) - T(.E) 0;27(>. - n tor

and if

~

0 ;;;;::

=

>. ; ; : t 14 then T(E) - T(.E) 56T - 0;15(>. -

~

or

Furthermore, from f 14 to perhaps 'Y' 10 we have exactly T(E) - T(.E)

=

3QT

as period of invisibility, where, however, the limit 'Y' 10 is very insecure. From there on to n 10 we know only that a decrease from 30r to 2W must take place.

D. Daily Motion No. 310 is the only text which gives positions of Mercury from day to day (not tithis!), undoubtedly for

actual lunar months. The text, though incomplete, covers a complete cycle of characteristic phenomena and so far as it can be judged, with essentially correct differences for the transformations 8 --+ Q, r--+}; for the corresponding part of the zodiac. The positions between the characteristic points are interpolated by means of a complicated scheme of first and second differences; cf. for details the commentary to No. 310 (p. 326). It is clear that the very same idea can be used for all planets to obtain from ephemerides for single phenomena new ephemerides for the continuous movement of the planet. In order to show the results for a longer section of the movement of Mercury, I have combined the elements for the years S.E. 145, 146 and 147 in No. 301 14 with the method in No. 310. The result is shown in the graph on Fig. 57c. 14 The errors mentioned in the critical apparatus and in the commentary to No. 301 were corrected for the present purpose.

r; {/

X

= /

..

:

(/

, 61

_;; .:/

6l I

Fig. 57c

§ 3.

300

VENUS

§ 3. THEORY OF VENUS Our knowledge of the theory of Venus is particularly fragmentary because of the small number and bad preservation of procedure texts as well as ephemerides dealing with this planet. On the other hand, a text like No. 812 leaves no doubt that a refined theory existed for Venus. Section 27 (p. 400 ff.) tells us, for example, that the evening appearance ( 8) and the next stationary point ('P) are 4, 19; 15° distant from each other if 8 occurs in Virgo; furthermore that last visibility ( Q) occurs after a retrograde motion of 6; 15°; finally, corrections for Q and reappearance as morning star (T) are given which depend on the zodiac and thus take into account the variability of the invisibility of Venus at inferior conjunction as function of its longitude. And many more details would be known if only one text like No. 812 had been fully preserved.

A. The Main Parameters From No. 400 we know the value of the mean synodic arc: (1)

~,\ =

3,35;30°

p

5 ~,\ = 17,57;30 '=" - 2;30°

mod. 6,0° ,

i.e., the longitudes decrease by 2;30° after 5 lines. Similarly we obtain from (4):

(6)

5 ~T =

=

49,25;5QT

=

(49,30 - 4;10r

1,39m -4;1QT = 8·12m +3m -4;1QT

Hence the date in line n + 5 is less by 4; 1QT than the date in line n. The month-name in line n + 5 is the same as in line n if this interval contains three intercalatory months but increases by one if only two intercalatory names are included - supposing, of course, that no change is needed because of the subtraction of 4;10 7 • The corresponding year-number is in general greater by 8 than the year number 5 lines before. 15

No. 420 obv.jrev. IV:

= ~ = 12•0 = 1;40,13, ... 3,35;30

(5)

Example:

which corresponds to a period {2a)

For the computation of an ephemeris of Venus it is of a great advantage to make use of the fact that five synodic periods are almost exactly the equivalent of eight sidereal rotations and therefore of eight years. More specifically, from (1) we obtain

obv. 25:

3,39

III 7 (intercalations: 3,41* 3,44*)

rev. -2:

3,47

IV 3 (intercalations: 3,47** 3,49* 3,52*)

rev. 3:

3,55

7,11

and (2b)

II= 12,0

z=

7,11

In other words, because of (6b) p. 283 (2c)

720 occurrences in 1151 years.

The actual mean distance travelled by Venus (or by the sun) between two consecutive occurrences of the same phenomenon is given by (3)

~s

=

~,\

+ 6,0

=

9,35;30°

For the synodic interval expressed in tithis the text uses (4)

~T = ~,\

+ 6,17;40 7 = 9,53;107

,

This relation would be obtained from the general relation (13b) on p. 286 by using the epact E = 11;3,4, ... This value is hardly significant because (4) is most likely only a rounded-off relation. Indeed, if one replaces 6,17;40 by 6,17;41,28,40 then one obtains € = 11;47 •

III

29

B. System A0 The simplest possible scheme for an ephemeris consists in the use of the mean synodic arc itself for the whole ecliptic. This procedure is attested in No. 400 for the first appearance of Venus as evening star. Comparison with the more refined systems shows that the use of mean motion can result in deviations of about 1oa and more in longitude and about the same amount in time. These inaccuracies are largely avoided if one proceeds with constant differences in steps of 5 synodic 15 Kugler (SSB II p. 203) formulates these rules inaccurately in' the form that the dates decrease by 4 days after 8 years.

§ 3.

301

VENUS

intervals which, as we have seen (p. 300), correspond to 8 years. This method occurs in the unfortunately very fragmentary text No. 401. The dates decrease by 4;5 days, a close approximation of the equivalent of 4;10 tithis required by formula (6) p. 300. It is perhaps not fully just to call these procedures, which are based on mean values only, a "system" of ephemerides. It would be possible, for instance, to interpret texts like Nos. 400 or 401 as having been computed for checking purposes.

C. Systems A1 and A2 From the fact that 5 synodic periods reduce longitudes by 2;30° and dates by 4;10'" (mod. 30) it follows that if we are given positions and dates in 5 consecutive lines all subsequent lines are known. The utilization of this property is common to both systems "A1 " and "A 2 ". The initial group of 5 lines, however, can no longer be obtained by subtraction of one constant difference from the first line. The difference between the two systems consists, first, in the arrangement within a group of 5 lines, and secondly, in the use of a decrease of

longitude after 5 lines by 2;30° in System A1, by 2;40° in System A 2• The procedure of System A1 which results in a decrease of 2;30° after 5 lines or 8 years agrees with System A 0 and formula (5) which in turn is equivalent with the fundamental period relation (2) p. 300. I see no reason for replacing the value 2;30° by 2;40° as is the case for System A 2 and it is still more surprising to see some columns of the same ephemeris (No. 420) computed with System A1 , some with System A 2 • On the other hand both systems assume a change in dates by -47 after 5 lines as compared with the theoretical value -4;10'" according to (6). For the computation of an ephemeris two types of rules should be known: (a) concerning the relation between a group of 5 consecutive longitudes and dates for a given phenomenon and (b) concerning the transition from one phenomenon to the next. Both questions can only be answered incompletely. The rules concerning consecutive positions can only be abstracted from fragmentary ephemerides and the procedure text colophon No. 821b written at the end of No. 420. The tables given here and on the following page, indicate

Sy.stem A,

-,1),

6l

)(

:!!:

b )(

.1-r-

61

)(

:e: 0 )(

'¥

3, ~-1, to•= 7 5 + + 3,30;30"' + 3, 3l "' 3,36)0 = + + 3,31,30 =

14; 10• 0;30 2 6,20

4,30

o•

10,0... =20"' + - 37 9,Z3 = IS 9 4S = 10, II :: + II + 7 10, 7 =

-

lj

3, 37;30°= 7 5 + 7, 3o• + 8,30 3,38;30 =

9

Y!f===

lJ

10,1• = 9,S1 =

l!f,... .1'

9,-11

-

,6),

961

!f=!:["l.]

.1' [o]

::=: )(

'r ~ [)(]

.11'"

BOl

~-"-[1711 }'[til

9.51 =

=>t

3,39;so·"' 7 5 + ?;.:.-o• + 11. 10 ' + 0;"10

3Jl.')~30

=.

3, 40;20 =

-

9,52 =

9.S9

= SS=

=

~;:so

+ 10;20

9,-i8' =20m-

9,S2

.1>.

1;30

+

-

-

-

,. I

,..,

.1r

II

9

2:.

3, 'ti;IO = ~ J I 30."10 I

9,

X

0;30

+ 13,30

zom

9,46 =

r 1' !.l [Jl]

-

3,28,30 = 3,43',30 =

= g

-

.3,29,30 =

.J'

-

/2r 8 I

- s - 8

)( 'l'

if 1(

Oll]f de

g

"ll'

b~

)(

...,..,

/J J( 9

Oll]p .... "1 J' ox::::

3', -BC=:>

~'

o~~+-~~~~~+-~~~--~+-4-~~~~

J
and::::::=, and a maximum of 12;10 in 915. These numbers probably give the retrogradations from to 1[1, The title of this text is"[...] the [... ] of Mars [...]". Fragments of the same text may be among Nos. 1013 to 1021.16

e

e

No. 805

u 180(10) Contents: Procedure text for Jupiter Arrangement: ? (only one side preserved) Provenance: U ruk [Warka photo] Photo: Pl. 248; Copy: Pl. 225 Section 1 Transcription

Beginning destroyed 1[...............] . 1,.,5,33[....... . 2[.•............ 1]5 zib 38,2 [....... . s[......... ta p]a en zib a-na .. [..... . 4[..•.••.......]43,30 lal [......... . Critical Apparatus

1: reading very uncertain. The first sign probably 1 or 4. Between zero and 30 a sign, perhaps 5 or 1. The 30 is followed by three vertical wedges; thus the last number is 33 or 36. 2 1]5: traces only. Commentary

Line 2 contains the number 38,2 which is known as the maximum of ~B for Jupiter, System B (Introduction p. 311 ). We also know that the apsidalline for the synodic movement of Jupiter passes through TIJ! 15 and }( 15 (cf. No. 812 Section 1), with the maximum in }( 15. In line 3 we may restore"[ ... from] t to}( .. ".The sign t, being 90° distant from }(, represents the region of mean synodic movement. The last line means either " ...]43,30 maximum" or" ... ]43,30 subtract". I see in the theory of Jupiter no parameter which might be meant here. Section 2 Transcription 5 [ ••• 6 [ •••

en] 9 gfr-tab iebertu(tur-tu) ta 9 [gfr-tab ..... . ta] 17 mul en 9 kusu qablitu(murub4-t[u) .... .

7[ • • • • • •]

qablitu(murub 4-tu) ip-pal-ka qablitu(murub 4-tu) [.....•

2 m]as dib .,4 tab rabitu(gal-tu) ip-pa[l-ka .... 9 [!d al-l]a 17 mul-mul dib 3,45l[al ..... . 10[..•.......... .]-ma tab ...... [ ....... . 11[...................] ... [............. .

8 [ •••

16

Cj. p. 448 ff.

No. 805

376

AND

No. 810,

SECTIONS

1

AND

2

Critical Apparatus

5 en] 9: traces of 9 only. 7 ip-pal-ka: reading of pal uncertain. 8 ip-pa[l-ka: restored from the parallel passage in line 7. 10

]-ma: perhaps sum]-ma.

11

] ... [: traces of two vertical wedges. Translation and Commentary

This section indicates the division of the ecliptic into arcs of different synodic motion, characteristic for Jupiter System A' (cf. Introduction p. 308) and the coefficients for the transition from one arc to the next. The following is a tentative restoration of lines 5 to 9: 9 Q15 to] 9 TTt, slow (arc). From 9 [TTl to 2 ~.medium (arc). From 2 ~to 17 ~ ,] (arc). From] 17 ~ to 9 QJ5, medium (arc). [(On the) slow (arc) add 30. (When) 9 TTl is passed over,] 7[(multiply by) 0;7,30 (and) add;] you will get the medium (arc). (On the) medium (arc) [add 33;45.] 8 [(When) 2] ~ is passed over, (multiply by) 0;4 (and) add; you will get the fast (arc). [(On the) fast (arc) add 36.] 9 [(When)] 17 ~ is passed over, (multiply by) 0;3,45 (and) sub[tract; you will get the fast (arc).] 5[From 6 [fast

The use of ip-pal-ka to indicate the result of an operation was found by Dr. Sachs. It also occurs in No. 812 Section 6 (below p. 394) in similar context.

§ 2. PROCEDURE TEXTS FROM BABYLON No. 810 BM 33869 ( = Rm.IV,431)

Contents: Procedure text for Jupiter Arrangement: 0 /R Provenance: Babylon [BM] Previously published: Kugler SSB I, Pl. 16 Photo: Pl. 250

Sections 1 and 2 Transcription

Section 1 Obv. 1[mul-b]abbar ta 9 kusu en 9 gir-tab 30 tab sd al-la 9 gir-tab dirig a-ra 1,7,3[0 DU] 2 ta 9 gir-tab en 2 mas 33,45 tab sd al-la 2 mas dirig a-ra 1,4 DU 3ta 2 mas en 17 mul-mul 36 tab sd al-la 17 mul-mul dirig a-ra 56,15 DU 4ta 17 mul-mul en 9 kusu 33,4[5 tab s]d al-la 9 kusu dirig a-ra 53,20 DU Section 2 5ta 9 kusu en 9 gfr-tab

~eljerti(tur-ti)

ta 9 gfr-tab (en] 2 maS qablitu(murub 4-tu 4) ta mas en 17 mul-mul

rabitu(gal-tu 4 ) 6ta 17 mul-mul en 9 kusu qablitu(murub4-tu4)

No. 810, SECTIONS 1 TO 3

377

Translation and Commentary Section 1 Obv. 1Jupiter. From 9 !Ei to 9 111._ add 30. The amount in excess of 9 111._ multiply by 1,7,30. 2 From 9 111._ to 2 ~ add 33,45. The amount in excess of 2 ~ multiply by 1,4. 3 From 2 ~ to 17 {l add 36. The amount in excess of 17 tl multiply by 56,15. 4 From 17 {l to 9 !Ei add 33,45. The amount in excess of 9 m5 multiply by 53,20. Section 2 5 From 9 !Ei to 9111._: slow (arc). From 9 111._ to 2 ~:medium (arc). From 2 ~to 17 Cl: fast (arc). 6 From 17 tl to 9 ®:medium (arc). These two sections give the fundamental parameters for System A' of the theory of Jupiter as described in the Introduction p. 308. The coefficients concerning the transition from one zone to the next are given here, however, in a different form. If a denotes the arc by which a preliminary position falls beyond a jumping point, then (3) p. 309 gives the coefficients c by which a must be multiplied in order to obtain the final arc a + ca beyond the jumping point. The present text does not give the coefficients c but the values of 1 + c = c' which can be expressed in the form (cf. p. 308)

w' w w'

1;7,30

w

=

0;56,15

w

w'

w W'

1;4 0;53,20 .

The arcs c' · a then give the arcs beyond the jumping points.

Section 3

Transcription Obv. 7ina feberti(tur-ti) ki samas ta 9 kusu en 9 gfr-tab sa me 12,[30 z]i ar i[gi] 30 me 8sa me 12,30 zi 3 itu-mes sa me 6,40 zi-ma US [4 itu-me)s sa me [4,10 zi-ma gur-ma us] 93 itu-mes sa me 6,23,20 zi 30 me ina panat(igi-at) su-su [sa m]e 12[,30 zi su)

Translation and Commentary In the slow (arc). With the sun, from 9 QD to 9 111._, per day 0;12,30 (is) the velocity. After (first) appearance (for) 30 days, per day 0;12[,30 (is) the velo)city. (For) 3 months, per day 0;6,40 (is) the velocity and the (first) stationary point (is reached). [(For) 4 months, per day 0;4,10 (is) the retrograde velocity, and the (second) stationary point (is reached).] (For) 3 months, per day 0;6,23,20 (is) the velocity. (For) 30 days before its disappearance, [per] day 0;12[,30 (is) the velocity, (and) disappearance (is reached)]. We operate here under the assumption that all phenomena of a complete synodic period fall into the slow arc from m5 9° to 111._ 9°. Then the daily velocities are given by the following scheme during invisibility and 30 days before and after: 0;12,30°/d 0;6,40 3 months before 1st stationary point ("!>): -0;4,10 4 months retrograde (from ri> to lJ'): 0;6,23,20 3 months after 2nd stationary point (lJ'): From the intervals where dates and velocities are given, we obtain a total direct motion of 23;45°. We furthermore know that in System A' the smallest synodic arc is assumed to be 30° (Section 1). Thus 6;15° remain to be covered during the period of invisibility where the velocity is given as 0;12,30°/d. Thus we obtain for the period of invisibility exactly 30 days (or tithis, if all "days" should actually be understood as tithis). A similar but not identical scheme is given in No. 813 Section 9 (cf. p. 407).

378

No. 810,

SECTIONS

4

AND

5

Section 4 Transcription Obv. 10ina qab[itu(murub 4 -tu,) ki samas ta 9 gfr-tab en 2 mas Stl me 14,[3,45 zi ar igi] 11 30 me sd me 14,3,45 zi 3 itu-mes sd me [7,30 zi-ma us 4 itu-mes sd me] 124,41,15 zi-ma gur-ma us 3 itu-mes sd me [7,11,15 zi 30 me ina ptintit(igi-at) su-su] 1 3fd me 14,3,45 zi su

Translation and Commentary In the medium tare). With the sun, from 9 Tll. to 2 ~ per day 0;14,[3,45 (is) the velocity. After (first) appearance] (for) 30 days, per day 0;14,3,45 (is) the velocity. (For) 3 months, per day [0;7,30 (is) the velocity and the (first) stationary point (is reached). (For) 4 months, per day] 0;4,41,15 (is) the retrograde velocity, and the (second) stationary point (is reached). (For) 3 months per day [0;7,11,15 (is) the velocity. (For) 30 days before its disappearance,] per day 0;14,3,45 (is) the velocity, (and) disappearance (is reached). On the medium arc, all velocities are increased in the ratio w':w else follows exactly the pattern of Section 3.

= 1;7,30 (cf. above Section 1, p. 377). Everything

Section 5 Transcription Rev. 1ina rabitu(gal-tu,) ki samtiSta 2 mas en 17 mul sd me 16,5[2,30 zi dr igi 30 me] 2sd me 9 zi 3 itu-mes sd me 5,37,30 zi-ma us [4 itu-mes sd me 8,37,30] 3zi-ma gur-ma us 3 itu-mes sd me 16,52,30 z[i 3]0 m[e ina ptintit(igi-at) su-su] 4sd me 16,52,30 zi su Critical Apparatus 2 9: sic, instead of 16,52,30. 2 5,37,30: sic, instead of 9. 2

8,37,30]: 5,37,30 would be correct.

3

16,52,30: sic, instead of 8,37,30.

Translation and Commentary Several numbers in this section are incorrect. The error can be simply explained as a cyclic permutation of all numbers involved, moving them one place ahead. In the following translation, the numbers are placed in their correct place. In the fast (arc). With the sun, from 2 ~to 17 ~,per day 0;16,5[2,30 (is) the velocity. After (first) appearance (for) 30 days,] per day 0;16,52,30 17 (is) the velocity. (For) 3 months, per day 0;9 18 (is) the velocity and the (first) stationary point (is reached). [(For) 4] mo[nths, per day 0;5,37,3019] (is) the retrograde velocity, and the (second) stationary point (is reached). (For) 3 month(s), per day 0;8,37,30 20 (is) the velocity. (For) [30] days [before its disappearance,] per day 0;16,52,30 (is) the velocity, (and) disappearance (is reached). All velocities in this section can be obtained from the corresponding velocities in the previous section by multiplication by 1;12

= ~ = :.

This shows that all numbers are wrong because 6: 5 is the ratio of the largest to the

smallest synodic arc and not of the largest to the medium synodic arc. Consequently the factor 1;12 should not 11

1s

Text: 9. Text: 5,37,30.

18

10

This is the corrected value. The text had most likely 8,37 ,30. Text: 16,52,30.

No. 810, SECTIONS 5 AND 6; No. 811, SECTION 1

379

have been applied to the velocities of the preceding section, which concerns the medium arc, but to Section 3. Thus we obtain the following list of velocities: during invisibility and 30 days before and after: Text: 0;16,52,30 3 months before 1st stationary point: 0;9 4 months retrograde: -0;5,37,30 3 months after 2nd stationary point: 0;8,37,30

correct: 0;15°/d 0;8 -0;5 0;7,40.

The time intervals are, of course, always the same because the synodic arc changes in the same ratio as the partial velocities.

Section 6

Transcription Rev.

ina qablitu(murub4-tu4) ki samaSta 17 mul en 9 kusu sa me 14,3,45 (zi) ar igi 30 me Sd me 14,[3,]45 zi 3 itu-mes sa me 7,30 zi 4 itu-mes sa me 74,41,15 zi-ma gur-ma us 3 itu-mes sd me 7,11,15 zi 8Sd 1[0,46,5]2,30 zi 30 me ina panat(igi-at) su-su sd me 14,3,45 [zi su] 5

6

Critical Apparatus 5 (zi): zi omitted by the text. 6 zi 4: sic, not zi-ma us 4 as in the preceding sections. On the remaining space of this tablet many traces of erased signs and numbers are visible.

Commentary This section concerns the second medium arc and is therefore identical with Section 4, with the exception of the proper change of zodiacal signs and degrees in the first line and a short addition at the beginning of line 8: "which (is) 1[0;46,5]2,30 (of) motion". As Kugler SSB I p. 144 remarked, this number gives the total progress during 3 months with the velocity 0;7,11,15 mentioned at the end of line 7.

No. 811 BM 33801 ( = Rm.IV,361) Contents: Procedure text for Jupiter, Saturn, and Mars Arrangement: 0 /R Provenance: Babylon [BM]

Colophon: Zs (p. 24) Photo: Pl. 250

Section 1

Transcription Obv. 1 [ mul-]babbar ta kusu en rfn 30 ta rfn en gir[ -tab 32] 2ina pa 34 t[a ma]s en bun [36] 3ina mul-mul 34,2[0 .... i]na mas-mas 32 [..... .] 4a-na 12 mu-mes 4 .. [...]33(?) ta mas-mas e[n pa ... ] 5ta mas en mul-mul 4,30 ki [....... .] 6 8 us ki su ana su lal ta ku[su] .... [...... .] 7ina pa 7,40 ina mas 9,30 ina gu [............ .] 8ina bun ina mul-mul U mas-mas 8,30 lal ina(?) .. [.....]

380

No. 811, SECTIONS 1 TO 3

Commentary The obverse of this text is very badly preserved and many readings and restorations are therefore uncertain. Lines 1, 2 and 3 describe a system of synodic arcs for Jupiter, leading from the minimum 30° to the maximum 36° by introducing two additional steps of 32° and 34° respectively. System A knows only two arcs for 30° and 36° respectively (cf. Introduction p. 307), System A' uses intermediate arcs of 33;45° (cf. Introduction p. 308). The distribution of synodic arcs in the present text is as follows: from

Qli

m

Tit

t

In

to~

30° [32] 34

from 1") to in tj

cr

[36] 34 ...

m II

32

The corresponding period is 10;58,22,56, ... , thus slightly greater than the period 10;51,40 in Systems A and A'. This result remains practically unchanged even if one had to restore fractions for the velocity in Tit or tj . What follows is almost completely unintelligible. Line 4 mentions the 12-year sidereal period of Jupiter. The subsequent numbers might indicate the increase in longitude during one 12-year period, depending on the initial zodiacal sign. Similar corrections for approximate sidereal periods are given in Section 2 for Saturn and in Section 3 for Mars. Corrections for different periods of Jupiter are known from No. 813 Section 1 (below p. 404). Nevertheless the details remain obscure. One correction seems to hold for the interval from II to t (line 4); another, of 4;30°, from ~ to II (line 5). Line 6 has a close parallel in rev. 2 and in No. 813 Section 28 (rev. III,13; below p. 417 f.). It seems to concern "longitudes (from) Q to Q". Lines 7 and 8 give the following values:

..........

in

t

In

1") -9;30

in =

-7;40

in cr (?) in tj II -8;30

[....]

perhaps for the period of 71 years (cf. No. 813 obv. I,6). In line 8 one may perhaps restore ina [7,]7 m[u-mes ki ana ki ikaS'sad] for the accurate period.

Section 2 Transcription Obv. 9genna ta a en gu ina 29 mu-mes [............] 105 tab ta zib-me en kusu 29 m[u-mes .........] Rest of obverse destroyed.

Critical Apparatus 9 genna: only a faint trace of the final vertical wedge of this sign is visible but the reading tur makes no sense.

Translation and Commentary "Saturn. From & to= during 29 years [for longitudes] add 5. From }( to Qli, 29 ye[ars for longitudes add 6]". The two arcs mentioned agree well enough with the arcs from & 10° to}( oo and from}( oo to & 10° in System A of Saturn, to show that we are dealing here with the same system. If one computes the increase in longitude during 29 years one obtains 5;12,30° if both beginning and end are located in the slow arc, and 6;15° for the fast arc.

Section 3 Transcription Rev., beginning destroyed. 1gir pa 10 mas [g]u [1]5 l[al zib-me bun 11,15] 2mul-mul ma8-mas 7,30 ki ana su lal ina [1,19 mu-mes]

No. 811, SECTION 3; No. 811a

381

kusu a 1,40 tab absin 0 rfn 2,15 tab gfr p[a 3,20] gu 5 zib-me bun 3,40 ina mul-mul [mas-mas 2,30] 5itti(ki)-su tab ina 4,44 mu-me ki[-su ikassad]

3

4mas

Critical Apparatus

1 [1]5: only lower part of 5 visible. 2

< su>: cf. the parallel in obv. 6 [Sachs]. Commentary

In line 5 appears the period of 4,44 years which is characteristic for Mars. Thus one can restore "in 4,44 years [it will reach its (original) place]" for the exact sidereal period. The preceding numbers give corrections which hold for smaller and less accurate periods. Two such periods of 47 and 79 years respectively were discovered by Kugler (SSB I p. 45). If we furthermore assume tha.t these corrections should be proportional to the synodic arcs in the respective zones we would obtain: [in 47 years: in

!Ei

61.

- so]

[n:li

~

- 6;40]

ll1_

t

V')=

in [1,19 years: in]

-10 [ -1]5

[)( or

-11;15]

l:5 II

- 7;30

!Ei

61.

n:v~

ll1_

t

+ 1;40° + 2;15 [+ 3;20]

}( cy>

+ 3;40

l:5 [II]

+[2;30]

m

(accurate: 2;13,20)

(accurate: 3;45)

From the basic relation for Mars 4,44 years = 2,13 occurrences = 2,31 rotations = 15,6,0° there follows 3P 5P

=

=

+ -1

22;10

=

22

syn. arcs

in 47 years

36;56,40

=

37 - - syn. arcs 18

in 79 years.

6

1

The corresponding corrections show exactly the ratio of -3 to 1 which we also find in the above list. This confirms our restorations. An explicit check for the correctness of the rules for the 47-year period can be made in the ephemerides Nos. 501 and 502 by comparing two positions 47 years apart, where one must only watch that both endpoints belong to the same interval. The first and the last line of No. 501 are exactly 79 years apart and lead from l:5 17;30 to l:5 20 as required by the above scheme.

No. 811a

BM 34676 ( = Sp.II,163 Contents: Procedure text for Mars Arrangement: 0 /R Provenance: Babylon [Sp.] Photo: Pl. 246; Copy: Pinches No. 138

+ Sp.II,304 + Sp.II,396 + Sp.II,491 + Sp.II,895)

No. 81la,

382

1

SECTIONS

TO

3

Sections 1 and 2 Transcription Obv. 21 Beginning destroyed.

Section 1 1• • • • • • • • • • • •

zib bu]n [6,3]0 [m]ul [m)as 7 gur an-n[a-a ...]

Section 2 2 [kusu a ..... absin rfn ..... gir-tab pa ...).,3[0] mas gu 10,45 zib bun 9,30 mul mas 10 an-na-[a . ..] Critical Apparatus Obv. 1 6,30: lower part visible. 2 ..] .,3[0]: the traces suggest ] 1,30 or ]4,30.

Commentary We find here lists of coefficients, valid for certain zodiacal signs which are arranged in pairs, characteristic for Mars, System A. Section 1 gives the retrogradations from ([) to e, following Scheme T (cf. Introduction p. 305). All that is preserved is the end: m )( and 'Y' m

~

and IT

6;30 7

retrograde

Section 2 does not mention retrogradation explicitly, but it seems plausible to expect the values for the remaining retrograde arc from (/) to lJf. From experience with Scheme S we know that the stretch from (/) to 1Jf should be the stretch from t1J to

~

e. The numbers given do not agree too well with this expectation: Text:

t

[..]30

In

Tll_ and

In

~ and=

10;45

and 'Y' and IT

9;30

In )( In

~

expected: 9;45

10

9 9;45 10;30

Section 3 Transcription 3 • • • • • • • • • • • • • • • • • • ] DUL-DU zi-ma bi-rit igi anti igi GIS-ma 23,37,52 4 • • • • • • • • • • • • • • • • • m]a u 4-mes qibi(e-bi) ta su anti igi ki bi-rit igi anti igi GIS-ma 5 • • • • • • • qibi(e-bi) t)a igi anti us ki bi-rit igi anti igi 1,53,13,13 itu-mes 6 • • • • • • • qibi( e-b]i) taus ma!Jritu(igi-tu) anti su ki bi-rit igi anti igi 3,56,44,35 Critical Apparatus DUL-DU: reading of DU doubtful.

Translation and Commentary subtract(?) (and?) the velocity, and(?) the distance (from one) appearance to the (next) appearance you compute(?) and 23;37,52 4• • • • • • • • • • • ] and predict the dates. From setting to rising: ... the distance between (one) appearance and the (next) appearance you compute(?) and 3• • • • • • • • • • • ]

21 Perhaps this side is the reverse of the tablet, the surface being slightly convex, as Mr. Wiseman kindly informs me.

No. 81la,

SECTIONS

3

TO

4a

383

s..... and predict the dates.] From rising to the (first) stationary point: ... the distance between (one) appearance and the (next) appearance (you compute and) 1,53;13,13 days (text: months) s..... and pre]dict [the dates.] From the first stationary point to setting: ... the distance between (one) appearance and the (next) appearance (you compute and) 3,56;44,35. The above translation is far from certain. Nevertheless the main trend seems to be clear. The number 23;37,52 or rather the number

(1)

c

=

12,23;37,521"

is known (cf. Introduction p. 306) to be the amount which must be added to the mean synodtc arc ll.\ in order to obtain the mean synodic time

(2)

LlT

=

ll.\

+

C •

A statement of this type must be contained in lines 3 and 4. Lines 4 to 6 describe the division of a complete synodic period into three sections: from Q to from to (/)' from (/) to Q such that

r,

r

(3) ~ r, i = 2 for r ~ ([), i = 3 for([)~ Q. The reason for going directly from the first stationary point(/) to Q is obviously the desire to avoid a negative ll.\ while allllT 's are necessarily positive. Instead of (1) and (2) we may also write

i = 1 for Q

(6,0 + ll.\) + c' c' = 6,23;37,52,. .

LlT

(3a)

=

This modification is motivated by the fact that Mars travels between two consecutive phenomena of the same kind not ll.\ degrees but 6,0 ll.\ degrees (cf. Introduction p. 302). Thus we see that

+

LlA

=

6,0

+ ll.\ =

6,48;43,18,30°

and c' = 6,23;37,52,.

are at the basis of the procedure of our text. In the second half of the present section, c' is divided into three components: c1 , c 2 , c 3 , which correspond to the segments Q ~ ~ ~ Q, respectively. Only the values of c2 and Ca are preserved.

r, r

(/), (/)

c 2 = 1,53;13,13,.

(4a)

C3

but c1 can be restored as c1

=

c' - (c 2

+c

= 3,56;44,35,.

3 ):

c1 = 33;40,4,. .

(4b)

This is confirmed by Sections 4 and 7.

Sections 4 and 4a Transcription

Section 4 7• • • • • • • • •

GAM 1,]50,40 DU-ma pap-pap gar-ma 33,40,4 sa pap-pap igi itti(ki)-su tab-ma qibi(e-bi)

Section 4a -]ma pap-pap tasakkan(gar-an) GAM 1,50,40 DU-ma ki pap-pap tab-ma ki u 4-mu tab-m[a ......]

8• • • • • • • • • • • • •

sa

igi

No. 81la,

384

SECTIONS

4

TO

6

Critical Apparatus 7sd

pap-pap: reading of pap-pap doubtful. Translation and Commentary

Section 4 you] multiply [by 0;1,]50,40 and put down the total; and 33;40,4 which you see(?) (as) the total you add to(?) it and predict (the date(?) ).

7•••••••••

Section 4a and put down the total. You multiply (it) by 0;1,50,40 and add (the result) to the total and add(?) it to(?) the day of the appearance a[ nd .....] These two Sections and Section 5 contain the rules for finding the dates of rand of