Advertisement

Babylonian Astronomy

  • Otto Neugebauer
Chapter
  • 875 Downloads
Part of the Studies in the History of Mathematics and Physical Sciences book series (HISTORY, volume 1)

Abstract

Delambre’s “Histoire de l’astronomie moderne,” published in 1821, begins as follows: Les recherches les plus exactes et les plus scrupuleuses n’ont pu jusqu’ici nous faire découvrir d’autre Astronomie que celle des Grecs. Partout nous retrouvons les idées d’Hipparque et de Ptolémée; leur Astronomie est celle des Arabes, des Persans, des Tartares, des Indiens, des Chinois, et celle des Européens jusqu’à Copernic.

Keywords

Lunar Eclipse Vernal Equinox Lunar Theory Nodal Zone Synodic Period 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    This is to be understood in a general sense. There exist texts which are copies of earlier tablets and probably all texts we do have are final copies of rough drafts. Nevertheless there will hardly ever arise problems as with mediaeval manuscript tradition. For biographies cf. Deimel [ 1920 ], Baumgartner [1894], and Schaumberger in Kugler-Schaumberger [1933], p. 97-100, respectively.Google Scholar
  2. 2.
    West of the Rhine, near Koblenz.Google Scholar
  3. 3.
    Epping-Strassmaier [1881]; cf. also Epping [1890].Google Scholar
  4. 4.
    Now part of ACT No. 122 (rev. X/XI lines 2 to 14 from SH 81-7-6, 277 in Fig. 1, p. 1315 ). Cf. also PI. IV.Google Scholar
  5. 5.
    Sp. 129 (=BM 34033); later published in Epping AB, pl. 1-3.Google Scholar
  6. 6.
    Epping-Strassmaier [1881], p. 285. The same could be said, 50 years later, about Babylonian mathematics.Google Scholar
  7. 7.
    Epping-Strassmaier [1881], p. 291 f.Google Scholar
  8. 8.
    Pl. II shows a page from Strassmaier’s notebooks, Pl. III gives one of his copies redrawn for the use of Epping and with later notes by Kugler; cf. now ACT No. 5 (Vol. I, p. 90 and photo Vol. III, pl. 255). The copies in the notebooks contain remarks in an oldfashioned shorthand (“Gabelsberger” which I had fortunately learnt in school) explaining the condition of the tablet and occasionally other remarks, e.g. at Sp. II, 604 (Pl. III left) “ein dickes Fragment einer ganz dicken Tafel, so:…” and “für P. Epping kopiert 1/3 93.” Sp. II, 604 is now joined with Sp. II, 453 in ACT, No. 70; cf. Pinches’ copy in Pinches-Sachs, LBAT, No. 51, p. [13].Google Scholar
  9. 9.
    Pannekoek [1916], p. 689, rediscovered by van der Waerden [1941], p. 28 (note 10); neither one realized the relation to Indian astronomy. Cf. Neugebauer [1947], p. 146. Cf. below p.358.Google Scholar
  10. 10.
    Thureau-Dangin, TU.Google Scholar
  11. 11.
    Signature VAT, i.e. “Vorderasiatische Tontafel”.Google Scholar
  12. 12.
    Schnabel, Ber.Google Scholar
  13. 13.
    ACT, No. 122.Google Scholar
  14. 14.
    Epping [ 1890 ], “Tafel A” and Epping AB, “Tablet A” and “Tablet C” without realizing the join; Kugler BMR, p. 12/13.Google Scholar
  15. 15.
    Cf. Neugebauer [1936], [1937, 1 to 3].Google Scholar
  16. 16.
    Seleucid” should always mean Seleucid and Parthian (if necessary even Roman).Google Scholar
  17. 17.
    Cf. PI. II, p. 1449.Google Scholar
  18. 18.
    Now published in Pinches-Sachs LBAT (1955). Sachs [1948] and Pinches-Sachs LBAT.Google Scholar
  19. 2.
    Below II C 3.Google Scholar
  20. 3.
    Alm. III, (Heiberg, p. 254, 10-13). Cf. also above p. 74; p. 118.Google Scholar
  21. 4.
    Cf. above p.350.Google Scholar
  22. 5.
    Originally a separation sign, here transcribed by a period. Cf. Neugebauer [1941] and ACT, p.4, p. 511; Aaboe-Sachs [1966], p. 3; Neugebauer-Sachs [1967], p. 210/12.Google Scholar
  23. 6.
    Cf. for colophons ACT I, p. 11 to 24.Google Scholar
  24. 7.
    On the lower edge of ACT, No. 122, shown Pl. IV; cf. also the copy ACT III, Pl. 221.Google Scholar
  25. 8.
    The error was detected by A. Sachs; cf. ACT I, p. 5, note 14 and Sachs [1948], p. 272, note 3.Google Scholar
  26. 9.
    Cf. ACT I, p. 5.Google Scholar
  27. 10.
    Cf., e.g., Epping-Strassmaier [1881], p.283, note 1.Google Scholar
  28. 11.
    Exceptions are the Kuyundjik (=Nineveh) Collection (K), DT, and Rm. Concordances for the astronomical texts are given in Pinches-Sachs, LBAT, p. XXXIXff.Google Scholar
  29. 1.
    Also full moons can be used, as in India.Google Scholar
  30. 2.
    Cf. below III 1 and VIA 2, 1.Google Scholar
  31. 3.
    The resulting “calendaric mean synodic month” is 29;31,50° long; cf. below p. 548.Google Scholar
  32. 4.
    Weidner [ 1935 ], p. 28/29 has shown that in the middle of the 12th cent. B.C. the same Assyrian month could coincide with seven different Babylonian months. This implies a rapidly shifting Assyrian calendar with respect to the essentially stable Babylonian calendar.Google Scholar
  33. 5.
    Cf. Parker, Calendars.Google Scholar
  34. 1.
    Parker-Dubberstein, BC, p. 1 ff.Google Scholar
  35. 2.
    Kugler, Sternk. II, p.422ff. and van der Waerden AA, p. 112 on the basis of four VIZ intercalations in similar positions within the 25 years from B.C. 527 to 503. Nobody takes it as evidence, however, for a cycle when four VIZ agree in the 16 years from B.C. 614 to 599.Google Scholar
  36. 3.
    Cf. below p. 620.Google Scholar
  37. 4.
    Such are the sources of the tabulation in Parker-Dubberstein, BC, p. 6, on which the following discussion is based (U=VIZ, A=XII2).Google Scholar
  38. 5.
    Parker-Dubberstein, BC, p. 6, cycle 14 year III. One should realize, however, that references to early intercalation may be untrustworthy. Aaboe-Sachs [1969], p. 21, n. 16 observed that two texts, both concerned with eclipses, assign the year Xerxes 18 (-467/6) a VIZ and a XII2, respectively.Google Scholar
  39. 6.
    The years in question are —445/4 and —426/5; Parker-Dubberstein BC, p. 6, year XVII in cycles 16 and 17. Cf. also below p. 364.Google Scholar
  40. 7.
    Diodorus XII, 36; cf. below p. 622.Google Scholar
  41. 8.
    This problem will be of interest, of course, in a later context (cf. below p. 542).Google Scholar
  42. 9.
    Cf. above p. 354.Google Scholar
  43. 10.
    This pattern was first discovered by Kugler, Sternk. I, p.212 (1907); cf. also Sternk. II, p.425. If we denote Parker-Dubberstein BC, p. 6 cycle p, year q by (p: q) then we have with (5)Google Scholar
  44. 12.
    Below p. 365.Google Scholar
  45. 1.
    Neugebauer [1947, 1] and [1948]; now ACT No. 199 (and Pl. 136) from Istanbul U 107 and U 124.Google Scholar
  46. 2.
    Cf. below p. 542.Google Scholar
  47. 3.
    Cf. above p. 356, Table 1, years [0:16], [0:19], and [1:2], respectively.Google Scholar
  48. 4.
    Cf. p. 355 (3 a).Google Scholar
  49. 5.
    Using Parker-Dubberstein BC, p. 40 (based on modern computations of first visibility of the moon). b Tuckerman, Tables I. Our discussion is only intended to produce an estimate for the significance of our dates but not an evaluation of their accuracy.Google Scholar
  50. For the accurate meaning of this statement cf. the next section (p. 360f.).Google Scholar
  51. 8.
    Below p. 395f, p. 500, etc.Google Scholar
  52. 9.
    Cf., e.g., Sûryasiddhânta I, 13.Google Scholar
  53. 1.
    Of non-ACT type, e.g. a text for Mercury (cf. Neugebauer [1948, 1], p. 212f.).Google Scholar
  54. 2.
    This was already suggested by Epping, AB, p. 151.Google Scholar
  55. 3.
    The three cases where the vernal equinox falls into a month I require, of course, a year number one higher than listed.Google Scholar
  56. 4.
    This pattern is confirmed by the large number of texts which give such dates. It is easy to show that the simple rule of always adding 11 from line to line in each column of Table 3 implies that these columns are derived from the truncated scheme (I), p. 360 by adding 3m 3′ from column to column and not 3m2;45,47,30′ which would be the difference for exactly equidistant cardinal points (cf. Neugebauer [1948, 1], p. 214ff.).Google Scholar
  57. 5.
    Kugler, Sternk. 1I, p. 606f.Google Scholar
  58. 6.
    Cf. above p. 354f.Google Scholar
  59. 7.
    Cf. the tabulation in Neugebauer [1948, 1], p. 218.Google Scholar
  60. 8.
    Aaboe-Sachs [1966], p. 11f.Google Scholar
  61. 9.
    Cf. below II A 6, 1 B (p. 424).Google Scholar
  62. 1.
    Our present O corresponds to 0,, p.1091.Google Scholar
  63. 2.
    Sachs [1952, 1].Google Scholar
  64. 3.
    I have arbitrarily given the year numbers of cycleGoogle Scholar
  65. 4.
    The dates shown remain the same for any other cycle.Google Scholar
  66. 5.
    Cf. above p.Google Scholar
  67. 6.
    We ignore here some errors in a text which gives correct dates for Q and 0 but day numbers one too low for F during the years S.E. 62= [3:6] to S.E. 69 = [3:13]. There is also some trouble with S.E. 189** =[9:19] where one finds II 22 for Q instead of the expected II 23 (cf. Table 4, cycle years 0 and 19).Google Scholar
  68. 7.
    Cf. above p. 357.Google Scholar
  69. 1.
    Our Tables 3 and 4.Google Scholar
  70. 2.
    Cf. above p. 40.Google Scholar
  71. 3.
    Cf. below p. 561.Google Scholar
  72. 4.
    Cf. p. 37, (4a).Google Scholar
  73. 5.
    The actual values are supposedly M~ 14;20” and m: 10;0h, hence M: m: 1,26 instead of 1;30 (Schaumberger, Erg., p. 377).Google Scholar
  74. 6.
    Above p.38.Google Scholar
  75. 7.
    Ptolemy, Geogr. V, 20, §6 (ed. Nobbe, p. 78).Google Scholar
  76. 1.
    Above p. 40f.Google Scholar
  77. 2.
    These schemes must, of course, satisfy the symmetry relations pi=p12, etc. Cf. above p. 35.Google Scholar
  78. 3.
    Cf., e.g., above p. 30.Google Scholar
  79. 4.
    Cf. below p. 545.Google Scholar
  80. 5.
    Kugler, Sternk. I, p. 172. Similar results were obtained by van der Waerden [ 1952 ], p. 222.Google Scholar
  81. 6.
    Including the fragment of a catalogue of stars, discovered by Sachs [1952, 2].Google Scholar
  82. 7.
    Huber [1958].Google Scholar
  83. 8.
    ACT No. 200, Sect. 2 (p. 187); No. 200b, Sect. 2 (p. 214); both texts for System A. For System B no procedure text is preserved. The corresponding part of Table 5 was first reconstructed from the applications by Kugler, Mondr., p. 99. Its derivation from rising times was given by Neugebauer [1936, 2].Google Scholar
  84. 9.
    ACT No.9 obv. III and IV, 12.Google Scholar
  85. 10.
    ACT No. 122 rev. II and III, 10.Google Scholar
  86. 11.
    In both figures I have omitted symmetric branches in order to avoid overcrowding of the graphs.Google Scholar
  87. 12.
    Actually we should say “setting time” instead of rising time, but by replacing A by A+180 one can always transform one problem into the other.Google Scholar
  88. 13.
    Cf. below II B 10, 2; also Neugebauer [1953].Google Scholar
  89. 14.
    Cf. below p. 727 ff. and p. 938.Google Scholar
  90. 15.
    Varâhamihira, Brhajjataka I, 19.Google Scholar
  91. 1.
    Above p. 368.Google Scholar
  92. 2.
    Above p. 57f.Google Scholar
  93. 3.
    Above p. 368.Google Scholar
  94. 4.
    The fact that the “Uruk scheme” for solstices and equinoxes (above p. 361) divides the year into four equal seasons is not decisive since there we are dealing with a purely calendaric scheme. Similarly the 19-year cycle does not reflect the level of the contemporary luni-solar theory.Google Scholar
  95. Cf. Bernsen [1969], p. 27, Fig. 2. An equal fit is obtainable for column A in System B with the parameters given in ACT I, p. 70.Google Scholar
  96. 1.
    The restriction p> 1/2 is only made for the sake of simplicity. It is satisfied in the most important cases which actually occur.Google Scholar
  97. 2.
    The value of P is about 14 mean synodic months; cf. below p. 476.Google Scholar
  98. 3.
    Cf. below p. 384 and Fig. 15 there.Google Scholar
  99. 4.
    A] denotes the greatest integer contained in A. Cf. above p. 373.Google Scholar
  100. 6.
    We make here use of the fact that we are dealing with strictly arithmetical patterns which contain no approximations or roundings.Google Scholar
  101. 7.
    Cf. (4a), p.374.Google Scholar
  102. 8.
    For other examples cf. below p. 392 fï:Google Scholar
  103. 9.
    ACT, p. 86 ff.Google Scholar
  104. 10.
    For examples cf. below p. 392 (2).Google Scholar
  105. 1.
    Table 6 is a transliteration of ACT No. 702 but omitting all details concerning readings and restorations for which see ACT II, p. 357f. and III, Pl. 207 and Pl. 249 (photo).Google Scholar
  106. 2.
    The year 151•* in obv. 11 is the year [7:19] in the 19-year cycle (cf. above p. 356).Google Scholar
  107. 3.
    Cf. ACT I, p. 20, colophon Z.Google Scholar
  108. 4.
    Cf., e.g., obv. V, 12 and rev. V, 7 with 311° and 18°, respectively.Google Scholar
  109. 8.
    Obviously ya+1=1/2d on a downgoing branch.Google Scholar
  110. 9.
    For the details of this part of the procedures cf. below p. 388.Google Scholar
  111. 10.
    Cf. below p. 397.Google Scholar
  112. 11.
    I have chosen ([1954]) this notation because it is short, adapted to mathematical symbols (e.g. 2(r)), and independent of the writer’s language. Purposely I did not follow Schoch (Ammiz., p. 103) who called Q and F of Venus “e last” and “m first”, respectively. Van der Waerden took up Schoch’s notation and speaks, e.g., about Mk= Morgenkehrpunkt for O. He also calls ([1957]) the Greek-letter phenomena “Kardinalpunkte” and replaces the commonly used term ephemeris by “Kardinaltafel” (probably because ephemeris literally should mean day by day positions).Google Scholar
  113. 12.
    Together they are represented as the Navagrahas (i.e. the Nine Demons). Cf., e.g., the sculptures shown in Sivaramamurti [ 1950 ], Pl. VIII C and IX A.Google Scholar
  114. 13.
    This also holds for ©; cf. p. 363.Google Scholar
  115. 14.
    Cf. II Intr.3.Google Scholar
  116. 15.
    Cf. II Intr. 3, 3.Google Scholar
  117. 16.
    Cf. II Intr. 4, 2.Google Scholar
  118. 17.
    Cf. the “Gates” of primitive ( Palestinian) lunar theory: Neugebauer [ 1964 ], p. 51 - 58.Google Scholar
  119. 18.
    Above p. 386.Google Scholar
  120. 19.
    Below p. 391.Google Scholar
  121. 20.
    Even at the great Islamic observatories of the latest period observational programs were not extended much beyond the shortest planetary periods, e.g., in Marâgha, in the 13th century, periods of 30 and 12 years, respectively; cf. Sayili, Observ. p. 204 and p. 276.Google Scholar
  122. 21.
    The “Era Nabonassar” is nowhere attested in cuneiform sources and is in all probability the invention of Greek astronomers (Hipparchus?) for purely astronomical purposes — much like the “julian days” in modern astronomy.Google Scholar
  123. 1.
    Cf. above p. 170 (1) and Fig. 158.Google Scholar
  124. 2.
    We shall return to the details below p.442.Google Scholar
  125. 3.
    Extant in “Normal Star Almanacs”; cf. below p. 555.Google Scholar
  126. 4.
    We shall see that these smaller periods need not to be exactly of the same character; the 71-year period, e.g., is used in the “Goal year texts” as restoring synodic phenomena, while the 83-year period is more accurate for sidereal returns; cf. below p. 554f.Google Scholar
  127. 5.
    Cf. also ACT II, p. 283.Google Scholar
  128. 6.
    Cf. p.151.Google Scholar
  129. 7.
    Boll [1898].Google Scholar
  130. 8.
    CCAG 1, p. 163, 19.Google Scholar
  131. 9.
    Cf. Tannery, Mém. Sci. 4, p. 265.Google Scholar
  132. 1.
    Above p. 375ff.Google Scholar
  133. 2.
    ACT 600; cf. for details ACT II, p. 339 and III, Pl. 176. The transliteration in Table 7 follows the same principle as in Table 6 for which cf. above p.380, note 1.Google Scholar
  134. 3.
    ACT, p. 17 colophon L.Google Scholar
  135. 4.
    Cf. p. 376 (3 b).Google Scholar
  136. 5.
    Note the “linear saw functions” formed by ω′, r, and A(0). Their period is found to be P0 = 7;12 = 36/5.Google Scholar
  137. 6.
    Above p. 391 (1 1).Google Scholar
  138. 1.
    Cf. above p. 389.Google Scholar
  139. 2.
    Above p. 380 (1a).Google Scholar
  140. 3.
    Above p. 381(1 b) and p. 390f. (l0a) and (11).Google Scholar
  141. 4.
    Again p. 380 (I a).Google Scholar
  142. 5.
    Cf. p. 389 (4).Google Scholar
  143. 6.
    Cf. note 3.Google Scholar
  144. 7.
    Above p. 378 (15 b).Google Scholar
  145. 8.
    Above p. 394 (6).Google Scholar
  146. 9.
    We always assume, of course, a mean velocity for the sun; the solar anomaly plays no role in the planetary theory.Google Scholar
  147. 10.
    Above p. 380 (l a), for M—µ.Google Scholar
  148. 11.
    Cf. below (p. 446) Jupiter System B with M = 38;2 µ = 33;8,45 for diGoogle Scholar
  149. 12.
    Differences and amplitudes are the same.Google Scholar
  150. 13.
    The mean value of these two differences is the expected d r - Al′ =11;27,20,37,30′ (cf. above p.395).Google Scholar
  151. 14.
    ACT 704, e.g., has a dr that follows AA at a distance of 0;1,42,30 of an interval.Google Scholar
  152. 1.
    For Mars and Venus this should be understood as AA+6,0°.Google Scholar
  153. 2.
    Cf. p. 365.Google Scholar
  154. 3.
    Cf.IC6andIC8.Google Scholar
  155. 1.
    The last two are now almost completely broken away.Google Scholar
  156. 2.
    Cf. above p. 391 (12) and note 4 there.Google Scholar
  157. 3.
    ACT No. 606, 600, 604, and 601, respectively.Google Scholar
  158. 4.
    Published by Aaboe-Sachs [1966], Text D.Google Scholar
  159. 5.
    Cf. p. 390 (10 a).Google Scholar
  160. 6.
    Cf. p. 392 (3). Restoration by Aaboe-Sachs [1966], p. 16. If the text began one line later it would be F at I.Google Scholar
  161. 7.
    This was discovered by P. Huber [1957], p. 277.Google Scholar
  162. 8.
    For the reconstruction of this pattern cf. ACT II, p. 312.Google Scholar
  163. 9.
    We shall also find it for Mars; cf. below p. 401 (7).Google Scholar
  164. 10.
    Ana ME-a Ma kur, or similar (Sachs).Google Scholar
  165. 1.
    Above p. 390 (10a).Google Scholar
  166. 2.
    Above p. 389 (4) and (6).Google Scholar
  167. 3.
    Cf. below IIA7,4B.Google Scholar
  168. 4.
    For the corresponding synodic arcs as function of A cf. below Fig. 40, p.1332.Google Scholar
  169. 5.
    Important details for this part of the theory of Mars will be discussed below p. 406ff.Google Scholar
  170. 6.
    For the other methods cf. below II A 7, 4 C and Fig. 41, p. 1332. Cf. p. 1320, Fig. 18.Google Scholar
  171. 1.
    Cf. ACT No. 800 and Sachs [1948], p. 283, respectively.Google Scholar
  172. 2.
    Cf. above p. 22, notes 7 to 9.Google Scholar
  173. 3.
    Including some variants for which cf. below p. 472.Google Scholar
  174. 4.
    Except for the scale the latter curves are the same as in Fig. 23, p.1322.Google Scholar
  175. 5.
    Fig. 25 is repeated from Fig. 238, p.1288 but changed to the scale of the present Fig. 24.Google Scholar
  176. Pliny, NH II, 77 (ed. Budé, Vol. 2, p. 33/34 and p. 165).Google Scholar
  177. 7.
    For the details cf. Neugebauer [1951].Google Scholar
  178. 8.
    Above p. 241 and p. 255.Google Scholar
  179. 9.
    Cf. Fig. 258, p. 1296.Google Scholar
  180. 1.
    Cf. for this whole episode Neugebauer [1951].Google Scholar
  181. 2.
    From Parker-Dubberstein, BC.Google Scholar
  182. 3.
    From Parker-Dubberstein: 147d and 121°, respectively.Google Scholar
  183. 4.
    Pinches-Sachs, LBAT No. 1019 (obv. 2, obv. 4, rev. 4′) [Sachs]. 5 Sachs [1948], p. 287.Google Scholar
  184. 1.
    Cf. above p. 400.Google Scholar
  185. 2.
    Cf. above p. 395 (3).Google Scholar
  186. 3.
    This is astronomically evident since the sun is in A and B in the same relative position to the planet; but it can also be seen from the preceding formulae that 1-70=6,0/(6,0+e)=(6,0+/1 1)/4r.Google Scholar
  187. 4.
    Cf. above p. 399 (1).Google Scholar
  188. 5.
    Cf., e. g., above p. 396 (5 b).Google Scholar
  189. 6.
    Cf. below p. 409 (16).Google Scholar
  190. 8.
    Cf. below IIA 7, 4.Google Scholar
  191. 9.
    Van der Waerden [1957], p. 52; also Anf. d. Astr., p. 190.Google Scholar
  192. 10.
    Scribal error in the text: omission of the final 5.Google Scholar
  193. 11.
    E.g. in Vitruvius, Archit. IX, 1, il; Pliny NH II, XII 59 (Jan-Mayhoff, p. 145); Paulus Alex., Apot. 15 (Boer, p. 31 f.); “Heliodorus”, Comm. 12 (Boer, p.19 f.) etc.; still in Copernicus, Comment. (Rosen TCT, p. 78; Swerdlow [ 1973 ], p. 480 ).Google Scholar
  194. 12.
    Cf. below p. 424.Google Scholar
  195. 1.
    Above p. 405.Google Scholar
  196. 2.
    Cf. also the excellent discussion of these methods by P. Huber [1957].Google Scholar
  197. 3.
    Cf. Neugebauer, MKT III, p. 83 s.v. Reihen; Neugebauer-Sachs, MCT, p. 100.Google Scholar
  198. 4.
    Cf. ACT I, p. 14f.Google Scholar
  199. 5.
    Cf. Neugebauer, MKT I, p. 103; first explained by Waschow [ 1932 ], p. 302f.Google Scholar
  200. 6.
    Cf., e.g., the relations between ß and G in the lunar System A (below II B 3, 2 B, p. 485).Google Scholar
  201. 7.
    Cf., e.g., column J in System B, obtained by summation of the linear zigzag function H (below II B 3, 5 B, p. 493).Google Scholar
  202. 1.
    Cf. ACT I, p. 2.Google Scholar
  203. 2.
    To facilitate the reading I have given all numbers with their complete sexagesimal order. The text omits all initial zeros as well as signs.Google Scholar
  204. 3.
    Using the Tuckerman Tables.Google Scholar
  205. 4.
    Cf. above p. 369.Google Scholar
  206. 5.
    Cf., e.g., above p. 398 (1) and p. 399 (4).Google Scholar
  207. 1.
    This date was suggested by Huber [1957], p. 276. The text is published in ACT No. 310.Google Scholar
  208. 2.
    The restorations are quite secure.Google Scholar
  209. 3.
    For details cf. ACT II, p. 326f.Google Scholar
  210. 4.
    The modern data are taken from the Tuckerman Tables. The curve of the text is not corrected for the difference of the zero points (cf. above p. 369), otherwise F would come near to the level of point 10. Using the same notation as with Jupiter.Google Scholar
  211. 5.
    Above p. 416.Google Scholar
  212. 6.
    Cf. above p.405; also ACT, p. 312.Google Scholar
  213. 1.
    Cf., e.g., above p. 391 (12).Google Scholar
  214. 2.
    Cf. above p. 389 (4) and (6).Google Scholar
  215. 3.
    Cf. above p. 382.Google Scholar
  216. 4.
    The slow rotation of all apsidal lines escaped notice in early astronomy.Google Scholar
  217. 1.
    Cf. below p. 423, Table 9.Google Scholar
  218. 2.
    Cf. below II A 7, 5.Google Scholar
  219. 3.
    Mercury has 2,40/1,36=1 +2/3; cf. Table 9.Google Scholar
  220. 4.
    This has been first observed by Aaboe [1965], p. 224.Google Scholar
  221. 5.
    Published by Aaboe-Sachs [1966], p. 9f. and p. 24f. (Texts G to J).Google Scholar
  222. 6.
    Cf. also above p. 409 (12) and (12a).Google Scholar
  223. 7.
    Cf. above p. 411 (22 b).Google Scholar
  224. 8.
    This distinction was established by Sachs (cf. Pinches-Sachs, LBAT, p. [XXV]).Google Scholar
  225. 9.
    Cf. above p. 391; also note 4 there.Google Scholar
  226. 10.
    Cf. also below II A 7, 4 A, p. 456.Google Scholar
  227. 11.
    ACT No. 811, Sect. 3 (p. 381).Google Scholar
  228. 1.
    We assume here and in the following that H is even (hence Z odd). This is convenient for our formulations but factually irrelevant. Nor is it essential for the following that the diameter in question be sidereally fixed.Google Scholar
  229. 2.
    They agree exactly with the parameters found in the ephemerides; cf. ACT II, p. 310f.Google Scholar
  230. 3.
    Cf. below IV D 1,1.Google Scholar
  231. 4.
    Repeated attempts to establish accurate dates for the origin of the two systems require too many explicit and implicit assumptions to be taken seriously.Google Scholar
  232. 1.
    Then under the directorship of Harald Bohr.Google Scholar
  233. 2.
    Neugebauer, Vorl. (1934).Google Scholar
  234. 3.
    Published as “Mathematische Keilschrift Texte” in QS A 3 in three volumes (1935, 1937).Google Scholar
  235. 4.
    Thureau-Dangin, TU (1922).Google Scholar
  236. 5.
    Schnabel, Ber. (1923), a rather chaotic publication.Google Scholar
  237. 6.
    Cf. above p. 374.Google Scholar
  238. 7.
    Cf. above p. 348.Google Scholar
  239. 8.
    The majority of Pinches’ copies are now available in Pinches-Sachs LBAT (1955).Google Scholar
  240. 9.
    Neugebauer [1936, 1].Google Scholar
  241. 10.
    London 1955. The omission of the date on the title page was overlooked by all concerned.Google Scholar
  242. 11.
    Van der Waerden [1957], p. 47.Google Scholar
  243. 12.
    Cf. above p. 427; van der Waerden. Anf. d. Astr., p. 187ff.Google Scholar
  244. 13.
    Cf. Aaboe [1965] and Aaboe-Sachs [1966].Google Scholar
  245. 14.
    Above p.427f.Google Scholar
  246. 15.
    Newcomb [1897].Google Scholar
  247. 16.
    It may be mentioned, however, that in Newcomb’s discussion also a linear diophantine equation stands in the center of the problem.Google Scholar
  248. 1.
    quote from the commentary to the edition of the Panca-Siddhântikâ by Neugebauer-Pingree, Vol. II.Google Scholar
  249. 2.
    We need not to discuss here the question as to whether this influence is due to hellenistic intermediaries or to Iranian contacts; cf. Pingree [1963, 1].Google Scholar
  250. 3.
    Pc: Sk. I, 15.Google Scholar
  251. 4.
    Pc.-Sk. II, 8 and XII, 5.Google Scholar
  252. 5.
    Cf. below p. 481 (Pc.-Sk. II, 2-6; III, 4-9); also V A 2,1 D 2.Google Scholar
  253. 6.
    Cf. below V A 2, 1.Google Scholar
  254. 7.
    Cf. Neugebauer-Pingree, Pc.-Sk. II, p. 109.Google Scholar
  255. 8.
    Pc: Sk. I1, p. 108; Chap. XVII, 58 (p. 125) gives a different sequence. The order chosen here in (1) is numerical; in this part of the text the order is the ordinary Indian one, i.e. the order of the weekdays.Google Scholar
  256. 9.
    Pc.-Sk. II, p. 115.Google Scholar
  257. 10.
    Cf. above p.420and p. 390f., as compared with Pc.-Sk. II, p. 112, Table 24.Google Scholar
  258. 11.
    Pc.-Sk. XVII, 64-80 (Vol. II, p. 126, Table 33).Google Scholar
  259. 12.
    Concerning the subsequent rules for obtaining the true from the mean positions cf the discussion in Pc.-Sk. II, p. 115ff. and p. 126ff.Google Scholar
  260. 13.
    The following is based on a suggestion made by A. Sachs.Google Scholar
  261. 14.
    Listed above p. 391 (11).Google Scholar
  262. 15.
    Pc.-Sk. Il, p. 126, Table 33 (where 53,12 for Mercury is a misprint for 55.12). For Venus 575= 215+360.Google Scholar
  263. 1.
    Cf. above p. 390 (l0a) and (11).Google Scholar
  264. 2.
    Van der Waerden (AA, p. 110 BA, p. 111) discusses a 589-year period which is, however, only the result of an incorrect restoration by Kugler of a broken passage; cf. Neugebauer-Sachs [1967], p. 206, n. 32.Google Scholar
  265. 1.
    Aaboe-Sachs [1966], p. 3f., Texts A and B.Google Scholar
  266. 2.
    Aaboe-Sachs [1966], p. 13 obv. column I, lines 1 to 5.Google Scholar
  267. 3.
    ACT Nos. 704 and 704a.Google Scholar
  268. 9.
    Cf. below p. 469.Google Scholar
  269. 1.
    ACT Nos. 700 to 709 and Nos. 801 and 802. For an example cf. above p. 381, Table 6 (ACT No. 702).Google Scholar
  270. 2.
    Cf. above p. 378.Google Scholar
  271. 3.
    Aaboe-Sachs [1966], p. 4, Table 3; there it was remarked that the procedure texts ACT Nos. 801 and 802 erroneously describe a motion as retrograde which is actually the direct motion 4′—Q. This correction was overlooked by van der Waerden (AA, p. 185, BA, p. 263) and amplified by an error of his own. arbitrarily increasing the numbers for the velocities in (11 b) for (1) O by 0; 10.Google Scholar
  272. 4.
    ACT Nos. 801 and 802; cf. the preceding note. Pc.-Sk. XVII, 19-20 (Neugebauer-Pingree II, p. 118). 6 Cf. below p. 791, Table 1.Google Scholar
  273. 5.
    Cf. below p. 964.Google Scholar
  274. 1.
    Cf. the diagram ACT II, p. XII.Google Scholar
  275. 2.
    Cf. below p. 446. For a comparison of the results with the actual facts cf. Aaboe [ 1958 ], p. 242 - 245.Google Scholar
  276. 3.
    Above p. 391 (12).Google Scholar
  277. 4.
    ACT No. 812, Sect. 10 (p. 395 f.) and No. 813, Sect. 20 (p. 414 ).Google Scholar
  278. 5.
    Kugler SSB I, p. 48, a passage which is part of Sachs LBAT No. 1593 (rev. 12ff.).Google Scholar
  279. 6.
    ACT No. 813, Sect. 20 gives the meaningless number 2,46,40 (instead of 2,58); the parallel passage in No. 814 is damaged. An emendation to 2,46 makes no sense because 2,46 =2 83 is not a new period.Google Scholar
  280. 7.
    That is to say: the sidereal longitude of a given phase will be /* +a after 12 years and 1* +ß after 71 years.Google Scholar
  281. 8.
    ACT No. 813, Sect. 1 and No. 814, Sect. 1; No. 812, Sect. 10 and No. 813, Sect. 20.Google Scholar
  282. 9.
    Cf. for details below II A 7, 3 A.Google Scholar
  283. 10.
    In our case 5 for the 12-year period, 6 for the 71-year period, among 391.Google Scholar
  284. 11.
    Starting with year 24 of Darius I, i.e.-497.Google Scholar
  285. 12.
    Cf. below p. 447.Google Scholar
  286. 13.
    Cf. the “checking rules” ACT, p. 307 and p. 309.Google Scholar
  287. 14.
    ACT No. 812, Sect. 2 and No. 813, Sect. 13.Google Scholar
  288. 15.
    Above p.439 and p. 395.Google Scholar
  289. 16.
    Above p. 395 (3).Google Scholar
  290. 17.
    Cf. below p. 446.Google Scholar
  291. 18.
    ACT No. 812, Sect. 2.Google Scholar
  292. 19.
    In the procedure text ACT No. 813, Sect. 1 (p. 403/404) the value of d is abbreviated to 6,42.Google Scholar
  293. 20.
    Cf. below p. 446.Google Scholar
  294. 1.
    ACT Nos. 600 to 608 and Nos. 609 to 614.Google Scholar
  295. 2.
    ACT No. 814, Sect. 2 (p. 424) seems to mention instead Op 12 and 1t 12.Google Scholar
  296. 3.
    Aaboe-Sachs [1966], p. 16-21.Google Scholar
  297. 4.
    Thus beginning at the boundary of the slow arc (cf. above (9)). The next line gives F at C91 and could also motivate the beginning and end for a number period.Google Scholar
  298. 5.
    Cf. above p.437f.Google Scholar
  299. 6.
    For an example cf. ACT, p. 308.Google Scholar
  300. 7.
    ACT No. 813, Sect. 14 to 16.Google Scholar
  301. 8.
    Cf. above p. 359.Google Scholar
  302. 9.
    Cf. ACT II, p. 310, No. 813, Sect. 7 and 8, No. 813 b, Sect. 3. Still another (six-zone) variant seems to be mentioned in the fragmentary Section 1 of No. 811.Google Scholar
  303. 10.
    Aaboe-Sachs [1966], p.8 and p.22f. (Text E).Google Scholar
  304. 11.
    Aaboe [1965], p. 223, p. 221.Google Scholar
  305. 12.
    Pc.-Sk XVII 9-11 (Neugebauer-Pingree II, p. 113).Google Scholar
  306. 13.
    Cf. above p. 438.Google Scholar
  307. 1.
    ACT No. 805, Sect. 1; No. 812, Sect 1; No. 813, Sect. 12, 21 and 22.Google Scholar
  308. 2.
    Cf. above p.444. Ptolemy found in the second century A.D. for Jupiter the apogee npl l; cf. above p. 179 (6).Google Scholar
  309. 1.
    Above p. 397.Google Scholar
  310. 2.
    Above p. 398 from Aaboe-Sachs [1966], Text D.Google Scholar
  311. 3.
    Above p. 398 (1).Google Scholar
  312. 4.
    Procedure text ACT No. 813, Sect. 1-2; also above p. 399 (4) and p. 405 (2) and ACT Nos. 610 and 611.Google Scholar
  313. 5.
    Below p.449.Google Scholar
  314. 6.
    ACT No. 813, Sect. 23 is probably only a garbled version of (2) for the slow arc. Also the Sect. 24 and 31 are marred by errors, Sect. 11 is incomplete.Google Scholar
  315. 7.
    Cf. above p. 398 (2).Google Scholar
  316. 8.
    The intervals on the fast arc are, of course, obtainable by using the factor 6/5. For the corresponding dates cf. below p.450.Google Scholar
  317. 9.
    Cf. above (3)Google Scholar
  318. 10.
    Cf. above p. 399 (3). An incorrectly computed ephemeris (No. 603) has only -5;55° and -7;10° for OGoogle Scholar
  319. 11.
    ACT No. 819b, Sect. 2.Google Scholar
  320. 12.
    Some insecure values of -4° -4°, -4° -5°, -3° -5;10° for Y-• O-7 are mentioned in ACT No. 813, Sect. 31, 24, 23, respectively. The fragmentary ephemeris No.612 seems to operate with -3;45° for b-• O on the slow arc.Google Scholar
  321. 13.
    Cf., e.g., above p. 405 (1).Google Scholar
  322. 14.
    Cf. above p. 443 (7 b).Google Scholar
  323. 16.
    ACT No. 813, Sect. 9, No. 810, Sect. 3, No. 812, Sect. 4, No. 818, Sect. 1 for the slow arc; No. 810, Sect. 4 and 6 for the medium arc. No. 810, Sect. 5 committed an error for the fast arc by using the ratio 6/5 with respect to the medium arc instead of the slow arc (cf. ACT, p. 378/9). Cf. also above p. 405 (3).Google Scholar
  324. 17.
    Cf., e.g., the position of the “apsidal line” (above p. 447).Google Scholar
  325. 18.
    Pc.-Sk. XVII 12-13, Neugebauer-Pingree II, p. 118, Table 26.Google Scholar
  326. 1.
    Jupiter, above II A 5, 3 A, Mercury II A 5, 3 B.Google Scholar
  327. 2.
    Huber [1957], p. 269-276. He combined ACT Nos. 652, 1015, 1016 in “Text A”, Nos. 650, 651, 653 in “Text B”, Nos. 1014, 1021, 1032 in “Text D”.Google Scholar
  328. 3.
    Cf. above p. 451 (18).Google Scholar
  329. 4.
    ACT No. 652, Obv. I. 10 to 15 = Huber, Text A, p. 292.Google Scholar
  330. 5.
    Cf. above p. 447 (2) and p. 450 (15).Google Scholar
  331. 6.
    Above p.448 (6) and p.450 (15).Google Scholar
  332. 7.
    Cf. the analogous situation for the retrograde arc. above p. 449 (9).Google Scholar
  333. 8.
    Cf., e.g., above II A 4 or II A 7, 3 C the proportionalitiy of (2), p. 447 and (18), p. 451.Google Scholar
  334. 9.
    Huber [1957], p. 274 says that the procedure texts give velocities in degrees per day, not in degrees per tithi. In the same paragraph, however, he declares that “months” in these texts are schematically reckoned as 30 days. I do not see how these two statements can be reconciled.Google Scholar
  335. 10.
    Huber [1957], p. 279-291, p. 298-303. Cf. above II A 5, 3 A.Google Scholar
  336. 1.
    Cf. above p. 399.Google Scholar
  337. 2.
    Making use of p. 422 (6).Google Scholar
  338. 3.
    This convenient formulation was first given by van der Waerden [1957], p. 47.Google Scholar
  339. 4.
    Cf above p. 179 (6): 625;30.Google Scholar
  340. 5.
    Above p. 426; also Sachs [1948], p. 283, Table IV and Kugler SSB I, p. 44.Google Scholar
  341. 6.
    ACT No. 501.Google Scholar
  342. 7.
    Cf. above p.442.Google Scholar
  343. 8.
    Cf. below p. 458.Google Scholar
  344. 9.
    Pc.-Sk. II, p. 112, Table 24.Google Scholar
  345. 10.
    The determination of the entry of a planet into consecutive signs (in direct or retrograde motion) is not unknown from Babylonian sources. The “Almanacs” and some of the “Normal Star Almanacs” regularly predict these dates; cf. Sachs [1948], pp. 277-282, p. 287. We also have a list of dates of entry for Mars during the year Philip Arrhidaeus 5 (-318/317), published Neugebauer-Sachs [1969], p. 94 Text H.Google Scholar
  346. 11.
    Van der Waerden [1972], p. 77-87 and BA, p. 316-320.Google Scholar
  347. 12.
    Cf. below p. 785, p. 788.Google Scholar
  348. 13.
    Above p. 408.Google Scholar
  349. 1.
    Cf. below p.1451, right upper corner on Pl. V, second row, second fragment from the right; published in ACT No. 510.Google Scholar
  350. 2.
    Centaurus 5 (1958), p. 246; van der Waerden BA, p. 274.Google Scholar
  351. 3.
    Cf. above p. 391 (11) or p. 407 (5) and p. 455 (2).Google Scholar
  352. 1.
    Above p. 400.Google Scholar
  353. 2.
    Above p. 425 (6).Google Scholar
  354. 3.
    Above p. 409 (12).Google Scholar
  355. 4.
    Above p. 411 (22).Google Scholar
  356. 5.
    Cf. above p. 411. n. 11 and below IV D 3, 4 or V A 3, 2.Google Scholar
  357. 6.
    Pc.-Sk. II, p. 127, Table 34.Google Scholar
  358. 7.
    Cf e.g., below Fig. 18 (p. 1320).Google Scholar
  359. 8.
    Described in detail above p. 401 and Fig. 20 (p. 1320).Google Scholar
  360. 9.
    Pc.-Sk. II, p. 120, Table 29.Google Scholar
  361. 10.
    Cf. above p.399 and Fig. 18, p. 1320.Google Scholar
  362. 11.
    Cf., e.g., Aaboe-Sachs [1966], p. 10, Table 9. 2 Cf. above p. 196; p. 193.Google Scholar
  363. 13.
    Above p. 459.Google Scholar
  364. 14.
    Pc.-Sk. II, p. 119 and p. 127.Google Scholar
  365. 15.
    Pc.-Sk. II, p. 120, Tables 27 to 29.Google Scholar
  366. 1.
    For the smallness of the eccentricity of the orbit of Venus cf. below p. 1443, Fig. 34.Google Scholar
  367. 2.
    This is also the goal-year period for Venus; cf. below p. 554 (1).Google Scholar
  368. 3.
    In contrast, e.g., to Jupiter, above p. 442. I think I was mistaken to seek evidence for approximate periods in the fragment ACT No. 815.Google Scholar
  369. 4.
    Neugebauer-Sachs [1967], p. 207.Google Scholar
  370. 5.
    The method is described in II A 4.Google Scholar
  371. 6.
    Cf., e.g., below p.462 (from ACT No. 410).Google Scholar
  372. 7.
    Cf. the tabulation in ACT, p. 301/302 and No. 821 b, p. 441 f.Google Scholar
  373. 8.
    We have only fragments of three ephemerides computed with System A,: ACT Nos. 410, 412, and 430 (for the restoration of No. 430 cf. van der Waerden [1957], p. 59).Google Scholar
  374. 9.
    This is the above mentioned text ACT No. 420+821 b (cf. p.462).Google Scholar
  375. 10.
    It seems to me not only pointless but seriously misleading to readers who are not in a position to control the primary sources to make such utterly fragmentary material the basis of far reaching historical conclusions and to formulate them as if they were well established results (van der Waerden [1957], p. 60 and again AA, p. 199, BA, p. 278).Google Scholar
  376. 11.
    Cf. below II B 10,2.Google Scholar
  377. 12.
    For one exception cf. below II C 3, p. 554.Google Scholar
  378. 13.
    Halma III, p. 22-25 for E, F, Q. For a graphical representation of these tables cf. below Fig. 125 to 127 (p. 1421 ff.). We have no way of testing 45 and T.Google Scholar
  379. 14.
    The values Si and S3 are found by interpolation between clima III and IV; they are easy to estimate in our Fig. 125.Google Scholar
  380. 15.
    Cf. ACT, p. 301.Google Scholar
  381. 16.
    Tabulated in ACT, p. 399.Google Scholar
  382. 17.
    The remaining data are lost. It should be remarked that intervals between evening setting and morning rising by definition cannot amount to integer days. All dates of this type are therefore subject to arbitrary interpretations, whether 1/2 day is included or excluded.Google Scholar
  383. 18.
    Alm. XIII, 8; cf. above I C 8, 3 A.Google Scholar
  384. 19.
    Cf. ACT, p. 399 for the time intervals, p. 397 for the corresponding arcs.Google Scholar
  385. 1.
    For some evidence for a reference to these points cf. below p. 471 and the Indian material (below p. 473). Cf. also above II A 5, 3 B for a possible contamination of Q and I with W and 0, respectively.Google Scholar
  386. 2.
    Cf. below p. 473.Google Scholar
  387. 3.
    Above p. 402.Google Scholar
  388. 4.
    Cf. below p.471.Google Scholar
  389. 5.
    Sachs [1948], p. 283, also ACT procedure text No. 800 (p. 363). It is probably only an accident that the periods of A, can be connected arithmetically with (3): At any rate such a relation has no astronomical meaning.Google Scholar
  390. 6.
    Cf. above p. 460 and p. 441.Google Scholar
  391. 7.
    Cf. Neugebauer-Sachs [1967], p. 206f.Google Scholar
  392. 8.
    Cf. below p. 899.Google Scholar
  393. 9.
    Cf. above II A 6.Google Scholar
  394. 10.
    The values for System A, are given in ACT, p. 290 f., for A2 on p. 296 f. The values of d are 129 for I′, 73 for â, 59 for E, 33 for Q.Google Scholar
  395. 11.
    Cf. below p. 470.Google Scholar
  396. 12.
    Cf. above p. 375.Google Scholar
  397. 1.
    Cf. the diagram ACT II, p. XII; similarly for the moon ACT I, p, XVI.Google Scholar
  398. 2.
    Cf. ACT, p. 317 f.Google Scholar
  399. 3.
    Cf. ACT, p. 319, B(S).Google Scholar
  400. 4.
    This text will be published by Aaboe-Henderson-Neugebauer-Sachs, restored from five fragments (BM 3665) The Mercury ephemeris is written on the reverse, at right angles to the writing on the obverse (which concerns lunar eclipses). Similar writing at right angles is found with the Venus ephemeris No. 430 and the Mars ephemeris No. 501 a, for a Mars ephemeris and solstices (AaboeSachs [1966], p. 11 f.), and for lunar theory and a list of unexplained numbers (Neugebauer-Sachs [1969], p. 96).Google Scholar
  401. 5.
    Cf. above p. 438.Google Scholar
  402. 6.
    This was discovered by A. Aaboe.Google Scholar
  403. 7.
    Instead of the first or third value in (14) or (17) one finds in the ephemeris decrements of 4;49,9,36,33,45 and 4;34,10 respectively.Google Scholar
  404. 8.
    Cf. above p.466.Google Scholar
  405. 9.
    This is confirmed by a comparison of the ephemeris with modern data, carried out by A. Sachs for the two historically possible periods covered by the ephemeris.Google Scholar
  406. 10.
    Cf. Fig. 23 p. 1322.Google Scholar
  407. 11.
    Cf. above II A 4.Google Scholar
  408. 12.
    Cf., e.g., ACT No. 300, col. I and II. For simplifying procedures cf. ACT, p. 293.Google Scholar
  409. 13.
    ACT No. 300a and 300b.Google Scholar
  410. 14.
    Cf. ACT, p. 298.Google Scholar
  411. 15.
    Cf. Fig. 21, p. 1321.Google Scholar
  412. 16.
    A copyist error made column II run from 43° up to 44;56° instead of down from 44° to 42;4°. The value 42=30° in =15° is suspect (34° would be better) but it is secured by the pattern of cols. IVVI-VIII and by the colophon (ACT No. 820a, p.438) of the ephemeris No. 301. For 0315°, No. 820a (obv. VI) has dr=24′ but No. 301 uses 25′ (cf. ACT, p. 321).Google Scholar
  413. 17.
    ACT Nos. 800c and d.Google Scholar
  414. 18.
    Mainly Nos. 301 and 302.Google Scholar
  415. 19.
    Cf. ACT, p. 293-295. In the case of II 15° there is some vacillation between dZ=46° and 45° (cf. ACT, p. 319). The value di=20′ at up 15° (from ACT No. 800d, col. 11/IV) is hardly correct since it is the only case for which dd> dt. The ephemeris No. 301 uses 30′ (ACT, p. 320) which is equally implausible; one expects 22′ or 23′.Google Scholar
  416. 20.
    Cf also the graph ACT, p. 299, Fig. 57c, derived from the ephemeris No.301 for the three years S.E. 145 to 147 (i.e. —166 to —163).Google Scholar
  417. 21.
    Cf. above p. 403.Google Scholar
  418. 22.
    Cf. ACT, p. 294.Google Scholar
  419. 23.
    Pc.-Sk II, p. 112, Table 24; cf. also above p. 466.Google Scholar
  420. 24.
    Pc.-Sk. II, p. 114.Google Scholar
  421. 25.
    Cf. Pc.-Sk. II, p. 122 and Fig. 67 on p. 123.Google Scholar
  422. 26.
    Compare Pc.-Sk. II, p. 124, Fig. 68 and ACT, p. 298f., Figs. 57a and 57 b.Google Scholar
  423. 27.
    Pc.-Sk. II, p. 128, Table 36.Google Scholar
  424. 28.
    Cf. above p.471.Google Scholar
  425. 1.
    Cf. below p. 516f.Google Scholar
  426. 2.
    Above p. 366 ff. and p. 371ff.Google Scholar
  427. 3.
    Below 11 B 8.Google Scholar
  428. 4.
    For details cf. ACT I, p. 43.Google Scholar
  429. 5.
    ACT No. 180ff.Google Scholar
  430. 6.
    Cf., e.g., the texts discussed in II A 5, 3 A and 3 B. This was suggested by Huber [1957], p. 276.Google Scholar
  431. This notation has its origin in the concept that the first day of the current month is the 30th day of the preceding one if the latter had been hollow; otherwise day 1 of month N follows day 30 of month N-1.Google Scholar
  432. 1.
    Cf. for these concepts above p. 375.Google Scholar
  433. 2.
    Cf. below II B 3, 1 and 2.Google Scholar
  434. 3.
    Cf. p. 475.Google Scholar
  435. 4.
    Below p.484. Later (p. 500) we shall see to what extent an exact relation F*.—.F can be reconstructed for System A.Google Scholar
  436. 5.
    ACT No. 120 rev. II gives F′, exactly parallel to F in the next column.Google Scholar
  437. 1.
    Cf. below lIB10.Google Scholar
  438. 2.
    Aaboe [1968], p. 30-34.Google Scholar
  439. 3.
    Cf. below p. 501 f.Google Scholar
  440. 4.
    Cf. also below p. 501; Neugebauer [1957, 1], p. 18f.Google Scholar
  441. 5.
  442. 1.
    ACT No. 194a, obv. II, 25.Google Scholar
  443. 2.
    Cf. below p. 492.Google Scholar
  444. 3.
    Above p. 478.Google Scholar
  445. 4.
    Manitius, p. 204 to 211. Cf. also below p. 602f.Google Scholar
  446. 5.
    Cf. Pc: Sk. I, p. 14. The Vasistha-Siddhânta is probably also the source of Pc.-Sk. XVII, 1-60 where one finds much undoubtedly Babylonian material for the theory of the planets (cf. above II A 7, 1 ).Google Scholar
  447. 6.
    Cf. below V A 2, I D 1.Google Scholar
  448. 7.
    Cf. below V A 2, 1 A.Google Scholar
  449. 8.
    For the details of Pc.-Sk. II, p. 18 f.Google Scholar
  450. 1.
    Cf. below p. 501 (9).Google Scholar
  451. 2.
    Above p.480.Google Scholar
  452. 1.
    Below p. 497 ff.Google Scholar
  453. The details will be described below in II B 3, 3 to 5.Google Scholar
  454. 1.
    Cf. above p. 478 (2).Google Scholar
  455. 2.
    Cf. below p. 492f.Google Scholar
  456. 3.
    Above p.69 (1).Google Scholar
  457. 4.
    E.g. with Birùni, with Maimonides, in the Hexapterygon, etc. As shown on p.480, about 363;5°.Google Scholar
  458. 6.
    Belonging to the interval from S.E. 179 to 210 (-132 to —101).Google Scholar
  459. 7.
    S.E. 235 (-76/75).Google Scholar
  460. 1.
    Below p. 498.Google Scholar
  461. 2.
    Above p.478 (5).Google Scholar
  462. 3.
    Since the number period of $ is 1,44,7m (cf. (2a)), i.e. more than 500 years, any date is uniquely determined within the historical limits of our material.Google Scholar
  463. 1.
    Cf. above p.478 (5).Google Scholar
  464. For a tabulation of the whole scheme cf. ACT, p. 60. In a subsequent section (Table 14, p. 509) we will find the same values of O(1) in column S, lines 18 to 27.Google Scholar
  465. 2.
    Cf. below p. 506.Google Scholar
  466. 3.
    ACT No. 204 rev. 9 (p. 249).Google Scholar
  467. 4.
    When I first collected the passages in which 2,13,20 is used like a noun (ACT, p. 212 and Neugebauer [1957], p. 18f.) I did not fully realize that “2,13,20” simply meant “0”. It was A. Sachs who finally clarified the situation which has a parallel in the use of “the 18” for the 18-year eclipse cycle (now called “Saros”).Google Scholar
  468. 5.
    Since 0 and G are not exactly in phase, all values of 0 near the maximum of G belong to the increasing branch of (cf. p. 1336, Fig. 50 b).Google Scholar
  469. 1.
    Cf. the table in ACT I, p. 60.Google Scholar
  470. 2.
    Below p.498f.Google Scholar
  471. 3.
    E.g. ACT No. 18, col. [— I] and V.Google Scholar
  472. 4.
    Cf. p.483.Google Scholar
  473. 5.
    Below p. 504.Google Scholar
  474. 1.
    Above p. 372.Google Scholar
  475. 2.
    Below p. 491.Google Scholar
  476. 2.
    This text has many more similar errors; cf. ACT I, p. 100.Google Scholar
  477. 1.
    Above p. 483 (3).Google Scholar
  478. 2.
    Above p. 483 (2).Google Scholar
  479. 3.
    Above p. 489.Google Scholar
  480. 4.
    Below p. 493.Google Scholar
  481. 5.
    Cf. below p.497.Google Scholar
  482. 1.
    Cf. above p.476.Google Scholar
  483. 2.
    The amplitudes of H and J in Fig. 53 are drawn in the correct ratio.Google Scholar
  484. 3.
    Cf. p. 374 (3). Cf. also Fig. 55, p.1339, inset.Google Scholar
  485. 4.
    Cf. above p.383.Google Scholar
  486. 5.
    It may be remarked that 11=0 actually occurs in our texts; cf. ACT No. 160 obv. I, 21=No. 161 obv. II, —3 (S.E. 124 I X ).Google Scholar
  487. 6.
    ACT No. 165 obv. III and IV, 12, 13 and 18 to 20.Google Scholar
  488. 7.
    Above p. 378 (15b).Google Scholar
  489. 8.
    Above p. 396 (5 b).Google Scholar
  490. 9.
    If one uses the rounded extrema (18) one finds instead of (19)Google Scholar
  491. 10.
    Cf. below p. 533 (2). We ignore here the period PA = 12;22,13,20 of the abbreviated column A. “ H and J are drawn in the same scale, except for the inset which shows the situation near Mi in greater detail.Google Scholar
  492. 11.
    The terminology found in the ephemerides is usually abridged to kur, nim, i;ú, du, respectively, cf. above p. 490 (I) and ACT, p. 80.Google Scholar
  493. 1.
    Section 12 in the now published text; cf. note 4.Google Scholar
  494. 2.
    This terminology is not of ancient origin; cf. Neugebauer [1937, 3], p. 241 to 245 and [1938, 1], p. 407 to 410 where I have shown that the interpretation of the word “Saros” as a name for an eclipse cycle originated first with Halley in 1691. The term became common among astronomers probably through Newcomb [1897], p. 7; to Schram this usage seemed to be new, or at least not well founded (Schram [1881], p. 182 note).Google Scholar
  495. 3.
    Cf. above p.486 (2).Google Scholar
  496. 4.
    Two larger pieces, BM 36705 and 36725, were joined by A. Sachs and form the basis of the publication Neugebauer [1957]. Several years later A. A.boe joined another small fragment, BM 37484, to the main parts; cf. Aaboe [ 1968 ], p. 35 - 38.Google Scholar
  497. 5.
    Above p. 487 (6).Google Scholar
  498. Due to van der Waerden, AA, p. 149.Google Scholar
  499. 1.
    Cf. above p.478.Google Scholar
  500. 2.
    O* is given to six sexagesimal places in contrast to the five digits of b. In order to compare the twoGoogle Scholar
  501. functions we norm the last digit of d* as minutes.Google Scholar
  502. 3.
    Cf. above p. 485.Google Scholar
  503. 4.
    The information for a “single value” must include, of course, the direction of the branch (increasing or decreasing) to which this value belongs.Google Scholar
  504. 5.
    Cf. ACT I, p. 119.Google Scholar
  505. 6.
    As far as can be seen there also seem to be errors obscuring this column. The correct value in No. 81 obv. VI, 6 would be 11,10,38,26,15 but not 11,12[.Google Scholar
  506. 7.
    Above p. 478 (4).Google Scholar
  507. 8.
    Above p.480; cf. also p.479.Google Scholar
  508. 9.
    Above p. 478 (5).Google Scholar
  509. 10.
    Sect. 2, line 15.Google Scholar
  510. 11.
    Below p. 506.Google Scholar
  511. 1.
    The following elegant derivation is due to Aaboe-Henderson [1975].Google Scholar
  512. 2.
    Cf. above p. 377f. (l0), (13) and (14).Google Scholar
  513. 3.
    Cf. for this concept above p. 433f.Google Scholar
  514. 4.
    This parameter is actually mentioned in the procedure text ACT No. 207, Sect. 7 (p. 250).Google Scholar
  515. 5.
    Above p.484 (2a).Google Scholar
  516. 6.
    Above p.488.Google Scholar
  517. 7.
    Cf. below 1I B 5, 3.Google Scholar
  518. 8.
    Cf. ACT No. 210, Sect. 3, line 8; the numbers are in part restored.Google Scholar
  519. 9.
    Cf. above p. 503 (9).Google Scholar
  520. 10.
    Cf. ACT No. 210, Sect. 3, line 18.Google Scholar
  521. 11.
    Cf., e.g., Ginzel, Hdb. I, p. 254.Google Scholar
  522. 12.
    Cf. below II B 7; also above p. 129.Google Scholar
  523. 13.
    This statement is found in a text to be published by Aaboe-Henderson-Neugebauer-Sachs [ 1975 ]. This relation was first suggested by van der Waerden (AA, p. 150, BA, p. 228); cf. also Aaboe [1968], p. 10.Google Scholar
  524. 2.
    Cf. above H B 3, 3.Google Scholar
  525. 3.
    Cf. Aaboe [1971], p. 10f., p. 11 f., p. 17, Fig. 3; [1969], p. 9, Fig. 1.Google Scholar
  526. 4.
    Aaboe [1968], p.28f.; refined [1969], p. 11.Google Scholar
  527. 5.
    An isolated case is an ephemeris for the years Philip Arrhidaeus 4 to 7 (-318 to —315) in which A and W follow the columns K and M. Cf. Aaboe [ 1969 ], p. 19.Google Scholar
  528. 6.
    ACT No. 55, restored by Aaboe [1971], p. 22. Similar texts are ACT Nos. 75 and 76. ACT No. 55 concerns S.E. 180 to 202 (_ —131 to —109).Google Scholar
  529. 7.
    It was van der Waerden who in AA, p. 152f. suggested that the computation of G should be based an a truncated function 0, but without giving any details. The actual existence of truncated functions F and 0 was established, and their working explained, by Aaboe [1968] (p. 8, Fig. 2 and p. 18, Fig. 4).Google Scholar
  530. 8.
    We need not specify the units of these intervals in the present context.Google Scholar
  531. 9.
    Cf. Fig. 50, p. 1336 and Fig. 61, p. 1342.Google Scholar
  532. 10.
    This is evident in the numerical values used for M′ and m′ of 1; cf. below p.508 (2). i Cf. above p.4781.Google Scholar
  533. 12.
    Above II B 4.Google Scholar
  534. 11.
    Cf. above p. 501 (12a) and (13a).Google Scholar
  535. 14.
    Cf. above p. 505 (2).Google Scholar
  536. 15.
    Cf. above p. 487 (6).Google Scholar
  537. 16.
    E.g. ACT, p. 60.Google Scholar
  538. 17.
    ACT No. 5 obv. VI, 15.Google Scholar
  539. 18.
    Neugebauer-Sachs [1969], p. 110. Cf. for the same text below p.548, p. 552f.Google Scholar
  540. 19.
    Cf. above p. 508 (4).Google Scholar
  541. 20.
    Aaboe [1971], p. 15f. and Table 5.Google Scholar
  542. 21.
    Cf. Aaboe [1969].Google Scholar
  543. 22.
    Cf. above p. 506 (4).Google Scholar
  544. 23.
    Cf. below p. 528 f.Google Scholar
  545. 24.
    Aaboe [1969], p. 12f.Google Scholar
  546. Here and in the following I reckon k as a positive quantity, i.e. I do not introduce signs for the directions of motions.Google Scholar
  547. 1.
    For the arguments on which this interpretation of the units of E is based cf. Neugebauer [ 1945 ]. Like all angular measurements these units also are originally measures of length, in this case such that 72 barley corns make tut=2/5 cubits (thus 1 cubit =180 te).Google Scholar
  548. 2.
    Hence parameters not expressly denoted as degrees are reckoned in §e.Google Scholar
  549. 3.
    Cf. above p. 372.Google Scholar
  550. 4.
    Cf. above p.514.Google Scholar
  551. 5.
    Assuming 29;31,50,8,20′ for the length of the mean synodic month one obtains —0;3,10,50°” as an estimate for the daily motion of nodes. The values (8a), (8 b), (8c) were first determined by van der Waerden by different arguments (AA, p. 145; BA, p. 216 to 220).Google Scholar
  552. 6.
    From ACT No. 6aa. Example 1 from Obv. 18/19 (correcting a scribal error in V, 18), Example 2 from Obv. 11/12.Google Scholar
  553. 7.
    In ACT No. 81 we have an ephemeris for the lunar latitude E* per tithi which has exactly the parameters required by our p, on the fast arc; cf. also ACT, p. 54f.Google Scholar
  554. 8.
    Cf. above p. 375 (6b).Google Scholar
  555. 9.
    For the numerical details of obtaining d and D from (11) cf. Neugebauer [1937, 3], p. 257f.Google Scholar
  556. 1.
    Cf. above p. 502 (2).Google Scholar
  557. 2.
    Cf. for, this term above p.486, n. 4.Google Scholar
  558. 3.
    Cf. above p. 503. (12).Google Scholar
  559. 4.
    Cf. below p. 523 (2c).Google Scholar
  560. 5.
    Cf. above p.502 (1).Google Scholar
  561. 6.
    Cf. Neugebauer [1937, 3], p.237f., or ACT, p.47 (1).Google Scholar
  562. 7.
    Cf. below p. 523 (2 a).Google Scholar
  563. 8.
    Heiberg I, p. 471, 32. For these tables in general cf. above I B 6, 1.Google Scholar
  564. 9.
    Cf. below p. 533 (1).Google Scholar
  565. 1.
    Cf. above p. 519 (10).Google Scholar
  566. 2.
    Cf. below 1I B 6.Google Scholar
  567. 3.
    Aaboe [1973] has studied this method in the wider frame of ancient astronomy.Google Scholar
  568. 4.
    The latitude function as drawn in Fig. 70 is, of course, arbitrary in the choice of the slope. S For the terminology cf. ACT, p. 123.Google Scholar
  569. 6.
    Cf. below p. 526f.Google Scholar
  570. 1.
    Cf., however, below p.551.Google Scholar
  571. It is possible that two consecutive values of E belong to this zone; then the value nearest to zero is chosen.Google Scholar
  572. 2.
    Cf. above p. 515.Google Scholar
  573. 3.
    E.g. ACT No. 10, col. II.Google Scholar
  574. 4.
    E.g. ACT No. 6, col. I.Google Scholar
  575. 1.
    Cf., e.g., ACT No. 100, col. V.Google Scholar
  576. 2.
    Cf. above p. 310 (3).Google Scholar
  577. 3.
    Cf. ACT No. 123, col. V.Google Scholar
  578. 4.
    Cf. above p. 516.Google Scholar
  579. 5.
    ACT No. 121 Rev. I and No. 123 Obv./Rev. VI, respectively.Google Scholar
  580. 6.
    For the determination of these parameters cf. Neugebauer [1937, 3], p. 303-313.Google Scholar
  581. 1.
    As Aaboe has shown ACT No. 55 is not an eclipse table, though related to the Saros; cf. above p. 506.Google Scholar
  582. 2.
    Cf. the graph in ACT p. 108 and the table in ACT III, Pl. 38.Google Scholar
  583. 3.
    Cf. above p. 505 (16).Google Scholar
  584. 4.
    Cf. above II B 4, 3 C 3.Google Scholar
  585. 5.
    Published Aaboe-Sachs [1969], p. 11 to 20.Google Scholar
  586. 6.
    To be discussed by Aaboe-Henderson-Neugebauer-Sachs [1975].Google Scholar
  587. 7.
    Cf. above p. 503 (5).Google Scholar
  588. 8.
    Cf. above p. 523 (1).Google Scholar
  589. 9.
    Dotted curve in Fig. 73, p. 1347.Google Scholar
  590. 10.
    Cf. Aaboe-Sachs [1969], p. 17, cols. III and IV.Google Scholar
  591. 11.
    Cf. above p. 524 (3).Google Scholar
  592. 12.
    Cf. below p. 688.Google Scholar
  593. 13.
    Cf. above p. 141 and Fig. 122, p. 1244.Google Scholar
  594. 1.
    Cf., e.g., above p. 396 (5 b), p. 439, p. 471 f., p. 524 (4 a).Google Scholar
  595. 2.
    Cf., e.g., above p. 378 (15 b) or p. 496 (20).Google Scholar
  596. 3.
    The interval (1) is not the Saros which would contain 1,49,45” (cf. above p. 503 (12)).Google Scholar
  597. 4.
    Cf., e.g., below p. 531, Table 16, column III. The same value is found in a great variety of later sources, e.g. in Vat. gr. 208 (fol. 64`); Pc.-Sk. IX, 11; al-KhwArizmi, al-Bâttani, Maimonides, Bar Hebraeus.Google Scholar
  598. 5.
    A year of 365;l5” would require i 0;59,8,15,16,…616Google Scholar
  599. 6.
    ACT Nos. 185 to 187, for S.E. 124 = —187/6.Google Scholar
  600. 7.
    Sachs-Neugebauer [1956].Google Scholar
  601. 8.
    Cf. above p. 513 (16).Google Scholar
  602. 9.
    Cf. above p. 483 (3).Google Scholar
  603. 10.
    Cf., e.g., 12;22,6,20 for the 19-year cycle (above p. 355) or P=12;22,8,53,20m in System B (below p.533 (2)).Google Scholar
  604. 11.
    Cf. below p. 543 (5 b).Google Scholar
  605. 12.
    Nallino, Batt. I, p.40; Pc: Sk. II, p. 24.Google Scholar
  606. 1.
    Cf. above p. 475.Google Scholar
  607. 2.
    Compare for the oblique ascensions p. 368 and Fig. 5, p. 1316.Google Scholar
  608. 3.
    Cf. Fig. 7, p. 1317.Google Scholar
  609. 4.
    ACT No. 135; cf. ACT, p. 277.Google Scholar
  610. 5.
    Cf. Pc.-Sk. I, p. 12 and p. 14.Google Scholar
  611. 1.
    Aaboe [ 1966 ]; Aaboe-Sachs [1969], p.7f.Google Scholar
  612. 2.
    Cf., e.g., p. 377.Google Scholar
  613. 3.
    To use Newcomb’s terminology; cf. above p. 434.Google Scholar
  614. 4.
    Almost all System A planetary schemes operate with integer 7cß; cf. above p. 423, Table 9.Google Scholar
  615. 1.
    Cf. the diagrams in ACT III, Pl. 140-150 which tell us much about the working of a real lunar calendar.Google Scholar
  616. 2.
    Cf. ACT, p. 206f. (from Nos. 200 and 202).Google Scholar
  617. 1.
    Cf. above p. 515 (3), p. 520 (l a); p. 523.Google Scholar
  618. 4.
    The inverse order (for full moons) is found in ACT No. 201, Sect. 6.Google Scholar
  619. 5.
    Cf. for details ACT, p. 83.Google Scholar
  620. 6.
    Schaumberger, Erg., p. 388 f.Google Scholar
  621. 1.
    The term na is also used for new moons where it means the time between sunset and moonset; cf. below p. 552.Google Scholar
  622. 2.
    Cf. for the details ACT No. 201, p. 234-236 and Figs. 48-50.Google Scholar
  623. 3.
    ACT No. 201, Sect. 1-4 (p. 223) and Sect. 6 (p. 240), respectively. We have no full moon version for the coefficients (8).Google Scholar
  624. 4.
    Above p. 537.Google Scholar
  625. 5.
    The term hab-rat for “disk” is also used for eclipse magnitudes; cf. below p. 550 and ACT, p. 197f. and p. 237f., Figs. 51 and 52.Google Scholar
  626. 6.
    Cf. below p. 546.Google Scholar
  627. 7.
    Cf. above p. 525, p. 527.Google Scholar
  628. 1.
    Cf. ACT, p. 67, Fig. 31.Google Scholar
  629. 2.
    ACT No. 180 for S.E. 120 to 125 (-191 to —185).Google Scholar
  630. 2.
    Cf. Sachs [1948], p. 283, p. 281.Google Scholar
  631. 3.
    Cf. above p. 502 (3).Google Scholar
  632. 4.
    This remark is due to Aaboe-Henderson [1975].Google Scholar
  633. 5.
    Cf. above p. 503 (5).Google Scholar
  634. 6.
    Above p. 502 (4).Google Scholar
  635. 7.
    Cf. above II Intr. 3, 2 and 3, 3.Google Scholar
  636. 8.
    Published Neugebauer-Sachs [1967], p. 183-190.Google Scholar
  637. 9.
    Cf. Sachs [1952, 1].Google Scholar
  638. 10.
    Cf. above p. 366.Google Scholar
  639. 11.
    Above p. 358.Google Scholar
  640. 12.
    Above p.529.Google Scholar
  641. 13.
    This is, incidentally, further evidence for the fact that the of the existence of precession.Google Scholar
  642. 14.
    Published in Neugebauer-Sachs [1967], p. 190.Google Scholar
  643. 15.
    The alignment of the text is not very carefully observed obviously was to have in the same line first r of year N, then I in S.E. 86, followed by Q in 88. Babylonian astronomers were not aware by the scribe. The original arrangement Q in N+ 1. The text as written ends withGoogle Scholar
  644. 16.
    Sachs [1948], p. 280, No. 26.Google Scholar
  645. 17.
    Cf. for this text below p.598. The shadow tables were published by Weidner [1924], repeated by van der Waerden [1951, 1] and AA, p. 80, BA, p. 84.Google Scholar
  646. 18.
    Weidner (and hence van der Waerden) misinterpreted the text by taking the phrase “ina 1 kùß” to refer to a gnomon of 1 cubit length. In fact this expression only means “reckoning in cubits” [Sachs].Google Scholar
  647. 19.
    Cf. Neugebauer, MKT I, p. 9.Google Scholar
  648. 20.
    Epping AB, p. 115.Google Scholar
  649. 21.
    Longitudes are drawn to scale, latitudes are exaggerated, but to scale relative to each other.Google Scholar
  650. 22.
    A catalogue of stars in the Handy Tables is similarly concerned only with latitudes between ± 10°; cf. below p. 1050.Google Scholar
  651. 23.
    For a list of the Normal Stars with their coordinates for —600, —300, and 0 cf. Sachs [1974], p. 46.Google Scholar
  652. 24.
    Sachs [1952, 2]; cf. also Huber [1958], p. 205 f.Google Scholar
  653. 25.
    The sequence of the still extant Diaries becomes relatively dense in the middle of the fourth century; cf. the graph in Sachs [1974], p. 47.Google Scholar
  654. 26.
    Cf. Sachs [1948] for the topics recorded in each class of these texts; also Sachs [1952, 2], p. 149.Google Scholar
  655. 27.
    Cf., e.g., below p. 807.Google Scholar
  656. 28.
    I investigated about a hundred cases, involving all Normal Stars, from a variety of texts put at my disposal by A. Sachs.Google Scholar
  657. 1.
    Above p.476.Google Scholar
  658. 2.
    Published Neugebauer-Sachs [1969], p. 92f. For F in System A cf. above p. 479 (7).Google Scholar
  659. 3.
    Cf. above p. 456, n. 10. For the years 4 to 7 of Philip we also have an uncanonical lunar ephemeris; cf. above p. 506 n. 5.Google Scholar
  660. 4.
    This was discovered by A. Aaboe in No. 200, Sect. 6, Obv. I, 24 and I, 31; cf. Aaboe-Henderson [ 1975 ].Google Scholar
  661. 5.
    Neugebauer-Sachs [1967], p. 199; also below p. 553.Google Scholar
  662. 6.
    Cf. above II B 4, 3 C 1.Google Scholar
  663. 7.
    From Neugebauer-Sachs [ 1969 ], Text K, Sect. 6 (p. 108-110). Cf. also below p. 552.Google Scholar
  664. 8.
    Compare also the truncation of F in System A, above p. 501.Google Scholar
  665. 9.
    Cf. above II B 3, 1.Google Scholar
  666. 10.
    Neugebauer-Sachs [1969], p. 110f. Cf. also above p. 511.Google Scholar
  667. 11.
    Cf. above p. 487 (6) and p. 501 (11).Google Scholar
  668. 12.
    Regularly reported in the “Diaries”; in a schematic form also in the “atypical” text discussed here (cf. Neugebauer-Sachs [1967], p. 203).Google Scholar
  669. 13.
    Neugebauer-Sachs [1967], p. 203.Google Scholar
  670. 14.
    Cf. above p. 535, n. 1.Google Scholar
  671. 15.
    Above p. 5 15 (8 b).Google Scholar
  672. 16.
    Neugebauer-Sachs [1967], p. 205.Google Scholar
  673. 17.
    Cf. above p.352, n. 3.Google Scholar
  674. 18.
    The periodicity of solar eclipses cannot be detected in empirical data from only one locality. Hence it is not surprising when a text concerned with lunar motion mentions only the periodicity of lunar eclipses; cf. Neugebauer-Sachs [ 1967 ], p. 205.Google Scholar
  675. 19.
    Cf. above p. 502 (1).Google Scholar
  676. 20.
    Cf. my discussion in [1937, 3], p. 248-253.Google Scholar
  677. 21.
    Cf. the tables discussed p. 525ff.Google Scholar
  678. 22.
    It is probably in connection with the determination of the duration of an eclipse that the colophon of a table of lunar eclipses (System B, ACT No. 220) gives a small table of solar velocities as function of the single zodiacal signs; cf. above p. 530.Google Scholar
  679. 23.
    ACT, p. 197, E (from TU 14).Google Scholar
  680. 24.
    Vitruvius, Arch. IX, II (Budé, p. 16f.); Schnabel, Ber., p. 258, Frg. 22.Google Scholar
  681. 25.
    Cf. ACT, p. 198.Google Scholar
  682. 26.
    ACT No. 204, Sect. 4 (p. 247/8); cf. also ACT No. 200, Sect. 9 (p. 196 ).Google Scholar
  683. 27.
    Above p. 522.Google Scholar
  684. 28.
    Above II B 6, 2 and p. 527.Google Scholar
  685. 29.
    Published Neugebauer-Sachs [ 1967 ], Text E (p. 203). This is the same text which gives 5° as well as 6° for the extremal latitude of the moon (cf. above p. 549 ).Google Scholar
  686. 30.
    Cf. below p. 667.Google Scholar
  687. 31.
    Cf. below p. 593.Google Scholar
  688. 32.
    Cf. above p. 88.Google Scholar
  689. 33.
    Above p. 539.Google Scholar
  690. 34.
    For details cf. Neugebauer-Sachs [ 1969 ], Text K (p. 96-111).Google Scholar
  691. 35.
    Cf. the graphs in Neugebauer-Sachs [1969], p. 103.Google Scholar
  692. 36.
    Cf. for this above p. 506ff.Google Scholar
  693. 2Neugebauer-Sachs [1967], [1969]; Aaboe-Henderson et al. [1975]. 2 Neugebauer-Sachs [ 1967 ], Text C.Google Scholar
  694. 3.
    Thus Artaxerxes I/Darius II.Google Scholar
  695. 4.
    Column AG in Table 14, above p. 509.Google Scholar
  696. 5.
    Kandalanu ruled only 22 years, thus 36 could only mean a regnal year if the text had been written during Kandalanu’s lifetime.Google Scholar
  697. 6.
    Neugebauer-Sachs [1967], Text F.Google Scholar
  698. 7.
    Terminology introduced by Sachs [1948], p. 282.Google Scholar
  699. 8.
    Cf., e.g., above II A 2, p. 391 or II A 6, 1 C, p. 426. LBAT, p. XXV.Google Scholar
  700. 10.
    Computed by A. Sachs.Google Scholar
  701. 11.
    Neugebauer-Sachs [1967], p. 206 (10).Google Scholar
  702. 12.
    Cf. Sachs [1948], p. 287.Google Scholar
  703. 13.
    Cf. Sachs [1974], p.49.Google Scholar
  704. 14.
    The Goal-Year texts are attested from about —250 on; cf. Sachs [1974], p. 49.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1975

Authors and Affiliations

  • Otto Neugebauer
    • 1
  1. 1.Brown UniversityProvidenceUSA

Personalised recommendations