A History of Ancient Mathematical Astronomy pp 345-555 | Cite as

# Babylonian Astronomy

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## 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
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## References

- 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.West of the Rhine, near Koblenz.Google Scholar
- 3.Epping-Strassmaier [1881]; cf. also Epping [1890].Google Scholar
- 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.Sp. 129 (=BM 34033); later published in Epping AB, pl. 1-3.Google Scholar
- 6.Epping-Strassmaier [1881], p. 285. The same could be said, 50 years later, about Babylonian mathematics.Google Scholar
- 7.Epping-Strassmaier [1881], p. 291 f.Google Scholar
- 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.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.Thureau-Dangin, TU.Google Scholar
- 11.Signature VAT, i.e. “Vorderasiatische Tontafel”.Google Scholar
- 12.Schnabel, Ber.Google Scholar
- 13.ACT, No. 122.Google Scholar
- 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.Cf. Neugebauer [1936], [1937, 1 to 3].Google Scholar
- 16.Seleucid” should always mean Seleucid and Parthian (if necessary even Roman).Google Scholar
- 17.Cf. PI. II, p. 1449.Google Scholar
- 18.Now published in Pinches-Sachs LBAT (1955). Sachs [1948] and Pinches-Sachs LBAT.Google Scholar
- 2.Below II C 3.Google Scholar
- 3.Alm. III, (Heiberg, p. 254, 10-13). Cf. also above p. 74; p. 118.Google Scholar
- 4.Cf. above p.350.Google Scholar
- 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
- 6.Cf. for colophons ACT I, p. 11 to 24.Google Scholar
- 7.On the lower edge of ACT, No. 122, shown Pl. IV; cf. also the copy ACT III, Pl. 221.Google Scholar
- 8.The error was detected by A. Sachs; cf. ACT I, p. 5, note 14 and Sachs [1948], p. 272, note 3.Google Scholar
- 9.Cf. ACT I, p. 5.Google Scholar
- 10.Cf., e.g., Epping-Strassmaier [1881], p.283, note 1.Google Scholar
- 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
- 1.Also full moons can be used, as in India.Google Scholar
- 2.Cf. below III 1 and VIA 2, 1.Google Scholar
- 3.The resulting “calendaric mean synodic month” is 29;31,50° long; cf. below p. 548.Google Scholar
- 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
- 5.Cf. Parker, Calendars.Google Scholar
- 1.Parker-Dubberstein, BC, p. 1 ff.Google Scholar
- 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
- 3.Cf. below p. 620.Google Scholar
- 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
- 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
- 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
- 7.Diodorus XII, 36; cf. below p. 622.Google Scholar
- 8.This problem will be of interest, of course, in a later context (cf. below p. 542).Google Scholar
- 9.Cf. above p. 354.Google Scholar
- 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
- 12.Below p. 365.Google Scholar
- 1.Neugebauer [1947, 1] and [1948]; now ACT No. 199 (and Pl. 136) from Istanbul U 107 and U 124.Google Scholar
- 2.Cf. below p. 542.Google Scholar
- 3.Cf. above p. 356, Table 1, years [0:16], [0:19], and [1:2], respectively.Google Scholar
- 4.Cf. p. 355 (3 a).Google Scholar
- 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
- For the accurate meaning of this statement cf. the next section (p. 360f.).Google Scholar
- 8.Below p. 395f, p. 500, etc.Google Scholar
- 9.Cf., e.g., Sûryasiddhânta I, 13.Google Scholar
- 1.Of non-ACT type, e.g. a text for Mercury (cf. Neugebauer [1948, 1], p. 212f.).Google Scholar
- 2.This was already suggested by Epping, AB, p. 151.Google Scholar
- 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
- 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
- 5.Kugler, Sternk. 1I, p. 606f.Google Scholar
- 6.Cf. above p. 354f.Google Scholar
- 7.Cf. the tabulation in Neugebauer [1948, 1], p. 218.Google Scholar
- 8.Aaboe-Sachs [1966], p. 11f.Google Scholar
- 9.Cf. below II A 6, 1 B (p. 424).Google Scholar
- 1.Our present O corresponds to 0,, p.1091.Google Scholar
- 2.Sachs [1952, 1].Google Scholar
- 3.I have arbitrarily given the year numbers of cycleGoogle Scholar
- 4.The dates shown remain the same for any other cycle.Google Scholar
- 5.Cf. above p.Google Scholar
- 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
- 7.Cf. above p. 357.Google Scholar
- 1.Our Tables 3 and 4.Google Scholar
- 2.Cf. above p. 40.Google Scholar
- 3.Cf. below p. 561.Google Scholar
- 4.Cf. p. 37, (4a).Google Scholar
- 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
- 6.Above p.38.Google Scholar
- 7.Ptolemy, Geogr. V, 20, §6 (ed. Nobbe, p. 78).Google Scholar
- 1.Above p. 40f.Google Scholar
- 2.These schemes must, of course, satisfy the symmetry relations pi=p12, etc. Cf. above p. 35.Google Scholar
- 3.Cf., e.g., above p. 30.Google Scholar
- 4.Cf. below p. 545.Google Scholar
- 5.Kugler, Sternk. I, p. 172. Similar results were obtained by van der Waerden [ 1952 ], p. 222.Google Scholar
- 6.Including the fragment of a catalogue of stars, discovered by Sachs [1952, 2].Google Scholar
- 7.Huber [1958].Google Scholar
- 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
- 9.ACT No.9 obv. III and IV, 12.Google Scholar
- 10.ACT No. 122 rev. II and III, 10.Google Scholar
- 11.In both figures I have omitted symmetric branches in order to avoid overcrowding of the graphs.Google Scholar
- 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
- 13.Cf. below II B 10, 2; also Neugebauer [1953].Google Scholar
- 14.Cf. below p. 727 ff. and p. 938.Google Scholar
- 15.Varâhamihira, Brhajjataka I, 19.Google Scholar
- 1.Above p. 368.Google Scholar
- 2.Above p. 57f.Google Scholar
- 3.Above p. 368.Google Scholar
- 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
- 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
- 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
- 2.The value of P is about 14 mean synodic months; cf. below p. 476.Google Scholar
- 3.Cf. below p. 384 and Fig. 15 there.Google Scholar
- 4.A] denotes the greatest integer contained in A. Cf. above p. 373.Google Scholar
- 6.We make here use of the fact that we are dealing with strictly arithmetical patterns which contain no approximations or roundings.Google Scholar
- 7.Cf. (4a), p.374.Google Scholar
- 8.For other examples cf. below p. 392 fï:Google Scholar
- 9.ACT, p. 86 ff.Google Scholar
- 10.For examples cf. below p. 392 (2).Google Scholar
- 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
- 2.The year 151•* in obv. 11 is the year [7:19] in the 19-year cycle (cf. above p. 356).Google Scholar
- 3.Cf. ACT I, p. 20, colophon Z.Google Scholar
- 4.Cf., e.g., obv. V, 12 and rev. V, 7 with 311° and 18°, respectively.Google Scholar
- 8.Obviously ya+1=1/2d on a downgoing branch.Google Scholar
- 9.For the details of this part of the procedures cf. below p. 388.Google Scholar
- 10.Cf. below p. 397.Google Scholar
- 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
- 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
- 13.This also holds for ©; cf. p. 363.Google Scholar
- 14.Cf. II Intr.3.Google Scholar
- 15.Cf. II Intr. 3, 3.Google Scholar
- 16.Cf. II Intr. 4, 2.Google Scholar
- 17.Cf. the “Gates” of primitive ( Palestinian) lunar theory: Neugebauer [ 1964 ], p. 51 - 58.Google Scholar
- 18.Above p. 386.Google Scholar
- 19.Below p. 391.Google Scholar
- 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
- 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
- 1.Cf. above p. 170 (1) and Fig. 158.Google Scholar
- 2.We shall return to the details below p.442.Google Scholar
- 3.Extant in “Normal Star Almanacs”; cf. below p. 555.Google Scholar
- 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
- 5.Cf. also ACT II, p. 283.Google Scholar
- 6.Cf. p.151.Google Scholar
- 7.Boll [1898].Google Scholar
- 8.CCAG 1, p. 163, 19.Google Scholar
- 9.Cf. Tannery, Mém. Sci. 4, p. 265.Google Scholar
- 1.Above p. 375ff.Google Scholar
- 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
- 3.ACT, p. 17 colophon L.Google Scholar
- 4.Cf. p. 376 (3 b).Google Scholar
- 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
- 6.Above p. 391 (1 1).Google Scholar
- 1.Cf. above p. 389.Google Scholar
- 2.Above p. 380 (1a).Google Scholar
- 3.Above p. 381(1 b) and p. 390f. (l0a) and (11).Google Scholar
- 4.Again p. 380 (I a).Google Scholar
- 5.Cf. p. 389 (4).Google Scholar
- 6.Cf. note 3.Google Scholar
- 7.Above p. 378 (15 b).Google Scholar
- 8.Above p. 394 (6).Google Scholar
- 9.We always assume, of course, a mean velocity for the sun; the solar anomaly plays no role in the planetary theory.Google Scholar
- 10.Above p. 380 (l a), for M—µ.Google Scholar
- 11.Cf. below (p. 446) Jupiter System B with M = 38;2 µ = 33;8,45 for diGoogle Scholar
- 12.Differences and amplitudes are the same.Google Scholar
- 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
- 14.ACT 704, e.g., has a dr that follows AA at a distance of 0;1,42,30 of an interval.Google Scholar
- 1.For Mars and Venus this should be understood as AA+6,0°.Google Scholar
- 2.Cf. p. 365.Google Scholar
- 3.Cf.IC6andIC8.Google Scholar
- 1.The last two are now almost completely broken away.Google Scholar
- 2.Cf. above p. 391 (12) and note 4 there.Google Scholar
- 3.ACT No. 606, 600, 604, and 601, respectively.Google Scholar
- 4.Published by Aaboe-Sachs [1966], Text D.Google Scholar
- 5.Cf. p. 390 (10 a).Google Scholar
- 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
- 7.This was discovered by P. Huber [1957], p. 277.Google Scholar
- 8.For the reconstruction of this pattern cf. ACT II, p. 312.Google Scholar
- 9.We shall also find it for Mars; cf. below p. 401 (7).Google Scholar
- 10.Ana ME-a Ma kur, or similar (Sachs).Google Scholar
- 1.Above p. 390 (10a).Google Scholar
- 2.Above p. 389 (4) and (6).Google Scholar
- 3.Cf. below IIA7,4B.Google Scholar
- 4.For the corresponding synodic arcs as function of A cf. below Fig. 40, p.1332.Google Scholar
- 5.Important details for this part of the theory of Mars will be discussed below p. 406ff.Google Scholar
- 6.For the other methods cf. below II A 7, 4 C and Fig. 41, p. 1332. Cf. p. 1320, Fig. 18.Google Scholar
- 1.Cf. ACT No. 800 and Sachs [1948], p. 283, respectively.Google Scholar
- 2.Cf. above p. 22, notes 7 to 9.Google Scholar
- 3.Including some variants for which cf. below p. 472.Google Scholar
- 4.Except for the scale the latter curves are the same as in Fig. 23, p.1322.Google Scholar
- 5.Fig. 25 is repeated from Fig. 238, p.1288 but changed to the scale of the present Fig. 24.Google Scholar
- Pliny, NH II, 77 (ed. Budé, Vol. 2, p. 33/34 and p. 165).Google Scholar
- 7.For the details cf. Neugebauer [1951].Google Scholar
- 8.Above p. 241 and p. 255.Google Scholar
- 9.Cf. Fig. 258, p. 1296.Google Scholar
- 1.Cf. for this whole episode Neugebauer [1951].Google Scholar
- 2.From Parker-Dubberstein, BC.Google Scholar
- 3.From Parker-Dubberstein: 147d and 121°, respectively.Google Scholar
- 4.Pinches-Sachs, LBAT No. 1019 (obv. 2, obv. 4, rev. 4′) [Sachs]. 5 Sachs [1948], p. 287.Google Scholar
- 1.Cf. above p. 400.Google Scholar
- 2.Cf. above p. 395 (3).Google Scholar
- 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
- 4.Cf. above p. 399 (1).Google Scholar
- 5.Cf., e. g., above p. 396 (5 b).Google Scholar
- 6.Cf. below p. 409 (16).Google Scholar
- 8.Cf. below IIA 7, 4.Google Scholar
- 9.Van der Waerden [1957], p. 52; also Anf. d. Astr., p. 190.Google Scholar
- 10.Scribal error in the text: omission of the final 5.Google Scholar
- 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
- 12.Cf. below p. 424.Google Scholar
- 1.Above p. 405.Google Scholar
- 2.Cf. also the excellent discussion of these methods by P. Huber [1957].Google Scholar
- 3.Cf. Neugebauer, MKT III, p. 83 s.v. Reihen; Neugebauer-Sachs, MCT, p. 100.Google Scholar
- 4.Cf. ACT I, p. 14f.Google Scholar
- 5.Cf. Neugebauer, MKT I, p. 103; first explained by Waschow [ 1932 ], p. 302f.Google Scholar
- 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
- 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
- 1.Cf. ACT I, p. 2.Google Scholar
- 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
- 3.Using the Tuckerman Tables.Google Scholar
- 4.Cf. above p. 369.Google Scholar
- 5.Cf., e.g., above p. 398 (1) and p. 399 (4).Google Scholar
- 1.This date was suggested by Huber [1957], p. 276. The text is published in ACT No. 310.Google Scholar
- 2.The restorations are quite secure.Google Scholar
- 3.For details cf. ACT II, p. 326f.Google Scholar
- 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
- 5.Above p. 416.Google Scholar
- 6.Cf. above p.405; also ACT, p. 312.Google Scholar
- 1.Cf., e.g., above p. 391 (12).Google Scholar
- 2.Cf. above p. 389 (4) and (6).Google Scholar
- 3.Cf. above p. 382.Google Scholar
- 4.The slow rotation of all apsidal lines escaped notice in early astronomy.Google Scholar
- 1.Cf. below p. 423, Table 9.Google Scholar
- 2.Cf. below II A 7, 5.Google Scholar
- 3.Mercury has 2,40/1,36=1 +2/3; cf. Table 9.Google Scholar
- 4.This has been first observed by Aaboe [1965], p. 224.Google Scholar
- 5.Published by Aaboe-Sachs [1966], p. 9f. and p. 24f. (Texts G to J).Google Scholar
- 6.Cf. also above p. 409 (12) and (12a).Google Scholar
- 7.Cf. above p. 411 (22 b).Google Scholar
- 8.This distinction was established by Sachs (cf. Pinches-Sachs, LBAT, p. [XXV]).Google Scholar
- 9.Cf. above p. 391; also note 4 there.Google Scholar
- 10.Cf. also below II A 7, 4 A, p. 456.Google Scholar
- 11.ACT No. 811, Sect. 3 (p. 381).Google Scholar
- 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
- 2.They agree exactly with the parameters found in the ephemerides; cf. ACT II, p. 310f.Google Scholar
- 3.Cf. below IV D 1,1.Google Scholar
- 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
- 1.Then under the directorship of Harald Bohr.Google Scholar
- 2.Neugebauer, Vorl. (1934).Google Scholar
- 3.Published as “Mathematische Keilschrift Texte” in QS A 3 in three volumes (1935, 1937).Google Scholar
- 4.Thureau-Dangin, TU (1922).Google Scholar
- 5.Schnabel, Ber. (1923), a rather chaotic publication.Google Scholar
- 6.Cf. above p. 374.Google Scholar
- 7.Cf. above p. 348.Google Scholar
- 8.The majority of Pinches’ copies are now available in Pinches-Sachs LBAT (1955).Google Scholar
- 9.Neugebauer [1936, 1].Google Scholar
- 10.London 1955. The omission of the date on the title page was overlooked by all concerned.Google Scholar
- 11.Van der Waerden [1957], p. 47.Google Scholar
- 12.Cf. above p. 427; van der Waerden. Anf. d. Astr., p. 187ff.Google Scholar
- 13.Cf. Aaboe [1965] and Aaboe-Sachs [1966].Google Scholar
- 14.Above p.427f.Google Scholar
- 15.Newcomb [1897].Google Scholar
- 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
- 1.quote from the commentary to the edition of the Panca-Siddhântikâ by Neugebauer-Pingree, Vol. II.Google Scholar
- 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
- 3.Pc: Sk. I, 15.Google Scholar
- 4.Pc.-Sk. II, 8 and XII, 5.Google Scholar
- 5.Cf. below p. 481 (Pc.-Sk. II, 2-6; III, 4-9); also V A 2,1 D 2.Google Scholar
- 6.Cf. below V A 2, 1.Google Scholar
- 7.Cf. Neugebauer-Pingree, Pc.-Sk. II, p. 109.Google Scholar
- 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
- 9.Pc.-Sk. II, p. 115.Google Scholar
- 10.Cf. above p.420and p. 390f., as compared with Pc.-Sk. II, p. 112, Table 24.Google Scholar
- 11.Pc.-Sk. XVII, 64-80 (Vol. II, p. 126, Table 33).Google Scholar
- 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
- 13.The following is based on a suggestion made by A. Sachs.Google Scholar
- 14.Listed above p. 391 (11).Google Scholar
- 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
- 1.Cf. above p. 390 (l0a) and (11).Google Scholar
- 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
- 1.Aaboe-Sachs [1966], p. 3f., Texts A and B.Google Scholar
- 2.Aaboe-Sachs [1966], p. 13 obv. column I, lines 1 to 5.Google Scholar
- 3.ACT Nos. 704 and 704a.Google Scholar
- 9.Cf. below p. 469.Google Scholar
- 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
- 2.Cf. above p. 378.Google Scholar
- 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
- 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
- 5.Cf. below p. 964.Google Scholar
- 1.Cf. the diagram ACT II, p. XII.Google Scholar
- 2.Cf. below p. 446. For a comparison of the results with the actual facts cf. Aaboe [ 1958 ], p. 242 - 245.Google Scholar
- 3.Above p. 391 (12).Google Scholar
- 4.ACT No. 812, Sect. 10 (p. 395 f.) and No. 813, Sect. 20 (p. 414 ).Google Scholar
- 5.Kugler SSB I, p. 48, a passage which is part of Sachs LBAT No. 1593 (rev. 12ff.).Google Scholar
- 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
- 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
- 8.ACT No. 813, Sect. 1 and No. 814, Sect. 1; No. 812, Sect. 10 and No. 813, Sect. 20.Google Scholar
- 9.Cf. for details below II A 7, 3 A.Google Scholar
- 10.In our case 5 for the 12-year period, 6 for the 71-year period, among 391.Google Scholar
- 11.Starting with year 24 of Darius I, i.e.-497.Google Scholar
- 12.Cf. below p. 447.Google Scholar
- 13.Cf. the “checking rules” ACT, p. 307 and p. 309.Google Scholar
- 14.ACT No. 812, Sect. 2 and No. 813, Sect. 13.Google Scholar
- 15.Above p.439 and p. 395.Google Scholar
- 16.Above p. 395 (3).Google Scholar
- 17.Cf. below p. 446.Google Scholar
- 18.ACT No. 812, Sect. 2.Google Scholar
- 19.In the procedure text ACT No. 813, Sect. 1 (p. 403/404) the value of d is abbreviated to 6,42.Google Scholar
- 20.Cf. below p. 446.Google Scholar
- 1.ACT Nos. 600 to 608 and Nos. 609 to 614.Google Scholar
- 2.ACT No. 814, Sect. 2 (p. 424) seems to mention instead Op 12 and 1t 12.Google Scholar
- 3.Aaboe-Sachs [1966], p. 16-21.Google Scholar
- 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
- 5.Cf. above p.437f.Google Scholar
- 6.For an example cf. ACT, p. 308.Google Scholar
- 7.ACT No. 813, Sect. 14 to 16.Google Scholar
- 8.Cf. above p. 359.Google Scholar
- 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
- 10.Aaboe-Sachs [1966], p.8 and p.22f. (Text E).Google Scholar
- 11.Aaboe [1965], p. 223, p. 221.Google Scholar
- 12.Pc.-Sk XVII 9-11 (Neugebauer-Pingree II, p. 113).Google Scholar
- 13.Cf. above p. 438.Google Scholar
- 1.ACT No. 805, Sect. 1; No. 812, Sect 1; No. 813, Sect. 12, 21 and 22.Google Scholar
- 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
- 1.Above p. 397.Google Scholar
- 2.Above p. 398 from Aaboe-Sachs [1966], Text D.Google Scholar
- 3.Above p. 398 (1).Google Scholar
- 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
- 5.Below p.449.Google Scholar
- 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
- 7.Cf. above p. 398 (2).Google Scholar
- 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
- 9.Cf. above (3)Google Scholar
- 10.Cf. above p. 399 (3). An incorrectly computed ephemeris (No. 603) has only -5;55° and -7;10° for OGoogle Scholar
- 11.ACT No. 819b, Sect. 2.Google Scholar
- 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
- 13.Cf., e.g., above p. 405 (1).Google Scholar
- 14.Cf. above p. 443 (7 b).Google Scholar
- 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
- 17.Cf., e.g., the position of the “apsidal line” (above p. 447).Google Scholar
- 18.Pc.-Sk. XVII 12-13, Neugebauer-Pingree II, p. 118, Table 26.Google Scholar
- 1.Jupiter, above II A 5, 3 A, Mercury II A 5, 3 B.Google Scholar
- 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
- 3.Cf. above p. 451 (18).Google Scholar
- 4.ACT No. 652, Obv. I. 10 to 15 = Huber, Text A, p. 292.Google Scholar
- 5.Cf. above p. 447 (2) and p. 450 (15).Google Scholar
- 6.Above p.448 (6) and p.450 (15).Google Scholar
- 7.Cf. the analogous situation for the retrograde arc. above p. 449 (9).Google Scholar
- 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
- 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
- 10.Huber [1957], p. 279-291, p. 298-303. Cf. above II A 5, 3 A.Google Scholar
- 1.Cf. above p. 399.Google Scholar
- 2.Making use of p. 422 (6).Google Scholar
- 3.This convenient formulation was first given by van der Waerden [1957], p. 47.Google Scholar
- 4.Cf above p. 179 (6): 625;30.Google Scholar
- 5.Above p. 426; also Sachs [1948], p. 283, Table IV and Kugler SSB I, p. 44.Google Scholar
- 6.ACT No. 501.Google Scholar
- 7.Cf. above p.442.Google Scholar
- 8.Cf. below p. 458.Google Scholar
- 9.Pc.-Sk. II, p. 112, Table 24.Google Scholar
- 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
- 11.Van der Waerden [1972], p. 77-87 and BA, p. 316-320.Google Scholar
- 12.Cf. below p. 785, p. 788.Google Scholar
- 13.Above p. 408.Google Scholar
- 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
- 2.Centaurus 5 (1958), p. 246; van der Waerden BA, p. 274.Google Scholar
- 3.Cf. above p. 391 (11) or p. 407 (5) and p. 455 (2).Google Scholar
- 1.Above p. 400.Google Scholar
- 2.Above p. 425 (6).Google Scholar
- 3.Above p. 409 (12).Google Scholar
- 4.Above p. 411 (22).Google Scholar
- 5.Cf. above p. 411. n. 11 and below IV D 3, 4 or V A 3, 2.Google Scholar
- 6.Pc.-Sk. II, p. 127, Table 34.Google Scholar
- 7.Cf e.g., below Fig. 18 (p. 1320).Google Scholar
- 8.Described in detail above p. 401 and Fig. 20 (p. 1320).Google Scholar
- 9.Pc.-Sk. II, p. 120, Table 29.Google Scholar
- 10.Cf. above p.399 and Fig. 18, p. 1320.Google Scholar
- 11.Cf., e.g., Aaboe-Sachs [1966], p. 10, Table 9. 2 Cf. above p. 196; p. 193.Google Scholar
- 13.Above p. 459.Google Scholar
- 14.Pc.-Sk. II, p. 119 and p. 127.Google Scholar
- 15.Pc.-Sk. II, p. 120, Tables 27 to 29.Google Scholar
- 1.For the smallness of the eccentricity of the orbit of Venus cf. below p. 1443, Fig. 34.Google Scholar
- 2.This is also the goal-year period for Venus; cf. below p. 554 (1).Google Scholar
- 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
- 4.Neugebauer-Sachs [1967], p. 207.Google Scholar
- 5.The method is described in II A 4.Google Scholar
- 6.Cf., e.g., below p.462 (from ACT No. 410).Google Scholar
- 7.Cf. the tabulation in ACT, p. 301/302 and No. 821 b, p. 441 f.Google Scholar
- 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
- 9.This is the above mentioned text ACT No. 420+821 b (cf. p.462).Google Scholar
- 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
- 11.Cf. below II B 10,2.Google Scholar
- 12.For one exception cf. below II C 3, p. 554.Google Scholar
- 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
- 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
- 15.Cf. ACT, p. 301.Google Scholar
- 16.Tabulated in ACT, p. 399.Google Scholar
- 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
- 18.Alm. XIII, 8; cf. above I C 8, 3 A.Google Scholar
- 19.Cf. ACT, p. 399 for the time intervals, p. 397 for the corresponding arcs.Google Scholar
- 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
- 2.Cf. below p. 473.Google Scholar
- 3.Above p. 402.Google Scholar
- 4.Cf. below p.471.Google Scholar
- 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
- 6.Cf. above p. 460 and p. 441.Google Scholar
- 7.Cf. Neugebauer-Sachs [1967], p. 206f.Google Scholar
- 8.Cf. below p. 899.Google Scholar
- 9.Cf. above II A 6.Google Scholar
- 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
- 11.Cf. below p. 470.Google Scholar
- 12.Cf. above p. 375.Google Scholar
- 1.Cf. the diagram ACT II, p. XII; similarly for the moon ACT I, p, XVI.Google Scholar
- 2.Cf. ACT, p. 317 f.Google Scholar
- 3.Cf. ACT, p. 319, B(S).Google Scholar
- 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
- 5.Cf. above p. 438.Google Scholar
- 6.This was discovered by A. Aaboe.Google Scholar
- 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
- 8.Cf. above p.466.Google Scholar
- 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
- 10.Cf. Fig. 23 p. 1322.Google Scholar
- 11.Cf. above II A 4.Google Scholar
- 12.Cf., e.g., ACT No. 300, col. I and II. For simplifying procedures cf. ACT, p. 293.Google Scholar
- 13.ACT No. 300a and 300b.Google Scholar
- 14.Cf. ACT, p. 298.Google Scholar
- 15.Cf. Fig. 21, p. 1321.Google Scholar
- 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
- 17.ACT Nos. 800c and d.Google Scholar
- 18.Mainly Nos. 301 and 302.Google Scholar
- 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
- 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
- 21.Cf. above p. 403.Google Scholar
- 22.Cf. ACT, p. 294.Google Scholar
- 23.Pc.-Sk II, p. 112, Table 24; cf. also above p. 466.Google Scholar
- 24.Pc.-Sk. II, p. 114.Google Scholar
- 25.Cf. Pc.-Sk. II, p. 122 and Fig. 67 on p. 123.Google Scholar
- 26.Compare Pc.-Sk. II, p. 124, Fig. 68 and ACT, p. 298f., Figs. 57a and 57 b.Google Scholar
- 27.Pc.-Sk. II, p. 128, Table 36.Google Scholar
- 28.Cf. above p.471.Google Scholar
- 1.Cf. below p. 516f.Google Scholar
- 2.Above p. 366 ff. and p. 371ff.Google Scholar
- 3.Below 11 B 8.Google Scholar
- 4.For details cf. ACT I, p. 43.Google Scholar
- 5.ACT No. 180ff.Google Scholar
- 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
- 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
- 1.Cf. for these concepts above p. 375.Google Scholar
- 2.Cf. below II B 3, 1 and 2.Google Scholar
- 3.Cf. p. 475.Google Scholar
- 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
- 5.ACT No. 120 rev. II gives F′, exactly parallel to F in the next column.Google Scholar
- 1.Cf. below lIB10.Google Scholar
- 2.Aaboe [1968], p. 30-34.Google Scholar
- 3.Cf. below p. 501 f.Google Scholar
- 4.Cf. also below p. 501; Neugebauer [1957, 1], p. 18f.Google Scholar
- 5.Sect. 4.Google Scholar
- 1.ACT No. 194a, obv. II, 25.Google Scholar
- 2.Cf. below p. 492.Google Scholar
- 3.Above p. 478.Google Scholar
- 4.Manitius, p. 204 to 211. Cf. also below p. 602f.Google Scholar
- 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
- 6.Cf. below V A 2, I D 1.Google Scholar
- 7.Cf. below V A 2, 1 A.Google Scholar
- 8.For the details of Pc.-Sk. II, p. 18 f.Google Scholar
- 1.Cf. below p. 501 (9).Google Scholar
- 2.Above p.480.Google Scholar
- 1.Below p. 497 ff.Google Scholar
- The details will be described below in II B 3, 3 to 5.Google Scholar
- 1.Cf. above p. 478 (2).Google Scholar
- 2.Cf. below p. 492f.Google Scholar
- 3.Above p.69 (1).Google Scholar
- 4.E.g. with Birùni, with Maimonides, in the Hexapterygon, etc. As shown on p.480, about 363;5°.Google Scholar
- 6.Belonging to the interval from S.E. 179 to 210 (-132 to —101).Google Scholar
- 7.S.E. 235 (-76/75).Google Scholar
- 1.Below p. 498.Google Scholar
- 2.Above p.478 (5).Google Scholar
- 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
- 1.Cf. above p.478 (5).Google Scholar
- 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
- 2.Cf. below p. 506.Google Scholar
- 3.ACT No. 204 rev. 9 (p. 249).Google Scholar
- 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
- 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
- 1.Cf. the table in ACT I, p. 60.Google Scholar
- 2.Below p.498f.Google Scholar
- 3.E.g. ACT No. 18, col. [— I] and V.Google Scholar
- 4.Cf. p.483.Google Scholar
- 5.Below p. 504.Google Scholar
- 1.Above p. 372.Google Scholar
- 2.Below p. 491.Google Scholar
- 2.This text has many more similar errors; cf. ACT I, p. 100.Google Scholar
- 1.Above p. 483 (3).Google Scholar
- 2.Above p. 483 (2).Google Scholar
- 3.Above p. 489.Google Scholar
- 4.Below p. 493.Google Scholar
- 5.Cf. below p.497.Google Scholar
- 1.Cf. above p.476.Google Scholar
- 2.The amplitudes of H and J in Fig. 53 are drawn in the correct ratio.Google Scholar
- 3.Cf. p. 374 (3). Cf. also Fig. 55, p.1339, inset.Google Scholar
- 4.Cf. above p.383.Google Scholar
- 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
- 6.ACT No. 165 obv. III and IV, 12, 13 and 18 to 20.Google Scholar
- 7.Above p. 378 (15b).Google Scholar
- 8.Above p. 396 (5 b).Google Scholar
- 9.If one uses the rounded extrema (18) one finds instead of (19)Google Scholar
- 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
- 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
- 1.Section 12 in the now published text; cf. note 4.Google Scholar
- 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
- 3.Cf. above p.486 (2).Google Scholar
- 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
- 5.Above p. 487 (6).Google Scholar
- Due to van der Waerden, AA, p. 149.Google Scholar
- 1.Cf. above p.478.Google Scholar
- 2.O* is given to six sexagesimal places in contrast to the five digits of b. In order to compare the twoGoogle Scholar
- functions we norm the last digit of d* as minutes.Google Scholar
- 3.Cf. above p. 485.Google Scholar
- 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
- 5.Cf. ACT I, p. 119.Google Scholar
- 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
- 7.Above p. 478 (4).Google Scholar
- 8.Above p.480; cf. also p.479.Google Scholar
- 9.Above p. 478 (5).Google Scholar
- 10.Sect. 2, line 15.Google Scholar
- 11.Below p. 506.Google Scholar
- 1.The following elegant derivation is due to Aaboe-Henderson [1975].Google Scholar
- 2.Cf. above p. 377f. (l0), (13) and (14).Google Scholar
- 3.Cf. for this concept above p. 433f.Google Scholar
- 4.This parameter is actually mentioned in the procedure text ACT No. 207, Sect. 7 (p. 250).Google Scholar
- 5.Above p.484 (2a).Google Scholar
- 6.Above p.488.Google Scholar
- 7.Cf. below 1I B 5, 3.Google Scholar
- 8.Cf. ACT No. 210, Sect. 3, line 8; the numbers are in part restored.Google Scholar
- 9.Cf. above p. 503 (9).Google Scholar
- 10.Cf. ACT No. 210, Sect. 3, line 18.Google Scholar
- 11.Cf., e.g., Ginzel, Hdb. I, p. 254.Google Scholar
- 12.Cf. below II B 7; also above p. 129.Google Scholar
- 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
- 2.Cf. above H B 3, 3.Google Scholar
- 3.Cf. Aaboe [1971], p. 10f., p. 11 f., p. 17, Fig. 3; [1969], p. 9, Fig. 1.Google Scholar
- 4.Aaboe [1968], p.28f.; refined [1969], p. 11.Google Scholar
- 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
- 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
- 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
- 8.We need not specify the units of these intervals in the present context.Google Scholar
- 9.Cf. Fig. 50, p. 1336 and Fig. 61, p. 1342.Google Scholar
- 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
- 12.Above II B 4.Google Scholar
- 11.Cf. above p. 501 (12a) and (13a).Google Scholar
- 14.Cf. above p. 505 (2).Google Scholar
- 15.Cf. above p. 487 (6).Google Scholar
- 16.E.g. ACT, p. 60.Google Scholar
- 17.ACT No. 5 obv. VI, 15.Google Scholar
- 18.Neugebauer-Sachs [1969], p. 110. Cf. for the same text below p.548, p. 552f.Google Scholar
- 19.Cf. above p. 508 (4).Google Scholar
- 20.Aaboe [1971], p. 15f. and Table 5.Google Scholar
- 21.Cf. Aaboe [1969].Google Scholar
- 22.Cf. above p. 506 (4).Google Scholar
- 23.Cf. below p. 528 f.Google Scholar
- 24.Aaboe [1969], p. 12f.Google Scholar
- 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
- 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
- 2.Hence parameters not expressly denoted as degrees are reckoned in §e.Google Scholar
- 3.Cf. above p. 372.Google Scholar
- 4.Cf. above p.514.Google Scholar
- 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
- 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
- 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
- 8.Cf. above p. 375 (6b).Google Scholar
- 9.For the numerical details of obtaining d and D from (11) cf. Neugebauer [1937, 3], p. 257f.Google Scholar
- 1.Cf. above p. 502 (2).Google Scholar
- 2.Cf. for, this term above p.486, n. 4.Google Scholar
- 3.Cf. above p. 503. (12).Google Scholar
- 4.Cf. below p. 523 (2c).Google Scholar
- 5.Cf. above p.502 (1).Google Scholar
- 6.Cf. Neugebauer [1937, 3], p.237f., or ACT, p.47 (1).Google Scholar
- 7.Cf. below p. 523 (2 a).Google Scholar
- 8.Heiberg I, p. 471, 32. For these tables in general cf. above I B 6, 1.Google Scholar
- 9.Cf. below p. 533 (1).Google Scholar
- 1.Cf. above p. 519 (10).Google Scholar
- 2.Cf. below 1I B 6.Google Scholar
- 3.Aaboe [1973] has studied this method in the wider frame of ancient astronomy.Google Scholar
- 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
- 6.Cf. below p. 526f.Google Scholar
- 1.Cf., however, below p.551.Google Scholar
- It is possible that two consecutive values of E belong to this zone; then the value nearest to zero is chosen.Google Scholar
- 2.Cf. above p. 515.Google Scholar
- 3.E.g. ACT No. 10, col. II.Google Scholar
- 4.E.g. ACT No. 6, col. I.Google Scholar
- 1.Cf., e.g., ACT No. 100, col. V.Google Scholar
- 2.Cf. above p. 310 (3).Google Scholar
- 3.Cf. ACT No. 123, col. V.Google Scholar
- 4.Cf. above p. 516.Google Scholar
- 5.ACT No. 121 Rev. I and No. 123 Obv./Rev. VI, respectively.Google Scholar
- 6.For the determination of these parameters cf. Neugebauer [1937, 3], p. 303-313.Google Scholar
- 1.As Aaboe has shown ACT No. 55 is not an eclipse table, though related to the Saros; cf. above p. 506.Google Scholar
- 2.Cf. the graph in ACT p. 108 and the table in ACT III, Pl. 38.Google Scholar
- 3.Cf. above p. 505 (16).Google Scholar
- 4.Cf. above II B 4, 3 C 3.Google Scholar
- 5.Published Aaboe-Sachs [1969], p. 11 to 20.Google Scholar
- 6.To be discussed by Aaboe-Henderson-Neugebauer-Sachs [1975].Google Scholar
- 7.Cf. above p. 503 (5).Google Scholar
- 8.Cf. above p. 523 (1).Google Scholar
- 9.Dotted curve in Fig. 73, p. 1347.Google Scholar
- 10.Cf. Aaboe-Sachs [1969], p. 17, cols. III and IV.Google Scholar
- 11.Cf. above p. 524 (3).Google Scholar
- 12.Cf. below p. 688.Google Scholar
- 13.Cf. above p. 141 and Fig. 122, p. 1244.Google Scholar
- 1.Cf., e.g., above p. 396 (5 b), p. 439, p. 471 f., p. 524 (4 a).Google Scholar
- 2.Cf., e.g., above p. 378 (15 b) or p. 496 (20).Google Scholar
- 3.The interval (1) is not the Saros which would contain 1,49,45” (cf. above p. 503 (12)).Google Scholar
- 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
- 5.A year of 365;l5” would require i 0;59,8,15,16,…616Google Scholar
- 6.ACT Nos. 185 to 187, for S.E. 124 = —187/6.Google Scholar
- 7.Sachs-Neugebauer [1956].Google Scholar
- 8.Cf. above p. 513 (16).Google Scholar
- 9.Cf. above p. 483 (3).Google Scholar
- 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
- 11.Cf. below p. 543 (5 b).Google Scholar
- 12.Nallino, Batt. I, p.40; Pc: Sk. II, p. 24.Google Scholar
- 1.Cf. above p. 475.Google Scholar
- 2.Compare for the oblique ascensions p. 368 and Fig. 5, p. 1316.Google Scholar
- 3.Cf. Fig. 7, p. 1317.Google Scholar
- 4.ACT No. 135; cf. ACT, p. 277.Google Scholar
- 5.Cf. Pc.-Sk. I, p. 12 and p. 14.Google Scholar
- 1.Aaboe [ 1966 ]; Aaboe-Sachs [1969], p.7f.Google Scholar
- 2.Cf., e.g., p. 377.Google Scholar
- 3.To use Newcomb’s terminology; cf. above p. 434.Google Scholar
- 4.Almost all System A planetary schemes operate with integer 7cß; cf. above p. 423, Table 9.Google Scholar
- 1.Cf. the diagrams in ACT III, Pl. 140-150 which tell us much about the working of a real lunar calendar.Google Scholar
- 2.Cf. ACT, p. 206f. (from Nos. 200 and 202).Google Scholar
- 1.Cf. above p. 515 (3), p. 520 (l a); p. 523.Google Scholar
- 4.The inverse order (for full moons) is found in ACT No. 201, Sect. 6.Google Scholar
- 5.Cf. for details ACT, p. 83.Google Scholar
- 6.Schaumberger, Erg., p. 388 f.Google Scholar
- 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
- 2.Cf. for the details ACT No. 201, p. 234-236 and Figs. 48-50.Google Scholar
- 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
- 4.Above p. 537.Google Scholar
- 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
- 6.Cf. below p. 546.Google Scholar
- 7.Cf. above p. 525, p. 527.Google Scholar
- 1.Cf. ACT, p. 67, Fig. 31.Google Scholar
- 2.ACT No. 180 for S.E. 120 to 125 (-191 to —185).Google Scholar
- 2.Cf. Sachs [1948], p. 283, p. 281.Google Scholar
- 3.Cf. above p. 502 (3).Google Scholar
- 4.This remark is due to Aaboe-Henderson [1975].Google Scholar
- 5.Cf. above p. 503 (5).Google Scholar
- 6.Above p. 502 (4).Google Scholar
- 7.Cf. above II Intr. 3, 2 and 3, 3.Google Scholar
- 8.Published Neugebauer-Sachs [1967], p. 183-190.Google Scholar
- 9.Cf. Sachs [1952, 1].Google Scholar
- 10.Cf. above p. 366.Google Scholar
- 11.Above p. 358.Google Scholar
- 12.Above p.529.Google Scholar
- 13.This is, incidentally, further evidence for the fact that the of the existence of precession.Google Scholar
- 14.Published in Neugebauer-Sachs [1967], p. 190.Google Scholar
- 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
- 16.Sachs [1948], p. 280, No. 26.Google Scholar
- 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
- 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
- 19.Cf. Neugebauer, MKT I, p. 9.Google Scholar
- 20.Epping AB, p. 115.Google Scholar
- 21.Longitudes are drawn to scale, latitudes are exaggerated, but to scale relative to each other.Google Scholar
- 22.A catalogue of stars in the Handy Tables is similarly concerned only with latitudes between ± 10°; cf. below p. 1050.Google Scholar
- 23.For a list of the Normal Stars with their coordinates for —600, —300, and 0 cf. Sachs [1974], p. 46.Google Scholar
- 24.Sachs [1952, 2]; cf. also Huber [1958], p. 205 f.Google Scholar
- 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
- 26.Cf. Sachs [1948] for the topics recorded in each class of these texts; also Sachs [1952, 2], p. 149.Google Scholar
- 27.Cf., e.g., below p. 807.Google Scholar
- 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
- 1.Above p.476.Google Scholar
- 2.Published Neugebauer-Sachs [1969], p. 92f. For F in System A cf. above p. 479 (7).Google Scholar
- 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
- 4.This was discovered by A. Aaboe in No. 200, Sect. 6, Obv. I, 24 and I, 31; cf. Aaboe-Henderson [ 1975 ].Google Scholar
- 5.Neugebauer-Sachs [1967], p. 199; also below p. 553.Google Scholar
- 6.Cf. above II B 4, 3 C 1.Google Scholar
- 7.From Neugebauer-Sachs [ 1969 ], Text K, Sect. 6 (p. 108-110). Cf. also below p. 552.Google Scholar
- 8.Compare also the truncation of F in System A, above p. 501.Google Scholar
- 9.Cf. above II B 3, 1.Google Scholar
- 10.Neugebauer-Sachs [1969], p. 110f. Cf. also above p. 511.Google Scholar
- 11.Cf. above p. 487 (6) and p. 501 (11).Google Scholar
- 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
- 13.Neugebauer-Sachs [1967], p. 203.Google Scholar
- 14.Cf. above p. 535, n. 1.Google Scholar
- 15.Above p. 5 15 (8 b).Google Scholar
- 16.Neugebauer-Sachs [1967], p. 205.Google Scholar
- 17.Cf. above p.352, n. 3.Google Scholar
- 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
- 19.Cf. above p. 502 (1).Google Scholar
- 20.Cf. my discussion in [1937, 3], p. 248-253.Google Scholar
- 21.Cf. the tables discussed p. 525ff.Google Scholar
- 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
- 23.ACT, p. 197, E (from TU 14).Google Scholar
- 24.Vitruvius, Arch. IX, II (Budé, p. 16f.); Schnabel, Ber., p. 258, Frg. 22.Google Scholar
- 25.Cf. ACT, p. 198.Google Scholar
- 26.ACT No. 204, Sect. 4 (p. 247/8); cf. also ACT No. 200, Sect. 9 (p. 196 ).Google Scholar
- 27.Above p. 522.Google Scholar
- 28.Above II B 6, 2 and p. 527.Google Scholar
- 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
- 30.Cf. below p. 667.Google Scholar
- 31.Cf. below p. 593.Google Scholar
- 32.Cf. above p. 88.Google Scholar
- 33.Above p. 539.Google Scholar
- 34.For details cf. Neugebauer-Sachs [ 1969 ], Text K (p. 96-111).Google Scholar
- 35.Cf. the graphs in Neugebauer-Sachs [1969], p. 103.Google Scholar
- 36.Cf. for this above p. 506ff.Google Scholar
- 2Neugebauer-Sachs [1967], [1969]; Aaboe-Henderson et al. [1975]. 2 Neugebauer-Sachs [ 1967 ], Text C.Google Scholar
- 3.Thus Artaxerxes I/Darius II.Google Scholar
- 4.Column AG in Table 14, above p. 509.Google Scholar
- 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
- 6.Neugebauer-Sachs [1967], Text F.Google Scholar
- 7.Terminology introduced by Sachs [1948], p. 282.Google Scholar
- 8.Cf., e.g., above II A 2, p. 391 or II A 6, 1 C, p. 426. LBAT, p. XXV.Google Scholar
- 10.Computed by A. Sachs.Google Scholar
- 11.Neugebauer-Sachs [1967], p. 206 (10).Google Scholar
- 12.Cf. Sachs [1948], p. 287.Google Scholar
- 13.Cf. Sachs [1974], p.49.Google Scholar
- 14.The Goal-Year texts are attested from about —250 on; cf. Sachs [1974], p. 49.Google Scholar

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