## Info

a Interval between alternate row entries of T: 62 + (n x 360). Interval between one set of alternate rows of x: 56 + (n x 360). Interval between other set of alternate rows of x: 6 + (n x 360).

Interval between single-row entries in columns T and S (Western Elongation): 245 days. Interval between single-row entries in columns S and x (near Superior Conjunction): 90 days. Interval between single-row entries in columns x and W (Eastern Elongation): 245 days. Interval between single-row entries in columns W and T (near Inferior Conjunction): 7 days.

a Interval between alternate row entries of T: 62 + (n x 360). Interval between one set of alternate rows of x: 56 + (n x 360). Interval between other set of alternate rows of x: 6 + (n x 360).

Interval between single-row entries in columns T and S (Western Elongation): 245 days. Interval between single-row entries in columns S and x (near Superior Conjunction): 90 days. Interval between single-row entries in columns x and W (Eastern Elongation): 245 days. Interval between single-row entries in columns W and T (near Inferior Conjunction): 7 days.

Table 7.2, which records the successive dates of the Venus configurations (in Neugebauer's 1957/1969 notation):

T = first appearance as a morning star S = last appearance as a morning star x = first appearance as evening star W = last appearance as evening star

See §2.4.3 and Figure 7.10 for the relationship between these instants and other planetary phenomena; for a full review of the Neugebauer planetary configuration notation, see §7.1.4, and for a newer interpretation of the original meanings of the table entries, see §7.1.3.2. As Pingree points out (Reiner and Pingree 1975, p. 17), all intervals are computed on the assumption that all months are 30 days long. The statements are arranged in such a way that each statement is one month and one day after the preceding statement, in an arithmetic progression. van der Waerden (1974, pp. 55-56) regards this as the first known application of an arithmetic progression to astronomy and describes it as "primitive." DHK, however, thinks that it is an interesting mnemonic tool embodying a complex and sophisticated series of statements. If one accepts that the dates are, indeed, referring to repetitive phenomena of Venus, then the interval between 2 Nissanu and 4 Simanu is not merely 62 days, but is 62 plus (n x 360), in which the solar cycle is close to an even multiple of the synodic period of Venus. Indeed, this interval holds for the entire first column (T, which marks the first visibility as morning star). DHK has found a periodicity linking the alternate pairs of dates of first visibility, viz., 2 Nissanu, 6 Abu, 10 Kislimu,... in one series, and 4 Simanu, 8 Tashritu, 12 Shabbatu,... in another, but the linkage between the two series, that is, how one gets from one series to another, is not clear. The interval in each case is

This interval is only ~2d more than 24 tropical years (each of 365d2422 length) or sidereal years (each of length 365d2564). The mean interval of 15 synodic periods of Venus (with Psyn = 583.92 MSD; see Table 2.9) is 8758.8 days. This is 5.2 days less than 8764 days. Going from 6 Abu as first appearance as Morning Star to 11 Duzu as first appearance as Evening Star, we have 335 days. However, the calculation from first appearance as Evening Star to the next first appearance as Morning Star does not start from 11 Duzu but from 5 Duzu, six days earlier. This is a good approximation to the correction demanded by the mean synodic period of Venus. It has been supposed from this table that the Babylonians calculated the mean synodic period of Venus as 587 days. Over 15 synodic periods, this would have amounted to an error of about 46 days, which surely would have been obvious to anyone doing systematic observations. We suggest that our interpretation of the table is more realistic, and that it is far from "primitive." There is still a great deal about this table that is unexplained, but we think that explanations that assume first-rate mathematical and observational skills are more likely to be correct than are those assuming incompetence. It seems to us clear from this table, and other evidence, that the Babylonians made extensive use of a 360-day calculating year with months shifting through all seasons. If the same set of month names was, indeed, being used in two chronologically different ways, interpretation of dates in particular texts may often be difficult. To maintain that there was still a third way compounds that difficulty, yet there is evidence favoring such a conclusion. van der Waerden (1974, p. 80) cites from Mul Apin (or mulApin), tablet 2, the following statement (Roman numerals indicate months; arabic numerals indicate days):

From XII 1 to II 30 the sun is in the path of Anu: Wind and storm. From III 1 to V 30 the sun is in the path of Enlil: Harvest and heat. From VI 1 to VIII 30 the sun is in the path of Anu: Wind and storm. From IX 1 to XI 30 the sun is in the path of Ea: Cold.

Mul Apin also says, repeatedly, that the spring equinox occurred on 15 Nisannu (Huber 1982, p. 9).

Furthermore, Mul Apin gives lists of heliacal risings for 36 stars or constellations and another list of intervals in days between risings of some of the most important of these stars, starting with Sirius. The statements of the two sources are in agreement if one assumes a year consisting of 12 months of 30 days each. However, it is very difficult to see how a 360-day year could have a generalized structure associating heliacal risings of stars with particular dates. Another interesting list gives the dates at which a number of stars culminated over Babylon. By making the two classes of observations together, it was possible to diminish the inaccuracies caused by missed data due to atmospheric conditions. The observer was supposed to begin observations on 20 Nisannu.

These texts on the sun and stars are consistent with each other, but are utterly unreasonable in their lack of correspondence with reality if "day" is taken in its normal meaning, or if the months are equated with lunations. Heliacal risings, whether of fixed stars or planets, would be shifting about 5 1/4 days a year. The statements about the Sun imply precise boundaries among the zones of Anu, Enlil, and Ea, but deny the inequality of the seasons, and would rapidly cease to be true with a 360-day year. However, the Sun moves slightly less than a degree a day. It would have been entirely natural to extend the meaning of "day" to mean, also, "a degree." Much technical vocabulary has arisen through defining a popular term more precisely, and somewhat differently. If "day" in the above statements is to be understood as "movement of the sun by 1 degree," the statements become astronomically reasonable.

With respect to the months, there are a number of statements that imply that months were lunations beginning with the first visibility of the moon after conjunction. There are also statements of omens that imply a different calendar structure. Thompson (1900, Omen No. 249, pp. lxxvii-lxxviii) says that the moon waned on the 27th and reappeared on the 30th. More remarkably, his numbers 119-172 deal with occasions when the Sun and Moon were seen together on the 12th, 13th, 14th, 15th, and 16th of the month. These can hardly refer to a system in which the moon was full on the 14th of each month.

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