## P

a A 13-month excerpt from Neugebauer's (1955/1983, Vol. I) interpretation of Tablet 120 from Babylon, transcribed here in decimal notation. b At the top is the first line of the original table in Neugebauer's notation. Due to missing material, many entries were recreated from other tablets. c Column explanations are as folows: Column T: The date of the syzygy in years and months of the Seleucid Era.

Column A: The solar velocity in degrees of longitude per mean synodic month. See text for details.

Column B: The longitude of the Moon (and Sun) at conjunction, measured in degrees from 8° from the start of the zodiacal sign. "hun" = Aries. The corresponding column on the reverse side bears the longitude of opposition, i.e., full moon. Column C: Length of daylight at Babylon. The unit H is "large hours," equal to 60° or 4 hours. Column D: Half the length of the night, in large hours. Note that D = 1/2 (6 - C).

Column Y: A quantity related to the lunar latitude, and a measure of the magnitude of eclipse if the lunar latitude is close to zero;measured in fingers (f) = 1/2 degree. Column DY contains the differences in units of 1/60 finger.

Column F': Lunar "velocity" represented by a linear zigzag function in degrees per large hour. Column F: Lunar "velocity" represented by a linear zigzag function in degrees per day.

Column G: An approximation to the length of the synodic month (over 29 days), computed with a linear zigzag function. Column H: Differences between the monthly corrections given in Column J;these differences form a linear zigzag function. Column J: Correction to the const. solar speed assumed for col. G;"lal" here means "subtract";"tab" means "add." Column K: The corrected time interval between successive syzygies: K + 29d = G + J + 28d; G and K are in large hours.

The fraction 18/60 = 3/10, so that the division is reduced to

= (12,20,128)3,13; 20) = (12,22,8,53;20) = 12d369136.

The number after the semicolon is the sexagesimal fraction of the place to the left. The result gives a slightly larger value for the solar year than was found for the step function of Tablet 5; the year length becomes 365d268, again a good approximation, with P = 12.369136, still assuming a lunar synodic period of 29d530589. Instead, though, supposing a year length of 365d25, we derive a lunation length of 29d529144; this yields a 223-lunation interval of 6584d9991, the residual (from 6585) of which is -00,03,04;41.

Perhaps we should rephrase the question: What synodic month and tropical year lengths are needed to recover the residual 17,46,40? The answer is 6585d2963/223 = 29d53048 for the period of the Moon, which gives a year-length of 12.369136 x 29d53048 = 365d266. Thus, both numbers are respectable and consistent with the derived periodicity. The corresponding year-length from our derived periodicity of Tablet 5 yields a year-length of 12.36889 x 29d53048 = 365d259 (fortuitously close to the length of the anomalistic year). Thus, the results of the analyses are not at all inconsistent with either the hypothesis that the number 17,46,40 represents the residual value of 223 lunations above a fixed interval of 6585 days or with the Mesopotamians possessing relatively accurate values for both the lunar and solar periods. Moreover, their tables indicated clearly the variable length of the synodic month.

The zigzag function is also present in the System A Tablet 5 discussed earlier, but in that case, the zigzag does not describe the assumed solar velocity. It is demonstrated in

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