(signs) (degrees) (arc-minutes) (arc-seconds)
The parentheses contain our interpretation. The isolated 5 on the top of the second column is somewhat reminiscent of Babylonian notation,22 although the example is nearly all in base-10 notation. Neugebauer's explication of the eclipse prediction procedure is as follows. First, a date must be known; how it is known is not clear, but a recurrent cycle could be used to predict the day of an eclipse (see Schlosser et al. 1991/1994, pp. 12-15 for examples of such "rules of thumb"). Although we cannot say if a trial-and-error type of method was used, it is certainly the case that if the result indicates that the cycle used was faulty, another attempt perhaps with a different cycle can be made. In any case, given the date,
(1) the true longitudes of the Sun and Moon (10 and 1M) are found for a given day;
(2) by extrapolation, the moment when 10 = 1M is found;
(3) by computation, the 1node is found for this instant;
(4) from a table (or some analog), the lunar latitude is found;
22 In Baylonian notation, if a number is divisible by 60, the integer part of the quotient precedes the remainder by a comma;thus, 121° is written 2,1. In the current example, 248 can be written 4,8 and the number (5 x 30°) + 29°58'13" can be written: 2,59;58,13°. Note the use of the semicolon to denote the fraction (58 + 13/60)/60.
(5) from the lunar velocity and the latitude, the magnitude of the eclipse is found; and
(6) the magnitude permits the track length across the shadow to be determined, and thus, with the rate of motion, the duration in the shadow cone is found, and the moments of first and last contact.
Step 1 involves computing the number of days since a particular but variable epoch, called ahargana, which is counted from the beginning of the (present) Kali yuga (3102 b.c.) and marks the moment when the Sun is at a vernal equinox. From this, the number of zodiacal signs (the Greco-Babylonian zodiacal signs) traversed by the Sun since the ahargana is worked out, and the number of days taken by the Sun to traverse these signs, is obtained from an encoded form of a rate table. The traversal time for each sign differs because of the eccentric orbit of Earth, which causes it to move most slowly at aphelion. Reflexively, the eastward motion of the Sun is slowest at this time (Jan-Feb in the current calendar), and this occurs when the Sun is in the sign of Gemini. Similarly, when the Earth moves most quickly, at perihelion, the Sun's reflexive eastward motion will be most rapid. The difference from one extreme to the other is slightly more than two days/month and, dividing the motion per sign by 30, amounts to 56'50"/day at apogee and 1°1'20"/day at perigee. Note that the motion per day is effectively 1°/day plus or minus a small correction. The informant apparently memorized a table of average corrections over eight-day intervals and thus corrected the Sun's longitude by the sum of the corrections, with some interpolation within the eight-day intervals for any excess over a multiple of eight. Thus, the date at which the Sun reached the beginning of the sign in which it is located at the time of the eclipse gives the number of days that the Sun must traverse within that sign; this difference in numbers of days since the Sun entered the last sign and the date of the eclipse gives the longitude within that sign: Taurus 19;6,48,5 (or 19°06'48" 5/60 in Taurus). This type of expression of longitude was used by the Greeks and Babylonians from whom the notation was borrowed.
The longitude of the Sun is only one part of what needs to be known. The position of the Moon is another. The number of days since an epoch when the Moon was at apogee and the rate of the Moon's motion on successive days are tabulated along with the sum of elapsed motion: the total travel in degrees of longitude since apogee. The length of an anomalistic month is 27d55455 (see §2.3.5 and Table 2.5) or roughly 27 5/9 days (in decimal notation, ~27d55556). The latter approximation is expressible without fractions by setting
In nine months, the error accruing from this approximation is only 0d00606. From §2.3.5, the anomalistic period is shorter than is the tropical period of the Moon because of the advancement (in the direction of the Moon's motion) of the line of apsides. The tropical period—the time taken for the Moon to move 360° of celestial longitude—is similar to the sidereal period over a nine-month interval; so the difference need not concern us here. The advance of the line
of apsides amounts to ~40°/year or 3.08°/month. Therefore, over a nine-month interval, the apogee will have slid forward 27°735. Thus, the correction to the anomalistic interval is applied and the longitude of the Moon determined, after some additional corrections.
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