System B

An example of the use of System B is found in the translation of Tablet 120 (Neugebauer 1983, Vol. III, Text No. 120, p. 68; excerpted in Neugebauer 1957/1969, pp. 110-112). A reconstruction of the first 12 columns of the obverse side of this tablet provides the data shown in Table 7.11. The first column (Col. A) is the date, as per System A texts; the second column (Col. B) gives the solar rate of motion in degrees during the month; the third column (C) gives the Sun's position on the ecliptic. Other columns are as indicated in Table 7.9 and in the notes to Table 7.11. A plot of the successive lines of data in Column A against month number (Figure 7.7) shows a characteristic zigzag pattern that indicates changing velocity month by month, but by the same constant value. Exceptions to the constancy are again found only in regions of turnaround, where the monthly difference changes sign and applies in mid-month. The equations for straight line segments are given in Table 7.12 for three line segments 1, 2, and 3, respectively, from which we derive the minima and maxima of the zigzag functions from the intersections of these line segments.

At minimum, x' = 2.516950, y' = 28.177685, and at maximum, x" = 8.701517, y" = 30.033055. From these, using Neugebauer's terms, we derive the amplitude (essentially y" - y'), D = 1.855370, the mean, m = 29.105370, and the period, P = 2D/|a| = 12.36913, where a = 0.03 is the slope. In the sexagesimal system, the arithmetic is more complicated, but instructive to do. Using Neugebauer's notation, the minimum is m = y' = 28,10,39,40 and the maximum is M = y" = 30,01,59,00, so that a direct manipulation of each place yields D = M - m = 1,51,19,20. Note that the treatment of the last place requires a borrowing of 60 from the column to the left. M + m = 58,11,98,40 = 58,12,38,40, and from this, we calculate the mean m = (M + m)/2 = 29,6,19,20. Because D = 1,51,19,20, and |a| = 0,18, we calculate

P = 2(1,51,19,20)/(0,18) = (2,102,38,40)/ (0,18) = (3,42,38,40)/(0,18).

19 The conversion is obtained by dividing the rightmost segment by 60, adding the result to the quantity to the left, dividing that by 60, and so on.

20 A close match to this modern value is determined for the period of Column G from Tablet 120 by Neugebauer (1955/1983, Vol. I, p. 78).

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 T(Date)
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