## Info

gu4 29 3.83639 su'

sig 1.236 sig 27 ...

a An excerpt from Neugebauer's (1955/1983, Vol. I) interpretation of Tablet 5, annotated and transcribed in decimal notation here. Units are given below each column header. b At the top is the first line of the original table in Neugebauer's notation. Because of the length of the hexadecimal digits, some numbers overflow their columns. c Columns are identified as follows (see text for a full description): Column T: The date of the syzygy in years and months of the Seleucid Era, which began in April, 311 b.c. (see §4.1.5). Column F: The "monthly variation." Related to the Saros. See text for details.

Column B: The longitude in degrees from the start of the zodiacal sign. "hun" = Aries, etc., as per the order in Table 2.4.

Column C: Length of daylight at Babylon. The unit H is "large hours," equal to 60° or 4 hours.

Column E: Latitude of the Moon's center, in units of the barleycorn (se) = 50 arc-sec or 0.01389 degrees.

Column F: Lunar "velocity" represented as a linear zigzag function measureed in degrees per day.

Column J: Correction to the const. solar speed assumed for col. G; "lal" means "subtract," i.e., all corrections in the J column in this table are negative. Column C': "lal" here means "subtract"; "tab" means "add." All quantities in this column are in degrees; divide by 60 to convert to large hours. Column K: The corrected length of the month: G + J + C'. Column M: Date of syzygy. dirig = month XII2, bar = month 1, etc.

Columns P1, P3: Durations of first and last visibility. These involve approximations to equatorial arcs computed from ecliptic longitude differences.

Figure 7.5. A plot of column II (B) against month number from a tablet from Babylon (No. 5), for the interval 146-148 Seleucid Era (166-165 to 164-163 b.c.). Thirty degrees have been added to the ordinate on each change in zodiacal sign of the columnar entries to show the progression of the Sun eastward among the stars.

Figure 7.5. A plot of column II (B) against month number from a tablet from Babylon (No. 5), for the interval 146-148 Seleucid Era (166-165 to 164-163 b.c.). Thirty degrees have been added to the ordinate on each change in zodiacal sign of the columnar entries to show the progression of the Sun eastward among the stars.

Figure 7.6. The differences between successive entries of Column B of Babylon Table 5 [viz., y(x +1) - y(x) + 30°], plotted here against month number, provide a measure of solar velocity in degrees per month. The plot shows the solar velocity as a step function, constant for a number of months, and then shifting to another constant velocity, characteristic of System A tablets. The intermediate values between the steps are merely differences between the end of one series of constant velocities and the beginning of the next; the actual change is sudden and occurs at a fraction of a month number between the two series.

Figure 7.6. The differences between successive entries of Column B of Babylon Table 5 [viz., y(x +1) - y(x) + 30°], plotted here against month number, provide a measure of solar velocity in degrees per month. The plot shows the solar velocity as a step function, constant for a number of months, and then shifting to another constant velocity, characteristic of System A tablets. The intermediate values between the steps are merely differences between the end of one series of constant velocities and the beginning of the next; the actual change is sudden and occurs at a fraction of a month number between the two series.

can be used to describe each of three lines; the coefficients a and b quantify the properties of the zigzag function. We will use subscripts to distinguish between the parameters and variables of the three equations. For the first decreasing segment, the slope, ai, is the difference between two successive values in Table 7.10, namely, -1,52,30 in sexagesimal notation, or -1.8750 in decimal notation.19 The intercept, b1, is easily calculated from the value of y at a given value of x. So, at x1 = 1, y1 = 25,18,45 = 25.3125. Therefore, b1 = 25,18,45 - (-1,52,30) x 1 = 27,11,15 (or 25.3125 +1.8750 x 1 = 27.1875).

This is the y-value at "month 0." Because the second line segment is horizontal, a2 = 0. Therefore, b2 = y2 = 18,8 = 18.1333. For the third line segment, for, say, x3 = 15, y3 = 11,11,15 = 11.1875. Again a3 = -1,52,30 = -1.8750; so b3 = y3 - a3 x x3 = 11,11,15 - (-1,52,30) x 15 = 11.1875 + 28.125 = 39.3125.

The intersections of any two line segments are found by setting the pairs of equations equal. From the first pair of lines, a1 x x1 + b1 = a2 x x2 + b2 so that the intersection values x' and y' are b2 - b1 -9.0542

a1 - a2 -1.8750 From (7.1), y = y1 = y2 = a2 x2 + b2 = 0 x (4.8289) +18.1333 = 18.1333.

The intersection of the second and third line segments is again at y = 18.13333, and the intersection values x" and y" are

a3 -1.87500

The difference between the two x intercepts is x"-x' = 11.29556 - 4.82889 = 6.46667.

If the segments of the function shown in Figure 7.6 were completely symmetric, we could multiply this interval by 2 and find the period. This is not the case, however, as can be seen by considering the next intersection: At y"' = y4 = 7.06667, y"" -b3 x" = 1-i± = 17.19778.

Hence,

Instead, the correct period is determined by taking the difference x" - x':

When this number (lunations per year) is multiplied by the mean length of the synodic month (29.53059d),20 the result is 365.261d, a reasonable approximation to the length of the year. On the other hand, dividing the quantity 12.36889 into a round 365.25-day year gives a lunar synodic period of 29d52973. With this period, 223 lunations amounts to 6585d1301, the decimal part of which is 7,48,24. This is not the value "17,46,40" for the residual from 6585 days noted earlier, but, in any case, the difference is small. The remainder 17,46,40 implies that Psyn = 6585.2963/223 = 29d53048. We next check the lunar periodicity in a System B tablet.

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