System A

van der Waerden (1974, p. 227) notes that the units of ® are the same as for column C, in large hours (60° or 4 hours) in the System A tablets. Column B gives the calculated longitude of the Moon in us, degrees of a zodiacal sign, based on the assumption of constant solar speed. Column C contains the length of the day in large hours during the month. Column E gives the latitude of the Moon in units of se (literally "barleycorn," 1/72 of an us or degree). Column F is the velocity of the Moon in degrees per day. Column G gives the length of the lunar month in large hours, assuming a constant solar velocity of 30°/month, but variable lunar velocity. Column J is a correction to be applied to column G for too high a velocity of the Sun during half of the year. The underlying assumption here is that the Sun moves according to a two-step function with either a high speed or a low speed. Column C expresses what Neugebauer calls the epoch of syzygy11 in civil time, which began at sundown. The unit is the us. Column K is the corrected length of the month, the sum of columns G, J, and C'. Column M gives the day, month, and large hours of syzygy (conjunction in this context). Column P is the time between sunset and

16 For detailed critical assessment of an early Babylonian origin for the "Saros" eclipse cycle, see Neugebauer (1957/1969, pp. 141-144). The significance of the interval of 223 synodic months is that it is equal to 242 draconitic months (see §2.3.4 and §5.2.2) and could be used to predict eclipses. This interval is not an integral number of days, and thus successive Saros eclipses may not be seen from the same place on Earth.

17 The term syzygy may refer to the situation when the centers of the Earth, Moon, and Sun are in a line, but Neugebauer and many others use the term in a slightly broader sense, to mean either the conjunction or opposition condition. In the first usage, a syzygy must result in an eclipse; this is certainly not the meaning in most of the tables.

moonset or between moonrise and sunrise. The designations and contents of some of the columns change if the ephemerides are for full rather than new moons. In addition, there are variants and additional columns, as our examples illustrate.

Some of the important features can be seen in Table 7.10, excerpted from Neugebauer's (1955/1983, Vol. III, Text No. 5, p. 10) interpretation of eight fragments of a tablet (No. 5) from Babylon (see Figure 7.4). The tablet deals with the interval 145-148 SE (Seleucid Era), corresponding to 166-165 to 164-163 b.c. This particular tablet provides one of the most complete examples of the usage of System A.

The structure of the table is revealed in plots of the columns against month number, n; the date column is designated by Neugebauer as Column 0. The first line marks the lunar conjunction that occurred at the 13th month of the previous year (145 S.E.), which we designate month 0. Each entry designates positions in terms of a length of arc along the ecliptic, measured in the zodiacal sign given in the text. Each sign is 30° long. This angular quantity, which cuneiform text scholars call the "longitude," is the forerunner of the much later celestial longitude (§2.3.3); however, the counting begins again at the start of each sign; so that the sign must be specified.18 A single value of the longitude is given for both Sun and Moon because they are in conjunction at each entry. In each successive entry, the Moon has completed a 360° sidereal period and ~30° more for a total revolution of ~390°, when it meets the Sun (which moved ~30° eastward in the course of the month) once again. A plot (Figure 7.5) of the continuously increasing values, i.e., column II (B) against month number with the addition of 30° to the column B entries at each change in zodiacal sign, shows merely the progression of the Sun eastward among the stars. Such a plot is less revealing than that of the differences between successive entries of Column B (Figure 7.6).

Each ordinate value is y(x + 1) - y(x) + 30°, plotted here against the month number, x, and is a measure of the solar velocity in degrees per month. It reveals the step function that characterizes the System A approach. Notice the constancy of the solar speed for an interval of months followed by a change to another constant velocity. 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. It is these instants that we would like to determine because the period of the function can be found by calculating the moments of intersection among three successive line segments of Figure 7.6. The equation for a straight line,

18 At this time in Babylonia, the vernal equinox was located at 10° of Aries in System A and 8° in System B (van der Waerden 1974, Vol. II, p. 215). In any case, as can be seen in Column B of Tablet 5 (Table 7.10), the tabulated quantities appear to be measured from the beginning of the signs: The longitude at date 2,28 III is given as 0,48,45 kusu = 0.8125° Cancer and is followed in month IV by 28,56,15 kusu = 28.9375° Cancer. In month V, the entry is 27,3,45 a = 27.0625° Leo. The difference between successive entries, after adding 30° to the later entry when needed, is 28.125°/month.

Table 7.10. Babylon Tablet 5: An example of System A.a

0 0