## The Step and Zigzag Functions

The construction of the lunar ephemerides from Systems A and B (§4.1.2) are extensively discussed by Neugebauer (1948,1983) and by van der Waerden (1974). Therefore, we merely summarize the two basic methods. The ephemerides are recorded on baked clay tablets, inscribed with reed stylus pens, and often cover both sides and even the edges of the tablets. The tablets record the successive dates, and associated data, of lunar conjunctions or oppositions.

One method involved a constant velocity for the Sun over a certain number of days (hence, over a certain region of the ecliptic), followed by another constant velocity for a similar number of days. When the longitude is plotted against month number, therefore, it describes a step function. Neugebauer (1957/1969, p. 114) refers to the theory behind this method as "System A."

A second method involved solar velocities that changed month by month, each value differing from the next by a constant value over a range of months. The latter differences changed sign after a certain number of intervals so that when these differences are plotted against time, they portray a zigzag pattern, hence, the name zigzag functions. Neugebauer refers to the theory behind this method as "System B."

The two methods differ in sophistication and accuracy, so that most scholars infer an earlier origin for the simpler System A. However, both were in use for the entire interval for which we have records: 250 to 50 b.c. (Neugebauer 1955/1983, Vol. 1, p. 42), and Neugebauer thought that elements of System B might even antedate System A.

Neugebauer (1957/1969, p. 115) expresses doubts that the simultaneous use of the two methods implies the existence of separate schools of astronomy, because they are found in both centers from which we have sufficiently attested data, Babylon and Uruk. On the other hand, in Astronomical Cuneiform Texts, Neugebauer (1955/1983, Vol. 1, p. 42) expresses less reservation about the idea. The origin and the founders of the schools, if this interpretation is correct, are unknown.15 In any case, zigzag and step functions may be found together in the same tablet (e.g., in Babylon Tablet 5, described below); the method in the tablets may be determined by whether the solar motion (revealed in Column B) is dual or multiple-valued.

The structure of the tablets is summarized in Table 7.9, adapted from Neugebauer (1955/1983, Vol. I, p. 43), with his notation. As a rule, the first column gives the date of either a conjunction of the Moon and Sun (astronomical new moon) or opposition (full moon) in the year of the Seleucid Era and the month (and sexagesimal fractions thereof). The contents of the other columns differ depending on the method. The columns are read from top to bottom, with parallel columns read from left to right, as in modern tables.

In general, the first column contains a date (the year in the Seleucid era and then each month in that year). In texts of System A, the second column gives a quantity, designated analogous to the lunar velocity. Fr. Kugler, a pioneer in the study of the Mesopotamian astronomical texts, thought

15 In Roman sources (Strabo, Pliny, and Vettius Valens), two names appear that are also found in tablets from Babylon: The name Nabu-rimmannu appears on a tablet displaying System A material, and the name Kidin (or Kidinnu) on another in System B. They were the basis for the belief that the individuals named were the founders of the respective schools. However, Neugebauer (1955/1983, p. 16) indicates that no Greek or Mesopotamian source explicitly names these two as the founders, that the tablets are from a late period, and that tablets from Uruk, which are older, do not mention them.

that this column recorded the lunar diameter, in units of the quarter-degree, a unit not attested outside the ® columns of System A texts (Neugebauer 1955/1983, Vol. I, p. 44). Subsequently, the mystery was solved through a Babylonian text of procedures discovered by Neugebauer (1957) and cited by van der Waerden (1974, p. 226). The text states

17,46,40 is the increase or decrease in 18 years.

Interpreting the "18 years" as an approximation to the Saros, van der Waerden (1974, pp. 226-229) argues that ® is the difference between a "Saros" period (223 synodic months)16 and the interval 6585 days. The Saros in its modern usage (not in its ancient usage of a fixed interval of 3600 years) is an interval of exactly 223 synodic months, but the actual length of the synodic month varies for reasons already discussed in §2.3.5 (the synodic period given in Table 2.6, 29d530589, being an average value of the length); hence, any true repetition of eclipses in a Saros "interval" must vary. With the mean value, 223Psyn equals 6585d32135, so that the residual, 0d.32135, in sexagesimal notation is 19,16,51. More to the point, however, it must be asked what value or values were used in Mesopotamia for the lunar synodic period. The answer to this question lies in the tablets. To find it, we must analyze them.

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