## The Reconciliation of Solar and Lunar Calendars

In this section, we deal with the attempts in Europe and the Near East to reconcile the motions of the Sun and Moon. The Mayan Calendar and its Mesoamerican variants, which dealt with the problem in an entirely different way, will be discussed in §12.

The basic calendar problem in this context is the fact that the tropical year17 is not an integral multiple of the synodic month: 365.24219878/29.530589 = 12.36826664. The remainder, 0d368266,..., is not easily dealt with. If we approximate the length of the synodic month by 29d5, however, the problem is more easily grasped. In this case, the following solution arises: 12 months x 29d5/month = 354d. This is 11d short of a year whose length is only 365d (such a year was used in ancient Egypt and was thus known as the Egyptian year18; see §8.1.4). In 3y, the deficit is 33d. The simplest solution is to add an additional month during some years in order to approximate a commensurability between the two periods. The process of adding an extra month to certain years is called intercalation. The problem in our example is that adding one month every three years would require that that intercalary month not be a synodic month interval, because it would be 33d long, not 29d5. This may not strike us as a terrific problem today, but to most in the modern world, the Moon is not a god or goddess to be appropriately served and honored, and the goal of achieving harmony with the workings of the cosmos is not a particularly strong driving force. The earliest known successful attempt at intercalation is that of Meton of Athens, in 431 b.c. The Metonic cycle is 19y long, an interval roughly equivalent to 235 lunations. The scheme was to employ 12y of 12 months each (giving 12 x 29d*5 = 354d), and 7y of 13 months each (giving 13 x 29d5 = 383d5) during this interval. In terms of total days, the 19y would contain

[12 x 354d] + [7 x 383.d5] = 4248d + 2684.5d = 6932.5d

= 18.9932 years of 365d each.

In terms of a slightly better approximation to the length of the tropical year, viz., 365d25, the interval 6932d5 amounts to 18y980; finally, with the value 365d24219878, we obtain 18y981. The Metonic cycle would seem to work best with the Egyptian year, although the Babylonians certainly knew the year to far better precision than this. The Metonic cycle actually works even better considering the actual accrual of time over the 19y interval:

= 4252.d 4048 + 2687.2836 = 6939.d688 or 19y 000248

of true tropical-year length. The resulting error is only 0d0902/cycle, which accumulates to 1d approximately 11 cycles later, a total span of ~210 tropical years. Even more

17 The period of time for the Sun to return to the same point on the ecliptic, for example, the vernal equinox. See Table 4.2 to distinguish this interval from other types of year.

18 Such a length may seem to be hopeless as far as keeping in step with seasons was concerned. Yet the 365d year was used in Egypt, Mesoamer-ica, and Hawaii. The Egyptians were aware of the slippage of their calendar with the seasons, however: The period of return to the same calendar date, the Sothic period, was 1460 years long. Over this interval, 532,900d elapsed. The slight variation of 1/4 day from year to year adds to 365d in this interval (see 8.1.2 for further discussion). A similar effect may have been noted in Mesoamerica (cf. 12.6).

remarkably, 6939d688 amounts to 18.999830 years of 365d25 length for an error of only -0.0620d/cycle, which accumulates to 1d in 16.129 cycles or ~306 Julian years, as the 365d25-years are known. It is important to state that although it is valuable to be able to see how "good" a cycle may be on very long time scales—for which modern values are useful—in order to see how it was actually used and thought about, it is extremely important to test such cycles in the systems for which they were designed, inaccuracies and imprecisions and all. With a mean synodic month length of 29d53, and a year length of 365d25, for example, the Metonic cycle works very well (the error is ~0d2). However, we have it on good authority (from Hipparchos, quoted by Ptolemy; cf. Toomer 1984, p. 139) that Meton used a value for the length of the year of (36514 + 1/76 d = 365d2631579 ...). With this value, we get in 19 years 6940d000, compared with the value above, assuming a 29d5-length month, namely, 6939d688—a very good approximation. However, see §7.3 for Hipparchos's contribution to the intercalation problem and improvements to the determined length of the year.

The Metonic cycle, or a variant of it, the Kallipic cycle,19 has been in wide use among Mediterranean cultures since its discovery. Neugebauer (1957/1969, p. 7) notes that Athens rejected Meton's proposal in the 5th century b.c., although it was incorporated into the Babylonian calendar. It was retained in the Seleucid empire of Babylonia and is still employed in both Jewish and East Indian religious calendars. Toomer (1984, pp. 337-340) has argued that Meton's purpose was not the reform of the Athenian calendar but the establishment of an astronomical chronology.

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