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That the cycle ultimately works would be proved by sample cycle 5 of Table 5.4, which shows a triple Saros—after 54y 1m 4d, an eclipse will be seen in roughly the same longitudes, at the same time of day, as well as fulfilling the other condi-tions.21 The triple Saros series is also known as the Exeligmos. It should be noted, though, that there are several Saros series occurring at any one time; thus, the predictability is considerably more complex than first meets the eye, and observations of eclipses alone are most unlikely to yield unambiguous predictions. On the other hand, it is certainly easy to spot when an eclipse is possible, merely by keeping track of the nodes of the Moon's orbit. The eclipses in a Saros series successively shift westward by about 1/3° with respect to the node, or, expressed in terms of time for the Moon to reach the node from the moment of syzygy, ~1/30 day. This and a slow latitude drift limit the length of any one Saros series to ~70 cycles, 15,610 lunations, or ~2100 years. In the case of solar eclipses, the descending-node series begins near the South Pole, and proceeds north; the ascending-node series begins near the North Pole, and proceeds South. In the case of lunar eclipses, the descending-node saros series begins with a penumbral eclipse of the Moon's northern limb; the ascending-node series begins with a penumbral eclipse of its southern limb.

Other historically important cycles include the Inex, the Tritos, the Thix, and the Fox. The periodicity of the Inex cycle was discovered in recent times by Simon Newcomb (although DHK thinks it probably was known in Mesoamer-ica). It involves 358 lunations (see cycle 4 in Table 5.4).

21 The latitude drift in the successive Saros will cause the eclipse to be only a partial one at a certain place on Earth, if the previous eclipse was total at that place.

According to Sivin (1969, pp. 39-40), the magnitude of an eclipse in this cycle can vary greatly from one eclipse to the next. The Inex cycle series is a very long one (23,000y or ~800 cycles), and unlike the Saros series, there is no gradual change in the longitude or latitude of the central track nor in the magnitude. It is especially difficult to track both very early and very late eclipses in the series.

The Tritos cycle (cycle 2 in Table 5.4) was used extensively in China. The term was coined by G. van den Bergh (1955); it is 1/3 of a Mayan cycle of 405 lunations and is equal to the difference between the Inex and Saros cycles. The magnitudes of eclipses in a series of this cycle will be large only for the middle 25-30 eclipses, and the overall length of the series is about that of the Saros. The Chinese of the Han dynasty recognized the Tritos cycle in a sense. Chinese astronomers of the Han dynasty (206 b.c.-220 a.d.) believed that there were 23 eclipses per 135 lunations, an incorrect number by about 6, but usually sufficient to predict a warning. This matter will be discussed further in §10.1.2.2.

The Thix and Fox cycles were important for the Maya. The names of these cycles were suggested by Smiley (1973). The Thix is a cycle of 25y630 = 9361d20, ~317 lunations, and ~36 Tzolkin (see §12.11). The Fox cycle is a triple Tritos cycle of 32X745 = 11,959d8, ~405 lunations, and ~46 Tzolkin.

An early prediction of an eclipse has been claimed for the Ionian astronomer Thales. Herodotus (5th century, b.c., Book I, §74) describes an eclipse that took place during a battle between the Medes and the Lydians:

In the sixth year of the war, which they had carried on with equal fortunes, an engagement took place in which it turned out that when the battle was in progress the day suddenly became night. This alteration of the day Thales the Milesian foretold to the Ionians, setting as its limit this year in which the change actually occurred. (Tr. G.S. Kirk in Kirk et al. 1983, pp. 81-82; cited in Panchenko 1994, p. 285)

The prediction would be meaningless if the event did not apply to an eclipse visible in the Mediterranean, because of the frequency of eclipses on a global scale. An eclipse was visible at the site of the battle on May 28, 585 b.c., but the vagueness of Herodotus's description and the lack of any other evidence for the existence of geographic predictability of eclipses among either Greeks or Babylonians at this time convinced Neugebauer (1969, p. 142) that the prediction was merely an unreliable story. Although earlier scholars had argued incorrectly that Thales could have used the Saros cycle to predict the eclipse, a more recent argument is that he might have had access to data from which the Exeligmos cycle could have been deduced. Dimitri Panchenko (1994) has made a case that the account of a prediction is probably accurate and that Thales was capable of making such a prediction, an argument made earlier by Hartner (1969) as well. First Panchenko notes that according to Diogenes Laertios,22 Thales's feat is mentioned by Heracleitos, Democritos, and Xenophon, so that it was widely attested and therefore deemed noteworthy in the ancient world. The source of Diogenes Laertios's informa tion was Eudemos of Rhodes, one of the earliest historians of science (see §7.3). Panchenko then argues that the text is usually misinterpreted. The phrase from Herodotus that Thales "set as its limit this year in which the change actually occurred"23 is most often interpreted to mean that Thales predicted merely the year of the eclipse. Panchenko, however, suggests that the word "limit" implies "no later than" the current year—the year of the prediction or of a series of years. He also argues that a respected figure such as Thales, mentioned even by Plato, would not have risked failure by casually predicting an eclipse nor of stating it in so vague a manner as to render the prediction meaningless. He speculates that during Thales's lifetime, astronomers from some of the Assyrian cities that were destroyed by a resurgent Babylon could well have taken refuge in Egypt, Assyria's ally, where Thales could have learned of records of previous eclipses, and where sufficient data might have been available to discover the exeligmos or triple saros cycle (last series in Table 5.4). As we have noted, the saros cycle could not have been used to predict similar eclipses in the same longitude region, and from a geographically limited database, it is unlikely that such a series could have been recorded. The 54.09-year interval would have been recorded in a database, if it extended back far enough, and an apparent, though not reliable, 27/28 year series of eclipses may have also played a role.

A difficulty with the argument is that an eclipse on May 28, 585 b.c. could not have been predicted, according to Panchenko, by any method currently known. He is forced, therefore, to suppose that the eclipse prediction was made in 585 b.c., shortly after the 585 eclipse, perhaps at a Pan-Ionian festival, which would require the battle to have occurred either on Sept. 21, 582 b.c. or on March 16,581 b.c., dates of total eclipses in the region. Panchenko favors the former because it was both earlier and of a larger magnitude than was the March eclipse. This in itself is an interesting prediction that present-day historians of science will need to attempt to verify. If Panchenko is correct, because the pan-Ionian festivals were held every four years, the prediction could have been made to assure the Ionians that such events were predictable, and the existence of eclipses in 582 and 581 b.c. could have been predicted from cycles deduced from Assyrian records.

Many atlases of eclipses are available with which ancient observations can be compared. For solar and lunar eclipses of the entire ancient world, only Oppolzer's Canon of Eclipses (1887/1962) has a nearly complete record. He lists 8000 solar and 5200 lunar eclipses. Schove (1984) states that the beginning and end points of Oppolzer's eclipses could have errors as large as 100 km on the polar plots provided. Oppolzer included a correction for the secular variation in the lunar longitude. It had been discovered by Laplace as early as 1786 that if the Earth's orbital eccentricity slowly decreased, the Moon would accelerate; subsequently, in 1853, J.C. Adams showed that the "acceleration" from this source would be ~6" (the "acceleration in this context is the

22 A noted 3rd-century b.c. historian of science, writing in Lives of 23 "oUpov ppo6é|ievo; eviautov toUtov, év t(fl §| Kai égeveto |

Eminent Philosophers (tr. Hicks 1938). |ieta|3o1r|."

coefficient of a term involving the square of the time, in Julian centuries). The slowing of the Earth's rotation and the magnitude of the tidal friction acceleration were not fully realized until the 20th century. The empirical corrections actually applied by Oppolzer and colleagues were described in his Syzygientafeln (1881), and are not provided in the Canon. P.V. Neugebauer (1931a,b) suggested that the error in timings amounted to only ~20 minutes in 700 b.c.

Other eclipse tables for selective purposes are:

(1) For Near East sites by P.V. Neugebauer and Hiller (1931) for the interval 4200 b.c. to 900 b.c. for solar and 3450 b.c. to 900 b.c. for lunar eclipses, and by Hudek and Mickler (1971) back to 3000 b.c.

(2) For the Mediterranean, by Ginzel (1899) for solar eclipses from 900 b.c. to 600 a.d.

(3) For the Far East, by Mucke and Meeus (1983) for the interval 2004 b.c. to 2526 a.d.; and by Newton (1977), whose lunar eclipse canon covers the years -1500 to -1000 of China, specifically. Also for the Far East, Stephenson and Houlden (1986) have solar eclipse maps—with modeled values of DT, for the interval 1500 b.c. to 1900 a.d., and Schove and Fletcher (1984) discuss sources of more localized maps. Liu and Fiala (1992) offer a modern and comprehensive canon of world lunar eclipses covering the interval -1500 to +3000 (with computer software to recreate the circumstances of eclipses on images of the Earth) with DT corrections. Their Table 3.3 contains corrections corresponding to the formulae of Stephenson and Morrison (1984) for the interval -1500 to +1500, but with a slightly different lunar tidal term (-23.895"/cy2 from the value -26"/cy2). This enables the user to subtract out the Stephenson and Morrison correction and to add other corrections, as improved values become available. Meeus and Mucke (1983) provide lunar eclipse tables for the interval 2003 b.c. to 2526 a.d.

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