On the epoch of the Antikythera mechanism and its eclipse predictor
Authors
Abstract
The eclipse predictor (or Saros dial) of the Antikythera mechanism provides a wealth of astronomical information and offers practically the only possibility for a close astronomical dating of the mechanism. We apply a series of constraints, in a sort of sieve of Eratosthenes, to sequentially eliminate possibilities for the epoch date. We find that the solar eclipse of month 13 of the Saros dial almost certainly belongs to solar Saros series 44. And the eclipse predictor would work best if the full Moon of month 1 of the Saros dial corresponds to May 12, 205 BCE, with the exeligmos dial set at 0. We also examine some possibilities for the theory that underlies the eclipse times on the Saros dial and find that a Babylonian-style arithmetical scheme employing an equation of center and daily velocities would match the inscribed times of day quite well. Indeed, an arithmetic scheme for the eclipse times matches the evidence somewhat better than does a trigonometric model.
1 Introduction
Lunar eclipses were predicted with reasonable accuracy by the Babylonians using the Saros cycle, involving a schematic distribution in the arrangement 8-8-7-8-7- (Steele 2000a). Here, each 8 represents a group or sequence of eight eclipses at 6-month intervals. And each 7 represents a sequence of seven eclipses at 6-month intervals. The last eclipse of any 8 or 7 group is separated from the first eclipse of the following group by only a 5-month interval. Thus, the locations of the 5-month intervals are indicated by the hyphens. The total number of lunar eclipses in one cycle is \(8 + 8 + 7 + 8 + 7 = 38\), and the duration of the cycle of 38 lunar eclipses is \(47 + 47 + 41 + 47 + 41 = 223\) synodic months. Of course, this is a cycle, so the starting point is arbitrary. Thus, one can also have lunar eclipses in the patterns 8-7-8-7-8- or 7-8-7-8-8-, for example. And in an actual sequence of predicted eclipses, it is by no means necessary to start at the beginning of an 8 or a 7 group. Solar eclipses were also predicted by the Babylonians using the same sort of 8-8-7-8-7- scheme. Perhaps we should refer to these as solar eclipse possibilities (EP), for in Babylonian astronomy there was no reliable way to determine which solar eclipses would actually be visible from Babylon.^{1}
The Antikythera mechanism (AM) is a Greek geared astronomical computing machine, built sometime between the late third century and the early first century BCE, which was recovered from an ancient shipwreck in 1901. A remarkable feature of a recent study by Freeth et al. (2008) is a demonstration that eclipses were predicted on the lower back dial of the mechanism by means of the Saros cycle. Only a small portion of the inscriptions on the Saros dial are preserved. Nevertheless, Freeth et al. were able to show that the predictive scheme is consistent with a Babylonian-style 8-8-7-8-7- pattern. The reconstruction is greatly aided by the presence of index letters in the glyphs for the eclipses. Each month box that bears an eclipse glyph is labeled by a Greek letter that shows where the glyph stands in the sequence. Thus, even though a large chunk of the dial is missing, it is possible to be sure exactly how many eclipse glyphs would have been carried by most of the missing part of the dial. Freeth et al. attempted a fit of the eclipses inscribed on the Antikythera mechanism, in an effort to determine which year or years might best correspond to the first year of the Saros dial. This would be one important approach, among several, to dating the mechanism. Other methods include archeological dating of the ship’s other cargo, radiocarbon dating of the ship’s timbers, and the dating of the Greek inscriptions on the mechanism by means of the forms of the letters. Of course, the first two of these do not directly give any information about the mechanism itself, while the third (dating by letter forms) has an uncertainty of perhaps as much as plus or minus a century. The Saros dial is practically the only astronomical feature of the mechanism that could possibly afford a more tightly constrained dating.
In their analysis of the mechanism’s Saros dial, Freeth et al. fitted the extant eclipse glyphs to modern computed eclipse data. A great deal of uncertainty is involved in deciding what to do about penumbral eclipses, easily predicted by modern theory, but rarely if ever mentioned by ancient astronomers. If they did not allow any penumbral eclipses to be counted as corresponding to glyphs on the mechanism, Freeth et al. found that they had no possible epoch dates. If, on the other hand, they allowed themselves the flexibility of counting an occasional penumbral eclipse as one of the Babylonian eclipse possibilities, they found they had over a hundred possible fits for epoch date. Clearly, one would like a result in between these two outcomes.
In this paper we shall offer a fresh analysis of the eclipses predicted on the Saros dial of the Antikythera mechanism, with the goal of establishing the epoch year for this feature of the mechanism. By epoch year we mean the starting year or “year 1” from which the ancient mechanic reckoned time for the purpose of the mechanism’s display. The epoch is not necessarily the same as the date of manufacture, though it would be surprising if they were very widely separated in time.
Our working hypothesis is that the Antikythera mechanism’s eclipse glyphs should be fitted, not to modern computations of eclipses, but to Babylonian practice. Such an analysis has the advantage of removing the spurious penumbral eclipses from consideration. Moreover, the 8-8-7-8-7- scheme is far from perfect: it can predict an eclipse that does not actually occur, fail to predict an eclipse that does occur, or predict an eclipse for the wrong month. Thus, it is a fallacy to suppose that the epoch of the Antikythera mechanism’s Saros dial could be reliably determined simply by matching its predictions against real eclipses. Although it is a priori likely that the Babylonian Saros scheme was the ultimate source for the predictive scheme represented on the mechanism, one should not simply assume this without scrutiny. We shall present evidence below that the eclipse prediction scheme on the Antikythera mechanism really does derive from the Babylonian example.
Of course, in parts of our analysis, we shall make considerable use of theoretically computed phenomena, such as nodal distances and eclipse times. We will show that it is possible to introduce constraints sequentially, as with a sort of “sieve of Eratosthenes,” to systematically restrict the number of possible solutions. The tools at our disposal include as follows: the distributions of the solar and lunar eclipse glyphs on the Antikythera mechanism, which may be fitted to Babylonian patterns in a number of different ways; the connection between the solar and lunar patterns, which may be compared with similar connections in the Babylonian material; the “omitted solar eclipses” (defined below), which may be used to classify the solar eclipses and to establish their places in the 8-8-7-8-7- sequence without ambiguity; nodal elongations, which may be used to narrow the range of fits; eclipse times, which may be used to restrict the solutions to a single Saros eclipse series, and even to rule out two fits of every three in that series. When these approaches are used in conjunction, it is possible to arrive at a single most likely epoch for the eclipse predictor.
2 Terminology
A single Saros series may persist for 1,200 years or more. At any moment in history, about 40 Saros series are in progress. Each eclipse can be labeled by its Saros series number in the standard catalogs of Espenak (Five Millennium Catalog of Solar Eclipses; Five Millennium Catalog of Lunar Eclipses). Saros series of solar eclipses are given even numbers if they occur near the descending node of the Moon’s orbit and odd numbers if they occur near the ascending node. When a new solar Saros series begins at the descending node, the new Moon occurs about \(18^{\circ }\) east of the descending node (and so below the ecliptic). A partial eclipse of the Sun will be visible from southern parts of the Earth—just the northern limb of the Sun will be grazed. At progressively later eclipses in the same series, the Sun will be centrally eclipsed, and then, gradually this Saros series will disappear off the south limb of the Sun (Espenak, Eclipses, and the Saros). Lunar Saros series have odd numbers if the eclipses occur with the Moon near the descending node of its orbit and even numbers if they occur with the Moon near the ascending node.
The reason for the finite lifetime of a Saros series is that the Saros relation 223 synodic months (sm) \(=\) 242 draconic months (dm) is not exact. A more precise correspondence is \(223\,\hbox {sm} = 241.99868\,\hbox {dm}\) (Smart 1977, p. 420). Thus, when 223 synodic months have elapsed, the time elapsed in draconic months is \(<\)242 by 0.00132 dm, which corresponds to \(0.475^{\circ }\) of mean motion with respect to the node. That is, from one eclipse to the next in the same Saros series, the position of the mean conjunction moves westward by about 0.475\(^{\circ }\). Now, the Sun’s ecliptic limits from the node for a solar eclipse to be possible vary from a maximum of about 18.4\(^{\circ }\) to a minimum of about 15.4\(^{\circ }\) (Smart 1977, p. 390). Let us take 17\(^{\circ }\) as a round value. The width of the nodal zone is thus 34\(^{\circ }\); the position of the mean conjunction recedes by about 0.475\(^{\circ }\) per Saros and so takes roughly 72 Saros cycles to cross the whole nodal zone, which amounts to about 1,300 years. If about 40 Saros series are in progress, on the average, at any one time, then a Saros series must end (and another must begin) roughly every \(1{,}300/40 = 33\) years or so. This is one reason why the Babylonian astronomers needed to re-calibrate their Saros eclipse scheme from time to time.
3 The lunar eclipse pattern on the Antikythera mechanism
In the first column of Table 1, the eclipse possibilities for one Saros period are labeled from 1 to 38. The next two columns illustrate the features of lunar pattern \(\alpha \). The month in which each EP falls is listed in column two, the months being numbered from the first cell of the Antikythera mechanism’s Saros dial. Cell numbers in bold type indicate surviving lunar eclipse inscriptions. The cell numbers written in parentheses are the other eclipse possibilities predicted by pattern \(\alpha \). The third column shows the grouping of eclipses into a 7-8-7-8-8- pattern. Note that, in this case, the first 7- group is not complete, as only six of the eclipses appear at the beginning of the chart. Alternatively, one may think of this 7- group as beginning with an eclipse at month 219 of the previous Saros cycle. The 5-month intervals are indicated by the heavy bars. The beginning of the double 8- group is indicated by the doubled heavy bar.
4 The solar eclipse pattern on the Antikythera mechanism
5 Britton, Steele, and alternative Steele rules for linking solar and lunar patterns
Babylonian practice favored linking a solar eclipse pattern to a lunar eclipse pattern, which could perhaps allow us to discard some solar patterns or, at least, to prefer the others. These links have been discussed by Britton and by Steele, with somewhat different results.
Britton (1989, pp. 21–24) described the relationship between lunar and solar eclipses in these terms: When the Moon pattern is 8-8-7-8-7, the Sun pattern is 7-8-7-8-8 (exactly the reverse) but starts three eclipse possibilities earlier.^{3} So, if we are given one lunar eclipse pattern, we automatically have the corresponding solar pattern. We will call this the Britton rule.
But Steele (2000a, p. 443) has proposed another rule: “There is in fact an even closer link between the lunar and solar Saroi, at least in the period after \(-\)250. From here until at least \(-\)70, it would seem that the solar and lunar Saroi have the same 8-7-8-7-8 distribution, with the solar Saros always starting 4 eclipse possibilities earlier than the lunar Saros.” So, according to Steele, when the Moon pattern is 8-7-8-7-8, the Sun pattern is the same, 8-7-8-7-8, but starts 4 EP earlier. We will call this the Steele rule.
According to Steele, the differing results may be accounted for by the different input information: “Britton’s rule is derived from a theoretical model of nodal elongation (proposed by Aaboe) and (I think) agrees with what we find on the so-called Saros Canon texts, which are a small group of theoretical texts from Babylon. The rule I gave is based upon actual preserved records of eclipse predictions and observations found in Babylonian texts. Thus, Britton’s rule is a theoretical rule (reflecting also a particular Babylonian theory), whereas my rule is a reflection of what the Babylonians did in practice” (John Steele, personal communication).
In practice, the Babylonians re-calibrated their Saros schemes from time to time, when they got too far out of synchrony with the eclipse phenomena. Re-calibration involves a discontinuity in the run of repeating 8-8-7-8-7- patterns (the details will be discussed below). Once a re-calibration had occurred, the scheme typically could run for 2–5 Saros cycles without readjustment. Often, the Babylonian re-calibration occurred a few Saros cycles later than we might judge to be ideal, which means that some time had to elapse before the Babylonian astronomers perceived that the errors in the eclipse predictions had built up sufficiently so that re-calibration was required. As we shall see, the successive Babylonian re-calibrations of the Saros provide a useful dating tool.
On rare occasions, it is possible that the scheme for solar eclipses was re-calibrated, but the scheme for lunar eclipses was left unchanged for some time.^{4} In such instances, what we shall call the alternative Steele rule would be in force, as it is effectively a different form of the Steele rule. When the alternative Steele rule is in effect, the solar and lunar Saros schemes have the same 8-7-8-7-8- distribution as one another, but with the solar scheme always starting 4 eclipse possibilities later than the lunar scheme. We do not know whether this was a genuine rule of Babylonian eclipse theory—it could have resulted simply from correcting the solar calibration but leaving the lunar calibration unchanged for a time. Historically, the alternative Steele rule seems to have been in effect from about \(-\)71 at least to the end of the first century BCE. It may also have happened that the alternative Steele rule was in effect from time to time for shorter stretches, as we do not know exactly when all the recalibrations were performed.^{5}
Babylonian tabulations of lunar and solar eclipses are typically found on different tablets—it is not common to find them in one document. Therefore, a Greek astronomer could also unwittingly arrive at the alternative Steele rule if he made use of a canon of solar eclipses and a canon of lunar eclipses that ran for an overlapping sequence of years, if the solar canon represented a post-recalibration sequence and the lunar canon represented a pre-recalibration sequence. We might expect it to be more likely that such a circumstance would obtain shortly after a Babylonian recalibration.
Britton rule If Lunar Pattern \(\alpha \) and Solar Pattern 7 are represented on the Antikythera mechanism, these would be connected by the Britton rule. If Lunar Pattern \(\beta \) and Solar Pattern 6 are represented on the mechanism, these would be connected by the Britton rule.
Steele rule If Lunar Pattern \(\alpha \) and Solar Pattern 6 are represented on the mechanism, these would be connected by the Steele rule. If Lunar Pattern \(\beta \) and Solar Pattern 2 are represented on the mechanism, these would be connected by the Steele rule.
Alternative Steele rule If Lunar Pattern \(\beta \) and Solar Pattern 7 are represented on the mechanism, these would be connected by the alternative Steele rule.
6 Node conventions: a basic link between solar and lunar patterns
A key historical fact is that in the Babylonian records stretching over hundreds of years, a solar eclipse that is considered to begin the 8-8- double group is generally an eclipse at the descending node of the Moon’s orbit. To confirm this, we have computed the argument of the latitude of the Moon (Moon’s mean distance from the ascending node)^{6} at each eclipse beginning a solar double 8- in the Babylonian eclipse record published by Steele. For actually occurring eclipses, we input the time of the midpoint of the eclipse, taken from the Espenak catalog. For eclipses predicted in the Babylonian Saros scheme but not actually occurring, we used the time of conjunction. This analysis confirms that we are dealing with a reliable Babylonian convention, carefully preserved across multiple recalibrations. The Saros cycle contains an even number (38) of solar EPs, so placing one Saros cycle after another does not disrupt the regular alternation between the ascending and descending nodes. As we shall see below, when re-calibrations of the Saros were required, this was usually done by inserting a 47-month cycle (with 8 solar EPs) between the ending of one Saros cycle and the beginning of another. Such a recalibration also did not disrupt the regular alternation between the nodes.
Some solar eclipses, first in the second 8- of a 8-8- | |
---|---|
Date | Solar Saros Series |
\(-\)351 May 3 | 42 |
\(-\)333 May 14 | 42 |
\(-\)239 Apr 25 | 44 |
\(-\)221 May 6 | 44 |
\(-\)181 Mar 15 | 46 |
\(-\)163 Mar 26 | 46 |
\(-\)105 Feb 14 | 48 |
\(-\)87 Feb 25 | 48 |
\(-\)29 Jan 15 | 50 |
\(-\)11 Jan 26 | 50 |
As we can see, the solar eclipse beginning the second 8- of an 8-8- always belongs to an even-numbered solar Saros series—and is therefore an eclipse at the descending node. The same is therefore true of the first eclipse of the first 8- of the double 8-. During the course of a recalibration, it is not always easy to decide where to put the divisions between the 8- groups and the 7- groups. This is because the eclipse records are spotty: Only isolated eclipse dates are actually preserved and only in a limited number of cases do we have an explicit statement that a particular eclipse was preceded by a 5-month interval. But whenever the Babylonian 8-8-7-8-7- scheme is running regularly, the pattern seems to hold, that the second 8- of a double 8- begins with a solar eclipse at the descending node—and this over the whole period that is possible for the construction of the Antikythera mechanism.
A similar examination of Steele’s list of Babylonian lunar eclipses shows that in the case of lunar eclipses, but only after about \(-\)250, the first eclipse of a double-8 group always occurs with the Moon at the ascending node of its orbit. Around \(-260\), a major recalibration occurred, in which the first lunar eclipse of an 8-8- moved from odd lunar Saros series to even ones. This was apparently connected with adoption of the Steele rule, which, as we saw above, went into effect around \(-250\). From the middle of the third century BCE down to the end of Steele’s table at the beginning of the first century CE (and so for the whole period in which the Antikythera mechanism might have been constructed), the first lunar eclipse of an 8-8- remained at the ascending node.
Moreover, this is a logical consequence of the Steele rule. Suppose that there were a cycle in which a solar eclipse fell at the start of a double-8 and that there was also a lunar eclipse this same month. Because it is the start of a solar double-8, we know the Moon was at the descending node at the solar eclipse; therefore, the Moon was at the ascending node at the lunar eclipse of the same month. But the lunar double-8 would start 4 EPs later or earlier (in the Steele or alternative Steele rules, respectively). 4 is an even number, so the Moon must also be at the ascending node that starts the lunar double-8.
Some lunar eclipses, first in the second 8- of a 8-8- | |
---|---|
Date | Lunar Saros Series |
\(-\)255 Mar 8 | 38 |
\(-\)237 Mar 20 | 38 |
\(-\)197 Jan 27 | 40 |
\(-\)179 Feb 7 | 40 |
\(-\)85 Jan 20 | 42 |
\(-\)67 Jan 30 | 42 |
Lunar Saros series have even numbers if they occur with the Moon near its ascending node. Thus, the lunar eclipses beginning the second 8- of an 8-8- have the Moon at the ascending node. The same is true of the lunar eclipses beginning the first 8-. Again, in the course of the recalibrations, things can be unclear. But when the system is running normally, each lunar 8-8- starts with an eclipse in which the Moon is at its ascending node.
Key are the months that have both solar and lunar eclipses: These are months 125, 131, 137, 172, 178, and 184. The requirements of the Babylonian conventions are highlighted in these months. Then, we may easily cross solar patterns against lunar patterns to see which combinations are permissible. We will use month 125 as our example, but any one of these 6 months would suffice. In month 125, solar pattern 7 requires the Moon to be at A for the solar eclipse; and lunar pattern \(\beta \) requires the Moon to be at D for the lunar eclipse of the same month. These are compatible requirements, since, in the 2 weeks from the lunar to the solar eclipse, the Moon will move to the opposite node. Thus, combination \(7\beta \) is permissible.
By contrast, combination \(1\alpha \) is ruled out: Solar pattern 1 requires the Moon to be at A for the solar eclipse of month 125, while lunar pattern \(\alpha \) requires the Moon also to be at A for the lunar eclipse of the same month. These are incompatible requirements.
By such examinations, it is easy to see that combinations \(1\alpha , 2\alpha , 3\beta , 4\alpha , 5\beta , 6\beta \), and \(7\alpha \) are all eliminated. Combinations \(1\beta , 2\beta , 3\alpha , 4\beta , 5\alpha , 6\alpha \), and \(7\beta \) are still possible.
We note that the Britton rule combinations \(6\beta \) and \(7\alpha \) are excluded. The Britton rule is not compatible with the Babylonian convention, attested in the Saros record from about \(-\)250 onward, of starting a solar 8-8- with the Moon at D for the solar eclipse and starting a lunar 8-8- with the Moon at A for the lunar eclipse.^{8} The node conventions, while less restrictive than the Steele rule, are compatible with seven possible solutions, including two (\(2\beta \) and \(6\alpha \)) consistent with the Steele rule and one \((7\beta )\) consistent with the alternative Steele rule.
7 The omitted solar eclipses as a tool for the classification of solar eclipses
In a Babylonian Saros scheme, equal numbers of solar and lunar eclipse possibilities are generated—38 each in a 223-month cycle. However, on the Antikythera mechanism, fewer solar eclipses were included than lunar eclipses. Freeth et al. (2008) argued convincingly that this is connected with the idea that if a new Moon occurred too far south of the node of the Moon’s orbit, the ancient astronomer may not have considered the eclipse to be visible. Although this practice has not yet been found in the Babylonian material, a similar idea is mentioned in Ptolemy’s Almagest (vi, 5), where it is connected with lunar parallax effects. According to Ptolemy, a lunar eclipse will generally occur if the mean Moon, when in opposition to the Sun, lies within \(15^{\circ }12^{\prime }\) of the node (Toomer 1984, p. 287). However, the cutoff criterion for solar eclipses is asymmetric: A solar eclipse will generally appear for the northern inhabited latitudes of the Earth if at the time of conjunction the center of the true Moon is no more than \(17^{\circ }41^{\prime }\) north of the node, or no more than \(8^{\circ }22^{\prime }\) south of the node, measured along the Moon’s inclined path (Toomer 1984, p. 286). (Ptolemy goes on to add that the cutoffs for the center of the mean Moon are: \(20^{\circ }41^{\prime }\) to the north of the node or \(11^{\circ }22^{\prime }\) to the south). Although the Almagest is several centuries later than the Antikythera mechanism, it is not unreasonable to suppose that similar rules governing solar eclipses were in place earlier. Freeth et al. showed that, for rather narrow ranges of the three cutoff parameters (limit of lunar, limit of northern solar, and limit of southern solar eclipses), the predictions of their generating scheme (based on the mean motions of the Saros cycle plus an asymmetric criterion for elongation from the node) can indeed match the evidence. These authors also remarked that “it does not appear to be possible to generate the Antikythera scheme [of solar eclipses] by a simple pattern of excisions from one of the Babylonian schemes” (Freeth et al. 2008, Supplementary Notes, p. 36). Finally, Freeth et al. experimented with schemes that incorporate the first anomalies of the Sun’s and Moon’s motion; but these did not lead to any real improvement in the simpler scheme based on mean motion and asymmetric cutoffs.
However, we suggest that since the original EPs simply follow the schematic 8-8-7-8-7- pattern, it may be questioned whether an elaborate month-by-month calculation of nodal elongation (whether based on mean or true positions) lies behind the pattern of excluded solar eclipses. This would have made the suppression of occasional solar eclipses far more work than the original listing of eclipse possibilities for the Sun and Moon. We shall show that a simple system for omission of unwanted solar eclipses is probably possible. The basic rule is: at each node, omit the southernmost eclipse in each “diagonal sequence.” (This term will be defined below).
8 Justification of the omission rule for the 47-month cycle
Mean solar eclipse possibilities at the descending node in a 47-month cycle
EP | Month number \(\hbox {N}_{{i}}\) | Time since \(\hbox {D}_{1}\)\( \Delta {t} = ({N}_{{i}} - {N}_{1})\) (51/47) | Angular distance from descending node \(\phi _{{d}} =\)\(-360 \hbox { frac}(\Delta {t}) + 13.4\) |
---|---|---|---|
\(\hbox {D}_{1}\) | 13 syn. mo. | 0 drac. mo. | \(13.40^{\circ }\hbox {N}\) |
\(\hbox {D}_{2}\) | 25 | 13 1/47 | \(5.74^{\circ }\hbox {N}\) |
\(\hbox {D}_{3}\) | 37 | 26 2/47 | \(1.92^{\circ }\hbox {S}\) |
\(\hbox {D}_{4}\) | 49 | 39 3/47 | \(9.58^{\circ }\hbox {S}\) |
\(\hbox {D}_{5}\) | 60 | 51 | \(13.40^{\circ }\hbox {N}\) |
The \(\hbox {D}_{3}\) conjunction will occur 26 2/47 draconic revolutions after \(\hbox {D}_{1}\) and thus \(2 \times 360^{\circ }/47\) counterclockwise from \(\hbox {D}_{1}\). The \(\hbox {D}_{4}\) conjunction will occur \(3 \times 360^{\circ }/47\) counterclockwise from \(\hbox {D}_{1}\), as shown in Table 4 and illustrated in Fig. 2. Thus, as we move to successive conjunctions at the same node, they occur progressively farther counterclockwise by steps of about \(7.66^{\circ }\), corresponding to time differences of 1/47 dm. But at \(\hbox {D}_{5}\), because there is only an 11-month interval (instead of a 12-month interval), the time difference is only \(11 \times 51/47\,\hbox {dm} = (12 - 3/47)\,\hbox {dm}\), corresponding to an angular shift of \(-22.98^{\circ }\), which exactly cancels the accumulated displacements of the three preceding steps. Thus, \(\hbox {D}_{5}\) occurs at the same position with respect to the node as \(\hbox {D}_{1}\), and the cycle of 47 complete synodic months starts over.
Mean solar eclipse possibilities at the ascending node in a 47-month cycle
EP | Month number \({N}_{{i}}\) | Time since \(\hbox {D}_{1}\)\( \Delta {t} = ({N}_{{i}} - {N}_{1})\) (51/47) | Angular distance from ascending node \(\phi _{{a}} = -360 \hbox { frac}(\Delta {t}) + 13.4 + 180\) |
---|---|---|---|
\(\hbox {A}_{1}\) | 19 syn mo. | 6 24/47 drac. mo. | \(9.57^{\circ }\hbox {S}\) |
\(\hbox {A}_{2}\) | 31 | 19 25/47 | \(1.91^{\circ }\hbox {S}\) |
\(\hbox {A}_{3}\) | 43 | 32 26/47 | \(5.75^{\circ }\hbox {N}\) |
\(\hbox {A}_{4}\) | 55 | 45 27/47 | \(13.41^{\circ }\hbox {N}\) |
\(\hbox {A}_{5}\) | 66 | 51 | \(9.57^{\circ }\hbox {S}\) |
9 Justification of the omission rule for the 8-8-7-8-7- Saros scheme
We explained the omission rule for solar eclipses on the basis of the 47-month eclipse cycle, as this is very simple. Now, we apply the same ideas to the more accurate Saros cycle. Here, the crucial parameter is \(s = 242/223 ({\approx }1.085202)\), as \(242\,\hbox {dm} = 223\,\hbox {sm}\).
Five successive 47-month cycles would have given us a total of 40 eclipses distributed over 235 months. The Saros cycle is an improvement that distributes 38 eclipses over 223 months. So now we must investigate the distribution of the eclipses between the two nodes in the 8-8-7-8-7- Saros scheme. It is convenient to begin reckoning from the first solar eclipse in the double-8. We call the month of this eclipse month 1, and we assign it to node D.
Mean solar eclipse possibilities at the descending node in the Saros scheme
Synodic month | Time w/r to D node (draconic months) | Angular distance from D node (\({}^{\circ }\)) |
---|---|---|
1 | \(-\)T | 360 T |
13 | \(-\)T \(+\) 5/223 | 360 (T \(-\) 5/223) |
25 | \(-\)T \(+\) 10/223 | 360 (T \(-\) 10/223) |
37 | \(-\)T \(+\) 15/223 | 360 (T \(-\) 15/223) |
48 | \(-\)T \(+\) 1/223 | 360 (T \(-\) 1/223) |
60 | \(-\)T \(+\) 6/223 | 360 (T \(-\) 6/223) |
72 | \(-\)T \(+\) 11/223 | 360 (T \(-\) 11/223) |
84 | \(-\)T \(+\) 16/223 | 360 (T \(-\) 16/223) |
95 | \(-\)T \(+\) 2/223 | 360 (T \(-\) 2/223) |
107 | \(-\)T \(+\) 7/223 | 360 (T \(-\) 7/223) |
119 | \(-\)T \(+\) 12/223 | 360 (T \(-\) 12/223) |
131 | \(-\)T \(+\) 17/223 | 360 (T \(-\) 17/223) |
142 | \(-\)T \(+\) 3/223 | 360 (T \(-\) 3/223) |
154 | \(-\)T \(+\) 8/223 | 360 (T \(-\) 8/223) |
166 | \(-\)T \(+\) 13/223 | 360 (T \(-\) 13/223) |
178 | \(-\)T \(+\) 18/223 | 360 (T \(-\) 18/223) |
189 | \(-\)T \(+\) 4/223 | 360 (T \(-\) 4/223) |
201 | \(-\)T \(+\) 9/223 | 360 (T \(-\) 9/223) |
213 | \(-\)T \(+\) 14/223 | 360 (T \(-\) 14/223) |
224 | \(-\)T | 360 T |
Mean solar eclipse possibilities at the ascending node in the Saros scheme
Synodic month | Time w/r to A node (draconic months) | Angular distance from A node (\({}^{\circ }\)) |
---|---|---|
7 | \(-\)T \(+\) 5/446 | 360 (\(-\)T \(+\) 5/446) |
19 | \(-\)T \(+\) 5/446 \(+\) 5/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 5/223) |
31 | \(-\)T \(+\) 5/446 \(+\) 10/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 10/223) |
43 | \(-\)T \(+\) 5/446 \(+\) 15/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 15/223) |
54 | \(-\)T \(+\) 5/446 \(+\) 1/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 1/223) |
66 | \(-\)T \(+\) 5/446 \(+\) 6/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 6/223) |
78 | \(-\)T \(+\) 5/446 \(+\) 11/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 11/223) |
90 | \(-\)T \(+\) 5/446 \(+\) 16/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 16/223) |
101 | \(-\)T \(+\) 5/446 \(+\) 2/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 2/223) |
113 | \(-\)T \(+\) 5/446 \(+\) 7/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 7/223) |
125 | \(-\)T \(+\) 5/446 \(+\) 12/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 12/223) |
136 | \(-\)T \(+\) 5/446 \(-\) 2/223 | 360 (\(-\)T \(+\) 5/446 \(-\) 2/223) |
148 | \(-\)T \(+\) 5/446 \(+\) 3/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 3/223) |
160 | \(-\)T \(+\) 5/446 \(+\) 8/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 8/223) |
172 | \(-\)T \(+\) 5/446 \(+\) 13/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 13/223) |
183 | \(-\)T \(+\) 5/446 \(-\) 1/223 | 360 (\(-\)T \(+\) 5/446 \(-\) 1/223) |
195 | \(-\)T \(+\) 5/446 \(+\) 4/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 4/223) |
207 | \(-\)T \(+\) 5/446 \(+\) 9/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 9/223) |
219 | \(-\)T \(+\) 5/446 \(+\) 14/223 | 360 (\(-\)T \(+\) 5/446 \(+\) 14/223) |
230 | \(-\)T \(+\) 5/446 | 360 (\(-\)T \(+\) 5/446) |
10 Elimination of solar patterns 1–6
Let us consider first the eclipses at the descending node. A descending line \(\backslash \) through a bold-faced number indicates a \(\hbox {D}_{4}\) eclipse that actually appears on the mechanism, contrary to the rule that such an eclipse should be omitted. Thus, in columns 3, 5, and 6, EP 137 and 184 are extant \(\hbox {D}_{4}\). These three cases must therefore be eliminated. In column 4, EP 131 and 178 are extant \(\hbox {D}_{4}\), so this case is eliminated. In column 1, EP 178 is an extant \(\hbox {D}_{4}\), so this case also is excluded. Now, in column 2, the descending line\(\backslash \)through indicates that a \(\hbox {D}_{1}\) is missing, contrary to the omission rule that a \(\hbox {D}_{1}\) should certainly not be omitted, so this case, too, is excluded. When all the cells crossed out by descending lines\(\backslash \)(indicating a violation of the omission rules at the descending node) are considered, it will be seen that Case 7 is the only survivor. But in Sect. 6, we showed that solar pattern 7 is only compatible with lunar pattern \(\beta \). Thus, we are left with \(7\beta \) as the only solution.
Some supporting evidence is also available from study of the eclipses at the ascending node. Violations of the omission rules at the ascending node are indicated by an ascending line /. It turns out that not enough of these eclipse inscriptions are preserved to provide as much discriminating power as do the eclipses at the descending node. However, cases 3, 5, and 6 are again ruled out—they each have an extant EP at an \(\hbox {A}_{1}\) (which should be omitted) in month 13. Case 6 is also excluded because the cell for month 190 clearly has no solar eclipse glyph, but is an A\(_{4}\) that, according to the omission rules, should not be omitted.
11 Empirical test of the omission rules
Given that an astronomer as cogent as Ptolemy adopted an omission rule for solar eclipses occurring too far south of the node, we might consider the possibility there is something reasonable in this strategy. Therefore, we used the Javascript Solar Eclipse Explorer, on the NASA Eclipse Web Site, to determine all the solar eclipses that were visible from Babylon between the years \(-\)400 and 0 (Espenak, Solar Eclipse Explorer). There were 149 of them.
For each of these eclipses, we calculated the Moon’s distance from the node at the moment of greatest eclipse. The most southern visible eclipse at the D node was \(6.02^{\circ }\) south of the node. The most southern visible eclipse at the A node was \(0.2^{\circ }\) south of the node. Now, in Fig. 4, the northernmost \(\hbox {A}_{1}\) eclipse is around \(7.7^{\circ }\) south of the node; and in Fig. 3, the northernmost \(\hbox {D}_{4 }\)eclipse is around \(9.3^{\circ }\) south of the node. Thus, it is clear that the southernmost eclipses actually visible at Babylon could not possibly be a \(\hbox {D}_{4}\) or \(\hbox {A}_{1}\). (The numerical boundaries of \(\hbox {D}_{4}\) and \(\hbox {A}_{1}\) in Figs. 3 and 4 can be modified a bit, by taking slightly different values for the parameter \(T\) in Tables 7 and 8; but the changes cannot be large, or the graphs will become grossly asymmetrical). There simply were no\({D}_{4}\)or\({A}_{1}\)solar eclipses visible from Babylon for 400 years. The omission rule would have performed very well.
12 Candidates for epoch
Table 10 displays the solar eclipses that are the candidates for the solar eclipse in Saros cell 13 of the Antikythera mechanism. Cell 13 carries, of course, the \(\hbox {D}_{1}\) eclipse that begins the solar 8-8- on the extant Saros dial. Dates shown in bold type come from Steele’s reconstruction of the Babylonian eclipse records: These are the dates of the first solar EP in a double 8- group. (These are, of course, all eclipses of type \(\hbox {D}_{1}\)). The heavy vertical bars mark the approximate location of Babylonian recalibrations of the Saros scheme. The extreme left-hand column gives the Saros series number of each of these eclipses. The shaded gray cells show solar eclipses of type \(\hbox {D}_{1}\) from the Espenak catalog (computed from modern theory) that could function well as the first eclipse of an 8-8-7-8-7- scheme.
Let us study how the Babylonian recalibrations occur. Take, for example, the successive Saros cycles beginning with \(-\)203 May 17, \(-\)185 May 28, and \(-\)167 June 7. Again, these are all dates of first solar eclipses in a double 8-. Since the Babylonian dates fall in the shaded cells, we can see that the Babylonian record is in good agreement with the phenomena for this stretch. But by year \(-\)149, solar Saros series 44 had ended—this was no longer an actually occurring eclipse. The Babylonians, however, continued to run their 8-8-7-8-7- scheme without modification for two more Saros cycles. Then, there was a change of calibration—with the eclipse of \(-\)109 April 28 now serving as the first eclipse of a double 8- (as opposed to the “eclipse” of \(-\)113 July 19, which has been abandoned).
13 Lunar and solar equations of center
On the Antikythera mechanism, eclipses of the same kind (solar or lunar) come at 5- or 6-month intervals. The mean synodic month \({T}_{\mathrm{S}}\) is about 29.53085 days^{11}, so a 5-month interval represents 147.654 days, or about 15.7 h over and above whole days. A 6-month interval represents 4.4 h over and above whole days. Thus, the local time of day of an eclipse should advance by either 15.7 h or by 4.4 h over its predecessor, owing to the mean motions alone. Departures of the recorded eclipse times from this simple pattern can be taken as signs that the solar and/or the lunar equation of center might be built into the predictive pattern. Solar and lunar equations could be embedded in the eclipse times in two different ways. If the eclipse times are the results of theoretical calculation, they should reflect whatever theory the ancient astronomer used to model the lunar anomaly, or solar anomaly, or both. The equation of center of the Moon could have been modeled by an epicycle or a linear zigzag function, for example. Or, if the eclipse times are based largely on observations (with some sort of interpolation for unobservable eclipses), they should reflect the actual equations of center of the Sun and Moon.
The mean motion of the Moon with respect to the Sun is roughly \(12.2^{\circ }/\hbox {day}\). Thus, the effect of the solar anomaly is to make the true conjunction come later than the mean conjunction by about \((2^{\circ })/(12.2^{\circ }/\hbox {day}) = 3.9\) h when the solar equation is positive and at its peak value. (Here, for simplicity, we take the relative velocity of the Moon and Sun to be constant. In Sect. 19, we shall see how to take its variability into account). The effect of the lunar anomaly is to make the true conjunction come earlier than the mean by about \((5^{\circ })/(12.2^{\circ }/\hbox {day}) = 9.8\) h when the lunar equation is positive and at its peak value. Alexander Jones made a study of the eclipse times in the glyphs of the Saros dial, successfully detected the lunar anomaly, and found that the Moon was very nearly at apogee at the full Moon of Saros cell 1. According to Jones, the solar anomaly did not show up (Jones, personal communication).
We must say a word about the form in which the eclipse times are inscribed on the mechanism. First, there is a notice of whether the eclipse is of the Sun or Moon. Then, the hour is given. For an eclipse of the Sun, if the eclipse takes place in the daytime, the hour is simply noted as a number; but if the eclipse takes place at night, the hour is given along with a notation for “night.” For an eclipse of the Moon, the convention is just the opposite: If the eclipse takes place in the night, the hour is noted, but no other notice is given; but if the eclipse takes place in the day, the hour is noted along with a notation for “day.”
1st hour of day | \(7^{\mathrm{h}}\) |
6th hour of day | 12 |
12th hour of day | 18 |
1st hour of night | 19 |
6th hour of night | 24 or 0 |
7th hour of night | 1 |
12th hour of night | 6 |
We do not know if this is really the correct way to treat the times. However, for days within a month of the equinox, and for times within a couple of hours of noon or midnight, the error of interpretation will be negligible. For other times of year and of the day, the error of interpretation could potentially rise to h or so (supposing, for instance, that the mechanism was built for the clime of h). In the search for the presence of a lunar or a solar equation (with amplitudes of about 10 and 4 h, respectively), the possibility of such small errors should not prevent us from proceeding.
Finally, we must point out that for some of the eclipses it is necessary to correct the glyph time by \({\pm }24\) h. Here is the reason. Let us begin from the lunar eclipse of month 20, which is marked on the mechanism as occurring at sixth hour of the night (\(=\)midnight, or \(0^{\mathrm{h}}\) in the 24-h clock system). Now, take some other eclipse, such as the lunar eclipse of month 137, which, according to the mechanism, takes place at the fifth hour of the day (\({=}11^{\mathrm{h}}\) in the 24-h system). Since the second eclipse takes place 117 synodic months later than the first one, from the mean motions alone the second eclipse should occur \(117 \times 29.53085^{\mathrm{d}} = 3455^{\mathrm{d}} 2.62^{\mathrm{h}}\) later than the first one. Thus, since the first eclipse occurred at \(0^{\mathrm{h}}\), we would expect the second eclipse to occur at \(2.62^{\mathrm{h}}\) on the basis of the mean motions alone. However, the advance in time between the two eclipses as engraved on the mechanism is \(11^{\mathrm{h}}\). Thus, the eclipse time has apparently advanced by \(11 - 2.62 = 8.38\) h more than we would expect on the basis of the mean motions. These 8.38 h are presumably due to the change in the solar and/or lunar equation. However, we do not really know that the time of eclipse has advanced by 8.38 h. If the second eclipse occurred \(3{,}454^{\mathrm{d}}\,11^{\mathrm{h}}\) after the first one, or \(3{,}455^{\mathrm{d}}\,11^{\mathrm{h}}\) after the first one, or \(3{,}456^{\mathrm{d}}\,11^{\mathrm{h}}\), the ancient mechanic would have in any case inscribed the time as the fifth hour of the day. But in the first case, the advance in time from the first eclipse, over and above the effect of the mean motions, would be \(8.38 - 24 = - 15.62^{\mathrm{h}}\); in the second case, the advance would be \(8.38^{\mathrm{h}}\); and in the third case, it would be \(8.38 + 24 = 32.38^{\mathrm{h}}\). That is, we cannot tell a priori whether the effect of the lunar and solar equations has been to advance the second eclipse by 8.38 h with respect to the mean motions, to advance it by 32.38, or to retard it by 15.62. Now, the maximum lunar equation of 10 h and the maximum solar equation of 4 h together amount to a maximum of 14 h. If the first eclipse occurred early by, say, 14 h and the second occurred 14 h late, the effect of the lunar and solar equations could shift the expected time of the second eclipse by up to 28 h, over and above the effect of the mean motions. Thus, the possibility of an advance by \(32.38^{\mathrm{h}}\) may probably be ruled out. However, we cannot tell whether the eclipse time advanced by \(8.32^{\mathrm{h}}\) or was retarded by \(15.62^{\mathrm{h}}\), over and above the effect of the mean motions.
Considerable headway can be made in determining which eclipses require a 24-h correction by comparing pairs or larger sets of eclipses for which the lunar mean anomaly is nearly the same. These would be eclipses separated by nearly a whole number of anomalistic months. Then, the effect of a change in the lunar equation is largely removed. Thus, the maximum possible time shift, over and above the effect of the mean motions, would be about 8 h—the maximum possible shift due to the solar anomaly acting alone. See “Appendix 4” for examples this sort of analysis.
Months and times of day for eclipses of the Moon (M) and Sun (S) engraved on the Saros dial of the Antikythera mechanism. These are the fundamental data used to investigate the lunar and solar equations
Data on Saros dial | In form for data analysis | ||||
---|---|---|---|---|---|
Eclipse | Hour | Time from opposition 1 (syn. mo.) | Time of day (h) | Cor. (h) | Adjusted time (h) |
1 Op | 0.0 | \(\overline{h} _0 \) | 0 | \(\overline{h} _0 \) | |
13S | 1 d | 12.5 | 7 | 7 | |
20M | 6 n | 19.0 | 0 | 0 | |
25S | 6 d | 24.5 | 12 | 12 | |
26M | 7 d | 25.0 | 13 | \(-\)24 | \(-\)11 |
78S | 1 d | 77.5 | 7 | 7 | |
79M | 10 d | 78.0 | 16 | \(-\)24 | \(-\)8 |
114M | 12 d | 113.0 | 18 | 18 | |
119S | 11 n | 118.5 | 5 | 5 | |
120M | 6 d | 119.0 | 12 | \(-\)24 | \(-\)12 |
125M | 8 d | 124.0 | 14 | 14 | |
125S | 3 d | 124.5 | 9 | 9 | |
131M | 2 n | 130.0 | 20 | \(-\)24 | \(-\)4 |
131S | 9 n | 130.5 | 3 | 3 | |
137M | 5 d | 136.0 | 11 | \(-\)24 | \(-\)13 |
137S | 12 d | 136.5 | 18 | 18 | |
172M | 6 n | 171.0 | 24 | 24 | |
172S | 12 d | 171.5 | 18 | \(-\)24 | \(-\)6 |
178M | 9 n | 177.0 | 3 | 3 | |
178S | 9 d | 177.5 | 15 | 15 | |
184M | 4 d | 183.0 | 10 | 10 | |
184S | 1 d | 183.5 | 7 | 7 | |
190M | 9 d | 189.0 | 15 | \(-\)24 | \(-\)9 |
Lunar and solar parameters deduced from the eclipse data in Table 11
\( (\hbox {h})\) | \(A_{\odot }\) (h) | \(\phi _{\odot }(^{\circ })\) | \(\overline{h}_0 \) (h) | ||
---|---|---|---|---|---|
Using all 22 eclipses | 9.73 | 191.1 | 3.30 | 151.0 | \(-\)8.83 |
Excluding 13S and 125M | 9.72 | 191.4 | 3.09 | 171.1 | \(-\)9.39 |
Excluding 13S, 120M, 125M, 184M | 9.11 | 184.7 | 3.03 | 180.4 | \(-\)9.41 |
12 lunar eclipses only | 9.84 | 206.0 | 3.27 | 171.1 | \(-\)7.96 |
All 22 eclipses, Solar 50 % weight | 9.80 | 193.6 | 3.41 | 149.5 | \(-\)8.64 |
Omit 13S, 120M, 125M and 184M, Solar 50 % weight | 9.04 | 184.8 | 3.26 | 175.5 | \(-\)9.28 |
The lunar mean anomaly (reckoned from perigee) at the opposition of month 1 comes out between \(184.7^{\circ }\) and \(206.0^{\circ }\). These are all reasonably near the apogee of the Moon’s orbit. That is, it seems that at the opposition of month 1, the Moon was near apogee. Here, we confirm a result of Alexander Jones. Jones conjectured that this might explain why the Saros dial beings with an empty, eclipseless month—the starting point was a date on which the Moon was at apogee. We certainly do not imagine that we can determine the Moon’s anomaly to better than \({\pm }2\) days’ worth of motion, i.e., about \(26^{\circ }\), so our results are compatible with Jones’ conjecture. While Jones found no evidence of a solar equation, we do find a robust solar signal. The solar anomaly (reckoned from perigee) comes out between \(149.5^{\circ }\) and \(180.4^{\circ }\) (depending on the subset of eclipses used). Notably, these are near the solar apogee. Thus, it seems possible that the ancient mechanic was using an arithmetical template for the anomalies, which started with both solar and lunar anomalies equal to zero (measured, in ancient fashion, from apogee).
14 Statistical test of the explanatory power of the solar fit
The fact that the lunar equation of center is present in the eclipse times inscribed on the AM seems indisputable. For the solar equation, the evidence is less clear-cut, so it is important to perform a statistical test to see how much faith should be placed in this result. Does the theoretical model with lunar equation plus solar equation provide a quantifiable explanatory advantage over a theoretical model with lunar equation only?
In this case, we have two models, a full model and a reduced model, with the reduced model nested within the full one. In the full model, there are five parameters—a clock calibration \(\bar{h}_0\) (which sets the time of day for the opposition of month 1), the amplitude and phase of the lunar equation of center, and the amplitude and phase of the solar equation. The reduced model is obtained from the full model by setting the amplitude of the solar equation of center equal to zero, in which case two parameters may be removed from the model. We shall use an \(F\)-test (named after the British statistician Ronald Fisher, who introduced a version of it in the 1920s) to assess whether the solar parameters contribute meaningfully. The null hypothesis for this type of \(F\)-test is that the full model offers no additional explanatory power over the reduced model. In other words, the null hypothesis is that the “true” underlying relationship between the recorded times of day for the eclipses and the explanatory parameters (amplitude and phase of the lunar equation of center, etc.) is that given by the reduced model; the full model with its additional parameters is unwittingly modeling as a systematic relationship some of what is actually random fluctuation.
\(F\)-test for the explanatory power of the solar equation of center
\(F\) | \(k_{\mathrm{F}} - k_{\mathrm{R}}\) | \(n - k_{\mathrm{F}}\) | \(p\) | |
---|---|---|---|---|
All 22 eclipses | 4.27 | 2 | 17 | 0.031 |
20 eclipses, excluding 13S and 125M | 5.92 | 2 | 15 | 0.013 |
18 eclipses, excluding 13S, 120M, 125M, 184M | 7.52 | 2 | 13 | 0.0068 |
15 Determination of the Saros series number of the solar eclipse of month 13
We have shown that the Saros dial of the AM entails that the lunar mean anomaly was around \(191^{\circ }\) (reckoned from perigee) at the opposition of month 1. Because we do not imagine that we can determine the anomaly to a better precision than about two days’ worth of motion, we shall take it to be \(191^{\circ }\,\pm \,26^{\circ }\). When used in conjunction with the Babylonian recalibrations, this provides a dating tool of considerable power. This is because of the discontinuous jump in the lunar mean anomaly produced by a Babylonian recalibration. A Saros cycle contains approximately a whole number of synodic months, draconic months, and anomalistic months. Thus, two eclipses in the same Saros series will have similar values of the lunar mean anomaly. But during a recalibration, a 47-month period is slipped in. The 47-month period contains approximately a whole number of synodic months and draconic months, but does not contain a whole number of anomalistic months. This is the source of the discontinuity of the lunar mean anomaly over a recalibration.
We compute from modern theory the value of the lunar mean anomaly at the middle of each solar eclipse belonging to these Saros series, for eclipses from the middle of the fourth to the middle of the first century BCE.^{15} We assume, for each in turn, that one of these eclipses is the eclipse of month 13. Then, it is an easy matter to roll the calculated lunar mean anomaly back by synodic months, to determine the value that the mean anomaly would have had at the opposition of month 1, if the eclipse being examined really were the solar eclipse of month 13.
16 Absolute time of eclipses
Candidates for solar eclipse 13, all from solar Saros series 44
Group | Candidates for eclipse 13 in solar Saros series 44 | Resulting date for opposition in month 1 |
---|---|---|
1 | \(-\)293 Mar 24 | \(-\)294 Mar 20 |
2 | \(-\)275 Apr 3 | \(-\)276 Mar 30 |
3 | \(-\)257 Apr 15 | \(-\)258 Apr 10 |
1 | \(-\)239 Apr 25 | \(-\)240 Apr 21 |
2 | \(-\)221 May 6 | \(-\)222 May 2 |
3 | \(-\)203 May 17 | \(-\)204 May 12 |
1 | \(-\)185 May 28 | \(-\)186 May 23 |
2 | \(-\)167 June 7 | \(-\)168 June 3 |
3 | \(-\)149 June 19 | \(-\)150 June 14 |
1 | \(-\)131 June 29 | \(-\)132 June 24 |
2 | \(-\)113 July 9 | \(-\)114 July 5 |
3 | \(-\) 95 July 19 | \(-\)96 July 16 |
17 Breakdown of the rule of 8 h
We have already alluded to the “rule of 8 h.” If the time of eclipse is given for each eclipse in a Saros cycle (let us call it cycle 0), the approximate time for the corresponding eclipse in the next Saros cycle (cycle 1) can be obtained by adding 8 h. Adding 8 more hours gives the approximate eclipse time in cycle 2. And then in cycle 3 (since one complete exeligmos consisting of 3 Saros cycles has elapsed), the eclipse time returns approximately to its value in cycle 0.
In Table 14, each eclipse of group 3 is still a candidate. Each candidate is separated from its group 3 neighbors by an exeligmos cycle. Can we find a way to select one single eclipse of group 3 and exclude the others? We can, by exploiting the breakdown of the rule of 8 h.
The deep, regular valley, with the well-defined minimum in Fig. 17 is again a confirmation that solar Saros 44 is the correct choice. For other Saros series, the minima are much shallower, and often there is no well-defined minimum at all. Examples of some of these other non-solutions are shown in Figs. 29 and 30.
9 lunar eclipses: 1.32 h
9 solar eclipses: 3.32 h
18 solar and lunar: 2.52 h
Steele (2000b, pp. 68–75) analyzes the accuracy of the predicted eclipse times surviving in Babylonian records. He distinguishes three categories for lunar eclipses: Category A, umbral lunar eclipses that were visible somewhere on the Earth’s surface, but not necessarily at the longitude of Babylon; Category B, penumbral lunar eclipses, and Category F, failed predictions. He finds that the average error is for Category A 1.31 h and for Category B 2.86 h. Steele distinguishes two categories for solar eclipses: Category A, solar eclipses that were visible at Babylon or would have been visible there if the Sun were above the horizon at the time of the eclipse (that is, eclipses that were visible from the latitude, but not necessarily from the longitude of Babylon); and Category B, eclipses that would not be visible at Babylon under any circumstance. He found for Category A an error of 2.01 h and for Category B an error of 3.55 h.
The accuracy of the times of the lunar eclipses on the AM is comparable to that of Babylonian lunar eclipse predictions of category A. This is presumptive evidence that the lunar eclipse times on the AM are the results of some kind of predictive scheme—and not directly of observation. (We will show this more explicitly in Sect. 19).
18 Analysis of the underlying lunar and solar models
- Moon
amplitude of equation of center curve: 8.88 h
“mean anomaly”^{20} at opposition of month 1: \(185.0^{\circ }\)
- Sun
amplitude of sawtooth function: 4.02 h
“Mean anomaly”^{21} at opposition of month 1: \(183.7^{\circ }\)
- \(\bar{h}_0\)
\(-\)9.39 h.
These are all comparable to the values shown in Table 12 for the sinusoidal fits. Visually, the quality of the Babylonian fit is good—and statistically this is borne out as well. The parabolic (for the Moon) and piecewise-linear (for the Sun) fits leave a sum of squared residuals of \(53.4\,\hbox {h}^{2}\). This is defined as: (glyph time minus time predicted by fit)\(^{2}\), summed over the 18 eclipses. For the sinusoidal fit, using the same 18 eclipses, the residuals come to \(57.0\,\hbox { h}^{2}\). The difference is due mostly to the better performance of the piecewise-linear solar theory. The difference in performance is not large enough to prove the use of Babylonian-style theories for handling the non-uniformity of motion, but this does seem a plausible conjecture, especially as there is evidence of the use of Babylonian solar theory of System A on the front side of the mechanism, in the differential graduation of the zodiac and Egyptian calendar scales (Evans et al. 2010). Moreover, given our likely epoch of \(-\)204, it is highly unlikely that trigonometric equations of center would have been available.
- Moon
amplitude of equation of center curve: 8.92 h
“mean anomaly” at opposition of month 1: \(185.3^{\circ }\)
- Sun
amplitude of sawtooth function: 3.99 h
“mean anomaly” at opposition of month 1: \(183.1^{\circ }\)
- \(\bar{h} _0\)
\(-\)9.44 h
19 Synthesis of eclipse times
Geometers from antiquity to the seventeenth century often maintained that a complete solution of a problem required two parts, characterized by different methods—analysis and synthesis. In Sect. 18, we applied analytical methods to the problem of the eclipse times to uncover information about the underlying model. We found that Babylonian-style arithmetical theories for the equations of center of the Moon and Sun would work somewhat better (in the sense of agreeing better with the inscriptions) than would an epicycle theory. Because of the early date of the likely epoch of the Saros dial, we have good reason to doubt that trigonometric functions or epicycles could have been involved. Moreover, we found in Sect. 13 that the lunar mean anomaly and the solar mean anomaly were near \(180^{\circ }\) (at apogee) at the full Moon of month 1. These results leave the model with little wiggle room.
We may now easily calculate the predicted times of the eclipses according to this scheme. We assume that the mean time of the eclipse is equal to the local clock time of the opposition of month 1, plus the advance in clock time that would result from the mean motions alone, plus a term , where each \(q\) is expressed in degrees and each \(v\) in degrees per day. Note that at the time of a given eclipse, \(v_{\odot }\) is just either one of two constant values, \(V_{\mathrm{F}}\) or \(V_{\mathrm{S}}\). For the Moon, is calculated from the formulas given above, i.e., .
Explanation: The full Moon of month 1 occurred at a certain time of day \(t_{1}\); both and \(q_{\odot }\) were then zero. At some later eclipse, if is again zero, the time of the eclipse will be greater than \(t_{1}\) owing to the effects of the mean motions alone (as explained in Sect. 13), thus producing an eclipse at a certain time of day \(t_{2}\). But if , the eclipse will occur earlier than \(t_{2}\). If , the eclipse will occur later than \(t_{2}\). Rather than dividing the difference in the equations of center by the constant mean relative velocity of the Moon and Sun , we divide by the relative velocity that actually holds at the moment of the mean eclipse. (Thus, in contrast to the procedure in the earlier, preliminary investigation, we are explicitly taking into account the variable speeds. This is now easy to do, since we now know the initial values of the mean anomalies).
Prediction of eclipse times using Babylonian-style equations of center and velocities
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|
Eclipse or opp. | Time of day from AM (h) | Fraction of anomalistic month | Fraction of year | Advance due to mean motions (h) | \(q_{\odot }\,({}^{\circ })\) | \(v_{\odot }\,({}^{\circ }/\hbox {d})\) | Predicted time of day (h) | ||
1 op | \(-\)9.40 | 0 | 0 | 0 | 0 | 0 | 11.86 | 0.96 | 14.60 |
20 M | 0 | 0.363229 | 0.53617 | 2.07 | \(-\)3.61 | 0.29 | 13.78 | 1.01 | 24.00 |
25 S | 12 | 0.257848 | 0.980851 | 12.14 | \(-\)4.53 | 0.17 | 13.22 | 0.96 | 11.95 |
26 M | 13 | 0.793722 | 0.021277 | 6.51 | 4.40 | \(-\)0.19 | 12.95 | 0.96 | 11.92 |
78 S | 7 | 0.060538 | 0.265957 | 15.38 | \(-\)1.93 | \(-\)1.90 | 12.18 | 1.01 | 6.05 |
79 M | 16 | 0.596413 | 0.306383 | 9.75 | 2.82 | \(-\)1.57 | 13.99 | 1.01 | 16.22 |
114 M | 18 | 0.107623 | 0.13617 | 23.67 | \(-\)3.07 | \(-\)1.23 | 12.43 | 0.96 | 18.11 |
119 S | 5 | 0.002242 | 0.580851 | 9.74 | \(-\)0.08 | 0.66 | 11.87 | 1.01 | 1.97 |
125 S | 9 | 0.432735 | 0.065957 | 14.18 | \(-\)2.11 | \(-\)0.60 | 14.14 | 0.96 | 7.54 |
131 M | 20 | 0.327354 | 0.510638 | 0.26 | \(-\)4.10 | 0.09 | 13.58 | 1.01 | 22.85 |
131 S | 3 | 0.863229 | 0.551064 | 18.63 | 3.61 | 0.41 | 12.58 | 1.01 | 2.61 |
137 M | 11 | 0.757848 | 0.995745 | 4.70 | 4.53 | 0.04 | 13.13 | 0.96 | 10.44 |
137 S | 18 | 0.293722 | 0.03617 | 23.07 | \(-\)4.40 | \(-\)0.33 | 13.41 | 0.96 | 21.52 |
172 M | 24 | 0.269058 | 0.825532 | 18.61 | \(-\)4.51 | 1.58 | 13.28 | 0.96 | 21.08 |
172 S | 18 | 0.804933 | 0.865957 | 12.98 | 4.32 | 1.21 | 12.89 | 0.96 | 21.33 |
178 M | 3 | 0.699552 | 0.310638 | 23.06 | 4.35 | \(-\)1.54 | 13.44 | 1.01 | 2.29 |
178 S | 15 | 0.235426 | 0.351064 | 17.43 | \(-\)4.52 | \(-\)1.21 | 13.10 | 1.01 | 14.61 |
184 S | 7 | 0.665919 | 0.83617 | 21.87 | 4.02 | 1.48 | 13.62 | 0.96 | 7.65 |
190 M | 15 | 0.560538 | 0.280851 | 7.94 | 1.93 | \(-\)1.78 | 14.17 | 1.01 | 15.78 |
We compute the sum of the squared residuals (SSR) by taking [column (10)–column (2)] \(^{2}\) and summing over all 18 eclipses. We minimize this sum with respect to the three free parameters, \(D\), \(V_{\mathrm{F}}\) and \(\bar{h} _0 \) (the time of day of the opposition of month 1). The results are shown in Figs. 24a, b, 25a, b. Thus, one may read off the values of , and \(v_{\odot }\) on these graphs and compare with the values given in Table 15. The results of the fits for the amplitudes of the equations of center are comparable to what we have seen before. The parameters of the fit are: \(\bar{h} _0 = {-}9.40\) h, \(D = 2.634^{\circ }/\hbox {day}\), and \(V_{\mathrm{F}}= 29.762^{\circ }/\hbox {sm}\), all quite reasonable. The value of \(D\) corresponds to an amplitude of \(4.54^{\circ }\) for the lunar equation of center. The value of \(V_{\mathrm{F}}\) corresponds to an amplitude of \(2.14^{\circ }\) for the sawtooth that is the solar equation of center function. These are in about the right range to function well in terms of actual phenomena. But they are a bit smaller than the corresponding values from standard Babylonian theory—another sign of adaptation by the designer of the Antikythera mechanism.
We have also tried an epicycle theory—in which the variable motions of both the Sun and the Moon are represented by epicycles. Again, we take both the solar and lunar mean anomaly to be \(180^{\circ }\) at the full Moon of month 1. We compute predicted times exactly according to the scheme described in this section, the only difference being the forms of the equation of center functions. So, again there are just three parameters to be varied—the radii of the solar and lunar epicycles and \(\bar{h} _0 \). We fit the parameters by requiring the sum of the squared residuals (SSR) in the eclipse times to be a minimum. The results are , where the radii of the deferent circles are taken to be unity.^{23} These parameters result in a graph that differs only very subtly from Fig. 26. Again, the epicycles \((\hbox {sum of squared residuals} = 58.0\,\hbox {h}^{2})\) fit the inscriptions only a little less well than does the Babylonian-style theory \((55.8\,\hbox {h}^{2})\). Thus, while a correction to the mean times based on equations of center and velocities appears quite plausible, it is not possible to reject the epicycles solely on the basis of the goodness of fit.
20 Longitude of best fit
Comparison of eclipse times on the Antikythera mechanism with the actual times of the eclipses in UT, for deducing the longitude of best fit
Eclipse | AM time (h) | Lunation number | Date (Espenak) | TD (Espenak) | \(\Delta \)T (s) (Espenak) | UT (h) | AM–UT (h) |
---|---|---|---|---|---|---|---|
20 M | 0 | \(-\)27237 | \(-\)203 Nov 25 | 2:18:04 | 12758 | \(-\)1.24 | 1.24 |
26 M | 13 | \(-\)27231 | \(-\)202 May 20 | 14:18:45 | 12752 | 10.77 | 2.23 |
79 M | 16 | \(-\)27178 | \(-\)198 Sep 1 | 18:01:27 | 12700 | 14.50 | 1.50 |
114 M | 18 | \(-\)27143 | \(-\)195 Jul 1 | 18:56:57 | 12666 | 15.43 | 2.57 |
131 M | 20 | \(-\)27126 | \(-\)194 Nov 16 | 0:22:07 | 12649 | 20.86 | \(-\)0.86 |
137 M | 11 | \(-\)27120 | \(-\)193 May 11 | 12:40:18 | 12643 | 9.16 | 1.84 |
172 M | 24 | \(-\)27085 | \(-\)190 Mar 11 | 0:44:45 | 12609 | 21.24 | 2.76 |
178 M | 3 | \(-\)27079 | \(-\)190 Sep 3 | 2:28:11 | 12604 | \(-\)1.03 | 4.03 |
190 M | 15 | \(-\)27067 | \(-\)189 Aug 23 | 17:49:16 | 12592 | 14.32 | 0.68 |
25 S | 12 | \(-\)27231 | \(-\)202 May 6 | 16:47:22 | 12752 | 13.25 | \(-\)1.25 |
78 S | 7 | \(-\)27178 | \(-\)198 Aug 18 | 5:59:42 | 12700 | 2.47 | 4.53 |
119 S | 5 | \(-\)27137 | \(-\)195 Dec 11 | 1:33:19 | 12660 | \(-\)1.96 | 6.96 |
125 S | 9 | \(-\)27131 | \(-\)194 Jun 6 | 11:04:05 | 12655 | 7.55 | 1.45 |
131 S | 3 | \(-\)27125 | \(-\)194 Nov 30 | 2:10:48 | 12649 | \(-\)1.33 | 4.33 |
137 S | 18 | \(-\)27119 | \(-\)193 May 27 | 1:16:17 | 12643 | 21.76 | \(-\)3.76 |
172 S | 18 | \(-\)27084 | \(-\)190 Mar 24 | 23:07:08 | 12609 | 19.62 | \(-\)1.62 |
178 S | 15 | \(-\)27078 | \(-\)190 Sep 18 | 14:44:22 | 12603 | 11.24 | 3.76 |
184 S | 7 | \(-\)27072 | \(-\)189 Mar 14 | 10:29:38 | 12597 | 6.99 | 0.01 |
21 Casting the net more broadly
Until now, we have assumed that the Saros dial eclipse pattern was inspired by Babylonian records, or, at least, was consistent with some Babylonian conventions. In this section, we will show that we arrive at exactly the same result without assuming knowledge of the details of Babylonian eclipse theory. In “Appendix 3,” we show that eclipse 13S is necessarily a \(\hbox {D}_{1}\). We will use just this datum plus the result of Sect. 13, based on the lunar and solar anomaly analysis (which is independent of any assumption of Babylonian influence), to show that, nevertheless, our candidate of \(-\)204 is still the only one possible. This section, therefore, could be understood as an alternative to Sect. 12 for those readers who question the dependence of the Saros dial on Babylonian conventions.
Therefore, we would not be able to exclude series 50 on the basis of the lunar anomaly alone. Nevertheless, series 50 may be ruled out in a number of ways. Perhaps, the easiest way to see this is to make a total squared error over a Saros (TSEOS) graph for Saros series 50, which is shown in Fig. 29. We let each \(\hbox {D}_{1}\) solar eclipse in Saros series 50 function (in turn) as a candidate for the eclipse of month 13. When we have assumed that a given \(\hbox {D}_{1}\) eclipse is 13S, it is an easy matter, just by counting months, to identify each other eclipse on the AM Saros dial with a particular eclipse in the Espenak catalog. But once we choose a series 50 eclipse for month 13, we must still specify whether to take the times on the Antikythera mechanism as we find them, or to add 8 h to them, or to add 16, corresponding to the three possible positions of the exeligmos dial. Then, for each choice that we make (of the eclipse at month 13 and of the position of the exeligmos dial), we calculate the total squared error over all 18 eclipses. To do this, we subtract the time of day of the actual eclipse (converted to local Aegean time) from the glyph time and square the difference. These differences are added up for all 18 eclipses. The result is plotted as a single point in Fig. 29. For each predicted and actually occurring eclipse, the glyph time was compared with the time of greatest eclipse, in UT. If an eclipse did not actually occur, the glyph time was compared with the time of true conjunction or true opposition, expressed in UT.
As can be seen, we obtain three curves. For example, if we suppose that the eclipse of \(-\)66 fell in month 13, we can either add 0 h to all the AM times (black point), add 8 h (gray) or add 16 h (white). These three points for \(-\)66 lie on three different curves. Points lying on a single curve are all consistent with one another as far as the calibration of the exeligmos dial goes, but assume different eclipses for month 13. The TSEOS graphs for series 50 show no well-defined minimum at all, and the total error is much higher than for Saros series 44. No calibration of the exeligmos dial is strongly preferred. Saros series 50 is rejected. (In “Appendix 7,” we also prove that Saros 50 is inconsistent with the requirements imposed by the solar and lunar equations on the time of year and time of day of the opposition of month 1).
For each \(\hbox {D}_{1}\) eclipse that is a candidate to be 13S, we of course have to do this three times, because in each run, we must choose whether to use all eighteen glyph times just as they come from the AM, or to add 8 h to all of them, or to add 16 h to all of them. In Fig. 30, solid black points indicate that the AM times are used directly, with nothing added to them. Gray points indicate that 8 h have been added to all the AM times. Hollow white points indicate that 16 h have been added to all the times.
22 Some confirming evidence
22.1 Date of the opposition of month 1 deduced from the solar anomaly
In establishing the epoch of the Saros dial, we used a sort of sieve of Eratosthenes to systematically remove possibilities, until we were left with a single date. In this approach, the lunar mean anomaly played a key role, as it allowed us to rule out all but solar Saros series 44 for the eclipse of month 13. But we did not need or make any use of the solar equation of center. The solar fit, however, provides important confirming evidence.
If we take the longitude of the Sun’s apogee to be about \(65.5^{\circ }\), which was Hipparchus’s value, and which, more importantly, was about right for the era of the Antikythera mechanism, the mean longitude of the Sun at the opposition of month 1 would be in the range \(35^{\circ }\)–\(66^{\circ }\).^{24} These longitudes correspond to dates of roughly April 26 to May 27 (Julian calendar). That is, according to the result of fitting the solar equation of center, the full Moon of month 1 should have been between late April and late May. This is in good agreement with the result of matching the absolute times of eclipses, which gave us the full Moon of \(-\)204 May 12 as the best fit for month 1. Thus, the surviving material is redundant—there is more than the minimum necessary for establishing an epoch. It is important that two different approaches lead to concordant results.
In looking at the TSEOS graphs of Figs. 17 or 18, one can see that \(-\)203 is favored for the solar eclipse of month 13, but are \(-\)257 and \(-\)149 so far from the bottom of the well that they must be excluded? Adopting either of these dates would give trouble with the requirement that opposition of month 1 lies between April 26 and May 27. For example, if we pick the eclipse of \(-\)149 June 19 for month 13, the full Moon of month 1 would be that of 150 June 14, which is too late in the year. Similarly, if we pick the eclipse of \(-\)257 April 15 for month 13, then the full Moon of month 1 would be that of \(-\)258 April 10, too early in the year. Thus, the interlocking requirements of the TSEOS graphs and the solar anomaly analysis provide a tightly constrained solution.
22.2 Value of \(\bar{h}_0\)
As mentioned in Sect. 13, the parameter \(\bar{h} _0 \), found by fitting the lunar and solar anomalies, represents the time of day of the opposition of month 1. Our result was between 14:35 and 16:02, in the local time appropriate to the Antikythera mechanism. We also found that the opposition of month 1 best corresponds to that of \(-\)204 May 12.
Consulting Espenak (Six Millennium Catalog of Phases of the Moon) on the NASA Eclipse Web Site, we find that the full Moon of \(-\)204 May 12 fell at 13:21 UT. For the meridian appropriate to the Antikythera mechanism, we use \(25.3^{\circ }\hbox { E}\), the longitude of best fit from Sect. 20. (Here are some representative longitudes that help define the Greek cultural zone: Alexandria \(29.9^{\circ }\hbox {E}\), Athens, \(23.7^{\circ }\), Syracuse \(15.3^{\circ }\hbox {E}\)). Converting from UT to “Antikythera mechanism time,” we find that the full Moon of \(-\)204 May 12 fell at \(13{:}21 + (25.3/15)^{\mathrm{h}} = \mathbf{15}{:}\mathbf{02}\). This would be the local time for \(25.3^{\circ }\hbox {E}\) longitude. We could add \(18^{\mathrm{m}}\) for Alexandria, or subtract \(40^{\mathrm{m}}\) for Syracuse and still be very close to the required zone. In short, the required time of the opposition of month 1 is a very good match to the opposition of \(-\)204 May 12.
22.3 The fiducial mark^{25}
Most of the moving parts of the mechanism were actuated by gears driven by a single input. However, one part had to be moved by hand. This is the Egyptian calendar ring, which was divided into the 12 months (30 days each) and five additional days of the Egyptian year. Because the Egyptian calendar year was always 365 days long, with no leap days, the calendar ring had to be displaced “by hand” by one day every 4 years. Beneath the Egyptian calendar ring is a circle of closely spaced holes drilled into the underlying plate. There was probably a little post (or posts) on the back of the calendar ring. The ring could therefore be pulled off, turned to the appropriate orientation for the year under consideration, and then plugged back in.
On the plate just outside the Egyptian calendar scale, near the beginning of the month of Payni, is an apparently engraved radial mark. Price (1974, pp. 19–20) argued that this was a fiducial mark for setting the Egyptian calendar ring for some initial date. But in his analysis Price assumed that the calendar ring is still in its original position and, when this led to impossible dates, that it was set at the correct day of the month, but the wrong month of the year. However, as is known, the Egyptian calendar ring is out of its proper position by several months for the epoch of the Antikythera mechanism, so no inference can be drawn from the day of the year that now happens to lie against the fiducial mark. But, since the mark is inscribed on the same plate as the zodiac, something interesting can be said about the zodiac degree corresponding to the mark. To be sure, the X-ray computed tomography (CT) scans show that the plate in the vicinity of the mark is badly cracked, so one could wonder whether this is a deliberately made mark or some sort of damage.
We point out that the fiducial mark is almost perfectly radial. (See Evans and Carman (2014), Figure 1 on p. 155). The radial direction of the mark supports the view that it is indeed associated with the scales. Like Price, we ask whether it might have been intended as the “\(t = 0\)” setting mark for the Egyptian calendar ring. Of course, it could be conceivable to prescribe the setting of the calendar ring without the use of a separate fiducial mark, if, for example, there were an inscription that said: for such and such a year, place 1 Thoth against a certain degree of the zodiac. We are lucky in the portion of the zodiac that is preserved, amounting to approximately a quadrant. For the Sun’s position on the first of Thoth fell in the extant portion of the zodiac between the years \(-\)424 and \(-\)71. This encompasses practically the whole range of possible dates for the construction of the Antikythera mechanism, except perhaps for a very few years at the most recent end of the interval, immediately before the shipwreck. Thus, if there were a calibration mark for the first of Thoth, it would almost certainly have to fall in the preserved portion of the zodiac. There is one, and only one, such mark visible in the CT and, as there is only one, it is likely the setting mark for the calendar scale. But we acknowledge that in the research community, opinion is divided about whether this mark is intentional or accidental.
Let us enquire for just which year the beginning of Thoth would be aligned with the fiducial mark. In the list below, for each year in column 1, column 2 indicates the Julian calendar date corresponding to 1 Thoth, taken from Bickerman (1980, pp. 115–112). Column 3 gives the longitude of the Sun calculated from modern theory for noon of 1 Thoth in the given year, at \(23^{\circ }\) East longitude. For \(\Delta {T}\) we used 3\(^{\mathrm{h}}\), which is appropriate for the years around \(-\)200.
Longitudes of the Sun (calculated from modern theory) at noon on the first day of Thoth, for geographical longitude \(23^{\circ }\hbox {E}\) | ||
---|---|---|
Year | 1 Thoth | Sun |
\(-\)197 | Oct. 12 | 195.2 |
\(-\)201 | Oct. 13 | 196.2 |
\(-\)205 | Oct. 14 | 197.2 |
\(-\)209 | Oct. 15 | 198.2 |
\(-\)213 | Oct. 16 | 199.1 |
\(-\)217 | Oct. 17 | 200.1 |
\(-\)221 | Oct. 18 | 201.1 |
Now, as we have seen, the middle of month 1 of the Saros dial fell in the month of May. But the first month of the Metonic dial is Phoinikaios, the first month of the civil year in the family of calendars related to the calendar of Corinth. Strong evidence requires this month to fall in August or September (Paul Iversen and John Morgan, personal communication; Iversen 2013; Morgan 2013). Thus, it is clear that the first month of the Saros dial and the first month of the Metonic dial do not correspond to the same month. This was a surprise to us, and at first a disappointment, as we had hypothesized that the two dials would begin at “month 1” together. This also would have had great simplifying advantages in the search for an epoch. But in fact the two pointers do not start off together, with each pointing to the first month of its spiral on “day 1.” As we have seen, there is a simple and plausible explanation for the choice guiding the structure of the Saros dial: At the full Moon of month 1, both the Moon and the Sun were at apogee. The first cell of the Metonic dial is, naturally enough, the first month of the civil year.
Now, does the fiducial mark correspond to the epoch of the Saros dial or to the epoch of the Metonic dial? The fiducial mark connects the (approximately) solar year of the Egyptians with the Sun’s motion around the zodiac. Since the Greek luni-solar year is also tied to the zodiac (with some sloshing back-and-forth in accordance with the 19-year Metonic cycle), the fiducial mark can be understood as providing a statement about calendars. In a way it provides the link between the Egyptian and the Greek calendar, through the intermediary of the zodiac. There is no reason why it should have anything to do with the Saros cycle directly.
Thus, we believe that the fiducial mark indicated the zodiac position of the first of Thoth in year 1 of a certain Metonic cycle. In this paper, we have given strong evidence for putting the first month of the Saros dial in \(-\)204. The fiducial mark (if such it is) suggests a Metonic cycle that began between \(-\)221 and \(-\)197. Of course, there is no reason for there to be a direct connection between a Metonic and a Saros cycle. They run with their own periods (19 years versus about 18 years and 11 days). And if one is constrained to start at the beginning of the civil year and the other to start when the Sun and Moon are both at apogee, it would require nearly a miracle to have them start out together.
23 Summary and closing discussion
If we assume that the solar and lunar eclipses were placed on the Saros dial in conformity with Babylonian 8-8-7-8-7- patterns, then 14 different reconstructions are consistent with the extant eclipse glyphs. If we invoke the Babylonian convention that the 8-8- of a solar Saros scheme should start at the descending node and that the 8-8- of a lunar Saros scheme should start with the Moon at the ascending node, and if we make use of the apparent fact that the southernmost solar eclipses of each diagonal sequence were omitted, these 14 combinations are reduced to a single solution, which we call \(7\beta \).
The eclipse predictor of the Antikythera mechanism works best if the full Moon of month 1 of the Saros dial corresponds to \(-\)204 May 12. This we shall refer to as the likely epoch of the eclipse predictor. Further, the exeligmos dial should read zero for the Saros cycle starting on \(-\)204 May 12.
The solar eclipse of month 13 belongs to solar Saros series 44 (this particular result is very strong, the strongest result of this investigation), and the eclipse predictor will work best if this is the eclipse of \(-\)203 May 17.
At the epoch, both the solar anomaly and the lunar anomaly were close to zero measured in the ancient way from apogee (or close to \(180^{\circ }\) if measured from perigee in the modern way). The lunar anomaly alone is enough to secure the dating. But the solar equation also implies that the first cell of the Saros dial corresponded roughly to May, which provides confirmation for the dating by means of the lunar anomaly.
Calendrical evidence implies that the first month of the Metonic dial probably corresponds to August or September. Thus, the first cell of the Metonic dial and the first cell of the Saros dial do not represent the same month.
The relation between the lunar and solar eclipses follows the alternative Steele rule. This suggests that the adaptor drew on a Babylonian solar eclipse list that had already moved over to a new calibration, but a lunar eclipse list that was still running on the previous calibration.
In Sect. 21, we were able to find a likely epoch date without making use of Table 10 (the list of eclipses historically used by the Babylonians to start a solar 8-8-). The epoch nevertheless agrees with the epoch we found using the Babylonian record. Some extra robustness is therefore lent to the epoch date determined.
It is plausible and perhaps likely, but not possible to prove statistically, that the corrections to the mean times of the eclipses were done on the basis of Babylonian-style “equations of center”—a quadratic form for the Moon and a piecewise-linear form for the Sun. Given the likely epoch of the eclipse predictor, it is reasonable to exclude a trigonometric equation of center, since the likely epoch is before the development of trigonometry. In any case, a method of prediction based on Babylonian-style equations of center and the associated daily velocities reproduces the eclipse times of the AM rather well. The eclipse times on the AM agree better with the results of this prediction scheme than they agree with the real times of the eclipses.
The geographical longitude of best fit (obtained by matching the AM eclipse times to the actual times of the eclipses) corresponds to the Aegean Sea with an uncertainty of about an hour of longitude.
What was the likely date of fabrication? The eclipse predictor works best for the years \(-\)204 to \(-\)186. It could run for a couple of Saros cycles before or after that, using the 8- and 16-h corrections from the exeligmos wheel, but the eclipse times would be less accurate. Was the machine fabricated in advance of \(-\)204, with the idea that it would be at its prime from \(-\)204 to \(-\)186? This would suggest a fabrication date somewhat before \(-\)204. Or were eclipse data compiled during the Saros of \(-\)204 to \(-\)186 (which would perhaps explain why the eclipses best fit this period)? This would imply a construction somewhat after \(-\)186. And, of course, we cannot exclude the possibility that a Greek mechanic used an old and outdated eclipse list, hoping that the 8-h rule would keep it relevant to his own day.
While it is not possible to be certain, there are pieces of evidence that bear on the issue. The solar eclipse predictor follows one calibration of the Saros scheme, but the lunar eclipse predictor was still running on an older calibration. This suggests a fabrication date reasonably close to a Babylonian recalibration of the Saros scheme. The approximate dates of these recalibrations are shown in Table 10—thus sometime around \(-\)261, around \(-\)203, or around \(-\)109. But the timing errors of the eclipses would be significantly larger around \(-\)261, and very much larger around \(-\)109. Again, \(-\)203 looks like the best candidate.
Also, it would not have been possible to construct an entire sequence of solar and lunar eclipse times based on observations over one Saros cycle (from \(-\)204 to \(-\)184, say), as too few of the eclipses would be visible from any one place; so this suggests prediction form theory, which means that there is no strong reason to prefer a date after \(-\)184. Moreover, for the Babylonians, observed times were frequently expressed in terms of the interval between the beginning of the eclipse and sunrise or sunset (Steele 2000b, pp. 57–58), often measured in terms of the uš (or “time degree,” which is one 360th of the day and night). And, according to Steele (2000b, p. 66), a typical Babylonian eclipse timing, running, e.g., for about \(2^{\mathrm{h}}40^{\mathrm{m}}\) has an accuracy of about \({\pm }24^{\mathrm{m}}\), which is quite a bit better than the typical timing errors for predicted eclipses. Thus, we can see that Babylonian records of observed eclipse times would not have been in a form that could have been easily or directly used. And, if the eclipse times of the AM were based directly on observations, the timing errors should be smaller than they are.
By contrast, as we have seen, the errors in the eclipse times are of about the right size to be consistent with prediction. Thus, while we cannot be certain, a plausible inference is that the eclipse times were predicted to run for the Saros cycle \(-\)204 to \(-\)184, which suggests a fabrication time close to \(-\)204, and possibly somewhat before. Finally, the fiducial mark suggests the mechanism was keyed to a Metonic cycle that began within about \({\pm }12\) years of \(-\)209.
There are more data extant on the Saros dial than the minimum needed to establish a likely epoch; the resulting crosschecks make the epoch date reasonably strong. Because the conclusions drawn in this paper are so tightly constrained, it would be easy to check them and to refute or refine them if more fragments of the Saros dial should ever come to light.
The Babylonians had no way to treat the lunar parallax. Some writers characterize also the lunar eclipses in a Saros scheme as “eclipse possibilities.” The advantage of this usage is that it acknowledges that the Babylonian astronomers knew that sometimes a predicted lunar eclipse does not occur or that sometimes one occurs in the wrong month. Still, the prediction of lunar eclipses was pretty accurate, so to describe these predictions as giving mere possibilities seems an unduly positivistic characterization of what the Babylonians were up to.
Original publication of the eclipse glyphs is then known: (Freeth et al. 2008). These (both the journal article and the online Supplementary Notes) contained some errors in the presentation of the eclipse data. These have been corrected in a new version of the Supplementary Notes posted at http://www.nature.com/nature/journal/v454/n7204/extref/nature07130-s1.pdf, to which readers should now refer. (Most of these errors were corrected by Beatriz Bandeira of the Universidad Nacional Tres de Febrero of Argentina). We have taken the times of the eclipses from the most recent analysis made by Alexander Jones in a still unpublished paper. They differ in some cases from the times published in the Supplementary Notes of Freeth et al. (2008). The following differences should be noted. We list first the eclipse cell and eclipse type, then the reading of Freeth et al., and finally Jones’s reading. 13S: day 1 or 4, day 1. 78S: night 1, day 1. 120M: day 8, day 6. 172M: no reading, night 6. 172S: no reading, day 12. 178S: most unclear, day 9. In all cases, we followed Jones’s readings. We excluded from the time analysis three eclipses: 120M because Jones’s reading is uncertain and 67S and 72H because the reading is incomplete. For S119 both Jones and Feeth et al. 2008 read \(\iota \)[ ] which could be 10, 11 or 12, so we assume the mean value, 11. A lunar eclipse in month 61 has recently been discovered (Anastasiou et al. 2014). Our eclipse analysis was completed before this discovery was made; but the new glyph fit right into the already reconstructed Babylonian pattern with no changes required.
That is, one modifies the solar Saros scheme without modifying the lunar scheme, in such a way that the solar scheme that began 4 EP earlier than the lunar one now begins 4 EP later than the lunar one. This could sound odd if one supposes that the solar schemes originated as modifications of the lunar schemes. Why would you recalibrate the solar and not the lunar if the solar is based on the lunar? But it seems that in the case of re-calibration the dependence is in the other direction. Steele (2000a, p. 443) suggests that one particular lunar recalibration, in \(-\)110, was unnecessary but nevertheless took place to coincide with a solar re-calibration. He also suggests that “Since solar eclipses can occur at greater nodal elongations than lunar eclipses, it was necessary to revise the solar Saros scheme more often than the lunar Saros.”
For example, Steele (2000a, p. 442) points out that the circumstances of the \(-\)110 recalibration are not completely clear.
For the Moon’s argument of latitude (the mean angular distance of the Moon from its ascending node), see (Meeus 2009, p. 338, Eq. 47.5). In our calculations, we used an Excel program, based on Meeus’s formula, written by Dennis Duke.
The groupings of eclipses into 8-8-7-8-7s are from Steele (2009, Table 4). The Saros series numbers for these eclipses come from Espenak (Five Millennium Catalog of Solar Eclipses).
However, the Britton rule could have been compatible with Babylonian practice before about \(-\)260: As we have seen, during the earlier period, while a solar 8-8- began with the Moon at D for the solar eclipse, the lunar 8-8- also began with the Moon at D for the lunar eclipse.
Cells 19 and 7 are, at least in theory, extant in fragments F and A, respectively. Cell 7 is almost impossible to see. It is also impossible to see Cell 8, or at least to read any letter. But cell 19 is clearer, and it seems reasonably clear that there is nothing there. While we cannot be sure from direct examination of these cells, the sequence of index letters is enough to decide the issue.
The maximum value of the Moon’s equation of center is actually about \(6.3^{\circ }\); but we are concerned with eclipses, i.e., with phenomena taking place at new and full Moon. In these situations, the equation of center is reduced by the evection by about \(1.3^{\circ }\); thus, the effective maximum equation of center at new and full Moons is about \(5^{\circ }\). We neglect higher order terms in the equation of center, the annual equation, and other small perturbations of the Moon’s longitude.
By the instant of opposition, we mean instant when the Moon’s mean longitude is equal to the Sun’s mean longitude \(+\) \(180^{\circ }\). \(\bar{h} _0 \) is the local time of day at which this happened. The time of true opposition would be \({h}_{0}\), which differs from \(\bar{h} _0\) due to the contributions of the lunar and solar equations.
For an introduction to the \(F\)-test, see (Ramsey and Schafer 2002, pp. 122–127, 280–285), with tables at pp. 720–727. A more mathematically detailed explanation is given in Fox (2008, pp. 200–202). Functions for calculating the p value from the F-statistic are provided in Excel and other commonly available programs.
For the middle of the eclipse, here, more precisely, is what we used: For eclipses that actually occurred, we used the time of greatest eclipse from the Espenak solar eclipse catalog. For eclipses predicted by the Babylonians which did not really occur, we use the time of true conjunction, from (Espenak, Six Millennium Catalog of Phases of the Moon).
But in the analysis in this paper, we use the length of the anomalistic month that is implicit in the Antikythera mechanism, which is (223/239) times the synodic month. For the synodic month, we use \(365.25^{\mathrm{d}} (19/235)\). This makes no practical difference: An anomalistic month of 27.556 days for System B and Geminus, 27.554 days implicit in the AM.
Our technique is the same as that used in Sect. 13. We compute the lunar and solar equations of center for the given models and assume that these need to be compensated for by the Moon and Sun running at their mean relative motion of \(12.2^{\circ }/\hbox {day}\). Once we know the locations of the perigees, it is possible to apply a more sophisticated reckoning, in which we take into account the variable speeds of the Moon and Sun. This is done in Sect. 19.
The mean anomaly does not have a clear-cut definition as an angle in this non-geometrical model. The parameter determined from the fit is the offset of the zero crossing from \(t = 0\) in Fig. 22, which is 0.01392 of an anomalistic period. Note that at \(t = 0\), the eclipses are slightly late and getting later; thus, the equation of center is slightly negative and getting more negative, i.e., we are near (and just after) apogee, by \(0.01392 \times 360^{\circ } = 5.0^{\circ }\), hence the value \(185.0^{\circ }\).
The mean anomaly does not have a clear-cut definition as an angle in this non-geometrical model. The parameter determined directly from the fit is the t-value of the most negative point in Fig. 23, which is 0.24084 of a year. The zero crossings of the graph are the moments when the equation of center is zero, which we associate with “perigee” and “apogee.” As can be seen, at \(t = 0\), the eclipses are very slightly early, and getting earlier. Hence, we are near, and just after, apogee. The zero crossing would have occurred at \({t} = -0.01028\) year. Hence, we take the mean anomaly to be \(0.01028 \times 360^{\circ }\) beyond apogee, i.e., \(183.7^{\circ }\). Note that in this Babylonian-inspired model, the maximum and minima are not uniformly spaced. However, the zero crossings are.
The Babylonian-style method described here is actually a pretty good predictor of eclipse times: Its RMS discrepancy with the true eclipse times is 1.51 h.
The maximum values of the equations of center corresponding to these epicycle radii are about \(1.75^{\circ }\) for the Sun and \(4.90^{\circ }\) for the Moon.
Recall that \(\bar{\lambda } =\bar{\alpha }+ \Pi \), where for the Sun we may put \(\Pi = 65.5^{\circ } - 180^{\circ }\). Thus, \(\bar{\lambda } =\bar{\alpha }- 114.5^{\circ }\). For the arbitrary value of the solar mean anomaly \(\bar{\alpha }\), put in any of the initial values of the mean solar anomaly \(\varphi _{\odot }\) from column 5 of Table 12. For example, if we use \(\varphi _{\odot }= 175.5^{\circ }\), we find \(\bar{\lambda }= 61.0^{\circ }\) for the mean longitude of the Sun at the opposition of month 1.
Acknowledgments
We are grateful to James Bernhard for a discussion of statistical issues. Alexander Jones generously shared preliminary results of his own unpublished research which disclosed the existence of a lunar equation of center embedded in the Antikythera mechanism eclipse times. This turned out to be crucial for our project—it is the lunar anomaly that allows one to unambiguously identify a particular solar Saros series. Dennis Duke gave valuable help in calculation and programming. We thank John Steele for answering our questions about the Babylonian Saros schemes. We thank Paul Iversen and John Morgan for their lively and helpful discussion of previous versions of the paper. The Antikythera Mechanism Research Project has, as always, been extremely generous in sharing images and data. James Evans expresses his thanks to the University of Puget Sound for research funds that helped make this work possible. Christián Carman would like to express his thanks for the support of Research Project PICT-2010-0319 of the Agencia Nacional de Promoción Científica y Tecnológica of Argentina. While we were in the last stages of completing our manuscript, an article by Tony Freeth was published that focuses on some of the same issues that we have studied (Freeth 2014), but it appeared too late for us to make any use of it. The work published here is independent of Dr. Freeth’s study.