THE COLIGNY CALENDAR AS A METONIC LUNAR CALENDAR

THE COLIGNy CALENDAR AS A METONIC LUNAR CALENDAR

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Helen T. McKAy

In history and folklore, the Celtic and other northern European peoples held the moon in great awe, and used its various phases as omens to regulate their own earthly activities.

The one thing that has always been accepted without question by scholars about the Coligny Calendar is that its focus is on the lunar cycle, and that its internal notations show patterns consistent both between months of the lunar year, and between days of the same month across the years, and so it follows that these notations contain a significance in relationship to the actual lunar phase at which they occur. This paper will accept that each month begins on the 'sixth day' of the new moon as recorded by Pliny the Elder 1 . This places day 1 of each month at the only point of the lunar cycle that can clearly be differentiated by eye, the quarter moon. This means that the calendar could be 'calculated', that is, set down ahead of time, rather than announced upon finally sighting the new moon, as many other calendars of its day were. It also means that everyone, farmer, king, or priest, could look to the sky and know when a month was starting. It places the full moon at day 8, and the opposite middle dark moon at day 8a (23), days 8 and 8a being the central days respectively of the two 15-day fortnights of each month on the calendar. The patterns of a number of internal daily notations are focused around these key central days of significance. For example, Mac Neill 2 noted that the N INIS R notation clusters around days 8 and 8a as part evidence for his assertion that the calendar was wholly lunar.

It makes no sense that a lunar calendar with complex and detailed notations focused on certain points of the lunar month, as the Coligny calendar is, should be allowed to slip out of sync with the actual moon's phases (lunations). The Coligny calendar from the start gives the appearance of being special in the very complexity of the web of daily notations which make best sense as tied to a point in the lunar phase.

Given the long history of the western European peoples watching and measuring the sun and moon since Neolithic times, we should expect the calendar to keep in sync with the lunar cycle with great precision. And it is this simple idea -that we should expect to be able to place the Coligny calendar within its cultural contextthat begins this paper.

In history

While the following paper has been written in a way that is designed to present the argument in a logical sequence to the reader, it might be of interest that this isn't the way it was initially worked. By the time I came to ask the question of just how closely the calendar tracks the lunations, I had already re-worked the internal notations, so I had a strong sense of how powerfully and precisely the internal patterns are made to track the internal points of the calendar month, and by this presumably the actual lunar synodic month. In consequence, it was highly likely that the Metonic cycle was also laid out on the calendar, as the Metonic cycle is a natural outcome of observation of the lunations with precision. I also, for example, already knew by this stage, that Intercalary One logically held 29 days, not the 30 that has until now been presumed, and that the anomalous IVOS run is the result of a copying error. When I put this corrected internal data into the calendar, the Metonic cycle simply appeared.

The debate over the exact shape of the lunar months and the calendar years, and the implications for its longer cycle, whether 19 years, 25 or 30, has raged for the last century. So many ideas and figures have been proposed, upheld, denied, and sometimes refined as the internal patterns of terms became better understood over time. I cannot present a detailed history here, I can but recommend Olmsted's 3 chapter on the fascinating history of research on the calendar.

A key aspect of the debate is that it has been difficult to perceive just how the calendar months could possibly track the lunations correctly, when every proposed shape of the calendar either allows the calendar months to slip out of sync with both the moon and the sun by having too many days in the 5-year cycle of the plaque, or, alternatively, in order to keep the calendar in sync with the moon at the 5-year mark that it must be made to hold at least two very short EQUOS months of 28 days, an idea which has never felt 'right' to anyone who has experienced the mathematical beauty and precision of this calendar, which otherwise only uses 29 -and 30-day months.

How best to sync the calendar and lunar months

To begin, I will examine the 5-year cycle on the calendar plaque that we have, to measure just how closely the calendar months fit with the actual 62 synodic months of the moon. This may sound straightforward except that we have three calendar months whose length we are not certain of, and so the possible lengths of these unknown months need to be again held up to scrutiny.

The calendar is organized into 5 years of 12 lunar months each, with 2 intercalary months, the first at the lead into year 1, and the second half way through year 3. This gives a total of 62 lunar months in 5 calendar years. 3. olmsteD, 1992, p. 31-51. If the 5 year period of the calendar is to stay in sync with the moon's phases overall it must contain 1 831 days, the closest possible day count to 1 830.94 (Fig. 1). Strangely, this count of 1 831 days has until now proven difficult to fit to the total day count of the calendar.

As a point to start this analysis, I have used the values as given by Olmsted 4 as an example of the closest fit to the average lunations per calendar year (Fig. 2). For the three months of unknown length, EQUOS in years 2 and 4 is given 28 and 29 respectively, and Intercalary One is given the traditional 30. However, this produces a total number of days of 1 832, one day too many. Using these values as a jumping off point, I will now look at the internal evidence of the relevant notations on the calendar, and examine how these calendar lunar lengths relate to the actual lunations.

year 1

year 2

year 3

year 4 year 5 intercal1 30 SAMON 30 30 30 30 30 DUMAN 29 29 29 29 29 RIUR 30 30 30 30 30 ANAG 29 29 29 29 29 OGRON 30 30 30 30 30 CUTIOS 30 30 30 30 30 intercal2 30 GIAMON 29 29 29 29 29 SIMUIS 30 30 30 30 30 EQUOS 30 28 30 29 30 ELEMB 29 29 29 29 29 AEDR 30 30 30 30 30 CANTL 29 29 29 29

The Length of EQUoS

We know with reasonable certainty the length of most calendar months. But the month EQUOS presents a problem as it shows an exceptional pattern to the usual MAT months of 30 days and ANM months of 29 days.

EQUOS is marked as ANM, the notation which is usually found on a month of 29 days, and EQUOS is indeed treated as an ANM month by the internal patterns allocated to individual terms. An example is the PRINI term, where PRINI LAG is assigned to EQUOS as it is to other ANM months. However, in years 1 and 5 we can see that EQUOS is marked as 30 days long. EQUOS in year 3, although its end is on a lacuna, can also be confidently said to contain 30 days, as the script attached to the second intercalary month which occurs in the middle of the third year, and which itself has 30 days, states that this year contains 385 (CCCLXXXV) days.

Most scholars have accepted that this means that EQUOS in years 2 and 4 must be shorter, probably 28 days long in year 2, and 28 or 29 days long in year 4. year 4 seems to require 29 days to keep its average calendar months reasonably in step with the lunar phases around it, although to give the calendar the right number of days over 5 years it would need to be reduced to 28 days, making year 4 much shorter than acceptable, and slipping noticeably against the actual lunar months.

Conversely, year 2 has so many 30-day months leading up to this point, that only a 28-day month seems to come close to keeping the calendar in sync with the months. However, the assignment of 28 days to a month is a glaring exception on a calendar where nothing else has been shown to be random or outside the normal patterns.

If we were to accept these proposed values of EQUOS as 30/28/30/29/30, this gives us a total count of 1 832 days, not 1 831 days, assuming that both intercalary months have 30 days. We seem to have a day too many. This difficulty of assigning reasonable lengths to EQUOS is taken as the jumping off point by Olmsted 5 for his theory that the calendar was allowed to go out of sync with both moon and sun, and that this then required extra computation to track, and that as time went on, the calendar was changed from a 30-year cycle to a 25-year cycle.

Zavaroni, on the other hand, looks at the individual lunations, and maintains EQUOS with 30 days across all years in order for it to always contain a full lunation. This idea has the serious consequence of allowing calendar months to very quickly slip out of phase with the lunar month and the solar year.

I recall here the reason of my thesis according to which Equos has always 30 days. Even if one does not consider the 'yearly lunation pattern' in Table 4 -from which one may infer that Equos' lunation starts 35.2 hours before the first day of the month -the Equos lunation would commonly start more than 24 hours before the end of Simivisonna because it follows a semester with 178 days and a month with 30 days which cause a large advance (2-08-29.53 x 7 = 1.29 days). Hence the Equos lunation is not wholly contained within the month because it always starts at least 24 hours before 5. olmsteD, 2001, p. 8-11. EC-42.indd 98 25/10/2016 12:01:00 the beginning of the first solar day of the month. This systematic "being out" of the beginning of Equos would be the reason why it was considered ANMAT 6 .

Implications of IVoS in ELEmBIU for the Length of EQUoS

The struggle to discover what were the lengths of EQUOS in years 2 and 4 is critical to the overall shape of the calendar and how closely it stays in sync with the actual lunations. One item that some scholars have pointed to is the issue of the IVOS run that extends over the end of EQUOS into the following month of ELEMBIU.

In all the IVOS runs, providing we allow for day 15a (the 30th day) to be virtual in a 29-day month, that is, a day marked on the calendar or used in a pattern but not representing a real day, then there is only one instance where the length of an IVOS run for any particular point in the year differs in length. There are many instances where the IVOS run is missing in part or full, but nothing else to suggest the IVOS runs differ from year to year at a particular point in their respective monthly runs.

The one IVOS run that has different lengths is the part of the IVOS run that sits at the end of EQUOS extending into the start of ELEMBIU. In the solar year after an intercalary month, this IVOS run is moved upwards to sit over the SIMIVISONNA/ EQUOS boundary (Fig. 3).

year 1 (post intercalary)

year 2 year 3 (post IC)

year 4 year 5 In year 1, SIMIVISONNA XI (day 26) has a copying error, with D AMB EQUI, which should be D AMB IVOS. Olmsted also marks this as an error 7 .

SIMIUISONNA

In year 1, the second part of the IVOS run is fully present, and shows an IVOS run from days 1 to 4, giving a total run of 9 days, 5 at the end of SIMIVISONNA and 4 at the start of EQUOS. But in year 2, the ELEMBIU part of the IVOS run appears to go from days 1 to 5. However, this is readily seen as a parallax copying error, as D AMB always sits on day 5, never day 6. This error can easily be rectified by deleting the error on day 5, and replacing it with day 6, to make it exactly as seen in year 1.

This error in year 2 has been proposed by scholars including Mac Neill, Pinault, and Olmsted 8 as an indication that EQUOS in year 2 has a lesser number of days than the 30 of years 1, 3, and 5, the idea being that IVOS maintains the same number of total days in its run, and so a 29-day EQUOS in year 2 would push the ELEMBIU part of the run out to a 5th day. However, to begin with, we should remember that we are expecting an EQUOS of 28 days in year 2 rather than 29 days, which would require two extra IVOS days, not one.

The copying error that we can see by the displacement of D AMB from its proper day 5 to day 6 in ELEMBIU year 2 strongly indicates that the IVOS term on that same day is likewise a displacement for the day above. It is also not the way IVOS runs act over the variable end of a month. As we see in the movement of CANTLOS 13a-14a, when this IVOS run is dragged upwards by the presence of an intercalary to EDRINNI (with 30 days instead of 29), then the run also takes in days 13a-14a-15a without changing the lower part of its run. In other words, it doesn't matter to the IVOS run how many 'real' days a month has, as it effectively counts DIVERTOMV as a 'virtual' 30th day anyway, and so maintains a constant value over all instances, and does not produce any change in the lower part of its run. This same treatment of the 30th day as always present even when only virtual can also be explicitly seen when the TII counters run up against the end of the month. Because of this consistent behaviour of patterns on the calendar, we have no reason to assume that a decrease in the length of EQUOS would result in the IVOS run into ELEMBIU being extended, as we can see this explicitly doesn't happen in any other similar situations.

Zavaroni appears to also support this line of argument in general:

Besides, the IVOS are never lent and if they were, EQUOS would rather lend 2 IVOS instead of one, in order to justify a 29 day EQUOS 9 .

For both the above reasons, a) that the extra day with IVOS is readily seen as a simple copying error, and b) that the extent of the IVOS run into the next month will not be affected by the length of the previous month, this extended IVOS run in year 2 cannot be used to indicate that EQUOS in year 2 is shorter than 30 days. Nor unfortunately, can it be used to show that the length of EQUOS is any value in particular. We must look for evidence elsewhere.

an Initial assessment Fig. 4 shows the sorted differences over the 5 years between the value in days of the end of each month cumulatively, and the value of an equivalent number of synodic lunar months. The chart shows that the calendar is almost entirely running ahead of the moon, often by up to two days. If, for example, we were expecting to see the full moon on a particular day of the month, then 2 days is visibly beyond the full moon, which seems unacceptable. There is also a small negative tail in the chart which is caused by having EQUOS of 28 days in year 2.

The chart does appear to indicate that the extra day which we still have over the target 1 831 is present from the start of the calendar. In other words, it suggests that an extra day is being allocated incorrectly by us to Intercalary One.

Horizontal axis shows difference in days, with 0 as the average lunation value.

Intercalary one with a length of 29 days

As well as the month of EQUOS in years 2 and 4, there is another month whose length is in question, Intercalary One. There is very little present in the fragments of this intercalary month, including the bottom section. If we were to say that Intercalary One was 29 days long, rather than the 30 days which has been generally accepted up until now, then we would have a lunar calendar that has the correct number of days to stay in sync with the 62 lunar cycles of the calendar over the five years of the plaque.

So why has Intercalary One been thought to have 30 days, rather than the necessary 29 days? There are a couple of reasons. The first is that Intercalary Two does indeed have 30 days, and it is easy to assume the same for both intercalary months because of their significance on the calendar.

The second reason is that line 3 of the top fragment of Intercalary One starts with MATV (then the fragment breaks), which has been taken to be an extension of the M(AT) of other months, even though the formula MID MonthName MAT is usually placed on one line, whereas here the MATV starts a new line and there is no other data available to confirm it is the MAT referring to the month's status. Unfortunately, because all the days of an intercalary month are either copied or brought up from other months, it has no internal MD/D notations of its own to provide supporting evidence as to whether or not the month is being used as a MAT month. Even should this be true, as we see from EQUOS, the naming of a month as MAT or ANM doesn't necessarily dictate its length under some circumstances.

It would be quite odd if the first month of the calendar, Intercalary One, which is arguably the most important month of all in regards to its position, would be marked overall as ANM(at), not-good or inauspicious. But even should the month be MAT, it does not necessarily follow that the month will hold 30 days.

It may not be correct logic to assume that the MAT/ANM status of a month is the same thing as having 30/29 days, as it is probably influenced by a number of things, only one of them being the even/odd dichotomy of Celtic lore. The use of even numbers as good or auspicious and odd numbers as inauspicious, remains in Celtic languages to this day. (Roman perceptions used the reverse of this dichotomy, with even numbers being considered inauspicious.) This use of even numbers as auspicious on the Coligny calendar is a clear indication that this calendar should be set in an older Celtic cultural and historical context than its manufacture date when Lugdunum was a major Romano-Gaulish city.

There is another reason why scholars have until now assumed Intercalary One to have 30 days. On the lowest Intercalary One fragment, there is a break fracture running after the last X of the list of day numbers in Roman numerals. It is questionable whether there is the downstroke of a V just visible after the X, giving XV (day 15a, i.e. the 30th day). This may be so, but it is very difficult to assert with any degree of certainty, and I personally cannot make it out. Even if this turns out to be correct, the day XV may have just been marked with the usual DIVERTOMV which occurs on the 'virtual' 30th day of ANM months of 29 'real' days length.

Usually an XV is not marked on the 30th day with DIVERTOMV, but sometimes it is, for example ANAGANTIOS year 4 reads 'XV DIVERTOMV'. So, even should this turn out to be XV for day 30, this does not guarantee that Intercalary One has 30 'real' days.

Conversely, is there any reason to consider that Intercalary One instead has only 29 days -apart from the necessity of keeping the calendar in sync with the moon?

The answer is yes. The internal terms assigned to each intercalary day are copied from a particular day and month which are dictated by the day number of the intercalary month together with the list of months which are systematically run down over the Intercalary month. Should Intercalary One have a 30th day, then this day's terms would need to be copied from day 15a (30) of CANTLOS. But, CANTLOS is an ANM month with only 29 days -there is no day there to be copied. Therefore it shouldn't be possible for Intercalary One to have a day 30.

Duval and Pinault (RIG, III, p. 389) proposed a solution which created terms in accord with the known patterns if extended into a 30th day. They did this on the basis that when the IVOS run of EDRINI, which passes over the end of a 29-day month is moved upwards after an intercalary month and then passes over the end of a 30-day month, it adds an IVOS on the 30th day as though this day was part of a pattern that includes a virtual day. This also happens when a TII counter triplet runs up against the end of a month, and again acts as though the 30th day is virtually present. It would then be possible for the terms of a day 30 to be reconstituted if Intercalary One had 30 days, but does this mean that Intercalary One had 30 days? In the case of the EDRINI IVOS the day is not added to the month, it is already there. Again, nothing on this calendar is made up to fix some gap in a pattern, nor is anything random, so it seems unlikely that the calendar would break with the pattern of having days which reflect and copy manifested month/days to add an extra day which is not elsewhere manifested.

If however the calendar does recreate terms for a 30th day of Intercalary One, there is one way that the calendar can still work as a calendar that accurately tracks the lunations, which is to assume that all EQUOS months are carved with 30 days, and then the necessary 29-day instances are determined on the fly, in this case with EQUOS in year 1 having only 29 days. This situation will be further examined when I look at the case for EQUOS being always carved with 30 days.

Another reason why Intercalary One should only hold 29 days is that if it holds 30 days, then by the end of the first SAMONIOS half of year 1, there would have been 5 months of 30 days and only two months of 29 days. At which point the calendar month would be already ahead of the lunations by the amount of nearly one and a half days (Fig. 5). And this misalignment would continue to pass down through the rest of the 5 year cycle. In other words, if Intercalary One has 30 days, it would throw the calendar's months significantly out of sync with the moon from the very start of the 5 year cycle. For the moment then, I will continue to now assume an Intercalary One value of 29 days.

Intercalary one and the metonic cycle

If the Coligny calendar is part of a larger cycle of years, then the Metonic cycle is a likely candidate. The Metonic cycle is 19 solar years long or 235 synodic lunar months, a total of 6 940 days. If we take 3 cycles of the Coligny calendar, 15 years, this contains 5 493 days, leaving 1 447 days for the remaining 4 years, and a year of 13 lunar months to drop. This is exactly the first year of the calendar that we now have -an intercalary of 29 days plus 12 lunar months of 355 days.

The amazing thing about this is that the Coligny plaque can be used for all years of a Metonic cycle, cycling over it four times, without ever replacing it with different bronze plaques. One way this can be arranged is shown in Fig. 6, by dropping the entire first year, including its 29-day intercalary, and then continuing to cycle through the next 19 years. Each box in Fig. 6 represents the 5-year Coligny plaque. If the first year of 12 lunar months and 29-day intercalary is dropped (highlighted in dark grey), then the 4 cycles of the 5-year calendar create a perfect example of a Metonic calendar. To summarise the case for Intercalary One being 29 days, rather than 30:

1. This brings the 5 year cycle into sync with the 1 831 days required by 62 lunations, and the calendar keeps in perfect time with the moon over a long period. 2. If the extra day was removed instead from EQUOS year 4, that would cause year 4 to have lunations significantly too short. 3. We have internal evidence that Intercalary One holds 29 days, as there is no day 30 in CANTLOS from which the intercalary month can borrow to form its day's notation at day 30. 4. No other internal evidence definitively proves that Intercalary One has 30 days. 5. A month of 29 days is needed to balance the run of months in the first half of year 1 which would otherwise hold 5 of the first 7 months as 30 day MAT months. A length of 30 days for the intercalary would push the calendar months out of sync immediately with respect to lunations. 6. An intercalary of 29 days is needed if the calendar is part of a Metonic cycle calendar. If Intercalary One has 29 days, then by ignoring the first year in the first cycle to give 4 years, followed by three 5-year cycles, we have a perfect Metonic cycle calendar, and a bronze plaque which displays all 19 years and never needs replacing by a different version. In other words, the Coligny calendar provides us with the complete calendar of the Gauls, not just a 5-year section as has been previously thought.

Early researchers, including Rhys, Fotheringham, and Orpen in the 1910s in particular made valiant attempts to lay out the calendar as a Metonic cycle 10 . This they did mainly because they had a strong sense of the precise beauty and intricacy of the calendar patterns, both internal and external. Mac Neill picked up this thread in 1926, using evidence that the calendar was primarily lunar and must track the moon accurately 11 . But the only way he could find to make the calendar Metonic was to assign 28 days to both EQUOS months with ends on lacunae in years 2 and 4, which appeared bizarre and unnecessary as the calendar could easily adjust itself with a series of 29and 30-day months in EQUOS if it needed to. For example, the sequence could have run 30/29/29/29/29. The use of a 28-day month in year 4 also creates a whole year around it where the calendar is running unacceptably fast compared to the lunations, gaining in that year nearly two days. The argument for the calendar being Metonic has not been seriously proposed since his 1926 paper, when it was finally realised that to create a Metonic calendar, one could cycle over the 5-year plaque three times, but in the four remaining years, it would require an extra day being somehow added to the calendar, and that this odd situation would require a separate plaque to be made. 10. olmsteD, 1992, p. 36-41. 11. olmsteD, 1992, yet it now seems that these researchers of the early decades of the calendar studies were indeed correct to expect a Metonic calendar -they were simply labouring with one piece of incorrect data, the assumption that Intercalary One held 30 days. Once this is corrected to 29 days, all previous concerns dissolve, we do not need any running adjustments to the calendar, and the one plaque can be used as it is for the entire Metonic 19 year cycle.

Syncing individual calendar months with lunations

So now we have a 5 year lunar calendar that has the correct number of days to keep it in alignment with 62 synodic moons. But, this is not good enough just by itself. If we are to use the internal notations on the calendar as indicative of a certain phase or day of the moon, then we must also keep each individual month as closely aligned to the synodic month as possible. That is, the lengths of months in the 5 year cycle must show a pattern that is quite rigidly controlled, so that each calendar month starts as closely as possible to the same point of a lunation.

The calendar is constrained to count in whole days, so the most it can be exact is within a day's difference either side of any point in the lunar phase. Within the Coligny calendar, we do have a very good indication of what the tolerance of this syncing might be. If we accept that day 1 of a month is the quarter moon, then the triplet 7-8-9 are the days around the full moon, while the triplet 7a-8a-9a are the three days of the dark moon, when the moon is lost in the glare of the sun. One indication of the importance of these triplets is that they are given special treatment by being moved to the previous month in the 12 months after an intercalary. The triplet of 7-8-9 gives us a wiggle room of approximately 1.5 days either side of the actual point of the full moon (midway through day 8) in which it is acceptable for the full moon to appear. Or expressed another way, 1 day either side of day 8. If the full moon appeared outside this range then the triplet 7-8-9 would no longer refer to the visible full moon phase -and presumably the activities normally associated with this triplet would no longer be appropriate for the full moon. Similarly, the dark moon triplet allows the same tolerance between the last appearance of the old moon and the first appearance of the new moon.

There is another crucial reason why the calendar would focus on the full moon and the opposite dark moon. It is these points where an eclipse of the sun or moon will occur. But if an eclipse was to occur outside the full moon or dark moon triplet of days, this would make the calendar symbolically unreliable.

Choosing the best fitting EQUoS values for years 2 and 4

The following table (Fig. 7) shows the way the calendar months track with the synodic months. EQUOS in years 2 and 4 are 28 and 29, this time using an Intercalary One of 29 days. The values are the difference between the accumulated calendar days at the end of each month, and the accumulated days of actual lunations. A negative value means that the calendar month is finishing ahead of its equivalent lunation. By comparison, if we were to keep the correct number of days for 62 lunar months, but swap the values for EQUOS in years 2 and 4, this gives the following chart of sorted differences (Fig. 9), which in comparison to Fig. 8 can be seen to push out past the tolerance of +1.5 and would see the full moon date of day 8 over two days later when the moon is visibly no longer full. In summary, we see that with Intercalary One of 29 days length, and with EQUOS of 30/28/30/29/30 that the Coligny calendar presents a pattern of calendar months which closely follows the 62 synodic lunar months of the 5 year cycle, maintaining itself within a day (a 1.5 day difference) either side of the exact lunar phase.

Checking the full moon tolerance

All the above tables are calculated using the end of a full calendar month. But there is one more thing to check before we can be sure that the above solution is correct, we must check whether the triplet 7-8-9 always contains the full moon.

Because we can't tell at what precise point in time the calendar is aligned exactly to the moon to begin its first year with, I have allocated the very first day 8 of Intercalary One as zero. By doing this, the difference values show very little difference from that of the pattern outlined above for the entire month. Difference values are between 0.91 and -1.41, indicating that the full moon will always occur within the triplet of days 7-8-9, and the dark moon within 7a-8a-9a.

Variable lunations

The above figures were calculated using the value of an average lunation of 29.530589 days. Actual lunations vary from this average value in a complex pattern which is dictated by the eccentric orbits of the moon and earth. Although most lunations fall within a few hours of the average value, an extreme lunation can vary by up to nearly 15 hours either side of the average. That is, 0.6 of a day. (Charts and discussions of the patterns of lunations can be viewed online at Bromberg 12 .)

This means that every so often, in cycles of about 111 months, the length of a lunation might be pushed out by up to nearly 15 hours either side of the average value. If we return to the chart of monthly differences (Fig. 8), we can see that the full moon would still occur easily within the 1.5 day tolerance on the positive side of the lunation, but if an extreme lunation of -15 hours occurred with one of the 4 most negative values, then the full moon would actually occur just over 2 days previously, which is nearly half a day outside our tolerance limits. This would be a very rare occurrence, but it could still happen. And it would not be a good thing if an eclipse happened during this period when the moon is moving between extreme lunations, and it happened to fall outside the eclipse triplets of full moon 7-8-9 and dark moon 7a-8a-9a.

These negative extremes occur when EQUOS is given 28 days in year 2, and this is another indication that assigning an exceptional value of 28 days to EQUOS is wrong. But it should be relatively easy to make a small adjustment to the EQUOS pattern to avoid these large negatives, by giving EQUOS 29 days in year 2 and removing a day from an adjoining year, either year 1 or year 3. The problem is that the calendar shows us that both years 1 and 3 have 30 days each. So is there a way to overcome this?

Variable EQUoS

What if the bronze plaque was created with EQUOS of 30 days carved across all years? This would then allow the astronomers to mark certain instances of EQUOS with the 30th day to ignore, simply by putting a special peg in its hole. And this pattern of when EQUOS was either 29 or 30 could then be adjusted according to how the actual lunations of that year are moving. For example, if a lunation is predicted to be in the extreme negative end, then allowing that year's EQUOS to be 29 days long would make for a very simple and flexible system of adjustment.

So, even though we can see that years 1, 3, and 5 on the calendar are carved with 30 days, it may not necessarily be the case that these months were normally considered to always contain 30 days, but could have been reduced to 29 days as required.

12. bromberg, 2011. And it also takes care of something else which has never felt right -that if EQUOS really did have 30 days in years 1, 3 and 5, then this requires EQUOS year 2 to only have 28 days. Which in a calendar that otherwise adjusts itself through a variation of 29 and 30 days, seems really very odd and uncomfortable indeed. But with a variable EQUOS the astronomers are free to allocate two 30 day months and three 29 day months to EQUOS in whatever pattern is appropriate across the five years.

If the Gaulish astronomers were creating a new calendar every 5 years, and making a new and expensive bronze plaque, this potential adjustment of EQUOS might be taken care of for each new cycle when deemed necessary. But there is a problem with that. The textual data of the Coligny calendar shows distinct signs of being copied from a much earlier calendar. Mistakes such as TRINO for PRINO show that the carvers no longer fully understood the meaning of these terms. There are also parallax mistakes where the correct notation has been miscopied onto the wrong line, and so on in a similar manner to any other copied manuscript, again suggesting that the Coligny calendar is a copy of a far older one.

But now it seems possible for the Gauls traditionally to have had just the one calendar, but with the flexibility built into it to readjust the values of EQUOS as required, by the simple device of allowing EQUOS to be carved with 30 days on all years, and a special peg put on EQUOS day 30 when it wasn't needed. Or, in other words, for EQUOS to be declared as 29 or 30, depending upon whatever adjustment was necessary.

This means that each 5-year cycle of the calendar will have as a norm two 30-day EQUOS and three 29-day EQUOS, unless a special adjustment is needed. This would eliminate the most negative differences in the graphs above -when EQUOS is given by us the unique and strange value of only 28 days.

I have looked at how best the different theoretical variations in the values of EQUOS would sit around an average lunation, and compared it to the values started with above, 30/28/30/29/30 (series 1). It turns out that there are only two likely candidate patterns that are better and that track the moon closely enough to work with the average lunation value (Fig. 10). Those values are 30/29/29/29/30 (series 2) and 29/30/29/29/30 (series 3). Series 2 has the values that track closest to the lunar average, and has the advantage of still holding 30 days in year 1, which is the value required for the year which needs to be dropped in a Metonic calendar. To understand the significance of the accuracy of the Coligny calendar, this accuracy of the calendar month lying within a day either side of the actual lunation is similar to the stunning accuracy claimed for the Mayan calendar.

EQUoS pattern From To average Total range

But in assuming that the plaque is carved with 30 days in EQUOS in all years, a wonderful flexibility is built into the calendar by which it can be finely adjusted according to current lunation patterns.

And it means that the bronze plaque could well be the only physical calendar that the Gauls ever needed, as it can be adjusted on the fly, simply by inserting a peg. It also implies that with only a 5-year calendar as base, it might now be just possible for a highly-trained person to memorise the calendar by heart, or to be able to use the patterns to constitute a day's notations on the fly. It would be a mental feat no doubt, but it could be done. So I end with another possible version of the shape of the calendar (Fig. 12), incorporating series 2. After all this, there is only one difference with the shape that I started with -EQUOS years 2 and 3 are now both 29 days long, rather than 28 and 30 respectively.

year 1

year 2

year 3

year 4 year 5 And I will now repeat here (Fig. 13), with EQUOS as 30/29/29/29/30, and Intercalary One as 29, the full Metonic cycle as laid out in the Coligny calendar by using 4 instances of the same 5-year plaque end to end, but skipping the very first year. There is one other idea that may be of relevance to the allocation of 29 or 30 days to an EQUOS month. A calendar is not the method by which any astronomer calculates precise time periods. Those calculations and observations are done outside the calendar in their own right. A calendar is merely a mechanism which helps to organize people's activities around the pre-calculated lunar and seasonal points. For instance, the date of Easter today is pre-calculated and then the date simply marked on a calendar, the calendar itself does not calculate Easter. In which case, it may be that the Coligny calendar's primary aim was to keep the lunar months as close as possible to a point which can be easily seen by the general population. To do this, there might be a rule applied each EQUOS, which, for example, might say that EQUOS ends and ELEMBIU starts with the first visible quarter moon after a known solar point, for example, the spring equinox. This wouldn't mean the astronomers didn't or couldn't calculate time periods precisely, as the general shape of the calendar clearly indicates they could. It may come down to what their culture considered was the primary aim of the calendar, a precise mathematical clock, or a calendar to regulate activities with the heavens and seasons.

Here I will return for a moment to the issue of the length of Intercalary One. While the pattern of whether EQUOS has 29 or 30 days may be flexible, it stills means that Intercalary One is best with 29 days, so that the first year of the calendar remains in close sync with the actual lunations. As was noted above in the discussion of the length of Intercalary One, the internal evidence strongly suggests that it has 29 days, although there still remains a small possibility that it holds 30 days. But if it holds 30 days, then the values of EQUOS in years 1 3 and 5 cannot be locked in at 30, as this will always cause the calendar to run under by a day for either a 19-or 30-year cycle. With an Intercalary One of 30 days, there is only one way to make the calendar track the lunations and perform accurately over the longer term, which is to have every EQUOS initially given 30 days and then to have some mechanism by which four out of every five EQUOS months are reduced to 29 days in three of the cycles and none in the fifth. Above all, EQUOS in year 1 must always have only 29 days to keep the calendar in sync with the lunations and to allow the calendar to be used to track either a 19-or 30-year cycle. The EQUOS values would usually run as 29/29/29/29/30 to keep the lunations in close sync, with an exception of an extra day in the final cycle. This messy solution is just possible, but it seems unlikely as it does require extra demands on the internal logic of this highly structured calendar.

The Saros eclipse cycle

The Saros cycle of 223 synodic months allows the prediction of a series of eclipses of the sun and moon. Eclipses are one of nature's most awesome events and most cultures of the world have invested them with mythology and fate, often inauspicious. But it is a feat of mathematics or observation to find patterns within the seemingly random nature of these events. If we take the first 18 years of the 19-year Metonic cycle on the calendar, that is ignoring year 1 in the first cycle over the plaque, then we have one exact Saros cycle. In real solar years, this is an odd number, 18 years and 11.5 days, but in months it is a round number of 223 synodic months, and so it is possible to lay this out on the Coligny lunar calendar. However, the fact that the first 18 calendar years amount to this exact number of lunar months required of the Saros cycle, well that can only be described as extraordinary.

The Venus cycle

The planet Venus traces a five-petal rose shape in the sky every 8 earth years. This path is also often represented by the five-pointed pentagram symbol. Each of the 5 Venus cycles takes 583.92 days. If we place the full 5-point Venus cycle over the first 8 calendar years of the Metonic cycle on the calendar, again ignoring year 1 in the first cycle as per the Metonic cycle, then the Venus cycle is marked out with the closest precision possible using lunar months as a base, the full cycle finishing merely 2.4 days before the 8 calendar years. Was this precision enough? There is nothing exceptional marked on the calendar 2 or 3 days before the end of the last month, CANTLOS. However, if we were to assume that these days were leapt over at the beginning of a Venus cycle, to begin the 8 year Venus cycle on the third day of the first month, then we find ourselves on a very significant day, SAMONIOS 3, the day which is involved in a unique three-way swap and gives rise to the notation TRINVX SAMONI. Again, this pattern for Venus is quite extraordinary and displays the mathematical beauty of a precision calendar which can include all these patterns, lunar, solar, Metonic, Saros, Venus -at the same time.

Comparing the metonic cycle with the thirty-year cycle

Pliny the Elder 13 related that the Gauls considered thirty years an age, which has led to the notion first put forward in 1898 by De Ricci 14 and largely accepted since then, that the 5-year plaque may be part of a 30-year long cycle. However, the debate about what is the long cycle of the Coligny calendar has been going on for the last century, main candidates being 30, 25 or 19 years.

Adjusting Intercalary One to 29 days has resulted in a calendar which follows a Metonic cycle of 19 years. But before accepting this as the long cycle of the Gaulish calendar, we need to revisit the idea of a 30-year cycle. It turns out that we can create a 30-year calendar by now positioning 6 of the 5-year plaques end to end, while skipping the Intercalary Two month in the last cycle (Fig. 14).

Again, we have a situation where the 5-year bronze plaque of the Coligny calendar is all that is ever needed. And, fascinatingly, this 30-year cycle contains two Metonic cycles -running from years 2 to 20, and years 7 to 25.

13. Pliny the Elder, ed. and transl. bostocK and riley, 1855. 14. olmsteD, 1992, p. 31. Both these calendars, 19-or 30-year, are superb lunar time charts, but which version were the Gauls using? To try to answer this requires further examination of what happens to the 19-or 30-year periods as they continue to cycle over time.

The moon

Because everything within the calendar is focused on the point of the lunar cycle, I am presuming that the prime directive of the calendar is always to keep in sync with the moon's phases. It is also worth noting that so far nothing at all on the calendar has been discovered that is purely a solar term, everything is at best luni-solar.

However, because the calendar is constrained to count in units of a day, it will always have difficulty tracking both the lunar and solar cycles as these are not exact multiples of a day. At the end of the 19-year Metonic cycle, the calendar has overrun the 62-month lunar point by 0.312 days, and this overrun will be continued into the next Metonic cycle. But this can be easily fixed, simply by removing a day roughly once every third Metonic cycle, or 61 years. And this can readily be done by reducing an EQUOS month from 30 back to 29 days.

Taking the 30-year calendar, we find that the lunar count is spectacularly accurate, in fact the most accurate point throughout the years, -0.15691 days. The 5-year plaque ends 0.0625 days ahead of the moon, which accumulates over the 6 x 5-year cycles, but which is then reduced by skipping a 30-day intercalary month (which brings both the moon and the sun back close to the point of synchronicity). The slight lunar difference at the end of each 30-year cycle requires a day to be added (by turning a 29-day EQUOS into a 30-day) roughly once every 191 years.

The sun

Because I am assuming the calendar must maintain itself with respect to the moon's phases, any drift against the solar year must wait until a full lunar month has accumulated before it can be adjusted.

At the end of the Metonic cycle, the sun finishes its 19-year course a mere 2 hours (0.086 days) before the last moon. Every 219 years this adds a day, for instance, SAMON day 1 would become SAMON day 2. But this drift of the solar year against the moon would be so slow that it would only require a lunar month to be skipped once every 6 536 years, a period which is no doubt longer than the calendar is old.

At the end of the 30-year cycle, the sun is still lacking 1.108781 days to finish its 30 solar years. This means that a lunar month will need to be added every 799 years to prevent a slippage against the solar year.

But note how a one day slippage per thirty 30-year cycles could be regarded as symbolically significant, even if not absolutely precise mathematically, and might indicate a further long term period of thirty 30-year periods, that is, 900 years, at which point this one day slippage could be adjusted.

The yearly swing

The individual lunar years of the Metonic calendar swing against the solar years within a range of 36 days in each cycle (-21.9 to +14). This swing would feel roughly similar to what we experience with our Easter. If we were to wait the full 6 536 years before an extra lunar month can be deleted, this swing would range over 51 days. But, if we were to propose that the calendar is only say 1 000 years old, its full range would so far have only added four and a half days, giving a total of about 40 days.

By comparison the 30-year calendar swings by 46 days (-45.43 to +1.27) in each cycle. Before it can be readjusted by the addition of a lunar month, it will swing through a total of 75 days. This is nearly twice the value of the Metonic swing, although it will take 799 years for this to happen.

This large solar swing of the 30-year calendar is perhaps the only significant disadvantage it has over the simple Metonic cycle.

If for example we were to peg the highest possible date for Beltaine at the summer solstice, then the Metonic swing would find Beltaine being celebrated between May 1 to June 21, a date suspiciously significant! But a 30-year calendar would find Beltaine between April 7 to June 21, being all but a fortnight shy of the quarter between the spring equinox and the summer solstice. While not unacceptable, this does suggest that we look closer at the possibility of the Coligny calendar being primarily a Metonic calendar. This would not necessarily mean that the Gauls didn't consider 30 years an age, only that the calendar itself would only cycle through the 19-year Metonic cycle. Ultimately, the question of a 19-or 30-year calendar remains open.

The next question

In summary, the Coligny calendar is a superb piece of time-keeping, tracking the synodic months within a tolerance of a day either side of the average lunation. With its external precision combined with layers upon layers of intricate internal patterns, it stands as a stunning accomplishment to this day.

In the end though, I haven't found a definitive answer to whether the calendar's long term cycle is 19 or 30 years. The Metonic cycle is slightly more mathematically precise in some ways, but not enough to rule out the 30-year cycle. The 30-year calendar has the beauty of including two Metonic cycles within it, and for this reason alone we might consider the 30-year cycle as a step beyond the simple Metonic cycle.

Where I seem to have ended up is the same place I started, with the question of how this calendar fits within the druidic philosophy and its cultural context. It seems that this study of culture, history and mythology must ultimately answer whether we have a 19-or 30-year calendar. We are told by Plutarch 15 that the Celts revered a deity manifesting as Saturn with its 30-year orbital period. Candidates served Cronus, called by the Celts the Nightwatchman, on the island of Ogygia for 30 years, where they learnt astronomy, geometry and natural philosophy, and became oracular, reporting prophesies as 'dreams of Cronus'. Is this a trace of why a cycle of 30 years is significant in this story of the calendar? We need to explore further.

yet there is definite beauty in the symmetry of calendar months with 30 days (the 29-day month having a 'virtual' 30th day), the period between intercalary months being 30 lunar months, an age of 30 years, perhaps even a symbolic 900-year cycle beyond that. Perhaps mathematics was not enough on its own, perhaps the calendar was also designed to mirror the wonder of the universe.