Post by dorienthomas on Oct 6, 2009 6:46:07 GMT -5
All calendars are an attempt at squaring the circle, since the year (365.242374 days) is not neatly divided by the month (29.530589). My analysis of the Coligny Calendar proceeds on the not unreasonable assumption that its creators' intention was to devise a system of calculating time as accurate as possible. My findings are that their end-product has an unexpected elegance, suggesting that the Ancient Celts were superb mathematicians.
Many descriptions of the Coligny Calendar are riddled with speculation presented as fact, so, to sweep all that away, here are the knowns: it is an inscribed and fragmentary Ancient Celtic bronze artefact covering a 5-year period. Its months alternate between 29 and 30 days in length. Each month has the word ATENOUX printed half-way through it. After each 30-month half-period there is an additional (or "intercalary") month inserted, of uncertain length, though one of them is apparently of 30 days, the other being too damaged to ascertain. (Some have postulated 29 days for the damaged intercalary, others 30. I hope to demonstrate that both are wrong.)
All arguments about what phase of the moon the Coligny month begins at, and there are several, are based on linguistic quibbles and have no bearing whatsoever upon the mathematics of the matter. For the sake of convenience I am taking a moon as a month and a month as a moon. Therefore ATENOUX in my analysis denotes the full moon, coming as it does in the middle of the Coligny month.
Several contemporary commentators have incidentally remarked upon the Celts' considering an "Age" to be of 30 years, though the relevance of this to the Coligny Calendar has never been explored. My findings make the 30-year Celtic Age central to an understanding of the calendar's inner workings.
Any English-Welsh dictionary will tell you that the Welsh word for 'fortnight' is 'pythefnos'. This is not at all accurate. 'Fortnight' is a contraction of 'fourteen nights', whereas 'pythefnos' is a contraction of 'pymtheg nos', which means 'fifteen nights'. It has even been suggested that this anomaly (which is also present in Irish and in French) is due to some primitive Celtic way of counting. Not so. The difference is explained by regarding the fortnight as predicate upon the seven-day week, whereas the pythefnos (cóicthiges, quinzaine) is clearly predicate upon the lunar month.
To our rural and, for the most part, illiterate ancestors the moon was the day-by-day calendar; the waxing and waning halves of the moon were their ready reckoner. Take, for illustration's sake, the period between the dark moon and the full. This comes to 14.765295 days. A fortnight after the dark moon thus falls 18 hours and 22 minutes (the best part of a day) short of the full moon, but the pythefnos only 5 hours and 38 minutes after it. If we think in terms of bi-monthly periods, it is clear that, for someone without a written calendar, the ability to count a cycle of 15 nights, 15 nights, 15 nights then 14 would prove a pretty good rule of thumb, its accuracy averaging out at 44 minutes per month - less than one and a half hours bi-monthly, whereas four fortnights are out by over three days. With this system it will be seen that we have arrived at the Coligny Calendar's basal alternation of 29- and 30-day months. The 15-night pythefnos, I aver, is at the root of the Celtic system of calculating time, and most decidedly not some primitive aberration. This is my first conclusion.
(The basic peasant requirement to count to fifteen may also explain the peculiar prominence of pymtheg in the traditional Welsh way of counting. After fifteen it proceeds 'one-on-fifteen', 'two-on-fifteen', - 'eighteen' is the curious 'two nines' - then 'four-on-fifteen' and after that 'twenty'.)
To reiterate, an alternation of 29-day and 30-day months averages out at 29.5 days to the month whereas the actual lunar month is 29.530589 days. The resultant 44 minute-per-month discrepancy would make ATENOUX arrive progressively earlier than the - easily visibly judgeable - full moon. It would take just under three years for this basal system to become a day out (which is considerably better than the fortnightly system's one-day discrepancy taking only three weeks, but was still not good enough for the Ancient Celts). However, if the two-and-a-half-yearly intercalary months are each a day longer than the others, i.e. 30 days and 31 instead of 29 and 30, this tiny discrepancy in the Celtic system will be even further reduced. A 5-year period will then consist of thirty 29-day months, thirty-one 30-day months and one 31-day month. This gives a mean of 29.532258 days, making the average Coligny month overlong by only 0.001669 of a day, which comes to 2 minutes and 24 seconds per month. With a 31-day intercalary it will take forty-eight-and-a-half years before ATENOUX is one day out of synchronisation with the full moon, and therefore visibly noticeable. My second conclusion is that the intercalary months alternate between 30 and 31 days.
The existence of a 31-day intercalary month also gives an average of precisely 366.2 days to the year, whereas the actual solar year is 365.242374 days. The Coligny year is consequently 22 hours and 59 minutes too long. This means the months will creep forward over the solar year by nearly a day per year. In order to keep the solstices and equinoxes - and therefore the seasons - within the months appropriate to them, one 30-day intercalary month needs to be dropped every 30 years. As we have seen, the Celts considered a 30-year period to be significant, calling it an "Age", which no-one has hitherto attempted to explain. A 30-year Age will then consist of one hundred and eighty 29-day months, one hundred and eighty-five 30-day months and six 31-day months. This gives an annual mean of precisely 365.2 days, making the average Coligny year short by only 0.042374 of a day, which comes to 1 hour and 1 minute per year, as well as further reducing the Coligny month's lunar discrepancy to 0.000408 of a day, which comes to 35 seconds per month. By my calculations this system would run for 198 years before requiring further adjustment. My third conclusion is that one 30-day intercalary month is dropped every 30-year Age.
To sum up:
Months alternate between 29 and 30 days in length.
After every 30th month an intercalary month is inserted.
Intercalary months alternate betweeen 30 and 31 days in length.
After every 30th year one 30-day intercalary month is omitted.
The resultant calendar does not only contrive to be true to both the moon and the sun, but succeeds in being both elegant and remarkably accurate.
(c) Dorien Thomas 2007-2011
REFERENCES
- Proceedings of the British Society, 4
- Zeitschrift für celtische Philologie, 23
- De Facie, Plutarch
- Geiriadur Prifysgol Cymru
- Foclóir Gaeilge-Béarla
- Oxford Hachette French Dictionary
- Encyclopaedia Britannica
Many descriptions of the Coligny Calendar are riddled with speculation presented as fact, so, to sweep all that away, here are the knowns: it is an inscribed and fragmentary Ancient Celtic bronze artefact covering a 5-year period. Its months alternate between 29 and 30 days in length. Each month has the word ATENOUX printed half-way through it. After each 30-month half-period there is an additional (or "intercalary") month inserted, of uncertain length, though one of them is apparently of 30 days, the other being too damaged to ascertain. (Some have postulated 29 days for the damaged intercalary, others 30. I hope to demonstrate that both are wrong.)
All arguments about what phase of the moon the Coligny month begins at, and there are several, are based on linguistic quibbles and have no bearing whatsoever upon the mathematics of the matter. For the sake of convenience I am taking a moon as a month and a month as a moon. Therefore ATENOUX in my analysis denotes the full moon, coming as it does in the middle of the Coligny month.
Several contemporary commentators have incidentally remarked upon the Celts' considering an "Age" to be of 30 years, though the relevance of this to the Coligny Calendar has never been explored. My findings make the 30-year Celtic Age central to an understanding of the calendar's inner workings.
Any English-Welsh dictionary will tell you that the Welsh word for 'fortnight' is 'pythefnos'. This is not at all accurate. 'Fortnight' is a contraction of 'fourteen nights', whereas 'pythefnos' is a contraction of 'pymtheg nos', which means 'fifteen nights'. It has even been suggested that this anomaly (which is also present in Irish and in French) is due to some primitive Celtic way of counting. Not so. The difference is explained by regarding the fortnight as predicate upon the seven-day week, whereas the pythefnos (cóicthiges, quinzaine) is clearly predicate upon the lunar month.
To our rural and, for the most part, illiterate ancestors the moon was the day-by-day calendar; the waxing and waning halves of the moon were their ready reckoner. Take, for illustration's sake, the period between the dark moon and the full. This comes to 14.765295 days. A fortnight after the dark moon thus falls 18 hours and 22 minutes (the best part of a day) short of the full moon, but the pythefnos only 5 hours and 38 minutes after it. If we think in terms of bi-monthly periods, it is clear that, for someone without a written calendar, the ability to count a cycle of 15 nights, 15 nights, 15 nights then 14 would prove a pretty good rule of thumb, its accuracy averaging out at 44 minutes per month - less than one and a half hours bi-monthly, whereas four fortnights are out by over three days. With this system it will be seen that we have arrived at the Coligny Calendar's basal alternation of 29- and 30-day months. The 15-night pythefnos, I aver, is at the root of the Celtic system of calculating time, and most decidedly not some primitive aberration. This is my first conclusion.
(The basic peasant requirement to count to fifteen may also explain the peculiar prominence of pymtheg in the traditional Welsh way of counting. After fifteen it proceeds 'one-on-fifteen', 'two-on-fifteen', - 'eighteen' is the curious 'two nines' - then 'four-on-fifteen' and after that 'twenty'.)
To reiterate, an alternation of 29-day and 30-day months averages out at 29.5 days to the month whereas the actual lunar month is 29.530589 days. The resultant 44 minute-per-month discrepancy would make ATENOUX arrive progressively earlier than the - easily visibly judgeable - full moon. It would take just under three years for this basal system to become a day out (which is considerably better than the fortnightly system's one-day discrepancy taking only three weeks, but was still not good enough for the Ancient Celts). However, if the two-and-a-half-yearly intercalary months are each a day longer than the others, i.e. 30 days and 31 instead of 29 and 30, this tiny discrepancy in the Celtic system will be even further reduced. A 5-year period will then consist of thirty 29-day months, thirty-one 30-day months and one 31-day month. This gives a mean of 29.532258 days, making the average Coligny month overlong by only 0.001669 of a day, which comes to 2 minutes and 24 seconds per month. With a 31-day intercalary it will take forty-eight-and-a-half years before ATENOUX is one day out of synchronisation with the full moon, and therefore visibly noticeable. My second conclusion is that the intercalary months alternate between 30 and 31 days.
The existence of a 31-day intercalary month also gives an average of precisely 366.2 days to the year, whereas the actual solar year is 365.242374 days. The Coligny year is consequently 22 hours and 59 minutes too long. This means the months will creep forward over the solar year by nearly a day per year. In order to keep the solstices and equinoxes - and therefore the seasons - within the months appropriate to them, one 30-day intercalary month needs to be dropped every 30 years. As we have seen, the Celts considered a 30-year period to be significant, calling it an "Age", which no-one has hitherto attempted to explain. A 30-year Age will then consist of one hundred and eighty 29-day months, one hundred and eighty-five 30-day months and six 31-day months. This gives an annual mean of precisely 365.2 days, making the average Coligny year short by only 0.042374 of a day, which comes to 1 hour and 1 minute per year, as well as further reducing the Coligny month's lunar discrepancy to 0.000408 of a day, which comes to 35 seconds per month. By my calculations this system would run for 198 years before requiring further adjustment. My third conclusion is that one 30-day intercalary month is dropped every 30-year Age.
To sum up:
Months alternate between 29 and 30 days in length.
After every 30th month an intercalary month is inserted.
Intercalary months alternate betweeen 30 and 31 days in length.
After every 30th year one 30-day intercalary month is omitted.
The resultant calendar does not only contrive to be true to both the moon and the sun, but succeeds in being both elegant and remarkably accurate.
(c) Dorien Thomas 2007-2011
REFERENCES
- Proceedings of the British Society, 4
- Zeitschrift für celtische Philologie, 23
- De Facie, Plutarch
- Geiriadur Prifysgol Cymru
- Foclóir Gaeilge-Béarla
- Oxford Hachette French Dictionary
- Encyclopaedia Britannica
