Metonic cycle

The Metonic cycle or enneadecaeteris (from Ancient Greek: ἐννεακαιδεκαετηρίς, from ἐννεακαίδεκα, "nineteen") is a period of almost exactly 19 years after which the lunar phases recur at the same time of the year. The recurrence is not perfect, and by precise observation the Metonic cycle defined as 235 synodic months is just 2 hours, 4 minutes and 58 seconds longer than 19 tropical years. Meton of Athens, in the 5th century BC, judged the cycle to be a whole number of days, 6,940. Using these whole numbers facilitates the construction of a lunisolar calendar.

Depiction of the 19 years of the Metonic cycle as a wheel, with the Julian date of the Easter New Moon, from a 9th-century computistic manuscript made in St. Emmeram's Abbey (Clm 14456, fol. 71r)
For example, by the 19-year Metonic cycle, the full moon repeats on or near Christmas day between 1711 and 2300.[1][2] A small horizontal libration is visible comparing their appearances. A red color shows full moons that are also lunar eclipses.

A tropical year is longer than 12 lunar months and shorter than 13 of them. In a Metonic calendar, 7 months are added over a cycle of 19 years to make up the necessary 235 (19×12+7 = 235).

Application in traditional calendars

In the Babylonian and Hebrew lunisolar calendars, the years 3, 6, 8, 11, 14, 17, and 19 are the long (13-month) years of the Metonic cycle. This cycle forms the basis of the Greek and Hebrew calendars. A 19-year cycle is used for the computation of the date of Easter each year.

The Babylonians applied the 19-year cycle from the late sixth century BC.[3]

According to Livy, the second king of Rome, Numa Pompilius (reigned 715–673 BC), inserted intercalary months in such a way that "in the twentieth year the days should fall in with the same position of the sun from which they had started".[4] As "the twentieth year" takes place nineteen years after "the first year", this seems to indicate that the Metonic cycle was applied to Numa's calendar.

Diodorus Siculus reports that Apollo is said to have visited the Hyperboreans once every 19 years.[5]

The Metonic cycle has been implemented in the Antikythera mechanism which offers unexpected evidence for the popularity of the calendar based on it.[6]

The (19-year) Metonic cycle is a lunisolar cycle, as is the (76-year) Callippic cycle.[7] An important example of an application of the Metonic cycle in the Julian calendar is the 19-year lunar cycle insofar as provided with a Metonic structure.[8] In the following century, Callippus developed the Callippic cycle of four 19-year periods for a 76-year cycle with a mean year of exactly 365.25 days.

Around AD 260 the Alexandrian computist Anatolius, who became bishop of Laodicea in AD 268, was the first to devise a method for determining the date of Easter Sunday.[9] However, it was some later, somewhat different, version of the Metonic 19-year lunar cycle which, as the basic structure of Dionysius Exiguus' and also of Bede's Easter table, would ultimately prevail throughout Christendom,[10] at least until in the year 1582, when the Gregorian calendar was introduced.

The Coligny calendar is a Celtic lunisolar calendar using the Metonic cycle. The bronze plaque on which it was found dates from c. AD 200, but the internal evidence points to the calendar itself being several centuries older, created in the Iron Age.

The Runic calendar is a perpetual calendar based on the 19-year-long Metonic cycle. It is also known as a Rune staff or Runic Almanac. This calendar does not rely on knowledge of the duration of the tropical year or of the occurrence of leap years. It is set at the beginning of each year by observing the first full moon after the winter solstice. The oldest one known, and the only one from the Middle Ages, is the Nyköping staff, which is believed to date from the 13th century.

The Bahá'í calendar, established during the middle of the 19th century, is also based on cycles of 19 solar years.

Hebrew calendar

A Small Maḥzor (Hebrew מחזור, pronounced [maχˈzor], meaning "cycle") is a 19-year cycle in the lunisolar calendar system used by the Jewish people. It is similar to, but slightly different in usage from, the Greek Metonic cycle (being based on a month of 29+13753⁄25920 days, giving a cycle of 6939+3575⁄5184 ≈ 6939.69 days[11]), and likely derived from or alongside the much earlier Babylonian calendar.[12]

Three ancient civilizations (Babylonia, China and Israel) used lunisolar calendars and knew of the rule of the intercalation from as early as 2000 BC. Whether or not the correlation indicates cause-and-effect relationship is an open question.[13][14]

Polynesia

It is possible that the Polynesian kilo-hoku (astronomers) discovered the Metonic cycle in the same way Meton had, by trying to make the month fit the year.[15]

Tidal Epoch

Sea level calculations also depend on the Metonic cycle.

https://tidesandcurrents.noaa.gov/publications/Understanding_Sea_Level_Change.pdf

Mathematical basis

The Metonic cycle is the most accurate cycle of time less than 100 years for synchronizing the tropical year and the lunar month, when the method of synchronizing is the intercalation of a thirteenth lunar month in a calendar year from time to time.[16]

Tropical year = 365.2422 days.[17]
365.2422 × 19 = 6,939.602 days (every 19 years)
Synodic month = 29.53059 days.[18]
29.53059 × 235 = 6,939.689 days (every 235 months)
19 years of 12 synodic months =
228 synodic months per cycle, 7 months short of the 235 months needed to achieve synchronization.

The traditional lunar year of 12 synodic months is about 354 days, approximately 11 days short of the solar year. Thus, every 2–3 years there is an accumulated discrepancy of approximately a full synodic month. In order to 'catch up' to this discrepancy, to maintain seasonal consistency, and to prevent dramatic shifts over time, seven intercalary months are added (one at a time), at intervals of every 2–3 years during the course of 19 solar years.

The difference between 19 solar years and 235 synodic months is only about two hours, or 0.087 days.

See also

Notes
  1. "Rare Full Moon on Christmas Day". NASA. 17 December 2015.
  2. Skilling, Tom (20 December 2015). "Ask Tom: How unusual is a full moon on Christmas Day?". Chicago Tribune.
  3. "The Babylonian Calendar". Mathematical Institute. Utrecht University. July 2021.
  4. Livy, Ab Urbe Condita, I, XIX, 6.
  5. Diodorus Siculus, Bibl. Hist. II.47.
  6. Freeth, Tony; Jones, Alexander; Steele, John M.; Bitsakis, Yanis (31 July 2008). "Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism" (PDF). Nature. 454 (7204): 614–7. Bibcode:2008Natur.454..614F. doi:10.1038/nature07130. PMID 18668103. S2CID 4400693. Retrieved 20 May 2014.
  7. Nothaft 2012, p. 168.
  8. McCarthy & Breen 2003, p. 17.
  9. Declercq 2000, pp. 65–66.
  10. Declercq 2000, p. 66.
  11. The month is 29.5 days and 793 "parts", where a part is 1/18 of a minute. Tøndering, Trine; Tøndering, Claus. "Calendar FAQ: the Hebrew calendar: New moon". This comes to 29+13753⁄25920 days. Multiplying this by 235 gives the length of the cycle.
  12. "Jewish religious year | Cycle, Holidays, & Facts | Britannica". www.britannica.com. Retrieved 14 November 2021.
  13. Watkins 1954.
  14. Hannah 2005.
  15. Johnson 2001, p. 238.
  16. Richards 1998, pp. 94–96.
  17. glossary 2022, s.v. year, tropical.
  18. Richards 2013, p. 587.

References
  • Declercq, Georges (2000). Anno Domini: The Origins of the Christian Era. Turnhout. ISBN 9782503510507.
  • "Glossary". The Astronomical Almanac Online!. Washington, DC: United States Naval Observatory. 2022. Archived from the original on 21 October 2020. Retrieved 30 March 2022.
  • Hannah, Robert (2005). Greek & Roman Calendars: Construction of Time in the Classical World. London: Duckworth.
  • Johnson, Rubellite Kawena (2001). Essays in Hawaiian Literature Part 1 Origin Myths and Migration traditions. author.
  • McCarthy, Daniel P.; Breen, Aidan (2003). The ante-Nicene Christian Pasch | De ratione paschali: The Paschal tract of Anatolius, bishop of Laodicea. Dublin: Four Courts Press. ISBN 9781851826971. OCLC 367715096.
  • Nothaft, C Philipp E. (2012). Dating the Passion: The Life of Jesus and the Emergence of Scientific Chronology (200–1600. Leiden: BRILL. ISBN 9789004212190.
  • Richards, E. G. (1998). Mapping Time: The Calendar and its History. Oxford University Press. ISBN 978-0192862051.
  • Richards, E. G. (2013). "Calendars". In Urban, Sean E.; Seidelmann, P. Kenneth (eds.). Explanatory Supplement to the Astronomical Almanac (3rd ed.). Mill Valley, CA: University Science Books. ISBN 978-1-891389-85-6.
  • Watkins, Harold (1954). Time Counts: The Story of the Calendars. New York: Philosophical Library.
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