Metonic cycle

The Metonic cycle or enneadecaeteris (from , from ἐννεακαίδεκα, "nineteen") is a period of approximately 19 years after which the phases of the moon recur at the same time of the year. The recurrence is not perfect, and by precise observation the Metonic cycle defined as 235 synodic lunar months is just 1 hour, 27 minutes and 33 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 luni-solar calendar.
A tropical year is longer than 12 lunar months and shorter than 13 of them. The arithmetical equation 12×12 + 7×13 = 235 allows it to be seen that a combination of 12 'short' years (12 months) and 7 'long' years (13 months) will be equal to 19 solar years.
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, and is used for the computation of the date of Easter each year.
The Babylonians applied the 19-year cycle since the late sixth century BC.
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." 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.
The Metonic cycle has been implemented in the Antikythera mechanism which offers unexpected evidence for the popularity of the calendar based on it.
The (19-year) Metonic cycle is a lunisolar cycle, as is the (76-year) Callippic cycle. 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. 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. 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, at least until in the year 1582, when the Gregorian calendar was introduced.
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 years.
In China, the traditional Chinese calendar used the Metonic cycle ever since the first known ancient China calendar. The cycle was continually used until the 5th century when it was replaced by a more accurate cycle.
Mathematical basis.
The importance of the tropical year for agriculture came to be realized much later than the adoption of lunar months for time keeping. However, it was recognized that the two cannot be easily coordinated over a short time span, so longer intervals were considered and the Metonic cycle was discovered as a rather good, but not perfect, schema. The currently accepted values are:
The difference is 0.086 days per cycle, which means that after around twelve cycles there will be a full day of delay between the month and the year. The error is actually one day every 219 years, or 12.4 parts per million. However, the Metonic cycle turned out to be very close to other periods:
Being close (to somewhat more than half a day) to 255 draconic months, the Metonic cycle is also an eclipse cycle, which lasts only for about 4 or 5 recurrences of eclipses. The Octon is of a Metonic cycle (47 synodic months, 3.8 years), and it recurs about 20 to 25 cycles.
This cycle seems to be a coincidence. The periods of the Moon's orbit around the Earth and the Earth's orbit around the Sun are independent, not to have any known physical resonance, and in fact the length of the month has been increasing over millions of years due to tidal acceleration. (An example of a non-coincidental cycle is the orbit of Mercury, with its 3:2 spin-orbit resonance.)
A lunar year of 12 synodic months is about 354 days, approximately 11 days short of the "365-day" solar year. Therefore, for a lunisolar calendar, every 2 to 3 years there is a difference of more than a full lunar month between the lunar and solar years, and an extra ("embolismic") month needs to be inserted (intercalation). The Athenians initially seem not to have had a regular means of intercalating a 13th month; instead, the question of when to add a month was decided by an official. Meton's discovery made it possible to propose a regular intercalation scheme. The Babylonians seem to have introduced this scheme around 500 BC, thus well before Meton.
Further details.
The Metonic cycle is related to two less accurate subcycles:
By combining appropriate numbers of 11-year and 19-year periods, it is possible to generate ever more accurate cycles. For example, combining 17 Metonic cycles with one 11-year cycle gives:
This gives an error of only about half an hour in 334 years, although this is subject to secular variation in the length of the tropical year and the lunation.
At the time of Meton, axial precession had not yet been discovered, and he could not distinguish between sidereal years (currently: 365.256363 days) and tropical years (currently: 365.242190 days). Most calendars, like the commonly used Gregorian calendar, are based on the tropical year and attempt to maintain the seasons at the same calendar times each year.