Metonic cycle

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.

For astronomy and calendar studies, the Metonic cycle or Enneadecaeteris (from Ancient Greek: ἐννεακαιδεκαετηρίς, "nineteen years") is a period of very close to 19 years that is nearly a common multiple of the solar year and the synodic (lunar) month. The Greek astronomer Meton of Athens (fifth century BC) observed that a period of 19 years is almost exactly equal to 235 synodic months and, rounded to full days, counts 6,940 days. The difference between the two periods (of 19 years and 235 synodic months) is only a few hours, depending on the definition of the year.

Considering a year to be ​119 of this 6,940-day cycle gives a year length of 365 + ​14 + ​176 days (the unrounded cycle is much more accurate), which is about 11 days more than 12 synodic months. To keep a 12-month lunar year in pace with the solar year, an intercalary 13th month would have to be added on seven occasions during the nineteen-year period (235 = 19 × 12 + 7). When Meton introduced the cycle around 432 BC, it was already known by Babylonian astronomers. A mechanical computation of the cycle is built into the Antikythera mechanism.

The cycle was used in the Babylonian calendar, ancient Chinese calendar systems (the 'Rule Cycle' 章) and the medieval computus (i.e., the calculation of the date of Easter). It regulates the 19-year cycle of intercalary months of the modern Hebrew calendar. The start of the Metonic cycle depends on which of these systems is being used; for Easter, the first year of the current Metonic cycle is 2014.

Mathematical basis

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 maintain the seasons at the same calendar times each year. Nineteen tropical years are about two hours shorter than 235 synodic months. The Metonic cycle's error is, therefore, one full day every 219 years, or 12.4 parts per million.

19 tropical years = 6,939.602 days (12 × 354-day years + 7 × 384-day years + 3.6 days).
235 synodic months (lunar phases) = 6,939.688 days (Metonic period by definition).
254 sidereal months (lunar orbits) = 6,939.702 days (19 + 235 = 254).
255 draconic months (lunar nodes) = 6,939.1161 days.

Note that the 19-year cycle is also close (to somewhat more than half a day) to 255 draconic months, so it is also an eclipse cycle, which lasts only for about 4 or 5 recurrences of eclipses. The Octon is ​15 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 believed to be independent, and not to have any known physical resonance. 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.

Other Languages
العربية: دورة ميتونية
беларуская: Метонаў цыкл
Deutsch: Meton-Zyklus
español: Ciclo metónico
français: Cycle métonique
한국어: 메톤 주기
հայերեն: Մետոնի ցիկլ
italiano: Ciclo metonico
Nederlands: Cyclus van Meton
日本語: メトン周期
norsk: Gyldentall
norsk nynorsk: Meton-syklus
polski: Cykl Metona
português: Ciclo metónico
русский: Метонов цикл
Simple English: Metonic cycle
slovenčina: Metónov cyklus
srpskohrvatski / српскохрватски: Metonski ciklus
svenska: Metons cykel
українська: Метонів цикл
Tiếng Việt: Chu kỳ Meton
中文: 默冬章