# Metonic cycle

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In astronomy and calendar studies, the Metonic cycle or Enneadecaeteris (from Greek words for nineteen years) is a period of very close to 19 years which is remarkable for being very nearly a common multiple of the tropical year and the synodic (lunar) month. The Greek astronomer Meton of Athens observed that a period of 19 tropical years is almost exactly equal to 235 synodic months, and rounded to full days counts 6940 days. The difference between the two periods (of 19 tropical years and 235 synodic months) is only 2 hours.

Taking a year to be 1/19th of this 6940-day cycle gives a year length of 365 + 1/4 + 1/76 days (the unrounded cycle is much more accurate), which is slightly 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. Meton introduced a formula for intercalation in circa 432 BC.

The cycle's most significant contemporary use is to help in flight planning (trajectory calculations and launch window analysis) for lunar spacecraft missions[citation needed] as well as serving as the basis for the Hebrew calendar's 19-year cycle. Another use is in computus, the calculation of the date of Easter.

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### Mathematical basis

19 tropical years differ from 235 synodic months by about 2 hours. The Metonic cycle's error is one full day every 219 years, or 12.4 parts per million.

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 a 1/5 of a Metonic cycle (47 synodic months, 3.8 years), and it recurs about 20 to 25 cycles.

This cycle appears to be a coincidence (although only a moderate one). The periods of the Moon's orbit around the Earth and the Earth's orbit around the Sun are believed to be independent, and have no 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 on average, 11 days short of the 365-day solar year. Therefore, in a lunisolar calendar, every 3 years or so there is a difference of more than a full lunar month between the lunar and solar years, and an extra (embolismic) month should be inserted (intercalation). The Athenians appear initially 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 appear to have introduced this scheme well before Meton, about 500 BCE.