For astronomy and calendar studies, the Metonic cycle or Enneadecaeteris (from Ancient Greek: εννεαδεκαετηρις, "nineteen years") is a period of very close to 19 years which is remarkable for being nearly a common multiple of the solar year and the synodic (lunar) month. The Greek astronomer Meton of Athens (fifth century BCE) observed that a period of 19 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 years and 235 synodic months) is only a few hours, depending on the definition of the year.
Considering a year to be 1⁄19 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 (235 = 19 × 12 + 7). Meton introduced the cycle in circa 432 BCE, but it was actually known earlier by Babylonian astronomers.
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), the medieval computus (i.e. the calculation of the date of Easter) and still regulates the 19-year cycle of intercalary months of the Hebrew calendar.
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 our Gregorian calendar, are based on the tropical year and maintain the seasons at the same calendar times each year. 19 tropical years are briefer than 235 synodic months by about 2 hours. The Metonic cycle's error is then one full day every 219 years, or 12.4 parts per million.
- 19 tropical years = 6939.602 days (12 × 354 day years + 7 × 384 day years + 3.6 days).
- 235 synodic months (lunar phases) = 6939.688 days (Metonic period by definition).
- 254 sidereal months (lunar orbits) = 6939.702 days (19 + 235 = 254).
- 255 draconic months (lunar nodes) = 6939.1161 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 1⁄5 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 do not 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 seem 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 seem to have introduced this scheme well before Meton, about 500 BCE.
Application in traditional calendars 
Traditionally, for 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 can be used to predict eclipses, forms the basis of the Greek and Hebrew calendars, and is used for the computation of the date of Easter each year.
The Chaldean astronomer Kidinnu (4th century BCE) knew of the 19-year cycle, but the Babylonians applied it since the late sixth century BCE. They measured the moon's motion against the stars, so the 235:19 relation may originally have referred to sidereal years, instead of tropical years as it has been used for various calendars.
The Runic calendar is a perpetual calendar based on the 19-year-long Metonic cycle. Also known as a Rune staff or Runic Almanac, it appears to have been a medieval Swedish invention. 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, 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.
Further details 
The Metonic cycle is related to two less accurate subcycles:
- 8 years = 99 lunations (an Octaeteris) to within 1.5 days, i.e. an error of one day in 5 years; and
- 11 years = 136 lunations within 1.5 days, i.e. an error of one day in 7.3 years.
By combining appropriate numbers of 11-year and 19-year periods, it is possible to generate ever more accurate cycles. For example simple arithmetic shows that:
- 687 tropical years = 250921.39 days;
- 8497 lunations = 250921.41 days;
giving an error of only about half an hour in 687 years (2.5 seconds a year), although this is subject to secular variation in the length of the tropical year and the lunation.
Meton of Athens approximated the cycle to a whole number (6940) of days, obtained by 125 long months of 30 days and 110 short months of 29 days. During the next 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.
See also 
- Attic calendar
- Bahá'í calendar
- Byzantine calendar
- Chinese calendar
- Hebrew calendar
- Saros cycle
- Runic calendar
- Julian day
- Mathematical Astronomy Morsels, Jean Meeus, Willmann-Bell, Inc., 1997 (Chapter 9, p. 51, Table 9.A Some eclipse Periodicities)