 Metonic cycle

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 solar year and the synodic (lunar) month. The Greek astronomer Meton of Athens 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.
Taking a year to be 1/19th of this 6940day 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 12month lunar year in pace with the solar year, an intercalary 13th month would have to be added on seven occasions during the nineteenyear period. Meton introduced the cycle in circa 432 BC but it was actually known earlier by Babylonian astronomers.
The cycle was used in the Babylonian calendar, the medieval computus (i.e. the calculation of the date of Easter) and still regulates the 19year cycle of intercalary months in the Hebrew calendar.
Contents
Mathematical basis
At the time of Meton, precession had not yet been discovered, and he could not make a distinction between sidereal years (currently: 365.256363 days) and tropical years (currently: 365.242190 days). Most calendars, like our Gregorian calendar, follow the seasons and are based on the tropical year. 19 tropical years are shorter 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
 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 19year 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 noncoincidental cycle is the orbit of Mercury, with its 3:2 spinorbit resonance.
A lunar year of 12 synodic months is about 354 days on average, 11 days short of the 365day 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 BC.
Application in traditional calendars
Traditionally (in the ancient Attic and Babylonian lunisolar calendars, as well as in the Hebrew calendar), the years 3, 6, 8, 11, 14, 17, and 19 are the long (13month) 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 in the computation of the date of Easter each year.
The Chaldean astronomer Kidinnu (4th century BC) knew of the 19year cycle, but the Babylonians may have learned of it earlier. 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 in various calendars.
The Runic calendar is a perpetual calendar based on the 19yearlong Metonic cycle. Also known as a Rune staff or Runic Almanac, it appears to have been a medieval Swedish invention. The calendar does not rely on knowledge of the length 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 in 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 11year and 19year 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. In the following century Callippus developed the Callippic cycle of four 19year periods for a 76year cycle with a mean year of exactly 365.25 days.
See also
 Antikythera mechanism
 Bahá'í calendar
 Byzantine calendar
 Chinese calendar
 Hebrew calendar
 Saros cycle
 Runic calendar
 Julian day
External links
References
 Mathematical Astronomy Morsels, Jean Meeus, WillmannBell, Inc., 1997 (Chapter 9, p. 51, Table 9.A Some eclipse Periodicities)
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Metonic cycle — Cycle Cy cle (s? k l), n. [F. ycle, LL. cyclus, fr. Gr. ky klos ring or circle, cycle; akin to Skr. cakra wheel, circle. See {Wheel}.] 1. An imaginary circle or orbit in the heavens; one of the celestial spheres. Milton. [1913 Webster] 2. An… … The Collaborative International Dictionary of English
Metonic cycle — Metonic Me*ton ic, a. [Cf. F. m[ e]tonique.] Pertaining to, or discovered by, Meton, the Athenian. [1913 Webster] {Metonic year} or {Metonic cycle}. (Astron.) See under {Cycle}. [1913 Webster] … The Collaborative International Dictionary of English
Metonic cycle — [mə tän′ik] n. [after Meton, Athenian astronomer (5th c. B.C. )] a period of about 19 years (almost 235 lunar revolutions), in which the phases of the moon repeat on the same dates as in the previous period: used for finding the date of Easter … English World dictionary
metonic cycle — noun , A particular approximate common multiple of the tropical year and the synodic month; in other words, the 19 year period over which the lunar phases occur on the same dates. According to the Metonic Cycle, a lunar calendar begins on the… … Wiktionary
Metonic cycle — /mi ton ik/, Astron. a cycle of 235 synodic months, very nearly equal to 19 years, after which the new moon occurs on the same day of the year as at the beginning of the cycle with perhaps a shift of one day, depending on the number of leap years … Universalium
Metonic cycle — /mətɒnɪk ˈsaɪkəl/ (say muhtonik suykuhl) noun a cycle of nineteen years, after which the new moon recurs on the same day of the year as at the beginning of the cycle. {named after the discoverer, Meton, 5th century BC Athenian astronomer. See ic} … Australian English dictionary
Metonic cycle — a unit of time equal to 19 years, used in astronomy in predicting the phases of the Moon. By coincidence, 19 years is equal to 6939.602 days and 235 lunar months is equal to 6939.689 days, just 125 minutes longer. As a result, the phases of… … Dictionary of units of measurement
Metonic cycle — [mɪ tɒnɪk] noun a period of 19 years (235 lunar months), after which the new and full moons return to the same day of the year. Origin named after Metōn, an Athenian astronomer of the 5th cent. BC … English new terms dictionary
metonic cycle — n. a period of 19 years (235 lunar months) covering all the changes of the moon s position relative to the sun and the earth. Etymology: Gk Meton, Athenian astronomer of the 5th c. BC … Useful english dictionary
Cycle — Cy cle (s? k l), n. [F. ycle, LL. cyclus, fr. Gr. ky klos ring or circle, cycle; akin to Skr. cakra wheel, circle. See {Wheel}.] 1. An imaginary circle or orbit in the heavens; one of the celestial spheres. Milton. [1913 Webster] 2. An interval… … The Collaborative International Dictionary of English