Metonic cycle

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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.

The Metonic cycle or enneadecaeteris (from Ancient Greek: ἐννεακαιδεκαετηρίς, from ἐννεακαίδεκα, "nineteen") is a period of almost exactly 19 years after which the lunar phases recur at the same time of the year. The recurrence is not perfect, and by precise observation the Metonic cycle defined as 235 synodic months is just 2 hours, 4 minutes and 58 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 lunisolar calendar.

A tropical year is longer than 12 lunar months and shorter than 13 of them. The arithmetic identity 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 almost exactly 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.^[3]

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."^[4] 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.^[5]

The Metonic cycle has been implemented in the Antikythera mechanism which offers unexpected evidence for the popularity of the calendar based on it.^[6]

The (19-year) Metonic cycle is a lunisolar cycle, as is the (76-year) Callippic cycle.^[7] 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.^[8] 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.^[9] 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,^[10] at least until in the year 1582, when the Gregorian calendar was introduced.

The Celts knew the Metonic cycle thousands of years ago, as evidenced by artifacts such as the Knowth Calendar Stone.^[11] It was almost certainly the basis for the 19-year so-called Celtic Great Year.

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 , 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 solar 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 more accurate determinations.^[12]

Hebrew calendar[]

A Small Mahzor (Hebrew מחזור, pronounced [maχˈzor], meaning "cycle") is a 19-year cycle in the lunisolar calendar system used by the Jewish people. It is similar to, but slightly different in usage with, the Greek Metonic cycle, and likely derived from or alongside the much earlier Babylonian calendar.^[13]

Three ancient civilizations (Babylonia, China and Israel) used lunisolar calendars and knew of the rule of the intercalation from as early as 2000 BC. Whether or not the correlation indicates cause-and-effect relationship is an open question.^{[14][15][verification needed]}

Polynesia[]

It is possible that the polynesian kilo-hoku (astronomers) discovered the 'Metonic cycle' in the same way Meton had, by trying to make the month fit the year. ^[16]

Mathematical basis[]

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The Metonic Cycle is the shortest cycle of time over which the synodic month and tropical year roughly synchronize, calculated by finding the least common multiple of the number of days in each.

Tropical year = 365.24219879 days.

365.24219879 x 19 = 6,939.602 days (every 19 years)

Synodic month = 29.53058868 days.

29.53058868 x 235 = 6,939.688 days (every 235 months)

The difference between 19 solar years and 235 lunar months is only about two hours, or 0.086 days per cycle, resulting in a full day of delay every 219 years (12.4 parts per million), roughly twelve cycles. This error is far smaller than that of previously calendrical schedules, though still imperfect.

19 years * 12 months = 228 months per cycle following the traditional 12 month schema, 7 months short of the 235 months that results in synchronization.

To solve this, seven intercalary months are added each nineteen year cycle to amount to the 235 months required. The lunar year is ~354 days, approximately 11 days short of the solar year. Thus, every 2-3 years there is a discrepancy of an approximate full lunar month. In order to 'catch up' to this discrepancy, the seven intercalary months are added at an interval of every 2-3 years in order to maintain seasonal consistency and prevent dramatic shifts over time.

Further details[]

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The Metonic cycle is related to two less accurate subcycles:

• 8 years = 99 lunations (an Octaeteris) minus approximately 1.591 days, i.e. a negative error of one day in 5 years; and
• 11 years = 136 lunations plus approximately 1.504 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, combining 17 Metonic cycles with one 11-year cycle gives:

• 334 tropical years ≈ 121990.89 days
• 4131 lunations ≈ 121990.86 days

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.

The Metonic cycle also aligns closely with other periods:

254 sidereal months (lunar orbits) = 6,939.702 days

255 draconic months (lunar nodes) = 6,939.1161 days.

20.021 eclipse years (40 eclipse seasons)

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 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 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.)

See also[]

• Octaeteris (8-year cycle of antiquity)
• Callippic cycle (76-year cycle from 330 BC)
• Hipparchic cycle (304-year cycle from 2nd century BC)
• Saros cycle of eclipses
• Attic and Byzantine calendar
• Julian day
• Date of Easter ("the Computus")

Notes[]

References[]

• Mathematical Astronomy Morsels, Jean Meeus, Willmann-Bell, Inc., 1997 (Chapter 9, p. 51, Table 9. A Some eclipse Periodicities)
• C. Philipp E. Nothaft (2012) Dating the Passion (The Life of Jesus and the Emergence of Scientific Chronology (200-1600), Leiden ISBN 9789004212190)
• Daniel P. Mc Carthy & Aidan Breen (2003) The ante-Nicene Christian Pasch De ratione paschali (The Paschal tract of Anatolius, bishop of Laodicea): Dublin (ISBN 9781851826971)
• Georges Declercq (2000) Anno Domini (The Origins of the Christian Era): Turnhout (ISBN 9782503510507)
• Rubellite Kawena Johnson (2001) ESSAYS IN HAWAIIAN LITERATURE Part I Origin Myths and Migration Traditions (p. 238)

External links[]

• Media related to Metonic cycle at Wikimedia Commons
• Eclipses, Cosmic Clockwork of the Ancients