# Metonic cycle

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The Metonic cycle or enneadecaeteris (from Ancient Greek : ἐννεακαιδεκαετηρίς, from ἐννεακαίδεκα, "nineteen") is a period of approximately 19 years after which the phases of the moon recur at the same time of the year. The recurrence is not perfect, and by precise observation the Metonic cycle defined as 235 synodic lunar months is just 1 hour, 27 minutes and 33 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 luni-solar calendar.

## Contents

A tropical year is longer than 12 lunar months and shorter than 13 of them. The arithmetical equation 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 equal to 19 solar years.

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 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 Nyköping staff, 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. [11]

## Mathematical basis

The importance of the tropical year for agriculture came to be realized much later than the adoption of lunar months for time keeping.[ citation needed ] However, it was recognized that the two cannot be easily coordinated over a short time span, so longer intervals were considered and the Metonic cycle was discovered as a rather good, but not perfect, schema. The currently accepted values are:

235 synodic months (lunar phases) = 6,939.688 days (Metonic period by definition).
19 tropical years = 6,939.602 days

The difference is 0.086 days per cycle, which means that after around twelve cycles there will be a full day of delay between the month and the year. The error is actually one day every 219 years, or 12.4 parts per million. However, the Metonic cycle turned out to be very close to 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 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 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.)

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.

## Further details

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
• 5776 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.

## Notes

1. "The Babylonian Calendar".
2. Livy, Ab Urbe Condita, I, XIX, 6.
3. Diodorus Siculus, Bibl. Hist. II.47.
4. Freeth, Tony; Jones, Alexander; Steele, John M.; Bitsakis, Yanis (31 July 2008). "Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism" (PDF). Nature. 454 (7204): 614–7. Bibcode:2008Natur.454..614F. doi:10.1038/nature07130. PMID   18668103 . Retrieved 20 May 2014.
5. Nothaft (2012) 168
6. Mc Carthy & Breen (2003) 17
7. Declecq (2000) 65-66
8. Declercq (2000) 66
9. 瞿曇悉達. "《古今历积年及章率》". 開元占經 (in Chinese). 第105卷.

## Related Research Articles

Intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or moon phases. Lunisolar calendars may require intercalations of both days and months.

A lunar calendar is a calendar based on the monthly cycles of the Moon's phases, in contrast to solar calendars, whose annual cycles are based only directly on the solar year. The most commonly used calendar, the Gregorian calendar, is a solar calendar system that originally evolved out of a lunar calendar system. A purely lunar calendar is also distinguished from a lunisolar calendar, whose lunar months are brought into alignment with the solar year through some process of intercalation. The details of when months begin varies from calendar to calendar, with some using new, full, or crescent moons and others employing detailed calculations.

A lunisolar calendar is a calendar in many cultures whose date indicates both the Moon phase and the time of the solar year. If the solar year is defined as a tropical year, then a lunisolar calendar will give an indication of the season; if it is taken as a sidereal year, then the calendar will predict the constellation near which the full moon may occur. As with all calendars which divide the year into months there is an additional requirement that the year have a whole number of months. In this case ordinary years consist of twelve months but every second or third year is an embolismic year, which adds a thirteenth intercalary, embolismic, or leap month.

A month is a unit of time, used with calendars, which is approximately as long as a natural period related to the motion of the Moon; month and Moon are cognates. The traditional concept arose with the cycle of Moon phases; such months (lunations) are synodic months and last approximately 29.53 days. From excavated tally sticks, researchers have deduced that people counted days in relation to the Moon's phases as early as the Paleolithic age. Synodic months, based on the Moon's orbital period with respect to the Earth-Sun line, are still the basis of many calendars today, and are used to divide the year.

In astronomy, the new moon is the first lunar phase, when the Moon and Sun have the same ecliptic longitude. At this phase, the lunar disk is not visible to the unaided eye, but its presence may be detected because it occults stars behind it.

Eclipses may occur repeatedly, separated by certain intervals of time: these intervals are called eclipse cycles. The series of eclipses separated by a repeat of one of these intervals is called an eclipse series.

The saros is a period of exactly 223 synodic months, approximately 6585.3211 days, or 18 years, 10, 11, or 12 days, and 8 hours, that can be used to predict eclipses of the Sun and Moon. One saros period after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, a near straight line, and a nearly identical eclipse will occur, in what is referred to as an eclipse cycle. A sar is one half of a saros.

A solar calendar is a calendar whose dates indicate the season or almost equivalently the apparent position of the Sun relative to the stars. The Gregorian calendar, widely accepted as a standard in the world, is an example of a solar calendar. The main other type of calendar is a lunar calendar, whose months correspond to cycles of Moon phases. The months of the Gregorian calendar do not correspond to cycles of the Moon phase.

As a moveable feast, the date of Easter is determined in each year through a calculation known as computus. Easter is celebrated on the first Sunday after the Paschal full moon, which is the first full moon on or after 21 March. Determining this date in advance requires a correlation between the lunar months and the solar year, while also accounting for the month, date, and weekday of the Julian or Gregorian calendar. The complexity of the algorithm arises because of the desire to associate the date of Easter with the date of the Jewish feast of Passover which, Christians believe, is when Jesus was crucified.

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Meton of Athens was a Greek mathematician, astronomer, geometer, and engineer who lived in Athens in the 5th century BC. He is best known for calculations involving the eponymous 19-year Metonic cycle which he introduced in 432 BC into the lunisolar Attic calendar. Euphronios says that Colonus was Meton's deme.

For astronomy and calendar studies, the Callippic cycle is a particular approximate common multiple of the year and the synodic month, that was proposed by Callippus during 330 BC. It is a period of 76 years, as an improvement of the 19-year Metonic cycle.

The Greek astronomer Hipparchus introduced two cycles that have been named after him in later literature.

Callippus was a Greek astronomer and mathematician.

The Ancient Macedonian calendar is a lunisolar calendar that was in use in ancient Macedon in the 1st millennium BC. It consisted of 12 synodic lunar months, which needed intercalary months to stay in step with the seasons. By the time the calendar was being used across the Hellenistic world, seven total embolimoi were being added in each 19-year Metonic cycle. The names of the ancient Macedonian Calendar remained in use in Syria even into the Christian era.

The Burmese calendar is a lunisolar calendar in which the months are based on lunar months and years are based on sidereal years. The calendar is largely based on an older version of the Hindu calendar, though unlike the Indian systems, it employs a version of the Metonic cycle. The calendar therefore has to reconcile the sidereal years of the Hindu calendar with the Metonic cycle's near tropical years by adding intercalary months and days at irregular intervals.

A total lunar eclipse will take place on March 3, 2026.

A total lunar eclipse will take place on May 6, 2069. The eclipse will be a dark one with the southern tip of the moon passing through the center of the Earth's shadow. This is the first central eclipse of Saros series 132.

A Small Mahzor 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.

## 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)