Metonic cycle

Encyclopedia

In astronomy

and calendar

studies, the

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 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 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 19-year cycle of intercalary months in the Hebrew calendar

.

s (currently: 365.256363 days) and tropical year

s (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.

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 (

). 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.

and Babylonian

lunisolar calendar

s, as well as in the Hebrew calendar

), 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 eclipse

s, forms the basis of the Greek and Hebrew calendars, and is used in the computation of the date of Easter

each year.

The Chaldea

n astronomer Kidinnu

(4th century BC) knew of the 19-year 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 year

s, instead of tropical year

s as it has been used in 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. 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.

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:

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 19-year periods for a 76-year cycle with a mean year of exactly 365.25 days.

Astronomy

Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

and calendar

Calendar

A calendar is a system of organizing days for social, religious, commercial, or administrative purposes. This is done by giving names to periods of time, typically days, weeks, months, and years. The name given to each day is known as a date. Periods in a calendar are usually, though not...

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 multipleLeast common multiple

In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...

of the solar year

Year

A year is the orbital period of the Earth moving around the Sun. For an observer on Earth, this corresponds to the period it takes the Sun to complete one course throughout the zodiac along the ecliptic....

and the synodic (lunar) month. The Greek

Greeks

The Greeks, also known as the Hellenes , are a nation and ethnic group native to Greece, Cyprus and neighboring regions. They also form a significant diaspora, with Greek communities established around the world....

astronomer

Astronomer

An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

Meton of Athens

Meton of Athens

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.The metonic calendar assumes...

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 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 the cycle in circa 432 BC but it was actually known earlier by Babylonian astronomers.

The cycle was used in the Babylonian calendar

Babylonian calendar

The Babylonian calendar was a lunisolar calendar with years consisting of 12 lunar months, each beginning when a new crescent moon was first sighted low on the western horizon at sunset, plus an intercalary month inserted as needed by decree. The calendar is based on a Sumerian precedecessor...

, the medieval computus

Computus

Computus is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age....

(i.e. the calculation of the date of Easter

Easter

Easter is the central feast in the Christian liturgical year. According to the Canonical gospels, Jesus rose from the dead on the third day after his crucifixion. His resurrection is celebrated on Easter Day or Easter Sunday...

) and still regulates the 19-year cycle of intercalary months in the Hebrew calendar

Hebrew calendar

The Hebrew calendar , or Jewish calendar, is a lunisolar calendar used today predominantly for Jewish religious observances. It determines the dates for Jewish holidays and the appropriate public reading of Torah portions, yahrzeits , and daily Psalm reading, among many ceremonial uses...

.

## Mathematical basis

At the time of Meton, precession had not yet been discovered, and he could not make a distinction between sidereal yearSidereal year

A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic. It was equal to at noon 1 January...

s (currently: 365.256363 days) and tropical year

Tropical year

A tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...

s (currently: 365.242190 days). Most calendars, like our Gregorian calendar

Gregorian calendar

The Gregorian calendar, also known as the Western calendar, or Christian calendar, is the internationally accepted civil calendar. It was introduced by Pope Gregory XIII, after whom the calendar was named, by a decree signed on 24 February 1582, a papal bull known by its opening words Inter...

, 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 yearTropical yearA tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...

s = 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 19-year cycle is also close (to somewhat more than half a day) to 255 draconic months, so it is also an eclipse cycle

Eclipse cycle

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.- Eclipse conditions :...

, 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

Resonance

In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

. An example of a non-coincidental cycle is the orbit of Mercury

Mercury (planet)

Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...

, 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

Lunisolar calendar

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...

, 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 (intercalationIntercalation

Intercalation 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.- Solar calendars :...

). 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 AtticAttic calendar

The Attic calendar is a hellenic calendar that was in use in ancient Attica, the ancestral territory of the Athenian polis. This article focuses on the 5th and 4th centuries BC, the classical period that produced some of the most significant works of ancient Greek literature. Because of the...

and Babylonian

Babylonian calendar

The Babylonian calendar was a lunisolar calendar with years consisting of 12 lunar months, each beginning when a new crescent moon was first sighted low on the western horizon at sunset, plus an intercalary month inserted as needed by decree. The calendar is based on a Sumerian precedecessor...

lunisolar calendar

Lunisolar calendar

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...

s, as well as in the Hebrew calendar

Hebrew calendar

The Hebrew calendar , or Jewish calendar, is a lunisolar calendar used today predominantly for Jewish religious observances. It determines the dates for Jewish holidays and the appropriate public reading of Torah portions, yahrzeits , and daily Psalm reading, among many ceremonial uses...

), 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 eclipse

Eclipse

An eclipse is an astronomical event that occurs when an astronomical object is temporarily obscured, either by passing into the shadow of another body or by having another body pass between it and the viewer...

s, forms the basis of the Greek and Hebrew calendars, and is used in the computation of the date of Easter

Computus

Computus is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age....

each year.

The Chaldea

Chaldea

Chaldea or Chaldaea , from Greek , Chaldaia; Akkadian ; Hebrew כשדים, Kaśdim; Aramaic: ܟܐܠܕܘ, Kaldo) was a marshy land located in modern-day southern Iraq which came to briefly rule Babylon...

n astronomer Kidinnu

Kidinnu

Kidinnu was a Chaldean astronomer and mathematician. Strabo of Amaseia called him Kidenas, Pliny the Elder Cidenas, and Vettius Valens Kidynas....

(4th century BC) knew of the 19-year 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 year

Sidereal year

A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic. It was equal to at noon 1 January...

s, instead of tropical year

Tropical year

A tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...

s as it has been used in various calendars.

The Runic calendar

Runic calendar

A Runic calendar is a perpetual calendar based on the 19 year long Metonic cycle of the Moon. Runic calendars were written on parchment or carved onto staves of wood, bone, or horn. The oldest one known, and the only one from the Middle Ages, is the Nyköping staff, believed to date from the 13th...

is a perpetual calendar

Perpetual calendar

A perpetual calendar is a calendar which is good for a span of many years, such as the Runic calendar.- General information :...

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. 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

Bahá'í calendar

The Bahá'í calendar, also called the Badí‘ calendar , used by the Bahá'í Faith, is a solar calendar with regular years of 365 days, and leap years of 366 days. Years are composed of 19 months of 19 days each, plus an extra period of "Intercalary Days"...

, 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 OctaeterisOctaeterisIn astronomy, an octaeteris is the period of eight solar years after which the moon phase occurs on the same day of the year plus one or two days....

) 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

Meton of Athens

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.The metonic calendar assumes...

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

Callippus

Callippus or Calippus was a Greek astronomer and mathematician.Callippus was born at Cyzicus, and studied under Eudoxus of Cnidus at the Academy of Plato. He also worked with Aristotle at the Lyceum, which means that he was active in Athens prior to Aristotle's death in 322...

developed the Callippic cycle

Callippic cycle

In 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 in 330 BC...

of four 19-year periods for a 76-year cycle with a mean year of exactly 365.25 days.

## See also

- Antikythera mechanismAntikythera mechanismThe Antikythera mechanism is an ancient mechanical computer designed to calculate astronomical positions. It was recovered in 1900–1901 from the Antikythera wreck. Its significance and complexity were not understood until decades later. Its time of construction is now estimated between 150 and 100...
- Bahá'í calendarBahá'í calendarThe Bahá'í calendar, also called the Badí‘ calendar , used by the Bahá'í Faith, is a solar calendar with regular years of 365 days, and leap years of 366 days. Years are composed of 19 months of 19 days each, plus an extra period of "Intercalary Days"...
- Byzantine calendarByzantine calendarThe Byzantine calendar, also "Creation Era of Constantinople," or "Era of the World" was the calendar used by the Eastern Orthodox Church from c. 691 to 1728 in the Ecumenical Patriarchate. It was also the official calendar of the Byzantine Empire from 988 to 1453, and in Russia from c...
- Chinese calendarChinese calendarThe Chinese calendar is a lunisolar calendar, incorporating elements of a lunar calendar with those of a solar calendar. It is not exclusive to China, but followed by many other Asian cultures as well...
- Hebrew calendarHebrew calendarThe Hebrew calendar , or Jewish calendar, is a lunisolar calendar used today predominantly for Jewish religious observances. It determines the dates for Jewish holidays and the appropriate public reading of Torah portions, yahrzeits , and daily Psalm reading, among many ceremonial uses...
- Saros cycleSaros cycleThe saros is a period of 223 synodic months , that can be used to predict eclipses of the Sun and Moon. One saros after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, and a nearly identical eclipse will occur, in what is referred to as an eclipse cycle...
- Runic calendarRunic calendarA Runic calendar is a perpetual calendar based on the 19 year long Metonic cycle of the Moon. Runic calendars were written on parchment or carved onto staves of wood, bone, or horn. The oldest one known, and the only one from the Middle Ages, is the Nyköping staff, believed to date from the 13th...
- Julian dayJulian dayJulian day is used in the Julian date system of time measurement for scientific use by the astronomy community, presenting the interval of time in days and fractions of a day since January 1, 4713 BC Greenwich noon...