For astronomy and calendar studies, the
Metonic cycle or
Enneadecaeteris (from Ancient Greek:
ἐννεακαιδεκαετηρίς, "nineteen years") is a period
of very close to 19 years that is nearly a common multiple of the
solar year and the synodic (lunar) month. The Greek astronomer Meton
of Athens (fifth century BC) observed that a period of 19 years is
almost exactly equal to 235 synodic months and, rounded to full days,
counts 6,940 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 6,940-day cycle gives a
year length of 365 + 1⁄4 + 1⁄76 days
(the unrounded cycle is much more accurate), which is about 11 days
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). When Meton introduced the cycle around 432 BC, it was already
known by Babylonian astronomers.
A mechanical computation of the cycle is built into the Antikythera
The cycle was used in the Babylonian calendar, ancient Chinese
calendar systems (the 'Rule Cycle' 章) and the medieval computus
(i.e. the calculation of the date of Easter). It regulates the 19-year
cycle of intercalary months of the modern Hebrew calendar. The start
Metonic cycle depends on which of these systems is being used;
for Easter, the first year of the current
Metonic cycle is 2014.
1 Mathematical basis
2 Application in traditional calendars
3 Further details
4 See also
5 External links
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 maintain the seasons at the same calendar times each
year. Nineteen tropical years are about two hours shorter than 235
synodic months. The Metonic cycle's error is, therefore, one full day
every 219 years, or 12.4 parts per million.
19 tropical years = 6,939.602 days (12 × 354-day years + 7 × 384-day
years + 3.6 days).
235 synodic months (lunar phases) = 6,939.688 days (Metonic period by
254 sidereal months (lunar orbits) = 6,939.702 days (19 + 235 = 254).
255 draconic months (lunar nodes) = 6,939.1161 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 not to 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
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.
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, which 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 Babylonians applied the 19-year cycle since the late sixth century
BC. As they measured the moon's motion against the stars, the 235:19
relationship may originally have referred to sidereal years, instead
of tropical years as it has been used for various calendars.
According to Livy, the king of Rome
Numa Pompilius (753-673 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. 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.
Apollo was said to have visited the Hyperboreans once every 19 years,
presumably at the high point of the cycle.
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, 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 years.
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
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 = 250,921.39 days;
8,497 lunations = 250,921.41 days.
This gives 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 approximated the cycle to a whole number (6,940) 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.
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 & Byzantine calendar
Eclipses, Cosmic Clockwork of the Ancients
^ Rare Full Moon on Christmas Day, NASA
^ Ask Tom: How unusual is a full moon on Christmas Day?
^ Livy, Ab Urbe Condita, I, XIX, 6.
Astronomy Morsels, Jean Meeus, Willmann-Bell, Inc., 1997
(Chapter 9, p. 51, Table 9.A Some eclipse Periodicities)
Ab urbe condita
Anno Domini / Common Era
Hindu units of time
Hindu units of time (Yuga)
Canon of Kings
Lists of kings
Pre-Julian / Julian
Old Style and New Style dates
Adoption of the Gregorian calendar
Astronomical year numbering
Chinese sexagenary cycle
ISO week date
Winter count (Plains Indians)
Geological history of Earth
Geological time units
Global Standard Stratigraphic Age (GSSA)
Global Boundary Stratotype Section and Point (GSSP)
Law of superposition
Amino acid racemisation
Terminus post quem
Ancient Greek astronomy
Hippocrates of Chios
Philip of Opus
Sosigenes of Alexandria
Sosigenes the Peripatetic
Theon of Alexandria
Theon of Smyrna
On Sizes and Distances
On Sizes and Distances (Hipparchus)
On the Sizes and Distances (Aristarchus)
On the Heavens
On the Heavens (Aristotle)
Circle of latitude
Deferent and epicycle
Medieval European science