For astronomy and calendar studies, the Metonic cycle or Enneadecaeteris is a
period of very close to 19 years that is remarkable for being nearly a common
multiple of the solar year and the synodic 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 6,940 days. The difference
between the two periods 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, 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. 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 mechanism.
The cycle was used in the Babylonian calendar, ancient Chinese calendar
systems and the medieval computus. It regulates the 19-year cycle of
intercalary months of the modern Hebrew calendar.
Mathematical basis At the time of Meton, axial precession
had not yet been discovered, and he could not distinguish between sidereal
years and tropical years. 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. 235 synodic months = 6,939.688 days.
254 sidereal months = 6,939.702 days. 255 draconic months = 6,939.1161 days.
Note that the 19-year cycle is also close 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, 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 month needs to be inserted. 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 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. Apollo was said to have visited the
Hyperboreans once every 19 years, presumably at the high point of the
cycle. 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.
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.
Further details The Metonic cycle is related to two less
accurate subcycles: 8 years = 99 lunations 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 = 250,921.39 days; 8,497 lunations = 250,921.41 days.
This gives an error of only about half an hour in 687 years, 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 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. See also
Octaeteris Callippic cycle
Hipparchic cycle Saros cycle of eclipses
Attic & Byzantine calendar Chinese calendar
Hebrew calendar Runic calendar
Julian day External links
Eclipses, Cosmic Clockwork of the Ancients
References Mathematical Astronomy Morsels, Jean
Meeus, Willmann-Bell, Inc., 1997