The Metonic cycle: nineteen years to tune the Sun and the Moon
Why 235 lunations almost equal 19 solar years: Meton, the Babylonians, Callippus, the golden number of Easter and the Hebrew calendar.
Every calendar that tries to follow both the phases of the Moon and the seasons runs into an arithmetical problem: the solar year does not contain a whole number of lunations. Twelve lunar months are too few, thirteen are too many. The most famous solution of antiquity is a nineteen-year cycle, which tradition links to the name of the Athenian astronomer Meton.
An almost perfect coincidence
Nineteen solar years last about 6,939.6 days, and 235 synodic months about 6,939.7: the difference is barely a couple of hours. This means that after nineteen years the lunar phases fall again on almost the same days of the solar year. If a full moon occurs on a certain date, nineteen years later another full moon will fall on the same date or on a nearby day.
In those nineteen years the 235 months are distributed into twelve years of twelve months and seven years of thirteen months. This is the scheme adopted by the lunisolar calendars that follow the cycle: seven intercalary months every nineteen years, placed in fixed positions, are enough to keep the feasts tied to the Moon in the right season for centuries.
Meton and Babylon
According to tradition Meton presented the cycle in Athens in 432 BC, the year in which, with his colleague Euctemon, he observed the summer solstice. It is not certain that he discovered it independently: Babylonian astronomers adopted a regular scheme of seven intercalations in nineteen years at roughly the same time. About a century later Callippus of Cyzicus proposed a cycle of 76 years, four Metonic cycles minus one day, to correct its small error.
The golden number and Easter
The Metonic cycle had a long life in the Christian calendar. To calculate Easter, which falls on the Sunday after the spring full moon, medieval computists needed to know the moons of every year. Each year was given a number from 1 to 19 indicating its position in the cycle: the golden number. According to a traditional explanation, the name comes from the custom of writing it in golden letters in calendars, but its origin is not documented with certainty.
The Hebrew calendar is also based on a nineteen-year cycle, in which years 3, 6, 8, 11, 14, 17 and 19 have an extra month, called Adar I. The traditional Chinese calendar, while following its own rules based on the solar terms, likewise produces a pattern very close to the Metonic rhythm, with seven months added roughly every nineteen years.
The year of Meton
For the Greeks the return of the stars to the same configuration also had a symbolic value, tied to the order of the cosmos. Diodorus Siculus, drawing on an older account of the Hyperboreans, relates that Apollo visited their island every nineteen years, the time in which the stars return to the same place in the sky, and adds that the Greeks called that period the «year of Meton». Some modern authors have linked the passage to Stonehenge, but this is a hypothesis without proof.
The Metonic cycle is an example of how a simple numerical observation can support entire religious and civil systems for millennia. Even today, whenever the date of Easter or of the Jewish New Year is fixed, an arithmetic is used that goes back to the astronomers of Athens and Babylon, who lived some two and a half millennia ago.