The Metonic cycle: nineteen years to reconcile the Sun and the Moon

Among the numbers that have shaped the history of calendars, few are as important as nineteen. In nineteen solar years almost exactly 235 lunar months go by: about 6939.6 days on one side, about 6939.7 on the other. After that period the phases of the Moon fall again on the same days of the year. The discovery of this relationship made it possible to build regular lunisolar calendars, and its name, the Metonic cycle, recalls an Athenian astronomer of the fifth century BC.

Meton and Euctemon

According to ancient sources, Meton of Athens, together with Euctemon, observed the summer solstice of 432 BC and proposed a cycle of 6940 days, divided into 235 months, 125 of thirty days and 110 of twenty-nine. The cycle comprised seven years of thirteen months and twelve years of twelve months. Diodorus Siculus, in the twelfth book of his Library of History, recalls the fame the cycle brought Meton among the Greeks.

The same scheme was used in Mesopotamia, where from about the fifth century BC intercalations follow a nineteen-year cycle. The relationship between the two traditions is debated: some scholars think of a transmission from Babylon to Greece, others of an independent discovery. In any case it was in the Greek world that the cycle received the name by which we still know it today.

Callippus and the Antikythera mechanism

A century later the astronomer Callippus of Cyzicus corrected Meton's cycle, which gave the year a length slightly greater than 365 and a quarter days. He combined four cycles into a period of seventy-six years and removed one day from it. The Callippic period, which began in 330 BC, became a reference for later astronomers, and Ptolemy still uses it in the Almagest to date ancient observations.

The Antikythera mechanism, the geared device recovered in the early twentieth century from a wreck off the Greek island and dated between the second and first centuries BC, shows how familiar these cycles were. On its back, a large spiral dial showed the 235 months of the Metonic cycle, while a smaller subsidiary dial was used to follow the Callippic cycle.

The golden number

In the Middle Ages the nineteen-year cycle became the central tool for calculating Easter. Each year received a number from 1 to 19, indicating its position in the lunar cycle: this is the golden number, so called, according to a traditional explanation, because it was written in gold in calendars. It is obtained by dividing the year by nineteen and adding one to the remainder; for 2026 it is 13.

In medieval liturgical calendars, golden numbers were written beside the days of the month on which new moons fell: knowing the golden number of the year was enough to know on which days the new Moon would fall. The system worked well, but the cycle's small error, about one day every three centuries, eventually shifted the calendar's moons away from the real ones.

The Gregorian reform of 1582 corrected this drift by introducing the system of epacts, but the nineteen-year cycle did not disappear. It still underlies the Hebrew calendar, appears in the Easter tables of liturgical books and survives in the formulas astronomers use to calculate the date of Easter. A numerical relationship observed some two and a half thousand years ago is still at work inside our calendars.