The problem the cycle solves
A lunar month is about 29.53 days and a solar year about 365.24 days. Twelve months fall about eleven days short of a year, as the Hijri calendar shows by letting the difference run. A calendar that wants months tied to the moon and years tied to the seasons must therefore add an extra month from time to time. The question is how often, and in what pattern.
The Metonic cycle gives a neat answer: over nineteen solar years, there are almost exactly 235 lunar months. It is named after Meton, an astronomer of Athens, who is said to have announced it in the fifth century BCE.
The arithmetic
Nineteen years of twelve months give 228 months. To reach 235 you need 7 more, so seven of the nineteen years carry an extra, thirteenth month: 228 plus 7 equals 235.
Checking against days, using round figures:
- 19 solar years of about 365.2422 days come to roughly 6,939.6 days.
- 235 lunar months of about 29.5306 days come to roughly 6,939.7 days.
The two totals differ by less than a tenth of a day, about two hours across nineteen years. That closeness is what makes the cycle so useful: after nineteen years the phases of the moon fall again on nearly the same dates of the solar year.
Not a Greek invention alone
The attribution to Meton is conventional, but nineteen-year patterns were noticed elsewhere. Babylonian astronomers were working with the same relationship around the same period or somewhat earlier, and it is often said that a regular nineteen-year scheme of intercalation, the insertion of extra months, was in place in Babylonia by about the fifth century BCE. Records are fragmentary, so the exact order of discovery is hard to state with confidence. Chinese astronomers also recognised the nineteen-year period, which they called the zhang, in the first millennium BCE. Some details of when each culture arrived at it remain uncertain.
The cycle is not exact, which led to refinements. The Greek astronomer Callippus proposed combining four Metonic cycles, 76 years, and removing a day to improve the match.
The Jewish calendar
The fixed Hebrew calendar uses a nineteen-year cycle in which seven years are leap years with an extra month, Adar I. The leap years are the 3rd, 6th, 8th, 11th, 14th, 17th and 19th of each cycle. That is seven in nineteen, matching the arithmetic above. The extra month keeps festivals such as Passover in spring. Without it, festivals tied to the seasons would slowly drift through the year.
The practice sits alongside the biblical cycle of years, in which every seventh year is observed as a rest year, discussed under sabbatical years. The seven in the sabbatical cycle and the seven leap years in nineteen are separate counts that happen to share a number.
The counting of days between Passover and Shavuot, described in counting the Omer, also depends on a calendar in which the dates of Passover remain in spring.
Easter and the Golden Number
In the Christian church, the date of Easter was fixed, in the early centuries, to depend on the spring equinox and the full moon. Computing it required a lunar calendar projected forward, and the Metonic cycle provided the framework. Easter tables, such as those associated with Dionysius Exiguus in the sixth century and the later work of Bede, relied on a nineteen-year cycle. Each year of the cycle was given a number from 1 to 19, known as the Golden Number, which is still used in the calculation of Easter in the traditional computus.
The rule for Easter, in its usual modern form, places the festival on the first Sunday after the ecclesiastical full moon that falls on or after 21 March. This ecclesiastical moon is calculated from tables and does not always match the actual astronomical full moon. Because it is calculated using the nineteen-year cycle, the Gregorian reform had to adjust the tables as well as the dates; that part of the story belongs with the 1582 reform.
Nineteen and its properties
Nineteen belongs to the prime numbers, so the cycle cannot be divided into shorter repeating sub-cycles. This is a mathematical property and not the reason for the cycle: the number 19 emerges from astronomical measurements and not from number theory. The relationship between 12 lunar months and a solar year is, in effect, a rational approximation to an irrational ratio, and 235 over 19 is among the best simple ones. Other, longer cycles exist, but nineteen is a short span that people could track in a lifetime.
Cycles in other calendars
The idea of leap months occurs well beyond the Metonic world. The Hijri calendar rejects it altogether, while the traditional Chinese calendar inserts leap months by rule, with seven in nineteen years as the long-run average. The Hindu calendar also adds an extra month at intervals. Babylonian clay-tablet records, written in a sexagesimal notation known as the Mesopotamian sexagesimal system, show that the scribes who recorded these corrections were working with the same arithmetic.
Seen across these examples, the Metonic cycle is a quiet example of how a pair of numbers, 19 and 235, tied together lunar and solar time and decided the rhythm of festivals for several religions. Its importance is practical, and any symbolism later attached to the number nineteen belongs to individual traditions.