Metonic Cycle - Mathematical Basis

Mathematical Basis

At the time of Meton, precession had not yet been discovered, and he could not make a distinction between sidereal years (currently: 365.256363 days) and tropical years (currently: 365.242190 days). Most calendars, like our Gregorian calendar, follow the seasons and are based on the tropical year. 19 tropical years are shorter than 235 synodic months by about 2 hours. The Metonic cycle's error is then one full day every 219 years, or 12.4 parts per million.

19 tropical years = 6939.602 days (12 × 354 day years + 7 × 384 day years + 3.6 days)

235 synodic months (lunar phases) = 6939.688 days (Metonic period by definition)

254 sidereal months (lunar orbits) = 6939.702 days (19 + 235 = 254)

255 draconic months (lunar nodes) = 6939.1161 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 a 1⁄₅ of a Metonic cycle (47 synodic months, 3.8 years), and it recurs about 20 to 25 cycles.

This cycle appears 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 appear to have no 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, in 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 appear initially 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 appear to have introduced this scheme well before Meton, about 500 BC.