The Metonic Cycle – 19 Years for Moon and Sun

Two celestial bodies that refuse to divide into one another: that is the founding problem of every lunisolar calendar. Twelve lunar months fall about eleven days short of a solar year. Anyone wanting to hold both in one calendar has to insert a thirteenth month from time to time. The question is when.

In 5th-century BC Athens someone found an answer that would outlast two and a half millennia.

19 years, 235 months, 6,940 days

The cycle rests on a plain observation: 19 solar years correspond almost exactly to 235 lunar months. Insert a leap month seven times across those 19 years, and Moon and Sun end up together again.

Friedrich Karl Ginzel, whose 1911 work on chronology is the basis here, reports the ancient astronomer Geminus. The Greeks had established through observation that nineteen years contain 6,940 days, or 235 months including the leap months, of which there are seven in the 19 years.[1] That implies a year of 365 5/19 days.

Modern figures show how well it lands: 235 synodic months run to 6,939.69 days, 19 tropical years to 6,939.60.[2] After 19 years the discrepancy comes to a little over two hours, about six minutes a year. More than two centuries pass before that grows into a whole day.

How the days were distributed

The finest part of the account concerns not the cycle itself but its implementation. 235 months had to be fitted into 6,940 days. Counting every month at 30 days gives 7,050, so 110 days had to go. Meton accordingly reckoned 110 hollow months of 29 days and 125 full months of 30.

Which days to drop was the real question, and the answer is elegant. Rather than mechanically striking every second month's last day, the Greeks spread the omissions evenly across the whole cycle: they divided 6,940 by 110 and got 63. A day had to be struck after each interval of 63 days.[3] What fell away was never simply the 30th of a month, but whichever day landed on the count of 63.

Ginzel notes explicitly where this improved on the older octaeteris: under Meton two full months could follow one another, which fits the actual motion of the Moon better. That measure, he writes, had not been observed in the eight-year period.[1]

The criticism came immediately

Antiquity named the cycle's weakness at once. Geminus judged that the months in it were rightly taken and the leap months arranged according to the celestial phenomena, but that the length of the year was not in accord with the heavens.[4]

He was right. Meton's year of 365 5/19 days runs about half an hour long. The cycle is excellent on the lunar side and wanting on the solar side. Both refinements that followed therefore worked on the length of the year, not on the lunar fit.

Callippus and Hipparchus

Callippus of Cyzicus arrived in Athens around 334 BC. He shortened Meton's year by 1/76 of a day, arriving at a clean 365¼. To do so he combined four Metonic periods into a 76-year cycle and struck one day: 27,759 instead of 27,760, spread across 940 lunar months.[5] The resulting mean month, in Ginzel's words, was "only 22 seconds too long against the astronomical" value. The cycle began in 330 BC.

Hipparchus of Nicaea went further around 125 BC. He recognised that even the supposedly unshakeable 365¼-day year ran long, by roughly one three-hundredth of a day. To reconcile his corrected year with the Callippic cycle, he quadrupled it into a period of 304 years with 111,035 days and 3,760 months.[6] He set this out in a work "On intercalary months and days," now lost and known only through Ptolemy.

That the error grows again with Callippus and Hipparchus is no regression. Both were correcting the length of the year, not the lunar fit. On the lunar side Meton remains the most accurate of the four.[2]

The 304 and the North

Here a connection emerges that surprises at first. The number 304 also appears in Norse tradition: Andreas Nordberg reads the legendary 300-year life of King Aun as folk memory of a 304-year cycle, derived as 16 times 19 years.[7] Hipparchus arrives at the same number by way of 4 times 76.

No dependence follows from this. 304 years fall out inevitably once you multiply the 19-year cycle until the remainder of days resolves. Two computational traditions can reach it independently, and a shared number proves no borrowing on its own. Claiming a connection would take more than that.

What survived

The Metonic cycle outlived its culture. It sits inside the calculation of Easter: the golden number (numerus aureus) marks a year's position within the 19-year lunar cycle, which medieval sources accordingly often call the paschal cycle, since it serves to fix the date of Easter.[8] The Golden Number still appears on medieval rune calendars such as the Norwegian primstav. The Hebrew calendar intercalates on the same rhythm.

The medieval computists ran into precisely the problem Meton had solved with his 63-day rule. Their cycle came to 6,940¼ days by calculation, while 19 Julian years hold only 6,939¾. One day was surplus and had to be suppressed; the point at which this happened was called the saltus lunae, the leap of the Moon.[9]

For the Germanic lunar calendar this means the 19-year cycle is no part of its original stock. There the reckoning ran on the octaeteris, eight years with three leap months. The Metonic cycle reached the North only with Christianisation, and the route can be named: the Easter rule of Dionysius Exiguus computes the full moons on a 19-year cycle, and it was spread above all by Bede.[10] The same author to whom we owe the Anglo-Saxon month names thus brought the 19-year cycle with him, displacing the older eight-year rhythm. For how differently cultures resolved the same discrepancy between Moon and Sun, see the survey on the history of lunar calendars.