The recurrence is not perfect, and by precise observation the Metonic cycle defined as 235 synodic months is just 2 hours, 4 minutes and 58 seconds longer than 19 tropical years.
[4] According to Livy, the second king of Rome, Numa Pompilius (reigned 715–673 BC), inserted intercalary months in such a way that "in the twentieth year the days should fall in with the same position of the sun from which they had started".
[6] The Metonic cycle has been implemented in the Antikythera mechanism which offers unexpected evidence for the popularity of the calendar based on it.
[10] However, it was some later, somewhat different, version of the Metonic 19-year lunar cycle which, as the basic structure of Dionysius Exiguus' and also of Bede's Easter table, would ultimately prevail throughout Christendom,[11] at least until in the year 1582, when the Gregorian calendar was introduced.
[12] The Metonic cycle is thought to be numerically encoded on the Berlin Gold Hat from central Europe, dating from c. 1000-800 BC.
The Bahá'í calendar, established during the middle of the 19th century, is also based on cycles of 19 solar years.
The mathematical logic is this: That duration is almost the same as 235 synodic months: Thus the algorithm is correct to 0.087 days (2 hours, 5 minutes and 16 seconds).