The Mathematics of Calendars

The day, month, and year are incommensurable — no whole number of days fits exactly into a year or a month. The solar year is approximately 365.2422 days. The Gregorian calendar approximates this with the rule: leap year if divisible by 4, except century years, unless divisible by 400. This gives 365.2425 days per year — an error of about 1 day per 3,300 years. The Hebrew calendar uses the 19-year Metonic cycle: 235 lunar months ≈ 19 solar years, an approximation known to Babylonian astronomers 2,500 years ago.