Intercalation is the insertion of a leap day, week or month into some calendar years to make the calendar follow the seasons or moon phases. Lunisolar calendars may require intercalations of both days and months.
The solar year does not have a whole number of days (it is about 365.24 days), but a calendar year must have a whole number of days. The only way to reconcile the two is to vary the number of days in the calendar year.
In solar calendars, this is done by adding to a common year of 365 days, an extra day ("leap day" or "intercalary day") about every four years, causing a leap year to have 366 days (Julian, Gregorian and Indian national calendars).
The Decree of Canopus, which was issued by the pharaoh Ptolemy III, Euergetes of Ancient Egypt in 239 BC, decreed a solar leap day system; an Egyptian leap year was not adopted until 25 BC, when the Roman Emperor Augustus successfully instituted a reformed Alexandrian calendar.
In the Julian calendar, as well as in the Gregorian calendar, which improved upon it, intercalation is done by adding an extra day to February in each leap year. In the Julian Calendar this was done every four years. In the Gregorian, years divisible by 100 but not 400, were exempted in order to improve accuracy. Thus, 2000 was a leap year; 1700, 1800, and 1900 were not.
Epagomenal days are days within a solar calendar that are outside any regular month. Usually five epagomenal days are included within every year (Egyptian, Coptic, Ethiopian and French Republican calendars), but a sixth epagomenal day is intercalated every four years in some (Coptic, Ethiopian and French Republican calendars).
The solar year does not have a whole number of lunar months (it is about 12.37 lunations), so a lunisolar calendar must have a variable number of months in a year. Regular years have 12 months, but embolismic years insert a 13th "intercalary" or "embolismic" month every second or third year. Whether to insert an intercalary month in a given year may be determined using regular cycles such as the 19-year Metonic cycle (Hebrew calendar and in the determination of Easter) or using calculations of lunar phases (Hindu lunisolar and Chinese calendars). The Buddhist calendar adds both an intercalary day and month on a usually regular cycle.
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