The date of Easter

The date of Easter is primarily used for liturgical purposes. Up to the 8th century AD there was no uniform method for determining the date of Easter but the method favoured by the Council of Nicaea in AD 325 gradually became the accepted method. The adoption of the Gregorian calendar necessitated some modifications to this scheme but it is still basically the same.

The simple standard definition of Easter is that it is the first Sunday after the Full Moon that occurs on or after the vernal equinox. If the full moon falls on a Sunday then Easter is the next Sunday.

Unfortunately this simple definition is not strictly speaking correct. The vernal equinox used is not the true equinox but an artificial one always assumed to be on 21 March. The full moon used is not the true full moon but an artificial construct based on the Metonic cycle (below).

The reasons for this are that the method is then independent of longitude on the Earth and is thus independent of time zone. It also allows the date of Easter to be calculated in advance regardless of the actual motion of the Earth around the Sun.

The method quoted here is valid for the determination of the date of Easter in Western Christian churches; the date used by the Eastern churches can be one, four or five weeks later.

The Metonic cycle

The Metonic cycle of 19 years is one in which the phases of the Moon repeat exactly. It is thus possible to have a 19-year cycle for the dates of Full or New Moon. In the Julian calendar this 19-year cycle can be fairly easily translated into a date for Easter via the calculation of two quantities called the Golden Number and the Dominical Letter. These can readily be found from appropriate tables and another table gives the date of Easter.

In the Gregorian calendar the calculation is complicated by the definition of which century years are leap years. These leap years mess up the simple Metonic cycle by altering the number of days in different periods of 19 years. The tabular method uses the Epact instead of the Golden Number.

The Epact

The Epact is the age of the Moon, diminished by one day, on 1 January in the Gregorian ecclesiastical calendar. From the involved rules for constructing the lunar calendar a table may be drawn up to give the Epact, which can vary between 0 and 29.

The Dominical letter in the Gregorian calendar has a cycle of 28 years within each century but the century leap years again throw this into disorder. There is an overall cycle of 400 years over which they repeat.

Algorithm to find the date of Easter

An algorithm to find the date of Easter which is valid from 1900 to 2099 has been derived by Carter as follows:

Calculate D="'225'" - 11(Y MOD 19).

If D is greater than 50 then subtract multiples of 30 until the resulting new value of D is
less than 51.

If D is greater than 48 subtract 1 from it.

Calculate E="'(Y" +' [Y/4] + D + 1) MOD 7. (NB Integer part of [Y/4])

Calculate Q="'D +'" 7 - E.

If Q is less than 32 then Easter is in March. If Q is greater than 31 then Q - 31 is its date in April.

For example, for 1998:
D = 225 - 11*(1998 MOD 19) = 225 - 11*3 = 192
D is greater than 50, therefore:
D = (192 - 5*30) = 42
E = (1998 + [1998/4] + 42 + 1) MOD 7="'2540'" MOD 7="'6'"
Q = 42 + 7 - 6="'43'"
Easter 1998="'43" -' 31="'12" April'