Calculating the date of Easter

The canonical rule is that Easter Sunday will fall on or after the day of the first Full Moon after the day of the vernal equinox. The day of the equinox is taken to be fixed at 21 March. For determining the feast, the Christian churches settled on a method to define a reckoned "ecclesiastic" Moon, rather than observations of the true Moon like the Jews did.

Tabular methods

Gregorian Calendar

This method for the computation of the date of Easter was introduced with the Gregorian calendar reform in 1582.

It first determines the epact[?] for the year, this is the age of the Moon on the last day of the preceding year. The epact can have a value from "*" (=0 or 30) to 29 days.

The (solar) year is always counted to have 365 days (and a bit). A lunar year of 12 months is counted to have 354 days, so the average lunation is 29+½ days (and a bit). So the lunar year is 11 days shorter than the solar year. As a consequence, to compute the epact for next year you have to add 11 to the epact of this year. Whenever the epact reaches or exceeds 30, then 30 has to be subtracted.

Now 19 tropical years are as long as 235 synodic months (Metonic cycle). So after 19 years the lunations should fall the same way in the solar years, so the epacts should repeat after 19 years. However, 19*11 = 209 modulo 30 = 29, not 0. So after 19 years the epact must be corrected by +1 day in order for the cycle to repeat. This is the so-called saltus lunae. The sequence number of the year in the 19-year cycle is called the Golden Number. The extra 209 or 210 days fill 7 so-called embolismic months, for a total of 19*12+7 = 235 lunations.

The epacts for the current (anno 2003) Metonic cycle are:

Year	1995	1996	1997	1998	1999	2000	2001	2002	2003	2004	2005	2006	2007	2008	2009	2010	2011	2012	2013
Golden Number	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
Epact	29	10	21	2	13	24	5	16	27	8	19	30	11	22	3	14	25	6	17

This table can be extended for previous and following 19-year periods, and is valid from 1900 to 2199.

The epacts are used to find the dates of New Moon in the following way. Write down a table of all 365 days of the year (the leap day is ignored). Then label all dates with a Roman number counting downwards, from "*" (= 0, or 30), "xxix" (29), downto "i" (1), starting from 1 January, and repeat this to the end of the year. However, in every second such period count only 29 days and label the date with xxv (25) also with xxiv (24). Finally, in addition add the label "25" to the dates that have "xxv" in the 30-day periods; but in 29-periods (which have "xxiv" together with "xxv") add the label "25" to the date with "xxvi".

Now if the epact is for instance 27 (Roman: xxvii), then there will be an ecclesiastic New Moon on every date that has the label "xxvii". The ecclesiastic Full Moon falls 14 days later.

Also label all the dates in the table with letters "A" to "G", starting from 1 January, and repeat to the end of the year. If for instance the first Sunday of the year is on 5 January, which has letter E, then every date with the letter "E" will be a Sunday that year. Then "E" is called the Dominical Letter for that year (from Latin: dies domini, day of the Lord). However, in leap years after 24 February the Sundays will fall on the previous letter of the alphabet, so leap years have 2 dominical letters: the first for before, the second for after the leap day.

Easter Sunday is the first Sunday on or after the first ecclesiastic Full Moon after 21 March.

This method of computation has several subtleties. It is independent of the leap day scheme for the solar year. Basically the Gregorian calendar still uses the Julian calendar with a leap day every 4 years, so a Metonic cycle of 19 years has 6940 or 6939 days with 5 or 4 leap days. Now the lunar cycle counts only 19*354+19*11=6935 days. By NOT labeling and counting the leap day with an epact number, but have the next New Moon fall on the same calendar date as without the leap day, the current lunation gets extended by a day, and the 235 lunations cover as many days as the 19 years.

The label "25" (as distinct from "xxv") is used as follows. Within a Metonic cycle, years that are 11 years apart have epacts that differ by 1 day. Now short months have the labels xxiv and xxv at the same date, so the New (and Full) Moons would fall on the same dates for these two years within the cycle. This is not actually possible. To avoid this, in years with a Golden Number larger than 11, the reckoned New Moon will fall on the date with the label "25" rather than "xxv"; in long months they are the same, in short ones this is the date which also has the label "xxvi". This does not move the problem to the pair "25" and "xxvi" because that would happen only in year 22 of the cycle, which lasts only 19 years and there is a saltus lunae in between.

The reason for moving around the epact label "xxv/25" rather than any other seems to be the following. In the Julian calendar the latest date of Easter was 25 April, and the Gregorian reform maintained that limit. Now a Full Moon on the equinox date, 21 March, implies a New Moon on 7 March, which has epact label xxiv. So years with epacts of xxiv would have their paschal New Moon on 6 April, and the Full Moon on 20 April: but then Easter could be as late as 26 April. So the paschal Full Moon must fall no later than 19 April, and the New Moon on 5 April, which has epact label xxv. So the short month must have two labels on 5 April, which happen to be xxiv and xxv. Then xxv has to be treated differently, as explained in the paragraph above.

As a consequence, 19 April is the date on which Easter falls most frequently in the Gregorian calendar: in about 3.87% of the years. 22 March is the least frequent, with 0.48%.

The Gregorian calendar has a correction to the solar year by dropping 3 leap days in 400 years (always in a century year). This is a correction to the length of the solar year, but should have no effect on the Metonic relation between years and lunations. Therefore the epact should be compensated for this by subtracting 1 in these century years. This is the so-called solar equation.

However, 19 years are a little shorter than 235 lunations. The difference accumulates to 1 day in about 310 years. Therefore in the Gregorian calendar, the epact gets corrected by adding 1 day 8 times in 2500 years, always in a century year: this is the so-called lunar equation.

The effect is that the Gregorian lunar calendar uses an epact table that is valid for a period of from 100 to 300 years. The epact table listed above is valid in the period 1900 to 2199.

Julian Calendar

The method for computing the date of the ecclesiastic Full Moon that was standard in Western Europe before the Gregorian calendar reform, made use of an uncorrected repetition of the 19-year Metonic cycle in combination with the Julian calendar. In terms of the method of the epacts discussed above, it effectively used a single epact table starting with an epact of * (0), which was never corrected. In this case, the epact was the age of the Moon on 22 March, the date after the vernal equinox. This repeats every 19 years, so there were only 19 possible dates for the ecclesiastic Full Moons after 21 March. The sequence number of a year in the 19-year cycle is called Golden Number, and it is given by:

GN = MOD(Y/19) + 1

so the remainder of the year count in the Christian Era divided by 19, plus 1.

This is the table:

Golden Number	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
Full Moon	5A	25M	13A	2A	22M	10A	30M	18A	7A	27M	15A	4A	24M	12A	1A	21M	9A	29M	17A

(M=March, A=April)

Easter Sunday is the first Sunday on or after these dates. So for a given date of ecclesiastic Full Moon, there are 7 possible Easter dates. The cycle of Sunday letters however does not repeat in 7 years: because of the interruptions of the leap day every 4 years, the full cycle in which weekdays recur in the calendar in the same way, is 4*7 = 28 years: the so-called Solar Cycle. So the Easter dates repeated in the same order after 4*7*19 = 532 years. This Paschal Cycle is called the Victorian Cycle, after Victorius of Aquitaine[?] who introduced it in 463 AD. It is also called the Dionysian cycle, after Dionysius Exiguus who prepared Easter tables based on this originally method that started in 532 AD. In fact Dionysius introduced the Christian era, that is the convention of counting years from the birth of Christ. Although it might appear that he put that event at the start of the 532-year cycle that preceded the one for which he made new tables, his tables covered only 95 years and he does not seem to have been aware of the full period. Venerable Bede (7th cy.) seems to have been the first who identified the Solar Cycle and explained the Paschal Cycle from the Metonic Cycle and the Solar Cycle.

Some links:

http://www.ortelius.de/kalender/east_en

http://www.davros.org/misc/easter

http://www.newadvent.org/cathen/05480b.htm

http://www.phys.uu.nl/~vgent/easter/eastercalculator.htm

http://hermes.ulaval.ca/~sitrau/calgreg/

http://www.bdl.fr/Granpub/Promenade/pages4/442

Gauss's algorithm

This algorithm for calculating the date of Easter Sunday has been first presented by the mathematician Carl Friedrich Gauss.

The number of the year is denoted by Y. In the following, mod denotes the remainder of integer division (e.g. 13 mod 5 = 3) Calculate first a, b and c:

 a = Y mod 19
 b = Y mod 4
 c = Y mod 7

Then calculate

 d = (19a + M) mod 30
 e = (2b + 4c + 6d + N) mod 7

For Julian calendar (used in eastern churches) M = 15 and N = 6, and for Gregorian calendar (used in western churches) M and N are from the following table:

   Years     M   N
 1583-1699  22   2
 1700-1799  23   3
 1800-1899  23   4
 1900-2099  24   5
 2100-2199  24   6
 2200-2299  25   0

If d+e < 10 then Easter is on (d+e+22)th of March, else on (d+e-9)th of April.

The following exceptions must be taken into account:

• If the date given by the formula is the 26th of April, Easter is on 19th of April.
• If the date given by the formula is the 25th of April, with d = 28, e = 6, and a > 10, Easter is on 18th of April.