Computus

Computus (Latin for computation) is calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age.

The canonical rule is that Easter Sunday is the first Sunday after the 14th day of the lunar month (the Full Moon) that falls on or after 21 March (the day of the vernal equinox; see also Ostara). 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.

Table of contents

1 History 2 Theory 3 Tabular methods 3.1 Gregorian Calendar 3.1.1 Details 3.2 Julian Calendar 3.3 External links 4 Algorithms 4.4 Gauss's algorithm 5 External links

History

Easter has been considered the most important Christian feast. Accordingly, the proper date of its celebration has been a cause of much controversy, at least as early as the meeting (ca. AD 154) of Anicetus, bishop of Rome and Polycarp, bishop of Smyrna. The problem is that the passion and resurrection of Jesus Christ occurred during the Jewish feast of Pesach, which they celebrate according to their lunisolar calendar. The Christians in late-roman times had to redefine this for the Julian calendar that was used at that time, which is a solar calendar.

At the First Council of Nicaea in AD 325, it was agreed that the Christians should use a common method to establish the date, independent from the Jews. Also they decided to celebrate it always on the dies Domini, Sunday, which was the day of the week on which Jesus was resurrected, and which has been the Christian holy day of the week for this reason (the Quartodecimans wished to follow the Jews and always celebrate it on the 14th day of the Moon, whatever day of the week that might be). However, they made not many decisions that were of practical use as guidelines for the computation, and it took several centuries before a common method was accepted throughout Christianity.

The method from Alexandria became authoritative. It was based on the epacts of a reckoned Moon according to the 19-year cycle. This was first used by bishop Anatolius of Laodicea (in present-day Turkey) ca. AD 277. The Alexandrians may have derived their method from a similar calendar, based on the Egyptian civil solar calendar, and used at that time by the influential Jewish community there; it may have survived in the Ethiopian computus. In Constantinople several computists were active over the centuries after Anatolius (and after the Nicaean Council), but their Easter dates coincided with those of the Alexandrians. A variation of the Alexandrian method, based on some interpretation of the edicts of the Council of Nicaea, was used in Rome by Dionysius Exiguus (who translated the edicts into Latin). Dionysius introduced the Christian Era (counting years from the birth of Christ) when he published new Easter tables in 525. This replaced a method based on a less accurate 84-year cycle introduced in Rome by one Augustalis in the 3rd century, and which was used in the British Isles in the Anglo-Saxon kingdom of Northumbria until the Synod of Whitby in 664, and by the Celtic Church as late as 768. Annianus of Alexandria around AD 410, and Victorius of Aquitania in Rome in AD 457 had devised a method which had a cycle of 532 years (28 × 19). This was later popularized by Bede. A simplified method, using Golden Numbers, was introduced by Abbo of Fleury in 988, and remained in use in Western Europe until the Gregorian calendar reform.

Theory

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 solar year is 11 days longer than the lunar year. Suppose a solar and lunar year start at the same day, with a crescent New Moon indicating the begin of a new lunar month on 1 January. At the start of the next solar year, already 11 days of the new lunar year have passed. After 2 years the difference has accumulated to 22: the start of the lunar months fall 11 days earlier into the solar calendar each year. These days in excess of the solar year over the lunar year are called epacts (Greek: epakta hèmerai). You have to add them to the day of the solar year to know the day in the lunar year. Whenever the epact reaches or exceeds 30, an extra (so-called embolismic or intercalary) month has to be inserted into the lunar calendar; then 30 has to be subtracted from the epact.

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 = 29 mod 30, not 0 mod 30. 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 extra 209 days fill 7 embolismic months, for a total of 19 × 12 + 7 = 235 lunations. The sequence number of the year in the 19-year cycle is called the Golden Number, and it is given by:

GN = MOD(Y/19) + 1

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

Tabular methods

Gregorian Calendar

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

First determine the epact for the year. The epact can have a value from "*" (=0 or 30) to 29 days. The first day of a lunar month is considered the day of the New Moon. The 14th day is considered the day of the Full Moon.

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	*	11	22	3	14	25	6	17
Paschal Full Moon	15M	3A	23M	11A	31M	18A	8A	28M	16A	5A	25M	13A	2A	22M	10A	30M	17A	7A	27M

(M=March, A=April)

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), down to "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). Treat the 13th period (last eleven days) as long though, and assign the labels "xxv" and "xxiv" to sequential dates (26 and 27 December respectively). Finally, in addition add the label "25" to the dates that have "xxv" in the 30-day periods; but in 29-day periods (which have "xxiv" together with "xxv") add the label "25" to the date with "xxvi". The distribution of the lengths of the months and the length of the epact cycles is such that each month starts and ends with the same epact label, except for February and for the epact labels xxv and 25 in July and August. This table is called the calendarium. If the epact for the year is for instance 27, then there is an ecclesiastic New Moon on every date in that year that has the epact label xxvii (27).

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). The Dominical Letter cycles backward one position every year. However, in leap years after 24 February the Sundays will fall on the previous letter of the cycle, so leap years have 2 Dominical Letters: the first for before, the second for after the leap day.

In practice for the purpose of calculating Easter, this need not be done for all 365 days of the year. For the epacts, you will find that March comes out exactly the same as January, so one need not calculate January or February. To also avoid the need to calculate the Dominical Letters for January and February, start with D for 1 March. You need the epacts only from 8 March to 5 April. This gives rise to the following table:

Label	March	DL	April	DL	|	Label	March	DL	April	DL
*	 1	D			|	xv	16	E	14	F
xxix	 2	E	 1	G	|	xiv	17	F	15	G
xxviii	 3	F	 2	A	|	xiii	18	G	16	A
xxvii	 4	G	 3	B	|	xii	19	A	17	B
xxvi	 5	A	 4	C	|	xi	20	B	18	C
25	 6	B	 4	C	|	x	21	C	19	D
xxv	 6	B	 5	D	|	ix	22	D	20	E
xxiv	 7	C	 5	D	|	viii	23	E	21	F
xxiii	 8	D	 6	E	|	vii	24	F	22	G
xxii	 9	E	 7	F	|	vi	25	G	23	A
xxi	10	F	 8	G	|	v	26	A	24	B
xx	11	G	 9	A	|	iv	27	B	25	C
xix	12	A	10	B	|	iii	28	C
xviii	13	B	11	C	|	ii	29	D
xvii	14	C	12	D	|	i	30	E
xvi	15	D	13	E	|	*	31	F

Example: 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 13 days later. From the above table this gives a New Moon on 4 March and 3 April and so a Full Moon on 17 March and 16 April.

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

In the example, this Paschal Full Moon is on 16 April. If the Dominical Letter is E, then Easter Sunday is on 20 April.

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 if the epacts 24 and 25 both occur within one Metonic cycle, then the New (and Full) Moons would fall on the same dates for these two years. This is not actually possible for the real Moon: the dates should repeat only after 19 years. 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 these 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: there is a saltus lunae in between that makes the New Moons fall on separate dates.

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 is compensated for this (partially - see epact) by subtracting 1 in these century years. This is the so-called solar equation.

However, 19 uncorrected Julian years are a little longer 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 , 8 times in 2500 (Gregorian) years, always in a century year: this is the so-called lunar equation. The first one was applied in 1800, and it will be applied every 300 years, except for an interval of 400 years between 3900 and 4300 which starts a new cycle.

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.

Details

This method of computation has several subtleties:

Every second lunar month has only 29 days, so one day must have two (of the 30) epact labels assigned to it. The reason for moving around the epact label "xxv/25" rather than any other seems to be the following. According to Dionysius (in his introductory letter to Petronius), the Nicene council on authority of Eusebius established that the first month of the ecclesiastic lunar year (the paschal month) should start from 8 March up to 5 April, and the 14th days fall from 21 March up to 18 April, so spanning a period of (only) 29 days. A New Moon on 7 March , which has epact label xxiv, has its 14th day (Full Moon) on 20 March, which is too early (before the equinox date). So years with an epact of xxiv would have their Paschal New Moon on 6 April, which is too late: the Full Moon would fall on 19 April, and Easter could be as late as 26 April. In the Julian calendar the latest date of Easter was 25 April, and the Gregorian reform maintained that limit. So the Paschal Full Moon must fall no later than 18 April, and the New Moon on 5 April, which has epact label xxv. So the short month must have its double epact labels on 5 April: xxiv and xxv. Then epact 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 relation between lunar and solar calendar dates is made 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. So the burden of synchronizing the calendar with the Moon (intermediate term accuracy) is shifted to the solar calendar, which may use any suitable intercalation scheme; all under the assumption that 19 solar years = 235 lunations (long term inaccuracy). A consequence is that the reckoned age of the Moon may be off by a day, and also that the lunations which contain the leap day may be 31 days long, which would never happen when the real Moon were followed (short term inaccuracies). This is the price for a regular fit to the solar calendar.

However, there is some protection of the lunar calendar against the errors of the solar calendar. The leap days are not inserted in an optimal way to keep the calendar synchronized to the solar year. The corrections to the leap day scheme are limited to century years, and add 2 nested intercalation cycles (100 and 400 years) around the 4-year cycle. Each cycle accumulates an error, and they add up to more than 2 days. So in the Gregorian calendar, the actual dates of the vernal equinox are scattered over a time window of about 53 hours around 20 March. This may be acceptable for a calendar period of a year, but is too much for a monthly period. By separating the "solar" from the "lunar equation", this jitter is not carried to the lunar calendar.

Besides the jitter in the solar calendar, there are also some flaws in the Gregorian lunar calendar. However, they do have no effect on the Paschal month and the date of Easter:

1. Lunations of 31 (and sometimes 28) days occur.
2. If a year with Golden Number 19 happens to have epact 19, then the last ecclesiastic New Moon falls on 2 December; the next would be due on 1 January. However at the start of the new year there is a saltus lunae which increases the epact by another unit, and the New Moon should have occurred on the previous day. So a New Moon is missed. The calendarium of the Missale Romanum takes account of this by assigning epact label "19" instead of "20" to 31 December of such a year. It happened every 19 years when the original Gregorian epact table was in effect, for the last time in AD 1690, and will not happen again until AD 8511.
3. If the epact of a year is "20", then there will be an ecclesiastic New Moon on 31 December. If that year falls before a century year, then in most cases there will be a "solar equation" correction which reduces the epact for the new year by 1: the resulting epact "*" means that another ecclesiastic New Moon is counted on 1 January; so formally a lunation of 1 day has passed. This will happen around the beginning of AD 4200.
4. Other borderline cases occur (much) later, and if the rules are followed strictly and these cases are not specially treated, they will generate successive New Moon dates that are 1, 28, 59, or (very rarely) 58 days apart.

A careful analysis shows that through the way they are used and corrected in the Gregorian calendar, the epacts are actually fractions of a lunation (1/30, also known as tithi) and not full days. See epact for a discussion.

The solar and lunar equations repeat after 4 × 25 = 100 centuries. In that period, the epact has changed by a total of -1 × (3/4) × 100 + 1 × (8/25) × 100 = -43 = 17 mod 30. This is prime to the 30 possible epacts, so it takes 100 × 30 = 3000 centuries before the epacts repeat; and 3000 × 19 = 57 000 centuries before the epacts repeat at the same Golden Number. This period has (5 700 000/19) × 235 + (-43/30) × (57 000/100) = 70 499 183 lunations. So the Gregorian Easter dates repeat in exactly the same order only after 5 700 000 years = 70 499 183 lunations = 2 081 882 250 days. However the calendar will have to be adjusted already after some millennia because of changes in the length of the vernal equinox year, the synodic month, and the day.

Julian Calendar

The method for computing the date of the ecclesiastic Full Moon that was standard for the Latin (catholic) church 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 counted on 22 March, the earliest acceptable date for Easter. 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. This was introduced by Abbo of Fleury in 988.

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 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 also called the Victorian Cycle, after Victorius of Aquitania who introduced it in Rome in AD 457. It is first known to have been used by Annianus of Alexandria ca. AD 410. It has also sometimes erroneously been called the Dionysian cycle, after Dionysius Exiguus who prepared Easter tables that started in AD 532; but he does not mention Victorius and appears to have used a modified method originally from Theophilus and St. Cyrillus from Alexandria. Venerable Bede (7th century) seems to have been the first who identified the Solar Cycle and explained the Paschal Cycle from the Metonic Cycle and the Solar Cycle.

External links

• The entry on epacts in the Catholic Encyclopedia of 1911: http://www.newadvent.org/cathen/05480b.htm
• An Easter calculator by Rob van Gent (with an extensive bibliography, and with useful links): http://www.phys.uu.nl/~vgent/easter/eastercalculator.htm
• A calendar page and calculator by Holger Oertel: http://www.ortelius.de/kalender/east_en.php
• A page from Clive Davros with a brief explanation, some more tables, and another algorithm: http://www.davros.org/misc/easter.html
• The original texts of the Gregorian calendar reform (in Latin), with translations into French by Rodolphe Audette: http://hermes.ulaval.ca/~sitrau/calgreg/
• An extensive calendar site (auf Deutsch) and calendar and Easter calculator by Nikolaus A. Bär: http://www.nabkal.de/gregkal.html
• An explanation (en Français), more tables, and an Easter calculator, from the Annuaire du Bureau des Longitudes: http://www.bdl.fr/Granpub/Promenade/pages4/442.html
• Gregorian Lunar Calendar: A table of the Gregorian New Moons for 1900-2199. http://home.telepath.com/~hrothgar/lunar_almanac.html. The same site also contains a table of Gregroian New Moons for the years 1700-1899: http://home.telepath.com/~hrothgar/lunar_almanac_1700f.html, a similar table for the years 2200-2299: http://home.telepath.com/~hrothgar/lunar_almanac_2200f.html, a calendar dividing the Gregorian lunar year 2003 divided into 30-day and 29-day lunar months: http://home.telepath.com/~hrothgar/ly2003.html, and a discussion of theNicene Council's Easter decision and of how some commentators exaggerate the scope of the Council's decision. http://home.telepath.com/~hrothgar/ttarchive2003.html#20030427

Algorithms

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.

External links

• Improved algorithm by Dr. Heiner Lichtenberg (in German)