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Computus (Latin for computation) is the 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 day is the first Sunday after the 14th day of the lunar month (the nominal full moon) that falls on or after 21 March (nominally the day of the vernal equinox). For determining the feast, Christian churches settled on a method to define a reckoned "ecclesiastical" full moon, rather than observations of the true Moon. Eastern Orthodox Christians calculate the fixed date of 21 March according to the Julian Calendar rather than the modern Gregorian Calendar, and use an ecclesiastical full moon that occurs 4 to 5 days later than the western ecclesiastical full moon.

In modern language, this definition is best described as: Easter Sunday is the Sunday following the Paschal Full Moon date. The Paschal Full Moon date is the Ecclesiastical Full Moon date following 20 March and, for the years AD 1900 to 2199, can be found in Tabular methods.

Contents 1 History 2 Theory 3 Tabular methods 3.1 Gregorian calendar 3.1.1 Details 3.1.2 British Calendar Act and Book of Common Prayer 3.2 Julian calendar 4 Algorithms 4.1 Note on Operations 4.2 Gauss algorithm 4.3 Anonymous Gregorian algorithm 4.4 Meeus Julian algorithm 5 See also 6 Notes 7 References 8 Further reading 9 External links

[edit] History

Easter is 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 (c. 154) of Anicetus, bishop of Rome, and Polycarp, bishop of Smyrna. According to Eusebius' Church History, quoting Polycrates of Ephesus^[1], churches in the Roman Province of Asia "always observed the day when the people put away the leaven", the 14th of the lunar month of Nisan. The rest of the Christian world at that time, according to Eusebius, held to "the view which still prevails," of always fixing Easter on Sunday. Eusebius does not say how the Sunday was decided. Other documents from the 3rd and 4th centuries, however, reveal that the customary practice was for Christians to consult their Jewish neighbors to determine when the week of Unleavened Bread would fall, and to set Easter on the Sunday that fell within that week.^[2][3]

By the end of the third century, however, some Christians had become dissatisfied with what they perceived as the disorderly state of the Jewish calendar. The chief complaint was that the Jewish practice sometimes set the 14th of Nisan before the spring equinox. This is implied by Dionysius, bishop of Alexandria in the mid-3rd century, who stated that "at no time other than the spring equinox is it legitimate to celebrate Easter" (Eusebius, Church History 7.20); and by Anatolius of Alexandria (quoted in Eusebius, Church History 7.32) who declared it a "great mistake" to set the Paschal lunar month when the sun is in the twelfth sign of the zodiac. And it was explicitly stated by Peter, bishop of Alexandria that "the men of the present day now celebrate [Passover] before the [spring] equinox...through negligence and error."^[4] Another objection to using the Jewish computation may have been that the Jewish calendar was not unified. Jews in one city might have a method for reckoning the Week of Unleavened Bread different from that used by the Jews of another city.^[5] Because of these perceived defects in the traditional practice, Christian computists began experimenting with systems for determining Easter that would be free of these defects. But these experiments themselves led to controversy, since some Christians held that the customary practice of holding Easter during the Jewish festival of Unleavened Bread should be continued, even if the Jewish computations were in error from the Christian point of view.^[6]

At the First Council of Nicaea in 325, it was agreed that the Christians should use a common method to establish the date, independent from the Jewish method.^[7] However, they made few 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 process of working out the details generated still further controversies.

The method from Alexandria became authoritative. In its developed form it was based on the epacts of a reckoned moon according to the 19-year cycle (a.k.a. the Metonic Cycle). Such a cycle was first proposed by Bishop Anatolius of Laodicea (in present-day Syria), c. 277.^[8] Alexandrian Easter tables were composed by Bishop Theophilus about 390 and within the bishopric of Cyril about 444. 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. Churches on the eastern frontier of the Byzantine Empire deviated from the Alexandrians during the sixth century, and now celebrate Easter on different dates from Eastern Orthodox churches four times every 532 years. The Alexandrian computus was converted from the Alexandrian calendar into the Julian calendar in Rome by Dionysius Exiguus, though only for 95 years. Dionysius introduced the Christian Era (counting years from the Incarnation of Christ) when he published new Easter tables in 525.^[9][10]

Dionysius's tables replaced earlier methods used by the Church of Rome. The earliest known Roman tables were devised in 222 by Hippolytus of Rome based on 8-year cycles. Then 84-year tables were introduced in Rome by Augustalis near the end of the 3rd century. These old tables were used in Northumbria until 664, and by isolated monasteries as late as 931. A modified 84-year cycle was adopted in Rome during the first half of the 4th century. Victorius of Aquitaine tried to adapt the Alexandrian method to Roman rules in 457 in the form of a 532-year table, but he introduced serious errors.^[11] These Victorian tables were used in Gaul (now France) and Spain until they were displaced by Dionysian tables at the end of the 8th century.

In the British Isles Dionysius's and Victorius's tables conflicted with older Roman tables based on an 84-year cycle. The Irish Synod of Mag Léne in 631 decided in favor of either the Dionysian or Victorian Easter and the northern English Synod of Whitby in 664 adopted the Dionysian tables. The Dionysian reckoning was fully described by Bede in 725.^[12] They may have been adopted by Charlemagne for the Frankish Church as early as 782 from Alcuin, a follower of Bede. The Dionysian/Bedan computus remained in use in Western Europe until the Gregorian calendar reform, which was mostly designed by Aloysius Lilius.

[edit] Theory

To each day in a calendar year, the Easter cycle implicitly assigns a lunar age, which is a whole number from 1 to 30. The moon's age starts at 1 and increases to 29 or 30, then starts over again at 1. Each period of 29 (or 30) days of the moon's age makes up a lunar month. With occasional exceptions, 30-day lunar months alternate with 29-day months. So a lunar year of 12 lunar months is reckoned to have 354 days. The solar year is 11 days longer than the lunar year. Supposing a solar and lunar year start on the same day, with a crescent new moon indicating the beginning of a new lunar month on 1 January, then the lunar year will finish first, and 11 days of the new lunar year will have already passed by the time the new solar year starts. After two years, the difference will have accumulated to 22: the start of lunar months fall 11 days earlier in the solar calendar each year. These days in excess of the solar year over the lunar year are called epacts (Greek: epakta hèmerai). It is necessary to add them to the day of the solar year to obtain the correct day in the lunar year. Whenever the epact reaches or exceeds 30, an extra (so-called embolismic or intercalary) month of 30 days has to be inserted into the lunar calendar; then 30 has to be subtracted from the epact.

Note that leap days are not counted in the schematic lunar calendar: The cycle assigns to the first day of March after the leap-day the same age of the moon that the day would have had if there had been no leap-day. The nineteen-year cycle (Metonic cycle) assumes that 19 tropical years are as long as 235 synodic months. So after 19 years the lunations should fall the same way in the solar years, and the epacts should repeat. However, 19 × 11 = 209 ≡ 29 (mod 30), not 0 (mod 30); that is, 209 divided by 30 leaves a remainder of 29 instead of being an even multiple of 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 or moon's leap. The extra 209 days fill seven 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 is given by the formula

GN = Y mod 19 + 1

That is, the remainder of the year number Y in the Christian era when divided by 19, plus one.^[13]

Using the method just described, a period of 19 calendar years is also divided into 19 lunar years of 12 or 13 lunar months each. In each calendar year (beginning on 1 January) one of the lunar months must be the first one within the calendar year to have its 14th day (its formal full moon) on or after 21 March. This lunar month is the Paschal or Easter-month, and Easter is the Sunday after its 14th day (or, saying the same thing, the Sunday within its third week.) The Paschal lunar month always begins on a date in the 29-day period from March 8 to April 5 inclusive. Its 14th day, therefore, always falls on a date between March 21 to April 18 inclusive, and the following Sunday then necessarily falls on a date in the range March 22 to April 25 inclusive. In the solar calendar Easter is called a moveable feast since its date varies within a 35-day range. But in the lunar calendar, Easter is always the 3rd Sunday in the Paschal lunar month, and is no more "moveable" than any holiday that is fixed to a particular day of the week and week within a month.

[edit] Tabular methods

[edit] Gregorian calendar

This method for the computation of the date of Easter was introduced with the Gregorian calendar reform in 1582.^[14]

The general method of working was given by Clavius in the Six Canons (1582), and a full explanation followed in his "Explicatio" (1603).

Easter Sunday is the Sunday following the Paschal Full Moon date. The Paschal Full Moon date is the Ecclesiastical Full Moon date following 20 March and can be found in this table:

Paschal Full Moon dates for the 300 years 1900 to 2199 AD (M=March A=April)

Remainder after dividing year by 19	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Paschal Full Moon date	14A	3A	23M	11A	31M	18A	8A	28M	16A	5A	25M	13A	2A	22M	10A	30M	17A	7A	27M

For example: 2038 AD divided by 19 gives a remainder of 5. PFM date is 18 April, a Sunday. Easter Sunday date is following Sunday, 25 April.

Historically, this method derives Paschal Full Moon dates by determining the epact for each 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 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[15]	29	10	21	2	13	24	5	16	27	8	19	*	11	22	3	14	25	6	17
Paschal Full Moon date[16]	14A	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 January 1, 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 (December 26 and 27, 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 civil calendar 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. The ecclesiastical new moons for any year are those dates at which the epact for the year is entered. If the epact for the year is for instance 27, then there is an ecclesiastical 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 February 24 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
*	1	D
xxix	2	E	1	G
xxviii	3	F	2	A
xxvii	4	G	3	B
xxvi	5	A	4	C
25	6	B	4	C
xxv	6	B	5	D
xxiv	7	C	5	D
xxiii	8	D	6	E
xxii	9	E	7	F
xxi	10	F	8	G
xx	11	G	9	A
xix	12	A	10	B
xviii	13	B	11	C
xvii	14	C	12	D
xvi	15	D	13	E
xv	16	E	14	F
xiv	17	F	15	G
xiii	18	G	16	A
xii	19	A	17	B
xi	20	B	18	C
x	21	C	19	D
ix	22	D	20	E
viii	23	E	21	F
vii	24	F	22	G
vi	25	G	23	A
v	26	A	24	B
iv	27	B	25	C
iii	28	C
ii	29	D
i	30	E
*	31	F

Example: If the epact is, for instance, 27 (Roman xxvii), then there will be an ecclesiastical new moon on every date that has the label "xxvii". The ecclesiastical full moon falls 13 days later. From the table above, this gives a new moon on 4 March and 3 April, and so a full moon on 17 March and 16 April.

Then Easter Day is the first Sunday after the first ecclesiastical full moon on or after 21 March. This definition uses “on or after 21 March” to avoid ambiguity with historic meaning of the word “after”. In modern language, this phrase simply means “after 20 March”. The definition of “on or after 21 March” is frequently incorrectly abbreviated to “after 21 March” in published and web-based articles, resulting in incorrect Easter dates.

In the example, this Paschal full moon is on 16 April. If the dominical letter is E, then Easter day 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 in the short months 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 that have epacts 25 and 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 three 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 correction or "solar equation" ("equation" being used in its medieval sense of "correction").

However, 19 uncorrected Julian years are a little longer than 235 lunations. The difference accumulates to one day in about 310 years. Therefore, in the Gregorian calendar, the epact gets corrected by adding 1 eight times in 2500 (Gregorian) years, always in a century year: this is the so-called lunar correction (historically 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 solar and lunar corrections work in opposite directions, and in some century years (for example, 1800 and 2100) they cancel each other. The result 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 for the period 1900 to 2199.

[edit] 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 the authority of Eusebius, established that the first month of the ecclesiastical lunar year (the Paschal month) should start between 8 March and 5 April inclusive, and the 14th day fall between 21 March and 18 April inclusive, thus 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 (not following 20 March). So years with an epact of xxiv, if the lunar month beginning on March 7 had 30 days, 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. The short month must therefore 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 four years, so a Metonic cycle of 19 years has 6940 or 6939 days with five or four 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 having the next new moon fall on the same calendar date as without the leap day, the current lunation gets extended by a day,^[17] 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.

From the perspective of those who might wish to use the Gregorian Easter cycle as a calendar for the entire year, there are some flaws in the Gregorian lunar calendar (also see D. Roegel (2004)^[18]). However, they 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 ecclesiastical 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 ecclesiastical new moon on 31 December. If that year falls before a century year, then in most cases there will be a solar correction which reduces the epact for the new year by one: the resulting epact "*" means that another ecclesiastical new moon is counted on 1 January; so formally a lunation of one 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 corrections 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 already have to have been adjusted after some millennia because of changes in the length of the vernal equinox year, the synodic month, and the day.

This raises the question why the Gregorian lunar calendar has separate solar and lunar corrections, which sometimes cancel each other; instead, the net 4×8 − 3×25 = 43 epact subtractions could be distributed evenly over 10,000 years (as has been proposed for example by Dr. Heiner Lichtenberg^[19]). Lilius' original work has not been preserved and Clavius does not explain this. The "solar corrections" approximately undo the effect of the Gregorian modifications to the leap days of the solar calendar, onto the lunar calendar: it (partially) brings the epact cycle back to the original Metonic relation between the Julian year and lunar month. The inherent mismatch between Sun and Moon in this basic 19-year cycle is then corrected every 3 or 4 centuries by the "lunar correction" to the epacts. However, the epact corrections occur at the beginning of Gregorian centuries, not Julian centuries, and therefore the original Julian Metonic cycle is not fully restored.

The drift in ecclesiastical full moons calculated by the Gregorian method compared to the true full moons is dominated by the gradual slowing of the Earth's rotation. Borkowski estimated that in the year 12,000 the Gregorian calendar would fall behind the tropical year by at least 8, but less than 12 days.^[20] The drift of full moons would be a similar amount.

[edit] British Calendar Act and Book of Common Prayer

The portion of the Tabular methods section above describes the historical arguments and methods by which the present dates of Easter Sunday were decided in the late 16th century by the Roman Catholic Church. In Britain, where the Julian Calendar was then still in use, Easter Sunday was defined, from 1662 to 1752 (in accordance with previous practice), by a simple Table of dates in the Anglican Prayer Book (decreed by the Act of Uniformity 1662). The Table was indexed directly by the Golden Number and the Sunday Letter, which (in the Easter section of the Book) were presumed to be already known.

For the British Empire and Colonies, the new determination of the Date of Easter Sunday was defined by the Calendar (New Style) Act 1750 with its Annexe. The method was chosen to give dates agreeing with the Gregorian Rule already in use elsewhere. It was required by the Act to be put in the Book of Common Prayer, and therefore it is the general Anglican Rule. The original Act can be seen in the British Statutes at Large 1765.^[21] The Annexe to the Act includes the definition: "Easter-day (on which the rest depend) is always the first Sunday after the Full Moon, which happens upon, or next after the Twenty-first Day of March. And if the Full Moon happens upon a Sunday, Easter-day is the Sunday after." The Annexe subsequently uses the terms "Paschal Full Moon" and "Ecclesiastical Full Moon", making it clear that they approximate to the real Full Moon.

The method is quite distinct from that described above in Gregorian calendar. For a general year, one first determines the Golden Number, then one uses three Tables to determine the Sunday Letter, a Cypher, and the date of the Paschal Full Moon, from which the date of Easter Sunday follows. A simpler Table can be used for limited periods (such as 1900-2199) during which the Cypher (which represents the effect of the Solar and Lunar Corrections) does not change. Clavius' details were employed in the construction of the method, but they play no subsequent part in its use.^[22][23]

J R Stockton shows his derivation of an efficient computer algorithm traceable to the Tables in the Prayer Book and the Calendar Act (assuming that a description of how to use the Tables is at hand), and verifies its processes by computing matching Tables.^[24]

[edit] Julian calendar

The method for computing the date of the ecclesiastical full moon that was standard for the Western Church before the Gregorian calendar reform, and is still used today by most Eastern Christians, 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 are only 19 possible dates for the Paschal Full Moon from 21 March to 18 April inclusive.

Because there are no corrections as there are for the Gregorian calendar, the ecclesiastical full moon drifts away from the true full moon by more than 3 days every millennium, and is already a few days later. As a result, the Eastern churches celebrate Easter one week later than the Western churches about 50% of the time. (The Eastern Easter is often 4 or 5 weeks later because the Julian 20 March is 13 days later than the Gregorian 20 March for years 1900 to 2099 AD.)

The sequence number of a year in the 19-year cycle is called its Golden Number. This term was first used in the computistic poem Massa Compoti by Alexander de Villa Dei in 1200. A later scribe added it to tables originally composed by Abbo of Fleury in 988.

This is the table of Paschal Full Moon dates for all Julian years from 326 AD:^[25]

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

(M=March, A=April)

Easter day is the first Sunday after these dates.

So for a given date of the ecclesiastical full moon, there are seven possible Easter dates. The cycle of Sunday letters, however, does not repeat in seven 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 Aquitaine, who introduced it in Rome in AD 457. It is first known to have been used by Annianus of Alexandria at the beginning of the 5th century. It has also sometimes erroneously been called the Dionysian cycle, after Dionysius Exiguus, who prepared Easter tables that started in AD 532; but he apparently did not realize that the Alexandrian computus which he described had a 532-year cycle, although he did realize that his 95-year table was not a true cycle. Venerable Bede (7th century) seems to have been the first to identify the solar cycle and explain the Paschal cycle from the Metonic cycle and the solar cycle.

In medieval western Europe, the dates of the Paschal Full Moon given above could be memorized with the help of a 19-line alliterative poem in Latin:^[26]

   Nonae Aprilis     norunt quinos
   octonae kalendae     assim depromunt.
   Idus Aprilis     etiam sexis,
   nonae quaternae     namque dipondio.
   Item undene     ambiunt quinos,
   quatuor idus     capiunt ternos.
   Ternas kalendas     titulant seni,
   quatuor dene     cubant in quadris.
   Septenas idus     septem eligunt,
   senae kalendae     sortiunt ternos,
   denis septenis     donant assim.
   Pridie nonas     porro quaternis,
   nonae kalendae     notantur septenis.
   Pridie idus     panditur quinis,
   kalendas Aprilis     exprimunt unus.
   Duodene namque     docte quaternis,
   speciem quintam     speramus duobus.
   Quaternae kalendae     quinque coniciunt,
   quindene constant     tribus adeptis.

The first half-line of each line gives the date of the Paschal Full Moon from the table above, for one year in the 19-year cycle. The second half-line gives the regular, or weekday displacement, of the day of the Paschal Full Moon from the weekday of March 24.

[edit] Algorithms

[edit] Note on Operations

When expressing Easter algorithms without using Tables, it has been customary to employ only the operations addition, subtraction, multiplication, division, modulo, and assignment (plus minus times div mod assign). That is compatible with the use of simple mechanical or electronic calculators. But it is an undesirable restriction for computer programming, where conditional operators and statements are always available. One can easily see how conversion from Day-of-March (22 to 56) to Day-and-Month (22 March to 25 April) can be done as
(if DoM>31) {Day=DoM-31, Month=Apr} else {Day=DoM, Month=Mar}.
More importantly, using such conditionals also simplifies the core of the Gregorian calculation.

[edit] Gauss algorithm

The mathematician Carl Friedrich Gauss presented this algorithm for calculating the date of the Julian or Gregorian Easter in 1800^[27][28] except for one step that he corrected in 1816.^[29] In 1800 he incorrectly stated p = floor (k/3). In 1807 he replaced the condition (11M + 11) mod 30 < 19 with the simpler a > 10. In 1811 he limited his algorithm to the 18th and 19th centuries only, and stated that 26 April is always replaced with 19 April and 25 April by 18 April. In 1816 he thanked his student P. Tittle for pointing out that p was wrong in 1800.^[30]

Expression	year = 1777
a = year mod 19	a = 10
b = year mod 4	b = 1
c = year mod 7	c = 6
k = floor (year/100)	k = 17
p = floor ((13 + 8k)/25)	p = 5
q = floor (k/4)	q = 4
M = (15 − p + k − q) mod 30	M = 23
N = (4 + k − q) mod 7	N = 3
d = (19a + M) mod 30	d = 3
e = (2b + 4c + 6d + N) mod 7	e = 5
Gregorian Easter is 22 + d + e March or d + e − 9 April	30 March
if d = 29 and e = 6, replace 26 April with 19 April
if d = 28, e = 6, and (11M + 11) mod 30 < 19, replace 25 April with 18 April
For the Julian Easter in the Julian calendar M = 15 and N = 6 (k, p, and q are unnecessary)

[edit] Anonymous Gregorian algorithm

"A New York correspondent" submitted this algorithm for determining the Gregorian Easter to the journal Nature in 1876.^[30][31] It has been reprinted many times, in 1877 by Samuel Butcher in The Ecclesiastical Calendar,^[32]:225 in 1922 by H. Spencer Jones in General Astronomy,^[33] in 1977 by the Journal of the British Astronomical Association,^[34] in 1977 by The Old Farmer's Almanac, in 1988 by Peter Duffett-Smith in Practical Astronomy With Your Calculator, and in 1991 by Jean Meeus in Astronomical Algorithms.^[35] The Gregorian Easter has been used since 1583 by the Roman Catholic Church and was adopted by most Protestant churches between 1753 and 1845. German Protestant states used an astronomical Easter based on the Rudolphine Tables of Johannes Kepler between 1700 and 1774, while Sweden used it from 1739 to 1844. This astronomical Easter was one week before the Gregorian Easter in 1724, 1744, 1778, 1798, etc.^[36][32]:153

Expression	Y = 1961	Y = 2009
a = Y mod 19	a = 4	a = 14
b = floor (Y / 100)	b = 19	b = 20
c = Y mod 100	c = 61	c = 9
d = floor (b / 4)	d = 4	d = 5
e = b mod 4	e = 3	e = 0
f = floor ((b + 8) / 25)	f = 1	f = 1
g = floor ((b − f + 1) / 3)	g = 6	g = 6
h = (19a + b − d − g + 15) mod 30	h = 10	h = 20
i = floor (c / 4)	i = 15	i = 2
k = c mod 4	k = 1	k = 1
L = (32 + 2e + 2i − h − k) mod 7	L = 1	L = 1
m = floor ((a + 11h + 22L) / 451)	m = 0	m = 0
month = floor ((h + L − 7m + 114) / 31)	month = 4 (April)	month = 4 (April)
day = ((h + L − 7m + 114) mod 31) + 1	day = 2	day = 12
Gregorian Easter	2 April 1961	12 April 2009

[edit] Meeus Julian algorithm

Jean Meeus, in his book Astronomical Algorithms (1991, p. 69), presents the following algorithm for calculating the Julian Easter in the Julian calendar. This is not the Gregorian Easter now used by Western churches. Before about 800 AD, other methods of calculating the Julian Easter existed. All churches place the vernal equinox on 21 March, in the Gregorian calendar for Western churches and in the Julian calendar for Eastern churches. To obtain the Eastern Orthodox Easter normally given in the Gregorian calendar, 13 days must be added between 1900 and 2099 inclusive. So the "vernal equinox" in the East is now on 3 April, and the Julian Easter can be on any date between 4 April and 8 May (1900–2099). Churches beyond the eastern frontier of the former Byzantine Empire use an Easter that differs four times every 532 years from this Easter.

Expression	Y = 2008	Y = 2009	Y = 2010
a = Y mod 4	a = 0	a = 1	a = 2
b = Y mod 7	b = 6	b = 0	b = 1
c = Y mod 19	c = 13	c = 14	c = 15
d = (19c + 15) mod 30	d = 22	d = 11	d = 0
e = (2a + 4b − d + 34) mod 7	e = 1	e = 4	e = 0
month = floor ((d + e + 114) / 31)	4 (April)	4 (April)	3 (March)
day = ((d + e + 114) mod 31) + 1	14	6	22
Easter Day (Julian calendar)	14 April 2008	6 April 2009	22 March 2010
Easter Day (Gregorian calendar)	27 April 2008	19 April 2009	4 April 2010

[edit] See also

• Easter
• Crucifixion eclipse
• Easter controversy
• Reform of the date of Easter
• Christian Zeller

[edit] Notes

[edit] References

• Blackburn, Bonnie, and Holford-Strevens, Leofranc. (2003). The Oxford Companion to the Year: An exploration of calendar customs and time-reckoning. (First published 1999, reprinted with corrections 2003.) Oxford: Oxford University Press.
• Borst, Arno (1993). The Ordering of Time: From the Ancient Computus to the Modern Computer Trans. by Andrew Winnard. Cambridge: Polity Press; Chicago: Univ. of Chicago Press.

• Clavius, Christopher (1603): Romani calendarij à Gregorio XIII. P. M. restituti explicatio. In the fifth volume of Opera Mathematica (1612). This [1] includes page images of the Six Canons and the Explicatio

• Constantine the Great, Emperor (325): Letter to the bishops who did not attend the first Nicaean Council; from Eusebius' Vita Constantini. English translations: [2] [3]
• Coyne, G. V., M. A. Hoskin, M. A., and Pedersen, O. (ed.) Gregorian reform of the calendar: Proceedings of the Vatican conference to commemorate its 400th anniversary, 1582-1982, (Vatican City: Pontifical Academy of Sciences, Specolo Vaticano, 1983).
• Dyonisius Exiguus (525): Liber de Paschate. On-line: (full Latin text) and (table with Argumenta in Latin, with English translation)
• Eusebius of Caesarea, The History of the Church, Translated by G. A. Williamson. Revised and edited with a new introduction by Andrew Louth. Penguin Books, London, 1989.
• Gibson, Margaret Dunlop, The Didascalia Apostolorum in Syriac, Cambridge University Press, London, 1903.
• Gregory XIII (Pope) and the calendar reform committee (1581): the Papal Bull Inter Gravissimas and the Six Canons. On-line under: "Les textes fondateurs du calendrier grégorien", with some parts of Clavius's Explicatio
• Schwartz, E., Christliche und jüdische Ostertafeln, (Abhandlungen der königlichen Gesellschaft der Wissenschaften zu Göttingen. Pilologisch-historische Klasse. Neue Folge, Band viii.) Weidmannsche Buchhandlung, Berlin, 1905.
• Stern, Sacha, Calendar and Community: A History of the Jewish Calendar Second Century BCE - Tenth Century CE, Oxford University Press, Oxford, 2001.
• Wallis, Faith., Bede: The Reckoning of Time, (Liverpool: Liverpool Univ. Pr., 1999), pp. lix-lxiii.
• Weisstein, Eric. (c. 2006) "Paschal Full Moon" in World of Astronomy. [4]

[edit] Further reading

• Mosshammer, Alden A. The Easter Computus and the Origins of the Christian Era. Oxford: Oxford University Press, 2008. ISBN 0-19-954312-7.

[edit] External links

• The Complete Works of Venerable Bede Vol. 6 (Contains De Temporibus and De Temporum Ratione.)
• The entry on epacts in the Catholic Encyclopedia of 1911
• The original texts of the Gregorian calendar reform (in Latin), with translations into French by Rodolphe Audette
• An Easter calculator by Rob van Gent (with an extensive bibliography, and with useful links)
• Ephemeris site of the Bureau des Longitudes with an Easter calculator
• A calendar page and calculator by Holger Oertel
• A page from Clive Feather with a brief explanation, some more tables, and another algorithm
• (German) An extensive calendar site (auf Deutsch) and calendar and Easter calculator by Nikolaus A. Bär
• Explanation of the Gregorian solar and lunar calendar, with improved procedures over the tabular method, by David Madore
• Gregorian Lunar Calendar: A table of the Gregorian New Moons for 1900-2199
  • A table of Gregorian New Moons for the years 1700-1899, from the same source
  • A similar table for the years 2200-2299
  • A calendar dividing the Gregorian lunar year 2003 into 30-day and 29-day lunar months
  • A discussion of the Nicene Council's Easter decision and of how some commentators exaggerate the scope of the Council's decision
• Improved algorithm by Dr. Heiner Lichtenberg (in German) (in English)
• Dionysius Exiguus' Easter table
• Mnemonic Computus Diagrams of Hands from manuscript in The British Library
• St. Gallen, Stiftsbibliothek, Codex Sangallensis 378 (11th century) p. 28. Contains the poem Nonae Aprilis norunt quinos.
• Towards a Common Date for Easter World Council of Churches (Faith and Order) and Middle East Council of Churches consultation; Aleppo, Syria; March 5 - 10, 1997
• A simple method for determining the date of Easter for all years 326 to 4099 A.D. by Ronald W. Mallen
• Text of the Calendar (New Style) Act 1750, British Act of Parliament introducing the Gregorian Calendar as amended to date. Contains tables for calculating Easter up until the year 8599. Contrast with the Act as passed.

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