Sunday, March 23, 2008

How to Calculate Easter, In Case You Needed to Know (Shamelessly plagiarized from Wikipedia)

Personally, I am still waiting on Mashiach. Rabbi Sneerson, as was the Sabbatai_Zevi, not to mention Mohammed, were all great personal dissapointments.


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 as the Jews did at the time. Eastern Orthodox Christians calculate the fixed date of 21 March according to the Julian Calendar rather than the modern Gregorian Calendar, and observe the additional rule that Easter may not precede or coincide with the first day of the Jewish Passover.

Easter controversy
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. The problem for Christians using the Roman civil Julian calendar, which is a solar calendar, was that the passion and resurrection of Jesus occurred during the Jewish feast of Passover, which Jews celebrate according to the Hebrew lunisolar calendar, and fixing the date by the Roman calendar would lead to the celebration of Easter at times unrelated to the Jewish observance of Passover.

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.[1] It was also decided to celebrate it always on the dies Domini, Sunday, the day of the week on which Jesus was resurrected, 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 Jewish month of Nisan, whatever day of the week that might be).[2] 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 method from Alexandria became authoritative. It was based on the epacts of a reckoned moon according to the 19-year cycle. Such a cycle was first used by Bishop Anatolius of Laodicea (in present-day Syria), c. 277. The Alexandrians may have derived their method from a similar calendar, based on the Egyptian civil solar calendar, used by the Jewish community there; it survives in the Ethiopian computus. 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.[3][4]

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 third century. These old tables were used in the British Isles 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 fourth 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.[5] These Victorian tables were used in Gaul (now France) and Spain until they were displaced by Dionysian tables at the end of the eighth 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 British Synod of Whitby in 664 adopted the Dionysian tables. The Dionysian reckoning was fully described by Bede in 725.[6] 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.

The solar year is reckoned to always have 365 days (excluding a small remainder). A lunar year of 12 months is reckoned to have 354 days, meaning the average lunation is 29½ days (excluding another small remainder). 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 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: they are a device to match the calendar year to the tropical year, and can be ignored when dealing with the relation between years and lunations. 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. 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.[7]

Tabular methods

Gregorian calendar
This method for the computation of the date of Easter was introduced with the Gregorian calendar reform in 1582.[8]

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 Metonic cycle are:

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

A table from Sweden to compute the date of Easter 1140-1671 according to the Julian calendar. Notice the runic writing.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.

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 equation.

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 one eight 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 solar and lunar equations work in opposite directions, and in some century years (for example, 1800 and 2100) they cancel each other. However, it is a bad idea to combine them and make more evenly spread and less frequent epact corrections, as will be explained below. The result of the correct procedure 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.

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 (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. 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%.

Distribution of the date of Easter for the complete 5,700,000 year cycle.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, 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 two nested intercalation cycles (100 and 400 years) around the four-year cycle. Each cycle accumulates an error, and they add up to more than two 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 equation" from the "lunar equation", this jitter is not carried to the lunar calendar. If we were to combine the solar and lunar equations and spread the net 4×8 - 3×25= 43 epact subtractions in 10,000 years evenly, then the solar jitter would also affect the lunar calendar, which would be unsatisfactory.

Besides the jitter in the solar calendar, there are also some flaws in the Gregorian lunar calendar (also see D.Roegel 2004 [1]). However, they have no effect on the Paschal month and the date of Easter:

Lunations of 31 (and sometimes 28) days occur.
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.
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 equation" 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.
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 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.

Julian calendar
The method for computing the date of the ecclesiastical full moon that was standard for the Roman (Catholic) church before the Gregorian calendar reform, and is still used today by 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 were only 19 possible dates for the ecclesiastical full moons after 21 March.

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:

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 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 fifth 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.


Gauss's algorithm
This algorithm for calculating the date of Easter Sunday was first presented by the mathematician Carl Friedrich Gauss:

The number of the year is denoted by Y; mod denotes the remainder of integer division (e.g., 13 mod 5 ≡ 3; see modular arithmetic). 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 the Julian calendar (used in Eastern churches), M = 15 and N = 6, and for the 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 the (d + e + 22)th of March, and is otherwise on the (d + e − 9)th of April.

The following exceptions must be taken into account:

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

Meeus/Jones/Butcher Gregorian algorithm
This algorithm for calculating the date of Easter Sunday is given by Jean Meeus in his book Astronomical Algorithms (1991), which in turn cites Spencer Jones in his book General Astronomy (1922) and also the Journal of the British Astronomical Association (1977). This algorithm also appears in The Old Farmer's Almanac (1977), p. 69. The JBAA cites Butcher's Ecclesiastical Calendar (1876).

The method is valid for all Gregorian years and has no exceptions and requires no tables.

Notation is as for the Gauss Algorithm above: all quotients are truncated to integers, thus 7 / 3 = floor(7 / 3) = 2 (not 2 1/3), and 7 mod 3 = 1.

Worked example
Year(Y) = 1961 Worked example
Year(Y) = 2008
a = Y mod 19 1961 mod 19 = 4 2008 mod 19 = 13
b = Y / 100 1961 / 100 = 19 2008 / 100 = 20
c = Y mod 100 1961 mod 100 = 61 2008 mod 100 = 8
d = b / 4 19 / 4 = 4 20 / 4 = 5
e = b mod 4 19 mod 4 = 3 20 mod 4 = 0
f = (b + 8) / 25 (19 + 8) / 25 = 1 (20 + 8) / 25 = 1
g = (b - f + 1) / 3 (19 - 1 + 1) / 3 = 6 (20 - 1 + 1) / 3 = 6
h = (19 × a + b - d - g + 15) mod 30 (19 × 4 + 19 - 4 - 6 + 15) mod 30 = 10 (19 × 13 + 20 - 5 - 6 + 15) mod 30 = 1
i = c / 4 61 / 4 = 15 8 / 4 = 2
k = c mod 4 61 mod 4 = 1 8 mod 4 = 0
L = (32 + 2 × e + 2 × i - h - k) mod 7 (32 + 2 × 3 + 2 × 15 - 10 - 1) mod 7 = 1 (32 + 2 × 0 + 2 × 2 - 1 - 0) mod 7 = 0
m = (a + 11 × h + 22 × L) / 451 (4 + 11 × 10 + 22 × 1) / 451 = 0 (13 + 11 × 1 + 22 × 0) / 451 = 0
month = (h + L - 7 × m + 114) / 31 (10 + 1 - 7 × 0 + 114) / 31 = 4 (April) (1 + 0 - 7 × 0 + 114) / 31 = 3 (March)
day = ((h + L - 7 × m + 114) mod 31) + 1 (10 + 1 - 7 × 0 + 114) mod 31 + 1 = 2 (1 + 0 - 7 × 0 + 114) mod 31 + 1 = 23
2 April 1961 23 March 2008

Meeus Julian algorithm
Jean Meeus, in his book Astronomical Algorithms (1991), also presents the following formula for calculating Easter Sunday in Julian years.

The method is valid for all Julian years and has no exceptions and requires no tables.

Notation is as for the Gauss Algorithm above: all values are integers, thus 7 / 3 = 2 (not 2 1/3), and 7 mod 3 = 1.

a = Y mod 4
b = Y mod 7
c = Y mod 19
d = (19 × c + 15) mod 30
e = (2 × a + 4 × b - d + 34) mod 7
month = (d + e + 114) / 31
day = ((d + e + 114) mod 31) + 1

No comments:

Post a Comment

Comments more than welcome!