Computus
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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 Sunday is the first Sunday after the 14th day of the lunar month (the nominal full moon) that falls on or after March 21 (nominally the day of the vernal equinox). For determining the feast, Christian churches settled on a method to define a reckoned "ecclesiastic" full moon, rather than observations of the true Moon as the Jews did.
HistoryEdit
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.
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] 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 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 .
Dionysius' 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. 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.
TheoryEdit
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 11 days of the new lunar year will have already passed at the start of the new solar year. 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, 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 = Y mod 19 + 1
i.e. the remainder of the year number Y in the Christian era when divided by 19, plus 1.^{[1]}
Tabular methodsEdit
Gregorian calendarEdit
This method for the computation of the date of Easter was introduced with the Gregorian calendar reform in 1582.^{[2]}
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 date | 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 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 |
---|---|---|---|---|
* | 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 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 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 however: 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 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 opposite, and in some century years (e.g. 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 in the period 1900 to 2199.
DetailsEdit
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 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.
Week tableEdit
Date | 01 | 02 | 03 | 04 | 05 | 06 | 07 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
08 | 09 | 10 | 11 | 12 | 13 | 14 | ||||||||||
15 | 16 | 17 | 18 | 19 | 20 | 21 | ||||||||||
22 | 23 | 24 | 25 | 26 | 27 | 28 | ||||||||||
Month | 29 | 30 | 31 | Year modulo 28 | Century mod 4 | Century mod 7 | ||||||||||
Apr Jul | Sun | Mon | Tue | Wed | Thu | Fri | Sat | 01 | 07 | 12 | 18 | 24 | 1600 2000 | 0 | 0500 1200 | 5 |
Sep Dec | Sat | Sun | Mon | Tue | Wed | Thu | Fri | 02 | 08 | 13 | 19 | 24 | 0600 1300 | 6 | ||
June | Fri | Sat | Sun | Mon | Tue | Wed | Thu | 03 | 08 | 14 | 20 | 25 | 1700 2100 | 1 | 0700 1400 | 0 |
Feb Mar Nov | Thu | Fri | Sat | Sun | Mon | Tue | Wed | 04 | 09 | 15 | 20 | 26 | 0800 1500 | 1 | ||
August | Wed | Thu | Fri | Sat | Sun | Mon | Tue | 04 | 10 | 16 | 21 | 27 | 1800 2200 | 2 | 0900 0200 | 2 |
May | Tue | Wed | Thu | Fri | Sat | Sun | Mon | 05 | 11 | 16 | 22 | 00 | 1000 0300 | 3 | ||
Jan Oct | Mon | Tue | Wed | Thu | Fri | Sat | Sun | 06 | 12 | 17 | 23 | 00 | 1900 2300 | 3 | 1100 0400 | 4 |
For determination of the day of the week (1 January 2000, Saturday)
- the day of the month: 1 ~ 31 (1)
- the month: 1 for January ~ 12 for December (1，Mon)
- the year: 00 ~ 99 mod 28 and italic for January or February in leap years (00 ~ Mon)
- the century mod 4 for the Gregorian calendar and mod 7 for the Julian calendar (20 or 0 ~ Sat).
For determination of the dominical letter of a year (2100 C ~ 2199 F)
- the century column: from the century row to Sun which is in the column and in the row (21 or 1)
- the dominical letter: Mon for A ~ Sun for G from the year row to the century column (00 ~ Wed for C, 15 for 99 ~ Sat for F).
AlgorithmsEdit
Note on operationsEdit
When expressing Easter algorithms without using tables, it has been customary to employ only the integer 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, as well as look-up tables, 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.
Gauss algorithmEdit
In 1800, the mathematician Carl Friedrich Gauss presented this algorithm for calculating the date of the Julian or Gregorian Easter and made corrections to one of the steps in 1816. 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 Peter Paul Tittel for pointing out that p was wrong in 1800.
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) |
Anonymous Gregorian algorithmEdit
"A New York correspondent" submitted this algorithm for determining the Gregorian Easter to the journal Nature in 1876. It has been reprinted many times, in 1877 by Samuel Butcher in The Ecclesiastical Calendar, in 1922 by H. Spencer Jones in General Astronomy, in 1977 by the Journal of the British Astronomical Association, 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. Because of the Meeus’ book citation, that is also called "Meeus/Jones/Butcher" algorithm:
Expression | Y = 1961 | Y = 2017 |
---|---|---|
a = Y mod 19 | a = 4 | a = 3 |
b = floor (Y / 100) | b = 19 | b = 20 |
c = Y mod 100 | c = 61 | c = 17 |
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 = 21 |
i = floor (c / 4) | i = 15 | i = 4 |
k = c mod 4 | k = 1 | k = 1 |
L = (32 + 2e + 2i − h − k) mod 7 | L = 1 | L = 4 |
m = floor ((a + 11h + 22L) / 451) | m = 0 | m = 0 |
month = floor ((h + L − 7m + 114) / 31) | month = 4 (April) | month = 4 (March) |
day = ((h + L − 7m + 114) mod 31) + 1 | day = 2 | day = 16 |
Gregorian Easter | 2 April 1961 | 16 March 2017 |
Meeus Julian algorithmEdit
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. To obtain the Eastern Orthodox Easter normally given in the Gregorian calendar, 13 days must be added to these Julian Easter dates between 1900 and 2099 inclusive as shown.
I found that this algorithm was not working for the year 2016. Can you explain it in this page?
R Sivaraman Founder Pie Mathematics Association
Expression | Y = 2012 | Y = 2013 | Y = 2014 | Y = 2015 |
---|---|---|---|---|
a = Y mod 4 | a = 0 | a = 1 | a = 2 | a = 3 |
b = Y mod 7 | b = 3 | b = 4 | b = 5 | b = 6 |
c = Y mod 19 | c = 17 | c = 18 | c = 0 | c = 1 |
d = (19c + 15) mod 30 | d = 8 | d = 17 | d = 15 | d = 4 |
e = (2a + 4b − d + 34) mod 7 | e = 3 | e = 0 | e = 1 | e = 4 |
month = floor ((d + e + 114) / 31) | 4 (April) | 4 (April) | 4 (April) | 3 (March) |
day = ((d + e + 114) mod 31) + 1 | 2 | 8 | 7 | 30 |
Easter Day (Julian calendar) | 2 April 2012 | 8 April 2013 | 7 April 2014 | 30 March 2015 |
Easter Day (Gregorian calendar) | 15 April 2012 | 21 April 2013 | 20 April 2014 | 12 April 2015 |
Other algorithms Edit
Faster and more compact algorithms for Gregorian Easter Sunday exist.
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