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Computus

From Academic Kids

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 21 March (nominally the day of the vernal equinox). For determining the feast, Christian churches settled on a method to define a reckoned "ecclesiastic" Moon, rather than observations of the true Moon like the Jews did.

Contents

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. The problem for Christians using the solar Julian 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. 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). 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. It is not known when the Church of Rome adopted Dionysius' tables, but it may have been as early as the sixth century. His tables were adopted in Britain by the Synod of Whitby in 664 and 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.

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. 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.

Theory

The solar year is always counted to have 365 days (excluding a small remainder). A lunar year of 12 months is counted to have 354 days, so the average lunation is 29 days (excluding another small remainder). So 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, 11 days of the new lunar year will have already passed by the commencement 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 hmerai). 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.

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

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:

Year199519961997199819992000200120022003 2004200520062007200820092010201120122013
Golden Number123456789 10111213141516171819
Epact2910212132451627 819*112231425617
Paschal Full Moon14A3A23M11A31M18A8A28M16A 5A25M13A2A22M10A30M17A7A27M

(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:

LabelMarchDLAprilDL
*  1D  
xxix  2E 1G
xxviii 3F 2A
xxvii  4G 3B
xxvi  5A 4C
25  6B 4C
xxv  6B 5D
xxiv  7C 5D
xxiii  8D 6E
xxii  9E 7F
xxi 10F 8G
xx 11G 9A
xix 12A10B
xviii 13B11C
xvii 14C12D
xvi 15D13E
xv 16E14F
xiv 17F15G
xiii 18G16A
xii 19A17B
xi 20B18C
x 21C19D
ix 22D20E
viii 23E21F
vii 24F22G
vi 25G23A
v 26A24B
iv 27B25C
iii 28C  
ii 29D  
i 30E  
* 31F  

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 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 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%.

Missing image
Easter_Distribution.png
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 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 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 Number123456789 10111213141516171819
Full Moon5A25M13A2A22M10A30M18A7A 27M15A4A24M12A1A21M9A29M17A

(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 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 who identified the Solar Cycle and explained the Paschal Cycle from the Metonic Cycle and the Solar Cycle.

Algorithms

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. In the following, 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 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, and is otherwise 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.

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). The latter 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 values are integers, thus 7 / 3 = 2 (not 2 1/3), and 7 mod 3 = 1.

Worked Example
Year(Y) = 1961
Worked Example
Year(Y) = 2000
a = Y mod 19 1961 mod 19 = 4 2000 mod 19 = 5
b = Y / 100 1961 / 100 = 19 2000 / 100 = 20
c = Y mod 100 1961 mod 100 = 61 2000 mod 100 = 0
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 * 5 + 20 - 5 - 6 + 15) mod 30 = 29
i = c / 4 61 / 4 = 15 0 / 4 = 0
k = c mod 4 61 mod 4 = 1 0 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 * 0 - 29 - 0) mod 7 = 3
m = (a + 11 * h + 22 * L) / 451 (4 + 11 * 10 + 22 * 1) / 451 = 0 (5 + 11 * 29 + 22 * 3) / 451 = 0
month = (h + L - 7 * m + 114) / 31 (10 + 1 - 7 * 0 + 114) / 31 = 4 (April) (29 + 3 - 7 * 0 + 114) / 31 = 4 (April)
day = ((h + L - 7 * m + 114) mod 31) + 1 (10 + 1 - 7 * 0 + 114) mod 31 + 1 = 2 (29 + 3 - 7 * 0 + 114) mod 31 + 1 = 23
2 April 1961 23 April 2000

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

External links

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