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Computus

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 Church|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 1900 to 2199 AD, can be found in #Tabular methods|Tabular methods.

History

Easter is the most important Christian feast. Accordingly, the proper date of its celebration has been a cause of much Easter controversy|controversy, at least as early as the meeting (c. 154) of Pope Anicetus|Anicetus, bishop of Rome, and Polycarp, bishop of Smyrna. According to Eusebius (''Church History'' 5.23), churches in the Roman Province of Asia had a custom of beginning the Easter festival on "the day when the people Jews 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.E. Schwartz,'' Christliche und jüdische Ostertafeln'', Berlin, 1905, p 104ff.Margaret Dunlop Gibson, ''The Didascalia Apostolorum in Syriac'', Cambridge University Press, London, 1903, p. 100. 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 is legitmate 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."Peter of Alexandria, quoted in the preface to the ''Chronicon Paschale'', Migne, ''PG'' 18, 512 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.Sacha Stern,'' Calendar and Community: A History of the Jewish Calendar Second Century BCE-Tenth Century CE'', Oxford University Press, 2001, pp. 72-79. 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.Epiphanius, Adversus Haereses 3.1.10, quotes a version of the ''Apostolic Constitutions'' used by the sect of the Audiani, which advises Christians not to do their own calculation, but to use the Jewish computation even if the Jewish computation is in error. 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.See "the letter from emperor Constantine to the absent bishops"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 Metonic cycle|19-year cycle (a.k.a. the Metonic Cycle). Such a cycle was first used by Bishop Anatolius of Laodicea (in present-day Syria), c. 277. Alexandrian Easter tables were composed by Bishop Theophilus of Alexandria|Theophilus about 390 and within the bishopric of Cyril of Alexandria|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 Coptic calendar|Alexandrian calendar into the Julian calendar in Rome by Dionysius Exiguus, though only for 95 years. Dionysius introduced the Anno Domini|Christian Era (counting years from the Incarnation of Christ) when he published new Easter tables in 525.See "''Liber de Paschate''"For confirmation of Dionysius's role see Blackburn & Holford-Strevens p. 794. 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 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 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.Blackburn & Holford-Strevens p. 793. 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 British Synod of Whitby in 664 adopted the Dionysian tables. The Dionysian reckoning was fully described by Bede in 725.Faith Wallis, ''Bede: The Reckoning of Time'', (Liverpool: Liverpool Univ. Pr., 1999), pp. lix-lxiii. 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.

Theory

The solar year is reckoned to always have 365 days (excluding a small remainder). To each day in the solar 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. Ordinarily 30-day lunar months alternate with 29-day months (exceptions will be noted later). 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|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 (modulo operation|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 numbers|Golden Number", and is given by the formula
- ''GN'' = ''Y'' mod 19 + 1 That is, the remainder of the year number ''Y'' in the Common era|Christian era when divided by 19, plus one."the Number of a year AD is found by adding one, dividing by 19, and taking the remainder (treating 0 as 19)." Blackburn & Holford-Strevens p. 810. 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 third week ''entirely after'' 21 March. Or, saying the same thing, one lunar month must be the first 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''.) In the solar calendar, Easter is a so-called moveable feast, which varies in its date from 22 March to 25 April. But in the lunar calendar, Easter is always the 3rd Sunday in the Paschal lunar month,and is no more "moveable" than American Thanksgiving.

Tabular methods

Gregorian calendar

This method for the computation of the date of Easter was introduced with the Gregorian calendar reform in 1582.See especially the first, second, fourth, and sixth canon, and the calendarium 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)

Meeus algorithm

Jean Meeus, in his book ''Astronomical Algorithms'' (1991, p. 69), presents the following formula 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. To obtain the Eastern Orthodox Easter normally given in the Gregorian calendar, 13 days must be added between 1900 and 2099 inclusive. Churches beyond the eastern frontier of the former Byzantine Empire use an Easter that differs four times every 532 years from this Easter.
- ''a'' = ''Y'' modulo operation|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'' = Floor and ceiling functions|floor ((''d'' + ''e'' + 114) / 31)
- ''day'' = ((''d'' + ''e'' + 114) mod 31) + 1

Notes

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.
- Constantine the Great, Emperor (325): Letter to the bishops who did not attend the first Nicaean Council; from Eusebius' ''Vita Constantini''. English translations: http://www.fordham.edu/halsall/basis/nicea1.txt http://ccel.org/fathers2/NPNF2-03/Npnf2-03-10.htm#P1155_247748
- 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): several bulls and canons. On-line under: "Les textes fondateurs du calendrier grégorien"
- Roegel, Denis (2004): ''The missing new moon of A.D. 16399 and other anomalies of the Gregorian calendar'' (unpublished draft) http://www.loria.fr/~roegel/articles/epact19.pdf.
- 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''. http://scienceworld.wolfram.com/astronomy/PaschalFullMoon.html