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As a
moveable feast A moveable feast or movable feast is an observance in a Christian liturgical calendar, borrowed from the Hebrew Lunisolar calendar, which therefore occurs on a different date (relative to the Roman Civil calendar, civil or solar calendar) in diffe ...
, the date of Easter is determined in each year through a calculation known as ''computus'' (
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

Latin
for 'computation').
Easter Easter,Traditional names for the feast in English are "Easter Day", as in the ''Book of Common Prayer''; "Easter Sunday", used by James Ussher''The Whole Works of the Most Rev. James Ussher, Volume 4'' and Samuel Pepys''The Diary of Samuel Pe ...

Easter
is celebrated on the first Sunday after the
Paschal full moonAn ecclesiastical full moon is formally the 14th day of the ecclesiastical lunar month (an ecclesiastical moon) in an ecclesiastical lunar calendar. The ecclesiastical lunar calendar spans the year with lunar months of 30 and 29 days which are inte ...
, which is the first
full moon The full moon is the lunar phase s in 2022 as viewed from the Southern Hemisphere The Southern Hemisphere is the half (hemisphere Hemisphere may refer to: * A half of a sphere As half of the Earth * A hemispheres of Earth, hemisphere o ...

full moon
on or after 21 March (a fixed approximation of the
March equinox The March equinox or northward equinox is the equinox An equinox is the instant of time when the plane of Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. About 29% of Earth's s ...
). Determining this date in advance requires a correlation between the
lunar month In lunar calendar A lunar calendar is a calendar based on the monthly cycles of the Moon's lunar phase, phases (Lunar month#Synodic month, synodic months, lunations), in contrast to solar calendars, whose annual cycles are based only directly o ...
s and the
solar year A tropical year (also known as a solar year or tropical period) is the time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past ...
, while also accounting for the month, date, and weekday of the Julian or
Gregorian calendar The Gregorian calendar is the calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a single, speci ...
. The complexity of the
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

algorithm
arises because of the desire to associate the date of Easter with the date of the Jewish feast of
Passover Passover, also called Pesach (; he, פֶּסַח '), is a major Jewish holiday Jewish holidays, also known as Jewish festivals or ''Yamim Tovim'' ( he, ימים טובים, , Good Days, or singular , in transliterated Translitera ...
which, Christians believe, is when Jesus was crucified. It was originally feasible for the entire Christian church to receive the date of Easter each year through an annual announcement by the
Pope The pope ( la, papa, from el, πάππας, translit=pappas, "father"), also known as the supreme pontiff () or the Roman pontiff (), is the bishop of Diocese of Rome, Rome, chief pastor of the worldwide Catholic Church, and head of state o ...

Pope
. By the early third century, however, communications in the
Roman empire The Roman Empire ( la, Imperium Rōmānum ; grc-gre, Βασιλεία τῶν Ῥωμαίων, Basileía tôn Rhōmaíōn) was the post-Republican Republican can refer to: Political ideology * An advocate of a republic, a type of governme ...

Roman empire
had deteriorated to the point that the church put great value in a system that would allow the clergy to determine the date for themselves, independently and consistently. Additionally, the church wished to eliminate dependencies on the
Hebrew calendar The Hebrew calendar ( Hebrew: , ), also called Jewish calendar, is a lunisolar calendar used today for Jewish religious observance, and as an official calendar of the state of Israel Israel (; he, יִשְׂרָאֵל; ar, إِس ...
, by deriving the date for Easter directly from the
March equinox The March equinox or northward equinox is the equinox An equinox is the instant of time when the plane of Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. About 29% of Earth's s ...
. In ''
The Reckoning of Time ''The Reckoning of Time'' ( la, De temporum ratione) is an Anglo-Saxon era treatise A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerne ...
'' (725),
Bede Bede ( ; ang, Bǣda , ; 672/326 May 735), also known as Saint Bede, The Venerable Bede, and Bede the Venerable ( la, Beda Venerabilis), was an English Benedictine The Benedictines, officially the Order of Saint Benedict ( la, Ordo Sa ...

Bede
uses ''computus'' as a general term for any sort of calculation, although he refers to the Easter cycles of Theophilus as a "Paschal ''computus''." By the end of the 8th century, ''computus'' came to refer specifically to the calculation of time. The calculations produce different results depending on whether the Julian calendar or the Gregorian calendar is used. For this reason, the
Catholic Church The Catholic Church, also known as the Roman Catholic Church, is the largest Christian church, with 1.3 billion baptised Baptism (from the Greek language, Greek noun βάπτισμα ''báptisma'') is a Christians, Christian ...

Catholic Church
and
Protestant churches Protestantism is a form of Christianity Christianity is an Abrahamic religions, Abrahamic Monotheism, monotheistic religion based on the Life of Jesus in the New Testament, life and Teachings of Jesus, teachings of Jesus, Jesus of Nazareth. I ...
(which follow the Gregorian calendar) celebrate Easter on a different date to the
Eastern Orthodox Churches The Eastern Orthodox Church, also called the Orthodox Church, is the List of Christian denominations by number of members, second-largest Christian church, with approximately 220 million baptised members. It operates as a Communion (Christ ...
(which follow the Julian calendar). It was the drift of 21 March from the observed equinox that led to the Gregorian reform of the calendar, to bring them back into line.


Background

Easter commemorates the
resurrection of Jesus The resurrection of Jesus ( gr, ανάσταση του Ιησού) is the Christianity, Christian belief that God in Christianity, God Resurrection, raised Jesus on the third day after Crucifixion of Jesus, his crucifixion, starting – or Pr ...
, which is believed to have occurred on the third day (inclusive) after
Passover Passover, also called Pesach (; he, פֶּסַח '), is a major Jewish holiday Jewish holidays, also known as Jewish festivals or ''Yamim Tovim'' ( he, ימים טובים, , Good Days, or singular , in transliterated Translitera ...
. In the Hebrew calendar, Passover occurs on the 14th of
Nisan ''Nisan'' (or ''Nissan''; he, נִיסָן, Hebrew language#Modern Hebrew, Standard ''Nisan'' Tiberian vocalization, Tiberian ''Nîsān'') in the Hebrew and the Babylonian calendars, is the month of the barley ripening and first month of spri ...
. Nisan is the first month of spring in the
northern hemisphere The Northern Hemisphere is the half of Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remain ...

northern hemisphere
, with the 14th corresponding to a full moon. Additionally, by the 2nd century, many Christians had chosen to observe Easter only on a Sunday. The Hebrew calendar is a
lunisolar A lunisolar calendar is a calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a single, specific day wit ...
one and does not have a simple relationship with the Christian calendars: it resynchronises with the solar year by intercalating a leap month every two or three years, before the
lunar new year Lunar New Year is the beginning of a calendar year whose months are moon cycles, based on the lunar calendar A lunar calendar is a calendar A calendar is a system of organizing days. This is done by giving names to periods of time, t ...

lunar new year
on 1
Nisan ''Nisan'' (or ''Nissan''; he, נִיסָן, Hebrew language#Modern Hebrew, Standard ''Nisan'' Tiberian vocalization, Tiberian ''Nîsān'') in the Hebrew and the Babylonian calendars, is the month of the barley ripening and first month of spri ...
. Later the Jews adopted the
Metonic cycle For example, by the 19-year Metonic cycle, the full moon repeats on or near Christmas day between 1711 and 2300. A small horizontal libration is visible comparing their appearances. A red color shows full moons that are also lunar eclipses. The ...

Metonic cycle
to predict future intercalations. A possible consequence of this intercalation is that 14 Nisan may occur before the equinox, which some third-century Christians considered unacceptable, although this cannot happen in the fixed calendar currently in use. Consequently, they decided to separate the dating of Easter from the Hebrew calendar. To do so, it was necessary to identify the first full moon following the March equinox. By the time of the
First Council of Nicaea The First Council of Nicaea (; grc, Νίκαια ) was a council of Christian bishops convened in the Bithynia Bithynia (; Koine Greek Koine Greek (, , Greek approximately ;. , , , lit. "Common Greek"), also known as Alexandrian dialec ...
, the
Church of Alexandria The Church of Alexandria in Egypt is the Christian Church headed by the Patriarch of Alexandria. It is one of the Pentarchy, original Apostolic Sees of Christianity, alongside Rome, Antioch, Constantinople and Jerusalem. Tradition holds that the C ...
had designated 21 March as an ecclesiastical date for the equinox, irrespective of actual astronomical observation. In 395, Theophilus published a table of future dates for Easter, validating the Alexandrian criteria. Thereafter, the ''computus'' would be the procedure of determining the first Sunday after the first ecclesiastical full moon falling on or after 21 March.


History

The earliest known Roman tables were devised in 222 by
Hippolytus of Rome Hippolytus of Rome (c. 170 – c. 235 AD) was one of the most important second-third century Christian theologians, whose provenance, identity and corpus remain elusive to scholars and historians. Suggested communities include Palestine, Egypt, A ...
based on eight-year cycles. Then 84-year tables were introduced in Rome by
Augustalis An augustalis or augustale, also agostaro, was a gold coin minted in the Kingdom of Sicily beginning in 1231. It was issued by Frederick II, Holy Roman Emperor, Frederick II, Holy Roman Emperor (from 1220) and King of Sicily (from 1198), and was m ...
near the end of the 3rd century. Although a process based on the 19-year Metonic cycle was first proposed by Bishop
Anatolius of Laodicea Anatolius of Laodicea (early 3rd century – July 3, 283), also known as Anatolios of Alexandria, became Bishop A bishop is an ordained, consecrated, or appointed member of the Clergy#Christianity, Christian clergy who is generally entrusted wi ...
around 277, the concept did not fully take hold until the Alexandrian method became authoritative in the late 4th century. The Alexandrian computus was converted from the Alexandrian calendar into the Julian calendar in Alexandria around AD 440, which resulted in a Paschal table (attributed to pope
Cyril of Alexandria Cyril of Alexandria ( grc, Κύριλλος Ἀλεξανδρείας; cop, Ⲡⲁⲡⲁ Ⲕⲩⲣⲓⲗⲗⲟⲩ ⲁ̅ also ⲡⲓ̀ⲁⲅⲓⲟⲥ Ⲕⲓⲣⲓⲗⲗⲟⲥ;  376 – 444) was the Patriarch of Alexandria The Patriarch ...

Cyril of Alexandria
) covering the years AD 437–531. This Paschal table was the source which inspired
Dionysius Exiguus Dionysius Exiguus (Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republ ...

Dionysius Exiguus
, who worked in Rome from about AD 500 to about AD 540, to construct a continuation of it in the form of his famous Paschal table covering the years AD 532–616. Dionysius introduced the
Christian Era The terms (AD) and before Christ (BC) are used to label or number years in the Julian and Gregorian calendar The Gregorian calendar is the calendar used in most of the world. It was introduced in October 1582 by Pope Gregory XIII as a ...
(counting years from the Incarnation of Christ) by publishing this new Easter table in AD 525. A modified 84-year cycle was adopted in Rome during the first half of the 4th century.
Victorius of AquitaineVictorius of Aquitaine, a countryman of Prosper of Aquitaine and also working in Rome, produced in AD 457 an Easter Cycle, which was based on the consular list provided by Prosper's Chronicle. This dependency caused scholars to think that Prosper h ...
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 Gaul ( la, Gallia) was a region of Western Europe Western Europe is the western region of Europe Europe is a continent A continent is any of several large landmasses. Generally identified by convention (norm), convention rat ...

Gaul
(now France) and Spain until they were displaced by Dionysian tables at the end of the 8th century. The tables of Dionysius and Victorius conflicted with those traditionally used in the British Isles. The British tables used an 84-year cycle, but an error made the full moons fall progressively too early. The discrepancy led to a report that Queen Eanfled, on the Dionysian system fasted on her
Palm Sunday Palm Sunday is a Christian moveable feast that falls on the Sunday before Easter. The feast commemorates Jesus' triumphal entry into Jerusalem, an event mentioned in each of the four canonical Gospels. Palm Sunday marks the first day of Holy Wee ...

Palm Sunday
while her husband
Oswy Oswiu, also known as Oswy or Oswig ( ang, Ōswīg) (c. 612 – 15 February 670), was King of Bernicia from 642 and of Northumbria Northumbria (; ang, Norþanhymbra Rīċe; la, Regnum Northanhymbrorum) was an early medieval Anglo-Saxon king ...
, king of Northumbria, feasted on his Easter Sunday. As a result of the Irish Synod of Magh-Lene in 630, the southern Irish began to use the Dionysian tables, and the northern English followed suit after the
Synod of Whitby In the Synod of Whitby in 664, King Oswiu of Northumbria Oswiu, also known as Oswy or Oswig ( ang, Ōswīg) (c. 612 – 15 February 670), was King of Bernicia from 642 and of Kingdom of Northumbria, Northumbria from 654 until his death. He is no ...
in 664. The Dionysian reckoning was fully described by
Bede Bede ( ; ang, Bǣda , ; 672/326 May 735), also known as Saint Bede, The Venerable Bede, and Bede the Venerable ( la, Beda Venerabilis), was an English Benedictine The Benedictines, officially the Order of Saint Benedict ( la, Ordo Sa ...

Bede
in 725. It may have been adopted by
Charlemagne Charlemagne ( , ) or Charles the Great ( la, Carolus Magnus; 2 April 748 – 28 January 814) was King of the Franks The Franks—Germanic-speaking peoples that invaded the Western Roman Empire in the 5th century—were first led by i ...

Charlemagne
for the Frankish Church as early as 782 from
Alcuin Alcuin of York (; la, Flaccus Albinus Alcuinus; 735 – 19 May 804) – also called Ealhwine, Alhwin, or Alchoin – was an English scholar, clergyman, poet, and teacher from York, Northumbria. He was born around 735 and became the ...
, a follower of Bede. The Dionysian/Bedan computus remained in use in western Europe until the Gregorian calendar reform, and remains in use in most Eastern Churches, including the vast majority of Eastern Orthodox Churches and Non-Chalcedonian Churches. The only Eastern Orthodox church which does not follow the system is the Finnish Orthodox Church, which uses the Gregorian. Having deviated from the Alexandrians during the 6th century, churches beyond the eastern frontier of the former Byzantine Empire, including the
Assyrian Church of the East The Assyrian Church of the East ( syc, ܥܕܬܐ ܕܡܕܢܚܐ ܕܐܬܘܖ̈ܝܐ, ʿĒḏtā ḏ-Maḏnḥā ḏ-ʾĀṯūrāyē, ar, كنيسة المشرق الآشورية), officially the Holy Apostolic Catholic Assyrian Church of the East ( sy ...
, now celebrate Easter on different dates from
Eastern Orthodox Churches The Eastern Orthodox Church, also called the Orthodox Church, is the List of Christian denominations by number of members, second-largest Christian church, with approximately 220 million baptised members. It operates as a Communion (Christ ...
four times every 532 years. Apart from these churches on the eastern fringes of the Roman empire, by the tenth century all had adopted the Alexandrian Easter, which still placed the vernal equinox on 21 March, although Bede had already noted its drift in 725 it had drifted even further by the 16th century. Worse, the reckoned Moon that was used to compute Easter was fixed to the Julian year by the 19-year cycle. That approximation built up an error of one day every 310 years, so by the 16th century the
lunar calendar A lunar calendar is a calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a single, specific day with ...
was out of phase with the real Moon by four days. The Gregorian Easter has been used since 1583 by the
Roman Catholic Church The Catholic Church, also known as the Roman Catholic Church, is the largest Christian church, with 1.3 billion baptised Baptism (from the Greek language, Greek noun βάπτισμα ''báptisma'') is a Christians, Christian ...

Roman Catholic Church
and was adopted by most
Protestant Protestantism is a form of that originated with the 16th-century , a movement against what its followers perceived to be in the . Protestants originating in the Reformation reject the Roman Catholic doctrine of , but disagree among themselves ...
churches between 1753 and 1845. German Protestant states used an astronomical Easter between 1700 and 1776, based on the ''
Rudolphine Tables The ''Rudolphine Tables'' ( la, Tabulae Rudolphinae) consist of a star catalogue and planetary tables published by Johannes Kepler in 1627, using observational data collected by Tycho Brahe (1546–1601). The tables are named in memory of Rudolf ...

Rudolphine Tables
'' of
Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer An astronomer is a in the field of who focuses their studies on a specific question or field outside the scope of . They observe s such as s, s, , s and ...

Johannes Kepler
, which were in turn based on astronomical positions of the Sun and Moon observed by
Tycho Brahe Tycho Brahe ( ; born Tyge Ottesen Brahe; 14 December 154624 October 1601) was a Danish astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. T ...

Tycho Brahe
at his
Uraniborg Uraniborg ( da, Uranienborg, sv, Uraniborg) was a Danish astronomical observatory and alchemy laboratory established and operated by Tycho Brahe. It was built on Hven, an island in the Øresund between Zealand (Denmark), Zealand and Scania, S ...

Uraniborg
observatory on the island of
Ven Ven may refer to: Places * Ven, Heeze-Leende, a hamlet in the Netherlands * Ven (Sweden) Ven ( da, Hven, older Swedish spelling Hven) is a small Swedish island in the Øresund Øresund or Öresund (, ; da, Øresund ; sv, Öresund ), com ...
, while Sweden used it from 1739 to 1844. This astronomical Easter was the Sunday after the full moon instant that was after the vernal equinox instant using Uraniborg time . However, it was delayed one week if that Sunday was the Jewish date Nisan15, the first day of Passover week, calculated according to modern Jewish methods. This Nisan15 rule affected two Swedish years, 1778 and 1798, that instead of being one week before the Gregorian Easter were delayed one week so they were on the same Sunday as the Gregorian Easter. Germany's astronomical Easter was one week before the Gregorian Easter in 1724 and 1744. Sweden's astronomical Easter was one week before the Gregorian Easter in 1744, but one week after it in 1805, 1811, 1818, 1825, and 1829. Two modern astronomical Easters were proposed but never used by any Church. The first was proposed as part of the
Revised Julian calendar The Revised Julian calendar, also known as the Milanković calendar, or less formally new calendar, is a calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months a ...
at a Synod in
Constantinople la, Constantinopolis ota, قسطنطينيه , alternate_name = Byzantion (earlier Greek name), Nova Roma ("New Rome"), Miklagard/Miklagarth (Old Norse Old Norse, Old Nordic, or Old Scandinavian is a stage of development of North Germa ...

Constantinople
in 1923 and the second was proposed by a 1997
World Council of Churches The World Council of Churches (WCC) is a worldwide Christian Christians () are people who follow or adhere to Christianity, a monotheistic Abrahamic religion based on the life and teachings of Jesus in Christianity, Jesus Christ. The words ...
Consultation in
Aleppo )), is an adjective which means "white-colored mixed with black". , motto = , image_map = , mapsize = , map_caption = , image_map1 ...

Aleppo
in 1997. Both used the same rule as the German and Swedish versions but used modern astronomical calculations and
Jerusalem Jerusalem (; he, יְרוּשָׁלַיִם ; ar, القُدس, ', , (combining the Biblical and common usage Arabic names); grc, Ἱερουσαλήμ/Ἰεροσόλυμα, Hierousalḗm/Hierosóluma; hy, Երուսաղեմ, Erusał ...

Jerusalem
time without the Nisan15 rule. The 1923 version would have placed the astronomical Easter one month before the Gregorian Easter in 1924, 1943, and 1962, but one week after it in 1927, 1954, and 1967. The 1997 version would have placed the astronomical Easter on the same Sunday as the Gregorian Easter for 2000–2025 except for 2019, when it would have been one month earlier.


Theory

The Easter cycle groups days into lunar months, which are either 29 or 30 days long. There is an exception. The month ending in March normally has thirty days, but if 29 February of a leap year falls within it, it contains 31. As these groups are based on the
lunar cycle The lunar phase or Moon phase is the shape of the Moon's directly sunlit portion as viewed from Earth. The lunar phases gradually change over a synodic month (about 29.53 days) as the Moon's orbital positions around Earth and Earth around th ...
, over the long term the average month in the lunar calendar is a very good approximation of the
synodic month In lunar calendar A lunar calendar is a calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a sing ...
, which is days long. There are 12 synodic months in a lunar year, totaling either 354 or 355 days. The lunar year is about 11 days shorter than the calendar year, which is either 365 or 366 days long. These days by which the solar year exceeds the lunar year are called
epact The epact ( la, epactae, from grc, ἐπακται ἡμεραι () = added days), used to be described by medieval Computus, computists as the age of a Lunar phase, phase of the Moon in days on 22 March; in the newer Gregorian calendar, however, ...
s ( grc-gre, ἐπακταὶ ἡμέραι, epaktai hēmerai, intercalary days). 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
intercalary month Intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or moon phases. Lunisolar calendar A lunisolar calendar is a calendar in many culture Cul ...
(or embolismic month) of 30 days must be inserted into the lunar calendar: then 30 must be subtracted from the epact.
Charles Wheatly Charles Wheatly (1686–1742) was an English clergyman, known for writings on the ''Book of Common Prayer''. Life He was born on 6 February 1686, the son of John Wheatly, a tradesman of London. His mother, whose maiden name was White, was a descend ...
provides the detail: Thus the lunar month took the name of the Julian month in which it ended. The nineteen-year 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, , not ; that is, 209 divided by 30 leaves a remainder of 29 instead of being a multiple of 30. So after 19 years, the epact must be corrected by one day for the cycle to repeat. This is the so-called ''saltus lunae'' ("leap of the moon"). The Julian calendar handles it by reducing the length of the lunar month that begins on 1 July in the last year of the cycle to 29 days. This makes three successive 29-day months. The ''saltus'' and the seven extra 30-day months were largely hidden by being located at the points where the Julian and lunar months begin at about the same time. The extra months commenced on 1 January (year 3), 2 September (year 5), 6 March (year 8), 3 January (year 11), 31 December (year 13), 1 September (year 16), and 5 March (year 19). The sequence number of the year in the 19-year cycle is called the " golden number", and is given by the formula :''GN'' = (''Y'' + 1) mod 19 That is, one is added to year number ''Y'' in the
Christian era The terms (AD) and before Christ (BC) are used to label or number years in the Julian and Gregorian calendar The Gregorian calendar is the calendar used in most of the world. It was introduced in October 1582 by Pope Gregory XIII as a ...
, the sum is divided by 19; the remainder is the golden number, with a remainder of 0 indicating the golden number is 19. The paschal or Easter-month is the first one in the year to have its fourteenth day (its formal
full moon The full moon is the lunar phase s in 2022 as viewed from the Southern Hemisphere The Southern Hemisphere is the half (hemisphere Hemisphere may refer to: * A half of a sphere As half of the Earth * A hemispheres of Earth, hemisphere o ...
) on or after 21 March. 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 8 March to 5 April inclusive. Its fourteenth day, therefore, always falls on a date between 21 March and 18 April inclusive, and the following Sunday then necessarily falls on a date in the range 22 March to 25 April inclusive. In the solar calendar Easter is called a
moveable feast A moveable feast or movable feast is an observance in a Christian liturgical calendar, borrowed from the Hebrew Lunisolar calendar, which therefore occurs on a different date (relative to the Roman Civil calendar, civil or solar calendar) in diffe ...
since its date varies within a 35-day range. But in the lunar calendar, Easter is always the third 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.


Tabular methods


Gregorian reform of the computus

As reforming the computus was the primary motivation for the introduction of the
Gregorian calendar The Gregorian calendar is the calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a single, speci ...
in 1582, a corresponding computus methodology was introduced alongside the new calendar. 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 on or after 21 March. The Gregorian method derives paschal full moon dates by determining the
epact The epact ( la, epactae, from grc, ἐπακται ἡμεραι () = added days), used to be described by medieval Computus, computists as the age of a Lunar phase, phase of the Moon in days on 22 March; in the newer Gregorian calendar, however, ...
for each year. The epact can have a value from * (0 or 30) to 29 days. It is the age of the moon (in days), i.e. the lunar date, on 1 January reduced by one day. In his book ''The Easter computus and the origins of the Christian era'' Alden A Mosshammer incorrectly states "Theoretically, the epact 30 = 0 represents the new moon at its conjunction with the sun. The epact of 1 represents the theoretical first visibility of the first crescent of the moon. It is from that point as day one that the fourteenth day of the moon is counted." The 14th day of the lunar month is considered the day of the
full moon The full moon is the lunar phase s in 2022 as viewed from the Southern Hemisphere The Southern Hemisphere is the half (hemisphere Hemisphere may refer to: * A half of a sphere As half of the Earth * A hemispheres of Earth, hemisphere o ...

full moon
. It is the day of the lunar month on which the moment of opposition ("full moon") is most likely to fall. The "new moon" is most likely to become visible (as a slender crescent in the western sky after sunset) on the first day of the lunar month. The conjunction of sun and moon ("new moon") is most likely to fall on the preceding day, which is day 29 of a "hollow" (29-day) month and day 30 of a "full" (30-day) month. Historically the paschal full moon date for a year was found from its sequence number in the Metonic cycle, called the golden number, which cycle repeats the lunar phase 1 January every 19 years. This method was abandoned in the Gregorian reform because the tabular dates go out of sync with reality after about two centuries, but from the epact method, a simplified table can be constructed that has a validity of one to three centuries. The epacts for the current Metonic cycle, which began in 2014, are: The epacts are used to find the dates of the 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 numeral Roman numerals are a numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other s ...
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, therefore, 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 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 when the epact for the year is entered. If the epact for the year is for instance 27, then there is an
ecclesiastical new moonAn ecclesiastical new moon is the first day of a lunar month (an ecclesiastical moon) in an ecclesiastical lunar calendar. Such months have a whole number of days, 29 or 30, whereas true synodic months can vary from about 29.27 to 29.83 days in len ...
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" is a Sunday that year. Then "E" is called the
dominical letterDominical letters or Sunday letters are a method used to determine the day of the week for particular dates. When using this method, each year is assigned a letter (or pair of letters for leap years) depending on which day of the week the year starts ...
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 fall on the previous letter of the cycle, so leap years have two 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, 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: Example: If the epact is 27 (xxvii), an ecclesiastical new moon falls on every date labeled ''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 one day. A month beginning on a date having labels xxiv and xxv written side by side has either 29 or 30 days. 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 possible for the real moon but is inelegant in a schematic lunar calendar; 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 falls on the date with the label ''25'' rather than ''xxv''. Where the labels ''25'' and ''xxv'' are together, there is no problem since they are the same. This does not move the problem to the pair "25" and "xxvi", because the earliest epact 26 could appear would be in year 23 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 tropical year by dropping three leap days in 400 years (always in a century year). This is a correction to the length of the tropical year, but should have no effect on the Metonic relation between years and lunations. Therefore, the epact is compensated for this (partially see
epact The epact ( la, epactae, from grc, ἐπακται ἡμεραι () = added days), used to be described by medieval Computus, computists as the age of a Lunar phase, phase of the Moon in days on 22 March; in the newer Gregorian calendar, however, ...
) by subtracting one 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 2,500 (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, the next is in 2100, and 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.


Details

This method of computation has several subtleties: Every other 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 Eusebius of Caesarea (; grc-gre, Εὐσέβιος τῆς Καισαρείας, ''Eusébios tés Kaisareías''; AD 260/265 – 339/340), also known as Eusebius Pamphili (from the grc-gre, Εὐσέβιος τοῦ Παμϕίλου) ...

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 7 March 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". 5 April must therefore have its double epact labels "xxiv" and "xxv". Then epact "xxv" must 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 6,940 or 6,939 days with five or four leap days. Now the lunar cycle counts only . 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 (creating a long-term inaccuracy). A consequence is that the reckoned age of the moon may be off by a day, and also that the lunations that contain the leap day may be 31 days long, which would never happen if the real moon were followed (short-term inaccuracies). This is the price of 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 (although 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, a increases the epact by another unit, and the new moon should have occurred on the previous day. So a new moon is missed. The of the ''
Missale Romanum The Roman Missal ( la, Missale Romanum) is the liturgical book Liturgy is the customary public worship Worship is an act of religion, religious wikt:devotion, devotion usually directed towards a deity. For many, worship is not about an emotio ...
'' takes account of this by assigning epact label "19" instead of "xx" to 31 December of such a year, making that date the new moon. It happened every 19 years when the original Gregorian epact table was in effect (for the last time in 1690), and next happens in 8511. # If the epact of a year is 20, an ecclesiastical new moon falls on 31 December. If that year falls before a century year, then in most cases, a solar correction 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 next happens in 4199–4200. # Other borderline cases occur (much) later, and if the rules are followed strictly and these cases are not specially treated, they 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 (, also known as a ''
tithi In Vedic timekeeping, a ''tithi'' is a uration of two faces of moon that is observed from earth known as ''milа̄lyа̄'' (𑐩𑐶𑐮𑐵𑐮𑑂𑐫𑐵𑑅, मिलाल्याः) in Newar language, Nepal Bhasa, or the time it t ...
'') and not full days. See
epact The epact ( la, epactae, from grc, ἐπακται ἡμεραι () = added days), used to be described by medieval Computus, computists as the age of a Lunar phase, phase of the Moon in days on 22 March; in the newer Gregorian calendar, however, ...
for a discussion. The solar and lunar corrections repeat after centuries. In that period, the epact has changed by a total of . This is prime to the 30 possible epacts, so it takes before the epacts repeat; and centuries before the epacts repeat at the same golden number. This period has . So the Gregorian Easter dates repeat in exactly the same order only after 5,700,000 years, 70,499,183 lunations, or 2,081,882,250 days; the mean lunation length is then 29.53058690 days. However, the calendar must already have been adjusted after some millennia because of changes in the length of the tropical 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. Lilius's original work has not been preserved, but his proposal was described in the circulated in 1577, in which it is explained that the correction system he devised was to be a perfectly flexible tool in the hands of future calendar reformers, since the solar and lunar calendar could henceforth be corrected without mutual interference. An example of this flexibility was provided through an alternative intercalation sequence derived from Copernicus's theories, along with its corresponding epact corrections. The "solar corrections" approximately undo the effect of the Gregorian modifications to the leap days of the solar calendar on the lunar calendar: they (partially) bring 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 three or four 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. While the net subtractions could be distributed evenly over 10,000 years (as has been proposed for example by ) if the corrections are combined, then the inaccuracies of the two cycles are also added and cannot be corrected separately. The ratios of (mean solar) days per year and days per lunation change both because of intrinsic long-term variations in the orbits, and because the rotation of the Earth is slowing down due to
tidal deceleration and the Moon The Moon is Earth's only proper natural satellite. At one-quarter the diameter of Earth (comparable to the width of Australia Australia, officially the Commonwealth of Australia, is a Sovereign state, sovereign count ...
, so the Gregorian parameters become increasingly obsolete. This does affect the date of the equinox, but it so happens that the interval between northward (northern hemisphere spring) equinoxes has been fairly stable over historical times, especially if measured in mean solar time (see, esp.) Also the drift in ecclesiastical full moons calculated by the Gregorian method compared to the true full moons is affected less than one would expect, because the increase in the length of the day is almost exactly compensated for by the increase in the length of the month, as tidal braking transfers angular momentum of the rotation of the Earth to orbital angular momentum of the Moon. The Ptolemaic value of the length of the mean synodic month, established around the 4th century BCE by the Babylonians, is (see Kidinnu); the current value is 0.46 s less (see
New moon In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses ...

New moon
). In the same historic stretch of time the length of the mean tropical year has diminished by about 10 s (all values mean solar time).


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 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 Anglicanism is a Western Christianity, Western Christian tradition that has developed from the practices, liturgy, and identity of the Church of England following the English Reformation. Adherents of Anglicanism are called ''Anglicans''; t ...

Anglican
Prayer Book A prayer book is a book containing prayer Prayer is an invocation An invocation (from the Latin verb ''invocare'' "to call on, invoke, to give") may take the form of: * Supplication, prayer Prayer is an invocation or act that seek ...

Prayer Book
(decreed by the
Act of Uniformity 1662 The Act of Uniformity 1662 (14 Car 2 c 4) is an Act of the Parliament of England The Parliament of England was the legislature A legislature is a deliberative assembly with the authority In the fields of sociology Sociology ...
). 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 what is now called the
Calendar (New Style) Act 1750 The Calendar (New Style) Act 1750 (24 Geo. II c.23), "An act for regulating the commencement of the year; and for correcting the calendar now in use", also known as Chesterfield's Act or (in American usage) the British Calendar ...
with its Annexe. The method was chosen to give dates agreeing with the Gregorian rule already in use elsewhere. The Act required that it be put in the
Book of Common Prayer A book is a medium for recording information Information is processed, organised and structured data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sense, data are a set of v ...

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''. 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 reform of the Computus. 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. The epact does not explicitly appear. Simpler tables 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's details were employed in the construction of the method, but they play no subsequent part in its use. 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.


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 three days every millennium. It 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 occasionally four or five weeks later because the Julian calendar is 13 days behind the Gregorian in 1900–2099, and so the Gregorian paschal full moon is sometimes before Julian 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 the golden number to tables originally composed by
Abbo of Fleury Abbo or Abbon of Fleury ( la, Abbo Floriacensis;  – 13 November 1004), also known as Saint Abbo or Abbon, was a monk and abbot of Fleury Abbey Fleury Abbey (Floriacum) in Saint-Benoît-sur-Loire, Loiret, France, founded in about 640, is o ...
in 988. The claim by the Catholic Church in the 1582
papal bull A papal bull is a type of public decree, letters patent, or charter issued by a pope of the Catholic Church. It is named after the leaden Seal (emblem), seal (''bulla (seal), bulla'') that was traditionally appended to the end in order to auth ...
''
Inter gravissimas ''Inter gravissimas'' (English: "Among the most serious...") was a papal bull issued by Pope Gregory XIII on 24 February 1582.
Inter gravissimas
'', which promulgated the Gregorian calendar, that it restored "the celebration of Easter according to the rules fixed by ... the great ecumenical council of Nicaea" was based on a false claim by Dionysius Exiguus (525) that "we determine the date of Easter Day ... in accordance with the proposal agreed upon by the 318 Fathers of the Church at the Council in Nicaea." The First Council of Nicaea (325) did not, however, provide any explicit rules to determine that date, but only wrote “all our brethren in the East who formerly followed the custom of the Jews are henceforth to celebrate the said most sacred feast of Easter at the same time with the Romans and yourselves he Church of Alexandriaand all those who have observed Easter from the beginning.” The medieval computus was based on the Alexandrian computus, which was developed by the
Church of Alexandria The Church of Alexandria in Egypt is the Christian Church headed by the Patriarch of Alexandria. It is one of the Pentarchy, original Apostolic Sees of Christianity, alongside Rome, Antioch, Constantinople and Jerusalem. Tradition holds that the C ...

Church of Alexandria
during the first decade of the 4th century using the Alexandrian calendar. The
eastern Roman Empire The Byzantine Empire, also referred to as the Eastern Roman Empire, or Byzantium, was the continuation of the Roman Empire The Roman Empire ( la, Imperium Rōmānum ; grc-gre, Βασιλεία τῶν Ῥωμαίων, Basileía tôn ...

eastern Roman Empire
accepted it shortly after 380 after converting the computus to the Julian calendar. Rome accepted it sometime between the sixth and ninth centuries. The British Isles accepted it during the eighth century except for a few monasteries.
Francia Francia, also called the Kingdom of the Franks ( la, Regnum Francorum), Frankish Kingdom, Frankland or Frankish Empire, was the largest History of the Roman Empire, post-Roman barbarian kingdom in Western Europe. It was ruled by the Franks du ...

Francia
(all of western Europe except Scandinavia (pagan), the British Isles, the
Iberian peninsula The Iberian Peninsula , ** * Aragonese Aragonese or Aragones may refer to: * Something related to Aragon, an autonomous community and former kingdom in Spain * the Aragonese people, those originating from or living in the historical region ...

Iberian peninsula
, and southern Italy) accepted it during the last quarter of the eighth century. The last Celtic monastery to accept it,
Iona Iona ( gd, Ì Chaluim Chille (IPA: iːˈxaɫ̪ɯimˈçiʎə, sometimes simply ''Ì''; sco, Iona) is a small island in the Inner Hebrides off the Ross of Mull on the western coast of Scotland. It is mainly known for Iona Abbey, though there a ...

Iona
, did so in 716, whereas the last English monastery to accept it did so in 931. Before these dates, other methods produced Easter Sunday dates that could differ by up to five weeks. This is the table of paschal full moon dates for all Julian years since 931: Example calculation using this table: The golden number for 1573 is 16 (; ). From the table, the paschal full moon for golden number 16 is 21 March. From the week table 21 March is Saturday. Easter Sunday is the following Sunday, 22 March. 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 four years, the full cycle in which weekdays recur in the calendar in the same way, is years, the so-called
solar cycle The solar cycle or solar magnetic activity cycle is a nearly periodic 11-year change in the Sun The Sun is the star A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its ...
. So the Easter dates repeated in the same order after years. This paschal cycle is also called the Victorian cycle, after Victorius of Aquitaine, who introduced it in Rome in 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 532; but he apparently did not realize that the Alexandrian computus he described had a 532-year cycle, although he did realize that his 95-year table was not a true cycle.
Venerable Bede Bede ( ; ang, Bǣda , ; 672/326 May 735), also known as Saint Bede, The Venerable Bede, and Bede the Venerable ( la, Beda Venerabilis), was an English Benedictine The Benedictines, officially the Order of Saint Benedict ( la, Ordo Sa ...

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 (14 Nisan) given above could be memorized with the help of a 19-line alliterative poem in Latin:
The first half-line of each line gives the date of the paschal full moon from the table above for each year in the 19-year cycle. The second half-line gives the ''ferial regular'', or weekday displacement, of the day of that year's paschal full moon from the ''concurrent'', or the weekday of 24 March. The ''ferial regular'' is repeated in Roman numerals in the third column.


"Paradoxical" Easter dates

Due to the discrepancies between the approximations of Computistical calculations of the time of the
mean There are several kinds of mean in mathematics, especially in statistics. For a data set, the ''arithmetic mean'', also known as arithmetic average, is a central value of a finite set of numbers: specifically, the sum of the values divided by ...
(northern hemisphere) vernal equinox and the lunar phases, and the true values computed according to astronomical principles, differences occasionally arise between the date of Easter according to computistical reckoning and the hypothetical date of Easter calculated by astronomical methods using the principles attributed to the Church fathers. These discrepancies are called "paradoxical" Easter dates. In his ''Kalendarium'' of 1474,
Regiomontanus Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) inc ...

Regiomontanus
computed the exact time of all
conjunctions ''Conjunctions'' is a biannual American literature, American literary journal based at Bard College. It was founded in 1981 and is currently edited by Bradford Morrow. Morrow received the PEN/Nora Magid Award for Magazine Editing in 2007. The jou ...

conjunctions
of the Sun and Moon for the longitude of
Nuremberg Nuremberg ( ; german: link=no, Nürnberg ; in the local East Franconian dialect: ''Nämberch'' ) is the second-largest city of the Germany, German States of Germany, state of Bavaria after its capital Munich, and its 518,370 (2019) inhabitants ...

Nuremberg
according to the
Alfonsine Tables 350px, Alfonsine tables The ''Alfonsine tables'' ( es, Tablas alfonsíes, la, tabulae alphonsinae), sometimes spelled ''Alphonsine tables'', provided data for computing the position of the Sun, Moon The Moon is Earth's only proper natur ...
for the period from 1475 to 1531. In his work he tabulated 30 instances where the Easter of the Julian computus disagreed with Easter computed using astronomical
New Moon In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses ...

New Moon
. In eighteen cases the date differed by a week, in seven cases by 35 days, and in five cases by 28 days. Ludwig Lange investigated and classified different types of paradoxical Easter dates using the Gregorian computus. In cases where the first vernal full moon according to astronomical calculation occurs on a Sunday and the Computus gives the same Sunday as Easter, the celebrated Easter occurs one week in advance compared to the hypothetical "astronomically" correct Easter. Lange called this case a negative weekly (hebdomadal) paraodox (H− paradox). If the astronomical calculation gives a Saturday for the first vernal full moon and Easter is not celebrated on the directly following Sunday but one week later, Easter is celebrated according to the computus one week too late in comparison to the astronomical result. He classified such cases a positive weekly (hebdomadal) paradox (H+ paradox). The discrepancies are even larger if there is a difference according to the vernal equinox with respect to astronomical theory and the approximation of the Computus. If the astronomical equinoctial full moon falls before the computistical equinoctial full moon, Easter will be celebrated four or even five weeks too late. Such cases are called a positive equinoctial paradox (A+ paradox) according to Lange. In the reverse case when the Computistical equinoctial full moon falls a month before the astronomical equinoctial full moon, Easter is celebrated four or five weeks too early. Such cases are called a negative equinoctial paradox (A− paradox). Equinoctial paradoxes are always valid globally for the whole earth, because the sequence of equinox and full moon does not depend on the geographical longitude. In contrast, weekly paradoxes are local in most cases and are valid only for part of the earth, because the change of day between Saturday and Sunday is dependent on the geographical longitude. The computistical calculations are based on astronomical tables valid for the longitude of Venice, which Lange called the Gregorian longitude. In the 21st and 22nd centuries negative weekly paradoxical Easter dates occur in 2049, 2076, 2106, 2119 (global), 2133, 2147, 2150, 2170, and 2174; positive weekly paradoxical dates occur in 2045, 2069, 2089, and 2096; positive equinoctial paradoxical dates in 2019, 2038, 2057, 2076, 2095, 2114, 2133, 2152, 2171, and 2190. In 2076 and 2133, 'double paradoxes (positive equinoctial and negative weekly) occur. Negative equinoctial paradoxes are extremely rare; they occur only twice until the year 4000 in 2353, when Easter is five weeks too early and in 2372, when Easter is four weeks too early.


Algorithms


Note on operations

When expressing Easter algorithms without using tables, it has been customary to employ only the integer operations
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

subtraction
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
,
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
, modulo, and assignment as it is compatible with the use of simple mechanical or electronic calculators. That restriction is undesirable for computer programming, where conditional operators and statements, as well as look-up tables, are 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 . More importantly, using such conditionals also simplifies the core of the Gregorian calculation.


Gauss's Easter algorithm

In 1800, the mathematician
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

Carl Friedrich Gauss
presented this algorithm for calculating the date of the Julian or Gregorian Easter. He corrected the expression for calculating the variable ''p'' in 1816. In 1800, he incorrectly stated . In 1807, he replaced the condition with the simpler . In 1811, he limited his algorithm to the 18th and 19th centuries only, and stated that 26 April is always replaced with 19 and 25 April by 18 April in the circumstances stated. In 1816, he thanked his student Peter Paul Tittel for pointing out that ''p'' was wrong in the original version. An analysis of the Gauss's Easter algorithm is divided into two parts. The first part is the approximate tracking of the lunar orbiting and the second part is the exact deterministic offsetting to obtain a Sunday following the full moon. The first part consists of determining the variable ''d'', the number of days (counting from 22 March) to the day after the full moon. The formula for ''d'' contains the terms 19''a'' and the constant M. ''a'' is the year's position in the 19-year lunar phase cycle, in which by assumption the moon's movement relative to earth repeats every 19 calendar years. In older times, 19 calendar years were equated to 235 lunar months (the Metonic cycle), which is remarkably close since 235 lunar months are approximately 6939.6813 days and 19 years are on average 6939.6075 days. The expression (19''a'' + M) mod 30 repeats every 19 years within each century as M is determined per century. The 19-year cycle has nothing to do with the '19' in 19''a'', it is just a coincidence that another '19' appears. The '19' in 19''a'' comes from correcting the mismatch between a calendar year and an integer number of lunar months. A calendar year (non-leap year) has 365 days and the closest one can come with an integer number of lunar months is days. The difference is 11 days, which must be corrected for by moving the following year's occurrence of a full moon 11 days back. But in modulo 30 arithmetic, subtracting 11 is the same as adding 19, hence the addition of 19 for each year added, i.e. 19''a''. The M in serves to have a correct starting point at the start of each century. It is determined by a calculation taking the number of leap years up until that century where ''k'' inhibits a leap day every 100 years and ''q'' reinstalls it every 400 years, yielding as the total number of inhibitions to the pattern of a leap day every four years. Thus we add to correct for leap days that never occurred. ''p'' corrects for the lunar orbit not being fully describable in integer terms. The range of days considered for the full moon to determine Easter are 21 March (the day of the ecclesiastical equinox of spring) to 18 April—a 29-day range. However, in the mod 30 arithmetic of variable ''d'' and constant ''M'', both of which can have integer values in the range 0 to 29, the range is 30. Therefore, adjustments are made in critical cases. Once ''d'' is determined, this is the number of days to add to 22 March (the day after the earliest possible full moon allowed, which is coincident with the ecclesiastical equinox of spring) to obtain the date of the day after the full moon. So the first allowable date of Easter is March 22 + d + 0, as Easter is to celebrate the Sunday after the ecclesiastical full moon, that is if the full moon falls on Sunday 21 March Easter is to be celebrated 7 days after, while if the full moon falls on Saturday 21 March Easter is the following 22 March. The second part is finding ''e'', the additional offset days that must be added to the date offset ''d'' to make it arrive at a Sunday. Since the week has 7 days, the offset must be in the range 0 to 6 and determined by modulo 7 arithmetic. ''e'' is determined by calculating . These constants may seem strange at first, but are quite easily explainable if we remember that we operate under mod 7 arithmetic. To begin with, ensures that we take care of the fact that weekdays slide for each year. A normal year has 365 days, but , so 52 full weeks make up one day too little. Hence, each consecutive year, the weekday "slides one day forward", meaning if 6 May was a Wednesday one year, it is a Thursday the following year (disregarding leap years). Both ''b'' and ''c'' increase by one for an advancement of one year (disregarding modulo effects). The expression thus increases by 6 – but remember that this is the same as subtracting 1 mod 7. To subtract by 1 is exactly what is required for a normal year – since the weekday slips one day forward we should compensate one day less to arrive at the correct weekday (i.e. Sunday). For a leap year, ''b'' becomes 0 and 2''b'' thus is 0 instead of 8 – which under mod 7, is another subtraction by 1 – i.e., a total subtraction by 2, as the weekdays after the leap day that year slide forward by two days. The expression 6''d'' works the same way. Increasing ''d'' by some number ''y'' indicates that the full moon occurs y days later this year, and hence we should compensate y days less. Adding 6''d'' is mod 7 the same as subtracting ''d'', which is the desired operation. Thus, again, we do subtraction by adding under modulo arithmetic. In total, the variable ''e'' contains the step from the day after the day of the full moon to the nearest following Sunday, between 0 and 6 days ahead. The constant ''N'' provides the starting point for the calculations for each century and depends on where 1 January, year 1 was implicitly located when the Gregorian calendar was constructed. The expression can yield offsets in the range 0 to 35 pointing to possible Easter Sundays on 22 March to 26 April. For reasons of historical compatibility, all offsets of 35 and some of 34 are subtracted by 7, jumping one Sunday back to the day of the full moon (in effect using a negative ''e'' of −1). This means that 26 April is never Easter Sunday and that 19 April is overrepresented. These latter corrections are for historical reasons only and have nothing to do with the mathematical algorithm. The offset of 34 is adjusted if (and only if) ''d'' = 28 and ''d'' = 29 elsewhere in the 19-year cycle. Using the Gauss's Easter algorithm for years prior to 1583 is historically pointless since the Gregorian calendar was not utilised for determining Easter before that year. Using the algorithm far into the future is questionable, since we know nothing about how different churches will define Easter far ahead. Easter calculations are based on agreements and conventions, not on the actual celestial movements nor on indisputable facts of history.


Anonymous Gregorian algorithm

"A New York correspondent" submitted this algorithm for determining the Gregorian Easter to the journal ''
Nature Nature, in the broadest sense, is the natural, physical, material world or universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter an ...
'' in 1876. It has been reprinted many times, e.g., in 1877 by Samuel Butcher in ''The Ecclesiastical Calendar'', in 1916 by Arthur Downing in '' The Observatory'', in 1922 by Harold Spencer Jones, H. Spencer Jones in ''General Astronomy'', in 1977 by the ''Journal of the British Astronomical Association'', in 1977 by Old Farmer's Almanac, ''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, this is also called "Meeus/Jones/Butcher" algorithm: In 1961 the ''New Scientist'' published a version of the ''Nature'' algorithm incorporating a few changes. The variable ''g'' was calculated using Gauss's 1816 correction, resulting in the elimination of variable ''f''. Some tidying results in the replacement of variable ''o'' (to which one must be added to obtain the date of Easter) with variable ''p'', which gives the date directly.


Meeus's Julian algorithm

Jean Meeus, in his book ''Astronomical Algorithms'' (1991, p. 69), presents the following algorithm for calculating the Julian Easter on the Julian Calendar, which is not the Gregorian Calendar used as the civil calendar throughout most of the contemporary world. To obtain the date of Eastern Orthodox Easter on the latter calendar, 13 days (as of 1900 through 2099) must be added to the Julian dates, producing the dates below, in the last row.


See also

*Christian Zeller *Crucifixion darkness *Reform of the date of Easter


References


Notes


Citations


Sources

* . * * * * * * * * * In the fifth volume of Opera Mathematica, Mainz, 1612. Opera Mathematica of Christoph Clavius includes page images of the Six Canons and the ''Explicatio'' (Go to page: Roman Calendar of Gregory XIII). *Constantine the Great, Emperor (325): Letter to the bishops who did not attend the first Nicaean Council; from Eusebius' ''Vita Constantini''. English translations
''Documents from the First Council of Nicea'', "On the keeping of Easter" (near end)
an

* * * *Dionysius Exiguus (525): ''Liber de Paschate''. On-line

an

* * *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. *Gregory XIII (Pope) and the calendar reform committee (1581): the Papal Bull ''Inter Gravissimas'' and the Six Canons. On-line under

, with some parts of Clavius's ''Explicatio''. * * * * * * * * * * * * * * * * * * * *


Further reading

*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. *Coyne, G. V., M. A. Hoskin, M. A., and Pedersen, O. (ed.)

', (Vatican City: Pontifical Academy of Sciences, Specolo Vaticano, 1983). *Gibson, Margaret Dunlop, ''The Didascalia Apostolorum in Syriac'', Cambridge University Press, London, 1903. *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.
Philip Schaff (ed.) Theodoret, Jerome, Gennadius, and Rufinius: historical writings
*Stern, Sacha, ''Calendar and Community: A History of the Jewish Calendar Second Century BCE – Tenth Century CE'', Oxford University Press, Oxford, 2001. *. *Walker, George W, ''Easter Intervals'', Popular Astronomy, April 1945, Vol. 53, pp. 162–178. *Walker, George W, ''Easter Intervals'' (Continued), Popular Astronomy, May 1945, Vol. 53, pp. 218–232. *Weisstein, Eric. (c. 2006)

in ''World of Astronomy''.


External links




The Complete Works of Venerable Bede Vol. 6
(Contains ''De Temporibus'' and ''De Temporum Ratione''.)



* [http://www.staff.science.uu.nl/~gent0113/easter/eastercalculator.htm An Easter calculator with an extensive bibliography, and with useful links]
Ephemeris site of the Bureau des Longitudes with an Easter calculator (valid between 325 and 2500)

A calendar page and calculator by Holger Oertel


* [http://www.nabkal.de/gregkal.html An extensive calendar site 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




* [http://www.e-codices.unifr.ch/en/csg/0378/28/medium St. Gallen, Stiftsbibliothek, Codex Sangallensis 378 (11th century) p. 28. Contains the poem ''Nonae Aprilis norunt quinos''.]
A simplified 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.
Computuslat
A database of medieval manuscripts containing Latin computistical algorithms, texts, tables, diagrams and calendars.

{{Time in religion and mythology Calendar algorithms Easter date Christian terminology