Date of Easter
As a moveable feast,[1][2] the date of Easter is determined in each year through a calculation known as computus (Latin for 'computation').[3] Easter is celebrated on the first Sunday after the Paschal full moon (a mathematical approximation of the first astronomical full moon, on or after 21 March – itself a fixed approximation of the March equinox). Determining this date in advance requires a correlation between the lunar months and the solar year, while also accounting for the month, date, and weekday of the Julian or Gregorian calendar.[4] The complexity of the algorithm arises because of the desire to associate the date of Easter with the date of the Jewish feast of Passover which, Christians believe, is when Jesus was crucified.[5]
This article is about how the date of Easter is calculated. For actual dates of Easter from 2015 to 2030, see List of dates for Easter.
It was originally feasible for the entire Christian Church to receive the date of Easter each year through an annual announcement by the pope. By the early third century, however, communications in the 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 yet consistently.[6] Additionally, the church wished to eliminate dependencies on the Hebrew calendar, by deriving the date for Easter directly from the March equinox.[7]
In The Reckoning of Time (725), 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.[8]
The calculations produce different results depending on whether the Julian calendar or the Gregorian calendar is used. For this reason, the Catholic Church and Protestant churches (which follow the Gregorian calendar) celebrate Easter on a different date from that of the Eastern and Oriental Orthodoxy (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.
History[edit]
The earliest known Roman tables were devised in 222 by Hippolytus of Rome based on eight-year cycles. Then 84-year tables were introduced in Rome by Augustalis near the end of the 3rd century.[a] Although a process based on the 19-year Metonic cycle was first proposed by Bishop Anatolius of Laodicea around 277, the concept did not fully take hold until the Alexandrian method became authoritative in the late 4th century.[b]
The Alexandrian computus was converted from the Alexandrian calendar into the Julian calendar in Alexandria around 440, which resulted in a Paschal table (attributed to pope Cyril of Alexandria) covering the years 437 to 531.[16] This Paschal table was the source which inspired Dionysius Exiguus, who worked in Rome from about 500 to about 540,[17] to construct a continuation of it in the form of his famous Paschal table covering the years 532 to 616.[18] Dionysius introduced the Christian Era (counting years from the Incarnation of Christ) by publishing this new Easter table in 525.[19][c]
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.[20] 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.
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.[21] The discrepancy led to a report that Queen Eanflæd, on the Dionysian system – fasted on her Palm Sunday while her husband Oswiu, king of Northumbria, feasted on his Easter Sunday.[22]
As a result of the Irish Synod of Magh-Lene in 630, the southern Irish began to use the Dionysian tables,[23] and the northern English followed suit after the Synod of Whitby in 664.[24]
The Dionysian reckoning was fully described by Bede in 725.[25] It 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, and remains in use in most Eastern Churches, including the vast majority of Eastern Orthodox Churches and Non-Chalcedonian Churches.[26] 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,[27] now celebrate Easter on different dates from Eastern Orthodox Churches 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.[d] 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 was out of phase with the real Moon by four days. The Gregorian Easter has been used since 1583 by the Roman Catholic Church and was adopted by most Protestant churches between 1753 and 1845.
German Protestant states used an astronomical Easter between 1700 and 1776, based on the Rudolphine Tables of Johannes Kepler, which were in turn based on astronomical positions of the Sun and Moon observed by Tycho Brahe at his Uraniborg observatory on the island of Ven, 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 (TT + 51m). However, it was delayed one week if that Sunday was the Jewish date Nisan 15, the first day of Passover week, calculated according to modern Jewish methods.[29]
This Nisan 15 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.[29] 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.[29]
Two modern astronomical Easters were proposed but never used by any Church. The first was proposed as part of the Revised Julian calendar at a Synod in Constantinople in 1923 and the second was proposed by a 1997 World Council of Churches Consultation in Aleppo in 1997. Both used the same rule as the German and Swedish versions but used modern astronomical calculations and Jerusalem time (TT + 2h 21m) without the Nisan 15 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.[30] 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.[31]
Algorithms[edit]
Note on operations[edit]
When expressing Easter algorithms without using tables, it has been customary to employ only the integer operations addition, subtraction, multiplication, division, 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 if (DoM > 31) {Day=DoM-31, Month=Apr} else {Day=DoM, Month=Mar}. More importantly, using such conditionals also simplifies the core of the Gregorian calculation.
Gauss's Easter algorithm[edit]
In 1800, the mathematician Carl Friedrich Gauss presented this algorithm for calculating the date of the Julian or Gregorian Easter.[71][72] He corrected the expression for calculating the variable p in 1816.[73] In 1800, he incorrectly stated p = floor (k/3) = ⌊k/3⌋. In 1807, he replaced the condition (11M + 11) mod 30 < 19 with the simpler a > 10. In 1811, he limited his algorithm to the 18th and 19th centuries only, and stated that 26 April is always replaced with 19 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.[74]