Computus (Latin for computation) is the calculation of the date of Easter in the Christian calendar. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Computation is a general term for any type of Information processing. A calculation is a deliberate process for transforming one or more inputs into one or more results with variable change The Gregorian calendar is the most widely used Calendar in the world today 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). Easter ( Greek: Πάσχα Pascha or Pasxa) is the most important religious feast in the Christian Liturgical year. In Lunar calendars a lunar month is the time between two identical syzygies ( New moons or Full moons. Full moon is a Lunar phase that occurs when the Moon is on the opposite side of the Earth from the Sun. Events 630 - Byzantine emperor Heraclius restores the True Cross to Jerusalem. An equinox is the event of the Sun passing over the Earth's equator in its annual cycle For determining the feast, Christian churches settled on a method to define a reckoned "ecclesiastical" full moon, rather than observations of the true Moon as the Jews did at the time. PLEASE TAKE NOTE************ Eastern Orthodox Christians calculate the fixed date of 21 March according to the Julian Calendar rather than the modern Gregorian Calendar. The Eastern Orthodox Church is the second largest single Christian Communion in the world Events 630 - Byzantine emperor Heraclius restores the True Cross to Jerusalem. The Julian calendar, a reform of the Roman calendar, was introduced by Julius Caesar in 46 BC and came into force in 45 BC (709 Ab urbe condita The Gregorian calendar is the most widely used Calendar in the world today

History

Main article: Easter controversy

Easter is the most important Christian feast. The Easter controversy is a series of controversies about the proper date to celebrate the Christian festival of Easter. Accordingly, the proper date of its celebration has been a cause of much controversy, at least as early as the meeting (c. The Easter controversy is a series of controversies about the proper date to celebrate the Christian festival of Easter. 154) of Anicetus, bishop of Rome, and Polycarp, bishop of Smyrna. The Bishop of Rome is the bishop of the Holy See, more often referred to in the Catholic tradition as the Pope. Saint Polycarp of Smyrna (ca 69 – ca 155 was a second century Bishop of Smyrna. This article is on the Ancient Greek city of Smyrna principally in connection with the ruins remaining to this day The problem for Christians using the Roman civil Julian calendar, which is a solar calendar, was that the passion and resurrection of Jesus occurred during the Jewish feast of Passover, which Jews celebrate according to the Hebrew lunisolar calendar, and fixing the date by the Roman calendar would lead to the celebration of Easter at times unrelated to the Jewish observance of Passover. The Julian calendar, a reform of the Roman calendar, was introduced by Julius Caesar in 46 BC and came into force in 45 BC (709 Ab urbe condita A solar calendar is a Calendar whose dates indicate the position of the earth on its revolution around the Sun (or equivalently the apparent position of the sun moving This article describes the Christian Passion For other meanings see Passion. Jesus of Nazareth (7–2 BC / BCE —26–36 AD / CE) Passover ( Hebrew, Yiddish: פֶּסַח Pesach, Tiberian: pɛsaħ Israeli: Pesah, Pesakh, Yiddish The Hebrew calendar (הלוח העברי ha'luach ha'ivri) or Jewish calendar is a Lunisolar calendar used by Jews for predominantly religious A lunisolar calendar is a Calendar in many Cultures whose date indicates both the Moon phase and the time of the solar Year.

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. The First Council of Nicaea, held in Nicaea in Bithynia (present-day İznik in Turkey) convoked by the Roman Emperor Constantine ^[1] It was also decided to celebrate it always on the dies Domini, Sunday, the day of the week on which Jesus resurrected, which has been the Christian holy day of the week for this reason (the Quartodecimans wished to follow the Jews and always celebrate it on the 14th day of the Jewish month of Nisan, whatever day of the week that might be). In Christianity, the Sabbath is generally a weekly religious Day of rest as ordained by one of the Ten Commandments (the third by Roman Catholic See also Easter controversy, Easter Quartodecimanism (derived from the Vulgate Latin: quarta decima, meaning fourteen The Hebrew calendar (הלוח העברי ha'luach ha'ivri) or Jewish calendar is a Lunisolar calendar used by Jews for predominantly religious This article is about the Jewish month of Nisan See Nissan Motors for the automobile manufacturer ^[2] However, they made few decisions that were of practical use as guidelines for the computation, and it took several centuries before a common method was accepted throughout Christianity.

The method from Alexandria became authoritative. It was based on the epacts of a reckoned moon according to the 19-year cycle (a. The epact (from Greek epaktai hèmerai = added days is a quantification of the difference between the solar and lunar calendars Metonic cycle or Enneadecaeteris in Astronomy and Calendar studies is a particular approximate common multiple of the Tropical year k. a. the Metonic Cycle). Such a cycle was first used by Bishop Anatolius of Laodicea (in present-day Syria), c. Saint Anatolius, (early 3rd century - July 3, 283) was Bishop of Laodicea in Syria, and was one of the foremost scholars of his 277. The Alexandrians may have derived their method from a similar calendar, based on the Egyptian civil solar calendar, used by the Jewish community there; it survives in the Ethiopian computus. The Ethiopian calendar ( Amharic: የኢትዮጵያ ዘመን አቆጣጠር ye'Ītyōṗṗyā zemen āḳoṭaṭer) also called the Ge'ez calendar, Alexandrian Easter tables were composed by Bishop Theophilus about 390 and within the bishopric of Cyril about 444. Theophilus of Alexandria (died 412 was Patriarch of Alexandria, Egypt from 385 to 412 Cyril of Alexandria (ca 378 - 444 was the Pope of Alexandria when the city was at its height of influence and power within the Roman Empire. 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 Eastern Orthodox Church is the second largest single Christian Communion in the world The Alexandrian computus was converted from the Alexandrian calendar into the Julian calendar in Rome by Dionysius Exiguus, though only for 95 years. The Coptic calendar, also called the Alexandrian calendar, is used by the Coptic Orthodox Church and still used in Egypt Dionysius Exiguus ( Dennis the Little or Dennis the Short, meaning humble (c Dionysius introduced the Christian Era (counting years from the Incarnation of Christ) when he published new Easter tables in 525. ^[3][4]

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. For places named after the saint see Saint-Hippolyte Saint Hippolytus of Rome (c Then 84-year tables were introduced in Rome by Augustalis near the end of the third century. The 3rd century is the period from 201 to 300 in accordance with the Julian calendar in the Christian / Common Era. These old tables were used in the British Isles until 664, and by isolated monasteries as late as 931. A modified 84-year cycle was adopted in Rome during the first half of the fourth century. As a means of recording the passage of Time, the 4th century (per the Julian calendar and Anno Domini / Common era) was that 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. Victorius of Aquitaine, a countryman of Prosper of Aquitaine and also working in Rome produced in 457 an Easter Cycle, which was based on the consular list provided ^[5] These Victorian tables were used in Gaul (now France) and Spain until they were displaced by Dionysian tables at the end of the eighth century. Gaul (Gallia was the Roman name for the region of Western Europe comprising present day northern Italy, France, Belgium, western The 8th century is the period from 701 to 800 in accordance with the Julian calendar in the Christian / Common Era.

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 Synod of Whitby was a seventh century Northumbrian Synod where King Oswiu of Northumbria ruled that his kingdom would calculate Easter and The Dionysian reckoning was fully described by Bede in 725. Bede (ˈbiːd (also Saint Bede, the Venerable Bede, or (from Latin Beda (beda (c ^[6] They may have been adopted by Charlemagne for the Frankish Church as early as 782 from Alcuin, a follower of Bede. Charlemagne (ˈʃɑrlɨmeɪn Carolus Magnus or Karolus Magnus meaning Charles the Great) (747 – 28 January 814 was King of the Franks from 768 to his Alcuin of York (Alcuinus or Ealhwine, nicknamed Albinus or Flaccus (c The Dionysian/Bedan computus remained in use in Western Europe until the Gregorian calendar reform, which was mostly designed by Aloysius Lilius. The Gregorian calendar is the most widely used Calendar in the world today Aloysius Lilius (c 1510 &ndash 1576 also Luigi Lilio or Luigi Giglio (or Aluise Lilio or Aloisius Lilius)was an Italian

Theory

The solar year is reckoned to always have 365 days (excluding a small remainder). A lunar year of 12 months is reckoned to have 354 days, meaning the average lunation is 29½ days (excluding another small remainder). In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of 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. This article is about the lunar phase for other uses see New Moon (disambiguation. New Year See also New Year The Ancient Romans began their consular year on January 1st since 153 BC 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). The epact (from Greek epaktai hèmerai = added days is a quantification of the difference between the solar and lunar calendars It is necessary to add them to the day of the solar year to obtain the correct day in the lunar year. Whenever the epact reaches or exceeds 30, an extra (so-called embolismic or intercalary) month has to be inserted into the lunar calendar; then 30 has to be subtracted from the epact. Intercalation is the insertion of a leap day week or month into some calendar years to make the calendar follow the seasons or moon phases

Note that leap days are not counted in the schematic lunar calendar: they are a device to match the calendar year to the tropical year, and can be ignored when dealing with the relation between years and lunations. Leap years Although the modern calendar counts a year as 365 days a complete revolution around the sun takes approximately 365 days and 6 hours The nineteen-year cycle (Metonic cycle) assumes that 19 tropical years are as long as 235 synodic months. Metonic cycle or Enneadecaeteris in Astronomy and Calendar studies is a particular approximate common multiple of the Tropical year A tropical year (also known as a solar year) is the length of time that the Sun takes to return to the same position in the cycle of seasons as seen from Earth The month is a unit of Time, used with Calendars which is approximately as long as some natural period related to the motion of the Moon; 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 (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. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers 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 Number", and is given by the formula

GN = Y mod 19 + 1

That is, the remainder of the year number Y in the Christian era when divided by 19, plus one. The golden numbers (sometimes capitalized are numbers assigned to each Year in sequence to indicate the year's position in a 19-year Metonic cycle. ^[7]

Tabular methods

Gregorian calendar

This method for the computation of the date of Easter was introduced with the Gregorian calendar reform in 1582. The Gregorian calendar is the most widely used Calendar in the world today ^[8]

First determine the epact for the year. The epact (from Greek epaktai hèmerai = added days is a quantification of the difference between the solar and lunar calendars The epact can have a value from "*" (=0 or 30) to 29 days. The first day of a lunar month is considered the day of the new moon. This article is about the lunar phase for other uses see New Moon (disambiguation. The 14th day is considered the day of the full moon. Full moon is a Lunar phase that occurs when the Moon is on the opposite side of the Earth from the Sun.

The epacts for the current Metonic cycle are:

(M=March, A=April)

This table can be extended for previous and following 19-year periods, and is valid from 1900 to 2199. The epact (from Greek epaktai hèmerai = added days is a quantification of the difference between the solar and lunar calendars

The epacts are used to find the dates of New Moon in the following way: Write down a table of all 365 days of the year (the leap day is ignored). Then label all dates with a Roman number counting downwards, from "*" (= 0 or 30), "xxix" (29), down to "i" (1), starting from January 1, and repeat this to the end of the year. However, in every second such period count only 29 days and label the date with xxv (25) also with xxiv (24). Treat the 13th period (last eleven days) as long, though, and assign the labels "xxv" and "xxiv" to sequential dates (December 26 and 27, respectively). Finally, in addition, add the label "25" to the dates that have "xxv" in the 30-day periods; but in 29-day periods (which have "xxiv" together with "xxv") add the label "25" to the date with "xxvi". The distribution of the lengths of the months and the length of the epact cycles is such that each month starts and ends with the same epact label, except for February and for the epact labels xxv and 25 in July and August. This table is called the calendarium. If the epact for the year is for instance 27, then there is an ecclesiastical New Moon on every date in that year that has the epact label xxvii (27).

Also label all the dates in the table with letters "A" to "G", starting from 1 January, and repeat to the end of the year. New Year See also New Year The Ancient Romans began their consular year on January 1st since 153 BC If, for instance, the first Sunday of the year is on 5 January, which has letter E, then every date with the letter "E" will be a Sunday that year. Events 1477 - Battle of Nancy: Charles the Bold is killed and Burgundy becomes part of France. Then "E" is called the Dominical letter for that year (from Latin: dies domini, day of the Lord). Dominical letters are letters A B C D E F and G assigned to days in a cycle of seven with the letter A always set against 1 January as an aid for finding the day of the Week The Dominical Letter cycles backward one position every year. However, in leap years after February 24 the Sundays will fall on the previous letter of the cycle, so leap years have 2 Dominical Letters: the first for before, the second for after the leap day.

In practice, for the purpose of calculating Easter, this need not be done for all 365 days of the year. For the epacts, you will find that March comes out exactly the same as January, so one need not calculate January or February. To also avoid the need to calculate the Dominical Letters for January and February, start with D for 1 March. Events 86 BC - Lucius Cornelius Sulla, at the head of a Roman Republic army enters in Athens, removing the Tyrant You need the epacts only from 8 March to 5 April. Events 1618 - Johannes Kepler discovers the third law of planetary motion. Events 456 - St Patrick returns to Ireland as a missionary bishop This gives rise to the following table:

A table from Sweden to compute the date of Easter 1140-1671 according to the Julian calendar. The Julian calendar, a reform of the Roman calendar, was introduced by Julius Caesar in 46 BC and came into force in 45 BC (709 Ab urbe condita Notice the runic writing.

Example: If the epact is, for instance, 27 (Roman xxvii), then there will be an ecclesiastical new moon on every date that has the label "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. Events 51 - Nero, later to become Roman Emperor, is given the title Princeps iuventutis (head of the youth Events 1043 - Edward the Confessor is crowned King of England. Events 45 BC - In his last victory Julius Caesar defeats the Pompeian forces of Titus Labienus and Pompey the Younger Events 1178 BC - A Solar eclipse may have marked the return of Odysseus, legendary King of Ithaca, to his kingdom

Then Easter Day is the first Sunday after the first ecclesiastical full moon on or after 21 March. Events 630 - Byzantine emperor Heraclius restores the True Cross to Jerusalem.

In the example, this Paschal full moon is on 16 April. Events 1178 BC - A Solar eclipse may have marked the return of Odysseus, legendary King of Ithaca, to his kingdom If the dominical letter is E, then Easter day is on 20 April. Events 1303 - The University of Rome La Sapienza is instituted by Pope Boniface VIII.

The label 25 (as distinct from "xxv") is used as follows: Within a Metonic cycle, years that are 11 years apart have epacts that differ by 1 day. Now short months have the labels xxiv and xxv at the same date, so if the epacts 24 and 25 both occur within one Metonic cycle, then in the short months the new (and full) moons would fall on the same dates for these two years. This is not actually possible for the real Moon: the dates should repeat only after 19 years. To avoid this, in years that have epacts 25 and with a Golden Number larger than 11, the reckoned new moon will fall on the date with the label "25" rather than "xxv". In long months, these are the same; in short ones, this is the date which also has the label "xxvi". This does not move the problem to the pair "25" and "xxvi," because that would happen only in year 22 of the cycle, which lasts only 19 years: there is a saltus lunae in between that makes the new moons fall on separate dates.

The Gregorian calendar has a correction to the solar year by dropping three leap days in 400 years (always in a century year). Leap years Although the modern calendar counts a year as 365 days a complete revolution around the sun takes approximately 365 days and 6 hours This is a correction to the length of the solar year, but should have no effect on the Metonic relation between years and lunations. Therefore the epact is compensated for this (partially—see epact) by subtracting 1 in these century years. The epact (from Greek epaktai hèmerai = added days is a quantification of the difference between the solar and lunar calendars This is the so-called solar equation.

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 one eight times in 2500 (Gregorian) years, always in a century year: this is the so-called lunar equation. The first one was applied in 1800, and it will be applied every 300 years except for an interval of 400 years between 3900 and 4300, which starts a new cycle.

The solar and lunar equations work in opposite directions, and in some century years (for example, 1800 and 2100) they cancel each other. However, it is a bad idea to combine them and make more evenly spread and less frequent epact corrections, as will be explained below. The result of the correct procedure is that the Gregorian lunar calendar uses an epact table that is valid for a period of from 100 to 300 years. The epact table listed above is valid for the period 1900 to 2199.

Details

This method of computation has several subtleties:

Every second lunar month has only 29 days, so one day must have two (of the 30) epact labels assigned to it. The reason for moving around the epact label "xxv/25" rather than any other seems to be the following: According to Dionysius (in his introductory letter to Petronius), the Nicene council, on the authority of 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. Events 1618 - Johannes Kepler discovers the third law of planetary motion. Events 456 - St Patrick returns to Ireland as a missionary bishop Events 630 - Byzantine emperor Heraclius restores the True Cross to Jerusalem. Events 1025 - Bolesław Chrobry is crowned in Gniezno, becoming the first King of Poland. A new moon on 7 March, which has epact label xxiv, has its 14th day (full moon) on 20 March, which is too early (before the equinox date). Events 161 - Roman Emperor Antoninus Pius dies and is succeeded by co-Emperors Marcus Aurelius and Lucius Verus Events 1600 - The Linköping Bloodbath takes place on Maundy Thursday in Linköping, Sweden. So years with an epact of xxiv would have their Paschal new moon on 6 April, which is too late: the full moon would fall on 19 April, and Easter could be as late as 26 April. Events 46 BC - Julius Caesar defeats Caecilius Metellus Scipio and Marcus Porcius Cato in the Battle of Thapsus Events 1012 - Martyrdom of Alphege in Greenwich London. 1529 - At the Second Diet of Speyer Events 1467 - The miraculous image in Our Lady of Good Counsel appear in Genazzano, Italy. In the Julian calendar the latest date of Easter was 25 April, and the Gregorian reform maintained that limit. Events 1607 - Eighty Years' War: The Dutch fleet destroys the anchored Spanish fleet at Gibraltar. So the Paschal full moon must fall no later than 18 April and the new moon on 5 April, which has epact label xxv. Events 1025 - Bolesław Chrobry is crowned in Gniezno, becoming the first King of Poland. Events 456 - St Patrick returns to Ireland as a missionary bishop The short month must therefore have its double epact labels on 5 April: xxiv and xxv. Events 456 - St Patrick returns to Ireland as a missionary bishop Then epact xxv has to be treated differently, as explained in the paragraph above.

As a consequence, 19 April is the date on which Easter falls most frequently in the Gregorian calendar: in about 3. Events 1012 - Martyrdom of Alphege in Greenwich London. 1529 - At the Second Diet of Speyer 87% of the years. 22 March is the least frequent, with 0. Events 238 - Gordian I and his son Gordian II are proclaimed Roman emperor. 48%.

Distribution of the date of Easter for the complete 5,700,000 year cycle.

The relation between lunar and solar calendar dates is made independent of the leap day scheme for the solar year. Basically the Gregorian calendar still uses the Julian calendar with a leap day every four years, so a Metonic cycle of 19 years has 6940 or 6939 days with five or four leap days. Leap years Although the modern calendar counts a year as 365 days a complete revolution around the sun takes approximately 365 days and 6 hours Now the lunar cycle counts only 19 × 354 + 19 × 11 = 6935 days. By not labeling and counting the leap day with an epact number, but having the next new moon fall on the same calendar date as without the leap day, the current lunation gets extended by a day, and the 235 lunations cover as many days as the 19 years. So the burden of synchronizing the calendar with the moon (intermediate-term accuracy) is shifted to the solar calendar, which may use any suitable intercalation scheme; all under the assumption that 19 solar years = 235 lunations (long-term inaccuracy). A consequence is that the reckoned age of the moon may be off by a day, and also that the lunations which contain the leap day may be 31 days long, which would never happen when the real Moon were followed (short-term inaccuracies). This is the price for a regular fit to the solar calendar.

However, there is some protection of the lunar calendar against the errors of the solar calendar. The leap days are not inserted in an optimal way to keep the calendar synchronized to the solar year. The corrections to the leap day scheme are limited to century years, and add two nested intercalation cycles (100 and 400 years) around the four-year cycle. Each cycle accumulates an error, and they add up to more than two days. So in the Gregorian calendar, the actual dates of the vernal equinox are scattered over a time window of about 53 hours around 20 March. An equinox is the event of the Sun passing over the Earth's equator in its annual cycle Events 1600 - The Linköping Bloodbath takes place on Maundy Thursday in Linköping, Sweden. This may be acceptable for a calendar period of a year, but is too much for a monthly period. By separating the "solar equation" from the "lunar equation", this jitter is not carried to the lunar calendar. If we were to combine the solar and lunar equations and spread the net 4×8 - 3×25 = 43 epact subtractions in 10,000 years evenly, then the solar jitter would also affect the lunar calendar, which would be unsatisfactory.

Besides the jitter in the solar calendar, there are also some flaws in the Gregorian lunar calendar (also see D. Roegel (2004) [1]). However, they have no effect on the Paschal month and the date of Easter:

1. Lunations of 31 (and sometimes 28) days occur.
2. If a year with Golden Number 19 happens to have epact 19, then the last ecclesiastical new moon falls on 2 December; the next would be due on 1 January. Events 1409 - The University of Leipzig opens 1755 - The second Eddystone Lighthouse is destroyed by fire New Year See also New Year The Ancient Romans began their consular year on January 1st since 153 BC However, at the start of the new year there is a saltus lunae which increases the epact by another unit, and the new moon should have occurred on the previous day. So a new moon is missed. The calendarium of the Missale Romanum takes account of this by assigning epact label "19" instead of "20" to 31 December of such a year. The Roman Missal ((Missale Romanum is the liturgical book that contains the texts and rubrics for the celebration of the Mass in the Roman Rite Events 406 – Vandals, Alans and Suebians cross the Rhine, beginning an invasion of Gallia. It happened every 19 years when the original Gregorian epact table was in effect (for the last time in AD 1690), and will not happen again until AD 8511.
3. If the epact of a year is "20", then there will be an ecclesiastical new moon on 31 December. Events 406 – Vandals, Alans and Suebians cross the Rhine, beginning an invasion of Gallia. If that year falls before a century year, then in most cases there will be a "solar equation" correction which reduces the epact for the new year by one: the resulting epact "*" means that another ecclesiastical new moon is counted on 1 January; so formally a lunation of one day has passed. New Year See also New Year The Ancient Romans began their consular year on January 1st since 153 BC This will happen around the beginning of AD 4200.
4. Other borderline cases occur (much) later, and if the rules are followed strictly and these cases are not specially treated, they will generate successive new moon dates that are 1, 28, 59, or (very rarely) 58 days apart.

A careful analysis shows that through the way they are used and corrected in the Gregorian calendar, the epacts are actually fractions of a lunation (1/30, also known as tithi) and not full days. In Vedic timekeeping, a tithi (also spelled thithi) is a Lunar day, or the time it takes for the longitudinal Angle between the See epact for a discussion. The epact (from Greek epaktai hèmerai = added days is a quantification of the difference between the solar and lunar calendars

The solar and lunar equations repeat after 4 × 25 = 100 centuries. In that period, the epact has changed by a total of −1 × (3/4) × 100 + 1 × (8/25) × 100 = −43 ≡ 17 mod 30. This is prime to the 30 possible epacts, so it takes 100 × 30 = 3000 centuries before the epacts repeat; and 3000 × 19 = 57,000 centuries before the epacts repeat at the same Golden Number. This period has (5,700,000/19) × 235 + (−43/30) × (57,000/100) = 70,499,183 lunations. So the Gregorian Easter dates repeat in exactly the same order only after 5,700,000 years = 70,499,183 lunations = 2,081,882,250 days. However, the calendar will already have to have been adjusted after some millennia because of changes in the length of the vernal equinox year, the synodic month, and the day.

The drift in ecclesiastical full moons calculated by this method compared to the true full moons is dominated by the gradual slowing of the earth's rotation. Borkowski estimated that in the year 12,000 the Gregorian calendar would fall behind the tropical year by at least 8, but less than 12 days. ^[11]. The drift of full moons would be a similar amount.

Julian calendar

The method for computing the date of the ecclesiastical full moon that was standard for the Roman (Catholic) church before the Gregorian calendar reform, and is still used today by Eastern Christians, made use of an uncorrected repetition of the 19-year Metonic cycle in combination with the Julian calendar. Families of churches Eastern Christians have a shared tradition but they became divided ( Schism) during the early centuries of Christianity in disputes about Metonic cycle or Enneadecaeteris in Astronomy and Calendar studies is a particular approximate common multiple of the Tropical year The Julian calendar, a reform of the Roman calendar, was introduced by Julius Caesar in 46 BC and came into force in 45 BC (709 Ab urbe condita 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. The epact (from Greek epaktai hèmerai = added days is a quantification of the difference between the solar and lunar calendars In this case, the epact was counted on 22 March, the earliest acceptable date for Easter. Events 238 - Gordian I and his son Gordian II are proclaimed Roman emperor. This repeats every 19 years, so there were only 19 possible dates for the ecclesiastical full moons on or after 21 March. Events 630 - Byzantine emperor Heraclius restores the True Cross to Jerusalem.

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 3 days every millennium, and 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 often 4 or 5 weeks later because the Julian March 21 is now 13 days later than the Gregorian March 21. )

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. Alexander of Villedieu was a French author and a mathematician A later scribe added it to tables originally composed by Abbo of Fleury in 988. Abbo of Fleury (in Latin Abbo Floriacensis) also known as Abbon or Saint Abbo' (c

This is the table:

(M=March, A=April)

Easter day is the first Sunday after these dates.

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 4 years, the full cycle in which weekdays recur in the calendar in the same way, is 4 × 7 = 28 years, the so-called solar cycle. So the Easter dates repeated in the same order after 4 × 7 × 19 = 532 years. This Paschal cycle is also called the Victorian cycle, after Victorius of Aquitaine, who introduced it in Rome in AD 457. Victorius of Aquitaine, a countryman of Prosper of Aquitaine and also working in Rome produced in 457 an Easter Cycle, which was based on the consular list provided It is first known to have been used by Annianus of Alexandria at the beginning of the fifth century. Annianus of Alexandria or Annianos was a monk who flourished in Alexandria during the Bishopric of Theophilus of Alexandria around the beginning The 5th century is the period from 401 to 500 in accordance with the Julian calendar in Anno Domini / Common Era. It has also sometimes erroneously been called the Dionysian cycle, after Dionysius Exiguus, who prepared Easter tables that started in AD 532; but he apparently did not realize that the Alexandrian computus which he described had a 532-year cycle, although he did realize that his 95-year table was not a true cycle. Dionysius Exiguus ( Dennis the Little or Dennis the Short, meaning humble (c 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. Bede (ˈbiːd (also Saint Bede, the Venerable Bede, or (from Latin Beda (beda (c

Algorithms

Gauss's algorithm

This algorithm for calculating the date of Easter Sunday was first presented by the mathematician Carl Friedrich Gauss:

The number of the year is denoted by Y; mod denotes the remainder of integer division (e. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Easter ( Greek: Πάσχα Pascha or Pasxa) is the most important religious feast in the Christian Liturgical year. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German g. , 13 mod 5 ≡ 3; see modular arithmetic). In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers Calculate first a, b, and c:

a = Y mod 19

b = Y mod 4

c = Y mod 7

Then calculate

d = (19a + M) mod 30

e = (2b + 4c + 6d + N) mod 7

For the Julian calendar (used in Eastern churches), M = 15 and N = 6, and for the Gregorian calendar (used in Western churches), M and N are from the following table:

If d + e < 10 then Easter is on the (d + e + 22)th of March, and is otherwise on the (d + e − 9)th of April. The Julian calendar, a reform of the Roman calendar, was introduced by Julius Caesar in 46 BC and came into force in 45 BC (709 Ab urbe condita The Gregorian calendar is the most widely used Calendar in the world today

The following exceptions must be taken into account:

• If the date given by the formula is 26 April, Easter is on 19 April. Events 1467 - The miraculous image in Our Lady of Good Counsel appear in Genazzano, Italy. Events 1012 - Martyrdom of Alphege in Greenwich London. 1529 - At the Second Diet of Speyer
• If the date given by the formula is 25 April, with d = 28, e = 6, and a > 10, Easter is on 18 April. Events 1607 - Eighty Years' War: The Dutch fleet destroys the anchored Spanish fleet at Gibraltar. Events 1025 - Bolesław Chrobry is crowned in Gniezno, becoming the first King of Poland.

Meeus/Jones/Butcher Gregorian algorithm

This algorithm for calculating the date of Easter Sunday is given by Jean Meeus in his book Astronomical Algorithms (1991), which in turn cites Spencer Jones in his book General Astronomy (1922) and also the Journal of the British Astronomical Association (1977). In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Jean Meeus (born 1928 is a Belgian Astronomer specializing in Celestial mechanics. Sir Harold Spencer Jones KBE ( March 29 1890 Kensington, London &ndash November 3 1960) was an English This algorithm also appears in The Old Farmer's Almanac (1977), p. The Old Farmer's Almanac is a reference book that contains Weather forecasts Tide tables planting charts astronomical data Recipes 69. The JBAA cites Butcher's Ecclesiastical Calendar (1876).

The method is valid for all Gregorian years and has no exceptions and requires no tables.

Notation is as for the Gauss Algorithm above: all quotients are truncated to integers, thus 7 / 3 = floor(7 / 3) = 2 (not 2 1/3), and 7 mod 3 = 1. In Mathematics, truncation is the term for limiting the number of digits right of the Decimal point, by discarding the least significant ones In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The

Meeus Julian algorithm

Jean Meeus, in his book Astronomical Algorithms (1991), also presents the following formula for calculating Easter Sunday in Julian years. Events 68 - Galba, Governor of Hispania, names himself legatus senatus populique Romani, breaking the line of Year 1961 ( MCMLXI) was a Common year starting on Sunday (link will display full calendar of the Gregorian calendar. Events 1174 - Jocelin, Abbot of Melrose, is elected Bishop of Glasgow. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Jean Meeus (born 1928 is a Belgian Astronomer specializing in Celestial mechanics.

The method is valid for all Julian years and has no exceptions and requires no tables.

Notation is as for the Gauss Algorithm above: all values are integers, thus 7 / 3 = 2 (not 2 1/3), and 7 mod 3 = 1.

a = Y mod 4

b = Y mod 7

c = Y mod 19

d = (19 × c + 15) mod 30

e = (2 × a + 4 × b - d + 34) mod 7

month = (d + e + 114) / 31

day = ((d + e + 114) mod 31) + 1

See also

• Crucifixion eclipse
• Easter controversy
• Reform of the date of Easter
• Christian Zeller

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: [2] [3]
• 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)
• 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) [4].
• 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. [5]

External links

• The entry on epacts in the Catholic Encyclopedia of 1911
• The original texts of the Gregorian calendar reform (in Latin), with translations into French by Rodolphe Audette
• An Easter calculator by Rob van Gent (with an extensive bibliography, and with useful links)
• Ephemeris site of the Bureau des Longitudes with an Easter calculator
• A calendar page and calculator by Holger Oertel
• A page from Clive Feather with a brief explanation, some more tables, and another algorithm
• (German) An extensive calendar site (auf Deutsch) 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
• Gregorian Lunar Calendar: A table of the Gregorian New Moons for 1900-2199
  • A table of Gregorian New Moons for the years 1700-1899, from the same source
  • A similar table for the years 2200-2299
  • A calendar dividing the Gregorian lunar year 2003 divided into 30-day and 29-day lunar months
  • A discussion of the Nicene Council's Easter decision and of how some commentators exaggerate the scope of the Council's decision
• Improved algorithm by Dr. Heiner Lichtenberg (in German)
• Dionysius Exiguus' Easter table
• Mnemonic Computus Diagrams of Hands from manuscript in The British Library
• Lutheran Table for Finding Easter/Pascha Dates
• Towards a Common Date for Easter World Council of Churches (Faith and Order) and Middle East Council of Churches consultation; Aleppo, Syria; March 5 - 10, 1997
• A simple method for determining the date of Easter for all years 326 to 4099 A.D.
• Text of the Calendar (New Style) Act 1750, British Act of Parliament introducing the Gregorian Calendar as amended to date. The German language (de ''Deutsch'') is a West Germanic language and one of the world's major languages. The World Council of Churches ( WCC) is an international Contains tables for calculating Easter up until the year 8599.