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In 325, at the Council of Nicæa the date of Easter was fixed by declaring that Easter should be the first Sunday after the first full moon occuring on or after the vernal equinox. This declaration was intended to be the end of a long dispute between eastern and western Christian communities.
Fixing the date of Easter exactly was almost impossible at that time, the vernal equinox and the full moons having had to be determined in advance. Therefore, Easter was computed using a cyclic calculation which made it necessary to fix the vernal equinox and to create a lunar calendar for predicting full moons.
The Council of Nicæa gave the task of calculating the date of Easter to the Church of Alexandria, which was to communicate the result to the pope for further distribution to the Christian communities. The computation described in this article is the one used by the Roman church since its introduction in the 6th century by Dionysius Exiguus. There are some slight differences between these rules and the original Alexandrian calculation, which have no effect on the date of Easter.
The Roman Empire, the western part of which eventually broke down under the pressure of "Barbarian" tribes in 476, covered a vast territory, which led to different dates of Easter celebrated by various communities. In the province of Asia, which was the north western part of what is now Turkey, Easter was celebrated on the Jewish 14 Nisan, regardless of the day of week. Many of that communities did so even after the Council of Nicæa. But, by the end of the 4th century, 14 Nisan was observed by some sectarians only.
The Churches of Alexandria and Rome had different points of view concerning the determination of Easter. In the 4th century, Easter was celebrated on different days several times. The Alexandrian Easter could fall on any day between 22 March and 25 April, while in Rome Easter was to fall between 25 March and 21 April. The Roman Church extended these so called Easter limits in 343, but the last differences were removed only in the beginning 6th century with the Easter tables of Dionysius Exiguus.
In Ireland, together with the Christian belief an 84 year cycle was introduced, which was later brought to Gaul and the Franks. In 729, this so called Irish-Gaulish Easter dispute was settled with the 532 year paschal cycle being introduced in Britain. In the Frankish empire the cycle of Victorius was adopted during the 5th century. The dates of Easter of the Victorian cycle sometimes differed from those of the Dionysian cycle. Nevertheless, both cycles consisted of 532 years. Under the rule of Charles the Great, in the end of the 8th century, the Dionysian cycle was introduced in the Frankish empire.
Considering the Julian calendar as being constantly aligned with the astronomical events, fixing the vernal equinox presented no problems. In the year 325, the astronomical vernal equinox occured around 21 March, to which it had already moved from 24 March since the introduction of the Julian calendar under Caesar. This led to 21 March being considered as the vernal equinox for computing Easter. Therefore, the earliest date of Easter became 22 March.
The calculation of the moon's phases was based on the so called Metonic cycle. This cycle is named after the Greek astronomer and mathematician Meton who lived in Athens in the 5th century BCE. Nevertheless, Babylonian astronomers had discovered earlier already that 235 (syndodic) lunar months have about the same length as 19 (tropical) years. Therefore it was concluded that the new moons must fall on the same dates every 19 years. To designate the years within this 19-year-cycle the so-called Golden Number was calculated. That is the remainder of the division of the number of the year by 19 increased by 1. For example, the year 1492 bore the Golden Number 11. The new moons of all years with the same Golden Number should fall on the same dates.
In 322 CE an astronomical new moon ocurred in the evening of 24 December. The Golden Number of that year(1) was 19. With the Christian ecclesiastical year beginning on Christmas Day (25 December), the start of each 19 year cycle was put on 24 December of years with the Golden Number 19.
Beginning with 24 December of a year with the Golden Number 19, cyclic lunar months of 30 and 29 days were counted, each 30-day-month being followed by a month of 29 days. 24 December was the first day of a 30-day-month. The last day of this month was 22 January, and on 23 January a month of 29 days began. A true lunar month being bit longer than 29,5 days, seven leap months of 30 days each were inserted in every 19 year cycle. The leap days of the solar leap years were inserted without counting them in the lunar calendar. In the last year of the 19 year cycle one day was ommitted to let the new cycle begin on 24 December. This was called saltus lunae. (2)
In this calendar, the mean length of a month was 29.53085 days. The true length of a lunar month, i. e. the time between two consecutive new moons, is 29.53059 days. The difference between the cyclic calendar and the true phases of the moon amounted to a whole day in about 310 years.(3)
Using the lunar calendar described above the new moons were determined. The following table shows the cyclic new moons of a 19 year Metonic cycle. Behind the names of the months, the days on which new moons occur in each of the 19 years, are given. The new moons of the seven leap months are given in bold letters. 24 December of year 19 was the cyclic new moon with which each 19-year-cycle began.
|Month Golden Number||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15||16||17||18||19|
The number of the day within the current lunar month indicated the moon's phase, the day of the new moon being counted as the first day of the month. The cyclic full moon was considered to occur on day 14 of the month. For example, the table above shows that in a year bearing the Golden Number 8 a new moon occurs on 4 February. The following (cyclic) full moon therefore takes place on 17 February. The first full moon occuring on or after 21 March was called Easter limit (terminus paschalis). For instance, in years with the Golden Number 7 the Easter limit was 30 March.
With the Easter limit determined, one had to find the Sunday following that day. This was done with the Dominical Letters. Every day of the year was marked with a letter, beginning with an A on 1 January. 2 January was given the letter B and so on until 7 January, which was marked with a G. On the next day, 8 January, the sequence of the letters was restarted, and this day got the letter A. This was done throughout the whole year, leaving the leap day aside. To find all Sundays of a year, it was only necessary to write down the letter of the first Sunday in that year. In 1307, 1 January was a Sunday, so in the tables this year was marked with the Dominical Letter A. Any day marked A was a Sunday in that year, too.
In leap years the insertion of a leap day resulted in a shift of the days of week by one day. Such years bore two Dominical Letters, the first of which represented the Sundays in January and February. The second Dominical Letter stood for the Sundays in the months from March to December. The year 1320, for example, had the Dominical Letters FE. The first Sunday of that year was therefore 6 January. All days of January and February that year, which were marked with the letter F, were Sundays, while from March until December all Sundays had an E written beside them.
To find Easter Sunday, the only thing to do was to go forward until the next day bearing the Dominical Letter of the year. In case of a leap year the second of the two letters was to be used.
As an illustration, Easter Sunday of 1311 is to be found. First, the Golden Number is computed with (1311 mod 19) + 1 = 1. The Dominical Letter of 1311 is C, and the first full moon on or after 21 March occurred on 5 April, which is the full moon following the new moon on 23 March. 5 April has the letter D, and the next day marked C is 11 April, which is Easter Sunday.
For reasons of convenience epacts were calculated. The epact was the moon's phase on 22 March. With a lunar year of 354 days being 11 days shorter than a Julian year of 365 days, the epact of one year was 11 units larger than the epact of the preceeding year. In the first year of the Metonic cycle the epacts was 0, while in the second its value was 11. The third year had the epact 22 and the fourth an epact of 3, the increase of the epacts being done modulo 30, as a lunar month had not more than 30 days. The epact of the last year of the lunar cycle was 18. The ommission of one day in the lunar calendar that year made the epact "jump" by 12 units, which explains the Latin "saltus lunae".(4)
With the epacts the determination of the Easter limit was done easily by going 14 days forward from 22 March and going back from there as many days as the value of the epact. If the resulting date lay before 22 March, one had to go forward another 30 days to find the Easter limit. It is obvious, that a certain Golden Number leads to a certain Easter limit. To find the following Sunday, the letter of the Easter limit could have been added. The result was a table like this one:- (M = March, A = April)
The foregoing rules imply that the date of Easter of a year is solely determined by the Golden Number and the Dominical Letter(s) of that year. As mentioned above, in leap years the second of the two Dominical Letters is to be used. In the following table the date of Easter could easily be found.
|1||9 A||10 A||11 A||12 A||6 A||7 A||8 A|
|2||26 M||27 M||28 M||29 M||30 M||31 M||1 A|
|3||16 A||17 A||18 A||19 A||20 A||14 A||15 A|
|4||9 A||3 A||4 A||5 A||6 A||7 A||8 A|
|5||26 M||27 M||28 M||29 M||23 M||24 M||25 M|
|6||16 A||17 A||11 A||12 A||13 A||14 A||15 A|
|7||2 A||3 A||4 A||5 A||6 A||31 M||1 A|
|8||23 A||24 A||25 A||19 A||20 A||21 A||22 A|
|9||9 A||10 A||11 A||12 A||13 A||14 A||8 A|
|10||2 A||3 A||28 M||29 M||30 M||31 M||1 A|
|11||16 A||17 A||18 A||19 A||20 A||21 A||22 A|
|12||9 A||10 A||11 A||5 A||6 A||7 A||8 A|
|13||26 M||27 M||28 M||29 M||30 M||31 M||25 M|
|14||16 A||17 A||18 A||19 A||13 A||14 A||15 A|
|15||2 A||3 A||4 A||5 A||6 A||7 A||8 A|
|16||26 M||27 M||28 M||22 M||23 M||24 M||25 M|
|17||16 A||10 A||11 A||12 A||13 A||14 A||15 A|
|18||2 A||3 A||4 A||5 A||30 M||31 M||1 A|
|19||23 A||24 A||18 A||19 A||20 A||21 A||22 A|
The Easter limits repeated after 19 years, while the days of the week fell on the same dates after 28 years in the Julian calendar. Therefore, the sequence of the dates of Easter repeated after 19 · 28 = 532 years. This cycle was called the Dionysian (after Dionysius Exiguus) or Victorian (after Victorius) cycle. The earliest possible date for Easter sunday is 22 March, the latest 25 April.
The Metonic cycle approximated the true phases of the moon only roughly. Because of the error increasing by one day every 310 years the true new moons soon occured earlier than the cyclic new moons, the difference amounting to almost 4 days in the 16th century. Therefore the Gregorian calendar reform included a new method of computing the date of Easter.
The leading architects of the Gregorian calendar are (Luigi Lilio or Giglio Ghiraldi) and Christoph Clavius (Christoph Klau). The lunar cycle used in the improved Easter calculation has 19 years as in the old calendar, but corrections are made in certain years.
The cyclic calculation of the moon is mainly attributed to Lilius. As in the Julian calendar there are lunar months with alternationg lengths of 30 days and 29 days. To determine the cyclic new moons the so-called Eternal Gregorian Calendar was designed, of which a description now follows. In that calendar, an epact (of the new style) is assigned to every day of the year with the exception of 29 February. 1 January has the epact 0 (Lilius wrote * instead of 0), a 30 day month beginning with that date. The days are counted backwards from 29 to 1. Thus, 2 Jan is marked with 29, 3 Jan with 28 etc. The last day of this lunar month is 30 January which receives the epact 1. 31 January is the start of a lunar month of 29 days. Because these months are one day shorter, the epacts 24 and 25 are put on the same date. All days are given an epact in the same manner (i. e. alternating months of 30 and 29 days). The result is the Eternal Gregorian Calendar as shown in the following table. (The marked numbers will be explained later.)
|4||27||26 25||27||26 25||25||23||23||21||20||19||18||17||4|
|5||26||25 24||26||25 24||24||22||22||20||19||18||17||16||5|
To determine the cyclic new moons of a year the epact of that year is used. This (yearly) epact must be distinguished from the (daily) epacts described above and contained in the Eternal Gregorian Calendar. The (yearly) epacts are following a 19-year-cycle like the old style epacts. At the time of the introduction of the Gregorian Calendar years with the Golden Number 1 bore the epact 1. The epact of a year increased by 11 compared to the preceeding year, the increase done modulo 30. The epacts of one cycle of 19 years constitute the epact cycle. The cycle valid at the introduction of the new calendar can be seen in the following table.
Combining epact cycle and Eternal Gregorian Calender the cyclic new moons of any year can be found quite easily by computing the Golden Number and finding the corresponding epact of the year. On any day marked with that epact in the Eternal Gregorian Calendar there is a cyclic new moon. The following cyclic full moon can be found by going forward 13 days.
So far the method in principal is the same as in the old calendar. To correct the error of the Julian calculation in the lengths of month and year further adaptions became necessary. Two points had to be taken into account. These were
These shifts are dealt with by adjusting the yearly epacts in certain years.
Caused by the different intervals of Sun Equation and Moon Equation in century years one of the Equations can occur alone or both can coincide, neutralising each other. In the latter case the epacts do not change. The following table shows the effects of the Sun and Moon Equations on the yearly epact of years with the Golden Number 1.
The yearly epacts resulting from these rules are compiled in the next table. Shown are the epacts depending on the Golden Number and the period in which a year occurs.
|E p a c t s|
The reform was to leave as much as possible unchanged compared to the Julian calendar. The shift of the epacts by the Sun and Moon Equations brought along a problem which had to be tackled with an exception. The date of Easter were to fall on dates between 22 March and 25 April, inclusively. These limits the reform intended to preserve. Another property of the Julian moon calculation which should remain untouched was that in every year of the 19-year cycle the cyclic new and full moons occured on different dates.
The Easter limits Lilius preserved by putting the epacts 24 and 25 on one day in the short lunar months. With the 25 on the day following the day with the daily epact 24 the Easter full moon would occur on 19 April. The resulting Easter Sunday would be 26 April which is outside the Easter limits.
The dates of the cyclic new moons presented a problem more intricate. The effect of the Sun and Moon Equations on the epacts makes possible the occurrence of both epacts (24 and 25) in one epact cycle. With these epacts put on the same dates in the Eternal Gregorian Calendar the same cyclic new moons would occur in two years of the 19-year cycle. This problem was solved by moving the daily epact 25 from the day with the epact 24 to the one with 26, i. e. 25 and 26 are put on one day instead of 24 and 25 whenever 24 and 25 occur in one epact cycle. This is marked in the table showing the Eternal Gregorian Calendar.
The error of the new moon calculation of the Gregorian calendar amounts to one day in about 70,000 years only. Of course, this has no practical meaning, as the error of the solar calendar itself as well as secular changes in the lengths of year and month will make necessary a correction until then already.
In 1700, Protestant countries of the Holy Roman Empire adopted the civil part of the Gregorian calendar, but did not compute Easter according to the Catholic algorithm. They used astronomical tables instead, which made Protestant and Catholic Easter fall on different days in 1724 and 1744. Finally, in 1776, the cyclic calculation was agreed upon between Protestants and Catholics in Germany, and the Gregorian calendar was introduced as "Common Imperial Calendar" ("Allgemeiner Reichskalender").
In Sweden, by ommitting the leap day in 1700, the calendar was one day ahead of the Julian calendar. The dates of Easter in Sweden were one unit higher than the Julian dates. In the years 1705, 1709, and 1711 Easter was celebrated a week earlier, the dates of Easter were six units less than those of the Julian calendar. In 1712, the Julian calendar was re-adopted by inserting 30 February, and the dates of Easter were those of the Julian calendar. In 1740 things became weird: Despite using the Julian calendar, Easter was celebrated together with the other Protestants. The dates of Easter in Sweden were out of the Easter limits (22 March until 25 April) sometimes. In 1742, for instance, the Swedish date of Easter was 14 March, which corresponded to the Gregorian 25 March.
The Swedish dates of Easter in the years 1700 until 1752 are given in the following table. The column "(Swed.)" shows the dates in the calendar valid in Sweden at the time, while the column "(Greg.)" shows the Gregorian date of the day Easter was celebrated in Sweden.
|Year||Easter Sunday in Sweden|
|1700||1 A||11 A|
|1701||21 A||1 Mai|
|1702||6 A||16 A|
|1703||29 M||8 A|
|1704||17 A||27 A|
|1705||2 A||12 A|
|1706||25 M||4 A|
|1707||14 A||24 A|
|1708||5 A||15 A|
|1709||18 A||28 A|
|1710||10 A||20 A|
|1711||26 M||5 A|
|1712...1739 Julian Easter Dates|
|1740||6 A||17 A|
|1741||22 M||2 A|
|1742||14 M||25 M|
|1743||3 A||14 A|
|1744||18 M||29 M|
|1745||7 A||18 A|
|1746||30 M||10 A|
|1747||22 M||2 A|
|1748||3 A||14 A|
|1749||26 M||6 A|
|1750||18 M||29 M|
|1751||31 M||11 A|
|1752||22 M||2 A|
The Protestants in Germany abolished their "astronomical" calculation of Easter in 1776, but Sweden continued to compute the date of Easter in that way until as late as 1844. In 1802, 1805, and 1818, in Sweden Easter was celebrated a week later.
Finland was a part of Sweden until 1809, when it fell to Russia. Until that year, the date of Easter in Finland was that of Sweden. When Finland was seized from Sweden by Russia, the Gregorian calendar was not abolished. Except of the years 1825, 1829, and 1845, when in Finland Easter Sunday was observed a week later, Gregorian Easter was celebrated.
The rules for determining Easter are quite intricate, and many mathematicians tried to find easy-to-use formulas for this problem over the centuries. Widely known are Carl Fiedrich Gauß's formulas for both the Julian and the Gregorian calendars. To calculate the Gregorian date of Easter it is necessary to use auxiliary values, and two exceptions must be observed.
In the August edition of the journal "Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde" (that may be in some way translated as "Monthly Correspondence for Advocating Earth and Celestial Lore") of the year 1800 the mathematician C F Gauß, then aged only 23 years, published Easter formulas. For a given year Y Gauß computes the date of Easter as follows.
|a||=||Y mod 19,|
|b||=||Y mod 4,|
|c||=||Y mod 7,|
|d||=||(19 · a + M) mod 30 und|
|e||=||(2 · b + 4 · c + 6 · d + N) mod 7.|
The date of Easter is (22 + d + e) March or (d + e - 9) April.
M und N are constant for the Julian calendar, whereas they vary in certain periods of time for the Gregorian calendar. For the latter there are two exceptions shown later.
For the Julian calendar we have M = 15 and N = 6, thus the formulas are
|a||=||Y mod 19|
|b||=||Y mod 4|
|c||=||Y mod 7|
|d||=||(19 · a + 15) mod 30|
|e||=||(2 · b + 4 · c + 6 · d + 6) mod 7|
The values of M and N for the Gregorian calendar are to be found in the following table.
Comparing the table of new style epacts and this table it can be seen, that M changes according to the epacts. N is increased in all century years not divisible by 400. M is changed modulo 30, N modulo 7.
The two exceptions are caused by the rules with which Lilius intended to retain the Easter limits and to avoid identical Paschal full moon dates within a 19-year-cycle of the Gregorian calendar. The exceptions are
Instances of the first exception are the years 1609, 1981, 2076, and 2133. The second exception occured in 1954 and will apply again in 2049 and 2106.
Gauß also gave formulas for the values of M and N, as follows:
|k||=||int(Y / 100)|
|p||=||int(k / 3)|
|q||=||int(k / 4)|
|M||=||(15 + k − p − q) mod 30 (see below)|
|N||=||(4 + k − q) mod 7|
However, these formulas are erroneous since they imply the application of the lunar equation every 300 years, not taking into account the 400-year-interval before the eighth application of the lunar equation at the end of each 2500-year-cycle. Thus the formulas start to yield incorrect values increasingly often starting with the year 4200.
J M Oudin dealed with Gauß's erroneous formulas in his treatise "Étude sur la date de Pâques", published in the "Bulletin astronomique" in 1940. Oudin introduced a correction to the formula for p (5). His formulas for p and the correction are
|x||=||int((k − 17) / 25),|
|p||=||int((k − x)/ 3).|
With M computed using this corrected p, Gauß's formulas yield correct Easter dates also after the year 4199.
In his treatise, Oudin called the number of the year m (for millésime) and gave these formulas for calculating the date of Easter.
|c||=||int(m / 100)|
|k||=||int((c − 17) / 25)|
|r||=||(15 + c − int(c / 4) − int((c − k) / 3) + 19 · (m mod 19)) mod 30|
|R||=||r − 1||, falls r = 29|
|=||r − 1||, falls r = 28 und (m mod 19) > 10|
|J||=||(3 · (m mod 7) + 5 · (m mod 4)+R + 2 - c + int(c / 4)) mod 7|
Easter sunday is (28 + R − J) March, or (R − J − 3) April of the year m.
In the "Nature" magazine of 20 April 1876, an anonymous author published a set of instructions to calculate the (Gregorian) date of Easter. Put into formulas they are as follows.
|a||=||Y mod 19|
|b||=||int(Y / 100)|
|c||=||Y mod 100|
|d||=||int(b / 4)|
|e||=||b mod 4|
|f||=||int((b + 8) / 25)|
|g||=||int((b − f + 1) / 3|
|h||=||(19 · a + b − d − g + 15) mod 30|
|i||=||int(c / 4)|
|k||=||c mod 4|
|l||=||(32 + 2 · e + 2 · i − h − k) mod 7|
|m||=||int((a + 11 · h + 22 · l) / 451)|
|n||=||int((h + l − 7 · m + 114) / 31)|
|o||=||(h + l − 7 · m + 114) mod 31|
n is the number of the month and o + 1 the number of the day on which Easter Sunday falls in the respective year Y. The algorithm does not need any auxiliary values.
Many Christian holidays depend on the date of Easter. Shrove tuesday is celebrated 47 days before Easter, Ash Wednesday 46 days before, while Ascension Day is 39 days after Easter Sunday. Whitmundy is 50 days after, Corpus Christi 60 days after Easter Sunday, except in the US, where it is observed 3 days later.
The Easter Calculator can be used to compute the date of Easter in any Julian or Gregorian year using Gauß's formulas for the Julian calendar and Oudin's algorithm for the Gregorian calendar. Note, that the year must be entered with all digits and cannot be abbreviated (as for instance 72 for 1972). Until the 8th century, many different dates of Easter were celebrated by several Christian churches. The Calculator gives the Dionysian date only.
In case of a year with another beginning than 1 January, this will be mentioned explicitly.
The algorithm described is the one used in the Occident. Bede ommitted the day in the last year of the lunar cycle by shortening the lunar month beginning on 27 October by a day.
The Alexandrians took this day from the lunar month beginning on 1 July, shortening this month to 29 days. With this, the last column in the table given later looks a bit different: July - 1,30; August - 28; September - 27; October - 26; November - 25.
A cycle of 19 years can contain 4 or 5 leap years. To compute the mean length of a cyclic lunar month one has to use a cycle of 4 · 19 = 76 years, which consists of 4 · 114 months with 30 days each, 4 · 114 months with 29 days each, 4 · 7 leap months with 30 days each, and the leap days of the 19 leap years within the cycle. Four days must be subtracted because of the saltus lunae (of which is carried out one in each of the four lunar cycles). This results in the four lunar cycles having 4 · (114 · 30 + 114 · 29 + 7 · 30) + 19 - 4 = 27759 days and 4 · 235 = 940 lunar months. Thus, the mean length of a lunar month is 27759 / 940 = 29,53085 days.
The error of a cyclic lunar month compared with a true one is about 22.5 seconds, which amounts to a whole day in about 3835 lunar months, or about 310 years.
The rule for computing the epact is: Decrease the number of the year by 1, multiply by 11 and subtract 30 as often as possible. That means, the multiplication is done modulo 30. This is the reason for the epact 0 of years with the Golden Number 1, in which a cyclic new moon occurs on 23 March, and 22 March is day 30 of the preceeding new moon.
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