Computus (Easter bill)

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The computus is an abbreviation for the Easter bill, the rule for calculating the annually changing Easter date . In a general sense, Computus means calculating with time.

The computists (Easter calculators) worked on behalf of the Pope . During the Gregorian calendar reform in 1582, the calendar was adjusted with high accuracy to the solar year and the lunar month ( lunation ) and the calculation rule for determining the Easter date was reformulated and published accordingly, so that the Easter date can be checked or determined without special prior knowledge. The mathematical discipline of computistics or computistics (Easter calculation), which was important in the Middle Ages , suddenly lost its importance.

In most countries, the public holiday regulation is now formally part of the sovereignty of the state, but for Easter and the holidays that are dependent on it, there is no deviation from the Computus result of the churches. In Germany, the Physikalisch-Technische Bundesanstalt prepares a non-binding Easter bill using a supplemented Gaussian Easter formula .

Computus board and perpetual calendar as a circular disc, valid for the Julian calendar

Easter date and Easter bill

The connection between Easter and the spring full moon comes from the beginnings of Christianity , when the Jewish lunisolar calendar was still used. The crucifixion of Jesus took place on the 14th day of the Jewish month of Nisan , which was the day of the spring full moon. This bond was already firmly agreed in the early Middle Ages. The transfer to the Julian calendar in detail was not formulated exactly, but clearly. The arithmetic tool used to correctly state the date of the spring full moon in advance in the Julian calendar , later in the Gregorian solar calendar, which moved through the spring month, was included in the term computus for arithmetic with time, which was created around the same time until modern times. The more precise term was computus paschalis. It wasn't until the computer absorbed the Latin root word computare that Computus became meaningless in its general meaning. It was only retained in its restriction to the Easter bill. Computus paschalis has been called Computus for short since then.

The anniversary of Jesus' death was a Friday, Good Friday . The third day, the day of his resurrection , was a Sunday. Both anniversaries wander through the seven days of the week. However, Christianity agreed that the day of death and the day of the resurrection should always be a Friday and a Sunday in commemoration, and determined the first Sunday after the spring full moon as Easter Sunday .

The Easter bill has to close from the first day of the spring full moon to the following Sunday. The only fixed determination is March 21st for the day of the beginning of spring as a sufficient approximation to the actual spring equinox . The future Easter dates calculated by the scholars ( computists , astronomers and mathematicians) were published as Easter tables in the Middle Ages . The result of the work could also be a perpetual calendar , with the help of which the Easter Sunday of a year could be determined individually.

Of several approaches to keep the calendar in accordance with the astronomical periods of the sun and moon, the one developed in Alexandria in the 3rd century prevailed, based on a cycle of 19 years - the lunar circle or Meton cycle . A cycle of 84 years was originally used in Rome, which is somewhat less precise. The Alexandrian-Dionysian approach was adopted by the Roman abbot Dionysius Exiguus in the 6th century and spread throughout the West . The services he acquired in determining the birth of Christ as the epoch (beginning) of the Christian era helped him. The learned English monk Beda Venerabilis implemented the computus, which is based on the Meton cycle, in the entire Christian western church in the 8th century and was the first to produce a complete Easter cycle for the years 532 to 1063. Abbo von Fleury calculated the Easter dates for the third Easter cycle from 1064 to 1595 . In 1582, shortly before the end of this period, the Gregorian calendar reform took place, in which the calendar and the algorithm for the Easter calculation were better adapted to the two underlying astronomical periods and new future Easter dates were published.

The computus in the Julian calendar

Computus
Julian
EP GZ date TB
23 16 March 21st  C.
22nd  5 March 22  D.
March 23  E.
20th 13 March 24th  F.
19th  2 25th March  G
26th of March  A.
17th 10 27th of March  B.
28th March  C.
15th 18th March 29  D.
14th  7th March 30  E.
 March 31  F.
12 15th  April 1st  G
11  4th  2nd of April  A.
 3rd of April  B.
 9 12  4. April  C.
 8th  1  April 5th  D.
 April 6th  E.
 6th  9  7th of April  F.
 April 8th  G
 4th 17th  9th April  A.
 3  6th 10th of April  B.
11 April  C.
 1 14th 12. April  D.
 0  3 April 13th  E.
April 14th  F.
28 11 April 15th  G
April 16  A.
26th 19th 17th April  B.
25th  8th April 18  C.
April 19th  D.
20th of April  E.
April 21  F.
April 22  G
April 23  A.
April 24th  B.
April 25  C.

Gregorian Computus
1900 to 2199
EP GZ date TB
23 March 21st  C.
22nd 14th March 22  D.
21st  3 March 23  E.
20th March 24th  F.
19th 11 25th March  G
18th 26th of March  A.
17th 19th 27th of March  B.
16  8th 28th March  C.
15th March 29  D.
14th 16 March 30  E.
13  5  March 31  F.
12  April 1st  G
11 13  2nd of April  A.
10  2  3rd of April  B.
 9  4. April  C.
 8th 10  April 5th  D.
 7th  April 6th  E.
 6th 18th  7th of April  F.
 5  7th  April 8th  G
 4th  9th April  A.
 3 15th 10th of April  B.
 2  4th 11 April  C.
 1 12. April  D.
 0 12 April 13th  E.
29  1 April 14th  F.
28 April 15th  G
27  9 April 16  A.
26th 17th April  B.
25th 17th April 18  C.
24  6th April 19th  D.
20th of April  E.
April 21  F.
April 22  G
April 23  A.
April 24th  B.
April 25  C.

Full moon date in the lunar circle

First of all, determine the day of the spring full moon. In a cycle (lunar circle) of 19 years, there is a fixed assignment of the full moon date to the calendar year. The full moon falls on 19 different days between March 21st and April 18th. The assignment between the calendar year and one of the 19 dates is made with the auxiliary variable golden number GZ, this is determined from the year number j according to the definition equation

GZ = (j + 1)  mod 19.
GZ = 0 *), 1,…, 17 or 18.
*) Instead of the zero, which they only got to know later, the computists wrote the divisor, here 19.

The golden number and the full moon date are written in pairs in a table, as in the two middle columns of the table on the left. According to the historical definition, April 5th belongs to GZ = 1. If the GZ is increased by 1, the date is to be set 11 days earlier, but if it falls below March 21, 19 days later instead. After 19 years, GZ = 1 applies again and the spring full moon is again on April 5th.

Dionysius chose the year 532 as the first year of a lunar circle, and found the new moon light on March 23rd. The 14th day after that (including March 23rd) was April 5th, which according to the method of the time was considered a full moon day.

Easter date in the solar circle

Because the full moon date can fall on any day of the week, but Easter is always on a Sunday, the date of the following Sunday must be determined. The days of the week are early by 1 calendar day from year to year and by 1 calendar day after a leap day. The assignment of the day of the week to a date is repeated in a solar circle of 28 years (= 7 · 4; 7 days of the week, 4-year switching period). You will initially receive a consecutive number from 0 to 27, the Sun Circle SZ .

SZ = (j + 9) mod 28; Result: SZ = 0 *), 1,…, 26 or 27.
*) Instead of the zero, which they only got to know later, the computists wrote the divisor, here 28.

The Sunday letter SB for each of these 28 years is the hallmark of the sun circle. The days of a year are assigned letters from A to G. January 1st gets the A, January 2nd the B and January 7th the G. On January 8th the next row starts again with A and so on. The assignment of the day letters to the weekday of a date is only valid for one year because it is well known that this does not consist of a whole number of weeks. For example, the first Sunday of the year always has a different date and thus a different day letter. Its day letter is called the Sunday letter of the year in question.

Sunday letter SB of a year
            first Sunday of the year on: 1. 2. 3. 4th 5. 6th 7th January
Sunday letter SB of this year: A. B. C. D. E. F. G

In a year without a leap day with SB = A, January 1st is Sunday, but also March 26th, April 2nd, ... and April 23rd. In a year with SB = C, January 3rd is Sunday, but also March 21st, March 28th, ... and April 25th. In the Computus table (left) the calendar days are provided with the day letter TB (last column). With the help of the Sunday letter, the possible Sundays for Easter can be identified (SB = A still April 9 and April 16; with SB = 3 still April 4 and April 11).

The assignment to the solar circle SZ is shown with the following list.

Sunday letter SB as a function of the sun circle SZ, Julian
SZ  0  1*  2  3  4th  5 *  6th  7th  8th  9 * 10 11 12 13 * 14th  15th  16  17 * 18th  19th  20th  21 * 22nd  23  24  25 * 26th  27 
SB A. F. E. D. C. A. G F. E. C. B. A. G E. D. C. B. G F. E. D. B. A. G F. D. C. B.

*) A leap year has two Sunday letters. When the leap day is inserted, the Sunday letter SB is shifted by another letter in the alphabet. The table only contains the second Sunday letter, which is relevant for Easter.

Use of the Julian computus table

1. The golden number GZ and the solar circle SZ are calculated.
2. With the value for SZ, you can find the Sunday letter SB in the SB list of SZ.
3. With the value for GZ you can find the full moon date in the Computus table (between March 21st and April 18th).
4. Easter Sunday is 1 to 7 days later. It is the calendar day whose day letter TB (Computus table, 4th column) is the same as the Sunday letter SB found under 2.

Example: year 1580
GZ = (1580 + 1) mod 19 = 4; SZ = (1580 + 9) mod 28 = 21 → SB = B
Spring full moon on April 2nd; Easter Sunday on April 3rd

The computus in the Gregorian calendar

Reform reasons

The determination of the Easter date in the Julian calendar is based on two simplifications. The counts of the lunar months on the one hand and solar years on the other hand are mutually synchronized via the moon circle, which was initially considered to be error-free. The following equation is used for this:

235 m = 19 j (m = lunar month (lunation) = 29.53059 d; j = solar year = 365.24219 d; d = day; the numerical values ​​are those that are considered correct today).

In the Julian calendar, 6,939.75 days are assigned to the lunar circle. If you
insert the correct values ​​for m and j, you get 19 j = 6,939.6016 d and 235 m = 6,939.6887 d, respectively.

This shows,

  • that the Julian calendar year with 365.25 days is about 0.0078 days (128 calendar years about one day) compared to the solar year: Inaccuracy 1)
    (calculation 365.25 - 365.2422 = 0.0078),
  • that 235 lunar months are about 0.0613 days too short for 19 calendar years (3.834 lunar months about one day for about 310 calendar years): Inaccuracy 2)
    (calculation 6939.75 - 6939.6887 = 0.0613).

The two inaccuracies led to the fact that after a few centuries the calendar year no longer coincided with the seasons, and that the Easter calculation became incorrect over time because of the incorrectly predicted spring full moon date.

In the 84-year cycle used in ancient Rome (84 Julian calendar years are equated to 30,681 days, 1,039 lunar months) the error is about five times larger: 812 lunar months are about one day too short for about 66 calendar years. That is why the 84-year method has rightly been superseded by the Alexandrian-Dionysian 19-year method.

The essence of Gregorian reform

The essence of the reform was that the counting scheme offered by the Julian calendar was generalized and thus made future-proof. The Gregorian calendar is not a fundamentally different, but a more flexible Julian calendar.

The time-calculating foundation - the lunar circle - will continue to be used for at least a century without correction. The corrections are made in secular years:

  • Inaccuracy 1) requires a correction of one day after about 128 years at the latest. The definition of not correcting three times every 100 years in 400 years and at the end of this period is the so-called solar equation . It is used on average about every 133 years.
  • Inaccuracy 2) requires a correction of one day after approximately 312 years at the latest. The definition of correcting seven times every 300 years in 2,500 years and the eighth time at the end of this period is the so-called lunar equation . It is applied every 312.5 years on average.

Correction of the calendar errors

Out of inaccuracy 1)

Due to the long calendar year, the reform in 1582 resulted in a delay of almost two weeks compared to the seasons. Ten days were omitted from the calendar (October 4, 1582 was immediately followed by October 15). This restored the situation at the time of the Council of Nicaea . The beginning of spring, which initially took place on March 23 ( Julius Caesar , 45 BC), had been postponed to March 21 (AD 325), which was set by the council as a fixed date for the Easter bill.

Control calculation: (1582−325) 0.0078 = 9.8 days.

Out of inaccuracy 2)

When the Computus was set up, the inaccuracy 2) was not known. It was assumed that 235 actual lunar months (lunations) were exactly (or sufficiently accurate) as long as 19 calendar years. At the time of the Reformation it was known that Easter could not be correctly determined not only because of the long calendar year, but also because of this inaccuracy. The accumulated error was about three days. The full moon dates in the calendar year 1582 were postponed to earlier by this difference.

example

GZ = 1, shift of the spring full moon from April 5th to April 2nd (or to April 12th after ten days were skipped).

The measure roughly coincided with the determination of the spring full moon and the synchronization of the computus with this date in the year 532 by Dionysius Exiguus .

Control calculation: (1582-532) · 0.0613 / 19 = 3.4 days.

Correction of the calendar year


Gregorian Computus
1900 to 2199
EP GZ date TB
23 March 21st  C.
22nd 14th March 22  D.
21st  3 March 23  E.
20th March 24th  F.
19th 11 25th March  G
18th 26th of March  A.
17th 19th 27th of March  B.
16  8th 28th March  C.
15th March 29  D.
14th 16 March 30  E.
13  5  March 31  F.
12  April 1st  G
11 13  2nd of April  A.
10  2  3rd of April  B.
 9  4. April  C.
 8th 10  April 5th  D.
 7th  April 6th  E.
 6th 18th  7th of April  F.
 5  7th  April 8th  G
 4th  9th April  A.
 3 15th 10th of April  B.
 2  4th 11 April  C.
 1 12. April  D.
 0 12 April 13th  E.
29  1 April 14th  F.
28 April 15th  G
27  9 April 16  A.
26th 17th April  B.
25th 17th April 18  C.
24  6th April 19th  D.
20th of April  E.
April 21  F.
April 22  G
April 23  A.
April 24th  B.
April 25  C.

The Julian calendar and its modified form, the Gregorian calendar, are so-called solilunar calendars, namely calendars with the “sun in the foreground” and the “moon in the background”. The fact that with the switching rule (solar equation) handled differently in secular years, the calendar year was better adapted to the solar year, is consequently better known than the application of the lunar equation.

The error between the Julian calendar year and the solar year was 0.0078 days. It has been reduced to 0.0003 days, an insignificant residual error that only becomes a day after about 3220 years.

Corrections to the full moon date

To better coordinate the predicted full moon date, in particular that of the first spring full moon, with the appearance of the actual full moon, was the task that was solved “in the background” in the mind of the public. Of the two tasks set for the reformers, it was the more demanding.

It is about eliminating the error from inaccuracy 2) . Due to the failure of the 3 leap days in 400 years (elimination of the error from inaccuracy 1) ), the underlying 19-year scheme, which is still to be used, for specifying the full moon dates is initially disrupted. The disruption is reversed by moving all full moon dates that follow a secular year without a leap day to a day later in the calendar. The solar equation is applied with the opposite sign with respect to the moon . Confusion can result if, without considering this reversal, only the application of the solar equation to the determination of the full moon date to be predicted is spoken of.

On the other hand, it is clear to speak of the application of the lunar equation if the error from inaccuracy 2) is eliminated. The shifting of the full moon date, carried out on the occasion of eight secular years selected within 2,500 years, takes place in each case one day earlier in the calendar (the other way around than when the fault was eliminated by the failed leap days).

The correction cycle started in 1800 and will continue in 2100. Between the year 3900 and the beginning of the next cycle in 4300 the leap is four centuries.

Effect of the new switching regulation on the Sunday letters

Every time the solar equation is used (that is, a missing leap day), the assignment between the solar circle SZ and the Sunday letter SB in the Gregorian calendar changes.

Sunday letter SB as a function of the sun circle SZ, Gregorian: 1582 to 3399
SZ  0  1*  2  3  4th  5 *  6th  7th  8th  9 * 10 11 12 13 * 14th  15th  16  17 * 18th  19th  20th  21 * 22nd  23  24  25 * 26th  27 
SB D. CB A. G F. ED C. B. A. GF E. D. C. BA G F. E. DC B. A. G FE D. C. B. AG F. E. 1582−1699 2500−2599
SB E. DC B. A. G FE D. C. B. AG F. E. D. CB A. G F. ED C. B. A. GF E. D. C. BA G F. 1700−1799 2600−2699
SB F. ED C. B. A. GF E. D. C. BA G F. E. DC B. A. G FE D. C. B. AG F. E. D. CB A. G 1800−1899 2700−2899
SB G FE D. C. B. AG F. E. D. CB A. G F. ED C. B. A. GF E. D. C. BA G F. E. DC B. A. 1900−2099 2900−2999
SB A. GF E. D. C. BA G F. E. DC B. A. G FE D. C. B. AG F. E. D. CB A. G F. ED C. B. 2100−2199 3000−3099
SB B. AG F. E. D. CB A. G F. ED C. B. A. GF E. D. C. BA G F. E. DC B. A. G FE D. C. 2200−2299 3100−3299
SB C. BA G F. E. DC B. A. G FE D. C. B. AG F. E. D. CB A. G F. ED C. B. A. GF E. D. 2300−2499 3300−3399

*) A leap year has two Sunday letters. When the leap day is inserted, the Sunday letter SB is shifted by another letter in the alphabet. The Sunday letter relevant to Easter in leap years is always the second, i.e. the one on the right.

Use of the Gregorian Computus Table, 1900-2199

  1. The golden number GZ and the solar circle SZ are calculated.
  2. With the value for SZ, the Sunday letter SB is found in the SB list of SZ.
  3. With the value for GZ you can find the full moon date (between March 21st and April 18th) in the Computus table.
        If April 19 or April 18 is determined, exception rules come into effect (see below: exception rules in the Gregorian calendar).
  4. Easter Sunday is 1 to 7 days later. It is the calendar day whose day letter TB (Computus table, 4th column) is the same as the Sunday letter SB found under 2.

Example: year 2009
GZ = (2009 + 1) mod 19 = 15; SZ = (2009 + 9) mod 28 = 2 → SB = D
Spring full moon on April 10th; Easter Sunday on April 12th

Exceptions in the Gregorian calendar

In the Julian calendar, the 19 full moon dates contained in the lunar circle were fixed. Due to the shifts in the Gregorian calendar, all 30 dates (length of a lunation, rounded up; full month) between March 21 and April 19 are possible over a long period. In the past, the latest Easter border was April 18th and Easter Sunday was April 25th. Now the calculation can also result in April 19th as the latest spring full moon. The latest Easter Sunday could be April 26th. The reform commission wanted to accommodate the skeptics of the new calendar and ruled out the extension until April 26th through an exception regulation.

  1. If the spring full moon is April 19th (currently with GZ = 6), and if this is a Sunday, the Easter border will be moved forward to April 18th. Easter Sunday is April 19th.
  2. If April 18th is determined with a GZ> 11 (currently with GZ = 17), and if this is a Sunday, the Easter border will be moved forward to April 17th. Easter Sunday is April 18th. Otherwise there would be Easter twice on April 25th within a 19 series, which did not occur in the Julian calendar.

Example for 1st rule: Year 1981
GZ = 6; SZ = 2 → SB = D → Easter border: Sunday, April 19th → Easter on April 19th (corrected border: April 18th)

Example for 2nd rule: Year 1954
GZ = 17, SZ = 3 → SB = C → Easter border: Sunday, April 18th → Easter without correction on April 25th (with corrected border = April 17th → April 18th)
In the year 1943, d. H. less than 19 years earlier, Easter was April 25th.
GZ = 6; SZ = 20 → SB = C → Easter border: Monday, April 19th → Easter on April 25th

The epacts

The original fixed assignment between the golden number GZ and the spring full moon has been lost. One has to shift GZ parallel to the (approximation) equations. This is what happened in the Gregorian Computus table (twice in the right margin). It applies to the period between 1900 and 2199. Compared to the original golden numbers GZ (left table), the shifted numbers GZ are 9 days later.

Control calculation: +7 (shift 1582) +3 (sun (an) equations 1700, 1800 and 1900) -1 (moon (an) equation 1800) = +9.

Both Computus tables begin with the Epakte EP, which was already known in the Middle Ages, but was only used more frequently as a result of the reform. It is popular because, unlike the golden number, it changes continuously. In the correction years the epact is changed by ± 1. This is called epact shift based on the physical shift of the golden numbers (shifting the GZ column in a Gregorian computus table) . If the lunar date is shifted to later, the epacts are reduced and vice versa. The annual value of the epacts is given in astronomical yearbooks next to the value of the golden number . However, "[...] it should be noted that the golden number cannot be dispensed with in the epact theory either." ( Bach )

Like the golden numbers, the epact series contains 19 values. It goes from EP = 29 to EP = 0, with 11 other gaps existing after each epact shift. The Julian series is fixed, among others it lacks EP = 29. (see the Computus table on the left, first column). By definition, the epact of a year is the age of the moon on the last day of the previous year. Counts from new light.

Example: full moon on January 1st (age 14 days), EP = 13.

The computus in Gaussian Easter formulas

Carl Friedrich Gauß (1777 to 1855) presented the computus, the algorithm of the Easter calculation, using modern mathematics. He wanted "with his rule very consciously to provide a practical aid that can be used by everyone without knowledge of the computus contained in it, compressed and veiled." ( Graßl )

Before Gauss, the computus was "[...] special art, [...] was at times [...] the only chapter in mathematics in university education [...] and, despite [...] alleged complications, has done far more than harm to mankind." ( Zemanek )

literature

Web links

Remarks

  1. ↑ See the title chosen by Arno Borst : Computus - Time and Number in the History of Europe . 3rd revised and expanded edition. Wagenbach, Berlin 2004.
  2. ^ In the Western Church according to the Gregorian calendar, in the Eastern Church (except in Finland) according to the Julian calendar
  3. ↑ Arno Borst: Computus - time and number in the history of Europe . 3rd revised and expanded edition. Wagenbach, Berlin 2004, ISBN 3-8031-2492-1 , p. 34.
  4. ↑ Arno Borst: Computus - time and number in the history of Europe . 3rd revised and expanded edition. Wagenbach, Berlin 2004, ISBN 3-8031-2492-1 , p. 134.
  5. ^ Heinz Zemanek : Calendar and Chronology . Munich 1990, p. 45.
  6. ^ Heiner Lichtenberg: The adaptable, cyclical, soliluneasre time counting system of the Gregorian calendar - a scientific masterpiece of the late Renaissance. In: Mathematical semester reports. Volume 50, 2003, p. 47.
  7. ^ Heinz Zemanek: Calendar and Chronology. Munich 1990, p. 29.
  8. ↑ Heiner Lichtenberg: The adaptable, cyclical, solilunear time counting system of the Gregorian calendar - a scientific masterpiece of the late Renaissance. In: Mathematical semester reports. Volume 50, 2003, p. 52.
  9. ↑ Joseph Bach: The Easter calculation in old and new times. In: Scientific supplement to the annual report of the Episcopal Gymnasium in Strasbourg i. E. 1907, pp. 34/35 , accessed October 26, 2012 .
  10. ↑ The reform commission accommodated the skeptics of the new calendar in this case too. All other Easter dates are possible twice within a 19 series both before and after the reform.
  11. ↑ Joseph Bach: The Easter calculation in old and new times. In: Scientific supplement to the annual report of the Episcopal Gymnasium in Strasbourg i. E. 1907, p. 36 , accessed October 26, 2012 .
  12. ↑ Alfons Graßl: The Gaussian Easter rule and its basics. In: Stars and Space. 4 (1993).
  13. ^ Heinz Zemanek: Calendar and Chronology. Munich 1990, ISBN 3-486-20927-2 , p. 35 u. P. 45.