Computus

The Computus is a short name for the Easter calculation, the procedure for calculating the variable annual Easter date. In a general sense Computus means reckoning with time.

The Computisten ( Easter calculator ) worked on behalf of the Pope. In the Gregorian calendar reform in 1582, the calendar with high accuracy to the solar year and the lunar month ( lunation ) was adapted and formulated the calculation rule for determining the date of Easter according to new and published, so that the date of Easter can be controlled or determined without any prior knowledge. The important medieval mathematical discipline of Computistik ( Easter calculation ) thereby lost abruptly important.

The Holiday regulation is now in most countries formally to the sovereignty of the state, for Easter and the holidays will depend on it but nowhere departing from Computus result of Churches. In Germany, the Physikalisch- Technische Bundesanstalt at a non-binding Easter calculation using a Gaussian Easter supplemented formula.

3.1 Reform reasons
3.2 The essence of the Gregorian reform
3.3 Correction of accrued calendar error 3.3.1 From inaccuracy 1)
3.3.2 From inaccuracy 2)
3.3.3 Example

Date of Easter and Easter calculation

The binding of the date of Easter in the spring full moon dates back to the beginnings of Christianity, as even the Jewish lunisolar calendar was used. The crucifixion of Jesus took place on the 14th day of the Jewish month of Nisan, which was the day of the full moon in spring. This binding was firmly agreed in the early Middle Ages. The transfer to the Julian calendar in detail was formulated not quite accurate, but clearly. The used computational tool in advance correctly specify the wandering through the spring months in the Julian and later the Gregorian solar calendar date of the vernal full moon was included to modern times in the resultant about the same time term Computus for Calculating with time. The more accurate term was computus paschalis. It was only when the computer is the Latin root word computare had absorbed, Computus was meaningless in its general sense. He survived only in its limitation to the Easter calculation. Computus paschalis has since been short Computus.

The death of Jesus was a Friday, Good Friday, the third day, the day of his resurrection, was a Sunday. Both days of hiking through the seven days of the week. Christendom agreed, however, that death and the Day of Resurrection in memory are always a Friday and Sunday, and measured the first Sunday after the full moon in spring than Easter Sunday.

The Easter calculation has found the first day of the spring full moon still too close to the following Sunday. The only firm determination on March 21, the day of the beginning of spring as a sufficient approximation to the actual spring equinox. The by scholars ( Computisten, astronomer and mathematician ) calculated future dates of Easter were issued in the Middle Ages as Easter tables. Work result could also be a perpetual calendar, with the help of which the Easter Sunday a year let individually determined.

Of several approaches to keep the calendar in line with the astronomical periods of sun and moon, sat down by the developed in Alexandria in the 3rd century, with a cycle of 19 years - the moon circle or was Metonzyklus - used. In Rome, a cycle of 84 years was used originally, which is slightly less accurate. The Alexandrian Dionysian policy was taken over by the Roman abbot Dionysius Exiguus in the 6th century and spread in the West. He was assisted by the services he is as an epoch (beginning) acquired the Christian era in the determination of the Nativity. The learned English monk Bede has enforced the based on the Metonzyklus Computus in the 8th century throughout the Christian West and the first church made a complete Easter cycle for the years 532-1063. The dates of Easter for the third Easter cycle from 1064 to 1595 calculated Abbo of Fleury. 1582, shortly before the end of this period, the Gregorian calendar reform took place in the calendar and the algorithm for the Easter calculation to better fit the two underlying astronomical periods and new future dates of Easter were published.

The Computus in the Julian calendar

Full Moon date in the Moon Circle

First day of the spring full moon is observed. In one cycle ( moon circles ) 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 21 March and 18 April. The association between calendar year and one of the 19 data occurs with the auxiliary quantity Golden Number GZ, this is determined from the number j years after the defining equation

Golden Number and full moon dates are written in pairs in a table, as in the two middle columns of the table at left. According to historical definition belongs to GZ = 1, April 5. When increasing from GZ by 1 the date is earlier to 11 days, in case of occurring below the March 21 but instead to put 19 days later. After 19 years applies again GZ = 1 and the spring full moon is again on April 5.

Dionysius chose the year 532 as the first year of a lunar circle, but he turned Mondneulicht firmly on March 23. The 14th day after (March 23 to be counted) was April 5, which was regarded as the full moon day in accordance with former method.

Easter date in Sonnenzirkel

Because the full moon date can fall on any day of the week, but Easter is always on a Sunday, the date for the following Sunday must be observed. The days of the week come early from year to year by one calendar day and after a leap again by 1 calendar day. The assignment of the weekday for a date is repeated in a Sonnenzirkel of 28 years ( = 7.4, 7 weekdays, 4-year -period ). You will get a serial number from 0 to 27, the Sonnenzirkel SZ.

Features within the solar circle is the Dominical Letter SB each of those 28 years. Divide the days of a year letter from A to G to. January 1 gets the A, January 2, B and 7th January, the G. On January 8, begins the next row with A, and so on. Allocation of letters to the days of the week a date is valid only for a year, because we know that this is not a whole number of weeks. For example, the first Sunday of the year is always a different date and thus another day letters. His letter days are called the dominical letter of that year.

In a year without leap with SB = A on January 1, Sunday, but also on March 26, April 2, ..., and on April 23. In a year with SB = C is on the 3rd Sunday in January, but on March 21, March 28, ... and on April 25. In the Computus table ( left) calendar days are provided with daily letters TB ( last column). With the help of the Sunday letter are the potential for Easter Sunday recognizable (SB = A or April 9 and April 16, when SB = 3 nor the April 4 and April 11 ).

The assignment to Sonnenzirkel SZ is shown with the following schedule.

*) A leap year has two dominical letters. With the insertion of the leap day of the Dominical Letter SB is postponed by another letter in the alphabet. The table contains only the second, relevant for Easter Sunday letters.

Use the Julian Computus table

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

The Computus in the Gregorian calendar

Reform reasons

The determination of the date of Easter in the Julian calendar takes place on the basis of two simplifications. The counts of lunar months and solar years the one hand, on the other hand are synchronized over the moon circles initially held for fault- free mutually exclusive. 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, and the numerical values are today regarded as correct ).

In the Julian calendar the moon circles are assigned to 6939.75 days. Substituting the appropriate values of m and j is one, one obtains 19 j = d, or approximately 235 6939.6016 6939.6887 m = d

Showing

That the Julian calendar year 365.25 days about 0.0078 days (128 calendar years one days approximately ) compared to the solar year is too long: imprecision 1) (Invoice from 365.25 to 365.2422 = 0.0078 ),
That 235 lunar months about 0.0613 days too short for 19 calendar years ( 3834 lunar months about a day for about 310 calendar years ) are: imprecision 2) (Invoice 6939.75 to 6939.6887 = 0.0613 ).

The two inaccuracies meant that the calendar year no longer coincided with the seasons after a few centuries, and that the Easter statement was incorrect because of wrong predicted spring full moon - date with the times.

When in ancient Rome employed 84 -year cycle (84 Julian calendar years to 30,681 days, 1,039 lunar months equal to ) the error is about five times larger: 812 lunar months are about one day for already about 66 calendar years too short. Therefore, the 84 - year - method right from the Alexandrian Dionysian 19 - years method was ousted.

The essence of the Gregorian reform

The essence of the reform was that the counting scheme, which offered the Julian calendar, was generalized and thus made fit for the future. The Gregorian calendar is not fundamentally different, but a flexibilised Julian calendar.

The time computational foundation - the moon circles - is also applied in the future is always at least a century without correction. The corrections made in Secular years:

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

Correction of accrued calendar error

From inaccuracy 1)

Due to long calendar year, almost two weeks late had arisen through the seasons until the reform in 1582. Was allowed ten days' calendar fail ( the October 4, 1582 immediately followed by October 15 ). Thus, the situation at the time of the Council of Nicaea was restored. The initially on 23 March ( Julius Caesar, 45 BC) held the beginning of spring, had at that time ( 325 AD) moved to March 21, which was established by the Council as a fixed date for Easter statement.

From inaccuracy 2)

When setting up the Computus the inaccuracy 2) was not known. It was assumed that 235 actual lunar months ( lunations ) exact (or accurate enough ) as long as 19 calendar years are. At the time of the Reformation it was known that Easter could not be determined correctly, not only because of the too long calendar year, but also because of this inaccuracy. The accumulated error was about three days. This difference, the full moon data has been moved in the calendar year 1582 earlier.

Example

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

The action coincided approximately with the determination of the spring full moon, and the synchronization of the Computus with this date in the year 532 by Dionysius Exiguus.

Correction of the calendar year

The Julian calendar and its modified form, the Gregorian calendar, are so-called solid lunar calendar, that calendar with the " sun on top" and the "moon in the background ." The fact that the secular years in different handled switching rule ( sun equation) was better adapted to the calendar year to the solar year, therefore, is also known as the application of the lunar equation.

The error between the Julian calendar year and the solar year was 0.0078 days. It was reduced to 0.0003 days, a negligible residual error, which accounts for approximately 3220 years after one day.

Corrections of the full moon date

The predicted full moon date, in particular that of the first spring full moon, in the future to better coordinate with the appearance of the actual full moon, was the awareness of the public " in the background " problem solved. From both tasks the reformers asked she was more sophisticated.

This involves the removal of the error from inaccuracy 2). Due to the failure of the three leap days in 400 years ( removal of the error from inaccuracy 1) ), but the underlying, more applicable 19 -year-old scheme for specifying the full moon data is first disturbed. The disorder is reversed by all full moon dates that follow a secular year without leap, to be postponed to the next day in the calendar. The sun equation is applied with respect to the moon quasi opposite sign. Confusion may result when spoken without regard to this reversal only by the application of solar equation to the determination of the predict, Full Moon date.

Clearly, however, is to speak of the application of the lunar equation, if the error from inaccuracy 2) is eliminated. The occasion of this eight selected within 2,500 years Secular years shift made of the Full Moon date occurs each ( vice versa as in the elimination of interference by the failed leap days ) on a day earlier on the calendar.

The correction cycle began in 1800 and continues in 2100. Between the years of 3900 and the beginning of the next cycle in the year 4300 of the jump is four centuries.

Impact of the new switching scheme on the Sunday letter

In each application, the Sun equation (ie, a failing leap day ) changes the mapping between Sonnenzirkel SZ and Sunday letters SB in the Gregorian calendar. For the years 1900-2099 following table applies:

From 2100 to 2199 applies because of the 2100 non-inserted leap day then a new table is all SB are shifted one place: to SZ = 0 belongs SB = A, and so on.

Use the Gregorian Computus table, 1900-2199

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

Exception Rules in the Gregorian calendar

In the Julian calendar the 19 contained in the Moon Circle Full Moon dates were fixed. Due to the shifts in the Gregorian calendar are excessive duration every 30 data ( duration of a lunation, rounded, full month) between 21 March and 19 April possible. Previously, the latest Easter border April 18, Latest Easter Sunday April 25. Now April 19 may arise as latest Spring full moon from the invoice. Latest Easter Sunday could be April 26. The Reform Commission wanted to meet the skeptics of the new calendar and closed by exception control the extension until April 26, from.

Example of Rule 1: 1981 GZ = 6; SZ = 2 → SB = D → limit Easter: Sunday, April 19 → Easter on April 19 (corrected limit: April 18 )

Example of Rule 2: 1954 GZ = 17, AN = 3 → SB = C → Easter limit: Sunday, April 18 → Easter without correction on April 25 ( with corrected boundary = April 17 → April 18 ) In 1943, ie less than 19 years earlier, Easter was already on 25 April. GZ = 6; SZ = 20 = C → SB → Easter limit: Monday, April 19 → Easter on April 25

The epact

The original fixed assignment between the Golden Number GZ and Spring full moon is lost. You have to move GZ equations parallel to the (An). This is in the Gregorian Computus table (standing twice on the right) to happen. It is valid for the period 1900-2199. Compared to the original Golden numbers GZ ( left table ) are the shifted numbers GZ nine days later.

Control account: 7 ( displacement 1582) 3 ( suns ( at ) equations in 1700, 1800 and 1900 ) -1 (Moon ( on ) equation 1800) = 9.

Both Computus tables begin epact EP, which was already known in the Middle Ages, but only came through the reform to frequent use. It is popular because they, unlike the Golden Number is continuously changing. In the correction years epact is changed by ± 1. This is called based on the physical displacement of the Golden Numbers ( shift of the GZ- column in a Gregorian Computus table) epacts shift. In postponing the moon date to a later reduces the epact and vice versa. The annual value of the epact is given in astronomical annals of the value of the Golden Number. However, it is " [ ... ] should be noted that even with the Epaktentheorie the golden number can not be dispensed with. " (Bach)

The epacts series contains as the Golden Numbers 19 values . It goes from 29 to EP = EP = 0, where after each shift epacts exist 11 different vulnerabilities. The Julian row is fixed, is missing in her under another EP = 29 (see above on the left edge standing Computus table, first column). By definition the epact one year is the age of the moon on the last day of the previous year. And will be counted Neulicht.

Example: Full moon on 1 January (Age 14 days), EP = 13

The Computus in the Gaussian formulas Easter

Carl Friedrich Gauss ( 1777-1855 ) has the Computus, the algorithm of the Easter calculation, represented by the means of modern mathematics. He wanted to " deliberately give his rule a practical tool in the hand, which can be applied without the knowledge of compressed in it and veiled computus by everyone present. " ( Grassl )

Before Gauss the Computus was " [ ... ] special art, [ ... ] was at times [ ... ] the only chapter of Mathematics of the University of Education [ ... ] and, despite [ ... ] alleged complication of humanity far more exploited than harm. " ( Zemanek )

Physikalisch-Technische Bundesanstalt Golden number (time) Heinz Zemanek

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