Easter cycle

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In two consecutive Easter cycles , the Easter dates are identical. Such a cycle consists of 532 Easter celebrations in the Julian calendar , or is 532 years long. In the Gregorian calendar , there are 5.7 million Easter celebrations or 5.7 million years. These two time intervals are called - deviating from the standard usage of the term cycle - the Julian and Gregorian Easter cycle.

The Julian Easter cycle

The lunar circle

The lunar circle is 19 years long. Every 19 years the spring full moon falls on the same calendar day.

The solar circle

The solar circle is 28 years long. Every 28 years the calendar days - including Sundays, one of which is Easter Sunday - have the same date again.

The solar circle is the smallest common multiple of the weekday circle and the leap year circle (7 × 4 = 28). Every seven days is the same day of the week again, and every four years (leap years) the days of the week are shifted by two days instead of one day in a normal year.

Julian Easter cycle

The smallest common multiple of the lunar and solar circles is the product 19 × 28 = 532. In the Julian calendar, every 532 years, 532 Easter are again distributed over the annual dates like the 532 Easter before. In the Julian calendar, the distribution of Easter among the annual dates in the calendar is repeated every 532 years .

The Gregorian Easter cycle

Basics about the changes in the Gregorian calendar compared to the Julian calendar are shown in the Easter calculation with the help of the computus .

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 with the help of the solar equation and the lunar equation . Applying these equations will make the solar circle longer. Another independent circle is added to the fundamental 19-year lunar circle.

The solar equation

The "solar equation" is the term used to describe the measure of not inserting a leap day in those secular years , the number of which cannot be divided by 400 without a remainder. It serves to better adapt the calendar year to the solar year . This changes the length of the calendar year from 365.25 days to 365.24250 days (the solar year in the old definition currently has 365.242375 days). The solar equation causes the epacts to decrease by 1 with each application , i.e. H. the moon phases are shifted back one day.

The extended solar circle

By applying the solar equation, the leap year circle has increased from 4 to 400 years. It also represents the circle of the sun, because a date falls on the same day of the week after 400 Gregorian calendar years.

Control: 400 years × 365.25 days / year - 3 days = 146,097 days = 20,871 weeks × 7 days / week

A multiplication of 400 × 7 is not required.

The lunar equation

The “lunar equation” is the name given to the measure of setting the predicted lunar dates eight times earlier in the calendar by one day each over a period of 2500 years. This approximately corrects the error that is contained in the fundamental 19-year lunar circle. The actual moon phases are shifted one day earlier in the Julian calendar in about 310 years. With the help of the lunar equation, this correction is made every 312.5 years on average (2500/8 = 312.5). Specifically, the lunar equation is applied seven times every 300 years and then once every 400 years in secular years. It came into effect for the first time after the Gregorian Reform in 1800. The next years of the lunar equation are: 2100, 2400, 2700, 3000, 3300, 3600, 3900, but only then again 4300. After that, the period of 2500 years begins again. The lunar equation increases the epacts by 1 with each application, i.e. H. the phases of the moon are corrected forward by one day.

An additional moon circle

By applying the lunar equation, the spring full moon no longer falls on just 19 calendar days between March 21 and April 18, but in the long term on all 30 calendar days of this period. April 19, which would be possible as a spring full moon in the Gregorian calendar (Epacts 24), is suppressed and postponed to April 18, otherwise April 26 would also be possible as the latest Easter date and April 25 would be the last possible Easter date as wanted to keep in the Julian calendar.

In 2500 years, the 19 possible lunar dates (epact table or epact series, see below) are shifted 8 times to an earlier calendar day each ( epact shift).

The fundamental 19-year lunar circle is only to be adapted to the counting scheme of the Julian calendar, which is done with the lunar equation. Applying the solar equation to improve the length of the calendar year disrupts this counting scheme. Therefore, if a leap day fails, the lunar date must be postponed by one day in the calendar. In the literature, the application of the solar equation is also abbreviated in this context, especially when describing the computus with the auxiliary variable "epacts". Confusion with their use to correct the length of the calendar year cannot be ruled out. In 400 years the 19 possible lunar dates will be postponed 3 times to a later calendar day (epact postponement).

It is now the smallest common multiple of the 2500 years and the 400 years in which the applications of the lunar equation or the solar equation are repeated. That's 10,000 years. In 10,000 years, the 19 possible lunar dates are shifted 43 times to a later calendar day each (epacts shift by using the solar equation and the lunar equation: 3x10000 / 400-8x10000 / 2500 = 75-32 = 43). There are 30 such periods to wait until the initial state is restored. The additional lunar circle is 300,000 years long (30 × 10,000).

In the secular years, neither of the two equations (e.g. year 1600, 2000), the solar equation alone (e.g. 1700, 1900, 2200, 2300) (epacts reduced by 1), the lunar equation alone (2400) (Epakte increases by 1) or both equations together (e.g. 1800, 2100) can be used. If both equations are used together, they compensate each other and the epact is not shifted. Unlike in the Julian calendar, in which the assignment of the golden number to the epact is always fixed, this results in various epact tables (maximum 30) that are valid for at least 100 years, and within which the assignment of the golden number to the epact remains constant. The golden number results from the remainder of the division (year number + 1) / 19 . Ginzel shows this very clearly. Complete overviews of the 30 possible epact tables (series) and their validity can be found e.g. B. with Clavius ​​or Coyne. Currently (from 1900 to 2199; 2000: no equation; 2100: compensation for the sun and moon equations) the following assignment applies:

Epact table (epact series)

Golden number Epacts
Julian
Epacts Gregorian
1583
1699
1700
1899
1900
2199
2200
2299
1 8th 1 0 29 28
2 19th 12 11 10 9
3 0 23 22nd 21st 20th
4th 11 4th 3 2 1
5 22nd 15th 14th 13 12
6th 3 26th 25th 24 23
7th 14th 7th 6th 5 4th
8th 25th 18th 17th 16 15th
9 6th 29 28 27 26th
10 17th 10 9 8th 7th
11 28 21st 20th 19th 18th
12 9 2 1 0 29
13 20th 13 12 11 10
14th 1 24 23 22nd 21st
15th 12 5 4th 3 2
16 23 16 15th 14th 13
17th 4th 27 26th 25th 24
18th 15th 8th 7th 6th 5
19th 26th 19th 18th 17th 16

The Gregorian Easter cycle

The distribution scheme for the date of Easter Sunday does not start again until all circles involved in its distribution start again on the same calendar day. The period of this scheme is the common multiple of the periods of the extended solar circle (400 years), the 19-year lunar circle (19 years) and the additional lunar circle (300,000 years).

In the Gregorian calendar, the distribution of Easter among the annual dates in the calendar is repeated every 5,700,000 years.

Control calculations using the Gaussian Easter formula

Carl Friedrich Gauß formulated the Oster algorithm as a set of algebraic formulas. In the following, a formula set supplemented with the exception rules is used (see A supplemented Easter formula ). In it the algorithm is conceptually fully formulated and can be fully evaluated with the help of a PC .

To determine the Easter date for year X, calculate the following values ​​in sequence:

 1. die Säkularzahl:                              K = X div 100
 2. die säkulare Mondschaltung:                   M = 15 + (3K + 3) div 4 − (8K + 13) div 25
 3. die säkulare Sonnenschaltung:                 S = 2 − (3K + 3) div 4
 4. den Mondparameter:                            A = X mod 19
 5. den Keim für den ersten Frühlingsvollmond:    D = (19A + M) mod 30
 6. die kalendarische Korrekturgröße:             R = D div 29 + (D div 28 − D div 29) (A div 11)
 7. die Ostergrenze:                             OG = 21 + D − R
 8. den ersten Sonntag im März:                  SZ = 7 − (X + X div 4 + S) mod 7
 9. die Entfernung des Ostersonntags von der
    Ostergrenze (Osterentfernung in Tagen):      OE = 7 − (OG − SZ) mod 7
10. das Datum des Ostersonntags als Märzdatum
    (32. März = 1. April usw.):                  OS = OG + OE

( div stands for an integer division, i.e. digits after the decimal point are truncated. mod stands for the non-negative remainder of the division in an integer division.) The above algorithm applies to the Gregorian calendar. For the Julian calendar, M  = 15 and S  = 0.

If you now replace the year number X by the year number X + 5,700,000 , the variables occurring in the algorithm change in the following way:

KK + 57,000
MM + 24,510
SS - 42.750

The other sizes A , D , R , OG , SZ , OE and OS do not change. (Reason: A : 5,700,000 is a multiple of 19. D : 24,510 is a multiple of 30. R, OG are then clear. SZ : 5,700,000 mod 7 = 5, (5,700,000 / 4) mod 7 = 3, 42.750 mod 7 = 1. OE and OS are clear again. Therefore, one gets the same Easter date again.

This shows the date of Easter that in any case always repeated every 5.7 million years.

But it still has to be investigated whether the Easter date is not repeated even after a fraction of this period. The number 5,700,000 is only divisible by the following prime numbers: 2, 3, 5 and 19. The Easter date could therefore also be every 5,700,000 / 2 years, every 5,700,000 / 3 years, every 5,700,000 / 5 years or Repeat every 5,700,000 / 19 years (and if so, then possibly also in even shorter periods that are dividers of these periods). The following calculation examples show that this is not the case.

a) The year 2010:

X = 2010, K = 20, M = 24, S = -13, A = 15, D = 9, R = 0, OG = 30, SZ = 7, OE = 5, OS = 35
Easter on April 4th ("March 35th"). The following examples are compared to this date:

b) The year 2,852,010 (= 2010 + 5,700,000 / 2):

X = 2,852,010, K = 28,520, M = 12,279, S = -21,388, A = 15, D = 24, R = 0, OG = 45, SZ = 7, OE = 4, OS = 49
Easter on April 18th ("March 49th"). The Easter dates are not repeated every 2,850,000 (= 5,700,000 / 2) years.

c) The year 1,902,010 (= 2010 + 5,700,000 / 3):

X = 1,902,010, K = 19,020, M = 8,194, S = -14,263, A = 15, D = 19, R = 0, OG = 40, SZ = 7, OE = 2, OS = 42
Easter on April 11th ("March 42nd"). The Easter dates are not repeated every 1,900,000 (= 5,700,000 / 3) years.

d) The year 1,142,010 (= 2010 + 5,700,000 / 5):

X = 1,142,010, K = 11,420, M = 4,926, S = -8,563, A = 15, D = 21, R = 0, OG = 42, SZ = 7, OE = 7, OS = 49
Easter on April 18th ("March 49th"). The Easter dates are not repeated every 1,140,000 (= 5,700,000 / 5) years.

e) The year 302.010 (= 2010 + 5.700.000 / 19):

X = 302.010, K = 3.020, M = 1.314, S = -2.263, A = 5, D = 29, R = 1, OG = 49, SZ = 7, OE = 7, OS = 56
Easter on April 25th ("March 56th"). The Easter dates are not repeated every 300,000 (= 5,700,000 / 19) years.

Thus, by refuting the counter-claim by a counterexample, it is shown that the Easter dates only repeat every 5,700,000 years.

literature

  • Friedrich Karl Ginzel : Handbook of mathematical and technical chronology. Volume 3: Calculation of the times of the Macedonians, Asia Minor and Syrians, the Teutons and Celts, the Middle Ages, the Byzantines (and Russians), Armenians, Copts, Abyssinians, the calculation of modern times, as well as additions to the three volumes. Hinrichs, Leipzig 1914.
  • Marcus Gossler: Term dictionary of chronology and its astronomical bases. With a bibliography. Second improved edition. University Library, Graz 1985 ( University Library Graz - Bibliographical Information 12).

Web links

Individual evidence

  1. a b Marcus Gossler: Term dictionary of chronology and its astronomical foundations , University Library Graz, 1981, p. 115
  2. a b Heiner Lichtenberg: The adaptable, cyclical, solilunear time counting system of the Gregorian calendar - a scientific masterpiece of the late Renaissance. Mathematical Semester Reports, Volume 50, 2003, p. 47
  3. a b The word part "equation" meant "correction" in the Middle Ages. See N. Dershowitz, EM Reingold: Calendrical Calculations. Cambridge University Press, 2008, ISBN 978-0-521-70238-6 , page 182
  4. ^ A b Friedrich Karl Ginzel: Handbook of the mathematical and technical chronology. Volume 3: Calculation of the times of the Macedonians, Asia Minor and Syrians, the Teutons and Celts, the Middle Ages, the Byzantines (and Russians), Armenians, Copts, Abyssinians, the calculation of modern times, as well as additions to the three volumes. Hinrichs, Leipzig 1914. Volume 3 , 1914, pp. 257-266 .
  5. Reduction of the epacts when applying the solar equation (in b))
  6. Christophorus Clavius: Romani Calendarii A Gregorio XIII. PM Restitvti Explicatio (Explicatio) . 1612, p. 132-133, 155 .
  7. Christophorus Clavius: Romani Calendarii A Gregorio XIII. PM Restitvti Explicatio (Explicatio). Retrieved January 28, 2018 (Latin).
  8. ^ Gregorian Reform of the Calendar . In: GV Coyne, MA Hoskin, O. Pedersen (eds.): Proceedings of the Vatican Conference to Commemorate its 400th Anniversary 1582-1982 . 1983.
  9. This 400 year long circle is already contained in whole numbers in the additional moon circle.
  10. Physikalisch-Technische Bundesanstalt : When is Easter?