Gaussian Easter formula

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The Gaussian Easter formula by Carl Friedrich Gauß allows the calculation of the Easter date for a given year. In this the complete algorithm of the Easter calculation is formulated. However, for the sake of clarity, the formula is noted as a set of equations to be calculated one after the other.

This set of equations applies generally to the Gregorian calendar , and after replacing two variable intermediate sizes with constant values, it also provides the Easter date in the Julian calendar .

The additional provision made in the Gregorian calendar reform that the last possible Easter Sunday is April 25th as before, Gauss did not incorporate into the Easter formula. In rare cases, the formula delivers April 26th as Easter Sunday. Gauss expressed the corresponding exception rules with regard to his formula - albeit in his own words - also only verbally.

background

Since the resolutions of the first Council of Nicaea in 325 and based on the work that Dionysius Exiguus began on behalf of Pope John I in 525 , Easter has been celebrated on the first Sunday after the spring full moon , Easter Sunday .

According to the decision, the day of the beginning of spring is March 21st. A full moon on March 21st is considered the earliest possible spring full moon. March 22nd is therefore the earliest calendar day on which Easter can fall. In the Julian calendar, the last possible Easter Sunday falls on April 25th. This limitation was retained in an additional provision in the Gregorian calendar. So there are 35 different Easter dates in both calendars. Easter has the character of a moving holiday . Easter plays a central role in the church year , as almost all movable Christian holidays such as Ash Wednesday , Ascension Day or Pentecost depend on it.

Traditional Easter bill

The algorithm used by Exiguus 525 for the Easter calculation has remained unchanged to this day. It was only expanded on the occasion of the Gregorian calendar reform in 1582. Gauss represented it using equations. Previously, the Easter calculation was carried out “by hand” with the help of tables and was called Computus pascalis , or Computus for short. Easter calculators like Exiguus were able to calculate Easter dates for any number of future years with the unique algorithm. This was only necessary again in 1582 after the calendar reform had taken place. Although Gauss worked more elegantly around 1800 than his predecessors, often referred to as computists, he placed himself in the long line of Easter calculators that had been in operation since Exiguus and always took on a job that was already done. In the late Middle Ages, the computus was at times the only mathematics chapter in university education.

Original versions by Gauss

div stands for an integer division (decimal places are cut off).

mod stands for the remainder of the division in an integer division.

Version from 1800

Carl Friedrich Gauß published his Easter formula for the first time in 1800. In the introduction he wrote: "The intention of this essay is [...] to give a purely analytical solution to this task [...] based only on the simplest calculation operations." He left at the time assumed that the lunar equation was to be applied regularly every 300 years.

Julian calendar Gregorian calendar
a = Jahr mod 19 
b = Jahr mod 4  
c = Jahr mod 7  
k = Jahr div 100
p = k div 3
q = k div 4
M = 15 M = (15 + k − p − q) mod 30
d = (19a + M) mod 30
N = 6  N = (4 + k − q) mod 7
e = (2b + 4c + 6d + N) mod 7
Ostern = (22 + d + e)ter März
(Der 32. März ist der 1. April usf.)
Ausnahmen: 1.) Falls  d=29  und  e=6,  dann Ostern=50

2.) Falls  d=28  und  e=6  und   (11M + 11) mod 30 < 19,  dann Ostern=49

Version from 1816

There is a handwritten addendum of unknown date (after 1807) in which Gauss took into account the change in the lunar equation that he had previously overlooked and made by the reformers . The correction was published in 1816 and only affects the variable p.

Julian calendar Gregorian calendar
a = Jahr mod 19 
b = Jahr mod 4  
c = Jahr mod 7  
k = Jahr div 100
p = (8k + 13) div 25
q = k div 4
M = 15 M = (15 + k − p − q) mod 30
d = (19a + M) mod 30
N = 6  N = (4 + k − q) mod 7
e = (2b + 4c + 6d + N) mod 7
Ostern = (22 + d + e)ter März
(Der 32. März ist der 1. April usf.)
Ausnahmen: 1.) Falls  d=29  und  e=6,  dann Ostern=50

2.) Falls  d=28  und  e=6  und  a>10,  dann Ostern=49

validity

The Gaussian Easter formula applies to any calendar year according to the Julian and Gregorian calendars , as long as the church rules for determining the Easter date are not changed, even if in some representations, limited tables give the impression or can arise that the validity is for certain years limited. However, the variables M and N change every 100 years; currently: M = 24 and N = 5.

A supplemented Easter formula

Although the Gaussian Easter formula elegantly describes the Easter algorithm in short, the specification of the latest Easter Sunday to April 25, contained in two exception rules , is not covered by the formula itself. A corresponding addition was given by Hermann Kinkelin in the 19th century , Christian Zeller wrote: "Incidentally, this exception can also be introduced into the formula itself [...]". The compact summary of the entire calculation only gained interest in the age of the PC, when the somewhat more complex calculation that you no longer had to carry out yourself played a smaller role than the clearer input in the form of a program.

Such a summary was presented again in 1997 by Heiner Lichtenberg, who also structured the formula again, which Gauss had demonstratively presented as "a purely analytical solution independent of those auxiliary terms [...]." It is shown below.

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

step meaning formula
1. the secular number K(X) = X div 100
2. the secular moon circuit M(K) = 15 + (3K + 3) div 4 − (8K + 13) div 25
3. the secular sunshade S(K) = 2 − (3K + 3) div 4
4th the lunar parameter A(X) = X mod 19
5. the germ for the first full moon in spring D(A,M) = (19A + M) mod 30
6th the calendar correction value R(D,A) = (D + A div 11) div 29
7th the Easter border OG(D,R) = 21 + D − R
8th. the first Sunday in March SZ(X,S) = 7 − (X + X div 4 + S) mod 7
9. the distance of Easter Sunday from the Easter border
(Easter distance in days)		 OE(OG,SZ) = 7 − (OG − SZ) mod 7
10. the date of Easter Sunday as the March date
(March 32 = April 1, etc.)		 OS = OG + OE

The above algorithm applies to the Gregorian calendar.

For the Julian calendar you set M = 15 and S = 0 and you also get a date in the Julian calendar as the result. This date can be converted into the Gregorian calendar used today using the following formula. You get the date of Easter of the Eastern Churches, as it is used, for example, in Greece: 

OS_Ost = OS + (X div 100) − (X div 400) − 2

Comparison: original formula - supplemented formula

The two full versions (Gauß and Lichtenberg, see above) for the Gregorian calendar are compared. The variable X is the calendar year.

Original formula supplemented formula
Gaussian cycle number a = X mod 19 A(X) = X mod 19 4th
b = X mod 4  
c = X mod 7  
k = X div 100 K(X) = X div 100 1.
p = (8k + 13) div 25  
q = k div 4  
Corr .: So- u. Mo equation:M = (15 + k - p - q) mod 30 M(K) = 15 + (3K + 3) div 4 - (8K + 13) div 25 2.
Corr .: solar equation N = (4 + k - q) mod 7
Distance to the moon: d = (19a + M) mod 30 D(A,M) = (19A + M) mod 30 5.
  S(K) = 2 - (3K + 3) div 4 3.
  R(D,A) = (D + A div 11) div 29 6th
  OG(D,R) = 21 + D - R 7th
  SZ(X,S) = 7 - (X + X div 4 + S) mod 7 8th.
Easter distance: e = (2b + 4c + 6d + N) mod 7 OE(OG,SZ) = 7 - (OG - SZ) mod 7 9.
Easter Sunday: = (22 + d + e) ter März OS = (OG + OE) ter März 10.
March 32nd is April 1st, etc. OS = 32 is April 1st, etc.

Exceptions

Calculation results in exceptional years

Exception rules

1981 - exception I. Year 1954 - exception II
original added original added
a = 5 A = 5 a = 16 A = 16
b = 1 b = 2
c = 0 c = 1
k = 19 K = 19 k = 19 K = 19
p = 6 p = 6
q = 4 q = 4
M = 24 M = 24
M = 24 M = 24
N = 5 N = 5
d = 29 D = 29 d = 28 D = 28
S = -13 S = -13
R = 1 R = 1
OG = 49 OG = 48
SZ = 1 SZ = 7
e = 6 OE = 1 e = 6 OE = 1
Easter Sunday
= March 57
= April 26 		Easter Sunday
= March 50
= April 19 		Easter Sunday
= March 56
= April 25 		Easter Sunday
= March 49
= April 18

Comments by Gauss on the exceptions

Gauss wrote four times about his method of determining Easter, three times about the handling of exceptions:

  • 1800: “If the Easter bill is April 26th, April 19th is always used. […] If the calculation gives d = 28, e = 6, and there is also the condition that 11M + 11 divided by 30 [sic] gives a remainder that is less than 19, Easter […] falls on the 18th April".
  • 1807: "only if the first remainder [note: the year mod 19] was not below 11" The second exception is formulated differently than 1800, but the effect is unchanged compared to the older formulation.
  • 1811: “If in the Gregor. Calendar the bill Easter on 26th. April gives, you always set the 19th. and if they are the 25th. brings, the 18th. ”Now the second exception is shown in an impermissibly shortened form. The complete edition contains a comment by Alfred Loewy on this error.
  • 1816: Gauss announced the essential correction due to the originally incorrectly assumed lunar equation, but no longer commented on the exceptions.

See also

literature

Web links

Wikibooks: Algorithm collection: Calendar: Holidays - Easter formula according to Lichtenberg  - Learning and teaching materials

Individual evidence

  1. ↑ Gauss himself spoke of “the simplest calculation operations”, see Gauß: Calculation of Easter , 1800, pp. 121–122
  2. ^ Arno Borst : Computus - Time and Numbers in the History of Europe , Wagenbach, 2004, ISBN 3-8031-2492-1 , p. 34
  3. ^ Arno Borst : Computus - time and number in the history of Europe , Wagenbach, 2004, ISBN 3-8031-2492-1 , p. 41
  4. ^ Heinz Zemanek : Calendar and Chronology , Munich, 1990, ISBN 3-486-20927-2 , p. 35 and p. 45
  5. ↑ Gauss: Calculation of Easter , 1800
  6. ^ Bear: The addendum to the Easter formula by CF Gauss in The Easter formula by CF Gauss
  7. ↑ a b Gauss: Correction to the essay: Calculation of Easter , 1816
  8. ^ Kinkelin : The calculation of the Christian Easter festival , 1870
  9. ^ Zeller : Calendar formulas , 1887
  10. ↑ Lichtenberg: On the interpretation of the Gaussian Easter formula and its exception rules , 1997
  11. ↑ It is used, for example, by the PTB , see When is Easter?
  12. ↑ Gauss: Calculation of the Easter Festival , 1800, pp. 121-122
  13. ↑ simplified form according to Kinkelin; near Lichtenberg:R(D,A) = D div 29 + (D div 28 − D div 29) (A div 11)
  14. ^ Gauss: Calculation of Easter , 1800, p. 129
  15. ↑ Gauß: Something else about the determination of Easter , 1807, end of Col. 594
  16. ↑ Gauss: A Easy Method to Find Easter Sunday , 1811, footnote on p. 274
  17. ^ Gauss: Works. Volume 11.1 , 1927, p. 200