Calculation of the date of Easter

the date of the festival of Easter is important as well in the religious life as in the civil life. In the religious life, because several festivals or celebrations are fixed according to this date of Easter in the Comput. In the civil life, that is due to the fact that, in very many countries, some of the religious Christian festivals are the occasion of fériés days; ; for example : Good Friday, Easter Monday, Whit Monday, Whit Monday in France or Ascension Day. Unfortunately, obtaining this date requires a certain number of calculations.

Dates contemporary of Easter: Sundays , and of the calendar grégorien.

Go back to Easter

This date is the subject of an astronomical definition of nature : in the year 325, the council of Nicée issued the rule suivante : Easter is celebrated the Sunday which follows the fourteenth day of the moon which reaches this age at March 21st or immediately afterwards. In other words, it is the first Sunday which follows or which coincides with the first full moon after the March 21st (marking the beginning of the Printemps). Before this Council, the festival followed variable rules, influenced by the date of Pessa' H.

In this definition, the expression “ age of the lune ” is not taken in its direction usuel : the age is counted in whole days, as from the day of the new moon. Moreover, it is the the moon of comput (known as also the moon Pascale ) which is used, and not the news the real Moon. The date depends neither on the astronomical tables, nor of a direct observation of the crescent (practical still of use for the observation of certain Muslim holidays). It is also noticed that it is not the real date of spring, but well on March 21st, which is used as goes back to reference (even if it falls often a March 21st, spring will have certain years one day or two of shift with this date).

By construction, Easter is located the March 22nd as soon as possible. That occurs when Pascale lunation begins the March 8th (the new moon) and that the March 22nd is one Sunday (posterior with the equinox which coincides this day with full moon of comput).

On the contrary the latest possible date is the April 25th. Indeed, in the Gregorian Comput, a lunation (the new moon) can begin on March 7th. The fourteenth day (last full moon of winter) then falls on March 20th and the following full moon (first full moon of spring) is on April 18th (29 days afterwards). If this April 18th is already Sunday, Easter fall next Sunday (25).

It is the same in the Comput Julien because, even if there is no possible new moon on March 7th in this Calendrier, a lunation which would begin the Julien on March 6th would last 30 days then and not 29.

Elements of the ecclesiastical comput

General information

Elements of the Comput. - the comput Julien uses two elements, the Sunday Lettre and the golden section astronomical. The Gregorian comput uses the Sunday letter and the épacte, but the knowledge of the golden section is also necessary, contrary to what is generally known as. Because the solar Cycle is an element whose employment is equivalent to that of the Sunday letter Julienne.

The Roman indiction does not have any more any role in the comput actuel ; it served in an old Christian chronology, based on a division of 15 years the successive cycle times as from the year 313 of the Christian era, and in which it indicated the one year old row in its cycle.

The value of these elements is given each year in the almanachs ; also one believed to have to publish it in the Knowledge of Times and the directory of the Office of longitudes. Because of this publication, the directory of the Office of longitudes devoted to the comput a periodic article, which was concluded naturally by calculation from the date from Easter. The text of this article was besides very appreciably the reproduction of the end of the Book IV|4 3rd edition of the Course of astronomy of H. Andoyer of 1923. It was about a tradition not justifiée : the elements of the comput do not have any scientific role, civil, or even monk. The real calculation of these days is more complicated than that of the Everlasting calendar published annually in Éphémérides astronomical of the directory of the Office of longitudes.

Form of calculation

Since the {{Romanian|I|1}} {{er}} council of Nicée, the date of Easter follows a given rule and thus was the subject of mathematical formulas such as those which are indicated hereafter. The tables of it do not result, however these calculations make it possible to easily find the dates of Easter and other festivals of the Comput. The demonstration of these formulas asks for only knowledge of arithmetic elementary. That does not mean that they are always simple for autant !

Example of calculation for the year 2006 :

It is necessary as a preliminary to calculate L according to the Sunday Lettre Gregorian (one poses L = 1 for the letter has , it is the case in 2006 for example, to L = 7 for the letter G ).
Then it is necessary to calculate E, the Gregorian épacte (E is worth zero per 2006) :

to keep the same formula in April, the épacte E is corrected ainsi :

if E = 24 and L = 4 (letter D), to then take E = - 5
if E = 25 and L = 3 (letter C), to then take E = - 4
if not, if E is worth 24 or beyond, to decrease E of 30 ;

To add the numbers E and L and to increment of 1 (0 + 1 + 1 = 2 for 2006).
To take the remainder in division by 7 ( 2 /7 = 0, but 0 × 7 = 0 instead of 2 , the remainder is worth thus 2).
To add 45 and withdraw E (2 + 45 - 0 = 47).

The result represents the Easter Day on the basis of at the beginning of March (1 = on March 1st). In fact the smallest value obtained possible east 22 for on March 22nd. If the result exceeds 31, it is that Easter fall in April (it is enough to cut off 31 on the date). For 2006, Easter thus fall on Sunday, April 16 (47 - 31 = 16).

Algorithm of Thomas O' Beirne

The first algorithm allotted to Gauss contained some errors. (In fact of error, it had been based on data limited in time. Its result is thus not guaranteed for all the periods. ) Here in any case an algorithm, initiated by Thomas O' Beirne, who has the merit to be right and to include all intermediate calculations of the Comput while leaving the year simply.

One recognizes there obviously parts of preceding calculation (golden section, etc .) even if the role of the sizes is not specified here. The algorithm is given in its simplified form because it is practical (the full version is in any event less interesting because gives place to calculations useless or that can be simplified; for example one could obtain the épacte by dividing directly X by 19). For the current period (1900 - 2099) there is way rapide :

That is to say M the year of calculation (let us take 2005 for example) :

One poses N = M - 1900 (one cuts off 1900 at the year, therefore N = 105 for our example) ;
One takes has , the remainder of N in division by 19 (105/19 = 5 but 5 × 19 = 95 instead of 105, it thus remains 10 ; = 10) has; ;
One calculates has × 7 + 1 (what gives for example 7 × 10 + 1 = 71) ;
One takes of it B , the result (whole) of division by 19 (71/19 = 3 thus B = 3) ;
One calculates (11 × a) - B + 4 (that is to say 11 × 10 - 3 + 4 = 111) ;
One takes of it C the remainder in division by 29 (111/29 = 3, but 3 × 29 = 87 instead of 111, it thus remains 111 - 87 = 24, therefore C = 24) ;
One calculates D the whole part of N/4 (105/4 = 26) ;
One calculates N - C + D + 31 (that is to say 105 - 24 + 26 + 31 = 138) ;
One takes of it E the remainder in division by 7 (138/7 = 19, but 19 × 7 = 133 instead of 138, it thus remains 138 - 133 = 5, therefore E = 5) ;
One calculates P = 25 - C - E (in the exemple : P = 25 - 24 - 5 = -4) ;
the date of Easter falls P days after March 31st (or before if P is negative). What means que :

for P = 1, April, in other words P positive corresponds directly to the day of the month of avril ;
for P = the 0 Easter Day is on March 31st, and
for P = - 1 on March 30th, in other words P negative must be added to 31 to obtain the day of the month of mars ;

: (For the year 2005, one finds P = -4, which wants to say that Easter is the Sunday the 31st - 4 = March 27th).

By remaking this calculation for the year 2006, one would find P = 16 and one would find the preceding result (Easter 2006 fall on April 16th).

Algorithm of Oudin

The advantage of this one is to be among those which require less operations. It is thus one of most interesting for calculation general practitioner (unbounded of century). It is presented in its form not simplified thus usable for any year after 1583 (after the appearance of the Gregorian Calendrier). The calculation of the date of Easter is far from being a so easy thing. One will take for example calculation in the current year ().

Divisions must always be whole (one removes the decimals).

G which represents the golden section decreased by 1: To divide the year by 19, to take the reste
of it (/19 = gold x19= and we need, therefore the variation is worth G=)
C and C_4 allows the follow-up of the leap years: to still divide the year by 100 then by 4
(/100 =C=
et /4 =C_4=)
E : To divide (8 X C + 13) by 25 without the décimales
(8x+13= /25 =E=)
H which depends on the épacte: to divide (19xG + C - C_4 - E + 15) by 30, to take the reste
of it (One takes the remainder of a division according to the same principle as for G: () /30= gold x30= and we need (), therefore the variation is worth H=)
K : to divide H by 28
(/28 = K =)
P : to divide 29 by (H+1)
(29/= P =)
Q : to divide (21-G) by 11
(21-= /11 =Q=)
I represents the number of days between the Pascale full moon and on March 21st: (KxPxQ - 1) X K + H
(xx-1= x= + = I=)
B : to divide the year by 4 and to remove the decimals, to add to it the année
(/4 = + =)
J1 : To add B + I + 2 + C_4 and to cut off C
(J1 =)
J2 calculates the day of the moon Pascale (0= Sunday the 1st =lundi… 6=samedi): to divide J1 by 7 and to take the reste.
of it (One always calculates the remainder of a division according to the same principle as with G and H, the result is J2=)

R the end result, finally: 28 + I - J2
(R=)

R represents the date of March, if it exceeds 31 one overflows over April (… 30 corresponds to March 30th, from March 31st to 31st, 32 at April 1st, 33 at April 2nd,…). To cut off 31 to obtain the date of April if necessary. (Easter thus fall it.)

Easter and Pessah'

It is often affirmed that the date of catholic and Protestant Easter (which are identical) and dates it from orthodoxe Easter would be delayed not to coincide with Jewish Passover. It of it is absolutely nothing, on the contrary. It is just that a difference in dates had sometimes occurred during the first centuries.

Indeed the coincidence of the two festivals was for the pagan philosophers a subject of derision with regard to the two religions which claimed to be opposed and seemed to follow the same liturgy. The celebrated events are different: Exodus of the Hebrews, brought back by the Old Testament and Résurrection of the Christ, brought back by the New , which places it during the week of Jewish Passover.

With the council of Nicée, a controversy rose: “The ones supported that it was necessary to follow the habit of the juifs ; the others claimed that time had to be examined exactly, and not to agree with people so far away from the grace of Écriture ” (Eusèbe, Life of the emperor Constantin , III|3 , 5).

A third solution prévalut : to eliminate any risk from coincidence with the others fêtes ; from where the rule of Nicée. The Jewish Passover is indeed fixed at the fourteenth day of the moon of Nissan , during which fall the equinox from spring. This month does not coincide inevitably with the fictitious Moon of the Christian comput. So that the required effect is not always reached.

Let us add that the rule of Nicée does not form part of the dogma and that it will be enough to an ecumenical council to modify it when the interest is sufficiently ressenti ; that possible modifications had been explicitly envisaged in the acts of the council of 325 ; finally that the rule itself was not promulgated at the time by the Pape but by the emperor Constantin, who was not baptized besides that with his bed of death, 10 years later. In fact, a declaration of the council of the Vatican II states that the Church would not be opposed to the principle of a fixed date for Easter.

Various styles of calculation

Orthodoxe calculation of Easter

Celtic calculation of Easter (or comput ancient)

See too

Internal bonds

To calculate the dates of the other festivals mobiles :

the Comput
the catholic liturgical Calendar

External bonds

Years classified by date of Easter
a calculation algorithm
Comparison of various algorithms (very documented)
Several algorithms with explanations (and punctually some typing errors in the formulas, damage)
On the site of IMCCE :

Explanations
a Gregorian everlasting calendar of the dates of Easter
Calculation per year

Nicée and the date of Easter
Easter Controversies
Table of Easter de Dionysius Exiguus

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