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Easter

What is Easter?

In the Christian world, Easter (and the days immediately preceding it) is the celebration of the death and resurrection of Jesus in (approximately) AD 30.

When is Easter? (Short answer)

Easter Sunday is the first Sunday after the first full moon on or after the vernal equinox.

When is Easter? (Long answer)

The calculation of Easter is complicated because it is linked to (an inaccurate version of) the Hebrew calendar.

Jesus was crucified immediately before the Jewish Passover, which is a celebration of the Exodus from Egypt under Moses. Celebration of Passover started on the 15th day of the (spring) month of Nisan. Jewish months start when the moon is new, therefore the 15th day of the month must be immediately after a full moon.

It was therefore decided to make Easter Sunday the first Sunday after the first full moon after vernal equinox. Or more precisely: Easter Sunday is the first Sunday after the “official” full moon on or after the “official” vernal equinox.

The official vernal equinox is always 21 March.

The official full moon may differ from the real full moon by one or two days.

(Note, however, that historically, some countries have used the real (astronomical) full moon instead of the official one when calculating Easter. This was the case, for example, of the German Protestant states, which used the astronomical full moon in the years 1700-1776. A similar practice was used in Sweden in the years 1740-1844 and in Denmark in the 1700s.)

The full moon that precedes Easter is called the Paschal full moon. Two concepts play an important role when calculating the Paschal full moon: The Golden Number and the Epact. They are described in the following sections.

The following sections give details about how to calculate the date for Easter. Note, however, that while the Julian calendar was in use, it was customary to use tables rather than calculations to determine Easter. The following sections do mention how to calculate Easter under the Julian calendar, but the reader should be aware that this is an attempt to express in formulas what was originally expressed in tables. The formulas can be taken as a good indication of when Easter was celebrated in the Western Church from approximately the 6th century.

The Golden Number of 2014:

1

What is the Golden Number?

Each year is associated with a Golden Number.

Considering that the relationship between the moon’s phases and the days of the year repeats itself every 19 years (as described in the section about astronomy), it is natural to associate a number between 1 and 19 with each year. This number is the so-called Golden Number. It is calculated thus:

GoldenNumber=(year mod 19) + 1

(Click here for a description of the operator ‘mod’.)

In years which have the same Golden Number, the new moon will fall on (approximately) the same date. The Golden Number is sufficient to calculate the Paschal full moon in the Julian calendar.

How does one calculate Easter then?

Under the Julian calendar the method was simple. If you know the Golden Number of the year, you can find the Paschal full moon in this table:

Golden
Number
Full moon Golden
Number
Full moon Golden
Number
Full moon
1 5April 818April 15 1April
225March 9 7April 1621March
313April 1027March 17 9April
4 2April 1115April 1829March
522March 12 4April 1917April
610April 1324March
730March 1412April

Easter Sunday is the first Sunday following the above full moon date. If the full moon falls on a Sunday, Easter Sunday is the following Sunday.

(In particular in the Eastern Orthodox Churches, an additional rule that Easter must come after the Jewish Passover has been used. As time went by, this rule became less important, as the error in the Julian calendar caused Easter to fall later in spring.)

Under the Gregorian calendar, things became much more complicated. One of the changes made in the Gregorian calendar reform was a modification of the way Easter was calculated. There were two reasons for this. First, the 19 year cycle of the phases of moon (the Metonic cycle) was known not to be perfect. Secondly, the Metonic cycle fitted the Gregorian calendar year worse than it fitted the Julian calendar year.

It was therefore decided to base Easter calculations on the so-called Epact.

The Epact of 2014:

29

What is the Epact?

Each year is associated with an Epact.

The Epact is a measure of the age of the moon (i.e. the number of days that have passed since an “official” new moon) on a particular date.

(In this context a “new moon” is the first visible crescent moon. In modern calendars, a “new moon” is the completely invisble moon. The two differ by approximately one day, and this difference must be kept in mind when comparing the new moon calculations presented here to reality.)

In the Julian calendar, the Epact is the age of the moon on 22 March.

In the Gregorian calendar, the Epact is the age of the moon at the start of the year.

The Epact is linked to the Golden Number in the following manner:

Under the Julian calendar, 19 years were assumed to be exactly an integral number of synodic months, and the following relationship exists between the Golden Number and the Epact:

Epact=(11 × (GoldenNumber – 1)) mod 30

If this formula yields zero, the Epact is by convention frequently designated by the symbol * and its value is said to be 30. Weird? Maybe, but people didn’t like the number zero in the old days.

Since there are only 19 possible golden numbers, the Epact can have only 19 different values: 1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, and 30.

In the Gregorian calendar reform, some modifications were made to the simple relationship between the Golden Number and the Epact.

In the Gregorian calendar the Epact should be calculated thus:

(Click here for a description of the symbols   and   and the operator ‘mod’.)

  1. Use the Julian formula:

    Epact = (11 × (GoldenNumber – 1)) mod 30

  2. Calculate the so-called “Solar Equation”:
    S =
     3 × century 
    4

    The Solar Equation is an expression of the difference between the Julian and the Gregorian calendar. The value of S increases by one in every century year that is not a leap year.

    (For the purpose of this calculation century=20 is used for the years 1900 through 1999, and similarly for other centuries, although this contradicts the normal rules.)

  3. Calculate the so-called “Lunar Equation”:
    L =
     8 × century + 5 
    25

    The Lunar Equation is an expression of the difference between the Julian calendar and the Metonic cycle. The value of L increases by one 8 times every 2500 years.

  4. Calculate the Gregorian epact thus:

    GregorianEpact = JulianEpact – S + L + 8

    The number 8 is a constant that calibrates the starting point of the Gregorian Epact so that it matches the actual age of the moon on new year’s day.

  5. Add or subtract 30 until GregorianEpact lies between 1 and 30.

In the Gregorian calendar, the Epact can have any value from 1 to 30.

Example: What was the Epact for 2011?

GoldenNumber = 2011 mod 19 + 1 = 17
1.JulianEpact = (11 × (17 – 1)) mod 30 = 26
2.S =
 3 × 20 
4
= 15
3.L =
 8 × 20 + 5 
25
= 6
4.GregorianEpact = 26 – 15 + 6 + 8 = 25
5.No adjustment is necessary

The Epact for 2011 was 25.

In 2014 Easter Sunday is:

20 April

How does one calculate Gregorian Easter then?

Look up the Epact in this table to find the date for the Paschal full moon:

EpactFull moon EpactFull moon EpactFull moon
112April 11 2April 2123March
211April 12 1April 2222March
310April 1331March 2321March
4 9April 1430March 2418April
5 8April 1529March 2518or 17 April
6 7April 1628March 2617April
7 6April 1727March 2716April
8 5April 1826March 2815April
9 4April 1925March 2914April
10 3April 2024March 3013April

Easter Sunday is the first Sunday following the above full moon date. If the full moon falls on a Sunday, Easter Sunday is the following Sunday.

An Epact of 25 requires special treatment, as it has two dates in the above table. There are two equivalent methods for choosing the correct full moon date:

  1. Choose 18 April, unless the current century contains years with an epact of 24, in which case 17 April should be used.
  2. If the Golden Number is > 11 choose 17 April, otherwise choose 18 April.

The proof that these two statements are equivalent is left as an exercise to the reader. (The frustrated ones may contact me for the proof.)

Example: When was Easter in 2011?

In the previous section we found that the Golden Number for 2011 was 17 and the Epact was 25. Looking in the table, we find that the Paschal full moon was either 17 or 18 April. By rule B above, we choose 17 April because the Golden Number > 11.

17 April 2011 was a Sunday. Easter Sunday must therefore have been the following Sunday, 24 April.

Isn’t there a simpler way to calculate Easter?

The formulas below are an attempt to boil down the information given in the previous sections. (Click here for a description of the symbols   and   and the operator ‘mod’.)

G = year mod 19
For the Julian calendar:
I = (19G + 15) mod 30
J = ( year + 
 year 
4
 + I ) mod 7
For the Gregorian calendar:
C =
 year 
100
H = ( C – 
 C 
4
 – 
 8C + 13 
25
 + 19G + 15 ) mod 30
I = H – 
 H 
28
 × (1 – 
 29 
H + 1
 × 
 21 – G 
11
)
J = ( year + 
 year 
4
 + I + 2 – C + 
 C 
4
) mod 7
Thereafter, for both calendars:
L = I – J
EasterMonth = 3 + 
 L + 40 
44
EasterDay = L + 28 – 31 × 
 EasterMonth 
4

This algorithm is based in part on the algorithm of Oudin (1940).[1]

People who want to dig into the workings of this algorithm, may be interested to know that

Gis the Golden Number–1.
His 23–Epact (modulo 30).
Iis the number of days from 21 March to the Paschal full moon.
Jis the weekday for the Paschal full moon (0=Sunday, 1=Monday, etc.).
Lis the number of days from 21 March to the Sunday on or before the Paschal full moon (a number between -6 and 28).

Isn’t there an even simpler way to calculate Easter?

If we confine ourselves to the years 1900-2099 and consider only the Gregorian calendar, the formulas of the previous section can be further simplified thus:

H = ( 24 + 19 × (year mod 19) ) mod 30
I = H – 
 H 
28
J = ( year + 
 year 
4
 + I – 13 ) mod 7
L = I – J
EasterMonth = 3 + 
 L + 40 
44
EasterDay = L + 28 – 31 × 
 EasterMonth 
4

Is there a simple relationship between two consecutive Easters?

Suppose you know the Easter date of the current year, can you easily find the Easter date in the next year? No, but you can make a qualified guess.

If Easter Sunday in the current year falls on day X and the next year is not a leap year, Easter Sunday of next year will fall on one of the following days: X–15, X–8, X+13 (rare), or X+20.

If Easter Sunday in the current year falls on day X and the next year is a leap year, Easter Sunday of next year will fall on one of the following days: X–16, X–9, X+12 (extremely rare), or X+19. (The jump X+12 occurs only once in the period 1800-2200, namely when going from 2075 to 2076.)

If you combine this knowledge with the fact that Easter Sunday never falls before 22 March and never falls after 25 April, you can narrow the possibilities down to two or three dates.

How frequently are the dates for Easter repeated?

The sequence of Easter dates repeats itself every 532 years in the Julian calendar. The number 532 is the product of the following numbers:

19 (the Metonic cycle or the cycle of the Golden Number)
28 (the Solar Cycle)

The sequence of Easter dates repeats itself every 5,700,000 years in the Gregorian calendar. Calculating this is not as simple as for the Julian calendar, but the number 5,700,000 turns out to be the product of the following numbers:

19 (the Metonic cycle or the cycle of the Golden Number)
400 (the Gregorian equivalent of the Solar Cycle)
25 (the cycle used in step 3 when calculating the Epact)
30 (the number of different Epact values)

What about Greek Orthodox Easter?

The Greek Orthodox Church does not always celebrate Easter on the same day as the Catholic and Protestant countries. The reason is that the Orthodox Church uses the Julian calendar when calculating Easter. This is the case even in the churches that otherwise use the Gregorian calendar.

When the Greek Orthodox Church in 1923 decided to change to the Gregorian calendar (or rather: a Revised Julian Calendar), they chose to use the astronomical full moon as the basis for calculating Easter, rather than the “official” full moon described in the previous sections. And they chose the meridian of Jerusalem to serve as definition of when a Sunday starts. However, except for some sporadic use in the 1920s, this system was never adopted in practice.

In calendrical calculations, we frequently use an operation call integer division.

Ordinarily we would say that, for example, 14 divided by 5 is 2.8. But when we use integer division, we discard the decimal fraction and simply state that 14 divided by 5 is 2.

We indicate that we use integer division by enclosing the division between the symbols   and   , for example thus:

  14/5   = 2

or thus:

 14 
5
= 2

When we perform integer division, the division leaves a remainder. In the case of 14 divided by 5, the remainder is 4: We can subtract 5 twice from 14, and this leaves us with 4.

We use the mathematical operator ‘mod’ to indicate the remainder. This is known as the modulo operator. We therefore have:

14 mod 5 = 4

If you want to use integer division in computer programs, you must realize that different programs use different notations for the operations. The following table shows how integer division and the modulo operator may be written:

Integer division Modulo
The Calendar FAQ   14/5   14 mod 5
Microsoft Excel (English) INT(14/5) MOD(14,5)
Visual Basic 14 \ 5 14 Mod 5
C, C#, C++ 14 / 5 14 % 5



References

^ [1] http://astro.nmsu.edu/~lhuber/leaphist.html, retrieved 20 September 2011, which contains a reprint of the “Explanatory Supplement to the Astronomical Almanac,” P. Kenneth Seidelmann, editor.