Whence Easter ?

Easter is the central point of the Christian calendar, with most other feast days and holidays (the moveable feasts) being based around it. As a result, it has a major role in the civil calendar as well. So it seems curious to the unaware that it appears to jump around the calendar so randomly.

The Bible tells us that the crucifixion took place at the time of the Passover feast, with the resurrection on the following Sunday. Since the Passover is basically on the first full moon of Spring, we get a definition for Easter:

Easter Sunday is the Sunday after the first full moon of Spring. The full moon may be on the first day of Spring (March 21st), but Easter will never be, and if the full moon is on a Sunday, Easter is on the following Sunday.

This rule seems fairly simple, but it has some nasty catches. Firstly, predicting the movements of the moon is far from simple, and determining the exact moment of the full moon even less so. If the moment is not got exactly right, the date of Easter could be wrong by a month (if the full moon is on the March 20th/21st boundary), or by a week (if it is on a Saturday/Sunday boundary), or both ! Then there is the question of where the full moon should be observed. The full moon could easily be on different days in different parts of the world. And when it was got wrong, it could be spectacularly wrong: the extreme case had an eclipse of the (full) moon on Easter Monday !

Over time the various authorities settled on a set of compromises; the most important of these were hammered out at the Council of Niceae in 325. The basic rules that were established are:

Julian Easter

Astronomers right back to Babylonian times have noticed that the cycles of the moon obey an interesting rule: 235 lunar months are almost exactly 19 years of 365.25 days. Therefore, if the dates of the full moons are recorded in 19 columns, we soon discover that each column will only ever contain two adjacent dates - two because the presence or absence of February 29th causes the moment of full moon to happen at 4 different times with 18 hours between the earliest and latest. For the sake of simplicity, it was tempting to ignore this detail and assign one date to each column, and this was soon done. The result was that in the Julian Calendar, which had leap years every four years, the rules for determing Easter were very simple:

  1. Divide the number of the year by 19, and determine the remainder. Add one: the result is called the golden number of the year.
  2. Look at a table which gives the date of the Paschal full moon for that golden number.
    Golden
    number
    Paschal
    full moon
    1 April  5
    2 March 25
    3 April 13
    4 April  2
    5 March 22
    6 April 10
    7 March 30
    Golden
    number
    Paschal
    full moon
    8 April 18
    9 April  7
    10 March 27
    11 April 15
    12 April  4
    13 March 24
    Golden
    number
    Paschal
    full moon
    14 April 12
    15 April  1
    16 March 21
    17 April  9
    18 March 29
    19 April 17
  3. Determine the day of the week that the Paschal full moon occurs on.
  4. Count forward up to 7 days to find Easter Sunday.

This rule is quite simple, and also has some nice properties: Easter Sunday recurs on the same date every 532 (4 * 7 * 19) years, all 19 possible dates for the Paschal full moon are equally likely, and there is a fixed and known range of dates for Easter Sunday. Therefore it is not surprising that it lasted about 1200 years.

The Problems

As is well known, the Julian Calendar has a major problem which became evident over the centuries: the year is not quite 365.25 days long, and so the dates of the year slowly drift relative to the seasons. The date of Easter also drifts, but the problem is less obvious. By the 16th century, however, it was getting to be embarassing, and so Pope Gregory XIII employed the astronomer Clavius to design a new calendar.

There are two major sources of error in the Julian Calendar.

  1. The year is slightly shorter than 365.25 days long; the error is, close enough, 0.0075 days per year, or 3 days in 4 centuries.
  2. 235 lunar months are not exactly 19 years of 365.25 days; the error is, close enough, 0.0032 days per year, or 8 days in 25 centuries.

These two discrepancies apply in opposite directions, but cause the real full moon to slowly drift relative to the Paschal one, at an average rate of 0.0043 days per year.

Gregorian Easter

As is well known, the Gregorian Calendar was introduced to handle these problems, and a number of days were omitted to bring the dates back into line with the seasons. In those countries obeying the Papal Bull completely (notably Italy, Poland, Portugal, and Spain), October 4th 1582 was followed by October 15th, while in most other Roman Catholic dominated parts of the world the change was made later that year or in 1583 or 1584. (The Protestant parts were suspicious of the innovation and delayed; in the British Empire September 2nd 1752 was followed by September 14th, and most other Protestant countries in Europe changed in the 18th century.) A new rule for leap years was introduced, stating that century years (1600, 1700, and so on) were only to be leap years when divisible by 400.

However, a much more important (to the church, at least) change made at the same time was to introduce a new set of rules for Easter. The basis behind these rules was to shift the 19 dates of the Paschal full moons at the start of each century:

Neither correction happens in 2000, but both happen - cancelling each other out - in 2100. Thus this shift of the Paschal full moons, having happened in 1900, does not happen again until 2200.

Clavius had to deal with one complication. A "natural" pattern of shifts would allow the Paschal full moon to be as late as April 19th, after which it would shift back to March 21st. However, it so happens that, while March 21st was one of the 19 dates used under the old system, April 19th wasn't, though April 18th was. So he had to find an adjustment that would avoid this date. Simply omitting this date, using a 29 day range instead of a 30 day one, wouldn't work. So instead, Clavius decided that, if the Paschal full moon would have fallen on April 19th, it should be "adjusted" to occur on April 18th instead. However, doing this meant that sometimes two golden numbers would both correspond to April 18th, which was viewed as undesirable (apart from anything else, published descriptions of the current system stated that the 19 dates were all different). Clavius noticed that, in this case, the "real" April 18th corresponded to a golden number of 12 or more, and decided that whenever one of these numbers fell on April 18th it would also be adjusted, to April 17th. The result of all this is that there are 30 possible sets of dates for the golden numbers, and in each set the 19 Paschal full moons all have different dates.

The Tables of the Book of Common Prayer

In those countries that used to be part of the British Empire, the above rules were brought into law as part of the Calendar (New Style) Act 1750 (24 Geo.2 c.23, also available here and here). This amended the tables at the front of the Book of Common Prayer (of the Church of England) to correspond to the new rules. Two tables are important. Firstly, Table II encodes the shifts that happen each century:

T A B L E II.

To find the Month and Days of the Month, to which the Golden Numbers ought to be prefixed in the Calendar in any given Year of our Lord, consisting of entire hundred Years, and in all the intermediate Years, betwixt that and the next hundredth Year following, look in the second Column of Table II. for the given Year, consisting of entire Hundreds, and Note the Number or Cypher which stands against it in the third Column; then, in table III. look for the same number in the Column under any given Golden Number, which when you have found, guide your Eye side-ways to the Left Hand, and in the first Column you will find the Month and Day, to which that Golden Number ought to be prefixed in the Calendar during that Period of one hundred Years.

The Letter B prefixed to certain hundredth Years in Table II. denotes those Years which are still to be accounted Bissextile or Leap Years in the new Calendar; whereas all the other hundredth Years are to be accounted only common Years.

1 2 3      1 2 3
Years
of our
Lord
Years
of our
Lord
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
4100
4200
4300
4400
4500
4600
4700
4800
4900
5000
5100
0
1
1
2
2
2
3
4
3
4
5
5
5
6
6
7
7
7
8
9
8
9
10
10
10
11
12
12
12
13
13
14
14
14
15
16
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
  B  
 
 
 
5200
5300
5400
5500
5600
5700
5800
5900
6000
6100
6200
6300
6400
6500
6600
6700
6800
6900
7000
7100
7200
7300
7400
7500
7600
7700
7800
7900
8000
8100
8200
8300
8400
8500
&c.
 
15
16
17
17
17
18
18
19
19
19
20
21
20
21
22
23
22
23
24
24
24
25
25
26
26
26
27
28
27
28
29
29
29
0
 
 

and then Table III gives the dates of the Paschal full moons for each possible shift pattern (the adjustments introduced by Clavius are clearly visible at the bottom):

T A B L E III.
Pascal Full
Moon
S The    G O L D E N    N U M B E R S.
i ii iii iv v vi vii viii ix x xi xii xiii xiv xv xvi xvii xviii xix
March 21
March 22
March 23
March 24
March 25
C
D
E
F
G
8
9
10
11
12
19
20
21
22
23
0
1
2
3
4
11
12
13
14
15
22
23
24
25
26
3
4
5
6
7
14
15
16
17
18
25
26
27
28
29
6
7
8
9
10
17
18
19
20
21
28
29
0
1
2
9
10
11
12
13
20
21
22
23
24
1
2
3
4
5
12
13
14
15
16
23
24
25
26
27
4
5
6
7
8
15
16
17
18
19
26
27
28
29
0
March 26
March 27
March 28
March 29
March 30
A
B
C
D
E
13
14
15
16
17
24
25
26
27
28
5
6
7
8
9
16
17
18
19
20
27
28
29
0
1
8
9
10
11
12
19
20
21
22
23
0
1
2
3
4
11
12
13
14
15
22
23
24
25
26
3
4
5
6
7
14
15
16
17
18
25
26
27
28
29
6
7
8
9
10
17
18
19
20
21
28
29
0
1
2
9
10
11
12
13
20
21
22
23
24
1
2
3
4
5
March 31
April  1
April  2
April  3
April  4
F
G
A
B
C
18
19
20
21
22
29
0
1
2
3
10
11
12
13
14
21
22
23
24
25
2
3
4
5
6
13
14
15
16
17
24
25
26
27
28
5
6
7
8
9
16
17
18
19
20
27
28
29
0
1
8
9
10
11
12
19
20
21
22
23
0
1
2
3
4
11
12
13
14
15
22
23
24
25
26
3
4
5
6
7
14
15
16
17
18
25
26
27
28
29
6
7
8
9
10
April  5
April  6
April  7
April  8
April  9
D
E
F
G
A
23
24
25
26
27
4
5
6
7
8
15
16
17
18
19
26
27
28
29
0
7
8
9
10
11
18
19
20
21
22
29
0
1
2
3
10
11
12
13
14
21
22
23
24
25
2
3
4
5
6
13
14
15
16
17
24
25
26
27
28
5
6
7
8
9
16
17
18
19
20
27
28
29
0
1
8
9
10
11
12
19
20
21
22
23
0
1
2
3
4
11
12
13
14
15
April 10
April 11
April 12
April 13
April 14
B
C
D
E
F
28
29
0
1
2
9
10
11
12
13
20
21
22
23
24
1
2
3
4
5
12
13
14
15
16
23
24
25
26
27
4
5
6
7
8
15
16
17
18
19
26
27
28
29
0
7
8
9
10
11
18
19
20
21
22
29
0
1
2
3
10
11
12
13
14
21
22
23
24
25
2
3
4
5
6
13
14
15
16
17
24
25
26
27
28
5
6
7
8
9
16
17
18
19
20
April 15
April 16
April 17
April 17
April 18
G
A
B
B
C
3
4
5
 
6
14
15
16
 
17
25
26
27
 
28
6
7
8
 
9
17
18
19
 
20
28
29
0
 
1
9
10
11
 
12
20
21
22
 
23
1
2
3
 
4
12
13
14
 
15
23
24
25
 
26
4
5
6
7
 
15
16
17
18
 
26
27
28
29
 
7
8
9
10
 
18
19
20
21
 
29
0
1
2
 
10
11
12
13
 
21
22
23
24
 
April 18 C 7 18 29 10 21 12 13 24 5 16 27 8 19 0 11 22 3 14 25

The heading S is actually Sunday Letter written sideways.

Note that the old Easter can be calculated using code 23, provided it is remembered that these are Julian, not Gregorian, dates.

Calculating Easter

Tables have their uses, particularly for making it easy for the average person to determine the date of Easter, but there are many circumstances where an algorithm is more convenient. A number of such have been published, and rather fewer have turned out to be correct - even Gauss managed to make a mistake (forgetting that the lunar correction happens on a 2500 year cycle, not a 2400 year one). The following algorithm is simple to implement and gives the correct date in all circumstances; it is also quite easy to explain.

Divide to get Explanation
this by quot. rem.
year 19 a a + 1 is the golden number.
year 100 b c Split the year into century and remnant.
b 4 d e Find the place in, and the number of, 400 year cycles. 400 years is an exact number of weeks.
c 4 f g Find the number of leap years so far this century (ignoring the century year if it was one), and the number of ordinary years since.
8 * b + 13 25 h Determine the number of days to shift the full moons because of the lunar correction. The 8 ensures that we get a total of 8 days every 2500 years, and the 13 ensures that we start at the right point.
19 * a + b - d  - h + 15 30 j j encodes the unadjusted date of the Paschal full moon. b - d is the number of century years that are not leap years, and so b - d - h is 7 more than the number found from Table II. Meanwhile 19 * a gives the position of a specific number (8) in the appropriate column of Table III, and the constant 15 corrects for the two offsets 7 and 8.
a + 11 * j 319 m m will be 1 if the full moon needs to be adjusted back one day, and 0 otherwise.
2 * e + 2 * f - g  - j + m + 32 7 k k + 1 is the number of days from the Paschal full moon to Easter Sunday. The best way to see how it is derived is to first note that the days of the week repeat exactly every 400 years, and then to rewrite it as:
4 - [124 * e + 5 * f + g] - (j - m) + 7 * 92
The term in square brackets represents the day of the week for March 21st, the third term advances this to the Paschal full moon, the constant 4 represents Saturday, and the last term makes the result positive. Then reduce as much as possible modulo 7.
j - m + k + 90 25 month We now have j - m + k representing the date of Easter Sunday, with 0 being March 22nd; the constant 90 derives from 10 representing April 1st.
j - m + k + 19 + month 32 date And finally we get the date in a similar way; adding in the month allows us to skip April 0th seamlessly.

Sorry, but the form to try it yourself is broken at present.

A fixed Easter

Pope Gregory noted in his papal Bull that, although he had decided to keep it aligned (approximately) with the Passover, there was no reason why Easter should not lie on a fixed date. And in 1928 the UK Parliament passed the Easter Act. This states that, after due consultation with appropriate authorities, the Crown may issue an Order In Council making Easter Sunday always be the day after the second Saturday in April (as if the Paschal full moon were always on April 8th). Should this ever come to pass, these rules would become a thing of the past.


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