Graph distribution of Easter Sunday depending on the epact
Easter Sunday is the first Sunday of spring following the year’s paschal full moon. This paschal full moon is what the church considers the first full moon of spring (pascha comes from the Aramaic word for Passover). Since the Gregorian calendar has the 21st of March fixed as the start of spring, it also functions as the earliest possible date for the paschal full moon. With the 21st as the earliest possible date for this ecclesiastical full moon, if that date is also a Saturday, the 22nd of March marks the earliest possible day Easter Sunday to occur. Now, if the full moon narrowly misses the 21st of March, and arrives a day early on the 20th, the next lunar cycle will put the paschal full moon on the 18th of April, marking the latest possible date for it to occur. If that date also happens to land on a Sunday, Easter will not transpire until the 25th of April, which is the latest possible date for the holiday to appear. Here it’s worth pointing out that Easter Sunday and paschal full moons can never be the same date, since Easter Sunday is always the Sunday after a paschal full moon, never shared with it. Depending on the day of the week of this full moon, a Dominical letter is assigned (in some literature you may come across the less Latinate name “Sunday letter”). In the event of a leap year, the second letter is to be used, as Easter Sunday always occurs after any leap day.
Varying across 30 possible epacts (these are determined by a number value that corresponds to the phase of the moon at the start of the year), it takes 10,000 years until solar and monthly corrections all become cyclically repetitive enough to calculate all of the Easter Sundays in its 19 year-long cycle (see: Golden number). Furthermore, from here we find that it takes 5,700,000 years until the order finds a complete balance, able to repeat itself identically, inclusive of Easter Sundays. 10,000 years is the largest common multiple for necessary solar corrections (the second largest is 400 years), and 2,500 years is the longest for making lunar corrections (see the Easter Sunday Calculation page in the Gregorian calendar for more detail). In comparison, a mere 400 years is all it takes for Dominical letters to be repeated in the same order. We can make a simple table that shows the occurrence of all the Dominical letters in this 400-year cycle, which provides us with the following table. As mentioned above, for leap years, we are only interested in the second letter, as Easter Sunday is always after a leap day.
| A | B | C | D | E | F | G |
|---|---|---|---|---|---|---|
| 56 | 58 | 56 | 58 | 57 | 57 | 58 |
Here we see that the probability a Dominical letter in a given year will be A is 56/400. Because there are 30 possible epacts, the probability that the year will have one of the given epacts is 1/30. From this, it follows that 5,700,000 × 56/400/30 = 26,600 times throughout the Easter cycle. The same applies to the Dominical letter C. Using a similar calculation, we get the occurrence of Dominical letters B, D and G as 27,750 times. Finally, with occurrences for Sunday letters E and F, we get 27,075 times.
We can make a graph where every calendar date of Easter Sunday is assigned with the Gregorian epact in which it occurs. In the first column, all possible calendar dates for Easter Sunday are listed, with their accompanying Dominical letters in parentheses. In the second column, the total number of Easter Sundays throughout the whole cycle of Easter are listed, their sum total adding up to 5,700,000. In the rectangular boxes, the Gregorian epact numbers are listed. The rectangle’s width (and also its color) show how many times the epact occurs for a given date throughout the Easter cycle.
- A,C: 26600×
- E,F: 27075×
- B,D,G: 27550×
This much is rather simple, but there is an exception that can make things become a bit complicated. As we know, the paschal full moon can occur between 21 March and 18 April, which gives us a total of 29 days, but the year’s epact is 30. The simplest solution is to allow an ecclesiastical full moon for epact 24 to occur on 19 April, but then Easter Sunday would occur on the 26th of April, which exceeds our cut-off date, and is therefore unacceptable. Therefore, in the case of epact 24, the ecclesiastical full moon moves one day back. Normally, this doesn’t make any difference. For example, if the paschal full moon of epact 24 happens on a Friday, it would move one day back to Thursday, leaving Easter Sunday unaffected. However, if the paschal full moon of epact 24 happens to occur on a Sunday, then by moving it one day back (on Saturday) the date of Easter Sunday falls back a whole week. That is why there is an extra epact 24 on April 19, and thanks to that, this Easter Sunday is also the most common possible date in the entire Easter cycle. The last time it happened was in 1981 and the next time it will happen is in 2076.
Due to this solution, the last date of Easter Sunday (25 April) can occur in epacts 25 and 24. Once again, this was a problem, because in one cycle of the golden number, this cut-off date could appear twice and the Church refused to accept that as a solution. We know that these two epacts will meet in one cycle of the golden number, but only if the golden number for a given year is greater than or equal to 12. And therefore, it is in these cases, that the date of the paschal full moon is shifted back one day. The reason for changing the date of Easter Sunday back by one week back to 18 April is for the same reason as previously mentioned in epact 24. The last time this happened was in 1954, and the next time it will happen again in 2049.
So, this gives us 8 possibilites from the 19 golden number values to consider. Years when Easter Sunday falls on 25 April, the Sunday letter C, sees a reduction in occurrences by 26,600 × 8/19 = 11,200. And the number of occurrences of Easter Sunday on April 18 will increase by that amount (this year also has Sunday letter C). The split epact 25 is highlighted on this chart.
Just for fun, we could show how the above graph would look like without these two exceptions, including the possibility of Easter Sunday on 26 April. However, this is only a hypothetical calendar. Nothing like this actually exists, but it certainly makes the chart a little easier to create.