Talk:Date of Easter

Tables of dates

Some of the lists available on the internet are this one from the US Census Bureau, this one from the Astronomical Society of South Australia and this one from T Larsen The first one looks like a reliable source even if the others are less so. They all give the date of Easter 1750 as 29 March. As this one from Petko Yotov's website shows, that is the date of the Catholic/Western Easter in the Gregorian calendar. As this 18th-century Book of Common Prayer shows, Easter in that year was actually celebrated on 15 April (Julian calendar) in England (and associated places - I do not want to get into a discussion here about Scotland, Wales, Ireland and the colonies). I note that 29 March in places which observed the Gregorian calendar was the same day as 18 March in places which observed the Julian calendar. As that date is too early for Easter, the Julian Easter fell a lunar month later on 15 April Julian, which was also 26 April Gregorian. Would it be worth noting this somewhere in the article? Alekksandr (talk) 21:06, 20 March 2022 (UTC)

Epact in Julian calendar

I don't understand the epact in the Julian calendar. According to what our article says (and the Swedish table with runes), the epact for golden number equal to 1 must be 8, which gives Paschal full moon on April 5. But then how did the Gregorian get to epact=29 when goldenn number=1? It decreased by 1 in 1700 and 1900, so it was 31, that is 1, at the time of the reform. It should have been 10 less than what it was before the reform, so that should have been 11, not 8. This is confirmed by the tables "Tabula æquationis cycli epactarum perpetui" and "Tabella cycli epactarum perpetua" in Canon 2 of Clavius. The table also seems to say that the initial epact was increased in AD 320, 800, 1100, and 1400. The initial epact was 8 back before AD 800. So what's goin' on? Are the Eastern Orthodox still using the value from before AD 800, in spite of what the table of Clavius says? Did the Roman Catholics do lunar corrections in AD 800, 1100, and 1400 which the Easterners didn't do? Eric Kvaalen (talk) 11:25, 12 May 2022 (UTC)

It's a long article. Please rewrite your questions so that every time you mention what our article says, someone reading your question can easily find the exact spot in the article that you are referring to. Jc3s5h (talk) 13:42, 12 May 2022 (UTC)

@Jc3s5h: The fact that the epact for golden number equal to 1 must be 8 is based on the fact that the ecclesiastical full moon is given as 5 April in the table introduced by the sentence "This is the table of paschal full moon dates for all Julian years since 931". If the full moon is on April 5, then the new moon, 13 days earlier, is on March 23, which, as shown in the table higher up, corresponds to epact viii (8). That's the only place where I referred to what our article says. Eric Kvaalen (talk) 08:16, 22 May 2022 (UTC)

@AstroLynx and Jc3s5h: According to this 1980 article (p. 157), "By the 16th century, the actual equinox was falling on the 11th March (10 days before the assumed equinox) and to return the equinox to the 21st March, 10 days were dropped from the calendar. Also the actual full moons were falling four days after the calculated full moons and a similar adjustment was made in the Lunar Calendar to correct this error." So apparently in the Gregorian reform they decreased the epact by 7 (from 8 to 1 for golden number 1, or from 4 to 27 for 1583) rather than by 10.

Frequencies of the dates of Easter

@AstroLynx and John Maynard Friedman: AstroLynx, you ask why I think the original graph was wrong, giving the frequencies of the dates of Easter. Very simple -- the reason that April 19 is more commonn than most of the others is that April 17 is twice as common as other dates as the date of the ecclesiastical full moon, because that's the date if the epact is either 25 or 26, whereas for other dates there's only one epact which gives that date. Now, Easter will fall on April 19 when the ecclesiastical full moon is on April 12, 13, 14, 15, 16, 17, or 18. Since the 17th is twice as common (in the long run), it's as though April 19 gets eight "votes", whereas a date such as April 16 gets only seven (April 9, 10, 11, 12, 13, 14, and 15). But the same thing goes for April 18! Easter is on the 18th when the ecclesiastical full moon is on April 11, 12, 13, 14, 15, 16, or 17. Again, since the 17th is twice as common as others, April 18 gets eight "votes" rather than seven. So both April 18 and April 19 are about eight sevenths as common as the dates from March 28 to April 17. The reason the ratio is not exactly 8/7 is that the date of Easter depends both on the epact of the year in question and on its dominical letter (actually the dominical letter for the latter part of the year, after February), and the seven dominical letters are not all equally common. In a 400-year cycle, two of them occur 56 times, two occur 57 times, and three occur 58 times. Easter falls on April 18 only when the dominical letter is C, which occurs 14% of the time (56/400), whereas it falls on April 19 only when the dominical letter is D, which occurs 14.5% of the time. So April 19 is slightly more common than April 18. (I actually made a mistake in my spreadsheet and had the dominical letters shifted by one, so my graph was off, but I have now corrected it.) Eric Kvaalen (talk) 08:16, 22 May 2022 (UTC)

This is all nice and well but the original graph is already properly sourced by two references. Several more could be added but I do not think that this will be necessary. AstroLynx (talk) 08:29, 22 May 2022 (UTC)

@AstroLynx: Are you referring to the 1944 article in Popular Astronomy and the 1980 article in The Irish Astronomical Journal? I can't access them. Can you? Eric Kvaalen (talk) 13:24, 22 May 2022 (UTC)

When I browse to the links provided by Eric Kvaalen at 13:24 UTC I see, on the right side of the window, a box that says "FULL TEXT SOURCES" with two icons, which offer two presentations of the scanned images of the articles. Jc3s5h (talk) 13:41, 22 May 2022 (UTC)

@AstroLynx and Jc3s5h: All right, thanks. I understand now, based on the 1980 article. There's a complication that we don't have in our Wikipedia article. In fact, the ecclesiastical full moon falls on April 17 only 27/19 as often as on most of the other dates, and on April 18 it falls 30/19 times as often as on those other dates. So for Easter, April 18 gets only 7 and 8/19 "votes", April 19 gets 8, and April 25 gets 30/19. That gives the distribution shown in the graph presently in the article. Eric Kvaalen (talk) 16:49, 22 May 2022 (UTC)

@AstroLynx and Jc3s5h: I want to redo the edit I did a couple weeks ago, but more correctly. I'll say here what I want to do so that it won't get reverted:

I want to add the complication (which I mentioned above) that is missing from our article, having to do with when the epact is 25.

I want to shorten the paragraph which I myself added earlier about the accuracy. I hadn't noticed that the accuracy is discussed further down, and in a more precise way.

I want to put in my graph of the distribution of the date of Easter during the period 1900 to 2199, because it's much different from the long-range distribution. People may get the impression that April 19 is more common than other dates NOW, which is not true. (One could also make a graph for the long-term distribution if we were to stay with the present series of apacts, but in fact we will stop using the present series in 2200.)

I want to add the following (corrected) explanation of the long-term distribution:

However, the distribution is quite different over the long term. The date of Easter in a given year depends only on the epact for the year, its golden numer, and its dominical letter, which tells us which days are Sundays (more precisely, the dominical letter for the part of the year after February, which is different in leap years form the letter for January and February). (The golden number only matters when the epact is 25, as explained earlier.) If we go forward 3,230,000 years from a particular year, we find a year at the same point in the 400-year Gregorian cycle and with the same golden number, but with the epact augmented by 1. Therefore, in the long term, all thirty epacts are equally likely. On the other hand, the dominical leters do not all have the same frequency  – years with the letters A and C (at the end of the year) occur 14% of the time each, E and F occur 14.25% of the time, and B, D, and G occur 14.5% of the time. Taking into consideration the complication having to do with epact 25, this gives the distribution shown in the second graph. April 19 is the most common because when the epact is 25 the ecclesiastical full moon falls on April 17 or 18 (depending on the golden number), and it also falls on these dates when the epact is 26 or 24, respectively. There are seven days on which the full moon can fall, including April 17 and April 18, in order for Easter to be on April 19. As a consequence, 19 April is the date on which Easter falls most frequently in the Gregorian calendar, with a frequency of 29/750 (about 3.867% of the years). April 18 is the second most common, because of the extra times that the full moon is on the 17th. 22 March is the least frequent, with a frequency of 29/6000 (0.467%).

I will use the 1980 article as reference.

Eric Kvaalen (talk) 11:40, 31 May 2022 (UTC)

@AstroLynx: You reverted my edit with the comment "reverted (original reseach - we only describe the current situation, unsourced suggestions for improvements do not belong here". I did not make suggestions for improvement! I did what I wrote above, and gave you two weeks to react. Why did you revert my whole edit? That's not correct behavior. Eric Kvaalen (talk) 15:24, 17 June 2022 (UTC)

I don't know what AstroLynx would say, but I have issues with several parts of the edit.

• Describing the distribution for the complete 5,700,000 cycle as "ill-defined" is wrong. The rules are well known.
• "the present system of calculating the date of Easter will not be accurate over many thousands of years" is true, but involves advanced topics such as the increasing length of the day and advanced orbit calculations. Such a statement should not be given without a reliable source. The article contains the statement "This corresponds to an error of less than a day in the phase of the moon over 10,000 years, but in fact the length of a day is changing (as is the length of a synodic month), so the system is not accurate over such periods." This statement should have a citation and I have added a citation needed template. I need to check if the other numbers in that paragraph have appropriate supporting citations.
• "It would be possible to get a similar level of accuracy with a much shorter period.<ref>For example, a period of 18,000 years would be possible by resetting the golden number to 1 when the year is divisible by 18,000." There are far to many cranks constantly trying to get the calendars they invented into Wikipedia. The no original research policy should be enforced with the highest degree of strictness and vigor when it comes to editor suggestions for calendar improvements. Jc3s5h (talk) 15:38, 17 June 2022 (UTC)

On the first point, what I mean is that the frequency distribution depends on which mapping from golden number to epact is in force. Each such mapping last for between 100 and 300 years. So the frequency distribution is "constantly" changing. The only way to get a true frequency distribution, as I wrote in the next paragraph, is by looking at the whole 5.7-million-year cycle. But that of course is ridiculous beccause the system will certainly not be used for anything like that much time! So even though the mathematical problem is well defined, the actual frequency distribution is not well defined.

On the second point, it's well known that the system is not accurate for many thousands of years -- as you admit yourself. It's easy to find a reference in one of our other articles. What I don't like is when people revert an edit because of statements they know are true, just because there is not a reference, instead of improving the article by finding a reference. I remember a guy once who wouldn't let me write a formula in a less ambiguous way, just because I didn't have an explicit reference! I think he knew that I was right.

On the third point, I didn't mean to "get a calendar I invented into Wikipedia". I was just motivating the next sentence, by saying (or implying) that it's not because the cycle is so long (millions of years) that it's not accurate in the long term -- we could use a much shorter cycle like 18,000 years but it would still be too long to be accurate. I put that example of an 18,000-year cycle in a footnote just so people can check whether the statement is true that there are shorter cycles with similar accuracy to that of the system we use. It's easy to check that it gives a month length very close to the present true value (from "Year 0" to AD 18000 there are 6574365 days and 222629 lunations, giving an average length of 29.53058676 days).

When I wrote what I wrote here yesterday I thought AstroLynx had reverted the edit I had described on this Talk page, but in fact he just reverted the subsequent edit I made in which I clarified some things and corrected a couple sentences. I hate it when people revert a whole edit just because of one little thing they don't like. And then they often start arguing about other things and we get bogged down in arguing instead of improving the article!

Eric Kvaalen (talk) 06:19, 18 June 2022 (UTC)

I propose to delete the details section

This section has been tagged as needing more sources for nearly two years. I don't know if it was ever correct, but the absence of sources makes it impossible for editors patrolling this page for vandalism or incompetence to quickly assess if a new edit is correct or not.

The following error discredits the whole section and warrants its removal:

So the Gregorian Easter dates repeat in exactly the same order only after 5,700,000 years, 70,499,183 lunations, or 2,081,882,250 days; the mean lunation length is then 29.53058690 days.

But Richards (2013, full citation in article) page 599 states:

The entire calendar involves a cycle 5700000 years containing 2081882250 days, which are equated to 70499175 lunations.

So the unknown editor Tom Peters who added these numbers in 2003 disagrees with the reliable source about the number of lunations in a cycle.

In addition to being wrong, the whole section is rambling and has no clear point.

It appears Richards should have put "of" between "cycle" and "5700000" but that's how it is in the book. Jc3s5h (talk) 16:03, 17 June 2022 (UTC) with the editor who made the original edit added 18 June 2022 21:44 UTC.

The numbers cited in this section are found in NAAE1931 (p. 744) and ESAA1992 (p. 582) and are also found in the various editions of Calendrical Calculations (CC1, p. 54; CC2, p. 122; CC3, p. 117; CC4, p. 148).

They could all be copying these numbers from each other but it is also possible that Richards made an error here. AstroLynx (talk) 10:38, 19 June 2022 (UTC)

The earliest reference to these numbers appears to be a 1837 treatise on the Gregorian Easter reckoning by Magnus Georg Paucker. AstroLynx (talk) 10:52, 19 June 2022 (UTC)

My name was mentioned so I need to jump in. I was one of the first authors of this page who put a lot of time into this.

First, about the number of 70499175 lunations in the 5.7 Myr Gregorian cycle as mentioned in the 3rd edition (from 2013 - after I wrote this page) of the Explanatory Supplement to the Nautical Almanac p.599 . It was 70499183 in all earlier sources mentioned here; the change is not explained. I speculate that it is the truncated value obtained by dividing the 2081882250 days in 5.7M Gregorian years, by 29,53059 which is a rounded value for the current mean length of the synodic month: so it would NOT be the actual number of lunations counted in the Gregorian calendar, which is what the section on this page is about. In any case the last two editions (1992, 2013) of the Expl.Suppl. are very poorly edited and are full of errors, often in the numbers, sometimes very serious: the list of errata is long (https://uscibooks.aip.org/explanatory-supplement-to-the-astronomical-almanac-3rd-edition-list-of-errata/). So much for relying on authorative sources in print. The computation of the number of lunations in the Gregorian calendar is explained on this Wiki page, so rely on your brain instead.

User:Jc3s5h questions the relevance of these details. But people have been excommunicated over the details of the computus. Easter dates jump around all over the solar calendar, and it is of general interest to do stats on the dates and explain why this happens. The old Julian computus had a cycle of only 532 years; the Gregorian cycle is much longer and needs explanation. The Gregorian lunar calendar is a complex beast not generally well understood. Misguided statements picked up from secondary literature pop up all the time. The information on this page is not readily obtained elsewhere - hidden in Latin or otherwise obscure sources. I think an encyclopedia must be a compendium of the best available sources. That it is complex or not relevant to every user is a poor argument. I have a PhD in science, but many of the mathematics pages in Wikipedia are beyond me - but I do not go around deleting them because I don't understand them or find them not useful.

That said, elaborating on the exact number of lunations in the 5.7Myr Gregorian cycle: the overall computation assumes that the net epact changes make a correction of a neat lunation of 30 days that can be subtracted from the total. They do not. In 2003 I wrote a program to follow the Gregorian rules exactly, and evaluated them over the 5.7 million year period. The epact changes sometimes cause lunations of 1, 59, or 58 days, which screws up the statistics. Lunations of 28 and 31 days also occur, those of 31 days frequently by including a leap day. The epact 19 with Golden Number 19, which happened to appear in the first Golden Number table after the Gregorian reform, causes a lunation of 59 days in dec./jan.; this happens 10000 times over the 5.7Myr cycle. The calendar reformers cared enough to fix this with an exception rule, even though it has no influence on the date of Easter. This is explained on this Wiki page. It is in the canon and if they cared enough, we should care to put it in Wikipedia.

With that exception rule, I counted 70500000 lunations. The number of 70499183 I can obtain when other exception rules are implemented, so that any remaining 1-day lunations are absorbed into a preceding or following lunation; and the remaining 58- and 59-day lunations are somehow split into two lunations. So as far as I can tell, with the documented canonical rules, the Gregorian lunar calendar counts 70500000 lunations and not 70499183, and the mean lunation length would be 29.53024468 days - much too short. But the books say 70499183 lunations. Saying otherwise would be "original research" - I get that accusation a lot.

Tom Peters (talk) 17:30, 11 August 2022 (UTC)

I am very much opposed to deleting that section, which I found very interesting. And it's obvious that we have the correct number of lunations in 5,700,000 years -- if there were no solar and lunar corrections, it would bee 5,700,000/19*235, which is 70,500,000, but every 10,000 years there are corrections amounting to 43 unit changes of epact (as explained in our article), or 43 lunations every 300,000 years. There are 19 such periods in 5,700,000 years, so the number of lunation in total is 70,500,000 minus 43*19, in other words 70,499,183! Eric Kvaalen (talk) 20:08, 20 June 2022 (UTC)

Astrolynx has cast considerable doubt on the number of lunations from Richards 2013. Eric Kvaalen has provided an explanation to support the number of lunations, although I am not able to follow it in detail. It turns out calendar days per lunation rounds to 29.53059 no matter which is correct, so no value change is needed in the article. But I have changed the citation to Richards 2013 just before the phrase "This corresponds to an error of less than a day" to a citation to Dershowitz and Reingold 2008.

I also checked Richards 1998 and it does not address this issue.

I still feel the details section is a mess and we should get rid of it. Jc3s5h (talk) 20:44, 20 June 2022 (UTC)

How can I say this? If you don't like reading about the details, then find something else to do! :) Really, don't get rid of things that other people appreciate. I do agree that there are a couple things that need fixing, which is what I tried to do last week. ````— Preceding unsigned comment added by Eric Kvaalen (talk • contribs) 11:59, 21 June 2022 (UTC)

I refer you to Wikipedia:What Wikipedia is not, especially the section Wikipedia is not an indiscriminate collection of information. What are the things about the date of Easter that might be worthy of putting in an encyclopedia?

• The significance of Easter to Christians and why they care about the date it's celebrated
• Rules Christians considered important for acceptable Easter dates
• Various methods of calculation in various areas and sects
• How this calculation was one of the main calculations done in Europe for a number of centuries
• How developments in mathematics and the import of mathematical ideas from Arabia and India affected the calculation

But how is the distribution of Easter dates relevant? Where is the evidence that how frequently Easter occurs on certain dates is important to Christians, or anyone else affected by Easter (such as candy makers)? Jc3s5h (talk) 13:37, 21 June 2022 (UTC)

I installed a version of Lisp on my computer and loaded the calendar package from Calendrical Calculations Ultimate edition (that is, 4th edition). So far I have compared 1 January 1600 with 1 January 5,701,600 and found, as expected, that January 1 falls on a Friday for both years, and Easter falls on 2 April for both years. I also confirmed that when I compute the Rata Die for 1 January 5,701,600 and subtract the Rata Die for 1 January 1600 I obtain 2,081,882,250 which is what the sources indicate it should be. I have not yet computed the number of lunations. Jc3s5h (talk) 13:20, 23 June 2022 (UTC)

If you go to the Book of Common Prayer, there is a helpful table which allows you to compute the Sunday Letter for any year. You add one quarter (omitting fractions) and the number at the top of the column (in this case 2), divide by 7 and note the remainder. The Sunday Letter is then found against that remainder. In the case of 1600, the calculation is 1600 + 400 + 2 = 2002. 2002/7 = 286 remainder 0. Sunday Letter is A. But that's not the whole story. Although irrelevant to the calculation of Easter, in January and February of a leap year (such as 1600) the Sunday Letter is one up in the series, i.e. B. So 2 January (against which B is marked) was Sunday and 1 January was Saturday, not Friday. For Easter, A is marked against 2 April, confirming your date. 82.46.116.18 (talk) 10:37, 10 August 2022 (UTC)

my english is poor ...

But in my opinion the literal meaning of

In 1807 ... stated that 26 April is always replaced with 19 and 25 April by 18 April in the circumstances stated.

is that there were TWO errors in the original formulation (that in p and the miss of this statement). Perhaps he simplified a previous statement.

Is it known the reason for the restriction to those two centuries? It seems a partial fix of the error in p.

The info in these few lines is not time-ordered. If you give me safe info on the above two points, I will propose here a time-ordered form. pietro 151.29.59.56 (talk) 08:39, 3 July 2022 (UTC)

thanks

I do not know who has removed the vandalism that prevented my italian translation of the gauss method. In any case, thanks. pietro. 151.29.59.56 (talk) 22:37, 14 July 2022 (UTC)