21 March 2008

Easter's early this year. Deal with it.

Family holidays ruined by earliest Easter in 90 years (from the Daily Mail).

About halfway down, a formula is given:

It may look daunting to non-mathematicians but the fiendishly complex formula used to work out when Easter actually falls is:

((19*t+u-w-(u-(u+8)\25)+1)\3)+15)mod30)+(32+2*x+2*y-(19*t+u-w- (u-(u+8)\25)+1)\3)+15)mod30)-z)mod7)-7*(t+11*(19*t+u-w(u- (u+8)\25)+1)\3)+15)mod30)+22*(32+2*x+2*y-(19*t+u-w-(u- (u+8)\25)+1)\3)+15)mod30)-g)mod7)+114)\31


Um, do you understand that formula? I think I know why some of the numbers are there -- the 31 at the end probably has something to do with the length of months, the 7 with the length of weeks, and the 19 with the Metonic cycle. Also, any sane mathematician wouldn't write the formula like that. First, there are repeated subexpressions like that ((u + 8) \ 25 + 1); I'd just call that by some other name and be done with it. Second, the formula just sits there in the middle of the article; this gives people the idea that mathematicians are freaks of nature who think in formula. What do the variables mean?

If you're curious, there is an algorithm at the Calendar FAQ. Easter is the first Sunday after the first (computed) full moon on or after the vernal equinox (calculated, and assumed to be March 21). The algorithm reflects this. First, assume that the Metonic cycle, which says that lunar phases repeat every 19 solar years, is exactly correct in the Julian calendar. (The algorithm was invented back when the Julian calendar was used.) Then make two corrections, one for the fact that the Julian calendar includes leap years that the Gregorian doesn't (years divisible by 100 but not 400) and one for the fact that the Metonic cycle's a bit off. (The expression "(u+8)\25" in the formula above comes from the second correction.) This gives the date of the full moon. Presumably if you've gotten this far you already know what the days of the week are.

Anyway, the cycle of Easter dates repeat themselves every 5,700,000 years. The cycle of epacts (which encode the date of the full moon) in the Julian calendar repeat every nineteen years. There are two corrections made to the epact, each of which depend only on the century; one repeats (modulo 30, which is what matters) every 120 centuries, the other every 375 centuries, so the air of them repeat every 300,000 years. The days of the week are on a 400-year cycle, which doesn't matter because that's a factor of 300,000. So the Easter cycle has length the least common multiple of 19 and 300,000, which is 5,700,000.

This whole computation is known as the computus (Latin for "computation"; I guess it was just that important at the time). Not surprisingly, Gauss had an algorithm which is much easier. Let Y be the current year. Then take:

a = Y mod 19

b = Y mod 4

c = Y mod 7

d = (19a + M) mod 30

e = (2b + 4c + 6d + N) mod 7

where M and N are constants depending on the century that don't look that hard to calculate, and which I assume are the corrections I alluded to above; the Wikipedia article gives them in a table. Then Easter falls on the d+e+22 of March or the d+e-9 of April, with certain exceptions which move it up a week when this algorithm gives a very late date for Easter. Basically, d finds the date of the full moon (so M is something like the epact) and e find the day of the week. In the case of this year you get a = 13, b = 0, c = 6; a table gives M = 24, N = 5 for this century, so d = 1, e = 0, and Easter is on the 23rd of March.

As for when Easter usually falls, well, go back to the original description: Easter is the date of the first Sunday after the first full moon on or after March 21. To me this seems like adding two random variables -- the number of days between March 21 and the first full moon, which is roughly uniformly distributed over [0, 29], and the number of days between that moon and the next Sunday, which is uniformly distributed over [1, 7]. There are 210 ordered pairs in ([0, 29] × [1, 7]). One of them sums to 1, giving an Easter date of March 22 in about one year out of 210. Two sum to 2, giving an Easter date of March 23 in two years out of 210. Three sum to 3 (March 24), ..., six sum to 6 (March 27). Seven sum to each of 7 through 30, giving Easter dates of each of March 28 through April 20 in seven years out of 210. Six sum to 31, giving April 21 in six years out of 210, ..., one sums to 36, giving April 26 in one year out of 210.

Indeed, this is basically what computations show, except that for some reason, when the methods given above call for Easter to be on April 26 it gets moved up to April 19. But basically the distribution of Easter dates is just a convolution of two uniform distributions! The Wikipedia article on the computus has a nice graph.

And I have no sympathy for the people quoted in that article. They've known this was coming since 1752, when the UK changed over to the Gregorian calendar. (It perhaps says something about me that I have more sympathy for the bakeries with lots of Irish patrons that are unhappy because Easter was only six days after St. Patrick's day this year.)

27 comments:

unapologetic said...

I guess it was just that important at the time

Yes, it was that important. The entire western world was built around the Catholic Church, and the Church was centered around the paschal celebration. This is what the religion is all about, and it's essential to know when it's supposed to happen.

Did you think Egyptian and Mayan calendar systems were secular? This is why we have timekeeping at all: religious observances and rituals. We do something periodically, and we need to organize the periods.

This Busy Monster said...

Well, it's not quite fair to attribute all timekeeping to religion alone. Whether you are a believer or not, the sun comes up on a regular basis, the seasons happen on a regular basis and the animals migrate on a regular basis. If you are interested in eating on a regular basis, you would likely take note of these things, even keep track of them.

williaty said...

Primitive timekeeping is about agriculture. When do we plant? When do we harvest? When do we burn the fields? When will the livestock give birth? All of these events had religious significance to the people, so they also had festivals associated with them. Add on the "gee, it's been a while since we've had a party and we're getting kind of depressed" festivals, and you have a full calendar of events that repeats itself with the seasons.

When the new religion on the block starts taking over from the old, the new religion goes "darn, we're getting our butts kicked on the festival issue" and suddenly the new religion has things to do on or about the same time. The most obvious being Christmas camping on Yule and Easter camping on Oestra.

As far as your incredibly arrogant comment that "the entire western world was built around the Catholic Church", well, reduce that to "the Catholic Church took over most of central Europe" and we have something to discuss. I think that, you know, the Romans, perhaps, might have something to say about the parts of Europe they built. And the Celts. And the Norse. And the Jews. And...

Andrew said...

except that for some reason, when the methods given above call for Easter to be on April 26 it gets moved up to April 19

The reason for that is that the latest date that Easter can be is April 25.
Consider the rule that Easter is the date of the first Sunday after the first full moon on or after March 21: that full moon can be no later than April 18 (i.e. March 20 + 29 days), and Easter can be at most a week after this full moon (when it falls on a Sunday,) giving a latest possible date of April 25.

unapologetic said...

As far as your incredibly arrogant comment that "the entire western world was built around the Catholic Church", well, reduce that to "the Catholic Church took over most of central Europe"

You're entirely right, I oversimplified. Mea culpa (damn, still can't escape its influence).

The western world was not founded on the Church, but by a certain point in history it was the social institution, and everything revolved around it. I didn't mean that it was the foundation, but that it was simply the most important thing around to pretty much everyone.

Anonymous said...

Surely it would be easier to calculate a date for the resurrection of Christ, and just use that date every year! How difficult could it be? (I'm terrified of the answers I'm going to get to this post!)

Anonymous said...

Surely it would be easier to calculate a date for the resurrection of Christ, and just use that date every year! How difficult could it be?

Totally easy: it's exactly the 14th day of Nisan in the Hebrew calendar, during the holiday of Passover.

Anonymous said...

Good one, anon. This is how the Orthodox Church calculates Easter.

RickMeasham said...

No, that's not how Orthodox Easter is calculated. From wikipedia:

Eastern Orthodox Christians calculate the fixed date of 21 March according to the Julian Calendar rather than the modern Gregorian Calendar, and observe the additional rule that Easter may not precede or coincide with the first day of the Jewish Passover.

I think it's a suggestion that has a lot of merit. I'll bring it up next time I'm chatting with the pope. After all .. surely calculating Passover can't be too hard. Right?

unapologetic said...

Additional complication: if the basic Passover calculation says that it will land on the Sabbath, then they delay another month (like this year). The Catholic version doesn't use that rule. That's why Passover this year is next month, instead of right now.

Isabel Lugo said...

John,

from my understanding (although that may be wrong) the reason Passover and Easter are occasionally separated by a month is not the one you give. Rather, it's that the Hebrew calendar includes leap months (some years are twelve months, some thirteen), and the lunar calendar implied by the Easter computations also includes leap months, and sometimes the leap months don't line up perfectly.

There also are certain rules about which day of the week Rosh Hashanah is allowed to fall on, though; those might be what you're thinking of.

Eric Jablow said...

The Hebrew calendar is arranged so Yom Kippur (Tishri 10) cannot fall on Friday or Sunday, and so Hoshana Rabbah (Tishri 21) , the seventh day of Sukkot, cannot fall on Shabbat. Having a major fast day immediately precede Shabbat would mean that no one could prepare a festive meal, or any meal actually for Shabbat. Having the major fast day follow Shabbat would cause other problems. Hoshana Rabbah has important ceremonies that can't be done on Shabbat.

Anton Erasmuson said...

Some of the problem comes from the method of calculating a leap year. It would be simpler to skip one year in 128. That way, the solar year averages out at 365.2421875 days, an error of about 2 seconds a year. Leap seconds which are erratic are of about that order. To institute it, skip as leap years years divisible by 128. That would make 2048 the next skip year. While you are at it, add 5000 to the current year to get rid of the silly AD BC thing. So we would now be in 7008 HE (Historical era.)

Robert said...

It might as well me be mentioned that this "Easter campaign on Oestra" is deceptive. While the name of the feast in English is clearly derived from the older pagan observance, the Latin original in use since the first century is "Paschalis", which is derived from the Hebrew Passover. Granted, there have been arguments that even the Passover may have at one time been agriculturally based, this simple fact still obliterates your claim on the Easter front because by the time of the evangelisation of Northern Europe and Britain, the calculation of Easter was already fixed to a central authority for the Universal Church.

As for Christmas, yes the Church did in fact choose to celebrate the Solemnity of the Incarnation on top of the Winter Solstice, where Yule previously fell. However, the decision was not so much made to replace a pagan celebration (convenient as this is) but for the symbolic significance it presents. The Solstice is the longest, deepest and darkest night of the year. What night could be more appropriate to represent the birth of God made Man than the point at which the light returns?

Anonymous said...

You all realize of course that if Jesus rose from the grave on a particular day of the year, it would be that day of the year and not based on the moon at all.

PS - We have written proof that India, Babylon and Egypt all celebrated the spring equinox and the first moon after long before Christinity existed and before the old Testament was written down. We can also suspect that other civilizations did as well because it is an important agricultural time.

unapologetic said...

You all realize of course that if Jesus rose from the grave on a particular day of the year, it would be that day of the year and not based on the moon at all.

What do you mean by "a particular day of the year"? According to the Hebrew calendar, Nisan 14 is a particular day.

Anonymous said...

Your formula is wrong. Check out the parenthesis, is bad-balanced! Computer programmers have a good eye for it. =P.

Anonymous said...

Good catch, anon. It's not even remotely resembling anything like being close to balanced!

Anonymous said...

As fas as I know, Christ died 14 Nisan (a Friday) and was resurrected on the third day, 16 Nisan (a Sunday).

Anonymous said...

What puzzles me (and is not answered in the OP, nor in wikipedia article) is which timezone is used in computus. I mean that the full moon occurs at an exact moment in time, which has two dates, depending on the timezone. This can skip the day of Easter for one week, but AFAIK we still celebrate it on the same Sunday, no matter what timezone we're in.

I'm pretty sure that Nicaean council used local time of the neigborhood of Jesus, which I believe currently uses UTC+2, or perhaps Greece (Nicae?), that currently falls also on UTC+2. But perhaps this has changed during the power of Catholic church, maybe to local time of Vatican (currently UTC+1). However, I'm pretty sure computus doesn't use the UTC, which would probably be selected, if the whole thing was invented today.

Isabel Lugo said...

anon 3:27 pm:

there's no need for the computus to use a time zone, because the calculation makes no reference to times of day. (Similarly, though the Hebrew calendar is lunar it doesn't explicitly make reference to a time zone -- although there is a time of the full moon calculated there, and I suspect it's at least meant to match up with Jerusalem time.) I suspect that if for some reason it were decided to use an astronomical reckoning to compute Easter, you'd see Jerusalem-based reckoning as well.

On a related note, the Iranian calendar has its new year at the midnight between the two solar noons on either side of the vernal equinox, which quite clearly requires a time zone; they use Tehran time.

Tigri said...

"except that for some reason, when the methods given above call for Easter to be on April 26 it gets moved up to April 19"

AND

"Easter is the date of the first Sunday after the first full moon on or after March 21."

See "some reason" ?

Anonymous said...

Better the Catholics than secular/atheist/pagan.

What exactly is your problem? Think there's no God or something along those lines? Then who, including you, cares what your problem is? Doesn't matter right...

Russo said...

Not to mention "(u+8)\25"... do you ever go to the street market and ask for 1\2 Kg of rice? I blame Bill Gates for switching the slahes.. :-P

Sean Henderson said...

Mind your parentheses Michael. You make reference to "subexpressions like that ((u + 8) \ 25 + 1);". It's actually (u- (u+8)\25)+1). I understand the intent of the statement, though.

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