On Thursday, I posted about the difference between the Gregorian calendar (used by the Western churches) and the Julian calendar (used by most Eastern Orthodox churches). There’s a 13-day difference between them for the period between 1900 and 2100, so any given day shows up 13 days later on the Julian calendar than on the Gregorian calendar. So someone with a Julian calendar would celebrate October 12 on a day that shows up as October 25 on our Gregorian calendars.

Then, on Friday, I posted about the definition of Easter: the Sunday after the full moon after the spring equinox, and how that definition came to be. I brought up an initial puzzle: since the equinox and the full moon can be determined by observing astronomical phenomena, and we all have the same Sundays, why should Easter be any different for anyone? Sure, there are different calendars, but why should calendars be relevant for feasts that are defined in terms of astronomically objective things?

(Footnote: Yesterday’s post on “Western Christians, Orthodox Christians, and Jews: the coming convergence” was an April Fool’s joke, but contrary to what some might have thought, the previous posts weren’t just an elaborate setup — I didn’t even get the idea of doing an April Fool’s joke until the very morning of April 1.)

One answer was that “spring equinox” doesn’t mean spring equinox. The actual spring equinox could be on March 19, 20, or 21. But for Easter calculation purposes, we define the spring equinox to be March 21. This has the advantage that we don’t need to calculate every year, and Easter is guaranteed to be in spring (which apparently people thought was important). The disadvantage is that, by pinning the equinox to a calendar date rather than an astronomical phenomenon, you introduce a 13-day difference between the West and East based on the difference in their calendars. The Orthodox won’t consider that the Easter-calculation spring equinox happens until April 3 (even though they’ll have observed an actual spring equinox by then). So if a full moon happens to fall between March 21 and April 3, Western Christians will use that full moon to calculate Easter, while the Orthodox will use the next full moon, which introduces a difference of about a month (the lunar cycle is about 29.5 days).

Now we’ll talk about what “full moon” means. You won’t be surprised to learn that “full moon” doesn’t mean full moon. It’s an “ecclesiastical full moon,” which is defined as being 14 days after the “ecclesiastical new moon,” which is different than the actual new moon. Why do this? Well, you want to be able to figure out when Easter falls ahead of time rather than when you actually observe a new moon, so you want to be able to have tables to predict the new moon. Hopefully the tables correspond with astronomical reality, but the Council of Nicaea also thought it was important that all Christians celebrate Easter on the same date, and people in different places in the world might observe a new moon on slightly different days. So, in practice, we determine the ecclesiastical new moon by means of tables, and different tables (with different accuracies) are responsible for disagreements in calculating the date of Easter.

This stuff is so complicated that I’ll only describe the basic gist — there’s a big section on how to do the actual calculations in the Wikipedia article on the Computus. Consider a lunar month of 29.5 days. We could define a lunar year as consisting of 12 lunar months, with lengths alternating between 29 and 30 days. That gives us a year of 29.5 x 12 = 354 days. That’s 11 days shorter than a solar (non-leap) year.

So suppose that in 2006 we know the full moon falls on April 13. That means that, in 2007, there’s going to be a full moon on April 2, 11 days before April 13. And in 2008 (we’re ignoring leap years for this example), the full moon would be 11 days before April 2, which would be March 22. And in 2009, the full moon would be 11 days before that: March 11. That’s too early because it’s now before the spring equinox. So we add 30 days and get April 10. Then for 2010, we subtract 11 again and get March 30. And so on.

Then we introduce a ton of corrections. A correction for leap years (and the special Gregorian rule about leap years). A correction for the “adding 30 days” part from the previous paragraph, when a lunar month is only 29.5 days. A correction for the lunar month’s being slightly off from 29.5 days (it’s actually 29.53059 days). All this gets us a table predicting when ecclesiastical full moons (i.e., 14 days after ecclesiastical new moons) fall.

This works for most years. But comparing this Easter table with an actual table of full moons, we see that, for the period 2005-2013, it’s off in 2008 (Mar. 22 vs. Mar. 21), 2009 (Apr. 10 vs. Apr. 9), 2010 (Mar. 30 vs. Mar. 29), and 2012 (Apr. 7 vs. Apr. 6).

This particular Easter table is based on the Metonic cycle, named after Meton of Athens, who figured it out in the 5th century BC. This method is still how they figure out when to intercalate months in the Jewish (lunar) calendar. The monk Dionysius Exiguus published an adaptation of Metonic cycles into the Julian calendar in 525 (along the way he invented the practice of counting years [incorrectly] from the birth of Christ). His tables replaced earlier (less accurate) tables, for instance by Victorius of Aquitaine. It took a while for the Dionysian tables to replace the Victorian tables everywhere, and this difference is responsible for the differences in Easter calculation between the Irish church and the Roman church that were resolved at the Synod of Whitby in Northumbria in 664.

How do the Orthodox do it? They just use the Metonic cycle without any corrections, which departs from the full moon by more than three days every millennium. Now it’s a couple of days off, and it’s going to keep getting worse. Now some of the time, there’s no Sunday between the Western and Orthodox ecclesiastical full moons. So this doesn’t make any difference in terms of Easter. But the rest of the time, there is a Sunday between them, so Western Easter Sunday will be a week earlier than Orthodox Easter Sunday. Add that to the possible four-week error (actually, more than four weeks because of the 29.5-day lunar month), and you get a difference between Western and Orthodox Easters of 0, 1, 4, or 5 weeks.

Note, though, that the difference between the spring equinoxes will keep getting worse. And the difference between the full moons based on the Metonic cycles will also keep getting worse. You can see the results in Table 1 on this website, showing the differences between Western and Orthodox Easter. From 1583 to the 2000s, the differences are 0, 1, 4, or 5 weeks. After 2100, the four-week difference drops out, and we only have differences of 0, 1, or 5 weeks. By the 2400s, the differences will be 0, 1, 5, or 6 weeks. By the 2700s, the two Easters will never overlap, and we’ll have differences of 1, 2, 5, or 6 weeks.