When The Hell Is Easter? -- An Introduction to the Computus

This is more or less exactly how I feel after writing this article.

Photo by Andrea Allen. (License: Creative Commons Attribution)

Tomorrow is Easter, perhaps the most important fest and event in the Christian Liturgical Year. Easter Sunday marks the celebration of Jesus' resurrection in the first century AD, viewed by Christians as a triumph over death and the moment of salvation for humankind. As Christendoms most important feast, Easter is also one of its oldest; records of Easter celebrations stretch back to 154 AD, slightly more than a century after the crucifixion.

But though histories provide a detailed account of Easters in years past, they fail us in the smaller measures of time. At present, Easter Sunday is celebrated in a widely varying range between the end of March and the end of April. Earlier Christian traditions celebrated the feast earlier in the months, with Easter falling between early April and early May. Over the centuries Christians have used differing calendars ranging from the Hebrew to the Julian, and the Gregorian - each placing Easter on a different day of the year.

A Brief History of Time (with apologies to Mr. Hawking)

Though the Bible provides an excellent account of the crucifixion of Jesus it does so without a frame of reference. The Crucifixion, biblically, follows the Last Supper, which is itself a celebration of Passover. Passover, however, is fixed in the Hebrew Calendar as a six day feast, beginning on the 15th day of Nisan and ending on the 21st. Problematically, the Hebrew calendar is lunar rather than solar, and has little correlation whatsoever to the standard Gregorian calendar.

The early Church, in a portent of policy and structure to come, responded bureaucratically. During the First Council of Nicaea (325 AD), Christians agreed that a common method should be used to calculate the day of Easter and, moreover, that this date should always fall on a Sunday. Following Nicaea, the Church used several methods for the calculation as it spread throughout the Roman Empire and Eastern Roman Empire. By the 3^rd century AD, Easter was calculated based upon the Alexandrian method, itself derived from the Egyptian civil solar calendar¹. By the 6^th Century, the Byzantine Method arose in churches along the fringes of the Byzantine Empire, itself a remnant of the Eastern Roman Empire².

In 535 AD the Emperor Dionysus introduced the "Christian Era," instituting the custom of enumerating dates from the birth of Jesus, and giving rise to the modern nomenclature of AD (Anno Domini -- In the year of [Our] Lord). The Dionysian reforms also included "Easter tables" which provided a equivalency between Passover, the Hebrew Calendar, and the decisions at Nicaea.

Dionysius' tables remained in use throughout the Western Church until the Gregorian Calendar reforms which altered the official structure of the year and made the tables unusable.

The Gregorian System

The introduction of the Gregorian calendar in 1582 made the Dionysian tables obsolete and necessitated a new method for calculating the proper day for the feast of Easter. This is the method used today and accounts for the wide variation of the date upon which Easter falls.

Step 1 - Determine the Epact

The Epact is the offset in days between the lunar and solar year. A lunar year is 11 days shorter than the solar year so the Epact itself increases by eleven with every (solar) year. However, once the Epact exceeds 30, a lunar month (30 days) is removed meaning that the calculation of the Epact itself is a modulus operation. 2006 provides an excellent starting point for these calculations as the lunar and solar years began on the same day in 2006. Thus 2006 had an Epact of 0. In January 1, 2007 the new lunar year was already 11 days old, so the Epact for 2007 is 11. In 2008 the Epact will be 22, and in 2009 it will be 3 owing to the modulus nature of the calculation³.

Step 2 - Determine The Ecclesiastical New Moon

This is where it gets ugly. A listing of all 365 calendar days is made (leap days are ignored, they are a solar calendar convenience). Starting on January 1, dates are labeled with a Roman Numeral starting with a place-holder character (we'll call it *) and then descending from 29 (xxix) through 1 (i). In practice this generates a January listing with January 2 - 30 labeled as xxix through i respectively and January 1 and 31 labeled as *.

This is repeated through the end of the year, but exceptions apply. Starting with the second period (the one beginning on February 1), every other period consists of 29 rather than 30 days. In the 29 day periods the 25^th (xxv) and 24^th (xxiv) days fall on the same date, allowing for the shortened period. The thirteenth period is, as the math would suggest a "long" period. Finally, the label of 25 (without a Roman numeral) is added to any date labeled 26 (xxvi) within the even numbered (29 day) cycles.

The resulting migraine inducing table is called the calendarium. With it, ecclesiastical new moons may be predicted by looking for dates labeled with the value of the year's Epact.

Step 3 - Determine the Dominical Letter

In addition to roman numerals, the calendar year is also labeled with letters ranging from A to G starting, again, on January 1 and continuing through the end of the year. The letter which falls on the first Sunday of the year is the Dominical Letter, and all other days with that letter throughout the year will also be Sundays⁴. Dominical letters cycle backwards one day each year, however in leap years⁵ Sundays will move back one additional letter following February 24^th. If one happens to be a 12^th century priest in provincial France, the Dominical Letter system is helpful in navigating the tables and charts necessary to determine the mathematical structure of the year. In more modern times, however, these are largely irrelevant and, as such, may be safely ignored.

Step 3 - Wait for March 21... and then wait a little longer

Easter is a spring festival and as such must be celebrated in the spring. March 21 has nothing specifically to do with the actual Vernal Equinox, which marks the beginning of spring. It is, however, a simple enough approximation of the equinox and is thus used as a convenient demarcation of the beginning of spring. Starting from March 21, the first ecclesiastic Full Moon may be found by searching for a date that falls thirteen days after the ecclesiastic New Moon. Starting from this Full Moon, called the "Paschal Full Moon," the first following Sunday will be Easter Sunday.

A Few More Catches

The entire system, however, hinges upon the lunar and solar cycles lining up in a very specific manner. Problematically, they simply do not do this. The above cycles should create a nineteen (solar) year cycle, called the Metonic cycle, as 19 solar years should work out to the same number of days as 235 synodic⁶ months. This predicts that Epacts and the various enumerations above would repeat every 19 years. Unfortunately, the math is off and an increment of the Epact is necessary every 19 years for the cycle to repeat, though even this fails to address the issue. Epacts are adjusted in this manner on the first year of the Metonic cycle, or on years evenly divisible by 19.

Further complicating the matter, in any year valued eleven or higher in the Metonic cycle which also has an Epact of xxv the New Moon will fall on the label of "25" rather than xxv in the short months where that label exists.

The Metonic cycle itself has a two hour discrepancy over the course of its 19 year period. Though this is remarkably close, the result is a full day of error every 219 years, requiring more manual adjustment.

The "Solar Equation" further complicates the calculations. The Gregorian Calendar drops three "leap days" every 400 years (on the century year) and as such the Epact is reduced by one on these years as well.

Serving much the same purpose, the "Lunar Equation" forces an increment of the Epact some eight times in a 2500 year span, amounting to once every three centuries. This increment will be skipped, however, between 3900 and 4300 AD which marks the beginning of a new cycle.

The result of this and several other complications too specific to mention results in a 5,700,000 year cycle for the so-called "Computus" or method for calculating the date of Easter. In the long run, April 19 is the most common day upon which Easter may fall, making up just sort of 4% of the cycle. March 22 is the least common, accounting for less than half a percent. In some sense it is quite comforting to see that the Catholic Church has judged it worthwhile to plot out the dates of Easter stretching over a nearly 6 Million Year cycle. On the other, such revelations lend new insight into the tedium of life in 12^th century France⁷.

Works Cited

Wikipedia: Computus
Wikipedia: Easter
Wikipedia: Calendars
Wikipedia: Easter Controversy
Wikipedia: Passover
Wikipedia: Paschal Full Moon
Wikipedia: Epact
Wikipedia: Dominical Letter
Wikipedia: Golden Numbers
Wikipedia: Metonic Cycle

1. This method survives as the Ethiopian Computus.
2. This method survives in the Eastern Orthodox church, which today celebrates Easter on much the same days as the Roman Catholic Church and its Protestant decedents. The Eastern Orthodox Computus results in deviation from the Roman Catholic Church's method four times every 532 years
3. 2008's Epact is 22. 22+11 = 33. 33 - 1 Month (30 days) = 3 with a one month offset.
4. This is not nearly as clever as it sounds. A-G is just 1 through 7. Seven days in a week so this makes rather a lot of sense.
5. We're back to dealing with the solar calendar so they matter here.
6. Wikipedia defines a Synodic Month as "the average period of the moon's revolution with respect to the sun"
7. Note that 12^th Century France has nothing to do with any of this, it just makes an easy target for some very strange humor.

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