Easter date algorithms (c) Henk Reints
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When is Easter?

What is Easter?

Easter is the most important Christian Holyday, celebrating the resurrection of Jesus Christ.

Christian authorities have always wanted to celebrate this on the same day of the year as the true resurrection. But at the time and place of the crucifiction our "normal" calendar was not used, but the Hebrew calendar, which is still in use by the Jews today. The Hebrew calendar is based on the Moon, and each month starts at new moon, i.e. on the evening when the crescent is first visible (well, nowadays a computed [accurate] estimate thereof is used). To keep in synchronisation with the seasons, a leap month is added approximately once every 3 years. Due to this definition, the 15th day of every Hebrew month is a full moon day. On the Hebrew calendar, the first month of the year is always the month that starts on the new moon that is closest to the beginning of spring, this month is called Nisan. See www.tondering.dk/claus/calendar.html for more info on the Hebrew calendar (and there are many other resources on the web).

The Jews celebrate the exodus from Egypt (which was a true liberation) every year on 15 Nisan, which is called Pesach (meaning passover). This is probably celebrated on this first full moon in spring because this "rebirth of light" also reflects a new beginning, which the exodus was as well.

Then the Romans came and they crucified Jesus Christ on the day of Pesach, 15 Nisan, which was a Friday in that year, and He resurrected the next Sunday. Since Nisan is the first spring month, 15 Nisan is the first full moon in spring and the resurrection took place on the Sunday thereafter.

But, in the early years of Christianity, Easter dating was not accuratly defined and it was celebrated on different dates in different parts of the world. That's why in A.D. 325 the First Council of Nicaea decided the following:

Greek text copied from: Rodolphe Audette's website.
Please click here if the text below is not properly displayed using the Greek alphabet.

Transcription by HR:

English translation copied from: http://www.fordham.edu/halsall/basis/nicea1.txt
We further proclaim to you the good news of the agreement concerning the holy Easter, that this particular also has through your prayers been rightly settled; so that all our brethren in the East who formerly followed the custom of the Jews are henceforth to celebrate the said most sacred feast of Easter at the same time with the Romans and yourselves and all those who have observed Easter from the beginning.

English translation copied from: http://www.ewtn.com/library/COUNCILS/NICAEA1.HTM
We also send you the good news of the settlement concerning the holy Pasch, namely that in answer to your prayers this question also has been resolved. All the brethren in the East who have hitherto followed the Jewish practice will henceforth observe the custom of the Romans and of yourselves and of all of us who from ancient times have kept Easter together with you.

In practice this means Easter is:

the first Sunday after the first full moon in spring.

Notes:

  • Presumably, the Roman Christians already celebrated Easter on this date.
  • In fact this is of course the Sunday after Pesach, thus also avoiding that the Christian Easter would coincide with the Jewish Passover festival (which was absolutely unwanted).
  • Since nowadays both the Christian Easter dating and the Jewish calendar are based on mathematics only, leaving the exact moon phases and the seasons for what they are and using different approximations thereof, the modern Easter Sunday is not always the first Sunday after Pesach, it can be a month earlier or later.

Later, in A.D. 525 (see below), Dionysius Exiguus defined it as:

Easter is the Sunday following the first Luna XIV (the 14th day of the moon) that occurs on or after XII Kalendas Aprilis (21 March).

(kalendas means counting days backward starting with 1 on the first day of the given month, which is according to the Roman calendar, and the change from 15 Nisan to Luna 14 probably has to do with the fact that on the Hebrew calendar days start at sunset, whilst on the Christian A.D. calendar (which was also introduced by Dionysius) days start at midnight.

Ecclesiastical authorities also wanted to make accurate predictions of Easter for forthcoming years, but the moon and the seasons are not behaving as regular as one would like. Based on the astronomical knowledge of that time, some simplifications were made:

  • The begin of spring, also called the equinox, which literally means: "equal night" (i.e. night equal to day (length) ) was fixed on 21 March 00:00. Astronomically, the begin of spring is not a day but the single moment at which the Sun passes the equatorial plane from South to North. In about 3 of every 4 years this is on 20 March, and on average that is somewhere around 20.75 March. This makes the night from 20 to 21 March the "equal night" and 21 March the first full day in spring.
     

  • The Julian calendar (introduced by Julius Caesar) was used, having a leap day every 4 years, resulting in a year length of exactly 365.25 days (which is astronomically incorrect). This 0.25 day causes the moment of the equinox to shift forward, and when 21 March is reached, the leap day more or less resets the astronomical equinox to 20 March.
     

  • The long term behaviour of the moon was described using a 19 year cycle called the moon cycle or Metonic cycle. Meton, an ancient Greek, had discovered that in 19 tropical years there were nearly exactly 235 full moons.
     

  • The short term behaviour of the moon was described by rounding the average synodic month (i.e. the time between 2 full moons) to 30 days.

Based on these rules, a hypothetical first full moon in spring was calculated, called the Paschal Full Moon (PFM). This PFM may differ from the real full moon by a few days and it may even differ from the astronomical first full moon in spring by an entire month, an event that is called an Easter paradox. These calculations are only based on integer days, the time-of-day is ignored.

In fact, the theoretical approach of Easter dating was not introduced immediately in A.D. 325, but, as already mentioned, in A.D. 525 by Dionysius Exiguus (who also and at the same time introduced the Anno Domini year counting that we still use today) in his Liber de Paschate (for an English translation click here, see alsy my Dutch translation).

This Easter dating method introduced by Dionysius is called the Julian method, since it is based on the Julian calendar. This Julian method is still in use by the eastern European orthodox churches. Since nowadays those countries also use the Gregorian calendar (see next paragraph), this Julian Easter date is usually converted to the Gregorian calendar (by adding 13 days, the current difference between the two calendars), resulting in an Easter date that may even be in May.

In 1582, under Pope Gregory XIII (by the Papal Bull "Inter Gravissimas"), a correction was made to the calendar and to the method for determining the Easter date, to make them more accurately match long term astronomical observations. The so called Gregorian calendar has no leap day in years that can be divided by 100 unless they are also a multiple of 400, assuming a year length of 365.2425 days (click here for a quick calculation method with the Gregorian calendar). For Easter dating, the 19 year moon cycle was corrected with 8 days every 25 centuries, which leads to an average synodic month of 29.53059 days, which is accurate to less than a second! This astonishing accuracy (certainly at that time!) is achieved whilst using only very simple integer calculations with rather small numbers!

This Easter dating method introduced by Pope Gregory XIII is called the Gregorian method, since it uses the Gregorian calendar AND the new Gregorian easter dating method using the 8 days per 25 centuries moon cycle correction. It is in use by most western churches and nowadays it is the western standard.

This site explains some different algorithms (both Julian and Gregorian) one can use to find any Easter date.

Although some of the algorithms claim a validity since A.D. 1 (as mentioned on the specified source when and where I got it), from the above can easily be concluded that any calculated Easter date before the First Council of Nicaea (held in A.D. 325, probably from May to August) is just a "would be" date that most likely does not reflect a historical fact, since before that date other methods were used based on the Jewish Passover festivals, and obviously it is certainly incorrect before Christ's resurrection (as mentioned above, astronomical computations of first moon visibility make A.D. 23 the best candidate year for the crucifiction, being 15 Nisan on a Friday).

On the other "end of time" in the far future the validity of the algorithms should also not be considered as accurate. The Gregorian calendar assumes a year length of exactly 365.2425 days, but currently it is 365.24219 (leading to a error of 1 day every 3226 years). This means that around the 41st century another correction will be necessary by skipping a leap day.
The designers of the Gregorian calendar, Lilius and Clavius, where however convinced of the absolute accuracy of their calendar and they made tables describing calendars up to the year 800 000 000 (yes, eight hundred million !!! HERE, HERE, and HERE you can see the original tables!).

Another fact is also important for the far future: the length of the year is NOT constant. Due to tidal friction the Earth's rotation slowly decreases, thus slowly incrementing the absolute day length and shortening the year length as expressed in days.
Tidal forces also make the Moon accelerate in its orbit and because the universe contains more mass than only the Earth and the Moon there is a residual gravitational force pulling on the Moon (mainly by the Sun and Jupiter). Together these forces make the Moon distantiate itself from the Earth resulting in a very slow but steady increment of the absolute length of the synodic month. In the same way the Earth's orbit around the Sun is also growing very slowly, which makes the absolute length of the year to increase very slowly.

Although the given algorithms do not exactly keep sync with the real Moon over shorter periods of time, they do (or their designers did) attempt to keep synchronised for the long term era. Since the algorithms don't correct for these changing astronomical facts, they are not of eternal value. Predictions for the far future are at most a good guess, and of course nobody knows yet how people may decide about the methods of time keeping and Easter dating in the far future - maybe they won't even celebrate Easter by then, who knows? As already mentioned, if people want to keep the vernal equinox around 20 or 21 March, then the Gregorian calendar needs to skip one leap day around A.D. 4000, so don't plan an Easter holiday trip based on these calculations for years beyond A.D. 4100...

In all given algorithms the following applies:

DIV = integer division discarding any remainder, e.g. 13 DIV 5 = 2.
MOD = the remainder of an integer division, e.g. 13 MOD 5 = 3.

All algorithms listed on this site return the result as a day in March, even if that is greater than 31 (if you cannot do the month rollover to April all by yourself then you'd better leave this site now, you won't understand a bit of it...).

Julian methods return a result on the Julian calendar,
Gregorian methods return a result on the Gregorian calendar.
To convert a Julian date to the Gregorian calendar (any date since 5 October 1582) use:

        if year >= 1700
        then
            if month >= 3 then C = year DIV 100 - 16 else C = (year-1) DIV 100 - 16
            result = day + 10 + C - C DIV 4
        else result = day + 10
        finally apply month rollover if necessary.
Update 2022-05-31:
I thank Hans Klaver for drawing my attention to a mistake I had made on this over 20 years old page, I didn't mention the difference only increments after the leap day. That's what you get if you write untestable text instead of a computer program... I DID thoroughly test all javascript on this website. Please let me know if you find any other missed ache. (I know there is a small error in my Hebrew calendar conversion, but I have not yet found time to repair it).

When converting a Julian (Orthodox) Easter date to the Gregorian calendar,
the resulting Easter date can also be in May.

 

 
DISCLAIMER

The algorithms explained here and the Easter date calculator available on this web site are provided for demonstrating purposes only and to enjoy people who have interest or fun in the mathematics of Easter date calculations. The author or any other source mentioned cannot be held responsible for any consequences of typing errors, miscalculations or misinterpretations of the result. When planning activities based on an Easter (related) date, always consult other references to verify the correctness of the dates calculated here.

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Easter date algorithms (c) Henk Reints