Easter, Rosh Hashanah and Passover

In what follows, any time you see a number or calculation in
*square brackets* [], you should take the *next smallest
integer*. Thus, [4.333]=4. Usually this just means crossing
off the decimals to the right of the decimal point as in this
example, but if the number is *negative*, you must be careful.
For example, the next smallest integer to -3.0666 is -4, *not*
-3, so [-3.0666]=-4!

Remainder(x|y) means the* *remainder when you divide x by
y. It is *never negative,* and is defined in terms of the
[] operation as follows:

- Remainder(x|y) = x - y[x/y]

For example, in the year 1996 we get:

- Remainder(1996|19) = 1996 -19[1996/19] = 1996 - 19[105.05...] = 1996-19*105 = 1996 - 1995 = 1.

It is possible in at least one of the formulas below for x to be negative, and in this case you must be careful. Always remember that Remainder(x|y) cannot be less than zero. If you calculate a negative number for Remainder(x|y), you'll know you've made a mistake. For example, the following way to calculate Remainder(-2,30) is wrong:

Remainder(-2,30) = -2 - 30[-2/30] = -2 - 30[-0.0666] = -2 - 30x0 = -2

This is wrong, because [-0.0666] is not zero but is -1! So the correct calculation is as follows:

Remainder(-2,30) = -2 - 30[-2/30] = -2 - 30[-0.0666] = -2 - 30x(-1) = -2 + 30 = 28

The following rules, due to John Conway, allow you to calculate
the date of Easter for any year on the Gregorian or Julian calendar.

First, calculate the *Golden Number *G. This is fundamental
to the calculation of both the date of Easter and the Date of
Rosh Hashanah. It is intimately connected with the Metonic Cycle.
For any year Y, the Golden Number is defined as

- G = Remainder(Y|19) + 1.
*Don't forget to add the 1!!!*

For example, in the year 1996, the Golden Number was 2 because Remainder(1996|19)=1. Next, compute S, where

- S=Remainder((11G + C)|30), and in the 20th through the 22nd century, C=-6. A table for C and a rule for calculating C are given below.
**Important Note:**S must be nonnegative. If you get a negative number, add an appropriate multiple of 30 to make S between 0 and 29 (inclusive).

For example, in 1996, S=Remainder((22-6)|30)=16.

Then, the *Paschal Full Moon* falls on the date (March 50=April
19) - S, except that if this formula gives April 19, the Paschal
Full Moon falls on April 18 instead, and if the rule gives April
18, *and* if G is greater than or equal to 12, the Paschal
Full Moon falls on April 17.

Then Easter is the first Sunday that falls *after* this date
(if the date you calculated is a Sunday, then Easter is one week
later). You can use Conway's Doomsday Rule for Day of the Week
to determine the day of the week of the Paschal Full Moon.

For example, in 1996, the Paschal Full Moon falls on April 19-S
= April 19-16 = April 3. Conway's Doomsday
Rule tells us that April 3 is a Wednesday that year, so Easter
is the *next* Sunday, or April 7.

Here's how to determine C:

- In all Julian Calendar years, C=+3. This gives us Julian Easter. Note that Julian (Orthodox) Easter may not fall on the same actual day as Gregorian Easter, because the rule for Gregorian Easter is different. Julian Easter is celebrated by the Orthodox Church.
- In years 15xx, 16xx of the Gregorian Calendar, C=-4 (note the negative sign).
- In years 17xx, 18xx of the Gregorian Calendar, C=-5
- In years 19xx, 20xx, 21xx of the Gregorian Calendar, C=-6

For Gregorian dates in other centuries, where the year is Y=Hxx, calculate C as follows:

- C = [H/4] + [8(H+11)/25] - H (note the square brackets!)

The following rules are also due to John Conway. In the Gregorian year Y of the Common Era, Rosh Hashanah normally falls on September N, where

N + fraction = {[Y/100] - [Y/400] - 2} + 765433/492480*Remainder(12G|19) + Remainder(Y|4)/4 - (313Y+89081)/98496

Here, G is the Golden Number, and * means multiply. However, if certain conditions are satisfied, Rosh Hashanah is postponed by one or even two days, as follows:

- If the day calculated above is a Sunday, Wednesday, or Friday, Rosh
Hashanah falls on the next day (Monday, Thursday or Saturday,
respectively).

- If the calculated day is a Monday, and if the fraction is greater than
or equal to 23269/25920, and if Remainder(12G|19) is greater than 11,
Rosh Hashanah falls on the next day, a Tuesday.

- If it is a Tuesday, and if the fraction is greater than or equal to
1367/2160, and if Remainder(12G|19) is greater than 6, Rosh Hashanah
falls
*two days*later*,*on*Thursday*(NOT WEDNESDAY!!).

For example, in 1996, G=2. So the calculation gives 13.5239...
(check this for yourself using your calculator)! However, since
Doomsday is Thursday in 1996, September 5 and 12 are Thursdays,
so September 13 is Friday. By Rule 1, we must postpone by one
day, so Rosh Hashanah falls on Saturday, September 14. (It actually
*began* at sundown on the 13th.) Yom Kippur *began*
at sundown on the 22nd of September, 9 days after the beginning
of Rosh Hashanah.

A simplified formula for the date of Rosh Hashanah on the *Gregorian
calendar* for 1900-2099 is gotten by calculating

N + fraction = 6.057778996 + 1.554241797*Remainder(12G|19) + 0.25*Remainder(y|4) - 0.003177794*y,

where y=Y-1900. Use the same postponement rules (note that
23269/25920=0.898,
and 1367/2160=0.633). This method is easier to calculate using
a pocket calculator.

*Julian Calendar*

If you are given a date on the Julian calendar, the rules are the same, except that you must ignore the term in curly braces {} in the formula that gives the date of Rosh Hashanah. This term corrects for the difference between the Julian and Gregorian calendars.

Once you have determined the date of Rosh Hashanah, it is easy
to calculate the date of Passover in the same (Gregorian or Julian)
year. Suppose Rosh Hashanah falls on September M. In the example
for 1996, Rosh Hashanah fell on September 14, so M=14. If Rosh
Hashanah falls in October, add 30 to the October date. For example,
if Rosh Hashanah falls on October 3, then M=3+30=33 (September
33=October 3).

Count M days from March 21. That is the date of Passover. It actually
begins at sundown on the previous evening. In the example for
1996, 14 days after March 21 is April 4 (there are 31 days in
March), so Passover begins at sundown on April 3.

[Thanks to Alfred Pauson for correspondence which led to the improvement of this section.]

If you pay attention to the dates of Easter and Passover from
year to year, you will notice that although they usually fall
within a week or so of each other, on occasion Passover falls
about a month *after *(Gregorian) Easter. At the present
time, this happens in in the 3rd, 11th, and 14th years of the
Metonoic Cycle (i.e., when the Golden Number equals 3, 11, or
14). The reason for this discrepancy is the fact that although
the Metonic Cycle is very good, it is not perfect (as we've seen
in this course). In particular, it is a little off if you use
it to predict the length of the tropical year. So, over the centuries
the date of the vernal equinox, *as predicted by the Metonic
Cycle,* has been drifting to later and later dates. So, the
rule for Passover, which was originally intended to track the
vernal equinox, has gotten a few days off. In ancient times this
was never a problem since Passover was set by actual observations
of the Moon and of the vernal equinox. However, after Hillel II
standardized the Hebrew calendar in the 4th century, actual observations
of celestial events no longer played a part in the determination
of the date of Passover. The Gregorian calendar reform of 1582
brought the Western Church back into conformity with astronomical
events, hence the discrepancy.

Similarly, you will notice that in many years Gregorian Easter
(the one marked on all calendars) differs from Julian (Orthodox)
Easter, sometimes by a week, sometimes by a month. Again, this
is due to the different rules of calculation. A major difference
is that Orthodox Easter uses the old Julian calendar for calculation,
and the date of the Vernal Equinox is slipping later and later
on the Julian calendar relative to the Gregorian calendar (and
to astronomical fact). Also, the date of Paschal Full Moon for
the Julian calculation is about 4 days later than that for the
Gregorian calculation. At present, in 5 out of 19 years in the
Metonic Cycle--the years when the Golden Number equals 3, 8, 11,
14 and 19--Orthodox Easter occurs a month after Gregorian Easter.
In three of these years, Passover also falls a month after Gregorian
Easter (see above).

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