# AST 309--TIME Easter, Rosh Hashanah and Passover

### Calculation rules

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

### Easter

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!)

### Rosh Hashanah

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:

#### Postponement rules

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

2. 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.

3. 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.

#### Passover

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.]

### Final Comment on Easter and Passover

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). 