Calendar FAQ v1.6 -- Part 2

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2.9. What is Easter?
--------------------

In the Christian world, Easter (and the days immediately preceding it)
is the celebration of the death and resurrection of Jesus in
(approximately) AD 30.

2.9.1. When is Easter? (Short answer)
-------------------------------------

Easter Sunday is the first Sunday after the first full moon after
vernal equinox.

2.9.2. When is Easter? (Long answer)
------------------------------------

The calculation of Easter is complicated because it is linked to (an
inaccurate version of) the Hebrew calendar.

Jesus was crucified immediately before the Jewish Passover, which is a
celebration of the Exodus from Egypt under Moses. Celebration of
Passover started on the 14th or 15th day of the (spring) month of
Nisan. Jewish months start when the moon is new, therefore the 14th or
15th day of the month must be immediately after a full moon.

It was therefore decided to make Easter Sunday the first Sunday after
the first full moon after vernal equinox. Or more precisely: Easter
Sunday is the first Sunday after the *official* full moon on or after
the *official* vernal equinox.

The official vernal equinox is always 21 March.

The official full moon may differ from the *real* full moon by one or
two days.

(Note, however, that historically, some countries have used the *real*
(astronomical) full moon instead of the official one when calculating
Easter. This was the case, for example, of the German Protestant states,
which used the astronomical full moon in the years 1700-1776. A
similar practice was used Sweden in the years 1740-1844 and in Denmark
in the 1700s.)

The full moon that precedes Easter is called the Paschal full
moon. Two concepts play an important role when calculating the Pascal
full moon: The Golden Number and the Epact. They are described in the
following sections.

The following sections give details about how to calculate the date
for Easter. Note, however, that while the Julian calendar was in use,
it was customary to use tables rather than calculations to determine
Easter. The following sections do mention how to calcuate Easter under
the Julian calendar, but the reader should be aware that this is an
attempt to express in formulas what was originally expressed in
tables. The formulas can be taken as a good indication of when Easter
was celebrated in the Western Church from approximately the 6th
century.

2.9.3. What is the Golden Number?
---------------------------------

Each year is associated with a Golden Number.

Considering that the relationship between the moon's phases and the
days of the year repeats itself every 19 years (as described in
section 1), it is natural to associate a number between 1 and 19
with each year. This number is the so-called Golden Number. It is
calculated thus:
        GoldenNumber = (year%19)+1

New moon will fall on (approximately) the same date in two years
with the same Golden Number.

2.9.4. What is the Epact?
-------------------------

Each year is associated with an Epact.

The Epact is a measure of the age of the moon (i.e. the number of days

that have passed since an "official" new moon) on a particular date.

In the Julian calendar, 8 + the Epact is the age of the moon at the
start of the year.  
In the Gregorian calendar, the Epact is the age of the moon at the
start of the year.

The Epact is linked to the Golden Number in the following manner:

Under the Julian calendar, 19 years were assumed to be exactly an
integral number of synodic months, and the following relationship
exists between the Golden Number and the Epact:

        Epact = (11 * (GoldenNumber-1)) % 30 

If this formula yields zero, the Epact is by convention frequently
designated by the symbol * and its value is said to be 30. Weird?
Maybe, but people didn't like the number zero in the old days.

Since there are only 19 possible golden numbers, the Epact can have
only 19 different values: 1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20,
22, 23, 25, 26, 28, and 30.


The Julian system for calculating full moons was inaccurate, and under
the Gregorian calendar, some modifications are made to the simple
relationship between the Golden Number and the Epact.

In the Gregorian calendar the Epact should be calculated thus (the
divisions are integer divisions, in which remainders are discarded):
        
1) Use the Julian formula:
      Epact = (11 * (GoldenNumber-1)) % 30

2) Adjust the Epact, taking into account the fact that 3 out of 4
   centuries have one leap year less than a Julian century:
        Epact = Epact - (3*century)/4

   (For the purpose of this calculation century=20 is used for the
   years 1900 through 1999, and similarly for other centuries,
   although this contradicts the rules in section 2.10.2.)

3) Adjust the Epact, taking into account the fact that 19 years is not
   exactly an integral number of synodic months:
        Epact = Epact + (8*century + 5)/25

   (This adds one to the epact 8 times every 2500 years.)

4) Add 8 to the Epact to make it the age of the moon on 1 January:
        Epact = Epact + 8

5) Add or subtract 30 until the Epact lies between 1 and 30.

In the Gregorian calendar, the Epact can have any value from 1 to 30.

Example: What was the Epact for 1992?

GoldenNumber = 1992%19 + 1 = 17
1) Epact = (11 * (17-1)) % 30 = 26
2) Epact = 26 - (3*20)/4 = 11
3) Epact = 11 + (8*20 + 5)/25 = 17
4) Epact = 17 + 8 = 25
5) Epact = 25

The Epact for 1992 was 25.


2.9.5. How does one calculate Easter then?
------------------------------------------

To find Easter the following algorithm is used:

1) Calculate the Epact as described in the previous section.

2) For the Julian calendar: Add 8 to the Epact. (For the Gregorian
   calendar, this has already been done in step 5 of the calculation of
   the Epact). Subtract 30 if the sum exceeds 30.

3) Look up the Epact (as possibly modified in step 2) in this table to
   find the date for the Paschal full moon:

     Epact   Full moon     Epact   Full moon     Epact   Full moon
     -----------------     -----------------     -----------------
        1    12 April        11     2 April        21    23 March
        2    11 April        12     1 April        22    22 March
        3    10 April        13    31 March        23    21 March
        4     9 April        14    30 March        24    18 April
        5     8 April        15    29 March        25    18 or 17 April
        6     7 April        16    28 March        26    17 April
        7     6 April        17    27 March        27    16 April
        8     5 April        18    26 March        28    15 April
        9     4 April        19    25 March        29    14 April
       10     3 April        20    24 March        30    13 April

4) Easter Sunday is the first Sunday following the above full moon
   date. If the full moon falls on a Sunday, Easter Sunday is the
   following Sunday.

An Epact of 25 requires special treatment, as it has two dates in the
above table. There are two equivalent methods for choosing the correct
full moon date:

A) Choose 18 April, unless the current century contains years with an
   epact of 24, in which case 17 April should be used.

B) If the Golden Number is > 11 choose 17 April, otherwise choose 18 April.

The proof that these two statements are equivalent is left as an
exercise to the reader. (The frustrated ones may contact me for the
proof.)

Example: When was Easter in 1992?

In the previous section we found that the Golden Number for 1992 was
17 and the Epact was 25. Looking in the table, we find that the
Paschal full moon was either 17 or 18 April. By rule B above, we
choose 17 April because the Golden Number > 11.

17 April 1992 was a Friday. Easter Sunday must therefore have been 19 April.

2.9.6. Isn't there a simpler way to calculate Easter?
-----------------------------------------------------

This is an attempt to boil down the information given in the previous
sections (the divisions are integer divisions, in which remainders are
discarded):

G = year % 19

For the Julian calendar:
    I = (19*G + 15) % 30
    J = (year + year/4 + I) % 7

For the Gregorian calendar:
    C = year/100
    H = (C - C/4 - (8*C+13)/25 + 19*G + 15) % 30
    I = H - (H/28)*(1 - (H/28)*(29/(H + 1))*((21 - G)/11))
    J = (year + year/4 + I + 2 - C + C/4) % 7

Thereafter, for both calendars:
L = I - J
EasterMonth = 3 + (L + 40)/44
EasterDay = L + 28 - 31*(EasterMonth/4)


This algorithm is based in part on the algorithm of Oudin (1940) and
quoted in "Explanatory Supplement to the Astronomical Almanac",
P. Kenneth Seidelmann, editor.

People who want to dig into the workings of this algorithm, may be
interested to know that
    G is the Golden Number-1
    H is 23-Epact (modulo 30)
    I is the number of days from 21 March to the Paschal full moon
    J is the weekday for the Paschal full moon (0=Sunday, 1=Monday,
      etc.)
    L is the number of days from 21 March to the Sunday on or before
      the Pascal full moon (a number between -6 and 28)


2.9.7. Is there a simple relationship between two consecutive Easters?
----------------------------------------------------------------------

Suppose you know the Easter date of the current year, can you easily
find the Easter date in the next year? No, but you can make a
qualified guess.

If Easter Sunday in the current year falls on day X and the next year
is not a leap year, Easter Sunday of next year will fall on one of the
following days: X-15, X-8, X+13 (rare), or X+20.

If Easter Sunday in the current year falls on day X and the next year
is a leap year, Easter Sunday of next year will fall on one of the
following days: X-16, X-9, X+12 (extremely rare), or X+19. (The jump
X+12 occurs only once in the period 1800-2099, namely when going from
2075 to 2076.)

If you combine this knowledge with the fact that Easter Sunday never
falls before 22 March and never falls after 25 April, you can
narrow the possibilities down to two or three dates.


2.9.8. How frequently are the dates for Easter repeated?
--------------------------------------------------------

The sequence of Easter dates repeats itself every 532 years in the
Julian calendar. The number 532 is the product of the following
numbers:

        19 (the Metonic cycle or the cycle of the Golden Number)
        28 (the Solar cycle, see section 2.4)

The sequence of Easter dates repeats itself every 5,700,000 years in
the Gregorian calendar. The number 5,700,000 is the product of the
following numbers:

        19 (the Metonic cycle or the cycle of the Golden Number)
        400 (the Gregorian equivalent of the Solar cycle, see section 2.4)
        25 (the cycle used in step 3 when calculating the Epact)
        30 (the number of different Epact values)


2.9.9. What about Greek Easter?
-------------------------------

The Greek Orthodox Church does not always celebrate Easter on the same
day as the Catholic and Protestant countries. The reason is that the
Orthodox Church uses the Julian calendar when calculating Easter. This
is case even in the churches that otherwise use the Gregorian calendar.

When the Greek Orthodox Church in 1923 decided to change to the
Gregorian calendar (or rather: a Revised Julian Calendar), they chose
to use the astronomical full moon as seen along the meridian of
Jerusalem as the basis for calculating Easter, rather than to use the
"official" full moon described in the previous sections. However,
except for some sporadic use the 1920s, this system was never adopted
in practice.

2.10. How does one count years?
-------------------------------

In about AD 523, the papal chancellor, Bonifatius, asked a monk by the
name of Dionysius Exiguus to devise a way to implement the rules from
the Nicean council (the so-called "Alexandrine Rules") for general use.

Dionysius Exiguus (in English known as Denis the Little) was a monk
from Scythia, he was a canon in the Roman curia, and his assignment
was to prepare calculations of the dates of Easter. At that time it
was customary to count years since the reign of emperor Diocletian;
but in his calculations Dionysius chose to number the years since the
birth of Christ, rather than honour the persecutor Diocletian.

Dionysius (wrongly) fixed Jesus' birth with respect to Diocletian's
reign in such a manner that it falls on 25 December 753 AUC (ab urbe
condita, i.e. since the founding of Rome), thus making the current era
start with AD 1 on 1 January 754 AUC.

How Dionysius established the year of Christ's birth is not known,
although a considerable number of theories exist. Jesus was born under
the reign of king Herod the Great, who died in 750 AUC, which means
that Jesus could have been born no later than that year. Dionysius'
calculations were disputed at a very early stage.

When people started dating years before 754 AUC using the term "Before
Christ", they let the year 1 BC immediately precede AD 1 with no
intervening year zero. 

Note, however, that astronomers frequently use another way of
numbering the years BC. Instead of 1 BC they use 0, instead of 2 BC
they use -1, instead of 3 BC they use -2, etc.

See also the following section.

In this section I have used AD 1 = 754 AUC. This is the most likely
equivalence between the two systems. However, some authorities state
that AD 1 = 753 AUC or 755 AUC. This confusion is not a modern one, it
appears that even the Romans were in some doubt about how to count
the years since the founding of Rome.

2.10.1. Was Jesus born in the year 0?
-------------------------------------

No. There are two reasons for this:
        - There is no year 0.
        - Jesus was born before 4 BC.

The concept of a year "zero" is a modern myth (but a very popular one).
Roman numerals do not have a figure designating zero, and treating zero
as a number on an equal footing with other numbers was not common in
the 6th century when our present year reckoning was established by
Dionysius Exiguus (see the previous section). Dionysius let the year
AD 1 start one week after what he believed to be Jesus' birthday.

Therefore, AD 1 follows immediately after 1 BC with no intervening
year zero. So a person who was born in 10 BC and died in AD 10,
would have died at the age of 19, not 20.

Furthermore, Dionysius' calculations were wrong. The Gospel of
Matthew tells us that Jesus was born under the reign of king Herod the
Great, and he died in 4 BC. It is likely that Jesus was actually born
around 7 BC. The date of his birth is unknown; it may or may not be 25
December.

2.10.2. When does the 21st century start?
-----------------------------------------

The first century started in AD 1. The second century must therefore
have started a hundred years later, in AD 101, and the 21st century must
start 2000 years after the first century, i.e. in the year 2001.

This is the cause of some heated debate, especially since some
dictionaries and encyclopaedias say that a century starts in years
that end in 00.

Let me propose a few compromises:

Any 100-year period is a century. Therefore the period from 23 June 1997
to 22 June 2097 is a century. So please feel free to celebrate the
start of a century any day you like!

Although the 20th century started in 1901, the 1900s started in
1900. Similarly, we can celebrate the start of the 2000s in 2000 and
the start of the 21st century in 2001.

Finally, let's take a lesson from history:
When 1899 became 1900 people celebrated the start of a new century.
When 1900 became 1901 people celebrated the start of a new century.
Two parties! Let's do the same thing again!

2.11. What is the Indiction?
----------------------------

The Indiction was used in the middle ages to specify the position of a
year in a 15 year taxation cycle. It was introduced by emperor
Constantine the Great on 1 September 312 and abolished [whatever that
means] in 1806.

The Indiction may be calculated thus:
        Indiction = (year + 2) % 15 + 1

The Indiction has no astronomical significance.

The Indiction did not always follow the calendar year. Three different
Indictions may be identified:

1) The Pontifical or Roman Indiction, which started on New Year's Day
   (being either 25 December, 1 January, or 25 March).
2) The Greek or Constantinopolitan Indiction, which started on 1 September.
3) The Imperial Indiction or Indiction of Constantine, which started
   on 24 September.

2.12. What is the Julian Period?
--------------------------------

The Julian period (and the Julian day number) must not be confused
with the Julian calendar. 

The French scholar Joseph Justus Scaliger (1540-1609) was interested
in assigning a positive number to every year without having to worry
about BC/AD. He invented what is today known as the "Julian Period".

The Julian Period probably takes its name from the Julian calendar,
although it has been claimed that it is named after Scaliger's father,
the Italian scholar Julius Caesar Scaliger (1484-1558).

Scaliger's Julian period starts on 1 January 4713 BC (Julian calendar)
and lasts for 7980 years. AD 1997 is thus year 6710 in the Julian
period. After 7980 years the number starts from 1 again.

Why 4713 BC and why 7980 years? Well, in 4713 BC the Indiction (see
section 2.11), the Golden Number (see section 2.9.3) and the Solar
Number (see section 2.4) were all 1. The next times this happens is
15*19*28=7980 years later, in AD 3268.

Astronomers have used the Julian period to assign a unique number to
every day since 1 January 4713 BC. This is the so-called Julian Day
(JD). JD 0 designates the 24 hours from noon UTC on 1 January 4713 BC
to noon UTC on 2 January 4713 BC.

This means that at noon UTC on 1 January AD 2000, JD 2,451,545 will start.

This can be calculated thus:
        From 4713 BC to AD 2000 there are 6712 years.
        In the Julian calendar, years have 365.25 days, so 6712 years
        correspond to 6712*365.25=2,451,558 days. Subtract from this
        the 13 days that the Gregorian calendar is ahead of the Julian
        calendar, and you get 2,451,545.

Often fractions of Julian day numbers are used, so that 1 January AD
2000 at 15:00 UTC is referred to as JD 2,451,545.125.

Note that some people use the term "Julian day number" to refer to any
numbering of days. NASA, for example, use the term to denote the
number of days since 1 January of the current year.

2.12.1. What is the modified Julian day?
----------------------------------------

Sometimes a modified Julian day number (MJD) is used which is
2,400,000.5 less than the Julian day number. This brings the numbers
into a more manageable numeric range and makes the day numbers change
at midnight UTC rather than noon.

MJD 0 thus falls on 17 Nov 1858 (Gregorian) at 00:00:00 UTC.

3. The Hebrew Calendar
----------------------

The current definition of the Hebrew calendar is generally said to
have been set down by the Sanhedrin president Hillel II in
approximately AD 359. The original details of his calendar are,
however, uncertain.

The Hebrew calendar is used for religious purposes by Jews all over
the world, and it is the official calendar of Israel.

The Hebrew calendar is a combined solar/lunar calendar, in that it
strives to have its years coincide with the tropical year and its
months coincide with the synodic months. This is a complicated goal,
and the rules for the Hebrew calendar are correspondingly fascinating.

3.1. What does a Hebrew year look like?
---------------------------------------

An ordinary (non-leap) year has 353, 354, or 355 days.
A leap year has 383, 384, or 385 days.
The three lengths of the years are termed, "deficient", "regular",
and "complete", respectively.

An ordinary year has 12 months, a leap year has 13 months.

Every month starts (approximately) on the day of a new moon.

The months and their lengths are:

          Length in a      Length in a     Length in a
Name      deficient year   regular year    complete year
-------   --------------   ------------    -------------
Tishri          30              30              30
Heshvan         29              29              30
Kislev          29              30              30
Tevet           29              29              29
Shevat          30              30              30
(Adar I         30              30              30)
Adar II         29              29              29
Nisan           30              30              30
Iyar            29              29              29
Sivan           30              30              30
Tammuz          29              29              29
Av              30              30              30
Elul            29              29              29
-------   --------------   ------------    -------------
Total:      353 or 383      354 or 384      355 or 385

The month Adar I is only present in leap years. In non-leap years
Adar II is simply called "Adar".

Note that in a regular year the numbers 30 and 29 alternate; a
complete year is created by adding a day to Heshvan, whereas a
deficient year is created by removing a day from Kislev.

The alteration of 30 and 29 ensures that when the year starts with a
new moon, so does each month.

3.2. What years are leap years?
-------------------------------

A year is a leap year if the number year%19 is one of the following:
0, 3, 6, 8, 11, 14, or 17.

The value for year in this formula is the 'Anno Mundi' described in
section 3.8.

3.3. What years are deficient, regular, and complete?
-----------------------------------------------------

That is the wrong question to ask. The correct question to ask is: When
does a Hebrew year begin? Once you have answered that question (see
section 3.6), the length of the year is the number of days between
1 Tishri in one year and 1 Tishri in the following year.

3.4. When is New Year's day?
----------------------------

That depends. Jews have 4 different days to choose from:

1 Tishri:  "Rosh HaShanah". This day is a celebration of the creation
           of the world and marks the start of a new calendar
           year. This will be the day we shall base our calculations on
           in the following sections.

15 Shevat: "Tu B'shevat". The new year for trees, when fruit tithes
           should be brought.

1 Nisan:   "New Year for Kings". Nisan is considered the first month,
           although it occurs 6 or 7 months after the start of the
           calendar year.

1 Elul:    "New Year for Animal Tithes (Taxes)".

Only the first two dates are celebrated nowadays.

3.5. When does a Hebrew day begin?
----------------------------------

A Hebrew day does not begin at midnight, but at sunset (when 3 stars
are visible).

Sunset marks the start of the 12 night hours, whereas sunrise marks the
start of the 12 day hours. This means that night hours may be longer
or shorter than day hours, depending on the season.

3.6. When does a Hebrew year begin?
-----------------------------------

The first day of the calendary year, Rosh HaShanah, on 1 Tishri is
determined as follows:

1) The new year starts on the day of the new moon that follows the last
   month of the previous year.

2) If the new moon occurs after noon on that day, delay the new year
   by one day. (Because in that case the new crescent moon will not be
   visible until the next day.)

3) If this would cause the new year to start on a Sunday, Wednesday,
   or Friday, delay it by one day. (Because we want to avoid that
   Yom Kippur (10 Tishri) falls on a Friday or Sunday, and that
   Hoshanah Rabba (21 Tishri) falls on a Sabbath (Saturday)).

4) If two consecutive years start 356 days apart (an illegal year
   length), delay the start of the first year by two days.

5) If two consecutive years start 382 days apart (an illegal year
   length), delay the start of the second year by one day.

Note: Rule 4 can only come into play if the first year was supposed
to start on a Tuesday. Therefore a two day delay is used rather that a
one day delay, as the year must not start on a Wednesday as stated in
rule 3.

3.7. When is the new moon?
--------------------------

A calculated new moon is used. In order to understand the
calculations, one must know that an hour is subdivided into 1080
'parts'.

The calculations are as follows:

The new moon that started the year AM 1, occurred 5 hours and 204
parts after sunset (i.e. just before midnight on Julian date 6 October
3761 BC).

The new moon of any particular year is calculated by extrapolating
from this time, using a synodic month of 29 days 12 hours and 793
parts.

3.8. How does one count years?
------------------------------

Years are counted since the creation of the world, which is assumed to
have taken place in 3761 BC. In that year, AM 1 started (AM = Anno
Mundi = year of the world).

In the year AD 1997 we will witness the start of Hebrew year AM 5758.

4. The Islamic Calendar
-----------------------

The Islamic calendar (or Hijri calendar) is a purely lunar
calendar. It contains 12 months that are based on the motion of the
moon, and because 12 synodic months is only 12*29.53=354.36 days, the
Islamic calendar is consistently shorter than a tropical year, and
therefore it shifts with respect to the Christian calendar.

The calendar is based on the Qur'an (Sura IX, 36-37) and its proper
observance is a sacred duty for Muslims.

The Islamic calendar is the official calendar in countries around the
Gulf, especially Saudi Arabia. But other Muslim countries use the
Gregorian calendar for civil purposes and only turn to the Islamic
calendar for religious purposes.

4.1. What does an Islamic year look like?
-----------------------------------------

The names of the 12 months that comprise the Islamic year are:

1. Muharram                          7. Rajab
2. Safar                             8. Sha'ban
3. Rabi' al-awwal (Rabi' I)          9. Ramadan
4. Rabi' al-thani (Rabi' II)        10. Shawwal
5. Jumada al-awwal (Jumada I)       11. Dhu al-Qi'dah
6. Jumada al-thani (Jumada II)      12. Dhu al-Hijjah

(Due to different transliterations of the Arabic alphabet, other
spellings of the months are possible.)

Each month starts when the lunar crescent is first seen (by an actual
human being) after a new moon.

Although new moons may be calculated quite precisely, the actual
visibility of the crescent is much more difficult to predict. It
depends on factors such a weather, the optical properties of the
atmosphere, and the location of the observer. It is therefore very
difficult to give accurate information in advance about when a new
month will start.

Furthermore, some Muslims depend on a local sighting of the moon,
whereas others depend on a sighting by authorities somewhere in the
Muslim world. Both are valid Islamic practices, but they may lead to
different starting days for the months.

4.2. So you can't print an Islamic calendar in advance?
-------------------------------------------------------

Not a reliable one. However, calendars are printed for planning
purposes, but such calendars are based on estimates of the visibility
of the lunar crescent, and the actual month may start a day earlier or
later than predicted in the printed calendar.

Different methods for estimating the calendars are used.

Some sources mention a crude system in which all odd numbered months
have 30 days and all even numbered months have 29 days with an extra
day added to the last month in 'leap years' (a concept otherwise
unknown in the calendar). Leap years could then be years in which the
number year%30 is one of the following: 2, 5, 7, 10, 13, 16, 18, 21,
24, 26, or 29. (This is the algorithm used in the calendar program of
the Gnu Emacs editor.)

Such a calendar would give an average month length of 29.53056 days,
which is quite close to the synodic month of 29.53059 days, so *on the
average* it would be quite accurate, but in any given month it is
still just a rough estimate.

Better algorithms for estimating the visibility of the new moon have
been devised. One such algorithm is implemented in a program called
'Islamic Timer' by professor Waleed A. Muhanna. Interested readers may
find the program on the World Wide Web at
http://www.cob.ohio-state.edu/facstf/homepage/muhanna/IslamicTimer.html

4.3. How does one count years?
------------------------------

Years are counted since the Hijra, that is, Mohammed's flight to
Medina, which is assumed to have taken place 16 July AD 622 (Julian
calendar). On that date AH 1 started (AH = Anno Hegirae = year of the Hijra).

In the year AD 1997 we have witnessed the start of Islamic year AH 1418.

Note that although only 1997-622=1376 years have passed in the
Christian calendar, 1417 years have passed in the Islamic calendar,
because its year is consistently shorter (by about 11 days) than the
tropical year used by the Christian calendar.

--- End of part 2 ---

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