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From: claus@tondering.dk (Claus Tondering)
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Subject: Calendar FAQ, v. 2.7 (modified 15 January 2005) Part 2/3
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Summary: This posting contains answers to Frequently Asked Questions about
the Christian, Hebrew, Persian, Islamic, Chinese and various
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FREQUENTLY ASKED QUESTIONS ABOUT
CALENDARS
Part 2 of 3
Version 2.7 - 15 January 2005
Copyright and disclaimer
------------------------
This document is Copyright (C) 2005 by Claus Tondering.
E-mail: claus@tondering.dk. (Please include the word
"calendar" in the subject line.)
The document may be freely distributed, provided this
copyright notice is included and no money is charged for
the document.
This document is provided "as is". No warranties are made as
to its correctness.
Introduction
------------
This is the calendar FAQ. Its purpose is to give an overview
of the Christian, Hebrew, Persian, and Islamic calendars in
common use. It will provide a historical background for the
Christian calendar, plus an overview of the French
Revolutionary calendar, the Maya calendar, and the Chinese
calendar.
Comments are very welcome. My e-mail address is given above.
Index:
------
In part 1 of this document:
1. What Astronomical Events Form the Basis of Calendars?
1.1. What are equinoxes and solstices?
2. The Christian Calendar
2.1. What is the Julian calendar?
2.1.1. What years are leap years?
2.1.2. What consequences did the use of the Julian
calendar have?
2.2. What is the Gregorian calendar?
2.2.1. What years are leap years?
2.2.2. Isn't there a 4000-year rule?
2.2.3. Don't the Greek do it differently?
2.2.4. When did country X change from the Julian to
the Gregorian calendar?
2.3. What day is the leap day?
2.4. What is the Solar Cycle?
2.5. What is the Dominical Letter?
2.6. What day of the week was 2 August 1953?
2.7. When can I reuse my 1992 calendar?
2.8. What is the Roman calendar?
2.7.1. How did the Romans number days?
2.9. What is the proleptic calendar?
2.10. Has the year always started on 1 January?
2.11. Then what about leap years?
2.12. What is the origin of the names of the months?
In part 2 of this document:
2.13. What is Easter?
2.13.1. When is Easter? (Short answer)
2.13.2. When is Easter? (Long answer)
2.13.3. What is the Golden Number?
2.13.4. How does one calculate Easter then?
2.13.5. What is the Epact?
2.13.6. How does one calculate Gregorian Easter then?
2.13.7. Isn't there a simpler way to calculate Easter?
2.13.8. Isn't there an even simpler way to calculate
Easter?
2.13.9. Is there a simple relationship between two
consecutive Easters?
2.13.10. How frequently are the dates for Easter repeated?
2.13.11. What about Greek Orthodox Easter?
2.13.12. Did the Easter dates change in 2001?
2.14. How does one count years?
2.14.1. How did Dionysius date Christ's birth?
2.14.2. Was Jesus born in the year 0?
2.14.3. When does the 3rd millennium start?
2.14.4. What do AD, BC, CE, and BCE stand for?
2.15. What is the Indiction?
2.16. What is the Julian period?
2.16.1. Is there a formula for calculating the Julian
day number?
2.16.2. What is the modified Julian day number?
2.16.3. What is the Lilian day number?
2.17. What is the correct way to write dates?
3. The Hebrew Calendar
3.1. What does a Hebrew year look like?
3.2. What years are leap years?
3.3. What years are deficient, regular, and complete?
3.4. When is New Year's day?
3.5. When does a Hebrew day begin?
3.6. When does a Hebrew year begin?
3.7. When is the new moon?
3.8. How does one count years?
4. The Islamic Calendar
4.1. What does an Islamic year look like?
4.2. So you can't print an Islamic calendar in advance?
4.3. How does one count years?
4.4. When will the Islamic calendar overtake the Gregorian
calendar?
4.5. Doesn't Saudi Arabia have special rules?
5. The Persian Calendar
5.1. What does a Persian year look like?
5.2. When does the Persian year begin?
5.3. How does one count years?
5.4. What years are leap years?
In part 3 of this document:
6. The Week
6.1. What is the origin of the 7-day week?
6.2. What do the names of the days of the week mean?
6.3. What is the system behind the planetary day names?
6.4. Has the 7-day week cycle ever been interrupted?
6.5. Which day is the day of rest?
6.6. What is the first day of the week?
6.7. What is the week number?
6.8. How can I calculate the week number?
6.9. Do weeks of different lengths exist?
7. The French Revolutionary Calendar
7.1. What does a Republican year look like?
7.2. How does one count years?
7.3. What years are leap years?
7.4. How does one convert a Republican date to a Gregorian one?
8. The Maya Calendar
8.1. What is the Long Count?
8.1.1. When did the Long Count start?
8.2. What is the Tzolkin?
8.2.1. When did the Tzolkin start?
8.3. What is the Haab?
8.3.1. When did the Haab start?
8.4. Did the Mayas think a year was 365 days?
9. The Chinese Calendar
9.1. What does the Chinese year look like?
9.2. What years are leap years?
9.3. How does one count years?
9.4. What is the current year in the Chinese calendar?
10. Frequently Asked Questions about this FAQ
10.1. Why doesn't the FAQ describe calendar X?
10.2. Why doesn't the FAQ contain information X?
10.3. Why don't you reply to my e-mail?
10.4. How do I know that I can trust your information?
10.5. Can you recommend any good books about calendars?
10.6. Do you know a web site where I can find information
about X?
11. Date
2.13. 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.
2.13.1. When is Easter? (Short answer)
--------------------------------------
Easter Sunday is the first Sunday after the first full moon after
vernal equinox.
2.13.2. 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 Paschal
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 calculate 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.
2.13.3. 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 mod 19)+1
In years which have the same Golden Number, the new moon will fall on
(approximately) the same date. The Golden Number is sufficient to
calculate the Paschal full moon in the Julian calendar.
2.13.4. How does one calculate Easter then?
-------------------------------------------
Under the Julian calendar the method was simple. If you know the
Golden Number of the year, you can find the Paschal full moon in this
table:
Golden Golden Golden
Number Full moon Number Full moon Number Full moon
------------------ ------------------ ------------------
1 5 April 8 18 April 15 1 April
2 25 March 9 7 April 16 21 March
3 13 April 10 27 March 17 9 April
4 2 April 11 15 April 18 29 March
5 22 March 12 4 April 19 17 April
6 10 April 13 24 March
7 30 March 14 12 April
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.
But under the Gregorian calendar, things became much more complicated.
One of the changes made in the Gregorian calendar reform was a
modification of the way Easter was calculated. There were two reasons
for this. First, the 19 year cycle of the phases of moon (the Metonic
cycle) was known not to be perfect. Secondly, the Metonic cycle fitted
the Gregorian calendar year worse than it fitted the Julian calendar
year.
It was therefore decided to base Easter calculations on the so-called
Epact.
2.13.5. 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, the Epact is the age of the moon on 22 March.
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)) mod 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.
In the Gregorian calendar reform, some modifications were 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:
JulianEpact = (11 * (GoldenNumber-1)) mod 30
2) Calculate the so-called "Solar Equation":
S = (3*century)/4
The Solar Equation is an expression of the difference between the
Julian and the Gregorian calendar. The value of S increases by one
in every century year that is not a leap year.
(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 2.14.3.)
3) Calculate the so-called "Lunar Equation":
L = (8*century + 5)/25
The Lunar Equation is an expression of the difference between the
Julian calendar and the Metonic cycle. The value of L increases by
one 8 times every 2500 years.
4) Calculate the Gregorian epact thus:
GregorianEpact = JulianEpact - S + L + 8
The number 8 is a constant that calibrates the starting point of
the Gregorian Epact so that it matches the actual age of the moon
on new year's day. Actually, this constant should have been 9, but
8 was probably chosen as a safety precaution; the calculation was
known to be inaccurate, and the sentiment was that it was better to
celebrate Easter too late than too early.
5) Add or subtract 30 until GregorianEpact 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 mod 19 + 1 = 17
1) JulianEpact = (11 * (17-1)) mod 30 = 26
2) S = (3*20)/4 = 15
3) L = (8*20 + 5)/25 = 6
4) GregorianEpact = 26 - 15 + 6 + 8 = 25
5) No adjustment is necessary
The Epact for 1992 was 25.
2.13.6. How does one calculate Gregorian Easter then?
-----------------------------------------------------
Look up the Epact 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
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.
2.13.7. 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 mod 19
For the Julian calendar:
I = (19*G + 15) mod 30
J = (year + year/4 + I) mod 7
For the Gregorian calendar:
C = year/100
H = (C - C/4 - (8*C+13)/25 + 19*G + 15) mod 30
I = H - (H/28)*(1 - (29/(H + 1))*((21 - G)/11))
J = (year + year/4 + I + 2 - C + C/4) mod 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) as
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 Paschal full moon (a number between -6 and 28)
2.13.8. Isn't there an even simpler way to calculate Easter?
------------------------------------------------------------
If we confine ourselves to the years 1900-2099 and consider only the
Gregorian calendar, the formulas of the previous section can be
further simplified thus:
H = (24 + 19*(year mod 19)) mod 30
I = H - H/28
J = (year + year/4 + I - 13) mod 7
L = I - J
EasterMonth = 3 + (L + 40)/44
EasterDay = L + 28 - 31*(EasterMonth/4)
(Again, the divisions are integer divisions, in which remainders are
discarded.)
2.13.9. 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 1800-2200, 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.
2.13.10. 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. Calculating this is not as simple as for the
Julian calendar, but the number 5,700,000 turns out to be 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)
2.13.11. What about Greek Orthodox 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 the basis for calculating Easter,
rather than the "official" full moon described in the previous
sections. And they chose the meridian of Jerusalem to serve as
definition of when a Sunday starts. However, except for some sporadic
use the 1920s, this system was never adopted in practice.
2.13.12. Did the Easter dates change in 2001?
---------------------------------------------
No.
At a meeting in Aleppo, Syria (5-10 March 1997), organised by the
World Council of Churches and the Middle East Council of Churches,
representatives of several churches and Christian world communions
suggested that the discrepancies between Easter calculations in the
Western and the Eastern churches could be resolved by adopting
astronomically accurate calculations of the vernal equinox and the
full moon, instead of using the algorithm presented in section 2.13.6.
The meridian of Jerusalem should be used for the astronomical
calculations.
The new method for calculating Easter should have taken effect from
the year 2001. In that year the Julian and Gregorian Easter dates
coincided (on 15 April Gregorian/2 April Julian), and it would
therefore be a reasonable starting point for the new system.
However, the Eastern churches (especially the Russian Orthodox Church)
are reluctant to change, having already experienced a schism in the
calendar question. So nothing will happen in the near future.
If the new system were introduced, churches using the Gregorian
calendar will hardly notice the change. Only once during the period
2001-2025 would these churches note a difference: In 2019 the
Gregorian method gives an Easter date of 21 April, but the proposed
new method gives 24 March.
Note that the new method makes an Easter date of 21 March possible.
This date was not possible under the Julian or Gregorian algorithms.
(Under the new method, Easter will fall on 21 March in the year 2877.
You're all invited to my house on that date!)
2.14. 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 (see
section 2.14.1 for a couple of theories). 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 section 2.14.2.
The earliest uses of BC dating are found in the works of the Venerable
Bede (673-735).
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.
2.14.1. How did Dionysius date Christ's birth?
----------------------------------------------
There are quite a few theories about this. And many of the theories
are presented as if they were indisputable historical fact.
Here are two theories that I personally consider likely:
1. According to the Gospel of Luke (3:1 & 3:23) Jesus was "about
thirty years old" shortly after "the fifteenth year of the reign of
Tiberius Caesar". Tiberius became emperor in AD 14. If you combine
these numbers you reach a birthyear for Jesus that is strikingly
close to the beginning of our year reckoning. This may have been
the basis for Dionysius' calculations.
2. Dionysius' original task was to calculate an Easter table. In the
Julian calendar, the dates for Easter repeat every 532 years (see
section 2.13.10). The first year in Dionysius' Easter tables is AD
532. Is it a coincidence that the number 532 appears twice here? Or
did Dionysius perhaps fix Jesus' birthyear so that his own Easter
tables would start exactly at the beginning of the second Easter
cycle after Jesus' birth?
2.14.2. 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). In our calendar, 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, as described in section 2.14, our year reckoning was
established by Dionysius Exiguus in the 6th century. Dionysius let
the year AD 1 start one week after what he believed to be Jesus'
birthday. But Dionysius' calculations were wrong. The Gospel of
Matthew tells us that Jesus was born under the reign of king Herod the
Great, who 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.
2.14.3. When did the 3rd millennium start?
------------------------------------------
The first millennium started in AD 1, so the millennia are counted in
this manner:
1st millennium: 1-1000
2nd millennium: 1001-2000
3rd millennium: 2001-3000
Thus, the 3rd millennium and, similarly, the 21st century started on
1 Jan 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. Furthermore, the change 1999/2000 is obviously much
more spectacular than the change 2000/2001.
Let me propose a few compromises:
Any 100-year period is a century. Therefore the period from 23 June 2004
to 22 June 2104 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, the 21st century started in 2001, but the 2000s started in
2000.
2.14.4. What do AD, BC, CE, and BCE stand for?
----------------------------------------------
Years before the birth of Christ are in English traditionally
identified using the abbreviation BC ("Before Christ").
Years after the birth of Christ are traditionally identified using the
Latin abbreviation AD ("Anno Domini", that is, "In the Year of the
Lord").
Some people, who want to avoid the reference to Christ that is implied
in these terms, prefer the abbreviations BCE ("Before the Common Era"
or "Before the Christian Era") and CE ("Common Era" or "Christian Era").
2.15. 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 ceased to be used in
1806.
The Indiction may be calculated thus:
Indiction = (year + 2) mod 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.
2.16. 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 2005 is thus year 6718 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 2.15), the Golden Number (see section 2.13.3) 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
started.
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, uses the term to denote the
number of days since 1 January of the current year, counting 1 January
as day 1.
2.16.1. Is there a formula for calculating the Julian day number?
-----------------------------------------------------------------
Try this one (the divisions are integer divisions, in which remainders
are discarded):
a = (14-month)/12
y = year+4800-a
m = month + 12*a - 3
For a date in the Gregorian calendar:
JD = day + (153*m+2)/5 + y*365 + y/4 - y/100 + y/400 - 32045
For a date in the Julian calendar:
JD = day + (153*m+2)/5 + y*365 + y/4 - 32083
JD is the Julian day number that starts at noon UTC on the specified
date.
The algorithm works fine for AD dates. If you want to use it for BC
dates, you must first convert the BC year to a negative year (e.g.,
10 BC = -9). The algorithm works correctly for all dates after 4800 BC,
i.e. at least for all positive Julian day numbers.
To convert the other way (i.e., to convert a Julian day number, JD,
to a day, month, and year) these formulas can be used (again, the
divisions are integer divisions):
For the Gregorian calendar:
a = JD + 32044
b = (4*a+3)/146097
c = a - (b*146097)/4
For the Julian calendar:
b = 0
c = JD + 32082
Then, for both calendars:
d = (4*c+3)/1461
e = c - (1461*d)/4
m = (5*e+2)/153
day = e - (153*m+2)/5 + 1
month = m + 3 - 12*(m/10)
year = b*100 + d - 4800 + m/10
2.16.2. What is the modified Julian day number?
-----------------------------------------------
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 started on 17 Nov 1858 (Gregorian) at 00:00:00 UTC.
2.16.3. What is the Lilian day number?
--------------------------------------
The Lilian day number is similar to the Julian day number, except that
Lilian day number 1 started at midnight on the first day of the
Gregorian calendar, that is, 15 October 1582.
The Lilian day number was invented by Bruce G. Ohms of IBM in 1986. It
is named after Aloysius Lilius mentioned in section 2.2.
2.17. What is the correct way to write dates?
---------------------------------------------
The answer to this question depends on what you mean by "correct".
Different countries have different customs.
Most countries use a day-month-year format, such as:
25.12.1998 25/12/1998 25/12-1998 25.XII.1998
In the U.S.A. a month-day-year format is common:
12/25/1998 12-25-1998
International standard ISO-8601 mandates a year-month-day format,
namely either 1998-12-25 or 19981225. This format is gaining
popularity in some countries.
In all of these systems, the first two digits of the year are
frequently omitted:
25.12.98 12/25/98 98-12-25
This confusion leads to misunderstandings. What is 02-03-04? To most
people it is 2 Mar 2004; to an American it is 3 Feb 2004; and to a
person using the international standard it would be 4 Mar 2002.
If you want to be sure that people understand you, I recommend that
you
* write the month with letters instead of numbers, and
* write the years as 4-digit numbers.
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 mod 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-calendar day does not begin at midnight, but at either sunset
or when three medium-sized stars should be visible, depending on the
religious circumstance.
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 occurs about
354 days (or 384 days if the previous year was a leap year) after
1 Tishri 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.
Note that 18:00 Jerusalem time (15:39 UTC) is used instead of sunset in
all these calculations.
3.8. How does one count years?
------------------------------
Years are counted since the creation of the world, which is assumed to
have taken place in the autumn of 3760 BC. In that year, after less
than a week belonging to AM 1, AM 2 started (AM = Anno Mundi = year of
the world).
In the year AD 2005 we shall witness the start of Hebrew year AM 5766.
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 see section 4.5). 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 as 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 mod 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, and a number of computer programs with this purpose
exist.
4.3. How does one count years?
------------------------------
Years are counted since the Hijra, that is, Mohammed's emigration to
Medina in AD 622. On 16 July (Julian calendar) of that year, AH 1
started (AH = Anno Hegirae = year of the Hijra).
In the year AD 2005 we shall witness the start of Islamic year AH 1426.
Note that although only 2005-622=1383 years have passed in the
Christian calendar, 1425 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.
4.4. When will the Islamic calendar overtake the Gregorian calendar?
--------------------------------------------------------------------
As the year in the Islamic calendar is about 11 days shorter than the
year in the Christian calendar, the Islamic years are slowly gaining
in on the Christian years. But it will be many years before the two
coincide. The 1st day of the 5th month of AD 20874 in the Gregorian
calendar will also be (approximately) the 1st day of the 5th month of
AH 20874 of the Islamic calendar.
4.5. Doesn't Saudi Arabia have special rules?
---------------------------------------------
[For civil (but not religious) purposes,] Saudi Arabia doesn't rely on
a visual sighting of the crescent moon to fix the start of a new
month. Instead they base their calendar on a calculated astronomical
moon.
Since 2002 (AH 1423) the rule has been as follows: If on the 29th day
of an Islamic month,
* the geocentric conjunction (that is, the new moon as seen from
the centre of the earth) occurs before sunset,
* the moon sets after the sun, and
* the weather is nice,
then the next day will be the first of a new month; otherwise the next
day will be the last (30th) of the current month.
The times for the setting of the sun and the moon are calculated for
the coordinates of Mecca.
[The rules above are taken from http://www.jas.org.jo/sau.html.
The English text on that web page does not include the statement about
the weather having to be nice. This does, however, appear to be a
requirement according to the Arabic text found on the same web page.
If anybody can confirm or refute this, please let me know.]
5. The Persian Calendar
-----------------------
The Persian calendar is a solar calendar with a starting point that
matches that of the Islamic calendar. Apart from that, the two
calendars are not related. The origin of the Persian calendar can be
traced back to the 11th century when a group of astronomers (including
the well-known poet Omar Khayyam) created what is known as the Jalaali
calendar. However, a number of changes have been made to the calendar
since then.
The current calendar has been used in Iran since 1925 and in
Afghanistan since 1957. However, Afghanistan used the Islamic calendar
in the years 1999-2002.
5.1. What does a Persian year look like?
----------------------------------------
The names and lengths of the 12 months that comprise the Persian year
are:
1. Farvardin (31 days) 7. Mehr (30 days)
2. Ordibehesht (31 days) 8. Aban (30 days)
3. Khordad (31 days) 9. Azar (30 days)
4. Tir (31 days) 10. Day (30 days)
5. Mordad (31 days) 11. Bahman (30 days)
6. Shahrivar (31 days) 12. Esfand (29/30 days)
(Due to different transliterations of the Persian alphabet, other
spellings of the months are possible.) In Afghanistan the months are
named differently.
The month of Esfand has 29 days in an ordinary year, 30 days in a leap
year.
5.2. When does the Persian year begin?
--------------------------------------
The Persian year starts at vernal equinox. If the astronomical vernal
equinox falls before noon (Tehran true time) on a particular day, then
that day is the first day of the year. If the astronomical vernal
equinox falls after noon, the following day is the first day of the
year.
5.3. How does one count years?
------------------------------
As in the Islamic calendar (section 4.3), years are counted since
Mohammed's emigration to Medina in AD 622. At vernal equinox of that
year, AP 1 started (AP = Anno Persico/Anno Persarum = Persian year).
Note that contrary to the Islamic calendar, the Persian calendar
counts solar years. In the year AD 2005 we shall therefore witness
the start of Persian year 1384, but the start of Islamic year 1426.
5.4. What years are leap years?
-------------------------------
Since the Persian year is defined by the astronomical vernal equinox,
the answer is simply: Leap years are years in which there are 366 days
between two Persian new year's days.
However, basing the Persian calendar purely on an astronomical
observation of the vernal equinox is rejected by many, and a few
mathematical rules for determining the length of the year have been
suggested.
The most popular (and complex) of these is probably the following:
The calendar is divided into periods of 2820 years. These periods are
then divided into 88 cycles whose lengths follow this pattern:
29, 33, 33, 33, 29, 33, 33, 33, 29, 33, 33, 33, ...
This gives 2816 years. The total of 2820 years is achieved by
extending the last cycle by 4 years (for a total of 37 years).
If you number the years within each cycle starting with 0, then leap
years are the years that are divisible by 4, except that the year 0 is
not a leap year.
So within, say, a 29 year cycle, this is the leap year pattern:
Year Year Year Year
0 Ordinary 8 Leap 16 Leap 24 Leap
1 Ordinary 9 Ordinary 17 Ordinary 25 Ordinary
2 Ordinary 10 Ordinary 18 Ordinary 26 Ordinary
3 Ordinary 11 Ordinary 19 Ordinary 27 Ordinary
4 Leap 12 Leap 20 Leap 28 Leap
5 Ordinary 13 Ordinary 21 Ordinary
6 Ordinary 14 Ordinary 22 Ordinary
7 Ordinary 15 Ordinary 23 Ordinary
This gives a total of 683 leap years every 2820 years, which
corresponds to an average year length of 365 683/2820 = 365.24220
days. This is a better approximation to the tropical year than the
365.2425 days of the Gregorian calendar.
The current 2820 year period started in the year AP 475 (AD 1096).
This "mathematical" calendar currently coincides closely with the
purely astronomical calendar. In the years between AP 1244 and 1531
(AD 1865 and 2152) a discrepancy of one day is seen twice, namely in
AP 1404 and 1437 (starting at vernal equinox of AD 2025 and 2058).
However, outside this period, discrepancies are more frequent.
--- End of part 2 ---