Section 2 of 2 - Prev - Next

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


2.15.2. 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.


2.15.3. 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 is named after Aloysius Lilius mentioned in
section 2.2.


2.16. 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 3761 BC. In that year, AM 1 started (AM = Anno
Mundi = year of the world).

In the year AD 2003 we witness the start of Hebrew year AM 5764.


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 2003 we have witnessed the start of Islamic year AH 1424.

Note that although only 2003-622=1381 years have passed in the
Christian calendar, 1423 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?
---------------------------------------------

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 1999 (1420 AH) the rule has been as follows: On the 29th day of
an Islamic month, the times when the sun and the moon set are
compared. If the sun sets before the moon, the next day will be the
first of a new month; but if the moon sets before the sun, 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.


5. The Persian Calendar
-----------------------

The Persian calendar is a solar calendar with a starting point that
matches that of the Islamic calendar. Its origin 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 2003 we have therefore witnessed
the start of Persian year 1382, but the start of Islamic year 1424.


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 ---

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