• Fossil calendars: 420 million years ago, a lunar month was only 9 short days.
• Julian Day Number (JDN) Counting days in the simplest of all calendars.
• The Week has not always been a period of seven days.
• Egyptian year of 365 days: Back to the same season after over 1500 years.
• Heliacal rising of Sirius: Sothic dating.
• Coptic Calendar: Reformed Egyptian calendar based on the Julian year.
• The Julian Calendar: Year starts March 25. Every fourth year is a leap year.
• Anno Domini: Counting roughly from the birth of Jesus Christ.
• Easter Day is defined as the first Sunday after the Paschal full moon.
• The Gregorian Calendar: Multiples of 100 not divisible by 400 aren't leap years.
• Zoroastrian Calendar.
• The Zodiac:  Zodiacal signs and constellations.  Precession of equinoxes.
• The Muslim Calendar:  The Islamic (Hijri) Calendar (AH = Anno Hegirae).
• The Jewish Calendar:  An accurate lunisolar calendar, set down by Hillel II.
• The Chinese Calendar.
• The Japanese Calendar.
• Mayan System(s):  Haab (365), Tzolkin (260), Round (18980), Long Count.
• Indian Calendar:  The Sun goes through a zodiacal sign in a solar month.
• Post-Gregorian Calendars:  Painless  improvements to the secular calendar.

Related articles on this site:

• Scientific year = 31557600 atomic seconds  (» Julian year of  365.25 solar days).
• Mean Gregorian month: either 30.436875 days, or 30.4587294742534...
• Switching calendars: Newton was not born the year Galileo died.
• Roman Numerals.
• Nemesis: A distant companion to the Sun could explain extinction periodicity.

Before universal calendars became dominant, dates were often recorded with respect to the beginnings of reigns.  Recovering the global chronology from such records can be a major source of headaches for historians, who may need the help of classic works like "L'Art de vérifier les dates des faits historiques" ("On the Art of Verifying the Dates of Historical Events", first published by the Benedictines in 1750).

Here's how to express the 89^th day of the year 1956 (CE) in various calendars.  Note the "double dating" [sic] in the Julian calendar for dates between January 1 and March 24, due to the fact that the New Year "Old Style" (O.S.) started on March 25.

• International ISO 8601 format:   1956-03-29
• ISO Week Date:   1956-W13-3   (Thursday, week 13, 1956).
• Gregorian:   Thursday, March 29, 1956 CE (= Common Era).
• Julian:   Thursday, March 16, 1955/1956 AD (= Anno Domini).
• Julian (Roman style):   A.D. XVII KAL. APR. MMDCCIX A.U.C.
• French revolutionary:   9 Germinal, an 164 (Nonidi, Décade I).
• Coptic:   Ptiou, 20 Paremhat, 1672 AM (= Anno Martyrum).
• Hebrew:   Yom hamishi, 17 Nisan, 5716 AM (= Anno Mundi).
• Islamic:   Yaum al-hamis, 16 Sha'ban, 1375 AH (= Anno Hegirae).
• Julian Day (at noon):   2435562 JD.
• Modified Julian Day Number (since midnight):   35561 MJDN.
• Mayan:   7 Cumku 5 Cauac (Long Count:  12.17.2.7.19).

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(2003-01-03)   Fossil Calendars
Over long periods, calendar ratios do change.

Modern nautilus shells invariably show about 30 daily growth lines between their chamber partitions, called septa, whose development is synchronized with the actual lunar month (currently about 29.5305889 days).

Nautiloids first appeared about 420 million years ago, when the solar day was only about 21 hours [1 hour = 3600 atomic SI seconds].  The fossil record shows that the earliest nautiloids had only 9 growth lines between septa:  420 million years ago, there were only about 9 days (of 21 hours) in a lunar month !

The distance to the Moon was only 40% of what it is today, so the apparent diameter of the Moon was about 2½ times what it is now.  Total solar eclipses were more common than partial eclipses today.

Even the rate of recession of the Moon does not remain constant over the ages.  The strength of tidal effects is strongly dependent on the configuration of the continents (and/or the ocean floor) which is extremely variable over geological time periods.  Currently, the Moon recesses from the Earth at the comparatively rapid rate of 38.2(7) mm per year (Dickey et al., 1994).  The paleontological study of so-called tidally laminated sediments (also called tidal rhythmites) has shown conclusively that this recession speed has varied greatly, but it was typically much slower in the distant past.

If this wasn't so, the Earth-Moon system couldn't have formed at the time indicated by radioactive dating (about 4½ billion years ago).  Some models explain the formation of the Earth-Moon system by a collision of the young Earth with an object 10 times smaller than itself.

A weaker tidal braking in the past would seem like a paradox at first, since a closer Moon should have produced stronger tides.  However, this general trend could be more than compensated by the large differences in the heights of the tides around different configurations of land masses.  This effect is commonly observed when comparing different coastlines, and it would dominate globally as the continents drift.  Everything seems to indicate that tidal effects are currently way above average, so the current rate of recession is a poor indication of what happened in the distant past.

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(2002-12-30)   JD = Julian Day [Number] & Absolute Time
Counting days and converting days to absolute time...

From a scientific perspective, a calendar is not about measuring time, it's about counting actual solar days.  No amount of averaging will ever be able to equate the two concepts over long periods of time, because the rotation of the Earth on its own axis is steadily slowing down (due to tidal braking):  The average length of a day currently increases by about 2 milliseconds each century.  This observation is a fairly recent discovery which affects the continuing accuracy of any calendar whose structure is based on some definite value of the solar year and/or the lunar month expressed in actual days.  (The scientific day unit of precisely 86400 atomic SI seconds is not directly relevant to calendars.)

This flaw is not present in the Julian Day numbering scheme, arguably the simplest of all calendars, because no attempt is made at counting anything but days --not years, not months, just days.  However, the lengthening of the astronomical day may not be neglected when absolute time differences (in atomic seconds) are to be obtained from calendar dates, in this or any other calendar.

The following definition of the Julian Day Number (JDN) has been given in 1997 by the 23rd International Astronomical Union General Assembly:

The Julian day number associated with the solar day is the number assigned to a day in a continuous count of days beginning with the Julian day number 0 assigned to the day starting at Greenwich mean noon on 1 January 4713 BC, Julian proleptic calendar -4712.

The JDN is thus a proper calendar, a well-defined method for counting days, fully specified by the JDN assigned to some specific day in a known calendar.  It's closely related to the Julian Date (JD), which is a continuous measure of time obtained by adding to the JDN the fraction of a day elapsed since noon GMT.

When it's more convenient to have days starting at midnight (GMT), it's best to use the so-called Modified Julian Date (MJD) which is equal to the Julian Date minus 2400 000.5.  In other words, the Modified Julian Date is the number of solar days elapsed since midnight UTC on Wednesday November 17, 1858 (which is Julian Day 240000).

This numbering scheme was invented aound 1583, in the wake of the Gregorian reform, by Joseph Justus Scaliger (1540-1609).  Scaliger put the origin in 4713 BC because this year predates all our recorded history, and it can be construed as a common beginning to the following three noteworthy cycles:

• The 28 year cycle of the Julian calendar.
  The pattern of weekdays and leap years repeats with a 28 year cycle in the Julian Calendar (it's 400 years with the modern Gregorian calendar). 
• The 19 year Metonic cycle.
  19 tropical years (about 6939.602 days) are only two hours short of 235 lunar months (about 6939.688 days).  For any reasonably accurate solar calendar, a given phase of the Moon will thus occur [nearly] at the same calendar date after a period of 19 years. 
• The 15 year Roman indiction cycle.
  This tax cycle was only abolished in 1806.  It had been introduced on September 1, 312, by Constantine the Great (c.274-337), the founder of Constantinople (modern Istanbul) and the first Roman emperor to become a Christian (baptized on his deathbed).  The indiction number was used as a calendrical era (e.g., "third year of the fourth indiction").

The lowest common multiple of these is a period of 7980 years, which is known as Scaliger's Julian period.  Scaliger reportedly named the thing after his late father (Julius Caesar Scaliger, 1484-1558), so the etymological connection with the Julian calendar (named after emperor Julius Caesar) is an indirect one.

Counting Seconds:

Leap Seconds   |   Julian Date

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(2003-01-21)   The 7-Day Week and its Ancestors
The fundamental social cycle has not always been a week of 7 days.

Second only to the natural daily rythm, a regular man-made cycle of 4 to 10 days has always governed human activity everywhere, throughout recorded history (and probably well before that).  This period has not always been the familiar week of 7 days, though.  Here are some examples:

Name	Days	When / What Circumstances	Who / Where
Decan	10	Antiquity	Egypt
Week	7	Antiquity	Jews, Persians
	8	Antiquity	Rome
	7	Since 1st Cent. AD / Persian Astrology Since AD 321 (officially) / Christianity
	9	Until 1385 (officially)	Pagan Lithuania
Décade	10	1793-10-24 to 1806-01-01	France
	5	1929 to 1931	Soviet Union
	6	1931-09-01 to 1940-06-26
Week	7	Modern Times	Worldwide

 

Etymology

	Celestial	French	English	Norse
0	Sun	Dimanche (Lord's day)	Sunday
1	Moon, Lune	Lundi	Monday
2	Mars	Mardi	Tuesday	Tiw's day
3	Mercury	Mercredi	Wednesday	Woden's day
4	Jupiter	Jeudi	Thursday	Thor's day
5	Venus	Vendredi	Friday	Freya's day
6	Saturn	Samedi	Saturday

Our seven-day week   |   Seven Day week   |   The week   |   360-day artifact   |   Days of the Week

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(2002-12-29)   The Egyptian Calendar & Calendar Creep
This early solar calendar paved the road for its successors.

The ancient Egyptian civilization lasted longer than any other.  It had a solar calendar whose year consisted of 12 months of 30 days (3 decans of 10 days each) and 5 additional "yearly days" (epagomenes), for a total of 365 days.

The myth was that Nut, goddess of the Sky, was separated from her lover Geb, god of the Earth, and cursed with barrenness:  She could not give birth in "any month of the year".  Thoth, moon-god of time and measure, decided to help Nut and Geb.  In a game of dice with the reigning gods, he won 5 extra days not belonging to any particular month, which Nut used to produce 5 children, including Isis and Osiris.

Egyptian astronomers knew that a period of 365 days was about ¼ day short of an actual tropical year, but an intercalary day was never added, and the calendar was allowed to drift through the seasons.

A drift of a fixed calendar date through the seasons is a flaw of a solar calendar called calendar creep.  The Egyptian calendar had a severe case of this, but it was originally designed to match the 3 seasons of the Nile (4 months each):

• Akhet :   "Inundation".
• Proyet, Peret, or Poret :   "Emergence", "Winter", or "Growing Season".
• Shomu or Shemu :   "Harvest", "Summer", or "Low Water".

Although the Egyptian months have specific names (tabulated below, in our discussion of the modern Coptic calendar), they are commonly denoted by their ranks within those fictitious calendar "seasons", whose own names are either transliterated or translated:  Third month of Akhet, first month of Harvest, etc.

The astronomical event which was once observed to mark the beginning of the actual  Inundation (as opposed to the calendrical one) was the so-called heliacal rising of Sirius, the brightest star in the sky.  This is to say that Sirius, the Dog Star, rises with the Sun at that time of year still known as the Dog Days of Summer (Sirius belongs to the "Great Dog" constellation, Canis Major).

1461 Egyptian years are equal to 1460 years of 365¼ days (the length of what would become the Julian year).  This period of 533265 days has been dubbed a Sothic period, because Sothis is the Greek name of Sirius, called Sopdet [spdt] by the Egyptians.  The Egyptian civilization lived through several such cycles...  (It has been reported that ancient Egyptians also had another "sacred" calendar based on a year of 365¼ days, but we found no evidence to support this claim.)

A period of 533265 days does not quite bring the Egyptian calendar back to the same point with respect to the actual seasons, because an actual tropical year is not exactly equal to 365.25 days:  It's now closer to 365.2422 days, which would imply a period of 1508 Egyptian years (1507 tropical years) between successive returns of the Egyptian calendar to the same seasonal point.

However, the braking effect of the tides continuously increases the length of the day (whereas the duration of any flavor of astronomical year is much steadier).  As longer days mean fewer days in a year, the number of days in a year decreases with time, and was thus slightly greater in the past than now:  In 3000 BC, the tropical year was about 365.24265 days, which would roughly reduce the ancient value of the above cycle down to 1505 Egyptian years (1504 tropical years).

The Egyptian Calendar System   |   The Egyptian Calendar   |   History of the Egyptian Calendar
Sun, Moon, and Sothis   |   Ancient Egyptian Calendars   |   Out Of Timelessness
Hieroglyphs: Numbers & Dates   |   Akhet = Inundation vs. Akhet = Eclipse or Horizon
Egyptian Astrology

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(2003-01-12)   Heliacal Rising of Sirius
"Sirius is the one consecrated to Isis, for it brings the water." --Plutarch

A heliacal rising of a star is defined as its appearance above the horizon just before sunrise.  In ancient times, the Egyptians observed that the heliacal rising of Sirius marked the yearly beginning of the Nile's floods.

Before the construction of the Aswan High Dam, the inundations of the Nile were a yearly phenomenon, caused by the summer rains over the Ethiopian highlands, which are drained by 2 of the 3 major tributaries of the Nile, the Blue Nile and the Black Nile.  The Blue Nile (Gihon) flows from lake Tana and joins the White Nile at Khartoum to form the Nile proper, whereas the Black Nile (Atbarah or Atbara) is the only tributary of the Nile after Khartoum.  The Black Nile is dry for most of the year, but in a few short months it provides over 20% of the Nile's total yearly volume of water, loaded with about 11 million tons of this black mud which once made Egypt fertile, but is now settling in Lake Nasser, behind the Aswan Dam.  The White Nile carries only half the total flow of the Blue Nile but it's much more regular.  It flows from Lake Victoria, under a succession of names.  The Kagera River flows into Lake Victoria, and has an upper branch, the Ruvyironza River of Burundi, whose source is now considered to be the ultimate source of the Nile.

The exact day when an heliacal rising is observed may depend on the longitude and latitude of the observer.  The altitude is somewhat relevant too (on the equator, a star rising due east would be seen from a 100 m cliff about 76.8 s earlier than from the beach).  The brightness of the star is important as well, since fainter objects disappear earlier at dawn.

Biblical Chronology   |   Sothic Dating   |   The Mysteries of Sirius
Keeping Track of Time in Ancient Egypt

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(2003-01-06)   Alexandrian Calendar & Coptic Calendar

To avoid most of the calendar creep described above, a reform of the Egyptian calendar was introduced at the time of Ptolemy III  (Decree of Canopus, in 238 BC) which consisted in the intercalation of a 6th epagomenal day every fourth year.  However, the reform was opposed by priests, and the idea was discarded until 25 BC or so, when Roman emperor Augustus formally reformed the calendar of Egypt to keep it forever synchronized with the newly introduced Julian calendar.  To distinguish it from the ancient Egyptian calendar, which remained in use by some astronomers until medieval times, this reformed calendar is known as the Alexandrian calendar and it's the basis for the religious Coptic calendar, which the Copts [the Christians from Egypt] are still using now.

The Coptic Orthodox Church was founded by St. Mark, author of the earliest Gospel and first Patriarch of the Coptic Church.  Saint Mark died a martyr, dragged with a rope around his neck through the streets of Alexandria, on Sunday May 8, AD 68.  The word Copt was originally synonymous with Egyptian, but it's now used to designate either a member of the Coptic Christian Church, or a person whose ancestry is from pre-Islamic Egypt.

Coptic years are counted from AD 284, the era of the Coptic martyrs, the year Diocletian became Roman Emperor (his reign was marked by tortures and mass executions of Christians).  The Coptic year is identified by the abbreviation "AM" (for Anno Martyrum) which is unfortunately also used for the unrelated Jewish year (Anno Mundi).  To obtain the Coptic year number, subtract from the Julian year number either 283 (before the Julian new year) or 284 (after it).

The table below shows the correspondence between the Coptic calendar and the Julian calendar.  For the period between 1901 and 2099 CE, the secular (Gregorian) date is obtained by adding 13 days to the Julian day shown in the table, so that the Coptic year actually starts on September 11, on most years.  The 7 months which precede the intercalation of a Julian February 29 actually start one day later (this is what the " + " signs in the table are reminders for).  Therefore, the Coptic year which starts just before a Julian leap year begins on August 30 in the Julian calendar, which corresponds to September 12 in the Gregorian calendar (every fourth year, from 1903 to 2095 CE).

For the usual Gregorian secular date between 1901 and 2099 CE, add 13 days to the Julian date shown (the 7 months before a Julian Feb. 29 start 1 day later).

	Days	Coptic Month (Egyptian Name)	First Day (Julian)	Nile Season
1	30	Thoth, Thot, Thout, Thuthy	August 29+	Inundation Akhet
2	30	Paophi, Paapi, Paopy	September 28+
3	30	Athyr, Hathor, Hathys, Athor	October 28+
4	30	Cohiac, Kiahk, Koiahk, Kiak, Choiach	November 27+
5	30	Tybi, Tobi, Tyby	December 27+	Emergence Proyet Peret, Poret
6	30	Mesir, Mechir, Menchir, Mekhir	January 26+
7	30	Phamenoth, Paremhat, Famenoth	February 25+
8	30	Pharmouthi, Paremoude, Parmuthy	March 27
9	30	Pachons, Pakhons	April 26	Summer Harvest "Low Water" Shomu Shemu
10	30	Payni, Paoni, Paony	May 26
11	30	Epiphi, Epip, Epipy, Epep	June 25
12	30	Mesori, Mesore	July 25
13	5 or 6	Epagomena, Little Month	August 24

Conversion between Coptic and Julian dates   |   Coptic Orthodox Calendar   |   Coptic Calendar

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(2002-12-22)   The Julian Calendar & Leap Years

The Julian calendar is still being used for religious purposes by some Eastern Orthodox churches, such as the Russian Orthodox church.

An early form of the Julian Calendar was introduced by Julius Caesar in 46 BC, on the advice of the Egyptian astronomer Sosigenes.  Officially, the first day of the Julian Calendar was the Kalends of Januarius, 709 AUC (January 1, 45 BC).  At first, there was a leap year every third year, but this was soon recognized to be a mistake:  In 8 BC, the calendrical reform of Augustus gave the months their modern names and lengths, and returned the calendar year back to the seasonal point intended by Julius Caesar.  This was done by shunning leap years until AD 8, which would be a leap year like every fourth year thereafter.  (5 BC, 1 BC and AD 4 were ordinary years.)

Historical Leap Years (before the Regular Julian Pattern)

43 BC	40 BC	37 BC	34 BC	31 BC	28 BC	25 BC	22 BC	19 BC	16 BC	13 BC	10 BC	AD 8	AD 4n
This is our interpretation of reports from Macrobius and others.  Some scholars dispute this.

The so-called proleptic Julian Calendar extends backward in time the regular pattern which has been in force since March of AD 4.  This may mean a discrepancy of several days from the historical calendar used between 45 BC and AD 4 and it's all but fictitious before that...

The Julian Calendar, before and after Augustus

	45 BC to 8 BC		After 8 BC
	Days	Month	Days	Month
I	31	Ianuarius	31	Ianuarius
II	29 or 30	Februarius	28 or 29	Februarius
III	31	Martius	31	Martius
IV	30	Aprilis	30	Aprilis
V	31	Maius	31	Maius
VI	30	Iunius	30	Iunius
VII	31	Quintilis / Iulius	31	Iulius
VIII	30	Sextilis	31	Augustus
IX	31	September	30	September
X	30	October	31	October
XI	31	November	30	November
XII	30	December	31	December

New Year's Day

Julius Caesar made the year start on January 1,  probably because this was the traditional beginning of the session in the Roman Senate (and the date when consuls used to be elected).  However, it seems that the popular use of the previous "March 1" system survived at least until the Augustan Age (27 BC-AD 14).  The "January 1" convention was not finally established (or restored) until the introduction of the Gregorian calendar.

March 1 used to be the beginning of the Roman year (it was the date when the elected consuls actually took office).  This explains the names of the months of September, October, November and December, which used to be the 7th, 8th, 9th and 10th months of the year.  In 153 BC, the Roman Senate had voted to have the new year coincide with the beginning of its own session, on January 1, but old habits kept prevailing among the people.  In the final(?) transition to the Julian year beginning on January 1, the abnormal duration of the year 46 BC (the so-called "year of confusion") should have helped, but apparently didn't...  The year 46 BC lasted 445 days from January to December, and March 1 of 46 BC was nearly at the same seasonal point as January 1 of 45 BC.

The most common convention in late medieval times was that the beginning of a new Julian year occurred on March 25.  This was the nominal date of the vernal equinox (it was the actual date of the equinox shortly before the calendar reform of Julius Caesar).  In medieval times, March 25 was thought of as the mythical anniversary of Creation.  For Christians, this is the Feast of the Annunciation, the Incarnation when Christ was conceived  (the alternate name Lady Day has a pagan origin, rooted in the Celtic tradition).  However, the Julian New Year has been celebrated at a variety of dates throughout history.  The following sketchy table is only meant to show the utter lack of universal conventions:

When does a calendar year start?  (sketchy data)

New Year's Day	When	Who / where
March 1	Until 222 BC (?)	Rome
March 15	222 BC - 153 BC (?)
January 1	Since 153 BC (and 45 BC)
March 25	Middle Ages
March 1	Until 800	France
March 25	800 to 996
Easter (See note below)	996 to 1566 "more Gallicano"
January 1	Since 1567 (or 1563)
November 1	Until AD 1179	Celts
December 25	7th Century thru 1338	England
March 25	Late Middle Ages	Europe
March 1	Until 1797	Venice
September 1	14th century thru 1918	Russia
January 1	Gregorian reform	Worldwide

Note :  When Easter was taken as the beginning of the year, there could be two days with the same date, at the beginning and at the end of some years.  The ambiguity used to be lifted by specifying "after Easter" of "before Easter".

Days of the Month

The Roman way of numbering days was used in Latin with the Julian calendar, even in the late Middle Ages.  Three special days were singled out each month:

• The Kalends:  First day of the month.   [Etymology of "calendar"]
• The Nones:  The 7th day of March, May, July, and October.
  The 5th day of the other months.  [The ninth day of the Ides, see below]
• The Ides:  The 15th day of March, May, July, and October.
  The 13th day of the other months.

The other days were counted backwards and inclusively, from the next such special day.  Thus, since March 13 was two days before the Ides of March, it was called the third day of the Ides of March.  Most of the month came after the Ides and was thus referred to the Kalends of the next month.  In a leap year, the intercalary day was inserted after February 23 (the seventh day of the Kalends of March) so there would be a day designated as bissextilis, being the "other sixth" day of the Kalends of March...  Leap years are thus still called bissextile.

The Orthodox Ecclesiastical Calendar   |   Gregorian/Julian Calendar Information
Changes in New Year's Day   |   Early Julian Calendar   |   Render unto Caesar   |   Julian Calendar

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(2002-12-31)   AD = Anno Domini   [CE = Common Era]

Dionysius Exiguus was a Russian monk who had been commissioned by pope St. John I  to work on calendrical matters, including the official computation of the date of Easter.  The story goes that he was confronted with the Coptic calendar in the course of his work with Alexandrian data.  He liked the idea of a continuous count of years based on a Christian milestone, but was disturbed by the choice of the Copts, who were honoring their greatest persecutor by counting from the year Diocletian became emperor (284 CE).  Dionysius had the idea to count years from a joyous event instead, the birth of Christ.  In 527, he formally declared that Jesus was born on December 25 in the year 753 AUC, equating the year 754 AUC with the year AD 1  (Anno Domini = Year of the Lord).

The historical record shows that the guess of Dionysius was off by several years:  Jesus was born during the census of Augustus (Luke 2:1) while Quirinius was governing Syria (Luke 2:2), under the reign of Herod the Great (Matthew 2:1).  Herod died in 750 AUC (4 BC), so Jesus was born at least 4 years earlier than Dionysius thought.  We don't know how Dionysius arrived at his incorrect result, but we may venture the guess that he took the Gospel of Luke too literally...

Jesus Himself began His ministry at about 30 years of age (Luke 3:23) after begin baptized by John, who began preaching in the 15th year of the reign of Tiberius Caesar (Luke 3:1).  As Tiberius became emperor in AD 14, the Gospel of Luke says that Jesus was baptized in AD 29 or AD 30, when he was about 30 (he was actually 34 or so).

The numbering scheme suggested by Dionysius may not have been popular until the time of the calendrical studies of the Venerable Bede (673-735) of Britain.

The Date of Christmas

Incidentally, this calendrical focus on the birth of Jesus turned Christmas into a major Christian festival, rivaling Easter.  The birth of Christ was hardly celebrated at all by early Christians, and different communities did so on different dates...  The choice of December 25 had been proposed by anti-pope Saint Hippolytus of Rome (170–236), but it was apparently not accepted until AD 336 or 364.  Dionysius emphatically quoted mystical justifications for this very choice:

March 25 was considered to be the anniversary of Creation itself.  It was the first day of the year in the medieval Julian Calendar and the nominal vernal equinox  (it had been the actual equinox at the time when the Julian calendar was originally designed).  Considering that Christ was conceived at that date turned March 25 into the Feast of the Annunciation which had to be followed, 9 months later, by the celebration of the birth of Christ, Christmas, on December 25...

There may have been more practical considerations for choosing December 25.  The choice would help substitute a major Christian holiday for the popular pagan celebrations around the winter solstice (Roman Saturnalia or Brumalia).  The religious competition was fierce.  In 274, Emperor Aurelian had declared a civil holiday on December 25  (Sol Invicta, the Unconquered Sun)  to celebrate the birth of Mithras, the Persian Sun-God whose cult predated Zoroastrianism and was then very popular among the Roman military...  Finally, joyous festivals are needed at that time of year, to fight the natural gloom of the season.  The Jews have Hanukkah, an eight-day festival beginning on the on the 25th day of Kislev.

Whatever the actual reasons were for choosing a December 25 celebration, the scriptures indicate that the birth of Jesus of Nazareth did not even take place around that time of year, since there were in the same country sherperds living out in the fields, keeping watch over their flock by night (Luke 2:8).  During cold months, shepherds brought their flocks into corals and did not sleep in the fields.  That's about all we know directly from scriptures, besides wild speculations.

Calendar History   |   The Bible's story of Christmas

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(2003-02-20)   The Date of Easter

According to Christian scriptures, Jesus Christ resuscitated on a Sunday that fell on the Jewish Spring festival of Pesach (Passover, Nissan 15), which is always near a full moon.  At the First Ecumenical Council of the Christian Church (held in Nicea, in 325 AD), it was decided to celebrate Easter on the Sunday following the so-called Paschal full moon:

The Paschal full moon is an arithmetical approximation to the first full moon after the vernal equinox.  John H. Conway points out that it may be expressed as follows in terms of the so-called Golden number (G) and Century term (C):

Paschal full moon (PFM)   =   (April 19, or March 50) - (C+11G) mod 30

... except  in two cases where the PFM is one day earlier than this, namely:

• When (C+11G) is 0 modulo 30,   PFM = April 18  (not April 19).
• When (C+11G) is 1 modulo 30, and G≥12,   PFM = April 17  (not 18).

Some famous algorithms, like the so-called Gauss formula, are wrong because they fail to incorporate these two exceptional cases (e.g., in 1981 the PFM was Saturday April 18, and Easter Sunday was April 19).  The Golden number (G) is the same for both Julian and Gregorian computations, but the Century term is constant (C = +3) in Julian computations:

• G = 1 + (Y mod 19)   in year Y   (Julian or Gregorian).
• C = -H + ëH/4û + ë8(H+11)/25û   with H = ëY/100û (Gregorian year Y)
  C is -4 from 1583 to 1699, -5 from 1700 to 1899, -6 from 1900 to 2199, -7 from 2200 to 2299.

As the Sunday following the PFM, Easter is one week after the PFM when the PFM happens to fall on a Sunday...  Note that you should work with the Julian calendar (C = +3) to find when Easter is celebrated by Orthodox churches.

(Erroneous) Gauss Formula   |   Bibliography on Easter Algorithms and the Computus   |   Easter Date
Date de Pâques - Comput (JavaScript)   |   How Easter Date is Determined   |   Easter Dating Method
How Passover became Easter

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(2002-12-22)   The Gregorian Calendar

The Gregorian calendar is like the above Julian calendar, except for its pattern of leap years.  Its Christian origins are all but forgotten, as it has now been adopted as a secular calendar by most modern nations.  A few countries are still officially using other traditional and/or religious calendars, but they all have to accomodate the Gregorian calendar, at least in an International context...

This calendar has been dubbed Gregorian because it was introduced under the authority of Gregory XIII, né Ugo Boncompagni (1502-1585), Pope from 1572 to 1585.  The Gregorian calendrical reform was engineered by astronomer Christopher Clavius to make the seasons correspond permanently to what they were under the Julian calendar in AD 325, at the time of the First Ecumenical Council of the Christian Church, the First Council of Nicea, when rules were adopted for the date of Easter.

The precise rules are rather involved, but Easter is usually the first Sunday after a full moon occurring no sooner than March 21, which was the actual date of the vernal equinox at the time of the First Council of Nicea.  Shortly before Julius Caesar reformed the calendar, the vernal equinox was occurring on the "nominal" date of March 25.  This was rightly discarded at Nicea, but the reason for the observed discrepancy was all but ignored (the actual tropical year is not quite equal to the Julian year of 365¼ days, so the date of the equinox keeps creeping back in the Julian calendar).  The Gregorian reform ensured that, for many centuries to come, the vernal equinox would occur around March 21 just like it did at the time of the Council of Nicea, so order would be restored to the computation of Easter...

The Council of Trent (1545-1563) had previously urged Pope Paul III to reform the calendar, and Clavius was one of several scientists who had been approached in the wake of that resolution.  Over 20 years later, Gregory XIII finally asked Clavius to lead a commission on the subject, which would be formally presided by Cardinal Guglielmo Sirleto (1514-1585), a contender for the papacy.

Building on the work of Luigi Lilio, this commission recommended dropping 10 calendar days immediately, and reducing the number of future leap years (to avoid a new drift of the calendar with respect to the seasons).  Thus, a Papal Bull (Inter Gravissimas) decreed that, October 4, 1582 would be followed by October 15.  Furthermore, future leap years would be multiples of 4 (as in the Julian calendar) except for years evenly divisible by 100 but not by 400 (so that 1600 and 2000 were indeed leap years).  This reduces the number of leap years to 97 (from 100 in the Julian scheme) for each Gregorian period of 400 years, or 146097 days (20871 weeks):   146097  =  303´365 + 97´366.

Interestingly, Inter Gravissimas was signed on February 24, 1582, although it bears a date of 1581 because the official year number used to change on March 25 before this very reform was enacted.
      Note also that Saint Teresa of Avila passed away in the night from Thursday October 4 to Friday, October 15, 1582.

Various countries adopted the "new" calendar only much later (see table below).  In particular, the earliest valid Gregorian date in England (and its American Colonies) is September 14, 1752, which followed September 2, 1752 (the difference between the two calendars had grown from 10 to 11 days by that time, because the year 1700 was not a leap year in the Gregorian calendar).

Gregorian Calendar   |   The Gregorian Calendar   |   Calendopaedia
Adoption of the Gregorian Calendar   |   When they adopted the modern calendar

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(2002-12-22)   Zoroastrian Calendar(s)

Zoroastrianism is a monotheist belief system based on righteousness (good thoughts, good words, good deeds).  When it was first preached in Persia by Zarathustra (c.628-c.551 BC), it was opposed to the prevalent cult of Mithras (which demanded sacrifices and advocated the consumption of narcotics and/or intoxicating beverages, then known as Haoma).  Some scholars have considered Zoroastrianism to be a precursor of Christianity.  Although Jews claim him as one of their own, it is generally believed that Zarathustra (or Zoroaster) was Indo-Iranian (Aryan).  He was most probably born in Mazar-I-Sharif (which is now in northern Afghanistan) and was "the son of Pourushaspa, of the Spitaman family".  Zoroaster is said to have given his very first teaching just after being born, in the form of an unusual laughter, telling believers that human life is worth living...

Zoroastrianism is still practiced by about 18 000 people in Iran, chiefly in Shiraz.  It is thriving in India (chiefly around Bombay) and Pakistan (chiefly in Karachi) among Parsis or Parsees, literally "Persians" whose ancestors fled Persia in the wake of the Arab conquest, and subsequent Islamization  ( 7th century AD).  The total number of Zoroastrians is currently estimated to be around 140 000.

The Zoroastrian calendar is based on months of 30 days and has the same basic structure as the ancient Egyptian calendar (and/or the modern Coptic calendar), including 5 extra days after the 12th month, the gatha days.

In the year 1006 CE, the first day of the Zoroastrian year (Noruz) occupied once again its original position at the vernal equinox.  (Incidentally, this would imply that the Zoroastrian calendar originated in 500 BC or so.)  It was then decided to intercalate a whole month every 120 years, to make the long-term average of the Zoroastrian year equal to 365¼ days, and avoid calendar creep with the exact same accuracy as the Julian calendar (in the long run, at least).  This unusual intercalation scheme may have been chosen for religious reasons, which made it difficult to have anything but 5 gatha days at the end of every year.

However, this rule was remembered only once, about 120 years later, and only by the Parsees of India, whose calendar (now called Shahanshahi or Shenshai) has been 30 days late ever since, relative to the original calendar (Qadimi or Kadmi) still kept by Iranian Zoroastrianists.  (Curiously, the discrepancy is said to have gone unnoticed until 1720.)  Both the Shenshai and Kadmi calendars are thus effectively variants of the Egyptian calendar, featuring a constant year of 365 days, without any intercalations.

The Fasli calendar, on the other hand, is a modern Zoroastrian calendar, designed in 1906, in strict alignment with the Gregorian calendar.  The Fasli year always starts on March 21 (the nominal Gregorian vernal equinox) and it consists of 12 months of 30 days and a 13th "month" of either 5 or 6 days.

Zoroastrianism was made the official religion of Persia by Shapur I, who reigned from 241 to 272, as the second king of the Sassanian Dynasty (AD 224-641).  Regnal years were then used with the Zoroastrian calendar.  The Persian empire was conquered by the Arabs after the battle of Nehavand in 641 CE, about 10 years after the coronation of the last of the Sassanids, Yazdegird III  [also known (?) as Yazdegerd, Yazdazard, or Yazdegar Sheheryar].

The era of this last Zoroastrian king is abbreviated YZ and has been continued up to the present time:  Year 1 YZ was 631 CE.

Zoroastrian Calendar   |   Zoroastrian Religious Calendar   |   History of the Zoroastrian Calendar
Iranian Calendars & Ancient Yazdgerdi Calendar   |   Persian and Zoroastrian Calendars
Culture of Iran

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(2002-12-22)   Signs of the Zodiac & Precession of Equinoxes

This calendar would be totally obsolete, if it was not for the fact that astrologers [fortune tellers] still use it.  In the last column of the table below is the correspondence with the modern Gregorian calendar which is most often used by "modern" astrologers...

Zodiacal Sign	Persian Month	First Day
Aries	Farvardin	March 21
Taurus	Ordibehesht	April 20
Gemini	Khordad	May 21
Cancer	Tir	June 22
Leo	Mordad	July 23
Virgo	Shahrivar	August 23
Libra	Mihr	September 23
Scorpio	Aban	October 23+
Sagittarius	Azar	November 22
Capricorn	Day	December 22
Aquarius	Bahman	January 20
Pisces	Esphand	February 19

About 2000 years ago, when this calendar was presumably devised, the eponymous constellation indicated the correct position of the Sun for the month corresponding to a given zodiacal sign.  Because of the so-called precession of equinoxes, this is no longer true at the present epoch.

In this context, it's important to maintain a clear distinction between 3 related concepts that are often confused: signs, constellations and houses:  Zodiacal signs are simply names given to months within the regular calendar year (synchronized with the tropical year) as tabulated above.  On the other hand, it's clear that 12 constellations were once defined which, unlike modern constellations, divided evenly the ecliptic (the apparent path of the Sun against the background of "fixed stars").  Such traditional constellations are best referred to as "houses".  We are not aware of any precise historical definition of the exact boundaries between houses (if you know better, let us know).  The 88 constellations of the entire celestial sphere do have precise modern definitions, but these are virtually irrelevant with respect to houses:  Not only are the modern zodiacal constellations of varying sizes, but there are 13 of them (the 13th zodiacal constellation being Ophiucus, the Serpent Bearer, which spans the ecliptic between Scorpio and Sagittarius).

The so-called vernal point is the position of the Sun at the spring equinox (it's at the intersection of the ecliptic and the current celestial equator).  It's also known as the "gamma point", after the Greek letter (g) traditionally used in various diagrams.  The precession of the Earth's axis of rotation makes this vernal point go a full circle around the Zodiac in about 26000 years.  Please, do not believe the many sources which tell you that this period is precisely 25920 years.  This would be the case only if the average yearly precession was exactly 50" (1°/ 72), because 25920 = 360 ´ 72.  The latest data available to us at this writing (MHB2000 nutation model) give an average yearly precession of 50.28792(2)", corresponding to a precession period of 25771.597(11) years [about 25772.126(11) Gregorian years].
 
Some ancient Babylonian astronomers must have known about this, but Hipparchus of Rhodes (190-120 BC) is credited for the first precise description of the phenomenon, which Copernicus would correctly attribute, in 1543, to the changing direction of the Earth's axis of rotation.  The actual dynamical reason for this precession was given by Isaac Newton in 1687:  The Earth "bulges at the Equator", and this oblateness implies that a distant body, like the Moon or the Sun, exerts a nonzero gravitational torque on the Earth, (except, ideally, in the rare symmetrical case when the Earth axis is precisely perpendicular to the direction of the body in question; for the Sun, this would be the configuration at either equinox).  This torque is always "trying" to reduce the tilt of the axis with respect to the direction of the body.  However, the Earth reacts like any rotating body would:  It changes its rotational axis toward the direction of the applied torque (the torque is a vector perpendicular to both the axis of rotation and the direction to the influencing body).  This causes a precession of the axis, instead of the naively expected reduction in tilt.
 
Traditionally, the time when the vernal point enters a new house marks the dawning of a new "age" (like the Age of Aquarius) which lasts for about 2148 years.  A poor definition of the traditional Zodiacal houses translates into a fuzzy beginning for each such age (a misalignment of 1° corresponds to an error of about 72 years).

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(2002-12-28)   Hegira Calendar   [AH = Anno Hegirae]
The Islamic calendar used by Muslims is the Hijri, or Hegira Calendar. 

The origin of the Muslim calendar is "1 Muharram 1 AH" (July 16, 622 CE), and predates by a few weeks the so-called "flight from Mecca" (Hajj, Latin: Hegira) which, according to Muslim tradition, actually took place in September 622.

The Hegira Calendar is purely lunar, without intercalations.  It was introduced in AD 639  (17 AH) by the second Caliph, `Umar ibn Al-KHaTTab (592-644).

Since an Islamic year (essentially, 12 lunar months) falls shorts of a tropical year by almost 11 days, the Islamic months are not related to the seasons at all.  Muslim festivals simply drift backwards through the seasons and return roughly to the same point after a period of 33 Islamic years (which happens to be about a week longer than 32 tropical years).

Traditionally, the beginning of a new Islamic month is defined locally from the time when the thin crescent of the young moon actually becomes visible again at dusk, a day or so after the new moon.  If the moon can't be observed for any reason, the new month is said to begin 30 days after the last one did.

However, printed Islamic calendars are based on a simple arithmetic prediction of moon sightings.  A regular cycle of 30 years is used, which includes 19 years of 354 days and 11 years of 355 days (modulo 30, the long years are: 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, and 29).  The average Islamic month is thus equal to 29.53055555... days, which is about 2.9 s shorter than the actual mean synodic lunar month of 29.530588853 days (it takes about 2428 tropical years to build up a discrepancy of a whole day). The standard Islamic year is tabulated below:

Number	Month Name	Days
1	MuHarram	30
2	Safar	29
3	Raby` al-awal	30
4	Raby` al-THaany	29
5	Jumaada al-awal	30
6	Jumaada al-THaany	29
7	Rajab	30
8	SHa`baan	29
9	RamaDHaan	30
10	SHawwal	29
11	Thw al-Qi`dah	30
12	Thw al-Hijjah	29 or 30

30 Islamic years (10631 days):

0	10	20
1	11	21
2	12	22
3	13	23
4	14	24
5	15	25
6	16	26
7	17	27
8	18	28
9	19	29

Hijri/Gregorian/Julian Converter   |   Gregorian-Hijri Converter   |   Islamic-Christian Converter
Hegira Calendar   |   Islamic Calendar(s)   |   July 16, 622   |   Islamic Calendar

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(2002-12-29)   Jewish Calendar   [AM = Anno Mundi]

The Jewish calendar is called lunisolar, because it uses lunar months (of either 29 or 30 days, following the phases of the Moon) while keeping the year roughly synchronized with the solar seasons through the regular intercalation of a 13th month:  In leap years, an extra month (Adar I, or Adar aleph) occurs just before the month when Purim is celebrated, the regular month of Adar (called Adar II, or Adar bet, in leap years).  This compensates for the fact that 12 lunar months are nearly 11 days short of a tropical year.

The Hebrew calendar is also known as the Hillel calendar, because its modern rules were set down (c. 359 CE) by the nasi (Sanhedrin president) Hillel II.

In the Judaic culture, a new calendar day begins in the evening:  Either at sunset, or when 3 stars should become visible [in clear weather].  The day is divided in 24 hours of 1080 "parts" (halakim) each; 10 seconds is 3 halakim (also spelled halaqim, chalakim or chalaqim, the singular form is helek, heleq, chalak or chalaq).  This helek of 3 1/3 seconds is further divided into 76 rega'im (the rega is 5/114 of a second, or about 43.386 ms).

The Jewish tradition gives the average duration of a lunar month to the nearest helek (there are 25920 halakim in a day): 29 days, 12 hours, 793 halakim (or 765433 halakim per month).  This value is said to date back to the time of Moses in the Sinai, and it's quite accurate:  The accepted modern value for the mean synodic lunar month is about 29.530588853 days, which translates into 29 days, 12 hours, 792.863 halakim (off by less than half a second).

Around 1500 BC, the day was about 70 ms shorter and the month was 1500 ms shorter.  Expressed in halakim and other fractions of a day, the month was then 29 days, 12 hours, 793.033 halakim.  This means that the traditional value was then 4 times more accurate than now... It was even entirely correct at some point (around 800 BC).

Years are counted since the mythical creation of the world, in 3761 BCE.  Jewish year numbers are best suffixed with "AM" (Anno Mundi; year of the world).

In each Metonic cycle of 19 years, there are 12 common years of 12 months, which may contain 353, 354 or 355 days.  The remaining 7 leap years have 13 months and contain 383, 384 or 385 days.  Modulo 19, the leap years are 0, 3, 6, 8, 11, 14, or 17.  Either type of year comes in three different lengths, called defective (H for Haser, 353 or 383 days), regular or normal (K for Kesidra, 354 or 384 days), and perfect or complete (S for Shalem, 355 or 385 days). 

The months are traditionally numbered as shown in the table below (Esther 2:16, 3:7, 3:12), but the year number changes on Rosh HaShanah ("Jewish New Year"), the first day of Tishri.  Formerly, the older "sacred year" started with the first day of Nissan (not Tishri), whereas the above convention applied only to the civil year.  Apparently, the former tradition faded away in the 3rd century (CE).

The names of the months are derived from the ancient Babylonian calendar, dating back to the days of the 70-year captivity to Babylon (c. 600 BC).

• Purim: Adar 14.
• Pesach (Passover): Nissan 15.
• Rosh Hashanah: Tishri 1.
• Yom Kippur: Tishri 10.
• Sukkot: Tishri 15.
• Hoshanah Rabbah: Tishri 22.
• Hanukkah, Chanukkah: Kislev 25.

The exact sequence of the different types of years is determined by 4 so-called rules of postponement (dehioth, dehiyyot) whose purpose is to prevent the celebration of some of the above festivals on certain days of the week.

encyclopedia.com   |   Jewish Calendar Tools   |   Judaism 101   |   Jewish Calendar
Jewish Calendar Rules   |   Measuring the Day   |   Jeroboam and the Hillel Calendar
Postponement Controversy in 921 CE   |   The Reformed Jewish Calendar
No Postponement in Temple Times   |   Is Anything Wrong with the Hillel Calendar?
Ancient Hebrew Calendar

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(2002-12-29)   Mayan Calendar  &  Long Count

The Mayan civil year, the haab consisted of 18 "months" (uinals), of 20 days each, and 5 extra days (which were believed to be unlucky ones), for the same total of 365 days as the Egyptian year.  The Mayas knew that the tropical year was closer to 365¼ days, but they chose to keep a constant number of days in each year, and shunned intercalary days (just like the ancient Egyptians).

The Mayan sacred year, the tzolkin, was a cycle of 260 days (the combination of a regular cycle of 13 numbers and of a regular cycle of 20 different signs).

When both calendars are used concurrently, a day is uniquely identified within any period of 18980 days known as a Mayan Calendar Round  (18980 is the lowest common multiple of 365 and 260; it's equal to 52 haabs or 73 tzolkins).

The synodic period of Venus is about 583.9214 days.  The Mayas estimated it to be 584 days, which happens to be 8/5 of their haab of 365 days.  Therefore, twice the above Calendar Round is a multiple of the Mayan value of the Venus period.  This period of 37960 days is the Mayan Venus Round, which is equal to 104 haabs, or 146 tzolkins, or [roughly] 65 synodic periods of Venus.

The Long Count

In addition to the above, the Mayas used a so-called Long Count to keep track of their historical events.  This was simply the number of days elapsed since the Mayan mythical creation of the World, using the following 5 units:

• A baktun is 144000 days (20 katuns).
• A katun is 72000 days (20 tuns).
• A tun is 360 days (18 uinals).
• A uinal is 20 days (20 kins).
• A kin is one day.

Each Mayan vigesimal "digit" could represent a number from 0 to 19, and a Long Count was expressed as a string of 5 such digits, usually transliterated as 5 numbers separated by dots ( baktuns.katuns.tuns.uinals.kins ).

For the Maya, a "Great Cycle" was 13 baktuns, or 1872000 days (exactly 7200 tzolkins, or over 5125 tropical years).  13 baktuns after its mythical beginning, the Mayan World comes to an end of sorts:  The Mayan tradition would simply reset the long count to 0.0.0.0.0 when it reaches 13.0.0.0.0, on December 21 (or 22), 2012 CE.

Calendopaedia   |   Introduction to the Mayan Calendar   |   Venus and the Mayan CR
Mayan/Christian Correlation   |   Critic of MD/JD Correlation   |   Calendars & the Long Count System

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(2003-01-01)   The Chinese Calendar & True Astronomical Motion

The Chinese calendar is an astronomical calendar, which explicitly depends on actual observations and/or delicate predictions of astronomical events.

It's currently used by about one fourth of the World's population (at least for traditional festivals).  Its modern form dates back to 1645 and is due to Father Schall (Johann Adam Schall von Bell, 1591-1666), a catholic missionary who was summoned to Peking in 1630 after the death of Father Terrentius (John Schreck) to take over the task of reforming the traditional Chinese calendar.

The Mathematics of the Chinese Calendar   |   The Chinese Calendar

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(2003-01-12)   Traditional Japanese Calendar

The latest periods in the traditional Japanese calendar system are called Edo, Meiji, Taisho, Showa and Heisei.  Starting with Meiji (1868-1912 CE), the period changes when the Emperor passes away, and years are numbered from the beginning of the period.  In the Edo period (1603-1868 CE), the Japanese calendar was based on its Chinese counterpart, with significant discrepancies due to the different longitudes used for critical observations.  Years were then named using the Chinese 12-year cycle  (Rat, Ox, Tiger, Hare, Dragon, Snake, Horse, Sheep, Monkey, Bird, Dog, Pig).  This tradition remains popular today, although Japan adopted the Gregorian calendar in 1873.

There was also a so-called Koki calendar based on a continuous count of years from the founding of the Japanese dynasty of Emperor Jimmu Tenno, in 660 BC.  The last two digits of this count were once used by the Japanese military for new or revised equipment.  This is why the "Zero" was so named, since this famous WW II  fighter plane ( Mitsubishi A6M ) appeared in 1940, Koki year 2600.

Japanese Calendar   |   Japan File   |   Lunar Calendar in Japan

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(2003-01-03)   The Indian Calendar & The Solar Month

The National Calendar of India was last reformed in 1957:  Its leap years coincide with those of the Gregorian calendar, but years begin at the vernal equinox and are counted from the Saka Era (the spring equinox of 79 CE).

Indian Calendars

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A minor issue is the inept pattern of month lengths, which the Gregorian calendar inherited from the calendar reform of Augustus.  Originally the Julian calendar was more rational:  It had long months of 31 days alternating with short months of 30 days, with the sole exception of one month obeying the regular pattern in leap years, but one day shorter in other years.  If it had not been for the Roman belief that even numbers were unlucky, the best scheme would have been to shorten a long month, but it was decided instead to shorten the short month of February (30 days in a leap year, 29 days otherwise).

(2003-01-10)   Post-Gregorian Calendars

There are intercalation patterns of leap years which could make the Gregorian calendar even more accurate in the very long term, while being consistent with the Gregorian rules for dates of the past (back to 1582 CE) and the near future.  However, proposals for such millenarian rules must be carefully evaluated in the framework presented here.

The Gregorian year is currently the best calendar approximation there is to the tropical year (which governs our seasons).  In a Gregorian cycle of 400 years, there are 97 leap years of 366 days and 303 regular years of 365 days, which makes the mean Gregorian year equal to 365.2425 days.

Although the issue is entirely irrelevant to calendar design, note that the above "mean" year is less than the "time-average" of a Gregorian year:  If we record daily the length of the current year in days (365 or 366) over a complete Gregorian cycle of 146097 days, the number 365 will be recorded 110595 times, whereas 366 will be recorded 35502 times, which makes the "time-average" exceed 365.2430029 days...

A solar calendar should be engineered to make the long-term ratio of the number of days to the number of elapsed calendar years (365.2425 for the Gregorian calendar) as close as possible to the observed number of days in a tropical year, which is slightly less than 365.2422.  At first, it would seem easy to reform the Gregorian calendar (by dropping a leap year once in a great while) in order to make the mean calendar year closer to this target number.

For example, if a rule were added to turn into ordinary years the years divisible by 3200 (which are leap years according to Gregorian rules), the mean calendar year would become 365.2421875 days.
      At least two other ideas have been floating around which are not as good as this one (because they are rooted in somewhat obsolete 19th century data).  The most popular one may have appeared around 1834 and is usually attributed either to Mary Somerville (after whom a college has been named at Oxford) or to John Herschel (1792-1871, son of the discoverer of Uranus).  It consists in turning multiples of 4000 into ordinary years, so the mean calendar year would become 365.24225 days.  Another idea (which is incompatible with Gregorian rules, except between 1601 and 2799) states that multiples of 100 should be leap years only when equal to either 200 or 600, modulo 900.  This rule would put 365.242222... days in a mean calendar year  (incidentally, just as if multiples of 3600 were made regular years).  In 1923, the Greeks switched from the Julian calendar and may have adopted this rule (we can only hope they'll recant before 2800).

However, all such efforts may be misguided, since the above target is a moving one (mainly because tidal braking keeps making our days longer).  To put it bluntly, a millenarian rule for leap years could be all but obsolete before coming into play, as long as it remains based only on the current number of days in a tropical year...  Let's see what the actual numbers are:

The definition of the ephemeris seconds makes the instantaneous value of the tropical year "at epoch 1900.0" exactly equal to 31556925.9747 ephemeris seconds.  Since the definition of the modern SI second was precisely engineered to make it virtually indistinguishable from an ephemeris second, we may as well take the above as the exact duration of the 1900.0 tropical year, in SI seconds.

There are exactly 86400 ephemeris seconds in an ephemeris day (by definition of the latter), but this ephemeris day is an abstract unit of time, which is irrelevant to the calendar structure.  What we need is a precise estimate of the 1900.0 duration of a mean solar day, because actual solar days is what calendars are meant to count.  In fact, for historical reasons, the mean solar day was precisely equal to 86400 seconds around 1820 or 1826, and has been increasing at a rate of roughly 2 ms per century ever since.  In this context, a "second" (s) is an SI second, a unit now defined in atomic terms, which is virtually indistinguishable from the ephemeris second (it's not the solar second, which is defined as 1/86400 of the mean solar day, whose variable duration we are evaluating).  All told, the mean solar day of 1900.0 would have been about 86400.0016 s.

Calendar Reform   |   Astronomical Leap Year Rule   |   Vernal Equinox 1788-2211