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Final Answers
© 2000-2014   Gérard P. Michon, Ph.D.

Calendars

I've been on a calendar,
but I've never been on time
.
 Marilyn Monroe  (1926-1962).
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Related articles on this site:

Related Links (Outside this Site)

Calendrica (Conversion Applet)   |   Today's Calendar and Clock Page
Religious Calendars   |   New Year Celebrations Galore   |   1751 = 282 days
Equation of Time   |   Time Systems   |   Time   |   Time Scales  by  Steve Allen
Historical Values of Delta T  by  Fred Espenak  (NASA).
Time by David Madore   |   Epochs and Eras   |   Milestones in Solar Astronomy
The Metonic Cycle and the Saros   |   Calendars trhough the Ages
Calendars in Singapore by Helmer Aslaksen
Mything Links  by Kathleen Jenks   |   Calendars  by Bill Hollon.
The Calendar Zone  by Janice McLean   |   Calendars  by L. E. Doggett.
Claus Tøndering's FAQ   |   The Christian Calendar  by Claus Tøndering.
Easter, Rosh Hashanah and Passover  (Conway's rules)  by William H. Jefferys.

Wild Speculations :

Dating creation  (Wikipedia)  |  Mayan EasterEnoch Calendar  by John P. Pratt
God's Calendar  by  Wade Cox  (Christian Churches of God, 1996-2000).
Bible Exposed  by  John Howard  (thatch343)
 
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Chronology & Calendars

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 89th 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 (until sunset):   Yom hamishi, 17 Nisan, 5716 AM (= Anno Mundi).
  • Islamic (til sunset):   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 (since sunrise):   7 Cumku 5 Cauac (Long Count:  12.17.2.7.19).

Nautilus (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.

Modern Calendrical Ratios

Precise astronomical formulas have been devised at the  Bureau des Longitudes  (Paris, France)  which give the number of days in a synodic lunar month or in a tropical year.  (Those are valid for millenia  but would fail for geological periods.)  Let's quote the lunar model of Michelle Chapront-Touzé & Jean Chapront  (1988)  and the orbital elements of  Jacques Laskar (1986) :

month/day   =   29.5305888531 + 0.00000021621 T - 3.64E-10 T2
year/day   =   365.2421896698 - 0.00000615359 T - 7.29E-10 T2 + 2.64E-10 T3

In both formulas,  T  is the time expressed as the number of Julian centuries of 3652425 "atomic" days  (i.e., 3155760000 s )  elapsed since 2000.0.

Those are just  mean  values:  The actual number of days between consecutive new moons may differ from the above by as much as 0.3  (i.e., 7 hours).  The above average number of days in a tropical year may differ by several minutes from the actual number of days observed from one vernal equinox to the next.

Ocean tides provide a braking mechanism which slows down the spinning of the Earth about its axis, thereby increasing the duration of the day.  Total angular momentum is a conserved mechanical quantity, so whatever is lost in the spin of the Earth goes  (mainly)  to the orbital angular momentun of the Moon around the Earth.  A lesser transfer of angular momentum affects the orbital motion of the Earth around the Sun.  Those two effects contribute, respectively, to an increase in the absolute durations of the month and the year.

By itself, tidal braking would increase the length of a day at a rate of  2.3 ms  per century.  However, observed historical eclipses show the actual increase to be only  1.7 ms per century.

The difference  (i.e.,  an additional  decrease  of 0.6 ms per century)  has been attributed to a reduction in the oblateness of the Earth since the last  ice age.  There may be a periodic oscillation in the shape of the Earth, in addition to its secular  (irrevocable)  decay.

As the Earth spins less rapidly, its buldge at the equator must be reduced.  Indeed, it's been observed that, in the main, major earthquakes tend to redistribute mass so as to reduce the equatorial buldge.

The sands of time and tidal friction  by  Leslie V. Morrison and F. Richard Stephenson   (1998)


(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 2.3 milliseconds per 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 start 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  (0:00 UTC)  on Wednesday November 17, 1858  (JD = 2400000.5, JDN=2400000).

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.  Conversely, if we use a perfect  lunar  calendar and estimate the solar year to be 235/19 lunar months, we'll drift away from the solar seasons at a rate of less than half a day per century  (this is essentially how the Jewish calendar is constructed). 
  • 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:

 Come back later, we're
 still working on this one...

Leap Seconds   |   Julian Date


(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:

NameDaysWhen / What CircumstancesWho / Where
Decan10AntiquityEgypt
Week7AntiquityJews, Persians
 8AntiquityRome
7Since 1st Cent. AD / Persian Astrology
Since AD 321 (officially) / Christianity
 9Until 1385 (officially)Pagan Lithuania
Décade101793-10-24 to 1806-01-01France
 51929 to 1931Soviet
Union
61931-09-01 to 1940-06-26
Week7Modern TimesWorldwide
 
Etymology
Number CelestialFrenchEnglishNorse
0SunDimanche (Lord's day)Sunday 
1Moon, LuneLundiMonday 
2MarsMardiTuesdayTiw's day
3MercuryMercrediWednesdayWoden's day
4JupiterJeudiThursdayThor's day
5VenusVendrediFridayFreya's day
6SaturnSamediSaturday 

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


Eye of HorusEye of Ra (2002-12-29)   Egyptian Calendar & Calendar Creep
This solar calendar paved the road for its successors.

When Moses was alive, these pyramids were a thousand years old.  Here began the history of architecture.  Here, people learned to measure time by a calendar, to plot the stars by astronomy and chart the Earth by geometry.  Here, they developed that most awesome of all ideas—the idea of eternity.
Walter Cronkite  (1916-)  CBS evening news (1962-1981).

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


(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.

 Come back later, we're
 still working on this one...

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


(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.

 Ankh
 Crux Ansata 
 Ansate 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).
DaysCoptic Month (Egyptian Name)First Day (Julian) Nile Season
130Thoth, Thot, Thout, ThuthyAugust 29+ Inundation
Akhet
Akhet
230Paophi, Paapi, PaopySeptember 28+
330Athyr, Hathor, Hathys, AthorOctober 28+
430Cohiac, Kiahk, Koiahk, Kiak, Choiach November 27+
530Tybi, Tobi, TybyDecember 27+ Emergence
Proyet
Peret, Poret

Poret
630Mesir, Mechir, Menchir, MekhirJanuary 26+
730Phamenoth, Paremhat, FamenothFebruary 25+
830Pharmouthi, Paremoude, ParmuthyMarch 27
930Pachons, PakhonsApril 26 Summer
Harvest
"Low Water"
Shomu
Shemu

Shomu
1030Payni, Paoni, PaonyMay 26
1130Epiphi, Epip, Epipy, EpepJune 25
1230Mesori, MesoreJuly 25
135 or 6Epagomena, Little MonthAugust 24

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


(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.  Julius 
Caesar

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
For  AD 4,  this interpretation of reports from Macrobius and others is disputed by some scholars.
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 BCAfter 8 BC
DaysMonth DaysMonth
I31Ianuarius31Ianuarius
II29 or 30Februarius28 or 29Februarius
III31Martius31Martius
IV30Aprilis30Aprilis
V31Maius31Maius
VI30Iunius30Iunius
VII31Quintilis / Iulius31Iulius
VIII30Sextilis31Augustus
IX31September30September
X30October31October
XI31November30November
XII30December31December

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 DayWhenWho / where
March 1Until 222 BC (?)Rome
March 15222 BC - 153 BC (?)
January 1Since 153 BC (and 45 BC)
March 25Middle Ages
March 1Until 800France
March 25800 to 996
Easter
(See note below)
996 to 1566
"more Gallicano"
January 1Since 1567 (or 1563)
November 1Until AD 1179Celts
December 257th Century thru 1338England
March 25Late Middle AgesEurope
March 1Until 1797Venice
September 114th century thru 1918Russia
January 1Gregorian reformWorldwide

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


(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 guess of Dionysius may have been 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).  In 1583, Scaliger argued that 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 result, but we may venture the guess that he simply took the Gospel of Luke 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 may have been 34 or so).

The original task of Dionysius was to prepare a table giving the dates of Easter starting with AD 532.  In the Julian calendar, such a table has a periodicity of 532 years, so that it was tempting to place the birth of Christ at the beginning of the previous cycle.  Either that or Dionysius guessed the birth of Christ first, by some other argument, and then chose to have his tables start with the second cycle.

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 nativity 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


(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...
   Pope
 Gregory XIII

This calendar has been dubbed Gregorian because it was introduced under the authority of pope 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 (down 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).

Some Official Transitions to the Gregorian Calendar
CountryLast Julian DateThe Next Day ...
Italy, Poland, Portugal, SpainOctober 4, 1582October 15, 1582
France, LotharingiaDecember 9, 1582December 20, 1582
Luxembourg December 14, 1582December 25, 1582
Holland, Brabant, FlandersDecember 21, 1582January 1, 1583
Austria, BohemiaJanuary 6, 1584January 17, 1584
Hungary (popular use since 1584)October 21, 1587November 1, 1587
Denmark, NorwayFebruary 18, 1700March 1, 1700
Cities:  Pisa, Florence, Venice(?)December 20, 1750January 1, 1751
England & British dominionsSeptember 2, 1752September 14, 1752
Sweden (1700-1712: Julian+1)February 17, 1753March 1, 1753
Japan[ Japanese calendar ]January 1, 1873
Alaska [crossed date line!]October 6, 1867October 18, 1867
Egypt18751875
China[ Chinese calendar ]January 1, 1912
Soviet UnionJanuary 31, 1918February 14, 1918
GreeceFebruary 15, 1923March 1, 1923
Romania (in use since 1919)September 30, 1924October 14, 1924

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


(2007-05-26)   Counting the days between two Gregorian dates
How to go back and forth between a Gregorian date and a straight count.

We present a way to go from a count of days to a Gregorian date and vice-versa, using only simple arithmetic formulas.  This makes it easy to determine how many days there are between two distant dates.

Making Mincemeat of Monthly Irregularities :

The key trick to deal with the not-so-regular pattern of varying numbers of day per month is to  pretend  that the year starts on March 1  (as it did when the months got their current Latin names).  This makes it possible to work out the following simple formula, which I designed back in July 1978.

Counting Days and/or Months from the previous March 1st
Mar.Apr.MayJuneJulyAug. Sep.Oct.Nov.Dec.Jan.Feb.
m 012345 67891011
 Nm  0316192122153184 214245275306337
Day  N  corresponds to month number   m  =  floor ( (N+0.5) / 30.6 ).
Conversely, the first day of month  m  is   Nm  =  ceiling ( 30.6 m - 0.5 ).

For the record, here's a full analysis which determines exactly how far one can wander away from the values 30.6 and 0.5 which appear in the above formulas  (In 1978, I wanted to obtain safe  binary  fixed-point values.)

If N is the number of days (from 0 to 365) elapsed since the previous March 1, we want the number of months elapsed (0 to 11) to be  floor ((N+y)/x).

The result will be correct for March if an N from 0 to 30 gives a result of 0, which means that (N+y)/x is between 0 (included) and 1 (excluded) when N is between 0 and 30.  This is true if and only if y is 0 or more and 30+y is less than x.  With similar constraints for the other months, we have a total of 24 inequalities to satisfy.  However, only 4 (or 5) of those are "critical", as they define the inside of a small quadrilateral in the (x,y) plane where all 24 inequalities are satisfied.  (The fifth "critical" inequality is  y < 6x-183.  It corresponds to the last day of August.  Its constraining line grazes the following  convex  solution quadrilateral  at corner B.)

 domain of 
 acceptable parameters

Upper boundary (excluded):
y < x - 30   (last day of March)   AB
y < 11x - 336   (last day of January)   BC
 
Lower boundary (included):
y ≥ 4x - 122   (first day of July)   CD
y ≥ 9x - 275   (first day of December)   DA

A = (30.625, 0.625)     B = (30.6, 0.6)
C =  ( 30 4/7 , 2/7 )     D = (30.6, 0.4)

Corners A, B and C are excluded.  The point D is included, so the value  y = 0.4  is barely acceptable with  x = 30.6.  The  best decimal value  is, of course, the middle of BD, namely:   x = 30.6   and   y = 0.5.  For low-level binary routines (my original concern, in 1978)  we may retain  y = 0.5  and use any value of  x  between 673/22 = 30.59090909... and 551/18 = 30.611111...  This is the interval represented by the red line in the above diagram.  In binary numeration:

11110.10010111010001011101000...  At least.
11110.10011100011100011100011...  At most.
11110.10011  (hex 3d3) is thus the coarsest usable value.

(In other words, we may use  30+19/32  = 30.59375  instead of  30.6.)
With  y = 0.5,  1/x should be between 18/551 and 22/673.  In binary, this is:

0.000010000101110011101100...     At least.
0.000010000101111001010101...     At most.
      10000101111   is the coarsest usable binary value.

If we multiply that 11-bit integer (42F in hexadecimal) by one plus twice the number of days, we obtain the month number by discarding the lower 16 bits of the product.  So, the following piece of 68000 assembly language turns a number of days  (0 to 365)  from the lower 16-bit word of D0 (a 32-bit register) into the corresponding  0-11  month number  (the other half of D0 becomes junk).

E340       N2MONTH  ASL.W   #1,D0     Multiply by 2
5240                ADDQ.W  #1,D0     Add 1 (i.e., add 0.5)
C0FC 042F           MULU.W  #$42F,D0  Multiply by 1/30.6
4840                SWAP    D0        Get integer result
                    RTS

Conversely, if the lower half of D0 contains a month number (0 to 11) we may obtain the day number (0 to 337) of the first day of that month, using the code:

C0FC 03D3  MONTH2N  MULU.W  #$3D3,D0  Multiply by 30.6
0640 0010           ADDI.W  #$10,D0   Add 0.5 (= 1.0 - 0.5)
EA48                LSR.W   #5,D0     Get integer result
                    RTS

If I may say so, I'm proud of my younger self for pioneering this, almost 30 years ago (time flies).  I just had a nice time retracing my own footsteps, as my 1978 notes are lost  (I did remember the quadrilateral's shape and the 30.6 value).  Halmos

Complete Conversion Algorithms :

With the issue of months out of the way, other Gregorian calendrical computations are straightforward if we consistently put exceptions at the  end of their respective periods, just like we put February at the end of each year in the above...  A leap year (366 days) is at the end of an olympiad of 1461 days.  A short olympiad of 1460 days (no leap year) is at the end of a normal century (36524 days).  A long century (36525 days = Julian century) is at the end of each Gregorian period of 400 years (exactly 146097 days).

With those conventions, everything falls into place nicely if we start counting days from what would have been the Gregorian date March 1 of year 0,  if  the Gregorian scheme had been in place back then  (in the proleptic Julian calendar actually used for that period of history, "year 0" is called "1 BC" or "1 BCE").  Because of our original trick  (which made it so easy to count months within a year)  we merely have to increment the year for the months of January and February so they belong to the same year as the following month of March, as is the modern usage...  That's all there is to it !

The Modified Julian Day Number  is 0 for November 17, 1858  which came  678881 days  after the above  arithmetically convenient  origin.  Therefore, we'll use that offset in an actual implementation which turns our counting of days into straight conversions to and from MJDN dating.

 Gregorian date, as a TI-92 function. 
 NOT valid for early Julian dates !

The screenshot at right shows how this can be implemented on an handheld calculator, like the TI-92, TI-89 or  Voyage 200  from Texas Instruments.  The  gdate  function takes an integer (although it also allows fractional numbers) interpreted as MJDN  and returns the corresponding Gregorian date in the format used by the calculator itself to read its own real-time clock  [ when  getDate ( )  is called ]  namely a list of the form  { year  month  day }.

MJDN to Gregorian
conversion, with early
proleptic Julian dates

 Gregorian date, as a TI-92 function. 
 VALID for early Julian dates too.
 

As the Gregorian calendar is  never  used for dates before October 15, 1582  (a negative MJDN of -100840)  we  must  modify the above to use the  proleptic  Julian calendar for all earlier dates  (by skipping Gregorian century rules and using an offset matching the Julian calendar).  The proper code shown at left can accomodate any switch date:  Simply replace -100840 by the MJDN of the earliest Gregorian date acceptable to  you,  if it's not October 15, 1582.

   Julian date, as a TI-92 function.
 Julian day, as a TI-92 function.
 Gregorian day, as a TI-92 function.
For Gregorian to Julian conversions, it is useful to have a version of the above which  never  switches to the Gregorian calendar.  We call it  jdate  (for "Julian date") and the simple code for it is given by the screenshot at right.

The two functions, dubbed  jday  and  day,  do the opposite of the above, namely they take a Julian or Gregorian date (respectively) and return the corresponding day number  (MJDN).

We do allow months outside of the 1-12 range to denote months of the previous or following year(s).  Likewise, the number of days can be outside the 1-31 range and  may be fractional. (Fractional years or months are  not  allowed.)  Year 0 is 1 BC, Year -1 is 2 BC, Year -2 is 3 BC, etc.  All of this is compatible with astronomical standards ;-)

The function  day  calls  jday  when it finds an MJDN that's below the earliest acceptable Gregorian date  (again, you may change the -100840 value to the switching MJDN of your choice).  Therefore,  day  correctly interprets  {1582,10,14}  as a deprecated Julian date, 9 days after {1582,10,15}.  Nice.

The above four functions present great computational flexibility.  For example:

date ( jday ( {yyyy,mm,dd} ))   obtains a Gregorian date from a Julian one.
jdate ( day ( {yyyy,mm,dd} ))   obtains a Julian date from a Gregorian one.
date ( day ( {yyyy,mm,dd} ))    puts a "generalized" date in standard form.
day({y2,m2,d2}) - day({y1,m1,d1})   is the difference in days between two dates.
jday({Y,1,1}) - day({Y,1,1})   is the Julian lag at the beginning of year Y.
day(getDate()) - day({1956,3,29})   is my current age, in days.
date(10000 + day({1956,3,29}))   is when I was 10,000 days old (Aug. 15, 1983).
date(20000 + day({1956,3,29}))   is when I'll be 20,000 days old (Dec. 31, 2010).
date(-2400000.5)   is  {-4712,1,1.5}.  That's Julian date 0.0  (defined by the IAU).
date(0)   is  {1858,11,17}  namely  MJD = 0.0  (Nov. 17, 1858 at 0:00 GMT).

The day of the week (0=Sunday, 1=Monday, 2=Tuesday, 3=Wednesday, 4=Thursday, 5=Friday, 6=Saturday) is given by the equivalent expressions:

mod ( 3 + day ({yyyy, mm, dd}) , 7 )
 mod ( 3 + jday ({yyyy, mm, dd}) , 7 )
 mod ( 3 + hday ({yyyy, mm, dd}) , 7 )

There's almost no legitimate need for projecting the Gregorian scheme into the distant past (before 1582) as the proleptic Julian calendar is universally used for that purpose by astronomers and historians alike.  The one useful purpose for a "pure" Gregorian scheme  (as first presented in our introductory gdate screenshot)  would be to find out the correct seasonal date for a yearly celebration of some event that happened well before the Gregorian calendar ever existed...

This would be similar to what George Washington did when he adjusted his own birthday to a Gregorian date  (February 22, 1732)  although it had first been recorded as February 11, in the Julian calendar used at the time in Great-Britain and in the "American Colonies".  In what would become the U.S., the switch occurred on  September 14, 1752,  when Washington was a young adult.

Of course, at the time of Washington's birth, the Gregorian calendar was already legitimate  somewhere else,  This is why the above date and jday functions are sufficient to check Washington's computation.


(2007-06-01)   How Old is the Moon?
The average synodic month is  29.530588853 days.

For calendrical purposes, we may consider only the  average  motion of the Moon based on the above period.

In traditional  lunar calendars,  a month starts with the actual observation of the thin crescent of a  new moon,  which typically takes places a day or two after the astronomical new Moon  (when the Moon is invisible).

We may define the latter as halfway between two full moons and take the middle of a total lunar eclipse as an "accurate"  full moon.  If we use for that purpose the lunar eclipse of March 3, 2007 (which started at 22:43 and ended at 23:58 UTC) we obtain the following formula for the  age of the Moon, in days:

mod ( 10.8927 + n ,  29.530588853 )   ®   moon (n)

That's how a  TI-92  function may be defined whose argument is the number  n  (the Modified Julian Date)  which is featured prominently in the previous article.  This is to be used jointly with the calendrical functions presented there.

 Crescent 
 Moon

The Islamic Month

The arithmetical version of the  Islamic calendar  is based on a cycle of 10631 days, divided into 360 months.  This yields an average month of :

10631 / 360   =   29.530555555... days

That's about  2.8769 s  short, compared to the astronomical average.  It would take about 2428 (tropical) years to build up a discrepancy of a whole day.

 Star of 
 David

The Jewish Month

The  arithmetical  Jewish calendar (the Hillel calendar) is based on the following estimate of the time between consecutive new moons:

765433 / 25920   =   29.530594135802469135802469... days

This is about 0.4564 s  short, compared to the astronomical average.  It would take about 15305 years to build up a discrepancy of a whole day.


Incidentally, the (long term) average of the Hillel year is obtained by multiplying the above by 235/19 (the Metonic approximation to the number of synodic lunar months in a tropical year).  This boils down to  35975351 / 98496  or:

365.246822205977907732293697205977907732293697... days

That's longer than the tropical year (at epoch  1900.0) by 399.4639 s.  Thus, the Hillel calendar drifts with respect to the solar seasons at a rate of about one day in  216 years  (more precisely, 3 days in 649 years, 7 days in 1514 years, 31 days in 6705 years or 69 days in 14924 years).  The Jewish Spring festival of Passover is moving toward the Summer at that rate.  On the  average,  Passover now occurs more than one week later (with respect to the solar seasons) than it did in the times of Hillel II.  The Jews are thus facing a problem similar to the 10-day offset which was bugging Christian authorities before the Gregorian reform of 1582.

The synchronization of the Jewish calendar with the seasons is not nearly as critical as its synchronization with the lunar cycle (a new moon occurs near the beginning of every month).  The large effect of intercalary months on Jewish festivals drowns the tiny drift of those festivals, at a rate of less than half a day per century...


 Christian  
 Cross (2003-02-20)   The Date of Easter
The resurrection of Christ is celebrated on Easter Sunday, reckoned as the Sunday following the  Paschal Full Moon.

According to Christian tradition, Jesus Christ was crucified on a Friday which fell just before the festival of  Passover  (15 Nissan)  which is always near a full moon.  The 14th of Nissan actually fell on a Friday on the following Julian dates:

  • 7th of April in AD 30.
  • 3rd of April in AD 33.

In 1733, Sir Isaac Newton had argued for the next year (AD 34) but this would only be possible with a Jewish calendrical rule of postponement that was not yet enforced at that time!  On the other hand, the baptism of Christ is clearly stated to have occurred during the 15th year of Tiberius Caesar (Luke 3:1).  which corresponds to AD 29 in the Julian calendar.  As the ministry of Christ covered 3 full years from that point on,  the day of the Crucifixion would be firmly established to be April 3rd of AD 33.  A partial lunar eclipse was visible at moonrise from Jerusalem on that very day, so that  the Moon appeared like blood.

Following J.K. Fotheringham (1910) who reconstructed the Jewish calendar astronomically (just like Newton had done, two centuries earlier) some scholars advocate AD 30 as the date of the Crucifixion.  This is obtained by reconciling an "age of Christ" of 33 years  (resulting from the speculations of Dionysius popularized by Bede)  with the belief, spread by Scaliger in 1583, that  King Herod  died in 4 BC...

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...

You should work entirely within the Julian calendar (C = +3) to find when Easter is celebrated by Orthodox churches.  If it doesn't take place on the same Sunday, such a celebration normally occurs 1, 4 or 5 weeks after the Gregorian date of Easter.  However, this will not always be so in the distant future as the calendars drift apart and the Julian  Pascal Full Moon  is no longer a good approximation of an actual full moon.  The pattern is first broken in 2437, when Gregorian Easter occurs on March 22, whereas the Julian version would be scheduled 6 weeks later, on May 3  (that's April 17, in the Julian calendar).  The Gregorian reform was precisely engineered to avoid this slow creep of Easter toward summertime.

(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  |  Passover: the Jewish and Christian versions  |  Ecclesiastical Calculator

Ecclesiastical Calendar :

For Christians, Fixed Holidays occur at fixed dates in the Gregorian calendar  (or in the Julian calendar for Orthodox churches)  whereas Moveable Holidays depend on the date of Easter Sunday (as computed above).

  • Lent  is the period of 40 days between Ash Wednesday and Easter.
  • The  Advent  is the period from Advent Sunday to Christmas (Dec. 25).
Fixed Christian Holidays
DateNameCelebration
6 JanuaryEpiphanyAdoration of the Magi
2 FebruaryCandlemas40th day of Christmas
25 MarchAnnunciation9 months before Christmas (Lady Day)
24 JuneSt. John the BaptistSummer Solstice
15 AugustAssumptionAssumption of Mary
29 SeptemberMichaelmasAll Angels  (Fall Equinox)
31 OctoberHalloweenAll Hallows Eve
1 NovemberAll HallowsAll Saints
2 NovemberAll SoulsEverybody
4th Sunday bef. XmasAdvent SundayBeginning of Advent
25 DecemberChristmasNativity of Jesus of Nazareth
 
Moveable Christian Holidays
DateNameCelebration
70 days before EasterSeptuagesima10 weeks before Easter
40 days before EasterShrove TuesdayLast day before Lent  (Mardi Gras)
39 days before EasterAsh WednesdayBeginning of Lent (40-day fast)
7 days before EasterPalm SundayBeginning of Holy Week
Thursday before EasterMaundy ThursdayLast Supper  (Eucharist)
Friday before EasterGood FridayCrucifixion of Jesus Christ
(Sunday after PFM)EasterDay of Resurrection
1 day after EasterEaster MondaySecond day of Easter
39 days after EasterAscension ThursdayAscension of Jesus Christ
49 days after EasterPentecost SundayHoly Spirit upon the Apostles  (Whitsun)
50 days after EasterWhit MondayPentecost Monday
56 days after EasterTrinity Sunday8 weeks after Easter

"Ordinary times" are counted from Trinity and end with Advent Sunday.

The Courts of England and Wales divide their year into 4 terms whose names are borrowed from the above ecclesiastical calendar:  Hilary (celebrated on January 14), Easter, Trinity and Michaelmas.  So do British universities with slightly different term names, as summarized below.  In 2004, Newcastle decided to drop traditional names in favor of  "culturally neutral"  ones  (Autumn, Spring, Summer)  like most American universities  (Fall, Spring and Summer quarters).  The traditional British academic year starts with the  Michaelmas term.

  • Michaelmas Term:  From October to December.
  • Lent (Cambridge), Epiphany or Hilary Term (Oxford):  January to March.
  • Easter Term (Trinity Term in Oxford only):  From April to June.
  • Trinity Term (judicial system only):  From June to September.


 Crescent Moon (2002-12-28)   Hegira Calendar   [AH = Anno Hegirae]
The Islamic calendar is called  Hijri  (or Hijrah calendar).

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

The numbering of years from the date of the Hegira was introduced in AD 639  (17 AH) by the second Caliph, 'Umar ibn Al-KHaTTab (592-644).  The monthly Islamic calendar itself had already been in use since  AD 631  (10 AH) as the Quran prescribes a  lunar  calendar  without  embolismic months  (9:36-37).

Before  10 AH, a long forgotten "Arabian calendar" was probably used, which was similar to the Jewish calendar and had an intercalary month, now and then, in order to compensate for the steady drift of the lunar cycle with respect to the solar seasons.

Since an Islamic year (12 lunar months) falls shorts of a tropical year by almost 11 days, the Islamic calendar isn't related to the seasons.  Muslim festivals simply drift backwards and return roughly to the same seasonal point after a period of 33 Islamic years (which is 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.

Tabular Islamic Calendars :

Printed Islamic calendars are most commonly based on standard  arithmetic  predictions of moon sightings.  We present the most common of 8 extant variants.

Such schemes were devised by Muslim asronomers  after the eighth century CE.  Historians use this routinely to convert an Islamic date to a Gregorian one  (unless a knowledge of the day of the week allows a precise synchronization with the relevant local observational calendar).

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 NameDays
1MuHarram30
2Safar29
3Raby' al-awal30
4Raby' al-THaany29
5Jumaada al-awal30
6Jumaada al-THaany29
7Rajab30
8SHa'baan29
9RamaDHaan30
10SHawwal29
11Thw al-Qi'dah30
12Thw al-Hijjah29 or 30
  30 Islamic years
(10631  days)
01020
11121
21222
31323
41424
51525
61626
71727
81828
91929
 Crescent 
 Moon

Important Islamic Celebrations  
DateArabicEnglish
10 MuHarramAshura Remembrance of Muharram
12 Raby' al-awalMaulidun-Nabi Birth of the Prophet
27 RajabLaylatul-Mi'raj Night of Ascension
15 SHa'baan  Laylatul-Bara'ah  
(Shabi-Baraat)
Night of Record
1 RamaDHaanRamaDHaan   Fast of Ramadan (first day)  
27 RamaDHaanLaylatul-Qadr Night of Power
1 SHawwalEid al-Fitar Breaking of the Fast
  10 Thw al-Hijjah   Eid al-Adha Sacrifice Festival  [3½ days]

Hijri Calendrical Formulas :   (2007-06-22)

The average Islamic year (12 months) is  10631 / 30 = 354.36666... days.  If day 0 (zero) is the first day of the above cycle of 30 Islamic years, then the number of the year to which day N belongs equals  floor ((30N+k)/10631)  provided  k  is between  26  (included)  and  27  (excluded).

Such a choice of  k  ensures that two critical inequalities are satisfied:  For year 16 to be longer than year 15, day 5669 must belong to year 16 and not 15.  This requires  k  to be  at least  16*10631-30*5669 = 26.  On the other hand, for year 26 to be longer than year 27, day 9567 must be in year 26 rather than 27, which implies that  k  must be strictly less than  27*10631-30*9567 = 27.  These two inequalities are sufficient to satisfy the 60 constraints imposed by the entire 30-year pattern.

Using  k = 26,  we obtain a formula  (valid for an  indefinite  number of 30-year Islamic cycles)  which gives the Islamic year Y corresponding to day N :

Y   =   floor ( [ 30 N + k ] / 10631 )

Conversely, the number NY corresponding to the first day of Islamic year Y is:

NY   =   ceiling ( [ 10631 Y - k ] / 30 )

Subtracting this quantity from the original day number (N) we obtain a number N' from 0 to 354 within the Islamic year.  From this number N', the Islamic month is not difficult to obtain.  Conversely, we may also get the number of the first day of the month.  The number within the month is obtained by subtracting that from N'.

All this can be embodied into two computer routines which convert a day number to an Islamic date (hdate) and vice-versa (hday).  For compatibility with the similar routines for the Gregorian and Julian calendars, we count days from the MJDN origin, with an offset of 451915 days.  The following implementations are for the TI-92, TI-89 and  Voyage 200  handheld calculators.  

 Hijri date, as a TI-92 function.    Hijri day, as a TI-92 function.

hday({y2,m2,d2}) - hday({y1,m1,d1}) is the number of days between two Hijri dates.
hdate ( hday ( {yyyy,mm,dd} )) puts a "generalized" Hijri date in standard form.
date ( hday ( {yyyy,mm,dd} )) obtains a Gregorian date from a Hijri date.
hdate ( day ( {yyyy,mm,dd} )) obtains a Hijri date from a Gregorian one.

Competing Variants of the Tabular Islamic Calendar :

The aforementioned date of  Friday  July 16, 622 CE  is by far the most common starting point of the Hegira calendar, but it may also be reckoned from  Thursday  July 15, 622 CE.  This so-called "Thursday" calendar would be obtained with an offset of 451916 days (instead of 451915) in both of the above routines.

There are no fewer than  30  regular  intercalatory patterns which would be based on the same 30-year period (of 10631 days) as above.  Apparently, only four of those have ever been advocated (as tabulated below).

The four extant intercalatory patterns agree that years 2, 5, 13, 21 and 24  (modulo 30)  are "long" years of 355 days, but disagree on some of the remaining 6 long years in the 30-year cycle of 10631 days.  This amounts to different values of  k  for the calendrical formulas introduced above  (with k = 26) :

The  8  Extant Variants of the Tabular Islamic Calendar
Long Years (modulo 30) kminkmax  Friday   Thursday
 2, 5, 7, 10, 13, 15, 18, 21, 24, 26, 29 25< 26IcIa
 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29  26< 27 IIcIIa
 2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 29 29< 30IIIcIIIa
 2, 5, 8, 11, 13, 16, 19, 21, 24, 27, 0 1< 2IVcIVa

To make the above calendrical functions match your favorite variation:

  • Change the constant 26 (which appears 3 times) into 25, 26, 29 or 1.
  • Use 451915 for a Friday calendar, or 451916 for a Thursday calendar.

The above numbering of the 8 extant variants of the Tabular Islamic Calendar follows the classification given by Robert Harry van Gent, who calls "civil" (c) the tabular calendar based on the usual starting point of  July 16, 622 CE  and "astronomical" (a) the "Thursday" calendar based on a July 15 starting point.

Ic - Kushyar ibn Labban  (AD 971-1029).
- Ulugh Beg  (AD 1393-1449).
- Convertisseur de dateMinistère des Habous et des Affaires Islamiques.
- Gregorian-Hijri Dates Converter, by Waleed Muhanna.
Ia - Microsoft's algorithm  (misleadingly called the Kuwaiti algorithm).
- Islam Online's Date Converter.
- Al-Islam's Agenda-Date Converter  (Ministry of Islamic Affairs, Saudi Arabia).
IIc - Gnu Emacs editor  (courtesy of Dershowitz and Reingold).
- Calendrica, by Edward M. Reingold and Nachum Dershowitz.
- Numericana, by Gérard P. Michon.
- Calendar Magic, by Alex Balfour.
- Today's Date, by Doug Zongker.
- Java Calendar Conversions, by Mark E. Shoulson.
- Fourmilab's Calendar Converter, by John Walker.
- Conversion of Islamic and Christian dates, by Johannes Thomann.
- Muslim Holidays.  Dates of Religious and Civil Holidays Around the World.
- Hijri/Gregorian/Julian Converter, by Tarek Maani.
- The Islamic Calendar, by Claus Tøndering.
IIIa - Fatimid Calendar.  Misri calendar.  Bohra calendar.
- Date Exchange, by Sualeh Fatehi.
- Hijri Calendar  (Dawoodi Bohra Version).
IVa - Ahmad ibn 'Abdallah Habash al-Hasib al-Marwazi  (d. ca. AD 870).
- Abu Arrayhan Muhammad ibn Ahmad al-Biruni  (AD 973-1048).
- Elias of Nisibis or Elias bar Senaya,  Patriarch Elias I of Tirhan  (1028-1049).

Many of the above authors do point out that such arithmetic approximations are not a substitute for the actual observational Islamic calendar sanctioned by religious authorities.  Reingold and Dershowitz  (Calendrica)  also provide an "observational" [sic] Islamic calendar, based on more precise astronomical computations to better predict the religious beginning of each Islamic month.

Crescent Moon Visibility and the Islamic Calendar (USNO)   |   Islamic Calendar(s)
The Umm al-Qura Calendar of Saudi Arabia   |   July 16, 622   |   Islamic Calendar


 Star of  David (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 (embolismic) month:  In leap years, this 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 the first version of its modern rules was established (in 358-359 CE) under the authority of Hillel II, president  (nasi)  of the Great Sanhedrin  (the highest Jewish court of law, which existed until the rabbinic patriarchate was abolished  c. 425 CE).  Before this  arithmetical  calendar was established, the Sanhedrin was issuing a monthly ruling to determine the beginning of the month, based (at least in part) on eyewitness accounts of actual sightings of the thin crescent of the new moon.

A Jewish day begins in the evening.  For calendrical computations,  Rambam time  is used which begins  (0h)  precisely a quarter of a day after high noon (solar time) in Jerusalem.  The calendar computed for Jerusalem is simply applied to other parts of the World according to local time.

The Rambam is how most students of Judaica call Rabbi Moshe ben Maimon (1135-1204) a prolific scholar born in Spain, also known as Moses Maimonides.  He developed the 13 Principles of Faith and authored the  Mishneh Torah,  an extensive code of Jewish Law whose 14th volume (written c. 1178 CE) is entitled  Hilchot Kiddush HaChodesh  ("Sanctification of the New Moon")  and deals with calendrical matters.

The day is divided into  24  fixed  hours of 1080 "parts" (halakim) each.  An interval of 10 seconds is thus 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 subdivided into 76 rega'im.  The rega is thus 5/114 of a second, or about 43.386 ms  (that unit of time is not used in calendrical computations).

The helek is equal to the ancient Babylonian  she  (barleycorn of time)  which is the 72nd part of the main Babylonian unit of time, the degree of time (itself equal to 4 modern minutes or 1/360 of a mean solar day, which is the time it takes the Sun to move one angular degree).

The Jewish tradition  (Mesorah)  gives the average duration of a lunar month to the nearest  helek  (there are 25920 halakim in a day) :

29 days, 12 hours and 793 halakim

That value, of  765433 halakim per month, is said to date back to the time of Moses in the Sinai, but it's also given prosaically in Ptolemy's Almagest with an attribution to Hipparchus of Rhodes (190-120 BC) using the Babylonian sexagesimal fractional notation (which originated in the Seleucid era, after 312 BC) whereby 1'  is 1/60 of a day, 1''  is 1/60 of that (1/3600 of a day) etc.

29 days  31' 50'' 8''' 20''''     [ NB:  1 helek  =  8''' 20'''' ]

That duration remains a whole number of halakim although the notation is capable of a precision 500 times greater.  We may infer that Ptolemy and/or Hipparchus were quoting the value recorded  to the nearest helek  by ancient astronomers.

That traditional value is less than half a second  (456.4 ms)  above the modern mean synodic lunar month value of about 29.530588853 days,  which equals  29 days, 12 hours, 792.863 halakim.  In Babylonian terms, this would be:

29 days  31' 50'' 7''' 11'''' ½

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 simple 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.  In other words,  Y  is a leap year if and only if

mod ( 12 Y - 2 , 19 )   >   11

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 in Babylon (c. 600 BC).

The ancient names shown in italics are obsolete.
NumberMonth Name(s) HKSSeason
1Nissan, Nisan, Abib30March-April
2Iyar, Ziv29April-May
3Sivan30May-June
4Tammuz29June-July
5Av, Ab30July-August
6Elul29August-Sept.
7Tishri, Tishrei, Ethanim    [New Year]30Sept.-Oct.
8Cheshvan, Heshvan, Marheshvan, Bul 292930Oct.-Nov.
9Kislev 293030Nov.-Dec.
10Tevet, Tebet, Tebeth 29Dec.-January
11Shevat, Sebat, Shebat30January-Feb.
12Adar I   (leap years only) 30Feb.-March
12 or 13Adar  (Adar II in leap years)29March-April

Shabbat  is a time of weekly rest which lasts about 25 hours, from Friday evening to Saturday evening.  The beginning and end of  Shabbat  is a function of  local  solar time.  Shabbat  begins on Friday evening, 18 minutes before sunset  (Sheqiya)  itself defined as the time when the center of the Sun is  50"  (i.e., an angle of 5°/6)  below the horizon.  Shabbat  ends Saturday evening a few minutes after nightfall  (tzeit hacokhavim, the birth of stars)  at a time often said to be when 3 stars should become visible (in clear wheather).  This is computed as the time when the center of the Sun is  8.5 °  below the horizon.  The same rules are used to define the beginning and the end of any Jewish festival  (Yom Tov).

The first day of any Jewish month is a minor festival  (Rosh Hodesh)  except, of course, for the beginning of Tishri  (Rosh HaShanah)  which marks the beginning of the Jewish year.  Rosh HaShanah  is a strongly observed 2-day  celebration.  The 3 "pilgrimage festivals"  (Sukkot, Passover and Shavuot)  were occasions for mass pilgrimages to the Temple in Jerusalem before its destruction (in 70 CE).  Passover is second only to  Yom Kippur  in traditional observance.

Main Jewish Festivals
DateHebrewEnglish
1 TishriRosh HaShanah
(Yom Teruah)
Jewish New Year  (2 days)
(Day of Trumpets)
10 TishriYom KippurDay of Atonement
15 TishriSukkot Feast of Tabernacles  (first day)
21 TishriHoshanah RabbahSeventh day of Sukkot
22 TishriShemini AzeretEighth Day of Assembly
Simchat Torah(celebrated 23 Tishri outside Israel)
25 Kislev Hanukkah, Chanukkah Festival of Lights  (8 days)
14 AdarPurimJewish  Mardi Gras
(15 Adar)(Shushan Purim)  (Purim is postponed in walled cities)  
15 NissanPessah, PesachPassover  (7 days)
 27* Nissan Yom HaShoahHolocaust Remembrance Day
18 IyarLag B'Omer 
 6 Sivan ShavuotFestival of Weeks
* Moved from a Friday to the preceeding Thursday and from a Sunday to the next Monday.

The above table is for celebration of Jewish festivals in Israel.  The tradition is that most Jewish holidays are extended by one additional day for the Jewish Diaspora outside Israel.  Exceptions include Rosh HaShanah  (2 days for everybody)  and  Yom Kippur  (one day for everybody).

The reason for this general rule dates back to the times when the Jewish calendar was not yet arithmetical (before 359 CE).  Far from Jerusalem, the calendrical decision of the Sanhedrin for the current month might not be known early enough to allow the celebrations to take place on the "correct" day, so they were held for two days instead.  For Rosh Hashanah, which occurs at the very beginning of the month, the Sanhedrin's decision could not reach anybody in time (as messengers wouldn't be dispatched during holidays) so, everybody celebrates for two days.  On the other hand, the observance of  Yom Kippur for two days would be too much of a hardship...  So, there's  only one  "Day of Atonement" for everybody!

Jewish Calendrical Formulas :   (2007-06-24)

First, we treat  Jewish embolismic months  with the method we used for Islamic embolismic days.  Namely, we observe that the pattern of Jewish leap years is such that months can be numbered continously  (starting with number  0  for Tishri of Jewish Year 0)  so that the year  Y  to which month  X  belongs is simply:

Y   =   floor ( (19 X + k) / 235 )       where   5 ≤ k < 6

Conversely, the number  X  of the  first month  (Tishri) of year  Y  is the  least  X  which verifies the above equation, namely:

X   =   ceiling ( (235 Y - k) / 19 )     [let's use k=5]

The  precise  key to the Jewish calendar is  Molad Tishri, namely the exact time  M  of the new moon which occurs on  Rosh HaShanah  (or slightly before, as explained below)  expressed in days and fraction of a day to the nearest helek  (in Rambam time at Jerusalem).  A new moon  (molad)  is defined as the time when the  apparent longitudes  of the Moon and the Sun coincide.

The Hillel calendar is based on an arithmetical approximation to  M,  whereby consecutive new moons are exactly  765433/25920  days apart  ( 29d 12h 793 ).

The periodic pattern must be anchored at some  reference Molad.  In practice, Tracey R. Rich recommends one of the following [equivalent] starting points:

  • Y = 5558 :   X = 68744.   (1797-09-21) + 12487 / 25920  (= 11h 607)
  • Y = 5661 :   X = 70018.   (1900-09-24) + 11889 / 25920  (= 11h 9)
  • Y = 5759 :   X = 71230.   (1998-09-21) + 13965 / 25920  (= 12h 1005)

So calibrated, the  molad  corresponding to month number  X  is found to be:

M   =    765433 X  +  8255
Vinculum
25920

For X=13, this does give a value  M = 384 + 5604/25920  (namely, 5h 204)  for  Molad Tishri  of Year 1,  known as  Molad Tohu  (the molad of creation).  For X=25  (Molad Tishri  of Year 2)  we obtain a  whole  number of hours  (14h).  This "coincidence" would normally happen only once in 1080 years, but we're told that the calendar was actually cooked to make it so...  Since Tishri of Year 2 used to be counted in the first year AM  (when the  sacred  year started with Nissan)  it was thought that the "first Molad Tishri" ought to be a round number.

Our previous calendrical formula turns the above expression into a formula giving the  Molad Tishri  of Jewish year  Y  (Anno Mundi)  namely:

M   =    765433 ceiling ( (235 Y - 5) / 19 )  +  8255
Vinculum
25920

In our straight count of days,  floor (M)  is the day number for  Rosh HaShanah,  except when otherwise specified by one of  4  so-called  "rules of postponement"  (dehhioth, dehioth, dehiyyot or dechiyot is the plural of dehhiah or dechiyah). 

  • Dechiyah #1 :   Molad Zakein   ("Old Molad")
    Molad Tishri  must occur before noon  (18h, Rambam time)  or else  Rosh HaShanah  is postponed to the next day.

The traditional origin for that rule was that the thin crescent of the young moon had to be observable at sunset on Rosh Hashanah.

  • Dechiyah #2 :   Lo A"DU Rosh   ("No Beginning on Alef-Dalet-Vav")
    Rosh HaShanah  is postponed to the following day if it would otherwise fall on Sunday, Wednesday or Friday.

The word  A"DU  is a mnemonic for the hebrew letters alef, dalet and vav, whose numerical values (1, 4 and 6) correspond to Sunday, Wednesday and Friday.  Rosh HaShanah  is only allowed to fall on a Monday, a Tuesday, a Thursday or a Saturday;  one of the "4 gates" through which the new year must be entered.

That dechiyah is designed to avoid certain days of the week for some festivals.  For example, the seventh day of the Feast of Tabernacles  (Hoshana Rabbah)  shouldn't fall on a Saturday since the ceremony of "beating the willow twigs" involves work not permitted on Shabbat.  That rules out Sunday for Rosh HaShanah, 20 days before.  Similarly, the Day of Atonement  (Yom Kippur)  would fall immediately before or after Shabbat if Rosh HaShanah was allowed to occur on a Wednesday or a Friday...

Outside of the months of Kislev, Tevet, Shevat and Adar I, every date is a fixed number of days away from either the prior  Rosh HaShanah  (for Tishri and Heshvan)  or the next one.  So, a given date outside those months can only fall on 4 days of the week.  For example, since Nissan 27 can't fall on a Saturday, there was no need to spell out that case in the law which made "Yom HaShoah" the day not adjacent to Shabbat closest to the 27th of Nissan  (cf. note in the above table).
  • Dechiyah #3 :   Gatarad   ("Tuesday, 9h, 204")
    Gatarad is a Hebrew mnemonic (Gimel-Teit-Reish-Dalet) for what this rule states, namely that  Rosh HaShanah  is to be postponed when Molad Tishri  of a 12-month year  falls on a Tuesday  (Gimel = 3 = Tuesday)  on or after 9 hours (Teit = 9) and 204 halakim  (Reish = 200, Dalet = 4).

  • Dechiyah #4 :   Betutkafot   ("Monday, 15h, 589")
    Again, Betutkafot is a Hebrew mnemonic for that rule, which states that  Rosh HaShanah  is to be postponed if Molad Tishri  of a year following a 13-month year  occurs on Monday (Beit = 2) on or after 15 hour (Teit-Vav = 9+6 = 15) and 589 halakim (Tav-Qof-Fe-Teit = 400+100+80+9 = 589).

 Come back later, we're
 still working on this one...

Torah Search Engine  (4torah.com)
encyclopedia.com   |   Jewish Calendar Tools   |   Judaism 101 (A Closer Look)   |   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   |   Controversy in 921 CE   |   Le calendrier juif   |   Wikipedia


 Faravahar 
 Symbol (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  Atash 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


(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 still use it.  In the last column of the table below is the Gregorian correspondence most often used by "modern" astrologers...

Zodiacal SignPersian MonthFirst Day
AriesFarvardinMarch 21
TaurusOrdibeheshtApril 20
GeminiKhordadMay 21
CancerTirJune 22
LeoMordadJuly 23
VirgoShahrivarAugust 23
LibraMihrSeptember 23
ScorpioAbanOctober 23+
SagittariusAzarNovember 22
CapricornDayDecember 22
AquariusBahmanJanuary 20
PiscesEsphandFebruary 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:  There are 13 (!) modern zodiacal constellations with uneven shares of the ecliptic.  The 13th zodiacal constellation is  Ophiuchus,  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).

(2009-08-17)   Iranian Calendar   [SH = Solar Hejri]
The Persian year  (Anno Persico  or  Anno Persarum).

Like the  lunar  Islamic calendar,  the current Iranian calendar counts its years from the  flight from Mecca  (July 622).  However, it is strictly a  solar  calendar  (the Iranian year begins at the Spring equinox)  whose offset with the Gregorian year remains constant at  622  (it's only  621  between the Gregorian New Year, January 1, and the  Persian New Year, Nowruz).

In March 1925, the Persian parliament enacted calendrical rules which revived the names of the ancient Persian names of the months without giving them their traditional zodiacal duration.  Instead, the first 6 months  (Farvardin to Shahrivar) have 31 days, the following 5 months  (Mehr to Bahman)  have 30 days and the last month (Esphand) has either 29 days or 30 days.

The SH calendar year begins at midnight between the two solar noons  (on the  Tehran standard meridian  at 51.5°E)  which bracket the  vernal equinox.

Wikipedia :   Iranian Calendar(s)


(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 Maya 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 Maya 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 Maya 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 ).

It has been argued that the Maya considered a "Great Cycle" to be 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, 2012 CE.

In other words, it seems that the Maya would only give the leading digit of a Long Count modulo 13...  We prefer to ignore that line of thought and advocate the use of leading elements beyond 13 for the Long Count as needed, in the near future.

The "5 digit" Long Count system goes beyond 13 baktuns witout any difficulty, at least until  19.19.19.17.19  (Thursday, October 12, 4772).  This gives scholars a couple of millenia to decide what's to be done at that point with the calendrical legacy from the Maya.  The next day (Friday the 13th ;-) would require some innovation, like a sixth "digit" as a coefficient of a counting unit larger than the  baktun.  The Maya themselves devised no less than three such units:  The piktun, kinchiltun and alautun, worth respectively 20, 400 and 8000 baktuns.  An extended 8-digit Long Count based on those 3 additional units would span more than 60 million future years...  By that time, the Sun will still be just as bright as today, but the human species will (most probably) be long gone.

Mayan Calendrical Formulas :

The regularity of the Mayan Long Count makes calendrical formulas trivial...  On a TI-92, TI-89 or Voyage 200  handheld calculator, the function  mayaday  which takes a Long Count (as a list of 5 numbers) and returns the corresponding MJDN can be given the following one-line definition:

(((x[1] 20 + x[2]) 20 + x[3]) 18 + x[4]) 20 + x[5] - 1815718   ®   mayaday (x)

 Mayan Long Count, 
 as a TI-92 function.

Conversely, the screenshot at right shows how to define the function  mayadate  which takes an MJDN and returns the corresponding  Long Count,  as a list of 5 numbers.

Use either function with the functions  day  or  date  to turn a  Long Count  into a Gregorian date, or vice-versa.

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


(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.

 Come back later, we're
 still working on this one...

The Mathematics of the Chinese Calendar   |   The Chinese Calendar


(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


(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).

 Come back later, we're
 still working on this one...

Indian Calendars


    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.

 Come back later, we're
 still working on this one...

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

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Geologic Time Scale

The approximate date when each interval begins is shown in the first column, in
terms of the number of years (a) or millions of years (Ma) before the present time.
BeginsEon EraPeriodEpochAge of...
11 000 a P
H
A
N
E
R
O
Z
O
I
C
 
E
O
N
Cenozoic Quaternary Holocene Human Civilizations
1.6 MaPleistoceneHominids, Mammoths
5 Ma Tertiary Pliocene Advanced Primates
24 MaMiocene Grazing and
Carnivorous
Mammals
38 MaOligocene
55 MaEocene
65 MaPaleocene Mammal Dominance
138 Ma Mesozoic Secondary Cretaceous Primates, Flowers
205 MaJurassic Birds, Cycads
245 MaTriassic Dinosaurs, Mammals
286 Ma Paleozoic P
r
i
m
a
r
y
Permian Pangea
325 MaCarboniferousPennsylvanian Reptiles
360 MaMississippian Fern Forests
410 MaDevonian Amphibians, Insects
435 MaSilurian Land Plants
500 MaOrdovician Fish, Chordates
570 MaCambrian Shellfish, Trilobites
900 Ma Precambrian
Era
Proterozoic
Eon
NeoproterozoicMarine Algae
1800 Ma MesoproterozoicEukariotic Cells
2500 Ma PaleoproterozoicFree Oxygen Buildup
3900 MaArchean Eon (Archeozoic) Prokariotic Cells
4550 MaHadean Time Prebiotic Molecules
 Cyanobacteria
Precambrian
 Trilobite
Primary
 Tyrannosaur
Secondary
 Chriacus
Tertiary
 AD 1976
Quaternary
visits since Jan. 11, 2003
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