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

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

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 over geological periods.)
Let's quote the lunar model of
Michelle Chapront-Touzé & Jean Chapront (1988)
and the orbital elements of
Jacques
Laskar (1986) :

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 rotation of
the Earth about its axis, thereby increasing the duration of the day.
Total angular momentum is a conserved quantity. So what's
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.

(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 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 around 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 can be construed as a common
beginning to the following three noteworthy cycles
(which repeat after 7980 years).

The 28 year cycle of the Julian calendar.
The pattern of weekdays and leap years repeat after a 28 years 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
(that's how the Jewish
calendar is built)._{ }

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.

(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:

(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 doesn't quite bring the Egyptian calendar
back to the same point in the actual cycle of the seasons,
because a tropical year isn't exactly equal to 365.25 days:
It's more like 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).
Longer days mean fewer days in a year;
the number of days in a year decreases with time, and it was thus 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).

(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.

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

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

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

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

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

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

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

Historical Leap Years (before the Regular Julian Pattern)

43 BC

40 BC

37 BC

34 BC

31 BC

28 BC

25 BC

22 BC

19 BC

16 BC

13 BC

10 BC

AD 8

AD 4n

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...

Yet, the proleptic Julian calendar is now universally used to record the
dates of ancient astronomical events, including
28 May 585 BC
the probable day of a total eclipse predicted by Thales
(the earliest such prediction on record).
That omen may have helped put an end to
a six-year war but little credence should be given to the popular interpretation of a report
by Herodotus
(itself written more than one century after the facts)
that a solar eclipse literally interrupted the
final battle.

The Julian Calendar, before and after Augustus

45 BC to
8 BC

After 8 BC

Days

Month

Days

Month

I

31

Ianuarius

31

Ianuarius

II

29 or 30

Februarius

28 or
29

Februarius

III

31

Martius

31

Martius

IV

30

Aprilis

30

Aprilis

V

31

Maius

31

Maius

VI

30

Iunius

30

Iunius

VII

31

Quintilis / Iulius

31

Iulius

VIII

30

Sextilis

31

Augustus

IX

31

September

30

September

X

30

October

31

October

XI

31

November

30

November

XII

30

December

31

December

New Year's Day

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

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

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

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,
until the late Middle Ages.
Three special days were singled out:

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
(i.e., always the _{ }ninth day of the Ides).

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.

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
Bede (673-735) in 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.

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

This calendar has been dubbed Gregorian because it was introduced under the
authority of 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).
For each Gregorian period of 400 years,
This reduces the number of leap years to 97 (down from 100 in the Julian scheme).
Thus:

400 Gregorian years = 146097 days
(303´365 + 97´366) = 20871 weeks

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 then, since 1700 wasn't a leap year
in the Gregorian calendar).

Some Official Transitions to the Gregorian 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 1^{st}

Mar.

Apr.

May

June

July

Aug.

Sep.

Oct.

Nov.

Dec.

Jan.

Feb.

m

0

1

2

3

4

5

6

7

8

9

10

11

N_{m}

0

31

61

92

122

153

184

214

245

275

306

337

Day N corresponds to month number
m = floor ( (N+0.5) / 30.6 ).
Conversely, the first day of month m is
N_{m} = 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 convexsolution quadrilateral at corner B.)

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).

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 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.

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
{ yearmonthday }.

MJDN to Gregorian conversion,
with early proleptic Julian dates

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.

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 for
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).

Let's define the latter as halfway between two
full moons and take the middle of a recent total lunar eclipse as
an "accurate" full moon.
Using the lunar eclipse of
March 3, 2007 (which started at 22:43 UTC and ended at 23:58) 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) prominently
featured in the previous article.
This is to be used jointly with the calendrical functions presented there.

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 of the astronomical average.
It would take about 2428 (tropical) years to build up a discrepancy of a whole day.

The Jewish Month

The arithmeticalJewish 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...

(2017-11-04, Beaver Moon) _{ } Names of the Full Moons in the Year
There are 12 or 13 full moons in each calendar year.

There's at least one full moon in each calendar month
(except, on rare occasions, in February).
Usually, there's just one.
The first full moons in calendar months bear the following names:

American Names of the Full Moons
(inherited from the Algonquian tribes)

Month

First Full Moon

January

Wolf Moon, Old Moon, Ice Moon

February

Snow Moon, Sorm Moon, Hunger Moon

March

Worm Moon, Death Moon, Crust Moon, Sap Moon

April

Pink Moon, Sprouting Grass Moon, Egg Moon, Fish Moon

May

Flower Moon, Hare Moon, Corn Planting Moon, Milk Moon

June

Strawberry Moon, Rose Moon, Hot Moon

July

Buck Moon, Thunder Moon, Hay Moon

August

Sturgeon Moon, Green Corn Moon, Grain Moon, Red Moon

September

Harvest Moon, Corn Moon, Barley Moon

October

Hunter's Moon, Travel Moon, Dying Grass Moon

November

Beaver Moon, Frost Moon

December

Cold Moon, Long Nights Moon, Oak Moon

Henry Porter Trefethen (1887-1957)
was the editor of 26 issues (for the years 1932 to 1957) of the popular
Maine Farmer's Almanac which was published from
1819 to 1972 (154 issues). He didn't try
to innovate in the 1932 issue but thereafter his Almanac clearly reflected his
own growing interest in Pagan lore and folklore (he was of Celtic ancestry).
In the 1933 issue, Trefethen introduced the names Harvest Moon and Hunter's Moon
for the full moons which occurred that year on Sept. 4 and Oct. 3, respectively
(he explained the names only in the 1934 issue).
The 1933 almanac also contained an essay on the Native American names for all the full moons occuring in various seasons
(the author being only identified by the initials C.G.F.).

Traditional rural societies recorded the full moons separately in each of the four seasons.

Trefethen's Names of the Full Moons

Winter

Spring

Summer

Autumn

First

Yule Moon

Egg Moon

Hay Moon

Harvest Moon

Middle

Wolf Moon

Milk Moon

Grain Moon

Hunter's Moon

Last

Lenten Moon

Honey Moon

Fruit Moon

Ice Moon

After the passing of Trefethen (1957) the editors of The Maine Farmer's Almanac
didn't apply his system reliably.
Note that this defunct almanac (1819-1972) bears no relation with the extant
Farmer's Almanac, which
is slightly older (1818) and moved
its offices to Lewiston, Maine, in 1955.

February :

Because the interval between two full moons (the lunar month of 29.530588853 days)
is greater than the duration of the month of February (28 or 29 days)
it's possible for February to contain no full moons.
This happens in February 2018, which is surrounded by two months with two full moons
(there's a new-style blue moon on January 31 and on March 31,
those two are separated by only 59 days, which is the minimum possible).
The same situation last happened in 1999. It will happen again in 2037.

(2017-11-06) _{ } Blue Moons
There are two definitions for what a calendrical blue moon is.

Blue Moon (New Style):
Second full moon in a calendar month.
Blue Moon (Old Style):
Third full moon in a season which has four.

In our Gregorian calendar, those two definitions are utterly incompatible.

Some linguists are content with a quick definition of once in a blue moon
(denoting the frequency of a rare event) and will call pedantic [sic]
the more precise calendrical definitions of what a lone blue moon is. I beg to differ.
The full story has delightful ramifications.

The locution blue moon was first used in print
by Pierce Egan (1772-1849)
in his 1821 book Life in London,
which begat a stage play
and a warm Christmas drink,
both called Tom and Jerry, also in 1821.
The drink inspired the titles of two series of cartoons:
Van Beuren's Tom and Jerry
(1931-1933) and MGM's hugely popular
Tom and Jerry cat-and-mouse cartoons by
Hanna and
Barbera (1940-1958, with ongoing spinoffs).

Life in London; or, The Day and Night Scenes of Jerry Hawthorn Esq. and
his Elegant Friend Corinthian Tom, accompanied by Bon Logic, the Oxonian,
in their Rambles and Sprees through the Metropolis. (by Pierce Egan, 1821)

[...] How's Harry and Ben? Haven't seen you this blue moon.

In 1821, Pierce Egan had to give an explanatory note for this early use.

The precise usage of blue moon to denote the second full moon in a month
is far more recent. It has been traced to an honest mistake made in 1946...

The board game Trivial Pursuit
(Genus II expansion pack #730077, 1986) was instrumental in the popularization of this new definition.
According to the publisher's own records, their source was a children's book which first appeared the previous year (1985):

What is a "blue moon"? When there are two full moons in a month,
the second one is called a blue moon. This is a rare occurrence.

The authors may well have heard about that in January 1980 on the
NPR
(National Public Radio) program StarDate produced by
Deborah Byrd (1951-)
as Byrd herself pointed out in the December 1990 issue of
Astronomy.
Her own source was a mistake made by the amateur astronomer
James Hugh Pruett (1886-1955)
in an article entitled Once in a Blue Moon published in the
March 1946 issue of Sky & Telescope.

Pruett had misinterpreted the prior definition of a blue moon
and became inadvertently responsible for the introduction of the simpler
new style version, by hastily issuing the following comment in print:

Seven times in 19 years there were --and still are-- 13 full moons in a year.
This gives 11 months with one full moon each and one with two.
This second in a month, so I interpret it, was called Blue Moon.

Incidentally, that summary is incorrect, since
February sometimes doesn't possess a full moon,
in which case there are two full moons in January and
two in March (that's still 13 full-moons for the whole year).
This happened in 1999, this is happening in 2018 and this will happen again in 2037.

The above words of Pruett lay dormant for more than thirty years in a yellowing copy of the
magazine at the Péridier Library
(in the Astronomy Department of
UT Austin).
That's where Deborah Byrd found them in the late 1970s.
She made this new style definition her own
and would naturally share it with her radio audience a couple of years later,
thus ushering it into folklore, before she--or anyone else--bothered
to chronicle its actual genesis.

The "source" quoted by Pruett in his 1946 essay
was just an earlier (1943) Sky and Telescope column by
Laurence LaFleur which referred to
the following comment given for the month of August 1937 in
The Maine Farmer's Almanac
(the very publication which had introduced the old style
of blue moons to the general public):

[...]
However, occasionally the moon comes full thirteen times in a year.
This was considered a very unfortunate circumstance, especially by the monks who had charge of the calendar.
It became necessary for them to make a calendar of thirteen months for that year,
and it upset the regular arrangement of church festivals.
For this reason thirteen came to be considered an unlucky number.
Also, this extra moon had a way of coming in each of the seasons so that it could not be given a name
appropriate to the time of year like the other moons.
It was usually called the Blue Moon.
There are seven Blue Moons in a cycle of nineteen years.
This year (1937) has a Blue Moon in August the same as 1918.
In 1934 and 1915 Blue Moons came in November.
The next Blue Moon will occur in May 1940 as it did in 1921.
There was a Blue Moon in February 1924.
In olden times the almanac makers had much difficulty in calculating the occurrence of the Blue Moon
and this uncertainty gave rise to the expression "Once in a Blue Moon."

Now, the first sentence of that passage is as misleading as can be.
By itself, that sentence could only be a prelude to a new style
definition! It's utterly irrelevant to the --then current-- old style definition;
especially for the year 1937, which only had 12 full moons!
The essay had indeed been prompted by the
full moon on 21 August 1937
which was an old style blue moon (since the next full moon
(on 20 September 1937)
would be the fourth full moon of the Summer of 1937).

Most of the above chain of events was unraveled by Phillip Hiscock who has published his
own account several times in the media, starting with a newspaper column he was prompted to write
on the occasion of the "Blue Moon" (new style) on
31 December 1990.

Dr. Hiscock is a self-described
folklorist at the Memorial University of Newfoundland Folklore and Language Archive
(MUNFLA).

His blue-moon communiqués culminated in a story
he wrote for the May 1999 issue of Sky & Telescope.
That article revived public interest in the matter and helped resurrect the old-style
definition. A newer version, dated 2012-08-24, was published
online
by Sky & Telescope.

Belewe Mone

In Old and Middle English, the adjective belewe meant either blue
or treacherous (to belewe meant to betray).
The earliest extant reference to a belewe mone
is found in a famous 1528 pamphlet by
William Barlow, Bishop of Chichester,
entitled The Treatyse of the Buryall of the Masse
but more commonly known by its first words, Rede Me and Be Nott Wrothe :

Rede Me and be Nott Wrothe,
for I Saye No Thinge But Trothe
[ ... ]
Yf they saye the mone is belewe,
We must believe that it is true,
Admittynge their interpretacion.

The three full moons in a regular season were called
first, middle and last.
Occasionally, the third full moon in a season wasn't the last one and
may thus have been perceived as misleading or belewe. That's one possible etymological
explanation for what may have been the origin of the term.

A dubious urban legend is that this extra full moon was printed in blue in some almanacs,
possibly including The Maine Farmer's Almanac (1819-1972)
presumably to draw attention to the fact that a given third full moon wasn't the last of the season.
I haven't seen any examples of that.

The average duration between two consecutive blue moons is slightly smaller
in the new style (month-based) system because there are more blue moons this way.
That's due to the fact that it's possible for the month of February to have no full moons
(thus increasing the number of months with two full moons in them).

In borderline cases, it's extremely complicated to predict astronomically the season a given full moon will belong to
(it even depends on the location on the surface of the Earth).
Thus, calendar makers had to rely on arithmetical approximations sanctioned by some central authority.
In particular, Christian monks had to determine the date of the first full moon after the vernal equinox.
This happens to be precisely what the Paschal full moon
is meant to approximate (in the computation of the date of Easter,
described in the next section).

Those almanac makers probably synchronized the other equinox and the two
solstices on the ecclesiastical spring equiquinox.
Before the calendrical reform of 1582, this was rather silly.
This goes a long way toward explaining the aforementioned complaint of Bishop Barlow about
those who were defining the belewe mone by some obscure arbitrary process...

At the time, the calendar was off by about ten days with respect to the seasons.
As farmers were still synchronizing their work with the lunar cycle,
mistakes in the naming of the full moons may have resulted in lesser crop yields.
The calendrical reform of 1582 wasn't just about church festivals;
it was beneficial to agriculture as well.

(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. (The correct date of the Crucifixion.)

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. Everything fits.

In 1910, J.K.
Fotheringham reconstructed the Jewish
calendar astronomically, like Newton had done, two centuries earlier:
He advocated AD 30 as the date of the Crucifixion,
to reconcile an "age of Christ" of 33 years
(speculated by Dionysius and
popularized by the Venerable 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 (1937-2020)
expressed it 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, then
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 those 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 currently
occurs 1, 4 or 5 weeks after the Gregorian date of Easter...

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 above 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.

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 (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).

"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.

(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 based on
standard arithmetic predictions of moon sightings.
We present the most common of
eight 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 relevant
local observational calendars).

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 29.53055555... days,
which is about 2.9 s shorter than the actual mean synodic lunar month
of 29.530588853 days
(it would take about 2428 tropical years to build up a discrepancy of a whole day).
The standard Islamic year is tabulated below:

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)
giving the Islamic year Y corresponding to day N:

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

Conversely, the number N_{Y} corresponding to the first day of year Y is:

N_{Y} = 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 200handheld calculators.

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

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.

- 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.

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.

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.

Number

Month Name(s)

H

K

S

Season

1

Nissan, Nisan,
Abib

30

March-April

2

Iyar,
Ziv

29

April-May

3

Sivan

30

May-June

4

Tammuz

29

June-July

5

Av, Ab

30

July-August

6

Elul

29

August-Sept.

7

Tishri, Tishrei,
Ethanim[New Year]

30

Sept.-Oct.

8

Cheshvan, Heshvan, Marheshvan,
Bul

29

29

30

Oct.-Nov.

9

Kislev

29

30

30

Nov.-Dec.

10

Tevet, Tebet, Tebeth

29

Dec.-January

11

Shevat, Sebat, Shebat

30

January-Feb.

12

Adar I (leap years only)

30

Feb.-March

12 or 13

Adar (Adar II in leap years)

29

March-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.

*
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

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

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).

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

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

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

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

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

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

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

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

(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 Sign

Persian Month

First Day

Aries

Farvardin

March 21

Taurus

Ordibehesht

April 20

Gemini

Khordad

May 21

Cancer

Tir

June 22

Leo

Mordad

July 23

Virgo

Shahrivar

August 23

Libra

Mihr

September 23

Scorpio

Aban

October 23+

Sagittarius

Azar

November 22

Capricorn

Day

December 22

Aquarius

Bahman

January 20

Pisces

Esphand

February 19

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

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

(2018-09-11) The Ethiopian calendar is a version of the Julian calendar.
In September of year N,
Enkutatash begins Ethiopian year N-7.

In the current Gregorian calaendar, Enkutatash
occurs on September 11, except when N+1 is a leap year
(in which case, it occurs on September 12).
So, September 11, 2001
was the first day of Ethiopian Year 1994.

This calendar is still the main calendar for civil life in
Ethiopia and is also observed by Erhiopians
established abroad. It's used to set the lithurgical calendar of several related religious communities:

Eritrean Orthodox Tewahedo Church.

Ethiopian Orthodox Tewahedo Church.

Eastern Catholic Churches.

Coptic Orthodox Church of Alexandria.

The Enkutatash celebration in September marks the beginning of a new Ethiopian year
whose number is 7 years behind the current Gregorian year.
For most of the year (between the Gregorian New Year and Enkutatash)
the two counts are 8 years apart.

The Ethiopian months are identical to the Copt months.

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 200handheld
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:

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
functionsday or date to turn a
Long Count into a Gregorian date, or vice-versa.

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

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

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

The 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.

(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).

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 exactly86400ephemeris 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.