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1. What astronomical events form the basis of calendars?
--------------------------------------------------------
Calendars are normally based on astronomical events, and the two most
important astronomical objects are the sun and the moon. Their cycles
are very important in the construction and understanding of calendars.
Our concept of a year is based on the earth's motion around the sun.
The time from one fixed point, such as a solstice or equinox, to the
next is called a "tropical year". Its length is currently 365.242190
days, but it varies. Around 1900 its length was 365.242196 days, and
around 2100 it will be 365.242184 days. (This definition of the
tropical year is not quite accurate, see section 1.1 for more
details.)
Our concept of a month is based on the moon's motion around the earth,
although this connection has been broken in the calendar commonly used
now. The time from one new moon to the next is called a "synodic
month", and its length is currently 29.5305889 days, but it
varies. Around 1900 its length was 29.5305886 days, and around 2100 it
will be 29.5305891 days.
Note that these numbers are averages. The actual length of a
particular year may vary by several minutes due to the influence of
the gravitational force from other planets. Similary, the time between
two new moons may vary by several hours due to a number of factors,
including changes in the gravitational force from the sun, and the
moon's orbital inclination.
It is unfortunate that the length of the tropical year is not a
multiple of the length of the synodic month. This means that with 12
months per year, the relationship between our month and the moon
cannot be maintained.
However, 19 tropical years is 234.997 synodic months, which is very
close to an integer. So every 19 years the phases of the moon fall on
the same dates (if it were not for the skewness introduced by leap
years). 19 years is called a Metonic cycle (after Meton, an astronomer
from Athens in the 5th century BC).
So, to summarise: There are three important numbers to note:
A tropical year is 365.2422 days.
A synodic month is 29.53059 days.
19 tropical years is close to an integral number of synodic months.
The Christian calendar is based on the motion of the earth around the
sun, while the months have no connection with the motion of the moon.
On the other hand, the Islamic calendar is based on the motion of the
moon, while the year has no connection with the motion of the earth
around the sun.
Finally, the Hebrew calendar combines both, in that its years are
linked to the motion of the earth around the sun, and its months are
linked to the motion of the moon.
1.1. What are Equinoxes and Solstices?
--------------------------------------
Equinoxes and solstices are frequently used as anchor points for
calendars. For people in the northern hemisphere:
- Winter solstice is the time in December when the sun reaches its
southernmost latitude. At this time we have the shortest day. The
date is typically around 21 December.
- Summer solstice is the time in June when the sun reaches its
northernmost latitude. At this time we have the longest day. The
date is typically around 21 June.
- Vernal equinox is the time in March when the sun passes the equator
moving from the southern to the northern hemisphere. Day and night
have approximately the same length. The date is typically around 20
March.
- Autumnal equinox is the time in September when the sun passes the
equator moving from the northern to the southern hemisphere. Day and
night have approximately the same length. The date is typically
around 22 September.
For people in the southern hemisphere these events are shifted half a
year.
The astronomical "tropical year" is frequently defined as the time
between, say, two vernal equinoxes, but this is not actually true.
Currently the time between two vernal equinoxes is slightly greater
than the tropical year. The reason is that the earth's position in its
orbit at the time of solstices and equinoxes shifts slightly each year
(taking approximately 21,000 years to move all the way around the
orbit). This, combined with the fact that the earth's orbit is not
completely circular, causes the equinoxes and solstices to shift with
respect to each other.
The astronomer's mean tropical year is really a somewhat artificial
average of the period between the time when the sun is in any given
position in the sky with respect to the equinoxes and the next time
the sun is in the same position.
2. The Christian calendar
-------------------------
The "Christian calendar" is the term I use to designate the calendar
commonly in use, although its connection with Christianity is highly
debatable.
The Christian calendar has years of 365 or 366 days. It is divided into
12 months that have no relationship to the motion of the moon. In
parallel with this system, the concept of "weeks" groups the days in
sets of 7.
Two main versions of the Christian calendar have existed in recent
times: The Julian calendar and the Gregorian calendar. The difference
between them lies in the way they approximate the length of the
tropical year and their rules for calculating Easter.
2.1. What is the Julian calendar?
---------------------------------
The Julian calendar was introduced by Julius Caesar in 45 BC. It was
in common use until the 1500s, when countries started changing to the
Gregorian calendar (section 2.2). However, some countries (for
example, Greece and Russia) used it into this century, and the
Orthodox church in Russia still uses it, as do some other Orthodox
churches.
In the Julian calendar, the tropical year is approximated as 365 1/4
days = 365.25 days. This gives an error of 1 day in approximately 128
years.
The approximation 365 1/4 is achieved by having 1 leap year every 4
years.
2.1.1. What years are leap years?
---------------------------------
The Julian calendar has 1 leap year every 4 years:
Every year divisible by 4 is a leap year.
However, this rule was not followed in the first years after the
introduction of the Julian calendar in 45 BC. Due to a counting error,
every 3rd year was a leap year in the first years of this calendar's
existence. The leap years were:
45 BC, 42 BC, 39 BC, 36 BC, 33 BC, 30 BC,
27 BC, 24 BC, 21 BC, 18 BC, 15 BC, 12 BC, 9 BC,
AD 8, AD 12, and every 4th year from then on.
There were no leap years between 9 BC and AD 8. This period without
leap years was decreed by emperor Augustus and earned him a place in
the calendar, as the 8th month was named after him.
It is a curious fact that although the method of reckoning years after
the (official) birthyear of Christ was not introduced until the 6th
century, by some stroke of luck the Julian leap years coincide with
years of our Lord that are divisible by 4.
2.1.2. What consequences did the use of the Julian calendar have?
-----------------------------------------------------------------
The Julian calendar introduces an error of 1 day every 128 years. So
every 128 years the tropical year shifts one day backwards with
respect to the calendar. Furthermore, the method for calculating the
dates for Easter was inaccurate and needed to be refined.
In order to remedy this, two steps were necessary: 1) The Julian
calendar had to be replaced by something more adequate. 2) The extra
days that the Julian calendar had inserted had to be dropped.
The solution to problem 1) was the Gregorian calendar described in
section 2.2.
The solution to problem 2) depended on the fact that it was felt that
21 March was the proper day for vernal equinox (because 21 March was
the date for vernal equinox during the Council of Nicaea in AD
325). The Gregorian calendar was therefore calibrated to make that day
vernal equinox.
By 1582 vernal equinox had moved (1582-325)/128 days = approximately
10 days backwards. So 10 days had to be dropped.
2.2. What is the Gregorian calendar?
------------------------------------
The Gregorian calendar is the one commonly used today. It was proposed
by Aloysius Lilius, a physician from Naples, and adopted by Pope
Gregory XIII in accordance with instructions from the Council of Trent
(1545-1563) to correct for errors in the older Julian Calendar. It
was decreed by Pope Gregory XIII in a papal bull in February 1582.
In the Gregorian calendar, the tropical year is approximated as
365 97/400 days = 365.2425 days. Thus it takes approximately 3300
years for the tropical year to shift one day with respect to the
Gregorian calendar.
The approximation 365 97/400 is achieved by having 97 leap years
every 400 years.
2.2.1. What years are leap years?
---------------------------------
The Gregorian calendar has 97 leap years every 400 years:
Every year divisible by 4 is a leap year.
However, every year divisible by 100 is not a leap year.
However, every year divisible by 400 is a leap year after all.
So, 1700, 1800, 1900, 2100, and 2200 are not leap years. But 1600,
2000, and 2400 are leap years.
(Destruction of a myth: There are no double leap years, i.e. no
years with 367 days. See, however, the note on Sweden in section
2.2.4.)
2.2.2. Isn't there a 4000-year rule?
------------------------------------
It has been suggested (by the astronomer John Herschel (1792-1871)
among others) that a better approximation to the length of the
tropical year would be 365 969/4000 days = 365.24225 days. This would
dictate 969 leap years every 4000 years, rather than the 970 leap
years mandated by the Gregorian calendar. This could be achieved by
dropping one leap year from the Gregorian calendar every 4000 years,
which would make years divisible by 4000 non-leap years.
This rule has, however, not been officially adopted.
2.2.3. Don't the Greek do it differently?
-----------------------------------------
When the Orthodox church in Greece finally decided to switch to the
Gregorian calendar in the 1920s, they tried to improve on the
Gregorian leap year rules, replacing the "divisible by 400" rule by
the following:
Every year which when divided by 900 leaves a remainder of 200
or 600 is a leap year.
This makes 1900, 2100, 2200, 2300, 2500, 2600, 2700, 2800 non-leap
years, whereas 2000, 2400, and 2900 are leap years. This will not
create a conflict with the rest of the world until the year 2800.
This rule gives 218 leap years every 900 years, which gives us an
average year of 365 218/900 days = 365.24222 days, which is certainly
more accurate than the official Gregorian number of 365.2425 days.
However, this rule is *not* official in Greece.
[I have received an e-mail indicating that this system is official in
Russia today. I'm investigating that. Information is very welcome.]
2.2.4. When did country X change from the Julian to the Gregorian calendar?
---------------------------------------------------------------------------
The papal bull of February 1582 decreed that 10 days should be dropped
from October 1582 so that 15 October should follow immediately after
4 October, and from then on the reformed calendar should be used.
This was observed in Italy, Poland, Portugal, and Spain. Other Catholic
countries followed shortly after, but Protestant countries were
reluctant to change, and the Greek orthodox countries didn't change
until the start of this century.
Changes in the 1500s required 10 days to be dropped.
Changes in the 1600s required 10 days to be dropped.
Changes in the 1700s required 11 days to be dropped.
Changes in the 1800s required 12 days to be dropped.
Changes in the 1900s required 13 days to be dropped.
(Exercise for the reader: Why is the error in the 1600s the same as
in the 1500s.)
The following list contains the dates for changes in a number of
countries.
Albania: December 1912
Austria: Different regions on different dates
5 Oct 1583 was followed by 16 Oct 1583
14 Dec 1583 was followed by 25 Dec 1583
Belgium: Different authorities say
14 Dec 1582 was followed by 25 Dec 1582
21 Dec 1582 was followed by 1 Jan 1583
Bulgaria: Different authorities say
Sometime in 1912
Sometime in 1915
18 Mar 1916 was followed by 1 Apr 1916
China: Different authorities say
18 Dec 1911 was followed by 1 Jan 1912
18 Dec 1928 was followed by 1 Jan 1929
Czechoslovakia (i.e. Bohemia and Moravia):
6 Jan 1584 was followed by 17 Jan 1584
Denmark (including Norway):
18 Feb 1700 was followed by 1 Mar 1700
Egypt: 1875
Estonia: January 1918
Finland: Then part of Sweden. (Note, however, that Finland later
became part of Russia, which then still used the
Julian calendar. The Gregorian calendar remained
official in Finland, but some use of the Julian
calendar was made.)
France: 9 Dec 1582 was followed by 20 Dec 1582
Germany: Different states on different dates:
Catholic states on various dates in 1583-1585
Prussia: 22 Aug 1610 was followed by 2 Sep 1610
Protestant states: 18 Feb 1700 was followed by 1 Mar 1700
Great Britain and Dominions (including what is now the USA):
2 Sep 1752 was followed by 14 Sep 1752
Greece: 9 Mar 1924 was followed by 23 Mar 1924
Hungary: 21 Oct 1587 was followed by 1 Nov 1587
Italy: 4 Oct 1582 was followed by 15 Oct 1582
Japan: Different authorities say:
19 Dec 1872 was followed by 1 Jan 1873
18 Dec 1918 was followed by 1 Jan 1919
Latvia: During German occupation 1915 to 1918
Lithuania: 1915
Luxemburg: 14 Dec 1582 was followed by 25 Dec 1582
Netherlands:
Brabant, Flanders, Holland, Artois, Hennegau:
14 Dec 1582 was followed by 25 Dec 1582
Geldern, Friesland, Zeuthen, Groningen, Overysel:
30 Nov 1700 was followed by 12 Dec 1700
Norway: Then part of Denmark.
Poland: 4 Oct 1582 was followed by 15 Oct 1582
Portugal: 4 Oct 1582 was followed by 15 Oct 1582
Romania: 31 Mar 1919 was followed by 14 Apr 1919
Russia: 31 Jan 1918 was followed by 14 Feb 1918
Spain: 4 Oct 1582 was followed by 15 Oct 1582
Sweden (including Finland):
17 Feb 1753 was followed by 1 Mar 1753 (see note below)
Switzerland:
Catholic cantons: 1583 or 1584
Zurich, Bern, Basel, Schafhausen, Neuchatel, Geneva:
31 Dec 1700 was followed by 12 Jan 1701
St Gallen: 1724
Turkey: 18 Dec 1926 was followed by 1 Jan 1927
USA: See Great Britain.
Yugoslavia: 1919
Sweden has a curious history. Sweden decided to make a gradual change
from the Julian to the Gregorian calendar. By dropping every leap year
from 1700 through 1740 the eleven superfluous days would be omitted
and from 1 Mar 1740 they would be in sync with the Gregorian
calendar. (But in the meantime they would be in sync with nobody!)
So 1700 (which should have been a leap year in the Julian calendar)
was not a leap year in Sweden. However, by mistake 1704 and 1708
became leap years. This left Sweden out of synchronisation with both
the Julian and the Gregorian world, so they decided to go *back* to
the Julian calendar. In order to do this, they inserted an extra day
in 1712, making that year a double leap year! So in 1712, February had
30 days in Sweden.
Later, in 1753, Sweden changed to the Gregorian calendar by dropping 11
days like everyone else.
2.3. What day is the leap day?
------------------------------
24 February!
Weird? Yes! The explanation is related to the Roman calendar and is
found in section 2.6.1.
From a numerical point of view, of course 29 February is the extra
day. But from the point of view of celebration of feast days, the
following correspondence between days in leap years and non-leap
years has traditionally been used:
Non-leap year Leap year
------------- ----------
22 February 22 February
23 February 23 February
24 February (extra day)
24 February 25 February
25 February 26 February
26 February 27 February
27 February 28 February
28 February 29 February
For example, the feast of St. Leander has been celebrated on 27
February in non-leap years and on 28 February in leap years.
The EU (European Union) in their infinite wisdom have decided that
starting in the year 2000, 29 February is to be the leap day. This
will affect countries such as Sweden and Austria that celebrate "name
days" (i.e. each day is associated with a name).
It appears that the Roman Catholic Church already uses 29 February as
the leap day.
2.4. What is the Solar Cycle?
-----------------------------
In the Julian calendar the relationship between the days of the week
and the dates of the year is repeated in cycles of 28 years. In the
Gregorian calendar this is still true for periods that do not cross
years that are divisible by 100 but not by 400.
A period of 28 years is called a Solar Cycle. The "Solar Number" of a
year is found as:
Solar Number = (year + 8) % 28 + 1
In the Julian calendar there is a one-to-one relationship between the
Solar Number and the day on which a particular date falls.
(The leap year cycle of the Gregorian calendar is 400 years, which is
146,097 days, which curiously enough is a multiple of 7. So in the
Gregorian calendar the equivalent of the "Solar Cycle" would be 400
years, not 7*400=2800 years as one might be tempted to believe.)
2.5. What day of the week was 2 August 1953?
--------------------------------------------
To calculate the day on which a particular date falls, the following
algorithm may be used (the divisions are integer divisions, in which
remainders are discarded):
a = (14 - month) / 12
y = year - a
m = month + 12*a - 2
For Julian calendar: d = (5 + day + y + y/4 + (31*m)/12) % 7
For Gregorian calendar: d = (day + y + y/4 - y/100 + y/400 + (31*m)/12) % 7
The value of d is 0 for a Sunday, 1 for a Monday, 2 for a Tuesday, etc.
Example: On what day of the week was the author born?
My birthday is 2 August 1953 (Gregorian, of course).
a = (14 - 8) / 12 = 0
y = 1953 - 0 = 1953
m = 8 + 12*0 - 2 = 6
d = (2 + 1953 + 1953/4 - 1953/100 + 1953/400 + (31*6)/12) % 7
= (2 + 1953 + 488 - 19 + 4 + 15 ) % 7
= 2443 % 7
= 0
I was born on a Sunday.
2.6. What is the Roman calendar?
--------------------------------
Before Julius Caesar introduced the Julian calendar in 45 BC, the
Roman calendar was a mess, and much of our so-called "knowledge" about
it seems to be little more than guesswork.
Originally, the year started on 1 March and consisted of only 304 days
or 10 months (Martius, Aprilis, Maius, Junius, Quintilis, Sextilis,
September, October, November, and December). These 304 days were
followed by an unnamed and unnumbered winter period. The Roman king
Numa Pompilius (c. 715-673 BC, although his historicity is disputed)
allegedly introduced February and January (in that order) between
December and March, increasing the length of the year to 354 or 355
days. In 450 BC, February was moved to its current position between
January and March.
In order to make up for the lack of days in a year, an extra month,
Intercalaris or Mercedonius, (allegedly with 22 or 23 days though some
authorities dispute this) was introduced in some years. In an 8 year
period the length of the years were:
1: 12 months or 355 days
2: 13 months or 377 days
3: 12 months or 355 days
4: 13 months or 378 days
5: 12 months or 355 days
6: 13 months or 377 days
7: 12 months or 355 days
8: 13 months or 378 days
A total of 2930 days corresponding to a year of 366 1/4 days. This
year was discovered to be too long, and therefore 7 days were later
dropped from the 8th year, yielding 365.375 days per year.
This is all theory. In practice it was the duty of the priesthood to
keep track of the calendars, but they failed miserably, partly due to
ignorance, partly because they were bribed to make certain years long
and other years short. Furthermore, leap years were considered unlucky
and were therefore avoided in time of crisis, such as the Second Punic
War.
In order to clean up this mess, Julius Caesar made his famous calendar
reform in 45 BC. We can make an educated guess about the length of the
months in the years 47 and 46 BC:
47 BC 46 BC
January 29 29
February 28 24
Intercalaris 27
March 31 31
April 29 29
May 31 31
June 29 29
Quintilis 31 31
Sextilis 29 29
September 29 29
October 31 31
November 29 29
Undecember 33
Duodecember 34
December 29 29
--- ---
Total 355 445
The length of the months from 45 BC onward were the same as the ones
we know today.
Occasionally one reads the following story:
"Julius Caesar made all odd numbered months 31 days long, and
all even numbered months 30 days long (with February having 29
days in non-leap years). In 44 BC Quintilis was renamed
'Julius' (July) in honour of Julius Caesar, and in 8 BC
Sextilis became 'Augustus' in honour of emperor Augustus. When
Augustus had a month named after him, he wanted his month to
be a full 31 days long, so he removed a day from February and
shifted the length of the other months so that August would
have 31 days."
This story, however, has no basis in actual fact. It is a fabrication
possibly dating back to the 14th century.
2.6.1. How did the Romans number days?
--------------------------------------
The Romans didn't number the days sequentially from 1. Instead they
had three fixed points in each month:
"Kalendae" (or "Calendae"), which was the first day of the month.
"Idus", which was the 13th day of January, February, April,
June, August, September, November, and December, or
the 15th day of March, May, July, or October.
"Nonae", which was the 9th day before Idus (counting Idus
itself as the 1st day).
The days between Kalendae and Nonae were called "the 4th day before
Nonae", "the 3rd day before Nonae", and "the 2nd day before
Nonae". (The 1st day before Nonae would be Nonae itself.)
Similarly, the days between Nonae and Idus were called "the Xth day
before Idus", and the days after Idus were called "the Xth day before
Kalendae (of the next month)".
Julius Caesar decreed that in leap years the "6th day before Kalendae
of March" should be doubled. So in contrast to our present system, in
which we introduce an extra date (29 February), the Romans had the
same date twice in leap years. The doubling of the 6th day before
Kalendae of March is the origin of the word "bissextile". If we
create a list of equivalences between the Roman days and our current
days of February in a leap year, we get the following:
7th day before Kalendae of March 23 February
6th day before Kalendae of March 24 February
6th day before Kalendae of March 25 February
5th day before Kalendae of March 26 February
4th day before Kalendae of March 27 February
3rd day before Kalendae of March 28 February
2nd day before Kalendae of March 29 February
Kalendae of March 1 March
You can see that the extra 6th day (going backwards) falls on what is
today 24 February. For this reason 24 February is still today
considered the "extra day" in leap years (see section 2.3). However,
at certain times in history the second 6th day (25 Feb) has been
considered the leap day.
Why did Caesar choose to double the 6th day before Kalendae of March?
It appears that the leap month Intercalaris/Mercedonius of the
pre-reform calendar was not placed after February, but inside it,
namely between the 7th and 6th day before Kalendae of March. It was
therefore natural to have the leap day in the same position.
2.7. Has the year always started on 1 January?
----------------------------------------------
For the man in the street, yes. When Julius Caesar introduced his
calendar in 45 BC, he made 1 January the start of the year, and it was
always the date on which the Solar Number and the Golden Number (see
section 2.9.3) were incremented.
However, the church didn't like the wild parties that took place at
the start of the new year, and in AD 567 the council of Tours declared
that having the year start on 1 January was an ancient mistake that
should be abolished.
Through the middle ages various New Year dates were used. If an
ancient document refers to year X, it may mean any of 7 different
periods in our present system:
- 1 Mar X to 28/29 Feb X+1
- 1 Jan X to 31 Dec X
- 1 Jan X-1 to 31 Dec X-1
- 25 Mar X-1 to 24 Mar X
- 25 Mar X to 24 Mar X+1
- Saturday before Easter X to Friday before Easter X+1
- 25 Dec X-1 to 24 Dec X
Choosing the right interpretation of a year number is difficult, so
much more as one country might use different systems for religious and
civil needs.
The Byzantine Empire used a year staring on 1 Sep, but they didn't
count years since the birth of Christ, instead they counted years
since the creation of the world which they dated to 1 September 5509 BC.
Since about 1600 most countries have used 1 January as the first day
of the year. Italy and England, however, did not make 1 January official
until around 1750.
In England (but not Scotland) three different years were used:
- The historical year, which started on 1 January.
- The liturgical year, which started on the first Sunday in advent.
- The civil year, which
from the 7th to the 12th century started on 25 December,
from the 12th century until 1751 started on 25 March,
from 1752 started on 1 January.
2.8. What is the origin of the names of the months?
---------------------------------------------------
January Latin: Januarius. Named after the god Janus.
February Latin: Februarius. Named after Februa, the purification
festival.
March Latin: Martius. Named after the god Mars.
April Latin: Aprilis. Named either after the goddess Aphrodite or
the Latin word "aperire", to open.
May Latin: Maius. Probably named after the goddess Maia.
June Latin: Junius. Probably named after the goddess Juno.
July Latin: Julius. Named after Julius Caesar in 44 BC. Prior
to that time its name was Quintilis from the word
"quintus", fifth, because it was the 5th month in the old
Roman calendar.
August Latin: Augustus. Named after emperor Augustus in 8
BC. Prior to that time the name was Sextilis from the
word "sextus", sixth, because it was the 6th month in the
old Roman calendar.
September Latin: September. From the word "septem", seven, because
it was the 7th month in the old Roman calendar.
October Latin: October. From the word "octo", eight, because it
was the 8th month in the old Roman calendar.
November Latin: November. From the word "novem", nine, because it
was the 9th month in the old Roman calendar.
December Latin: December. From the word "decem", ten, because it
was the 10th month in the old Roman calendar.
--- End of part 1 ---
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