2. The Christian Calendar

The ``Christian calendar'' is the term traditionally used to designate the calendar commonly in use, although it originated in pre-Christian Rome.

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 late 1500s, when countries started changing to the Gregorian calendar (section 2.2). However, some countries (for example, Greece and Russia) used it into the early 1900s, 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 365¹/₄ days = 365.25 days. This gives an error of 1 day in approximately 128 years.

The approximation 365¹/₄ 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, the 4-year 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.

Authorities disagree about whether 45 BC was a leap year or not.

There were no leap years between 9 BC and AD 8 (or, according to some authorities, between 12 BC and AD 4). This period without leap years was decreed by emperor Augustus in order to make up for the surplus of leap years introduced previously, and it 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 on 24 February 1582. This bull is named ``Inter Gravissimas'' after its first two words.

In the Gregorian calendar, the tropical year is approximated as 365⁹⁷/₄₀₀ 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 365⁹⁷/₄₀₀ 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 365⁹⁶⁹/₄₀₀₀ 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 Greeks 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 365²¹⁸/₉₀₀ 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.

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 the 1900s.

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. It is very strange that in many cases there seems to be some doubt among authorities about what the correct days are. Different sources give very different dates in some cases. The list below does not include all the different opinions about when the change took place.

Albania:

December 1912

Austria:

Different regions on different dates
Brixen, Salzburg and Tyrol:
        5 Oct 1583 was followed by 16 Oct 1583
Carinthia and Styria:
        14 Dec 1583 was followed by 25 Dec 1583
See also Czechoslovakia and Hungary

Belgium:

See the Netherlands

Bulgaria:

31 Mar 1916 was followed by 14 Apr 1916

Canada:

Different areas changed at different times.
Newfoundland and Hudson Bay coast:
        2 Sep 1752 was followed by 14 Sep 1752
Mainland Nova Scotia:
        Gregorian 1605 - 13 Oct 1710
        Julian 2 Oct 1710 - 2 Sep 1752
        Gregorian since 14 Sep 1752
Rest of Canada:
        Gregorian from first European settlement

China:

The Gregorian calendar replaced the Chinese calendar in 1912, but the Gregorian calendar was not used throughout the country until the communist revolution of 1949.

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:

31 Jan 1918 was followed by 14 Feb 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
Alsace: 5 Feb 1682 was followed by 16 Feb 1682
Lorraine: 16 Feb 1760 was followed by 28 Feb 1760
Strasbourg: February 1682

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
(Many local variations)

Great Britain and colonies:

2 Sep 1752 was followed by 14 Sep 1752

Greece:

[9 Mar 1924 was followed by 23 Mar 1924
(Some sources say 1916 and 1920)]

Hungary:

21 Oct 1587 was followed by 1 Nov 1587

Ireland:

As Great Britain

Italy:

4 Oct 1582 was followed by 15 Oct 1582

Japan:

The Gregorian calendar was introduced to supplement the traditional Japanese calendar on 1 Jan 1873.

Latvia:

During German occupation 1915 to 1918

Lithuania:

1915

Luxemburg:

14 Dec 1582 was followed by 25 Dec 1582

Netherlands:

Zeeland, Brabrant, and the ``Staten Generaal'':
        14 Dec 1582 was followed by 25 Dec 1582
Holland:
        1 Jan 1583 was followed by 12 Jan 1583
Limburg and the southern provinces (currently Belgium):
          20 Dec 1582 was followed by 31 Dec 1582
        or
          21 Dec 1582 was followed by 1 Jan 1583
Groningen:
        10 Feb 1583 was followed by 21 Feb 1583
        Went back to Julian in the summer of 1594
        31 Dec 1700 was followed by 12 Jan 1701
Gelderland:
        30 Jun 1700 was followed by 12 Jul 1700
Utrecht and Overijssel:
        30 Nov 1700 was followed by 12 Dec 1700
Friesland:
        31 Dec 1700 was followed by 12 Jan 1701
Drenthe:
        30 Apr 1701 was followed by 12 May 1701

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
[The Greek Orthodox parts of the country may have changed later.]

Russia:

31 Jan 1918 was followed by 14 Feb 1918
[In the eastern parts of the country the change may not have occurred until 1920.]

Scotland:

See Great Britain

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, 1584 or 1597
Protestant cantons:
        31 Dec 1700 was followed by 12 Jan 1701
(Many local variations)

Turkey:

Gregorian calendar introduced 1 Jan 1927

USA:

Different areas changed at different times.
Along the Eastern seaboard: With Great Britain in 1752.
Mississippi valley: With France in 1582.
Texas, Florida, California, Nevada, Arizona, New Mexico: With Spain in 1582
Washington, Oregon: With Britain in 1752.
Alaska: October 1867 when Alaska became part of the USA.

Wales:

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?

It is 24 February!

Weird? Yes! The explanation is related to the Roman calendar and is found in section 2.8.1.

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.

Many countries are gradually changing the leap day from the 24th to the 29th. This affects countries such as Sweden and Austria that celebrate ``name days'' (i.e. each day is associated with a name).

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) mod 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 is the Dominical Letter?

Each ordinary (non-leap) year is assigned a letter in the range A to G which describes what days of the year are Sundays. This letter is called the ``Dominical Letter'' (``Sunday Letter'') of the year.

It works in this manner: Assign the letter A to 1 January, B to 2 Jan, C to 3 Jan, ... G to 7 Jan, A to 8 Jan, B to 9 Jan, and so on, using the letters A to G and omitting the leap day.

In a year with Dominical Letter A, all days marked A are Sundays. In a year with Dominical Letter B, all days marked B are Sundays. And so on.

Leap years have two Dominical Letters, one which is used from the start of January until the leap day, and another one which is used for the rest of the year.

The Dominical Letters of 2008 (a leap year) are F and E.

2.6 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):

Example: On what day of the week was the author born?

My birthday is 2 August 1953 (Gregorian, of course).

$$\begin{array}{rl} a & = \frac{14 - 8}{12} = 0 \\ y & = 1953... ...+ 4 + 15 ) \bmod 7 \\ & = 2443 \bmod 7 \\ & = 0 \end{array}$$

I was born on a Sunday.

2.7 When can I reuse my 1992 calendar?

Let us first assume that you are only interested in which dates fall on which days of the week; you are not interested in the dates for Easter and other irregular holidays.

Let us further confine ourselves to the years 1901-2099.

With these restrictions, the answer is as follows:

• If year X is a leap year, you can reuse its calendar in year X+28.
• If year X is the first year after a leap year, you can reuse its calendar in years X+6, X+17, and X+28.
• If year X is the second year after a leap year, you can reuse its calendar in years X+11, X+17, and X+28.
• If year X is the third year after a leap year, you can reuse its calendar in years X+11, X+22, and X+28.

Note that the expression X+28 occurs in all four items above. So you can always reuse your calendar every 28 years.

But if you also want your calendar's indication of Easter and other Christian holidays to be correct, the rules are far too complex to be put to a simple formula. Sometimes calendars can be reused after just six years. For example, the calendars for the years 1981 and 1987 are identical, even when it comes to the date for Easter. But sometimes a very long time can pass before a calendar can be reused; if you happen to have a calendar from 1940, you won't be able to reuse it until the year 5280!

2.8 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 366¹/₄ 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
Januarius	29	29
Februarius	28	24
Intercalaris		27
Martius	31	31
Aprilis	29	29
Maius	31	31
Junius	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 names of the last four months are derived from the Latin words for nine, eleven, twelve, ten - in that order!

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 invented by the English-French scholar Johannes de Sacrobosco in the 13th century.

2.8.1 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 5th day before Nonae'', ``the 4th day before Nonae'', ``the 3rd day before Nonae'', and ``the day before Nonae''. (There was no ``2nd day before Nonae''. This was because of the inclusive way of counting used by the Romans: To them, Nonae itself was the first day, and thus ``the 2nd day before'' and ``the day before'' would mean the same thing.)

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
the 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.

2.9 What is the proleptic calendar?

The Julian calendar was introduced in 45 BC, but when historians date events prior to that year, they normally extend the Julian calendar backward in time. This extended calendar is known as the ``Julian Proleptic Calendar''.

Similarly, it is possible to extend the Gregorian calendar backward in time before 1582. However, this ``Gregorian Proleptic Calendar'' is not commonly used.

If someone refers to, for example, 15 March 429 BC, they are probably using the Julian proleptic calendar.

In the Julian proleptic calendar, year X BC is a leap year, if X-1 is divisible by 4. This is the natural extension of the Julian leap year rules.

2.10 Has the year always started on 1 January?

In some ways, yes. When Julius Caesar introduced his calendar in 45 BC, the year started on 1 January, and it was always the date on which the Solar Number and the Golden Number (see section 2.13.3) 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:

$\bullet$

1 Mar X to 28/29 Feb X+1

$\bullet$

1 Jan X to 31 Dec X

$\bullet$

1 Jan X-1 to 31 Dec X-1

$\bullet$

25 Mar X-1 to 24 Mar X

$\bullet$

25 Mar X to 24 Mar X+1

$\bullet$

Saturday before Easter X to Friday before Easter X+1

$\bullet$

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 starting 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:

$\bullet$

The historical year, which started on 1 January.

$\bullet$

The liturgical year, which started on the first Sunday in advent.

$\bullet$

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.

It is sometimes claimed that having the year start on 1 January was part of the Gregorian calendar reform. This is not true. This myth has probably started because in 1752 England moved the start of the year to 1 January and also changed to the Gregorian calendar. But in most other countries the two events were not related. Scotland, for example, changed to the Gregorian calendar together with England in 1752, but they moved the start of the year to 1 January in 1600.

2.11 Then what about leap years?

If the year started on, for example, 1 March, two months later than our present year, when was the leap day inserted?

[The following information is to the best of my knowledge true. If anyone can confirm or refute it, please let me know.]

When it comes to determining if a year is a leap year, since AD 8 the Julian calendar has always had 48 months between two leap days. So, in a country using a year starting on 1 March, 1439 would have been a leap year, because their February 1439 would correspond to February 1440 in the January-based reckoning.

2.12 What is the origin of the names of the months?

A lot of languages, including English, use month names based on Latin. Their origin is listed below. However, some languages (Czech and Polish, for example) use quite different names.

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. The origin of the name is unknown. It may come from the Latin word aperire (``to open''), or - more likely - from the ancient Latin prefix apter- (``subsequent'').

May

Latin: Maius. Named after the goddess Maia or the god Maius.

June

Latin: Junius. 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.

2.13 What is Easter?

In the Christian world, Easter (and the days immediately preceding it) is the celebration of the death and resurrection of Jesus in (approximately) AD 30.

2.13.1 When is Easter? (Short answer)

Easter Sunday is the first Sunday after the first full moon after vernal equinox.

2.13.2 When is Easter? (Long answer)

The calculation of Easter is complicated because it is linked to (an inaccurate version of) the Hebrew calendar.

Jesus was crucified immediately before the Jewish Passover, which is a celebration of the Exodus from Egypt under Moses. Celebration of Passover started on the 15th day of the (spring) month of Nisan. Jewish months start when the moon is new, therefore the 15th day of the month must be immediately after a full moon.

It was therefore decided to make Easter Sunday the first Sunday after the first full moon after vernal equinox. Or more precisely: Easter Sunday is the first Sunday after the ``official'' full moon on or after the ``official'' vernal equinox.

The official vernal equinox is always 21 March.

The official full moon may differ from the real full moon by one or two days.

(Note, however, that historically, some countries have used the real (astronomical) full moon instead of the official one when calculating Easter. This was the case, for example, of the German Protestant states, which used the astronomical full moon in the years 1700-1776. A similar practice was used in Sweden in the years 1740-1844 and in Denmark in the 1700s.)

The full moon that precedes Easter is called the Paschal full moon. Two concepts play an important role when calculating the Paschal full moon: The Golden Number and the Epact. They are described in the following sections.

The following sections give details about how to calculate the date for Easter. Note, however, that while the Julian calendar was in use, it was customary to use tables rather than calculations to determine Easter. The following sections do mention how to calculate Easter under the Julian calendar, but the reader should be aware that this is an attempt to express in formulas what was originally expressed in tables. The formulas can be taken as a good indication of when Easter was celebrated in the Western Church from approximately the 6th century.

2.13.3 What is the Golden Number?

Each year is associated with a Golden Number.

Considering that the relationship between the moon's phases and the days of the year repeats itself every 19 years (as described in chapter 1), it is natural to associate a number between 1 and 19 with each year. This number is the so-called Golden Number. It is calculated thus:

GoldenNumber = (year mod 19)+1

In years which have the same Golden Number, the new moon will fall on (approximately) the same date. The Golden Number is sufficient to calculate the Paschal full moon in the Julian calendar.

2.13.4 How does one calculate Easter then?

Under the Julian calendar the method was simple. If you know the Golden Number of the year, you can find the Paschal full moon in this table:

Easter Sunday is the first Sunday following the above full moon date. If the full moon falls on a Sunday, Easter Sunday is the following Sunday.

(In particular in the Eastern Orthodox Churches, an additional rule that Easter must come after the Jewish Passover has been used. As time went by, this rule became less important, as the error in the Julian calendar caused Easter to fall later in spring.)

Under the Gregorian calendar, things became much more complicated. One of the changes made in the Gregorian calendar reform was a modification of the way Easter was calculated. There were two reasons for this. First, the 19 year cycle of the phases of moon (the Metonic cycle) was known not to be perfect. Secondly, the Metonic cycle fitted the Gregorian calendar year worse than it fitted the Julian calendar year.

It was therefore decided to base Easter calculations on the so-called Epact.

2.13.5 What is the Epact?

Each year is associated with an Epact.

The Epact is a measure of the age of the moon (i.e. the number of days that have passed since an ``official'' new moon) on a particular date.

(In this context a ``new moon'' is the first visible crescent moon. In modern calendars, a ``new moon'' is the completely invisble moon. The two differ by approximately one day, and this difference must be kept in mind when comparing the new moon calculations presented here to reality.)

In the Julian calendar, the Epact is the age of the moon on 22 March.

In the Gregorian calendar, the Epact is the age of the moon at the start of the year.

The Epact is linked to the Golden Number in the following manner:

Under the Julian calendar, 19 years were assumed to be exactly an integral number of synodic months, and the following relationship exists between the Golden Number and the Epact:

Epact = (11×(GoldenNumber-1)) mod 30

If this formula yields zero, the Epact is by convention frequently designated by the symbol * and its value is said to be 30. Weird? Maybe, but people didn't like the number zero in the old days.

Since there are only 19 possible golden numbers, the Epact can have only 19 different values: 1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, and 30.

In the Gregorian calendar reform, some modifications were made to the simple relationship between the Golden Number and the Epact.

In the Gregorian calendar the Epact should be calculated thus (the divisions are integer divisions, in which remainders are discarded):

1. Use the Julian formula:

   JulianEpact = (11×(GoldenNumber-1)) mod 30

2. Calculate the so-called ``Solar Equation'':

   S = (3×century)/4

   The Solar Equation is an expression of the difference between the Julian and the Gregorian calendar. The value of $S$ increases by one in every century year that is not a leap year.

   (For the purpose of this calculation $century=20$ is used for the years 1900 through 1999, and similarly for other centuries, although this contradicts the rules in section 2.14.3.)

3. Calculate the so-called ``Lunar Equation'':

   L = (8×century + 5)/25

   The Lunar Equation is an expression of the difference between the Julian calendar and the Metonic cycle. The value of $L$ increases by one 8 times every 2500 years.

4. Calculate the Gregorian epact thus:
   
   $$GregorianEpact = JulianEpact - S + L + 8$$
   
   

   The number 8 is a constant that calibrates the starting point of the Gregorian Epact so that it matches the actual age of the moon on new year's day.

5. Add or subtract 30 until GregorianEpact lies between 1 and 30.

In the Gregorian calendar, the Epact can have any value from 1 to 30.

Example: What was the Epact for 1992?

$$\begin{array}{l@{\hspace{5mm}}l} & GoldenNumber = 1992 \bmod... ... = 25\\ 5.& \mathrm{No\ adjustment\ is\ necessary} \end{array}$$

The Epact for 1992 was 25.

2.13.6 How does one calculate Gregorian Easter then?

Look up the Epact in this table to find the date for the Paschal full moon:

Easter Sunday is the first Sunday following the above full moon date. If the full moon falls on a Sunday, Easter Sunday is the following Sunday.

An Epact of 25 requires special treatment, as it has two dates in the above table. There are two equivalent methods for choosing the correct full moon date:

A)

Choose 18 April, unless the current century contains years with an epact of 24, in which case 17 April should be used.

B)

If the Golden Number is $>11$ choose 17 April, otherwise choose 18 April.

The proof that these two statements are equivalent is left as an exercise to the reader. (The frustrated ones may contact me for the proof.)

Example: When was Easter in 1992?

In the previous section we found that the Golden Number for 1992 was 17 and the Epact was 25. Looking in the table, we find that the Paschal full moon was either 17 or 18 April. By rule B above, we choose 17 April because the Golden Number $>11$.

17 April 1992 was a Friday. Easter Sunday must therefore have been 19 April.

2.13.7 Isn't there a simpler way to calculate Easter?

This is an attempt to boil down the information given in the previous sections (the divisions are integer divisions, in which remainders are discarded):

This algorithm is based in part on the algorithm of Oudin (1940) as quoted in ``Explanatory Supplement to the Astronomical Almanac'', P. Kenneth Seidelmann, editor.

People who want to dig into the workings of this algorithm, may be interested to know that

$G$

is the Golden Number-1

$H$

is 23-Epact (modulo 30)

$I$

is the number of days from 21 March to the Paschal full moon

$J$

is the weekday for the Paschal full moon (0=Sunday, 1=Monday, etc.)

$L$

is the number of days from 21 March to the Sunday on or before the Paschal full moon (a number between -6 and 28)

2.13.8 Isn't there an even simpler way to calculate Easter?

If we confine ourselves to the years 1900-2099 and consider only the Gregorian calendar, the formulas of the previous section can be furthser simplified thus:

(Again, the divisions are integer divisions, in which remainders are discarded.)

2.13.9 Is there a simple relationship between two consecutive Easters?

Suppose you know the Easter date of the current year, can you easily find the Easter date in the next year? No, but you can make a qualified guess.

If Easter Sunday in the current year falls on day X and the next year is not a leap year, Easter Sunday of next year will fall on one of the following days: X-15, X-8, X+13 (rare), or X+20.

If Easter Sunday in the current year falls on day X and the next year is a leap year, Easter Sunday of next year will fall on one of the following days: X-16, X-9, X+12 (extremely rare), or X+19. (The jump X+12 occurs only once in the period 1800-2200, namely when going from 2075 to 2076.)

If you combine this knowledge with the fact that Easter Sunday never falls before 22 March and never falls after 25 April, you can narrow the possibilities down to two or three dates.

2.13.10 How frequently are the dates for Easter repeated?

The sequence of Easter dates repeats itself every 532 years in the Julian calendar. The number 532 is the product of the following numbers:

19 (the Metonic cycle or the cycle of the Golden Number)
28 (the Solar cycle, see section 2.4)

The sequence of Easter dates repeats itself every 5,700,000 years in the Gregorian calendar. Calculating this is not as simple as for the Julian calendar, but the number 5,700,000 turns out to be the product of the following numbers:

19 (the Metonic cycle or the cycle of the Golden Number)
400 (the Gregorian equivalent of the Solar cycle, see section 2.4)
25 (the cycle used in step 3 when calculating the Epact)
30 (the number of different Epact values)

2.13.11 What about Greek Orthodox Easter?

The Greek Orthodox Church does not always celebrate Easter on the same day as the Catholic and Protestant countries. The reason is that the Orthodox Church uses the Julian calendar when calculating Easter. This is the case even in the churches that otherwise use the Gregorian calendar.

When the Greek Orthodox Church in 1923 decided to change to the Gregorian calendar (or rather: a Revised Julian Calendar), they chose to use the astronomical full moon as the basis for calculating Easter, rather than the ``official'' full moon described in the previous sections. And they chose the meridian of Jerusalem to serve as definition of when a Sunday starts. However, except for some sporadic use in the 1920s, this system was never adopted in practice.

2.13.12 Did the Easter dates change in 2001?

No.

At a meeting in Aleppo, Syria (5-10 March 1997), organised by the World Council of Churches and the Middle East Council of Churches, representatives of several churches and Christian world communions suggested that the discrepancies between Easter calculations in the Western and the Eastern churches could be resolved by adopting astronomically accurate calculations of the vernal equinox and the full moon, instead of using the algorithm presented in section 2.13.6. The meridian of Jerusalem should be used for the astronomical calculations.

The new method for calculating Easter should have taken effect from the year 2001. In that year the Julian and Gregorian Easter dates coincided (on 15 April Gregorian/2 April Julian), and it would therefore be a reasonable starting point for the new system.

However, the Eastern churches (especially the Russian Orthodox Church) are reluctant to change, having already experienced a schism in the calendar question. So nothing will happen in the near future.

If the new system were introduced, churches using the Gregorian calendar will hardly notice the change. Only once during the period 2001-2025 would these churches note a difference: In 2019 the Gregorian method gives an Easter date of 21 April, but the proposed new method gives 24 March.

Note that the new method makes an Easter date of 21 March possible. This date was not possible under the Julian or Gregorian algorithms. (Under the new method, Easter will fall on 21 March in the year 2877. You're all invited to my house on that date!)

2.14 How does one count years?

In about AD 523, the papal chancellor, Bonifatius, asked a monk by the name of Dionysius Exiguus to devise a way to implement the rules from the Council of Nicaea (the so-called ``Alexandrine Rules'') for general use.

Dionysius Exiguus (in English known as Denis the Little) was a monk from Scythia, he was a canon in the Roman Curia, and his assignment was to prepare calculations of the dates of Easter. At that time it was customary to count years since the reign of emperor Diocletian; but in his calculations Dionysius chose to number the years since the birth of Christ, rather than honour the persecutor Diocletian.

Dionysius (wrongly) fixed Jesus' birth with respect to Diocletian's reign in such a manner that it falls on 25 December 753 AUC (ab urbe condita, i.e. since the founding of Rome), thus making the current era start with AD 1 on 1 January 754 AUC.

How Dionysius established the year of Christ's birth is not known (see section 2.14.1 for a couple of theories). Jesus was born under the reign of King Herod the Great, who died in 750 AUC, which means that Jesus could have been born no later than that year. Dionysius' calculations were disputed at a very early stage.

When people started dating years before 754 AUC using the term ``Before Christ'', they let the year 1 BC immediately precede AD 1 with no intervening year zero.

Note, however, that astronomers frequently use another way of numbering the years BC. Instead of 1 BC they use 0, instead of 2 BC they use -1, instead of 3 BC they use -2, etc.

See also section 2.14.2.

The earliest uses of BC dating are found in the works of the Venerable Bede (673-735).

In this section I have used AD 1 = 754 AUC. This is the most likely equivalence between the two systems. However, some authorities state that AD 1 = 753 AUC or 755 AUC. This confusion is not a modern one, it appears that even the Romans were in some doubt about how to count the years since the founding of Rome.

2.14.1 How did Dionysius date Christ's birth?

There are quite a few theories about this. And many of the theories are presented as if they were indisputable historical fact.

Here are two theories that I personally consider likely:

1. According to the Gospel of Luke (3:1 & 3:23) Jesus was ``about thirty years old'' shortly after ``the fifteenth year of the reign of Tiberius Caesar''. Tiberius became emperor in AD 14. If you combine these numbers you reach a birthyear for Jesus that is strikingly close to the beginning of our year reckoning. This may have been the basis for Dionysius' calculations.

2. Dionysius' original task was to calculate an Easter table. In the Julian calendar, the dates for Easter repeat every 532 years (see section 2.13.10). The first year in Dionysius' Easter tables is AD 532. Is it a coincidence that the number 532 appears twice here? Or did Dionysius perhaps fix Jesus' birthyear so that his own Easter tables would start exactly at the beginning of the second Easter cycle after Jesus' birth?

2.14.2 Was Jesus born in the year 0?

No.

There are two reasons for this:

• There is no year 0.
• Jesus was born before 4 BC.

The concept of a year ``zero'' is a modern myth (but a very popular one). In our calendar, AD 1 follows immediately after 1 BC with no intervening year zero. So a person who was born in 10 BC and died in AD 10, would have died at the age of 19, not 20.

Furthermore, as described in section 2.14, our year reckoning was established by Dionysius Exiguus in the 6th century. Dionysius let the year AD 1 start one week after what he believed to be Jesus' birthday. But Dionysius' calculations were wrong. The Gospel of Matthew tells us that Jesus was born under the reign of King Herod the Great, who died in 4 BC. It is likely that Jesus was actually born around 7 BC. The date of his birth is unknown; it may or may not be 25 December.

2.14.3 When did the 3rd millennium start?

The first millennium started in AD 1, so the millennia are counted in this manner:

1st millennium:	1-1000
2nd millennium:	1001-2000
3rd millennium:	2001-3000

Thus, the 3rd millennium and, similarly, the 21st century started on 1 Jan 2001.

This is the cause of some heated debate, especially since some dictionaries and encyclopaedias say that a century starts in years that end in 00. Furthermore, the change 1999/2000 is obviously much more spectacular than the change 2000/2001.

Let me propose a few compromises:

Any 100-year period is a century. Therefore the period from 23 June 2004 to 22 June 2104 is a century. So please feel free to celebrate the start of a century any day you like!

Although the 20th century started in 1901, the 1900s started in 1900. Similarly, the 21st century started in 2001, but the 2000s started in 2000.

2.14.4 What do AD, BC, CE, and BCE stand for?

Years before the birth of Christ are in English traditionally identified using the abbreviation BC (``Before Christ'').

Years after the birth of Christ are traditionally identified using the Latin abbreviation AD (``Anno Domini'', that is, ``In the Year of the Lord'').

Some people, who want to avoid the reference to Christ that is implied in these terms, prefer the abbreviations BCE (``Before the Common Era'' or ``Before the Christian Era'') and CE (``Common Era'' or ``Christian Era'').

2.15 What is the Indiction?

The Indiction was used in the middle ages to specify the position of a year in a 15 year taxation cycle. It was introduced by emperor Constantine the Great on 1 September 312 and ceased to be used in 1806.

The Indiction may be calculated thus:

Indiction = (year+2) mod 15 + 1

The Indiction has no astronomical significance.

The Indiction did not always follow the calendar year. Three different Indictions may be identified:

1. The Pontifical or Roman Indiction, which started on New Year's Day (being either 25 December, 1 January, or 25 March).
2. The Greek or Constantinopolitan Indiction, which started on 1 September.
3. The Imperial Indiction or Indiction of Constantine, which started on 24 September.

2.16 What is the Julian Period?

The Julian period (and the Julian day number) must not be confused with the Julian calendar.

The French scholar Joseph Justus Scaliger (1540-1609) was interested in assigning a positive number to every year without having to worry about BC/AD. He invented what is today known as the Julian Period.

The Julian Period probably takes its name from the Julian calendar, although it has been claimed that it is named after Scaliger's father, the Italian scholar Julius Caesar Scaliger (1484-1558).

Scaliger's Julian period starts on 1 January 4713 BC (Julian calendar) and lasts for 7980 years. AD 2008 is thus year 6721 in the Julian period. After 7980 years the number starts from 1 again.

Why 4713 BC and why 7980 years? Well, in 4713 BC the Indiction (see section 2.15), the Golden Number (see section 2.13.3) and the Solar Number (see section 2.4) were all 1. The next times this happens is 15×19×28=7980 years later, in AD 3268.

Astronomers have used the Julian period to assign a unique number to every day since 1 January 4713 BC. This is the so-called Julian Day (JD). JD 0 designates the 24 hours from noon UT on 1 January 4713 BC to noon UT on 2 January 4713 BC. (UT=Universal Time, roughly equivalent to GMT.)

This means that at noon UT on 1 January AD 2000, JD 2,451,545 started.

This can be calculated thus:

From 4713 BC to AD 2000 there are 6712 years.
In the Julian calendar, years have 365.25 days, so 6712 years correspond to 6712×365.25=2,451,558 days. Subtract from this the 13 days that the Gregorian calendar is ahead of the Julian calendar, and you get 2,451,545.

Often fractions of Julian day numbers are used, so that 1 January AD 2000 at 15:00 UT is referred to as JD 2,451,545.125.

Note that some people use the term ``Julian day number'' to refer to any numbering of days. NASA, for example, uses the term to denote the number of days since 1 January of the current year, counting 1 January as day 1.

2.16.1 Is there a formula for calculating the Julian day number?

Try this one (the divisions are integer divisions, in which remainders are discarded):

$JD$ is the Julian day number that starts at noon UT on the specified date.

The algorithm works fine for AD dates. If you want to use it for BC dates, you must first convert the BC year to a negative year (e.g., 10 BC = -9). The algorithm works correctly for all dates after 4800 BC, i.e. at least for all positive Julian day numbers.

To convert the other way (i.e., to convert a Julian day number, $JD$, to a day, month, and year) these formulas can be used (again, the divisions are integer divisions):

2.16.2 What is the modified Julian day number?

Sometimes a modified Julian day number (MJD) is used which is 2,400,000.5 less than the Julian day number. This brings the numbers into a more manageable numeric range and makes the day numbers change at midnight UT rather than noon.

MJD 0 thus started on 17 Nov 1858 (Gregorian) at 00:00:00 UT.

2.16.3 What is the Lilian day number?

The Lilian day number is similar to the Julian day number, except that Lilian day number 1 started at midnight of the first day of the Gregorian calendar, that is, 15 October 1582.

The Lilian day number was invented by Bruce G. Ohms of IBM in 1986. It is named after Aloysius Lilius mentioned in section 2.2.

2.17 What is the correct way to write dates?

The answer to this question depends on what you mean by ``correct''. Different countries have different customs.

Most countries use a day-month-year format, such as:

25.12.1998         25/12/1998         25/12-1998         25.XII.1998

In the U.S.A. a month-day-year format is common:

12/25/1998         12-25-1998

International standard ISO 8601 (see chapter 3) mandates a year-month-day format, namely either

1998-12-25 or 19981225.

This format is gaining popularity in some countries.

In all of these systems, the first two digits of the year are frequently omitted:

25.12.98         12/25/98         98-12-25

However, although the last form is frequently seen, it is not allowed by the ISO standard.

This confusion leads to misunderstandings. What is 02-03-04? To most people it is 2 Mar 2004; to an American it is 3 Feb 2004; and to a person using the international standard it could be 4 Mar 2002 (although a year specified with only two digits does not conform to the ISO standard).

If you want to be sure that people understand you, I recommend that you

• write the month with letters instead of numbers, and
• write the years as 4-digit numbers.