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Gregorian calendar

The Gregorian calendar is the most widely used calendar in the world today. It was first proposed by the Calabrian doctor Aloysius Lilius, and decreed by Pope Gregory XIII, after whom it was named, on 24 February 1582 by papal bull Inter gravissimas. It is a reform of the Julian calendar.

Years in the reformed calendar continue the numbering system of the Julian calendar, which are numbered from the traditional Incarnation year of Jesus, which has been labeled the "anno Domini" (AD) era, and is sometimes labeled the "common era" (CE), otherwise known as the "Christian Era".

The changes made by Gregory corrected the drift in the civil calendar which arose because the mean Julian calendar year (exactly 365 1/4 days) was slightly too long, causing the vernal equinox, and consequently the date on which Easter was being celebrated, to drift slowly forward in relation to the civil calendar and the seasons.

The Gregorian calendar system dropped 10 days to bring the calendar back into synchronization with the seasons and, to keep it there, adopted the following leap year rule:

Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100; the centurial years that are exactly divisible by 400 are still leap years. For example, the year 1900 was not a leap year; the year 2000 was a leap year.

In the Julian calendar, all years exactly divisible by 4 are leap years.

Description

The Gregorian solar calendar is an arithmetical calendar. It counts days as the basic unit of time, grouping them into years of 365 or 366 days; and repeats completely every 146,097 days, which fill 400 years, and which also happens to be 20,871 seven-day weeks. Of these 400 years, 303 (the "common years") have 365 days, and 97 (the leap years) have 366 days. This gives an average year length of exactly 365.2425 days, or 365 days, 5 hours, 49 minutes and 12 seconds.

A Gregorian year is divided into twelve months of irregular length:

No.	Name	Days
1	January	31
2	February	28 or 29
3	March	31
4	April	30
5	May	31
6	June	30
7	July	31
8	August	31
9	September	30
10	October	31
11	November	30
12	December	31

A calendar date is fully specified by the year (numbered by some scheme beyond the scope of the calendar itself), the month (identified by name or number), and the day of the month (numbered sequentially starting at 1).

Leap years add a 29th day to February, which normally has 28 days. Thus, the essential ongoing differentiating feature of the Gregorian calendar, as opposed to the Julian calendar, is that the Gregorian omits 3 leap days every 400 years. This difference would have been more noticeable in modern memory were it not for the fact that the year 2000 was a leap year in both the Julian and Gregorian calendar systems.

The intercalary day in a leap year is known as a leap day. Since Roman times 24 February (bissextile) was counted as the leap day , but nowadays 29 February is regarded as the leap day in most countries.

Although the calendar year runs from 1 January to 31 December, sometimes year numbers were based on a different starting point within the calendar. Confusingly, the term "Anno Domini" is not specific on this point, and actually refers to a family of year numbering systems with different starting points for the years. (See the section below for more on this issue.)

Lunar aspects

The Catholic church maintained a tabular lunar calendar, which was primarily to calculate the date of Easter, and the lunar calendar required reform as well. A perpetual lunar calendar was created, in the sense that 30 different arrangements (lines in the expanded table of epacts) for lunar months were created. One of the 30 arrangements applies to a century (for this purpose, the century begins with a year divisible by 100). When the arrangement to be used for a given century is communicated, anyone in possession of the tables can find the age of the moon on any date, and calculate the date of Easter.

History

Gregorian reform

The motivation of the Catholic Church in adjusting the calendar was to celebrate Easter at the time it thought the First Council of Nicaea had agreed upon in 325. Although a canon of the council implies that all churches used the same Easter, they did not. The Church of Alexandria celebrated Easter on the Sunday after the 14th day of the moon (computed using the Metonic cycle) that falls on or after the vernal equinox, which they placed on 21 March. However, the Church of Rome still regarded 25 March as the equinox (until 342) and used a different cycle to compute the day of the moon. In the Alexandrian system, since the 14th day of the Easter moon could fall at earliest on 21 March its first day could fall no earlier than 8 March and no later than 5 April. This meant that Easter varied between 22 March and 25 April. At Rome, Easter was not allowed to fall later than 21 April, this being the day of the Parilia or birthday of Rome and a pagan festival. The first day of the Easter moon could fall no earlier than 5 March and no later than 2 April. Easter was the Sunday after the 15th day of this moon, whose 14th day was allowed to precede the equinox. Where the two systems produced different dates there was generally a compromise so that both churches were able to celebrate on the same day. By the tenth century all churches (except some on the eastern border of the Byzantine Empire) had adopted the Alexandrian Easter, which still placed the vernal equinox on 21 March, although Bede had already noted its drift in 725—it had drifted even further by the sixteenth century.

Worse, the reckoned Moon that was used to compute Easter was fixed to the Julian year by a 19 year cycle. However, that approximation built up an error of one day every 310 years, so by the sixteenth century the lunar calendar was out of phase with the real Moon by four days.

The Council of Trent approved a plan in 1563 for correcting the calendrical errors, requiring that the date of the vernal equinox be restored to that which it held at the time of the First Council of Nicaea in 325 and that an alteration to the calendar be designed to prevent future drift. This would allow for a more consistent and accurate scheduling of the feast of Easter.

The fix was to come in two stages. First, it was necessary to approximate the correct length of a solar year. The value chosen was 365.2425 days in decimal notation. Although close to the mean tropical year of 365.24219 days, it is even closer to the vernal equinox year of 365.2424 days; this fact made the choice of approximation particularly appropriate as the purpose of creating the calendar was to ensure that the vernal equinox would be near a specific date (21 March). (See Accuracy).

The second stage was to devise a model based on the approximation which would provide an accurate yet simple, rule-based calendar. The formula designed by Aloysius Lilius was ultimately successful. It proposed a 10-day correction to revert the drift since Nicaea, and the imposition of a leap day in only 97 years in 400 rather than in 1 year in 4. To implement the model, it was provided that years divisible by 100 would be leap years only if they were divisible by 400 as well. So, in the last millennium, 1600 and 2000 were leap years, but 1700, 1800 and 1900 were not. In this millennium, 2100, 2200, 2300 and 2500 will not be leap years, but 2400 will be. This theory was expanded upon by Christopher Clavius in a closely argued, 800 page volume. He would later defend his and Lilius's work against detractors.

The 19-year cycle used for the lunar calendar was also to be corrected by one day every 300 or 400 years (8 times in 2500 years) along with corrections for the years (1700, 1800, 1900, 2100 et cetera) that are no longer leap years. In fact, a new method for computing the date of Easter was introduced.

In 1577 a Compendium was sent to expert mathematicians outside the reform commission for comments. Some of these experts, including Giambattista Benedetti and Giuseppe Moleto, believed Easter should be computed from the true motions of the sun and moon, rather than using a tabular method, but these recommendations were not adopted.

Lilius originally proposed that the 10-day correction should be implemented by deleting the Julian leap day on each of its ten occurrences during a period of 40 years, thereby providing for a gradual return of the equinox to 21 March. However, Clavius's opinion was that the correction should take place in one move and it was this advice which prevailed with Gregory. Accordingly, when the new calendar was put in use, the error accumulated in the 13 centuries since the Council of Nicaea was corrected by a deletion of ten days. The last day of the Julian calendar was Thursday, 4 October 1582 and this was followed by the first day of the Gregorian calendar, Friday, 15 October 1582 (the cycle of weekdays was not affected).

Adoption

Though Gregory's reform was enacted in the most solemn of forms available to the Church, in fact the bull had no authority beyond the Catholic Church and the Papal States. The changes which he was proposing were changes to the civil calendar over which he had no authority. The changes required adoption by the civil authorities in each country to have legal effect.

The Nicene Council of 325 sought to devise rules whereby all Christians would celebrate Easter on the same day. In fact it took a very long time before Christians achieved that objective (see Easter for the issues which arose). However, the bull Inter gravissimas became the law of the Catholic Church. It was not recognised, however, by Protestant Churches nor by Orthodox Churches and others. Consequently, the days on which Easter and related holidays were celebrated by different Christian Churches again diverged.

Adoption in Europe

Only four Catholic countries adopted the new calendar on the date specified by the bull. Other Catholic countries experienced some delay before adopting the reform; and non-Catholic countries, not being subject to the decrees of the Pope, initially rejected or simply ignored the reform altogether, although they all eventually adopted it. Hence, the dates 5 October 1582 to 14 October 1582 (inclusive) are valid dates in many countries.

Spain, Portugal, the Polish-Lithuanian Commonwealth, and most of Italy implemented the new calendar on Friday, 15 October 1582, following Julian Thursday, 4 October 1582. The Spanish and Portuguese colonies adopted the calendar later due to the slowness of communication. France adopted the new calendar on Monday, 20 December 1582, following Sunday, 9 December 1582. The Protestant Dutch provinces of Holland and Zeeland also adopted it in December of that year.

Most non-Catholic countries initially objected to adopting a Catholic invention, especially during the Counter-Reformation (of which Gregory was a leading proponent); some Protestants feared the new calendar was part of a plot to return them to the Catholic fold. In the Czech lands, Protestants resisted the calendar imposed by the Habsburg Monarchy. In parts of Ireland, Catholic rebels until their defeat in the Nine Years' War kept the "new" Easter in defiance of the English-loyal authorities; later, Catholics practising in secret petitioned the Propaganda Fide for dispensation from observing the new calendar, as it signalled their disloyalty.

Denmark, which then included Norway and some Protestant states of Germany, adopted the solar portion of the new calendar on Monday, 1 March 1700, following Sunday, 18 February 1700, due to the influence of Ole Rømer, but did not adopt the lunar portion. Instead, they decided to calculate the date of Easter astronomically using the instant of the vernal equinox and the full moon according to Kepler's Rudolphine Tables of 1627. They finally adopted the lunar portion of the Gregorian calendar in 1776. The remaining provinces of the Dutch Republic also adopted the Gregorian calendar in 1700.

Sweden's relationship with the Gregorian Calendar was a difficult one. Sweden started to make the change from the Julian calendar and towards the Gregorian calendar in 1700, but it was decided to make the (then 11-day) adjustment gradually, by excluding the leap days (29 February) from each of 11 successive leap years, 1700 to 1740. In the meantime, the Swedish calendar would be out of step with both the Julian calendar and the Gregorian calendar for 40 years; also, the difference would not be constant but would change every 4 years. This strange system clearly had great potential for endless confusion when working out the dates of Swedish events in this 40-year period. To make matters worse, the system was poorly administered and the leap days that should have been excluded from 1704 and 1708 were not excluded. The Swedish calendar (according to the transition plan) should now have been 8 days behind the Gregorian, but was still in fact 10 days behind. King Charles XII wisely recognised that the gradual change to the new system was not working, and he abandoned it.

However, rather than proceeding directly to the Gregorian calendar, it was decided to revert to the Julian calendar. This was achieved by introducing the unique date 30 February in the year 1712, adjusting the discrepancy in the calendars from 10 back to 11 days. Sweden finally adopted the Gregorian calendar in 1753, when Wednesday, 17 February was followed by Thursday, 1 March. Since Finland was under Swedish rule at that time, it did the same.

Britain and the British Empire (including the eastern part of what is now the United States) adopted the Gregorian calendar in 1752 by which time it was necessary to correct by 11 days. Wednesday, 2 September 1752 was followed by Thursday, 14 September 1752 to account for 29 February 1700 (Julian). Claims that rioters demanded "Give us our eleven days" were invented by the painter William Hogarth. After 1753, the British tax year in Britain continued to operate on the Julian calendar and began on 5 April, which was the "Old Style" new tax year of 25 March. A 12th skipped Julian leap day in 1800 changed its start to 6 April. It was not changed when a 13th Julian leap day was skipped in 1900, so the tax year in the United Kingdom still begins on 6 April.

In Alaska, the change took place when Friday, 6 October 1867 was followed again by Friday, 18 October after the US purchase of Alaska from Russia, which was still on the Julian calendar. Instead of 12 days, only 11 were skipped, and the day of the week was repeated on successive days, because the International Date Line was shifted from Alaska's eastern to western boundary along with the change to the Gregorian calendar.

In Russia the Gregorian calendar was accepted after the October Revolution (so named because it took place in October 1917 in the Julian calendar). On 24 January 1918 the Council of People's Commissars issued a Decree that Wednesday, 31 January 1918 was to be followed by Thursday, 14 February 1918.

The last country of Eastern Orthodox Europe to adopt the Gregorian calendar was Greece on Thursday, 1 March 1923, following Wednesday, 15 February 1923.

Adoption in Eastern Asia

The Republic of China (ROC) formally adopted the Gregorian calendar at its founding on 1 January 1912, but China soon descended into a period of warlordism with different warlords using different calendars. With the unification of China under the Kuomintang in October 1928, the Nationalist Government decreed that effective 1 January 1929 the Gregorian calendar would be used. However, China retained the Chinese traditions of numbering the months and a modified Era System, backdating the first year of the ROC to 1912; this system is still in use in Taiwan where this ROC government retains control. Upon its foundation in 1949, the People's Republic of China continued to use the Gregorian calendar with numbered months, but abolished the ROC Era System and adopted Western numbered years.

Japan replaced the traditional lunisolar calendar with the Gregorian calendar on 1 January 1873, but, like China, continued to number the months, and used reign names instead of the Common Era: Meiji 1=1868, Taisho 1=1912, Showa 1=1926, Heisei 1=1989, and so on. The "Western calendar" (西暦, seireki) using western year numbers, is also widely accepted by civilians and to a lesser extent by government agencies.

Korea started using the Gregorian calendar on 1 January 1896 due to Japanese influence. The lunisolar Korean calendar used immediately before that day was based on the lunisolar Chinese calendar.

Adoption by Orthodox Churches

Despite all the civil adoptions, none of the national Orthodox Churches recognized it. Instead, a Revised Julian calendar was proposed in May 1923 which dropped 13 days in 1923 and adopted a different leap year rule. There will be no difference between the two calendars until 2800. The Orthodox churches of Constantinople, Alexandria, Antioch, Greece, Cyprus, Romania, and Bulgaria adopted the Revised Julian calendar, so these New calendarists would celebrate the Nativity along with the Western churches on 25 December in the Gregorian calendar until 2800.

The Orthodox churches of Jerusalem, Russia, Serbia, Macedonia, Georgia, Poland and the Greek Old Calendarists did not accept the Revised Julian calendar. All these Old Calendarists continue to celebrate the Nativity on 25 December in the Julian calendar, which is 7 January in the Gregorian calendar until 2100.

All of the other Eastern churches, the Oriental Orthodox churches (Coptic, Ethiopian, Eritrean, Syrian, Armenian) and the Assyrian Church, continue to use their own calendars, which usually result in fixed dates being celebrated in accordance with the Julian calendar.

All Eastern churches continue to use the Julian Easter with the sole exception of the Finnish Orthodox Church, which has adopted the Gregorian Easter.

Timeline

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Difference between Gregorian and Julian calendar dates

Since the introduction of the Gregorian calendar, the difference between Gregorian and Julian calendar dates has increased by three days every four centuries:

Gregorian range	Julian range	Difference
From 15 October 1582 to 28 February 1700	From 5 October 1582 to 18 February 1700	10 days
From 1 March 1700 to 28 February 1800	From 19 February 1700 to 17 February 1800	11 days
From 1 March 1800 to 28 February 1900	From 18 February 1800 to 16 February 1900	12 days
From 1 March 1900 to 28 February 2100	From 17 February 1900 to 15 February 2100	13 days
From 1 March 2100 to 28 February 2200	From 16 February 2100 to 14 February 2200	14 days

Beginning of the year

Ancient Romans began their years on 1 January since 153 BC. During the Middle Ages, under the influence of the Christian Church, many countries moved the start of the year to one of several important Christian festivals — 25 December (the Nativity of Jesus), 1 March, 25 March (Annunciation), or Easter.

In common usage, 1 January was regarded as New Year's Day and celebrated as such, but from the 12^th century until 1751 the legal year in England began on 25 March (Lady Day). So, for example, the Parliamentary record records the execution of Charles I occurring in 1648, (as the year did not end until 24 March,) although modern histories adjust the start of the year to 1 January and record the execution as occurring in 1649.

Most Western European countries changed the start of the year to 1 January before they adopted the Gregorian calendar. For example Scotland changed the start of the Scottish New Year to 1 January in 1600 (this means that 1599 was a short year). England, Ireland and the British colonies changed the start of the year to 1 January in 1752, (so 1751 was a short year with only 282 days). Later that year in September the Gregorian calendar was introduced throughout Britain and the British colonies (See the section Adoption). These two reforms were implemented by the Calendar (New Style) Act 1750.

Country	Start numbered year on 1 January	Adoption of Gregorian Calendar
Venice	1522	1582
Holy Roman Empire	1544	from 1583
Spain, Portugal, and Southern Netherlands	1556	1582
Prussia	1559	1700
Denmark	1559	1700
Sweden	1559	1753
France	1564	1582
Lorraine	1579	1682
Dutch Republic	1583	from 1582
Scotland	1600	1752
Russia	1700	1918
Tuscany	1721	1750
Britain and British Empire except Scotland	1752	1752

Neither the papal bull nor its attached canons explicitly fix such a date, though it is implied by two tables of saint's days, one labeled 1582 which ends on 31 December, and another for any full year that begins on 1 January. It also specifies its epact relative to 1 January, in contrast with the Julian calendar, which specified it relative to 22 March. These would have been the inevitable result of the above shift in the beginning of the Julian year.

Dual dating

During the period between 1582, when the first countries adopted the Gregorian calendar, and 1923, when the last European country adopted it, it was often necessary to indicate the date of some event in both the Julian calendar and in the Gregorian calendar, for example, "10/21 February 1750/51", where the dual year accounts for some countries already beginning their numbered year on 1 January while others were still using some other date. Even before 1582, the year sometimes had to be double dated because of the different beginnings of the year in various countries. Woolley, writing in his biography of John Dee (1527–1608/9), notes that immediately after 1582 English letter writers "customarily" used "two dates" on their letters, one OS and one NS.

Old Style and New Style dates

"Old Style" (OS) and "New Style" (NS) are sometimes added to dates to identify which system is used in the British Empire and other countries that did not immediately change. Because the Calendar Act of 1750 altered the start of the year, and also aligned the British calendar with the Gregorian calendar, there is some confusion as to what these terms mean. They can indicate that the start of the Julian year has been adjusted to start on 1 January (NS) even though contemporary documents use a different start of year (OS); or to indicate that a date conforms to the Julian calendar (OS), formerly in use in many countries, rather than the Gregorian calendar (NS).

Use of dates from historical documents

There was a great deal of confusion when the calendar changed, and the confusion continues today. In some cases, historians did not differentiate between the years, forcing some researchers to guess between two years when interpreting the information. When "translating" dates from historical documents to current documents for dates that have been incorrectly double dated by historians, both years should be entered into present-day documents until a copy of the original primary source verifies which year was written in the official record. Often errors have been perpetuated from the early 19th century and still exist today. When translating dates from historical documents to current documents for dates that have been correctly double dated by historians, the standard practice is to enter the earlier year first, and the later year second.

Proleptic Gregorian calendar

The Gregorian calendar can, for certain purposes, be extended backwards to dates preceding its official introduction, producing the proleptic Gregorian calendar. However, this proleptic calendar should be used with great caution.

For ordinary purposes, the dates of events occurring prior to 15 October 1582 are generally shown as they appeared in the Julian calendar, with the year starting on 1 January, and no conversion to their Gregorian equivalents. The Battle of Agincourt is universally known to have been fought on 25 October 1415 which is Saint Crispin's Day.

Usually, the mapping of new dates onto old dates with a start of year adjustment works well with little confusion for events which happened before the introduction of the Gregorian Calendar. But for the period between the first introduction of the Gregorian calendar on 15 October 1582 and its introduction in Britain on 14 September 1752, there can be considerable confusion between events in continental western Europe and in British domains in English language histories. Events in continental western Europe are usually reported in English language histories as happening under the Gregorian calendar. For example the Battle of Blenheim is always given as 13 August 1704. However confusion occurs when an event affects both. For example William III of England arrived at Brixham in England on 5 November (Julian calendar), after setting sail from the Netherlands on 11 November (Gregorian calendar).

Shakespeare and Cervantes apparently died on exactly the same date (23 April 1616), but in fact Cervantes predeceased Shakespeare by ten days in real time (for dating these events, Spain used the Gregorian calendar, but Britain used the Julian calendar). This coincidence however has allowed UNESCO to make 23 April the World Book and Copyright Day.

Astronomers avoid this ambiguity by the use of the Julian day number.

For dates before the year 1, unlike the proleptic Gregorian calendar used in the international standard ISO 8601, the traditional proleptic Gregorian calendar (like the Julian calendar) does not have a year 0 and instead uses the ordinal numbers 1, 2, … both for years AD and BC. Thus the traditional timeline is 2 BC, 1 BC, AD 1, and AD 2. ISO 8601 uses astronomical year numbering which includes a year 0 and negative numbers before it. Thus the ISO 8601 timeline is -0001, 0000, 0001, and 0002.

Months of the year

English speakers sometimes remember the number of days in each month by the use of the traditional mnemonic verse:

Thirty days hath September,

April, June, and November.

All the rest have thirty-one,

excepting February alone,

which hath twenty-eight.

Leap year cometh one year in four,

in which February hath one day more.

For variations and alternate endings, see Thirty days hath September.

A language-independent alternative used in many countries is to hold up your two fists with the index knuckle of your left hand against the index knuckle of your right hand. Then, starting with January from the little knuckle of your left hand, count knuckle, space, knuckle, space through the months. A knuckle represents a month of 31 days, and a space represents a short month (a 28- or 29-day February or any 30-day month). The junction between the hands is not counted, so the two index knuckles represent July and August. This method also works by starting the sequence on the right hand's little knuckle, and continue toward to the left. You can also use just one hand; after counting the fourth knuckle as July, start again counting the first knuckle as August. A similar mnemonic can be found on a piano keyboard: starting on the key F for January, moving up the keyboard in semitones, the black notes give the short months, the white notes the long ones.

The Origins of English naming used by the Gregorian calendar:

January: Janus (Roman god of gates, doorways, beginnings and endings)
February: Februus (Etruscan god of death) Februarius (mensis) (Latin for "month of purification (rituals)" it is said to be a Sabine word, the last month of ancient pre-450 BC Roman calendar). It is related to fever.
March: Mars (Roman god of war)
April: "Modern scholars associate the name with an ancient root meaning 'other', i.e the second month of a year beginning in March.
May: Maia Maiestas (Roman goddess)
June: Juno (Roman goddess, wife of Jupiter)
July: Julius Caesar (Roman dictator) (month was formerly named Quintilis, the fifth month of the calendar of Romulus)
August: Augustus (first Roman emperor) (month was formerly named Sextilis, the sixth month of Romulus)
September: septem (Latin for seven, the seventh month of Romulus)
October: octo (Latin for eight, the eighth month of Romulus)
November: novem (Latin for nine, the ninth month of Romulus)
December: decem (Latin for ten, the tenth month of Romulus)

Week

In conjunction with the system of months there is a system of weeks. A physical or electronic calendar provides conversion from a given date to the weekday, and shows multiple dates for a given weekday and month. Calculating the day of the week is not very simple, because of the irregularities in the Gregorian system. When the Gregorian calendar was adopted by each country, the weekly cycle continued uninterrupted. So, using the original proposed adoption date, Thursday, 4 October 1582 would be followed by Friday 15 October.

Distribution of dates by day of the week

Since the 400-year cycle of the Gregorian calendar consists of a whole number of weeks, each cycle has a fixed distribution of weekdays among calendar dates. It then becomes possible that this distribution is not even.

Indeed, because there are 97 leap years in every 400 years in the Gregorian Calendar, there are on average 13 for each starting weekday in each cycle. This already shows that the frequency is not the same for each weekday, which is due to the effects of the "common" centennial years (1700, 1800, 1900, 2100, 2200 etc.).

The absence of an extra day in such years causes the following leap year (1704, 1804, 1904, 2104 etc.) to start on the same day of the week as the leap year twelve years before (1692, 1792, 1892, 2092 etc.). Similarly, the leap year eight years after a "common" centennial year (1708, 1808, 1908, 2108 etc.) starts on the same day of the week as the leap year immediately prior to the "common" centennial year (1696, 1796, 1896, 2096 etc.). Thus, those days of the week on which such leap years begin gain an extra year or two in each cycle. In each cycle there are:

Note that as a cycle, this pattern is symmetric with respect to the low Saturday value.

A leap year starting on Sunday means the next year does not start on Monday, so more leap years starting on Sunday means fewer years starting on Monday, etc. Thus the pattern of number of years starting on each day is inverted and shifted by one weekday: 58, 56, 58, 57, 57, 58, 56 (symmetric with respect to the high Sunday value).

The number of common years starting on each day is found by subtraction: 43, 43, 44, 43, 44, 43, 43.

The frequency of a particular date being on a particular weekday can easily be derived from the above (for dates in March and later, relate them to the next New Year).

Accuracy

The Gregorian calendar improves the approximation made by the Julian calendar by skipping three Julian leap days in every 400 years, giving an average year of 365.2425 mean solar days long, which has an error of about one day per 3300 years with respect to the mean tropical year, which in 2000 had a length of 365.24219 days (each day consisting of 86,400 SI seconds) but less than half this error with respect to the vernal equinox year of 365.24237 days. By either criterion, the Gregorian calendar is substantially more accurate than the one day in 128 years error of the Julian calendar (average year 365.25 days).

In the 19th century, Sir John Herschel proposed a modification to the Gregorian calendar with 969 leap days per 4000 years, instead of 970 leap days that the Gregorian calendar would insert over the same period. This would reduce the average year to 365.24225 days. Herschel's proposal would make the year 4000 common instead of leap. While this modification has often been proposed since, it has never been officially adopted.

On timescales of thousands of years, the Gregorian calendar falls behind the seasons because the slowing down of the Earth's rotation makes each day slightly longer over time (see tidal acceleration, leap second, and precession) while the year maintains a more uniform duration. Borkowski reviewed mathematical models in the literature, and found the results generally fall between a model by McCarthy and Babcock, and another by Stephenson and Morrison. If so, in the year 4000, the calendar will fall behind by at least 0.8, but less than 1.1 days. In the year 12,000 the calendar would fall behind at least 8, but less than 12 days.

Calendar seasonal error

This image shows the difference between the Gregorian calendar and the seasons.

The y-axis is "days error" and the x-axis is Gregorian calendar years.

Each point represents a single date on a given year. The error shifts by about a quarter of a day per year. Centurial years are ordinary years, unless they are divisible by 400, in which case they are leap years. This causes a correction on years 1700, 1800, 1900, 2100, 2200, and 2300.

For instance, these corrections cause 23 December 1903 to be the latest December solstice, and 20 December 2096 to be the earliest solstice—2.25 days of variation compared with the seasonal event.

Numerical facts

When leap years and common years are taken into account, there are a total of 14 possible Gregorian calendars.

When different dates of Easter are also taken into account, there are a total of 70 possible Gregorian calendars.

An average year is 365.2425 days = 52.1775 weeks = 8,765.82 hours = 525,949.2 minutes = 31,556,952 seconds. All these numbers are based on UT1, not UTC.

A common year is 365 days = 8,760 hours = 525,600 minutes = 31,536,000 seconds.

A leap year is 366 days = 8,784 hours = 527,040 minutes = 31,622,400 seconds.

Since 1971, some years may also contain one or more leap seconds, to account for cumulative irregularities in the Earth's rotation. So far, these have always been positive and have occurred on average once every 18 months.

The day of the year is somewhat inconvenient to compute, partly because the leap day does not fall at the end of the year. But the calendar exhibits a repeating pattern for the number of days in the months March through July and August through December: 31, 30, 31, 30, 31, for a total of 153 days each. In fact, any five consecutive months not containing February contain exactly 153 days.

The 400-year cycle of the Gregorian calendar has 146,097 days and hence exactly 20,871 weeks. So, for example, the days of the week in Gregorian 1603 were exactly the same as for 2003. The years that are divisible by 400 begin on a Saturday. In the 400-year cycle, more months begin on a Sunday (and hence have Friday 13) than any other day of the week (see above under #Week for a more detailed explanation of how this happens). 688 out of every 4800 months (or 172/1200) begin on a Sunday, while only 684 out of every 4800 months (171/1200) begin on each of Saturday and Monday, the least common cases.

A smaller cycle is 28 years (1,461 weeks), provided that there is no dropped leap year in between. Days of the week in years may also repeat after 6, 11, 12, 28 or 40 years. Intervals of 6 and 11 are only possible with common years, while intervals of 28 and 40 are only possible with leap years. An interval of 12 years only occurs with common years when there is a dropped leap year in between.

The Doomsday algorithm is a method by which you can discern which of the 14 calendar variations should be used in any given year (after the Gregorian reformation). It is based on the last day in February, referred to as the Doomsday.

The Rata Die is the number of days from 1 January A.D. 1 (counting that day as day 1) in the proleptic Gregorian calendar. For , , it is . It is 678,576 more than the Modified Julian date, and 1,721,425 less than the Julian date .

Day of week

Common years always begin and end on the same day of the week, since 364 (365 - 1) is a multiple of 7, the number of days in a week. For example, 2003 began on a Wednesday and ended on a Wednesday. Leap years end on the next day of the week from which they begin. For example, 2004 began on a Thursday and ended on a Friday.

Not counting leap years, any calendar date will move to the next day of the week the following year. For example, if your birthday fell on a Tuesday in 2002, it fell on a Wednesday in 2003. Leap years make things a little more complicated. 2004 was a leap year, so calendar days of March 1 or later in the year, moved two days of the week from 2003. However, calendar days occurring before March 1 do not make the extra day of the week jump until the year following a leap year. So, if your birthday is June 15, then it must have fallen on a Sunday in 2003 and a Tuesday in 2004. If, however, your birthday is February 15, then it must have fallen on a Saturday in 2003, a Sunday in 2004 and a Tuesday in 2005.

In any year (even a leap year), July always begins on the same day of the week that April does. Therefore, the only difference between a July calendar page and an April calendar page in the same year is the extra day July has. The same relationship exists between September and December as well as between March and November. Add an extra day to the September page and you've got December. Take a day away from the March page and you've got November. In common years only, there are additional matches: October duplicates January, and March and November duplicate February in their first 28 days. In leap years only, there is a different set of additional matches: July is a duplicate of January while February is duplicated in the first 29 days of August.

References

Blackburn, B. & Holford-Strevens, L. (1999). The Oxford Companion to the Year. Oxford University Press ISBN 0-19-214231-3. Pages 98-99.
Coyne, G. V., Hoskin, M. A., and Pedersen, O.(Eds.) (1983). Gregorian reform of the calendar: Proceedings of the Vatican conference to commemorate its 400th anniversary, 1582-1982. Vatican City: Pontifical Academy of Sciences, Specolo Vaticano.
Duncan, D. E. (1999). Calendar: Humanity's Epic Struggle To Determine A True And Accurate Year. Harper Perennial. ISBN 0-380-79324-5.
Gregory XIII. (2002). Inter Gravissimas (W. Spenser & R. T. Crowley, Trans.). International Organization for Standardization. (Original work published 1582)
Online Etymology Dictionary retrieved 23 August 2006
Seidelmann, P. K. (Ed.). (1992). Explanatory Supplement to the Astronomical Almanac. Sausalito, CA: University Science Books.