Gregorian calendar
The Gregorian calendar is the calendar used in most of the world.Dershowitz & Reingold 2008, p. 45. "The calendar in use today in most of the world is the Gregorian or new-style calendar designed by a commission assembled by Pope Gregory XIII in the sixteenth century." It is named after Pope Gregory XIII, who introduced it in October 1582. 

The calendar spaces leap years to make the average year 365.2425 days long, approximating the 365.2422-day tropical year that is determined by the Earth's revolution around the Sun. The rule for leap years is:

Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100, but these centurial years are leap years if they are exactly divisible by 400. For example, the years 1700, 1800, and 1900 are not leap years, but the years 1600 and 2000 are.Introduction to Calendars. (15 May 2013). United States Naval Observatory.

The calendar was developed as a correction to the Julian calendar,The Julian calendar assumed incorrectly that the average year is exactly 365.25 (365¼) days long. shortening the average year by 0.0075 days to stop the drift of the calendar with respect to the equinoxes.See Wikisource English translation of the (Latin) 1582 papal bull Inter gravissimas. To deal with the 10 days' difference (between calendar and reality) that this drift had already reached, the date was advanced so that 4 October 1582 was followed by 15 October 1582. There was no discontinuity in the cycle of weekdays or of the Anno Domini calendar era. The reform also altered the lunar cycle used by the Church to calculate the date for Easter (computus), restoring it to the time of the year as originally celebrated by the early Church. 

The reform was adopted initially by the Catholic countries of Europe and their overseas possessions. Over the next three centuries, the Protestant and Eastern Orthodox countries also moved to what they called the Improved calendar, with Greece being the last European country to adopt the calendar in 1923.Blegen n.d. To unambiguously specify a date during the transition period, (or in history texts), dual dating is sometimes used to specify both Old Style and New Style dates (abbreviated as O.S and N.S. respectively). Due to globalization in the 20th century, the calendar has also been adopted by most non-Western countries for civil purposes. The calendar era carries the alternative secular name of "Common Era".

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Description
+A year is divided into twelve months   Name  Length in days 1  January  31 2  February  28 (29 in leap years) 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

The Gregorian calendar is a solar calendar with 12 months of 28–31 days each. A regular Gregorian year consists of 365 days, but in certain years known as leap years, a leap day is added to February. Gregorian years are identified by consecutive year numbers.Clause 3.2.1 ISO 8601 A calendar date is fully specified by the year (numbered according to a calendar era, in this case Anno Domini or Common Era), the month (identified by name or number), and the day of the month (numbered sequentially starting from 1). Although the calendar year currently runs from 1 January to 31 December, at previous times year numbers were based on a different starting point within the calendar (see the "beginning of the year" section below).

In the Julian calendar, a leap year occurred every 4 years, and the leap day was inserted by doubling 24 February. The Gregorian reform omitted a leap day in three of every 400 years and left the leap day unchanged. However, it has become customary in the modern period to number the days sequentially with no gaps, and 29 February is typically considered as the leap day. Before the 1969 revision of the Roman Calendar, the Roman Catholic Church delayed February feasts after the 23rd by one day in leap years; Masses celebrated according to the previous calendar still reflect this delay.Richards 1998, p. 101

Calendar cycles repeat completely every 400 years, which equals 146,097 days. Of these 400 years, 303 are regular years of 365 days and 97 are leap years of 366 days. A mean calendar year is  days = 365.2425 days, or 365 days, 5 hours, 49 minutes and 12 seconds.The same result is obtained by summing the fractional parts implied by the rule: 

Gregorian reform

The Gregorian calendar was a reform of the Julian calendar. It was instituted by papal bull Inter gravissimas dated 24 February 1582 by Pope Gregory XIII, after whom the calendar is named. The motivation for the adjustment was to bring the date for the celebration of Easter to the time of year in which it was celebrated when it was introduced by the early Church. The error in the Julian calendar (its assumption that there are exactly 365.25 days in a year) had led to the date of the equinox according to the calendar drifting from the observed reality, and thus an error had been introduced into the calculation of the date of Easter. Although a recommendation of the First Council of Nicaea in 325 specified that all Christians should celebrate Easter on the same day, it took almost five centuries before virtually all Christians achieved that objective by adopting the rules of the Church of Alexandria (see Easter for the issues which arose).The last major Christian region to accept the Alexandrian rules was the Carolingian Empire (most of Western Europe) during 780–800. The last monastery in England to accept the Alexandrian rules did so in 931, and a few churches in southwest Asia beyond the eastern border of the Byzantine Empire continued to use rules that differed slightly, causing four dates for Easter to differ every 532 years.

Background
Because the date of Easter is a function the computus of the date of the (northern hemisphere) spring equinox, the Catholic Church considered unacceptable the increasing divergence between the canonical date of the equinox and observed reality. Easter is celebrated on the Sunday after the ecclesiastical full moon on or after 21 March, which was adopted as approximation to the March equinox.Richards 2013, p. 599. European scholars had been well aware of the calendar drift since the early medieval period.

Bede, writing in the 8th century, showed that the accumulated error in his time was more than three days. Roger Bacon in c. 1200 estimated the error at seven or eight days. Dante, writing c. 1300, was aware of the need of a calendar reform. The first attempt to go forward with such a reform was undertaken by Pope Sixtus IV, who in 1475 invited Regiomontanus to the Vatican for this purpose. However, the project was interrupted by the death of Regiomontanus shortly after his arrival in Rome.Ari Ben-Menahem, Historical Encyclopedia of Natural and Mathematical Sciences vol. 1 (2009), p. 863.
The increase of astronomical knowledge and the precision of observations towards the end of the 15th century made the question more pressing. Numerous publications over the following decades called for a calendar reform, among them two papers sent to the Vatican by the University of Salamanca in 1515 and 1578,Carabias Torres, 2012, p. 241
but the project was not taken up again until the 1540s, and implemented only under Pope Gregory XIII (r. 1572–1585).

Preparation
In 1545, the Council of Trent authorized Pope Paul III to reform the calendar, 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.

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.Ziggelaar (1983), pp. 211, 214. The reform adopted was a modification of a proposal made by the Calabrian doctor Aloysius Lilius (or Lilio).Moyer 1983.

Lilius's proposal included reducing the number of leap years in four centuries from 100 to 97, by making three out of four centurial years common instead of leap years. He also produced an original and practical scheme for adjusting the epacts of the Moon when calculating the annual date of Easter, solving a long-standing obstacle to calendar reform.

Ancient tables provided the Sun's mean longitude.See, for example,Tabule illustrissimi principis regis alfonsii, Prague 1401−4 (Latin). A full set of Alphonsine Tables (including tables for mean motions, conjunctions of Sun and Moon, equation of time, spherical astronomy, longitudes and latitudes of cities, star tables, eclipse tables).
For an example of the information provided see Jacques Cassini, Tables astronomiques du soleil, de la lune, des planètes, des étoiles fixes, et des satellites de Jupiter et de Saturne, Paris 1740, available at  (go forward ten pages to Table III on p. 10). The German mathematician Christopher Clavius, the architect of the Gregorian calendar, noted that the tables agreed neither on the time when the Sun passed through the vernal equinox nor on the length of the mean tropical year. Tycho Brahe also noticed discrepancies. The Gregorian leap year rule (97 leap years in 400 years) was put forward by Petrus Pitatus of Verona in 1560. He noted that it is consistent with the tropical year of the Alfonsine tables and with the mean tropical year of Copernicus (De revolutionibus) and Erasmus Reinhold (Prutenic tables). The three mean tropical years in Babylonian sexagesimals as the excess over 365 days (the way they would have been extracted from the tables of mean longitude) were 14,33,9,57 (Alfonsine), 14,33,11,12 (Copernicus) and 14,33,9,24 (Reinhold). All values are the same to two places (14:33) and this is also the mean length of the Gregorian year. Thus Pitatus' solution would have commended itself to the astronomers.Swerdlow (1986).

Lilius's proposals had two components. First, he proposed a correction to the length of the year. The mean tropical year is 365.24219 days long.Meeus and Savoie (1992). A commonly used value in Lilius's time, from the Alfonsine tables, is 365.2425463 days.Moyer (1983). p. As the average length of a Julian year is 365.25 days, the Julian year is almost 11 minutes longer than the mean tropical year. The discrepancy results in a drift of about three days every 400 years. Lilius's proposal resulted in an average year of 365.2425 days (see Accuracy). At the time of Gregory's reform there had already been a drift of 10 days since the Council of Nicaea, resulting in the vernal equinox falling on 10 or 11 March instead of the ecclesiastically fixed date of 21 March, and if unreformed it would have drifted further. Lilius proposed that the 10-day drift should be corrected by deleting the Julian leap day on each of its ten occurrences over a period of forty years, thereby providing for a gradual return of the equinox to 21 March.

Lilius's work was expanded upon by Christopher Clavius in a closely argued, 800-page volume. He would later defend his and Lilius's work against detractors. Clavius's opinion was that the correction should take place in one move, and it was this advice which prevailed with Gregory.

The second component consisted of an approximation which would provide an accurate yet simple, rule-based calendar. Lilius's formula was a 10-day correction to revert the drift since the Council of Nicaea, and the imposition of a leap day in only 97 years in 400 rather than in 1 year in 4. The proposed rule was that years divisible by 100 would be leap years only if they were divisible by 400 as well.

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 that are no longer leap years (i.e. 1700, 1800, 1900, 2100, etc.) In fact, a new method for computing the date of Easter was introduced.

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 10 days. The Julian calendar day Thursday, 4 October 1582 was followed by the first day of the Gregorian calendar, Friday, 15 October 1582 (the cycle of weekdays was not affected).

First printed Gregorian calendar
thumb|
Lunario Novo, Secondo la Nuova Riforma della Correttione del l'Anno Riformato da N.S. Gregorio XIII, printed in Rome by Vincenzo Accolti in 1582, one of the first printed editions of the new calendar.
A month after having decreed the reform, the pope (with a brief of 3 April 1582) granted to one Antoni Lilio the exclusive right to publish the calendar for a period of ten years. The Lunario Novo secondo la nuova riforma was printed by Vincenzo Accolti, one of the first calendars printed in Rome after the reform, notes at the bottom that it was signed with papal authorization and by Lilio (Con licentia delli Superiori... et permissu Ant(onii) Lilij). The papal brief was revoked on 20 September 1582, because Antonio Lilio proved unable to keep up with the demand for copies.Mezzi, E., and Vizza, F., Luigi Lilio Medico Astronomo e Matematico di Cirò, Laruffa Editore, Reggio Calabria, 2010, p. 14; p. 52, citing as primary references: Biblioteca Nazionale Centrale die Firenze, Magl. 5.10.5/a, ASV A.A., Arm. I‑XVIII, 5506, f. 362r.

Adoption 

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

The bull Inter gravissimas became the law of the Catholic Church in 1582, but it was not recognised by Protestant Churches, Eastern Orthodox Churches, Oriental Orthodox Churches, and a few others. Consequently, the days on which Easter and related holidays were celebrated by different Christian Churches again diverged.

On 29 September 1582, Philip II of Spain decreed the change from the Julian to the Gregorian calendar. This affected much of Roman Catholic Europe, as Philip was at the time ruler over Spain and Portugal as well as much of Italy. In these territories, as well as in the Polish–Lithuanian Commonwealth (ruled by Anna Jagiellon) and in the Papal States, the new calendar was implemented on the date specified by the bull, with Julian Thursday, 4 October 1582, being followed by Gregorian Friday, 15 October 1582. The Spanish and Portuguese colonies followed somewhat later de facto because of delay in communication."Pragmatica" on the Ten Days of the Year World Digital Library, the first known South American imprint, produced in 1584 by Antonio Ricardo, of a four-page edict issued by King Philip II of Spain in 1582, decreeing the change from the Julian to the Gregorian calendar.

Many Protestant countries initially objected to adopting a Catholic innovation; some Protestants feared the new calendar was part of a plot to return them to the Catholic fold. For example, the British could not bring themselves to adopt the Catholic system explicitly: the Annexe to their Calendar (New Style) Act 1750 established a computation for the date of Easter that achieved the same result as Gregory's rules, without actually referring to him.24 Geo. II Ch. 23, § 3.

Britain and the British Empire (including the eastern part of what is now the United States) adopted the Gregorian calendar in 1752. Sweden followed in 1753.

Prior to 1917, Turkey used the lunar Islamic calendar with the Hegira era for general purposes and the Julian calendar for fiscal purposes. The start of the fiscal year was eventually fixed at 1 March and the year number was roughly equivalent to the Hegira year (see Rumi calendar). As the solar year is longer than the lunar year this originally entailed the use of "escape years" every so often when the number of the fiscal year would jump. From 1 March 1917 the fiscal year became Gregorian, rather than Julian. On 1 January 1926 the use of the Gregorian calendar was extended to include use for general purposes and the number of the year became the same as in most other countries.

Adoption of the Gregorian Calendar

 Year Country/-ies/Areas  1582 Spain, Portugal, France, Poland, Italy, Catholic Low Countries, Luxemburg, and colonies 1584 Kingdom of Bohemia 1610 Prussia 1648 Alsace 1682 Strasbourg 1700 'Germany', Swiss Cantons, Protestant Low Countries, Norway, Denmark 1752 Great Britain and colonies 1753 Sweden and Finland 1873 Japan 1875 Egypt 1896 Korea 1912 China, Albania 1915 Latvia, Lithuania 1916 Bulgaria 1918 Russia, Estonia 1919 Romania, Yugoslavia1919 in the regions comprising the former Kingdoms of Serbia and Montenegro (present-day Kosovo, Montenegro, Serbia and Macedonia). The western and northern regions of what became Yugoslavia were already using the Gregorian calendar. For example, most of Slovenia adopted the Gregorian calendar at the same time as Austria in 1583. Coastal Croatia, which was at the time ruled by Venice, adopted the Gregorian calendar in 1582. Inland Croatia, ruled by the Habsburgs, adopted it in 1587 along with Hungary. The Gregorian calendar was used in Bosnia and Herzegovina since the 16th century by the Catholic population and was formally adopted for government use in 1878 following occupation by Austria-Hungary. 1923 Greece 1926 Turkey 2016 Saudi Arabia

Difference between Gregorian and Julian calendar dates
+Conversion from Julian to Gregorian dates.A more extensive list is available at Conversion between Julian and Gregorian calendars Gregorian range  Julian range  DifferenceFrom 15 October 1582to 28 February 1700From 5 October 1582to 18 February 170010 daysFrom 1 March 1700to 28 February 1800From 19 February 1700to 17 February 180011 daysFrom 1 March 1800to 28 February 1900From 18 February 1800to 16 February 190012 daysFrom 1 March 1900to 28 February 2100From 17 February 1900to 15 February 210013 daysFrom 1 March 2100to 28 February 2200From 16 February 2100to 14 February 220014 days

This section always places the intercalary day on  even though it was always obtained by doubling  (the bissextum (twice sixth) or bissextile day) until the late Middle Ages. The Gregorian calendar is proleptic before 1582 (assumed to exist before 1582), and the difference between Gregorian and Julian calendar dates increases by three days every four centuries (all date ranges are inclusive).

The following equation gives the number of days (actually, dates) that the Gregorian calendar is ahead of the Julian calendar, called the secular difference between the two calendars. A negative difference means the Julian calendar is ahead of the Gregorian calendar.Blackburn & Holford-Strevens (1999), p. 788.

where  is the secular difference and  is the year using astronomical year numbering, that is, use  for BC years.  means that if the result of the division is not an integer it is rounded down to the nearest integer. Thus during the 1900s, 1900/400 = 4, while during the −500s, −500/400 = −2.

The general rule, in years which are leap years in the Julian calendar but not the Gregorian, is:

Up to 28 February in the calendar being converted , add one day less or subtract one day more than the calculated value. Give February the appropriate number of days for the calendar being converted . When subtracting days to calculate the Gregorian equivalent of 29 February (Julian), 29 February is discounted. Thus if the calculated value is −4 the Gregorian equivalent of this date is 24 February.James Evans, The history and practice of ancient astronomy (Oxford: Oxford University Press, 1998) 169. .
Explanatory Supplement to The Astronomical Ephemeris and The American Ephemeris and Nautical Almanac (London: Her Majesty's Stationery Office, 1961) 417.

Beginning of the year
 Country  Start numbered yearon 1 January  Adoption ofGregorian calendar Denmark  Gradual change from13th to 16th centuriesHerluf Nielsen: Kronologi (2nd ed., Dansk Historisk Fællesforening, Copenhagen 1967), pp. 48–50.  1700 Holy Roman Empire (Catholic states)  1544  1583 Spain, Poland, Portugal  1556  1582 Holy Roman Empire (Protestant states)  1559  1700 Sweden  1559  1753 France  1564Le calendrier grégorien en France   1582 Southern Netherlands  1576Per decree of 16 June 1575. Hermann Grotefend, "Osteranfang" (Easter beginning), Zeitrechnung de Deutschen Mittelalters und der Neuzeit (Chronology of the German Middle Ages and modern times) (1891–1898)  1582 Lorraine  1579  1582Lorraine reverted to Julian in 1735 and adopted Gregorian again in 1760 Dutch Republic  1583  1582 Scotland  1600Blackburn & Holford-Strevens (1999), p. 784.John James Bond, Handy-book of rules and tables for verifying dates with the Christian era Scottish decree on pp. xvii–xviii.  1752 Russia  1700Roscoe Lamont, The reform of the Julian calendar, Popular Astronomy 28 (1920) 18–32. Decree of Peter the Great is on pp. 23–24.1918  Tuscany  1750Alexandre Dumas,  Storia del governo della Toscana: sotto La casa de'Medici.
Il calendario fiorentino.  1582Lorenzo Cattini, Legislazione toscana raccolta e illustrata, vol. 10, p. 208. Great Britain and the British Empireexcept Scotland  1752  1752 Venice  1797Fora Febraro.  1582
The year used in dates during the Roman Republic and the Roman Empire was the consular year, which began on the day when consuls first entered office—probably 1 May before 222 BC, 15 March from 222 BC and 1 January from 153 BC. The Julian calendar, which began in 45 BC, continued to use 1 January as the first day of the new year. Even though the year used for dates changed, the civil year always displayed its months in the order January to December from the Roman Republican period until the present.

During the Middle Ages, under the influence of the Catholic Church, many Western European countries moved the start of the year to one of several important Christian festivals—25 December (supposed Nativity of Jesus), 25 March (Annunciation), or Easter (France),Mike Spathaky Old Style and New Style Dates and the change to the Gregorian Calendar: A summary for genealogists while the Byzantine Empire began its year on 1 September and Russia did so on 1 March until 1492 when the new year was moved to 1 September.S. I. Seleschnikow: Wieviel Monde hat ein Jahr? (Aulis-Verlag, Leipzig/Jena/Berlin 1981, p. 149), which is a German translation of С. И. Селешников: История календаря и хронология (Издательство "Наука", Moscow 1977). The relevant chapter is available online here: История календаря в России и в СССР (Calendar history in Russia and the USSR). Anno Mundi 7000 lasted from  to . 

In common usage, 1 January was regarded as New Year's Day and celebrated as such,Tuesday 31 December 1661, The Diary of Samuel Pepys "I sat down to end my journell for this year, ..." but from the 12th century until 1751 the legal year in England began on 25 March (Lady Day).Nørby, Toke. The Perpetual Calendar: What about England Version 29 February 2000 So, for example, the Parliamentary record lists the execution of Charles I on 30 January as occurring in 1648 (as the year did not end until 24 March), although later 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) though in England the start of the tax year remained at 25 March (O.S.), 5 April (N.S.) until 1800, when it moved to 6 April. Later in 1752 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.Nørby, Toke. The Perpetual Calendar

In some countries, an official decree or law specified that the start of the year should be 1 January. For such countries a specific year when a 1 January-year became the norm can be identified. In other countries the customs varied, and the start of the year moved back and forth as fashion and influence from other countries dictated various customs.

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 labelled 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. The old date was derived from the Greek system: the earlier Supputatio Romana specified it relative to 1 January.

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.Benjamin Woolley, The Queen's Conjurer: The science and magic of Dr. John Dee, adviser to Queen Elizabeth I (New York: Henry Holt, 2001) p. 173

Old Style and New Style dates

"Old Style" (OS) and "New Style" (NS) are sometimes added to dates to identify which calendar reference system is used for the date given. In Britain and its colonies, where the Calendar Act of 1750 altered the start of the year,In Scotland the legal start of year had been moved to 1 January in 1600 (Mike Spathaky. Old Style New Style dates and the change to the Gregorian calendar). 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).Death warrant of Charles I web page of the UK National Archives. A demonstration of New Style meaning Julian calendar with a start of year adjustment.Spathaky, Mike Old Style New Style dates and the change to the Gregorian calendar. "increasingly parish registers, in addition to a new year heading after 24th March showing, for example '1733', had another heading at the end of the following December indicating '1733/4'. This showed where the New Style 1734 started even though the Old Style 1733 continued until 24th March. ... We as historians have no excuse for creating ambiguity and must keep to the notation described above in one of its forms. It is no good writing simply 20th January 1745, for a reader is left wondering whether we have used the Old or the New Style reckoning. The date should either be written 20th January 1745 OS (if indeed it was Old Style) or as 20th January 1745/6. The hyphen (1745-6) is best avoided as it can be interpreted as indicating a period of time."
The October (November) Revolution Britannica encyclopaedia, A demonstration of New Style meaning the Gregorian calendar.
Stockton, J.R. Date Miscellany I: The Old and New Styles "The terms 'Old Style' and 'New Style' are now commonly used for both the 'Start of Year' and 'Leap Year' [(Gregorian calendar)] changes (England & Wales: both in 1752; Scotland: 1600, 1752). I believe that, properly and historically, the 'Styles' really refer only to the 'Start of Year' change (from March 25th to January 1st); and that the 'Leap Year' change should be described as the change from Julian to Gregorian."

Proleptic Gregorian calendar

Extending the Gregorian calendar backwards to dates preceding its official introduction produces a proleptic calendar, which should be used with some 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. For example, the Battle of Agincourt is universally considered 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 that 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. Confusion occurs when an event affects both. For example, William III of England set sail from the Netherlands on 11 November 1688 (Gregorian calendar) and arrived at Brixham in England on 5 November 1688 (Julian calendar).

Shakespeare and Cervantes seemingly died on exactly the same date (23 April 1616), but Cervantes predeceased Shakespeare by ten days in real time (as Spain used the Gregorian calendar, but Britain used the Julian calendar). This coincidence encouraged 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 time line 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 time line is , 0000, 0001, and 0002.

Months
The Gregorian calendar continued to employ the Julian months, which have Latinate names and irregular numbers of days:

 January (31 days), from Latin , "Month of Janus",. the Roman god of gates, doorways, beginnings and endings
 February (28 days in common and 29 in leap years), from Latin , "Month of the Februa", the Roman festival of purgation and purification,.. cognate with fever, the Etruscan death god Februus ("Purifier"), and the Proto-Indo-European word for sulfur
 March (31 days), from Latin , "Month of Mars",. the Roman war god
 April (30 days), from Latin , of uncertain meaning. but usually derived from some form of the verb  ("to open") or the name of the goddess Aphrodite
 May (31 days), from Latin , "Month of Maia",. a Roman vegetation goddess whose name is cognate with Latin  ("great") and English major
 June (30 days), from Latin , "Month of Juno",. the Roman goddess of marriage, childbirth, and rule
 July (31 days), from Latin , "Month of Julius Caesar", the month of Caesar's birth, instituted in 44. as part of his calendrical reforms
 August (31 days), from Latin , "Month of Augustus", instituted by Augustus in 8 in agreement with July and from the occurrence during the month of several important events during his rise to power. 
 September (30 days), from Latin , "seventh month", of the ten-month Roman year of Romulus .
 October (31 days), from Latin , "eighth month", of the ten-month Roman year of Romulus .
 November (30 days), from Latin , "ninth month", of the ten-month Roman year of Romulus .
 December (31 days), from Latin , "tenth month", of the ten-month Roman year of Romulus .

Europeans sometimes attempt to remember the number of days in each month by memorizing some form of the traditional verse "Thirty Days Hath September". It appears in Latin,.
.  Italian,. and French,. and belongs to a broad oral tradition but the earliest currently attested form of the poem is the English marginalia inserted into a calendar of saints :..
.

   

thumb|The knuckle mnemonic for the days of the months of the year
Variations appeared in Mother Goose and continue to be taught at schools. The unhelpfulness of such involved mnemonics has been parodied as "Thirty days hath September/ But all the rest I can't remember". but it has also been called "probably the only sixteenth-century poem most ordinary citizens know by heart".. A common nonverbal alternative is the knuckle mnemonic, considering the knuckles of one's hands as months with 31 days and the lower spaces between them as the months with fewer days. Using two hands, one may start from either pinkie knuckle as January and count across, omitting the space between the index knuckles (July and August). The same procedure can be done using the knuckles of a single hand, returning from the last (July) to the first (August) and continuing through. A similar mnemonic is to move up a piano keyboard in semitones from an F key, taking the white keys as the longer months and the black keys as the shorter ones.

Weeks

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. For example, in the case of the few countries that adopted the reformed calendar on the date proposed by Gregory XIII for the calendar's adoption, Friday, 15 October 1582, the preceding date was Thursday, 4 October 1582 (Julian calendar).

Opinions vary about the numbering of the days of the week. ISO 8601, in common use worldwide, starts with Monday=1; printed monthly calendar grids often list Mondays in the first (left) column of dates and Sundays in the last. In North America, the week typically begins on Sunday and ends on Saturday.

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.Seidelmann (1992), pp. 580–581. This approximation has an error of about one day per 3,030 yearsUsing value from Richards (2013, p. 587) for tropical year in mean solar days, the calculation is  with respect to the current value of the mean tropical year. However, because of the precession of the equinoxes, which is not constant, and the movement of the perihelion (which affects the Earth's orbital speed) the error with respect to the astronomical vernal equinox is variable; using the average interval between vernal equinoxes near 2000 of 365.24237 daysMeeus and Savoie (1992), p. 42 implies an error closer to 1 day every 7,700 years. By any criterion, the Gregorian calendar is substantially more accurate than the 1 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 every 4000 years, instead of 970 leap days that the Gregorian calendar would insert over the same period.John Herschel, Outlines of Astronomy, 1849, p. 629. This would reduce the average year to 365.24225 days. Herschel's proposal would make the year 4000, and multiples thereof, common instead of leap. While this modification has often been proposed since, it has never been officially adopted.

On time scales of thousands of years, the Gregorian calendar falls behind the astronomical seasons. This is because the Earth's speed of rotation is gradually slowing down, which makes each day slightly longer over time (see tidal acceleration and leap second) while the year maintains a more uniform duration.

Calendar seasonal error
frameless|upright=3.65|Gregorian calendar seasons difference

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

The y-axis is the date in June and the x-axis is Gregorian calendar years.

Each point is the date and time of the June solstice in that particular 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 in the 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—about 2.35 days of variation compared with the seasonal event.

Proposed reforms
The following are proposed reforms of the Gregorian calendar:
 Holocene calendar
 International Fixed Calendar (also called the International Perpetual calendar)
 World Calendar
 World Season Calendar
 Leap week calendars
 Pax Calendar
 Symmetry454
 Hanke–Henry Permanent Calendar

See also

Calendar (New Style) Act 1750
Calendar reform
Conversion between Julian and Gregorian calendars
Doomsday rule
French revolutionary calendar
Hebrew calendar
Islamic calendar
Inter gravissimas in English – Wikisource
Julian day
History of calendars
 ISO 8601, an international standard for the representation of dates and times, which uses the Gregorian calendar (see Section 3.2.1).
List of adoption dates of the Gregorian calendar per country
List of calendars
Old Calendarists
Greek Old Calendarists
Revised Julian calendar (Milanković) – used in Eastern Orthodoxy
Precursors of the Gregorian reform
 Johannes de Sacrobosco, De Anni Ratione ("On reckoning the years"), c. 1235
 Roger Bacon, Opus Majus ("Greater Work"), c. 1267

Notes

Citations

References
Barsoum, Ignatius A. (2003). The Scattered Pearls. Piscataway: Georgias Press.
Blackburn, B. & Holford-Strevens, L. (1999). The Oxford Companion to the Year. Oxford University Press. .
Blackburn, B. & Holford-Strevens, L. (2003). The Oxford Companion to the Year: An exploration of calendar customs and time-reckoning, Oxford University Press.
Blegen, Carl W. (n.d.). "An Odd Christmas". Posted with an introduction by Natalia Vogeikoff-Brogan on 25 December 2013. From the Archivist's Notebook retrieved 1 April 2018.
Borkowski, K. M., (1991). "The tropical calendar and solar year", J. Royal Astronomical Soc. of Canada 85(3): 121–130.
Carabias Torres, A. M. (2012). Salamanca y la medida del tiempo. Salamanca: Ediciones Universidad de Salamanca.
Coyne, G. V., Hoskin, M. A., 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, Vatican Observatory (Pontificia Academia Scientarum, Specola Vaticana).
Dershowitz, D., Reingold, E. M. (2008). Calendrical Calculations, 3rd ed. Cambridge University Press.
Duncan, D. E. (1999). Calendar: Humanity's Epic Struggle To Determine A True And Accurate Year. HarperCollins. .
Gregory XIII. (2002 [1582]). Inter Gravissimas (W. Spenser & R. T. Crowley, Trans.). International Organization for Standardization.
Meeus, J. & Savoie, D. (1992). The history of the tropical year. Journal of the British Astronomical Association, 102(1): 40–42.
Morrison, L. V. & Stephenson, F. R. (2004). Historical values of the Earth's clock error ΔT and the calculation of eclipses. Journal for the History of Astronomy Vol. 35, Part 3, No. 120, pp. 327–336.
Moyer, Gordon (May 1982). "The Gregorian Calendar". Scientific American, pp. 144–152.
Moyer, Gordon (1983). "Aloisius Lilius and the Compendium Novae Rationis Restituendi Kalendarium". In Coyne, Hoskin, Pedersen (1983), pp. 171–188.
Pattie, T.S. (1976) "An unexpected effect of the change in calendar in 1752". British Library Journal.
Pedersen, O. (1983). "The Ecclesiastical Calendar and the Life of the Church". In Coyne, Hoskin, Pedersen (eds), Gregorian Reform of the Calendar: Proceedings of the Vatican Conference to Commemorate its 400th Anniversary. Vatican City: Pontifical Academy of Sciences, Specolo Vaticano, pp. 17–74.
Richards, E. G. (1998). Mapping Time: The Calendar and its History. Oxford U. Press.
Richards, E. G. (2013). "Calendars". In S. E. Urban and P. K. Seidelmann (eds.), Explanatory Supplement to the Astronomical Almanac 3rd ed. (pp. 585–624). Mill Valley CA: University Science Books. 
Seidelmann, P. K. (Ed.) (1992). Explanatory Supplement to the Astronomical Almanac. 2nd ed. Sausalito, CA: University Science Books.
Swerdlow, N. M. (1986). The Length of the Year in the Original Proposal for the Gregorian Calendar. Journal for the History of Astronomy Vol. 17, No. 49, pp. 109–118.
Walker, G. W. "Easter Intervals". Popular Astronomy June 1945, Vol. 53, pp. 162–178, 218–232.
Ziggelaar, A. (1983). "The Papal Bull of 1582 Promulgating a Reform of the Calendar". In Coyne, Hoskin, Pedersen (eds), Gregorian Reform of the Calendar: Proceedings of the Vatican Conference to Commemorate its 400th Anniversary. Vatican City: Pontifical Academy of Sciences, Specolo Vaticano, pp. 201–239.

External links
 
 Calendar Converter
 Inter Gravissimas (Latin and French plus English)
 History of Gregorian Calendar
 The Perpetual Calendar Gregorian Calendar adoption dates for many countries. 
 World records for mentally calculating the day of the week in the Gregorian Calendar 
 The Calendar FAQ – Frequently Asked Questions about Calendars
 Today's date (Gregorian) in over 800 more-or-less obscure foreign languages

 
Category:1582 establishments
Category:Articles which contain graphical timelines
Category:1582 establishments in Europe