Gregorian calendar

===Preparation===
In 1545, the [[Council of Trent]] authorised [[Pope Paul III]] to reform the calendar, requiring that the date of the [[March equinox|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 more consistent and accurate scheduling of the feast of Easter.

In 1577, a {{lang|la|Compendium}} was sent to expert mathematicians outside the reform commission for comments. Some of these experts, including [[Giambattista Benedetti]] and [[Giuseppe Moletti|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.{{sfnp|Ziggelaar|1983|pages=211, 214}} The reform adopted was a modification of a proposal made by the [[Calabria]]n doctor [[Aloysius Lilius]] (or Lilio).{{sfnp|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 [[Epact|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.{{efn|See, for example, {{lang|la|Tabule illustrissimi principis regis alfonsii}} (Prague 1401−4). 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).<ref>{{cite book |title=Tabule illustrissimi principis regis alfonsii |trans-title=The tablet of the most illustrious prince King Alphonsus |last=John of Saxony |url=https://archive.org/details/ljs174/page/n3/mode/2up |date=1401  |language=la}}</ref> For an example of the information provided see Jacques Cassini, {{lang|fr|Tables astronomiques du soleil, de la lune, des planètes, des étoiles fixes, et des satellites de Jupiter et de Saturne}}, Table III.<ref>{{cite book |title=Tables astronomiques du soleil, de la lune, des planètes, des étoiles fixes, et des satellites de Jupiter et de Saturne |language=fr |trans-title=Astronomical tables of the sun, the moon, the planets, the fixed stars, and the satellites of Jupiter and Saturn |first=Jacques |last=Cassini |date=1740 |location=Paris |publisher=Imprimerie Royale |page=T10 |url=https://archive.org/details/s2id11854200/page/10/mode/1up}}</ref>}} 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.<ref>{{cite book |last=Dreyer |first=J L E |url=https://books.google.com/books?id=CdzSAgAAQBAJ&pg=PA52 |title=Tycho Brahe |date=2014 |publisher=Cambridge University Press |isbn=978-1-108-06871-0 |location=Cambridge |page=52 |quote=He remarks that both the Alphonsine and the Prutenic Tables are several hours wrong with regard to the time of the equinoxes and solstices.}}</ref><ref>{{cite book |url=https://books.google.com/books?id=q2xxatst4OQC&pg=PA29 |last=North |first=J |title=The Universal frame: historical essays in astronomy, natural philosophy and scientific method |location=London |date=1989 |page=29 |isbn=978-0-907628-95-8 |quote=He noted on one occasion that the ''Alphonsine tables'' differed from the ''Prutenic'' by nineteen hours as to the time of the vernal equinox of 1588.}}</ref> 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 [[sexagesimal]]s as the excess over 365 days (the way they would have been extracted from the tables of mean longitude) were 0;14,33,9,57 (Alfonsine), 0;14,33,11,12 (Copernicus) and 0;14,33,9,24 (Reinhold).{{efn|For an explanation of this notation, see [[Sexagesimal#Notations]].}} In decimal notation, these are equal to 0.24254606, 0.24255185, and 0.24254352, respectively. All values are the same to two sexagesimal places (0;14,33, equal to decimal 0.2425) and this is also the mean length of the Gregorian year. Thus Pitatus's solution would have commended itself to the astronomers.{{sfnp|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.{{sfnp|Meeus|Savoie|1992}} A commonly used value in Lilius's time, from the Alfonsine tables, is 365.2425463 days.{{sfnp|Moyer|1983}} 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|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 that prevailed with Gregory.

The second component consisted of an approximation that 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 required revision because the astronomical new moon was, at the time of the reform, four days before the calculated new moon.{{sfnp|Richards|2013|page=599}} It was 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. The method proposed by Lilius was revised somewhat in the final reform.{{sfnp|Ziggelaar|1983|page = 220}}

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).
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====First printed Gregorian calendar====
[[File:Reforma Gregoriana del Calendario Juliano.jpg|thumb|
''Lunario Novo, Secondo la Nuova Riforma della Correttione del l'Anno Riformato da N.S. Gregorio XIII'',{{efn|name=Lunario|"New Almanac according to the new reform for the correction of the year, [as] reformed by His Holiness Gregory 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 {{lang|it|Lunario Novo secondo la nuova riforma}}{{efn|name=Lunario}} 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.<ref>{{cite book| last1=Mezzi |first1=E. |last2= Vizza | first2=F. | title= Luigi Lilio Medico Astronomo e Matematico di Cirò | publisher= Laruffa Editore |location = Reggio Calabria | year=2010 |pages=14, 52 |isbn=9788872214817 }} citing as primary references: {{lang|it|Biblioteca Nazionale Centrale di Firenze}}, Magl. 5.10.5/a, [[Vatican Apostolic Archive]] A.A., Arm. I‑XVIII, 5506, f. 362r.</ref>