NOTES on the CALENDAR ~~~~~~~~~~~~~~~~~~~~~ The intention of this document is to provide information on the calendar for computer programmers or others who may need to do calculations involving calendar dates, or who may need to verify that a given date validly represents some actual historical day. 1. TERMINOLOGY USED The following terms, when used in this document, have the meanings given here. Some further expansion of these definitions may be found in later sections. The terms are presented in alphabetical order. AD - Anno Domini (in the year of the Lord), prefixed to the number designating a year in the christian era. AND - a mathematical symbol (from the programming language PASCAL) meaning that the conditions to the left and the right must both be true. AUC - Ab Urbe Condita (from the foundation of the city), suffixed to the number designating a year in the Roman (republican and imperial) civil calendar. BC - Before Christ, suffixed to the number designating a year in the pre-christian era. Calendar - a system for numbering and counting the cycles of days, months and years. DIV - a mathematical symbol (from the programming language PASCAL) meaning integer division of the term to the left of DIV by the term to the right of DIV, where integer division means division with any remainder (however large) ignored; for example "a DIV b" means the integer quotient of `a' divided by `b', the remainder being ignored, or 13 DIV 7 = 1. Gregorian - the calendar now in use throughout the world and also known as the New (or New-style) Calendar. Julian - the calendar in use throughout Christendom until AD 1582 and in England (and other places) until AD 1752 and also known as the Old (or Old-style) Calendar. Leap year - any year which is one day longer than the standard year of the calendar in use. MOD - a mathematical symbol (from the programming language PASCAL) meaning the remainder which results from the division of the term to the left of MOD by the term to the right of MOD; for example "a MOD b" means the remainder of `a' divided by `b', the quotient being ignored, or 13 MOD 7 = 6. Month - the major subdivision of a year; in all calendars considered here there are twelve months in every year with varying numbers of days in each month - this document uses the modern English names of the months (sometimes abbreviated to the first three letters of the name) or their modern numbers (January = 1, February = 2, etc) throughout. OR - a mathematical symbol (from the programming language PASCAL) meaning that the conditions to the left and the right may either or both be true. Week - the seven-day cycle of days starting on Sunday and ending on Saturday; this is independent of the cycles of months and years and has Jewish and other origins - this document uses the modern English names of the days of the week. <> - a mathematical symbol (from the programming language PASCAL) meaning "is not equal to". * - a mathematical symbol (from the programming language PASCAL) meaning "multiplied by". ( ) - parentheses and other mathematical symbols not explicitly defined here have their normal mathematical meanings and significances. 2. BASIC ASSUMPTIONS The following statements are assumed to be facts - the interested reader may verify them from various reference sources. For brevity the sources are mostly not quoted here. 2.1 THE WEEK The seven-day cycle is of very ancient origin (Jewish and other). a) The individual days are numbered and named as follows. 1 Sunday 2 Monday 3 Tuesday 4 Wednesday 5 Thursday 6 Friday 7 Saturday b) At different times and in different places the names of the days may vary. c) This cycle has not been broken or in any way disturbed within the period of any calendar discussed here. Consequently any system for calculating the day of the week from the date must result in Friday when applied to 3rd April AD 33 (the traditional date of the Crucifixion, Good Friday). [It is neither important for the purpose of this document to believe in the historical accuracy of any traditional dates, nor in the truth or otherwise of any events. It is necessary to know what the traditions are and at least their mathematical significances. Such dates are the basis of any calendar and are used here for the purpose of calculation.] 2.2 THE ROMAN CIVIL CALENDAR In 46 BC Julius Caesar regularized the civil calendar of the Roman republic to make it operate as follows. a) Every year was divided into twelve months which were numbered as follows and to which we assign the following modern names. 1 January 2 February 3 March 4 April 5 May 6 June 7 July 8 August 9 September 10 October 11 November 12 December b) The years were numbered AUC (from the date of the foundation of the city of Rome). c) Every fourth year was to have one extra day in February. However, the Roman civil authorities responsible for the calendar mismanaged it after Caesar's death, keeping a leap year every third year, so by 8 BC three days had been gained. 2.3 THE JULIAN CALENDAR In 8 BC the Emperor Augustus adjusted the Roman civil calendar by dropping the days which had been gained - omitting leap years until the year we now know as AD 12. Thus AD 12 was the first leap year of the christian era. (At the same time Augustus gave the months their present numbers of days.) This calendar is known today as Julian after Julius Caesar who first devised it as mentioned in the foregoing section. From 8 BC the calendar operated as stated in (a)-(e) and has the continuing history given in (f).... a) Every year was divided into twelve months which were numbered and named as follows and each month had the following numbers of days: 1 January 31 days 2 February 28 days except in leap years when it had 29 days 3 March 31 days 4 April 30 days 5 May 31 days 6 June 30 days 7 July 31 days 8 August 31 days 9 September 30 days 10 October 31 days 11 November 30 days 12 December 31 days b) Every normal year had 365 days; every leap year had 366 days. c) Every year began on 1st January and ended on the following 31st December. d) A leap year occured in AD 12 and every fourth year thereafter (AD 4 and AD 8 were not leap years because of Augustus' adjustment). e) The years were still numbered AUC (from the date of the foundation of the city of Rome). f) With the christianization of the Roman empire this calendar continued to be used for all purposes throughout both eastern and western parts of the empire. It was the Emperor Constantine who first officially introduced the seven-day cycle (week) into the Roman system. It was unchanged then (and since) from the Jewish seven-day cycle. g) When the AD system of numbering the years was introduced calculations made at the time fixed that year as AD 532. AD 1 was reckoned to begin on 1st January when the infant Christ would have been seven days old. The year 754 AUC was reckoned as AD 1. The year in which Christ was conceived and born became known as 1 BC. For the purposes of calculation it may be convenient to regard that year as "AD 0". h) At some times and in some places within the currency of this christian form of the Julian calendar (particularly in England and its colonies at the time but not universally) the year was regarded as beginning on Lady Day so that 25th March was regarded as the first day of the year and the same year was regarded as ending on 24th March next following. This still meant that the conception and birth of Christ were in the year 1 BC. i) A leap year still occured every fourth year - when the AD year-number was exactly divisible by four. Mathematically the year AD `yr' is a leap year if and only if: (`yr' MOD 4) = 0 and, assuming the year to begin on 1st January, the February in that year had 29 days. In those places where the year was reckoned to begin on 25th March the February which had 29 days was the February previous to the 25th March on which the year was reckoned to begin. This meant that all places had a 29th February together although their reckoning of the AD year-number differed. For those places which started the year on 25th March one could calculate a year to be a leap year if and only if: ((`yr'+1) MOD 4) = 0 for the February in the year AD `yr' to have 29 days. The years AD 4 and AD 8 (starting 1st January) were special cases and were not leap years because of Augustus' adjustments. The calendar followed its own rules without irregularity from 1st March AD 8 (the day following the effectiveness of Augustus final adjustment) throughout its remaining period of use. j) This calendar was eventually found (by astronomical observation) to be inaccurate. By the sixteenth century it had gained ten days because too many years were being kept as leap years. k) As described in the next section (2.4) the Gregorian calendar was devised (with fewer leap years) at the instigation of the pope of the time and the last effective day of the Julian calendar in Rome was 4th October AD 1582 which was a Thursday. (The ten days were dropped without altering the weekly cycle of days and the next day was Friday 15th October AD 1582 on the Gregorian calendar.) At the same time the first day of the year was fixed as 1st January to avoid the former confusion. l) Some places (including England and its colonies at the time) continued to use the Julian calendar officially - not recognizing the authority of the pope. Some of these places (again including England and colonies) were those which started the year on 25th March - they continued to do so. 2.4 GREGORIAN CALENDAR This calendar replaced the Julian calendar, operates as stated in (a)-(d) and has the history given in (e).... a) Every year is divided into twelve months which are numbered and named as follows and each month has the following numbers of days: 1 January 31 days 2 February 28 days except in leap years when it has 29 days 3 March 31 days 4 April 30 days 5 May 31 days 6 June 30 days 7 July 31 days 8 August 31 days 9 September 30 days 10 October 31 days 11 November 30 days 12 December 31 days "Thirty days hath September, April, June and November; All the rest have thirty-one excepting February alone; Which hath twenty-eight days clear and twenty-nine in each leap year." b) Every normal year has 365 days; every leap year has 366 days. c) Every year begins on 1st January and ends on the following 31st December. d) A year is a leap year if and only if: i) the year-number is exactly divisible by 400; or ii) the year-number is not exactly divisible by 100 and is exactly divisible by 4. Mathematically the year AD `yr' is a leap year if and only if: ((`yr' MOD 400) = 0) OR (((`yr' MOD 100) <> 0)) AND ((`yr' MOD 4) = 0)) e) If the Gregorian calendar is projected back to 1st January AD 1 this is the same day as Julian 1st January AD 1 but the Gregorian leap year rule has no special cases. This means that the earliest day which has different dates in the two systems is Gregorian 29th February AD 4 which is Julian 1st March AD 4. In each calendar it is the first day after 28th February AD 4. f) This calendar was devised as a result of astronomical observation and calculation on behalf of Pope Gregory XIII whose name is associated with it. g) The first year of this calendar (AD 1) is reckoned to be the year following the birth of Christ - ie. the year identical with Julian AD 1, beginning on 1st January when Christ was seven days old. This meant that the AD year-numbers were not changed by the Gregorian reform of the calendar if the year is regarded as beginning on 1st January. h) It came into force in the Papal States (Rome and other parts of central Italy) and in some other parts of Europe (and their colonies at the time) on Friday Gregorian 15th October AD 1582. (The previous day had been Thursday Julian 4th October AD 1582.) Dates in the range 5th October AD 1582 through 14th October AD 1582 are therefore not valid in places which adopted the new calendar immediately. The weekly cycle of days was not altered. i) Many parts of Europe (including England) did not adopt it immediately. j) Most countries of the world have now adopted it at least for civil purposes. k) England and its colonies at the time brought this calendar into use for all purposes with effect from 14th September AD 1752 which was a Thursday. 2.5 THE CHANGE IN ENGLAND In England and its colonies of the time the change to the Gregorian system was effected as follows. a) Until AD 1751 (which began on 25th March) the Julian calendar was the official calendar although some events had already been (and continued to be) recorded with two year-numbers. For example, "3rd Feb. 1750/1" would record a day in the year which was still officially AD 1750 in England but was recognized as AD 1751 elsewhere. b) AD 1751 officially ended in England on 31st December and AD 1752 officially began on the following day, 1st January. c) By this time the difference between the English (Julian) and the more accurate (Gregorian) calendar was eleven days, so this was the number of days which had to be dropped. "Give us back our eleven days." d) In England the next day after Wednesday Julian 2nd September AD 1752 was officially made Thursday Gregorian 14th September AD 1752. Dates in the range 3rd September AD 1752 through 13th September AD 1752 are therefore not valid in England or any place which was its colony in AD 1752. e) The English financial year had previously begun on the 25th March with each new AD year. Because the financial community of the time refused to accept one year being shSEEN-BY: 5/0 140/1 7105/1 7106/20 22 46 SEEN-BY: 633/267 270 @PATH: 7106/22 140/1 106/2000 633/267