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[ Note from Psych0: I concatenated the three parts into one ] From: c-t@pip.dknet.dk (Claus Tondering) Newsgroups: sci.astro,soc.history,sci.answers,soc.answers,news.answers Subject: Calendar FAQ, v. 1.7 (modified 27 Mar 1997) Part 1/3 Approved: news-answers-request@MIT.EDU Followup-To: sci.astro,soc.history Summary: This posting contains answers to Frequently Asked Questions about the Christian, Hebrew, Islamic, and various historical calendars. Expires: 1 May 1997 00:00:00 GMT Archive-name: calendars/faq/part1 Posting-Frequency: monthly Last-modified: 1997/03/27 Version: 1.7 URL: http://www.pip.dknet.dk/~pip10160/calendar.html FREQUENTLY ASKED QUESTIONS ABOUT CALENDARS Part 1 of 3 Version 1.7 - 27 Mar 1997 Copyright and disclaimer ------------------------ This document is Copyright (C) 1997 by Claus Tondering. E-mail: c-t@pip.dknet.dk. The document may be freely distributed, provided this copyright notice is included and no money is charged for the document. This document is provided "as is". No warranties are made as to its correctness. Introduction ------------ This is the calendar FAQ. Its purpose is to give an overview of the Christian, Hebrew, and Islamic calendars in common use. It will provide a historical background for the Christian calendar, plus an overview of the French Revolutionary calendar and the Maya calendar. Comments are very welcome. My e-mail address is given above. I would like to thank - Dr. Monzur Ahmed of the University of Birmingham, UK, - Michael J Appel, - Jay Ball, - Chris Carrier, - Simon Cassidy, - Claus Dobesch, - Leofranc Holford-Strevens, - H. Koenig, - Marcos Montes, - James E. Morrison, - Waleed A. Muhanna of the Fisher College of Business, Columbus, Ohio, USA, - Paul Schlyter of the Swedish Amateur Astronomer's Society for their help with this document. Changes since version 1.6 ------------------------- A few of minor corrections. Section 2.12.1 ("Is there a formula for calculating the Julian day number?") added and the following section renumbered. Section 4.4 ("When will the Islamic calendar overtake the Gregorian calendar?") added. Writing dates and years ----------------------- Dates will be written in the British format (1 January) rather than the American format (January 1). Dates will occasionally be abbreviated: "1 Jan" rather than "1 January". Years before and after the "official" birth year of Christ will be written "45 BC" or "AD 1997", respectively. I prefer this notation over the secular "45 B.C.E." and "1997 C.E." The % operator -------------- Throughout this document the operator % will be used to signify the modulo or remainder operator. For example, 17%7=3 because the result of the division 17/7 is 2 with a remainder of 3. The text in square brackets --------------------------- Square brackets [like this] identify information that I am unsure about and about which I would like more information. Please write me at c-t@pip.dknet.dk. Index: ------ In part 1 of this document: 1. What astronomical events form the basis of calendars? 1.1. What are Equinoxes and Solstices? 2. The Christian calendar 2.1. What is the Julian calendar? 2.1.1. What years are leap years? 2.1.2. What consequences did the use of the Julian calendar have? 2.2. What is the Gregorian calendar? 2.2.1. What years are leap years? 2.2.2. Isn't there a 4000-year rule? 2.2.3. Don't the Greek do it differently? 2.2.4. When did country X change from the Julian to the Gregorian calendar? 2.3. What day is the leap day? 2.4. What is the Solar Cycle? 2.5. What day of the week was 2 August 1953? 2.6. What is the Roman calendar? 2.6.1. How did the Romans number days? 2.7. Has the year always started on 1 January? 2.8. What is the origin of the names of the months? In part 2 of this document: 2.9. What is Easter? 2.9.1. When is Easter? (Short answer) 2.9.2. When is Easter? (Long answer) 2.9.3. What is the Golden Number? 2.9.4. What is the Epact? 2.9.5. How does one calculate Easter then? 2.9.6. Isn't there a simpler way to calculate Easter? 2.9.7. Is there a simple relationship between two consecutive Easters? 2.9.8. How frequently are the dates for Easter repeated? 2.9.9. What about Greek Easter? 2.10. How does one count years? 2.10.1. Was Jesus born in the year 0? 2.10.2. When does the 21st century start? 2.11. What is the Indiction? 2.12. What is the Julian period? 2.12.1. Is there a formula for calculating the Julian day number? 2.12.2. What is the modified Julian day? 3. The Hebrew Calendar 3.1. What does a Hebrew year look like? 3.2. What years are leap years? 3.3. What years are deficient, regular, and complete? 3.4. When is New Year's day? 3.5. When does a Hebrew day begin? 3.6. When does a Hebrew year begin? 3.7. When is the new moon? 3.8. How does one count years? 4. The Islamic Calendar 4.1. What does an Islamic year look like? 4.2. So you can't print an Islamic calendar in advance? 4.3. How does one count years? 4.4. When will the Islamic calendar overtake the Gregorian calendar? In part 3 of this document: 5. The Week 5.1. What Is the Origin of the 7-Day Week? 5.2. What Do the Names of the Days of the Week Mean? 5.3. Has the 7-Day Week Cycle Ever Been Interrupted? 5.4. Which Day is the Day of Rest? 5.5. What Is the First Day of the Week? 5.6. What Is the Week Number? 5.7. Do Weeks of Different Lengths Exist? 6. The French Revolutionary Calendar 6.1. What does a Republican year look like? 6.2. How does one count years? 6.3. What years are leap years? 6.4. How does one convert a Republican date to a Gregorian one? 7. The Maya Calendar 7.1. What is the Long Count? 7.1.1. When did the Long Count Start? 7.2. What is the Tzolkin? 7.2.1. When did the Tzolkin Start? 7.3. What is the Haab? 7.3.1. When did the Haab Start? 7.4. Did the Maya Think a Year Was 365 Days? 8. Date 1. What astronomical events form the basis of calendars? -------------------------------------------------------- Calendars are normally based on astronomical events, and the two most important astronomical objects are the sun and the moon. Their cycles are very important in the construction and understanding of calendars. Our concept of a year is based on the earth's motion around the sun. The time from one fixed point, such as a solstice or equinox, to the next is called a "tropical year". Its length is currently 365.242190 days, but it varies. Around 1900 its length was 365.242196 days, and around 2100 it will be 365.242184 days. (This definition of the tropical year is not quite accurate, see section 1.1 for more details.) Our concept of a month is based on the moon's motion around the earth, although this connection has been broken in the calendar commonly used now. The time from one new moon to the next is called a "synodic month", and its length is currently 29.5305889 days, but it varies. Around 1900 its length was 29.5305886 days, and around 2100 it will be 29.5305891 days. Note that these numbers are averages. The actual length of a particular year may vary by several minutes due to the influence of the gravitational force from other planets. Similary, the time between two new moons may vary by several hours due to a number of factors, including changes in the gravitational force from the sun, and the moon's orbital inclination. It is unfortunate that the length of the tropical year is not a multiple of the length of the synodic month. This means that with 12 months per year, the relationship between our month and the moon cannot be maintained. However, 19 tropical years is 234.997 synodic months, which is very close to an integer. So every 19 years the phases of the moon fall on the same dates (if it were not for the skewness introduced by leap years). 19 years is called a Metonic cycle (after Meton, an astronomer from Athens in the 5th century BC). So, to summarise: There are three important numbers to note: A tropical year is 365.2422 days. A synodic month is 29.53059 days. 19 tropical years is close to an integral number of synodic months. The Christian calendar is based on the motion of the earth around the sun, while the months have no connection with the motion of the moon. On the other hand, the Islamic calendar is based on the motion of the moon, while the year has no connection with the motion of the earth around the sun. Finally, the Hebrew calendar combines both, in that its years are linked to the motion of the earth around the sun, and its months are linked to the motion of the moon. 1.1. What are Equinoxes and Solstices? -------------------------------------- Equinoxes and solstices are frequently used as anchor points for calendars. For people in the northern hemisphere: - Winter solstice is the time in December when the sun reaches its southernmost latitude. At this time we have the shortest day. The date is typically around 21 December. - Summer solstice is the time in June when the sun reaches its northernmost latitude. At this time we have the longest day. The date is typically around 21 June. - Vernal equinox is the time in March when the sun passes the equator moving from the southern to the northern hemisphere. Day and night have approximately the same length. The date is typically around 20 March. - Autumnal equinox is the time in September when the sun passes the equator moving from the northern to the southern hemisphere. Day and night have approximately the same length. The date is typically around 22 September. For people in the southern hemisphere these events are shifted half a year. The astronomical "tropical year" is frequently defined as the time between, say, two vernal equinoxes, but this is not actually true. Currently the time between two vernal equinoxes is slightly greater than the tropical year. The reason is that the earth's position in its orbit at the time of solstices and equinoxes shifts slightly each year (taking approximately 21,000 years to move all the way around the orbit). This, combined with the fact that the earth's orbit is not completely circular, causes the equinoxes and solstices to shift with respect to each other. The astronomer's mean tropical year is really a somewhat artificial average of the period between the time when the sun is in any given position in the sky with respect to the equinoxes and the next time the sun is in the same position. 2. The Christian calendar ------------------------- The "Christian calendar" is the term I use to designate the calendar commonly in use, although its connection with Christianity is highly debatable. The Christian calendar has years of 365 or 366 days. It is divided into 12 months that have no relationship to the motion of the moon. In parallel with this system, the concept of "weeks" groups the days in sets of 7. Two main versions of the Christian calendar have existed in recent times: The Julian calendar and the Gregorian calendar. The difference between them lies in the way they approximate the length of the tropical year and their rules for calculating Easter. 2.1. What is the Julian calendar? --------------------------------- The Julian calendar was introduced by Julius Caesar in 45 BC. It was in common use until the 1500s, when countries started changing to the Gregorian calendar (section 2.2). However, some countries (for example, Greece and Russia) used it into this century, and the Orthodox church in Russia still uses it, as do some other Orthodox churches. In the Julian calendar, the tropical year is approximated as 365 1/4 days = 365.25 days. This gives an error of 1 day in approximately 128 years. The approximation 365 1/4 is achieved by having 1 leap year every 4 years. 2.1.1. What years are leap years? --------------------------------- The Julian calendar has 1 leap year every 4 years: Every year divisible by 4 is a leap year. However, this rule was not followed in the first years after the introduction of the Julian calendar in 45 BC. Due to a counting error, every 3rd year was a leap year in the first years of this calendar's existence. The leap years were: 45 BC, 42 BC, 39 BC, 36 BC, 33 BC, 30 BC, 27 BC, 24 BC, 21 BC, 18 BC, 15 BC, 12 BC, 9 BC, AD 8, AD 12, and every 4th year from then on. There were no leap years between 9 BC and AD 8. This period without leap years was decreed by emperor Augustus and earned him a place in the calendar, as the 8th month was named after him. It is a curious fact that although the method of reckoning years after the (official) birthyear of Christ was not introduced until the 6th century, by some stroke of luck the Julian leap years coincide with years of our Lord that are divisible by 4. 2.1.2. What consequences did the use of the Julian calendar have? ----------------------------------------------------------------- The Julian calendar introduces an error of 1 day every 128 years. So every 128 years the tropical year shifts one day backwards with respect to the calendar. Furthermore, the method for calculating the dates for Easter was inaccurate and needed to be refined. In order to remedy this, two steps were necessary: 1) The Julian calendar had to be replaced by something more adequate. 2) The extra days that the Julian calendar had inserted had to be dropped. The solution to problem 1) was the Gregorian calendar described in section 2.2. The solution to problem 2) depended on the fact that it was felt that 21 March was the proper day for vernal equinox (because 21 March was the date for vernal equinox during the Council of Nicaea in AD 325). The Gregorian calendar was therefore calibrated to make that day vernal equinox. By 1582 vernal equinox had moved (1582-325)/128 days = approximately 10 days backwards. So 10 days had to be dropped. 2.2. What is the Gregorian calendar? ------------------------------------ The Gregorian calendar is the one commonly used today. It was proposed by Aloysius Lilius, a physician from Naples, and adopted by Pope Gregory XIII in accordance with instructions from the Council of Trent (1545-1563) to correct for errors in the older Julian Calendar. It was decreed by Pope Gregory XIII in a papal bull in February 1582. In the Gregorian calendar, the tropical year is approximated as 365 97/400 days = 365.2425 days. Thus it takes approximately 3300 years for the tropical year to shift one day with respect to the Gregorian calendar. The approximation 365 97/400 is achieved by having 97 leap years every 400 years. 2.2.1. What years are leap years? --------------------------------- The Gregorian calendar has 97 leap years every 400 years: Every year divisible by 4 is a leap year. However, every year divisible by 100 is not a leap year. However, every year divisible by 400 is a leap year after all. So, 1700, 1800, 1900, 2100, and 2200 are not leap years. But 1600, 2000, and 2400 are leap years. (Destruction of a myth: There are no double leap years, i.e. no years with 367 days. See, however, the note on Sweden in section 2.2.4.) 2.2.2. Isn't there a 4000-year rule? ------------------------------------ It has been suggested (by the astronomer John Herschel (1792-1871) among others) that a better approximation to the length of the tropical year would be 365 969/4000 days = 365.24225 days. This would dictate 969 leap years every 4000 years, rather than the 970 leap years mandated by the Gregorian calendar. This could be achieved by dropping one leap year from the Gregorian calendar every 4000 years, which would make years divisible by 4000 non-leap years. This rule has, however, not been officially adopted. 2.2.3. Don't the Greek do it differently? ----------------------------------------- When the Orthodox church in Greece finally decided to switch to the Gregorian calendar in the 1920s, they tried to improve on the Gregorian leap year rules, replacing the "divisible by 400" rule by the following: Every year which when divided by 900 leaves a remainder of 200 or 600 is a leap year. This makes 1900, 2100, 2200, 2300, 2500, 2600, 2700, 2800 non-leap years, whereas 2000, 2400, and 2900 are leap years. This will not create a conflict with the rest of the world until the year 2800. This rule gives 218 leap years every 900 years, which gives us an average year of 365 218/900 days = 365.24222 days, which is certainly more accurate than the official Gregorian number of 365.2425 days. However, this rule is *not* official in Greece. [I have received an e-mail indicating that this system is official in Russia today. I'm investigating that. Information is very welcome.] 2.2.4. When did country X change from the Julian to the Gregorian calendar? --------------------------------------------------------------------------- The papal bull of February 1582 decreed that 10 days should be dropped from October 1582 so that 15 October should follow immediately after 4 October, and from then on the reformed calendar should be used. This was observed in Italy, Poland, Portugal, and Spain. Other Catholic countries followed shortly after, but Protestant countries were reluctant to change, and the Greek orthodox countries didn't change until the start of this century. Changes in the 1500s required 10 days to be dropped. Changes in the 1600s required 10 days to be dropped. Changes in the 1700s required 11 days to be dropped. Changes in the 1800s required 12 days to be dropped. Changes in the 1900s required 13 days to be dropped. (Exercise for the reader: Why is the error in the 1600s the same as in the 1500s.) The following list contains the dates for changes in a number of countries. Albania: December 1912 Austria: Different regions on different dates 5 Oct 1583 was followed by 16 Oct 1583 14 Dec 1583 was followed by 25 Dec 1583 Belgium: Different authorities say 14 Dec 1582 was followed by 25 Dec 1582 21 Dec 1582 was followed by 1 Jan 1583 Bulgaria: Different authorities say Sometime in 1912 Sometime in 1915 18 Mar 1916 was followed by 1 Apr 1916 China: Different authorities say 18 Dec 1911 was followed by 1 Jan 1912 18 Dec 1928 was followed by 1 Jan 1929 Czechoslovakia (i.e. Bohemia and Moravia): 6 Jan 1584 was followed by 17 Jan 1584 Denmark (including Norway): 18 Feb 1700 was followed by 1 Mar 1700 Egypt: 1875 Estonia: January 1918 Finland: Then part of Sweden. (Note, however, that Finland later became part of Russia, which then still used the Julian calendar. The Gregorian calendar remained official in Finland, but some use of the Julian calendar was made.) France: 9 Dec 1582 was followed by 20 Dec 1582 Germany: Different states on different dates: Catholic states on various dates in 1583-1585 Prussia: 22 Aug 1610 was followed by 2 Sep 1610 Protestant states: 18 Feb 1700 was followed by 1 Mar 1700 Great Britain and Dominions (including what is now the USA): 2 Sep 1752 was followed by 14 Sep 1752 Greece: 9 Mar 1924 was followed by 23 Mar 1924 Hungary: 21 Oct 1587 was followed by 1 Nov 1587 Italy: 4 Oct 1582 was followed by 15 Oct 1582 Japan: Different authorities say: 19 Dec 1872 was followed by 1 Jan 1873 18 Dec 1918 was followed by 1 Jan 1919 Latvia: During German occupation 1915 to 1918 Lithuania: 1915 Luxemburg: 14 Dec 1582 was followed by 25 Dec 1582 Netherlands: Brabant, Flanders, Holland, Artois, Hennegau: 14 Dec 1582 was followed by 25 Dec 1582 Geldern, Friesland, Zeuthen, Groningen, Overysel: 30 Nov 1700 was followed by 12 Dec 1700 Norway: Then part of Denmark. Poland: 4 Oct 1582 was followed by 15 Oct 1582 Portugal: 4 Oct 1582 was followed by 15 Oct 1582 Romania: 31 Mar 1919 was followed by 14 Apr 1919 Russia: 31 Jan 1918 was followed by 14 Feb 1918 Spain: 4 Oct 1582 was followed by 15 Oct 1582 Sweden (including Finland): 17 Feb 1753 was followed by 1 Mar 1753 (see note below) Switzerland: Catholic cantons: 1583 or 1584 Zurich, Bern, Basel, Schafhausen, Neuchatel, Geneva: 31 Dec 1700 was followed by 12 Jan 1701 St Gallen: 1724 Turkey: 18 Dec 1926 was followed by 1 Jan 1927 USA: See Great Britain, of which it was then a colony. Yugoslavia: 1919 Sweden has a curious history. Sweden decided to make a gradual change from the Julian to the Gregorian calendar. By dropping every leap year from 1700 through 1740 the eleven superfluous days would be omitted and from 1 Mar 1740 they would be in sync with the Gregorian calendar. (But in the meantime they would be in sync with nobody!) So 1700 (which should have been a leap year in the Julian calendar) was not a leap year in Sweden. However, by mistake 1704 and 1708 became leap years. This left Sweden out of synchronisation with both the Julian and the Gregorian world, so they decided to go *back* to the Julian calendar. In order to do this, they inserted an extra day in 1712, making that year a double leap year! So in 1712, February had 30 days in Sweden. Later, in 1753, Sweden changed to the Gregorian calendar by dropping 11 days like everyone else. 2.3. What day is the leap day? ------------------------------ It is 24 February! Weird? Yes! The explanation is related to the Roman calendar and is found in section 2.6.1. From a numerical point of view, of course 29 February is the extra day. But from the point of view of celebration of feast days, the following correspondence between days in leap years and non-leap years has traditionally been used: Non-leap year Leap year ------------- ---------- 22 February 22 February 23 February 23 February 24 February (extra day) 24 February 25 February 25 February 26 February 26 February 27 February 27 February 28 February 28 February 29 February For example, the feast of St. Leander has been celebrated on 27 February in non-leap years and on 28 February in leap years. The EU (European Union) in their infinite wisdom have decided that starting in the year 2000, 29 February is to be the leap day. This will affect countries such as Sweden and Austria that celebrate "name days" (i.e. each day is associated with a name). It appears that the Roman Catholic Church already uses 29 February as the leap day. 2.4. What is the Solar Cycle? ----------------------------- In the Julian calendar the relationship between the days of the week and the dates of the year is repeated in cycles of 28 years. In the Gregorian calendar this is still true for periods that do not cross years that are divisible by 100 but not by 400. A period of 28 years is called a Solar Cycle. The "Solar Number" of a year is found as: Solar Number = (year + 8) % 28 + 1 In the Julian calendar there is a one-to-one relationship between the Solar Number and the day on which a particular date falls. (The leap year cycle of the Gregorian calendar is 400 years, which is 146,097 days, which curiously enough is a multiple of 7. So in the Gregorian calendar the equivalent of the "Solar Cycle" would be 400 years, not 7*400=2800 years as one might be tempted to believe.) 2.5. What day of the week was 2 August 1953? -------------------------------------------- To calculate the day on which a particular date falls, the following algorithm may be used (the divisions are integer divisions, in which remainders are discarded): a = (14 - month) / 12 y = year - a m = month + 12*a - 2 For Julian calendar: d = (5 + day + y + y/4 + (31*m)/12) % 7 For Gregorian calendar: d = (day + y + y/4 - y/100 + y/400 + (31*m)/12) % 7 The value of d is 0 for a Sunday, 1 for a Monday, 2 for a Tuesday, etc. Example: On what day of the week was the author born? My birthday is 2 August 1953 (Gregorian, of course). a = (14 - 8) / 12 = 0 y = 1953 - 0 = 1953 m = 8 + 12*0 - 2 = 6 d = (2 + 1953 + 1953/4 - 1953/100 + 1953/400 + (31*6)/12) % 7 = (2 + 1953 + 488 - 19 + 4 + 15 ) % 7 = 2443 % 7 = 0 I was born on a Sunday. 2.6. What is the Roman calendar? -------------------------------- Before Julius Caesar introduced the Julian calendar in 45 BC, the Roman calendar was a mess, and much of our so-called "knowledge" about it seems to be little more than guesswork. Originally, the year started on 1 March and consisted of only 304 days or 10 months (Martius, Aprilis, Maius, Junius, Quintilis, Sextilis, September, October, November, and December). These 304 days were followed by an unnamed and unnumbered winter period. The Roman king Numa Pompilius (c. 715-673 BC, although his historicity is disputed) allegedly introduced February and January (in that order) between December and March, increasing the length of the year to 354 or 355 days. In 450 BC, February was moved to its current position between January and March. In order to make up for the lack of days in a year, an extra month, Intercalaris or Mercedonius, (allegedly with 22 or 23 days though some authorities dispute this) was introduced in some years. In an 8 year period the length of the years were: 1: 12 months or 355 days 2: 13 months or 377 days 3: 12 months or 355 days 4: 13 months or 378 days 5: 12 months or 355 days 6: 13 months or 377 days 7: 12 months or 355 days 8: 13 months or 378 days A total of 2930 days corresponding to a year of 366 1/4 days. This year was discovered to be too long, and therefore 7 days were later dropped from the 8th year, yielding 365.375 days per year. This is all theory. In practice it was the duty of the priesthood to keep track of the calendars, but they failed miserably, partly due to ignorance, partly because they were bribed to make certain years long and other years short. Furthermore, leap years were considered unlucky and were therefore avoided in time of crisis, such as the Second Punic War. In order to clean up this mess, Julius Caesar made his famous calendar reform in 45 BC. We can make an educated guess about the length of the months in the years 47 and 46 BC: 47 BC 46 BC January 29 29 February 28 24 Intercalaris 27 March 31 31 April 29 29 May 31 31 June 29 29 Quintilis 31 31 Sextilis 29 29 September 29 29 October 31 31 November 29 29 Undecember 33 Duodecember 34 December 29 29 --- --- Total 355 445 The length of the months from 45 BC onward were the same as the ones we know today. Occasionally one reads the following story: "Julius Caesar made all odd numbered months 31 days long, and all even numbered months 30 days long (with February having 29 days in non-leap years). In 44 BC Quintilis was renamed 'Julius' (July) in honour of Julius Caesar, and in 8 BC Sextilis became 'Augustus' in honour of emperor Augustus. When Augustus had a month named after him, he wanted his month to be a full 31 days long, so he removed a day from February and shifted the length of the other months so that August would have 31 days." This story, however, has no basis in actual fact. It is a fabrication possibly dating back to the 14th century. 2.6.1. How did the Romans number days? -------------------------------------- The Romans didn't number the days sequentially from 1. Instead they had three fixed points in each month: "Kalendae" (or "Calendae"), which was the first day of the month. "Idus", which was the 13th day of January, February, April, June, August, September, November, and December, or the 15th day of March, May, July, or October. "Nonae", which was the 9th day before Idus (counting Idus itself as the 1st day). The days between Kalendae and Nonae were called "the 4th day before Nonae", "the 3rd day before Nonae", and "the 2nd day before Nonae". (The 1st day before Nonae would be Nonae itself.) Similarly, the days between Nonae and Idus were called "the Xth day before Idus", and the days after Idus were called "the Xth day before Kalendae (of the next month)". Julius Caesar decreed that in leap years the "6th day before Kalendae of March" should be doubled. So in contrast to our present system, in which we introduce an extra date (29 February), the Romans had the same date twice in leap years. The doubling of the 6th day before Kalendae of March is the origin of the word "bissextile". If we create a list of equivalences between the Roman days and our current days of February in a leap year, we get the following: 7th day before Kalendae of March 23 February 6th day before Kalendae of March 24 February 6th day before Kalendae of March 25 February 5th day before Kalendae of March 26 February 4th day before Kalendae of March 27 February 3rd day before Kalendae of March 28 February 2nd day before Kalendae of March 29 February Kalendae of March 1 March You can see that the extra 6th day (going backwards) falls on what is today 24 February. For this reason 24 February is still today considered the "extra day" in leap years (see section 2.3). However, at certain times in history the second 6th day (25 Feb) has been considered the leap day. Why did Caesar choose to double the 6th day before Kalendae of March? It appears that the leap month Intercalaris/Mercedonius of the pre-reform calendar was not placed after February, but inside it, namely between the 7th and 6th day before Kalendae of March. It was therefore natural to have the leap day in the same position. 2.7. Has the year always started on 1 January? ---------------------------------------------- For the man in the street, yes. When Julius Caesar introduced his calendar in 45 BC, he made 1 January the start of the year, and it was always the date on which the Solar Number and the Golden Number (see section 2.9.3) were incremented. However, the church didn't like the wild parties that took place at the start of the new year, and in AD 567 the council of Tours declared that having the year start on 1 January was an ancient mistake that should be abolished. Through the middle ages various New Year dates were used. If an ancient document refers to year X, it may mean any of 7 different periods in our present system: - 1 Mar X to 28/29 Feb X+1 - 1 Jan X to 31 Dec X - 1 Jan X-1 to 31 Dec X-1 - 25 Mar X-1 to 24 Mar X - 25 Mar X to 24 Mar X+1 - Saturday before Easter X to Friday before Easter X+1 - 25 Dec X-1 to 24 Dec X Choosing the right interpretation of a year number is difficult, so much more as one country might use different systems for religious and civil needs. The Byzantine Empire used a year staring on 1 Sep, but they didn't count years since the birth of Christ, instead they counted years since the creation of the world which they dated to 1 September 5509 BC. Since about 1600 most countries have used 1 January as the first day of the year. Italy and England, however, did not make 1 January official until around 1750. In England (but not Scotland) three different years were used: - The historical year, which started on 1 January. - The liturgical year, which started on the first Sunday in advent. - The civil year, which from the 7th to the 12th century started on 25 December, from the 12th century until 1751 started on 25 March, from 1752 started on 1 January. 2.8. What is the origin of the names of the months? --------------------------------------------------- January Latin: Januarius. Named after the god Janus. February Latin: Februarius. Named after Februa, the purification festival. March Latin: Martius. Named after the god Mars. April Latin: Aprilis. Named either after the goddess Aphrodite or the Latin word "aperire", to open. May Latin: Maius. Probably named after the goddess Maia. June Latin: Junius. Probably named after the goddess Juno. July Latin: Julius. Named after Julius Caesar in 44 BC. Prior to that time its name was Quintilis from the word "quintus", fifth, because it was the 5th month in the old Roman calendar. August Latin: Augustus. Named after emperor Augustus in 8 BC. Prior to that time the name was Sextilis from the word "sextus", sixth, because it was the 6th month in the old Roman calendar. September Latin: September. From the word "septem", seven, because it was the 7th month in the old Roman calendar. October Latin: October. From the word "octo", eight, because it was the 8th month in the old Roman calendar. November Latin: November. From the word "novem", nine, because it was the 9th month in the old Roman calendar. December Latin: December. From the word "decem", ten, because it was the 10th month in the old Roman calendar. --- End of part 1 --- 2.9. What is Easter? -------------------- In the Christian world, Easter (and the days immediately preceding it) is the celebration of the death and resurrection of Jesus in (approximately) AD 30. 2.9.1. When is Easter? (Short answer) ------------------------------------- Easter Sunday is the first Sunday after the first full moon after vernal equinox. 2.9.2. When is Easter? (Long answer) ------------------------------------ The calculation of Easter is complicated because it is linked to (an inaccurate version of) the Hebrew calendar. Jesus was crucified immediately before the Jewish Passover, which is a celebration of the Exodus from Egypt under Moses. Celebration of Passover started on the 14th or 15th day of the (spring) month of Nisan. Jewish months start when the moon is new, therefore the 14th or 15th day of the month must be immediately after a full moon. It was therefore decided to make Easter Sunday the first Sunday after the first full moon after vernal equinox. Or more precisely: Easter Sunday is the first Sunday after the *official* full moon on or after the *official* vernal equinox. The official vernal equinox is always 21 March. The official full moon may differ from the *real* full moon by one or two days. (Note, however, that historically, some countries have used the *real* (astronomical) full moon instead of the official one when calculating Easter. This was the case, for example, of the German Protestant states, which used the astronomical full moon in the years 1700-1776. A similar practice was used Sweden in the years 1740-1844 and in Denmark in the 1700s.) The full moon that precedes Easter is called the Paschal full moon. Two concepts play an important role when calculating the Pascal full moon: The Golden Number and the Epact. They are described in the following sections. The following sections give details about how to calculate the date for Easter. Note, however, that while the Julian calendar was in use, it was customary to use tables rather than calculations to determine Easter. The following sections do mention how to calcuate Easter under the Julian calendar, but the reader should be aware that this is an attempt to express in formulas what was originally expressed in tables. The formulas can be taken as a good indication of when Easter was celebrated in the Western Church from approximately the 6th century. 2.9.3. What is the Golden Number? --------------------------------- Each year is associated with a Golden Number. Considering that the relationship between the moon's phases and the days of the year repeats itself every 19 years (as described in section 1), it is natural to associate a number between 1 and 19 with each year. This number is the so-called Golden Number. It is calculated thus: GoldenNumber = (year%19)+1 New moon will fall on (approximately) the same date in two years with the same Golden Number. 2.9.4. What is the Epact? ------------------------- Each year is associated with an Epact. The Epact is a measure of the age of the moon (i.e. the number of days that have passed since an "official" new moon) on a particular date. In the Julian calendar, 8 + the Epact is the age of the moon at the start of the year. In the Gregorian calendar, the Epact is the age of the moon at the start of the year. The Epact is linked to the Golden Number in the following manner: Under the Julian calendar, 19 years were assumed to be exactly an integral number of synodic months, and the following relationship exists between the Golden Number and the Epact: Epact = (11 * (GoldenNumber-1)) % 30 If this formula yields zero, the Epact is by convention frequently designated by the symbol * and its value is said to be 30. Weird? Maybe, but people didn't like the number zero in the old days. Since there are only 19 possible golden numbers, the Epact can have only 19 different values: 1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, and 30. The Julian system for calculating full moons was inaccurate, and under the Gregorian calendar, some modifications are made to the simple relationship between the Golden Number and the Epact. In the Gregorian calendar the Epact should be calculated thus (the divisions are integer divisions, in which remainders are discarded): 1) Use the Julian formula: Epact = (11 * (GoldenNumber-1)) % 30 x 2) Adjust the Epact, taking into account the fact that 3 out of 4 centuries have one leap year less than a Julian century: Epact = Epact - (3*century)/4 (For the purpose of this calculation century=20 is used for the years 1900 through 1999, and similarly for other centuries, although this contradicts the rules in section 2.10.2.) 3) Adjust the Epact, taking into account the fact that 19 years is not exactly an integral number of synodic months: Epact = Epact + (8*century + 5)/25 (This adds one to the epact 8 times every 2500 years.) 4) Add 8 to the Epact to make it the age of the moon on 1 January: Epact = Epact + 8 5) Add or subtract 30 until the Epact lies between 1 and 30. In the Gregorian calendar, the Epact can have any value from 1 to 30. Example: What was the Epact for 1992? GoldenNumber = 1992%19 + 1 = 17 1) Epact = (11 * (17-1)) % 30 = 26 2) Epact = 26 - (3*20)/4 = 11 3) Epact = 11 + (8*20 + 5)/25 = 17 4) Epact = 17 + 8 = 25 5) Epact = 25 The Epact for 1992 was 25. 2.9.5. How does one calculate Easter then? ------------------------------------------ To find Easter the following algorithm is used: 1) Calculate the Epact as described in the previous section. 2) For the Julian calendar: Add 8 to the Epact. (For the Gregorian calendar, this has already been done in step 5 of the calculation of the Epact). Subtract 30 if the sum exceeds 30. 3) Look up the Epact (as possibly modified in step 2) in this table to find the date for the Paschal full moon: Epact Full moon Epact Full moon Epact Full moon ----------------- ----------------- ----------------- 1 12 April 11 2 April 21 23 March 2 11 April 12 1 April 22 22 March 3 10 April 13 31 March 23 21 March 4 9 April 14 30 March 24 18 April 5 8 April 15 29 March 25 18 or 17 April 6 7 April 16 28 March 26 17 April 7 6 April 17 27 March 27 16 April 8 5 April 18 26 March 28 15 April 9 4 April 19 25 March 29 14 April 10 3 April 20 24 March 30 13 April 4) Easter Sunday is the first Sunday following the above full moon date. If the full moon falls on a Sunday, Easter Sunday is the following Sunday. An Epact of 25 requires special treatment, as it has two dates in the above table. There are two equivalent methods for choosing the correct full moon date: A) Choose 18 April, unless the current century contains years with an epact of 24, in which case 17 April should be used. B) If the Golden Number is > 11 choose 17 April, otherwise choose 18 April. The proof that these two statements are equivalent is left as an exercise to the reader. (The frustrated ones may contact me for the proof.) Example: When was Easter in 1992? In the previous section we found that the Golden Number for 1992 was 17 and the Epact was 25. Looking in the table, we find that the Paschal full moon was either 17 or 18 April. By rule B above, we choose 17 April because the Golden Number > 11. 17 April 1992 was a Friday. Easter Sunday must therefore have been 19 April. 2.9.6. Isn't there a simpler way to calculate Easter? ----------------------------------------------------- This is an attempt to boil down the information given in the previous sections (the divisions are integer divisions, in which remainders are discarded): G = year % 19 For the Julian calendar: I = (19*G + 15) % 30 J = (year + year/4 + I) % 7 For the Gregorian calendar: C = year/100 H = (C - C/4 - (8*C+13)/25 + 19*G + 15) % 30 I = H - (H/28)*(1 - (H/28)*(29/(H + 1))*((21 - G)/11)) J = (year + year/4 + I + 2 - C + C/4) % 7 Thereafter, for both calendars: L = I - J EasterMonth = 3 + (L + 40)/44 EasterDay = L + 28 - 31*(EasterMonth/4) This algorithm is based in part on the algorithm of Oudin (1940) as quoted in "Explanatory Supplement to the Astronomical Almanac", P. Kenneth Seidelmann, editor. People who want to dig into the workings of this algorithm, may be interested to know that G is the Golden Number-1 H is 23-Epact (modulo 30) I is the number of days from 21 March to the Paschal full moon J is the weekday for the Paschal full moon (0=Sunday, 1=Monday, etc.) L is the number of days from 21 March to the Sunday on or before the Pascal full moon (a number between -6 and 28) 2.9.7. Is there a simple relationship between two consecutive Easters? ---------------------------------------------------------------------- Suppose you know the Easter date of the current year, can you easily find the Easter date in the next year? No, but you can make a qualified guess. If Easter Sunday in the current year falls on day X and the next year is not a leap year, Easter Sunday of next year will fall on one of the following days: X-15, X-8, X+13 (rare), or X+20. If Easter Sunday in the current year falls on day X and the next year is a leap year, Easter Sunday of next year will fall on one of the following days: X-16, X-9, X+12 (extremely rare), or X+19. (The jump X+12 occurs only once in the period 1800-2099, namely when going from 2075 to 2076.) If you combine this knowledge with the fact that Easter Sunday never falls before 22 March and never falls after 25 April, you can narrow the possibilities down to two or three dates. 2.9.8. How frequently are the dates for Easter repeated? -------------------------------------------------------- The sequence of Easter dates repeats itself every 532 years in the Julian calendar. The number 532 is the product of the following numbers: 19 (the Metonic cycle or the cycle of the Golden Number) 28 (the Solar cycle, see section 2.4) The sequence of Easter dates repeats itself every 5,700,000 years in the Gregorian calendar. The number 5,700,000 is the product of the following numbers: 19 (the Metonic cycle or the cycle of the Golden Number) 400 (the Gregorian equivalent of the Solar cycle, see section 2.4) 25 (the cycle used in step 3 when calculating the Epact) 30 (the number of different Epact values) 2.9.9. What about Greek Easter? ------------------------------- The Greek Orthodox Church does not always celebrate Easter on the same day as the Catholic and Protestant countries. The reason is that the Orthodox Church uses the Julian calendar when calculating Easter. This is case even in the churches that otherwise use the Gregorian calendar. When the Greek Orthodox Church in 1923 decided to change to the Gregorian calendar (or rather: a Revised Julian Calendar), they chose to use the astronomical full moon as seen along the meridian of Jerusalem as the basis for calculating Easter, rather than to use the "official" full moon described in the previous sections. However, except for some sporadic use the 1920s, this system was never adopted in practice. 2.10. How does one count years? ------------------------------- In about AD 523, the papal chancellor, Bonifatius, asked a monk by the name of Dionysius Exiguus to devise a way to implement the rules from the Nicean council (the so-called "Alexandrine Rules") for general use. Dionysius Exiguus (in English known as Denis the Little) was a monk from Scythia, he was a canon in the Roman curia, and his assignment was to prepare calculations of the dates of Easter. At that time it was customary to count years since the reign of emperor Diocletian; but in his calculations Dionysius chose to number the years since the birth of Christ, rather than honour the persecutor Diocletian. Dionysius (wrongly) fixed Jesus' birth with respect to Diocletian's reign in such a manner that it falls on 25 December 753 AUC (ab urbe condita, i.e. since the founding of Rome), thus making the current era start with AD 1 on 1 January 754 AUC. How Dionysius established the year of Christ's birth is not known, although a considerable number of theories exist. Jesus was born under the reign of king Herod the Great, who died in 750 AUC, which means that Jesus could have been born no later than that year. Dionysius' calculations were disputed at a very early stage. When people started dating years before 754 AUC using the term "Before Christ", they let the year 1 BC immediately precede AD 1 with no intervening year zero. Note, however, that astronomers frequently use another way of numbering the years BC. Instead of 1 BC they use 0, instead of 2 BC they use -1, instead of 3 BC they use -2, etc. See also the following section. In this section I have used AD 1 = 754 AUC. This is the most likely equivalence between the two systems. However, some authorities state that AD 1 = 753 AUC or 755 AUC. This confusion is not a modern one, it appears that even the Romans were in some doubt about how to count the years since the founding of Rome. 2.10.1. Was Jesus born in the year 0? ------------------------------------- No. There are two reasons for this: - There is no year 0. - Jesus was born before 4 BC. The concept of a year "zero" is a modern myth (but a very popular one). Roman numerals do not have a figure designating zero, and treating zero as a number on an equal footing with other numbers was not common in the 6th century when our present year reckoning was established by Dionysius Exiguus (see the previous section). Dionysius let the year AD 1 start one week after what he believed to be Jesus' birthday. Therefore, AD 1 follows immediately after 1 BC with no intervening year zero. So a person who was born in 10 BC and died in AD 10, would have died at the age of 19, not 20. Furthermore, Dionysius' calculations were wrong. The Gospel of Matthew tells us that Jesus was born under the reign of king Herod the Great, and he died in 4 BC. It is likely that Jesus was actually born around 7 BC. The date of his birth is unknown; it may or may not be 25 December. 2.10.2. When does the 21st century start? ----------------------------------------- The first century started in AD 1. The second century must therefore have started a hundred years later, in AD 101, and the 21st century must start 2000 years after the first century, i.e. in the year 2001. This is the cause of some heated debate, especially since some dictionaries and encyclopaedias say that a century starts in years that end in 00. Let me propose a few compromises: Any 100-year period is a century. Therefore the period from 23 June 1997 to 22 June 2097 is a century. So please feel free to celebrate the start of a century any day you like! Although the 20th century started in 1901, the 1900s started in 1900. Similarly, we can celebrate the start of the 2000s in 2000 and the start of the 21st century in 2001. Finally, let's take a lesson from history: When 1899 became 1900 people celebrated the start of a new century. When 1900 became 1901 people celebrated the start of a new century. Two parties! Let's do the same thing again! 2.11. What is the Indiction? ---------------------------- The Indiction was used in the middle ages to specify the position of a year in a 15 year taxation cycle. It was introduced by emperor Constantine the Great on 1 September 312 and abolished [whatever that means] in 1806. The Indiction may be calculated thus: Indiction = (year + 2) % 15 + 1 The Indiction has no astronomical significance. The Indiction did not always follow the calendar year. Three different Indictions may be identified: 1) The Pontifical or Roman Indiction, which started on New Year's Day (being either 25 December, 1 January, or 25 March). 2) The Greek or Constantinopolitan Indiction, which started on 1 September. 3) The Imperial Indiction or Indiction of Constantine, which started on 24 September. 2.12. What is the Julian Period? -------------------------------- The Julian period (and the Julian day number) must not be confused with the Julian calendar. The French scholar Joseph Justus Scaliger (1540-1609) was interested in assigning a positive number to every year without having to worry about BC/AD. He invented what is today known as the "Julian Period". The Julian Period probably takes its name from the Julian calendar, although it has been claimed that it is named after Scaliger's father, the Italian scholar Julius Caesar Scaliger (1484-1558). Scaliger's Julian period starts on 1 January 4713 BC (Julian calendar) and lasts for 7980 years. AD 1997 is thus year 6710 in the Julian period. After 7980 years the number starts from 1 again. Why 4713 BC and why 7980 years? Well, in 4713 BC the Indiction (see section 2.11), the Golden Number (see section 2.9.3) and the Solar Number (see section 2.4) were all 1. The next times this happens is 15*19*28=7980 years later, in AD 3268. Astronomers have used the Julian period to assign a unique number to every day since 1 January 4713 BC. This is the so-called Julian Day (JD). JD 0 designates the 24 hours from noon UTC on 1 January 4713 BC to noon UTC on 2 January 4713 BC. This means that at noon UTC on 1 January AD 2000, JD 2,451,545 will start. This can be calculated thus: From 4713 BC to AD 2000 there are 6712 years. In the Julian calendar, years have 365.25 days, so 6712 years correspond to 6712*365.25=2,451,558 days. Subtract from this the 13 days that the Gregorian calendar is ahead of the Julian calendar, and you get 2,451,545. Often fractions of Julian day numbers are used, so that 1 January AD 2000 at 15:00 UTC is referred to as JD 2,451,545.125. Note that some people use the term "Julian day number" to refer to any numbering of days. NASA, for example, use the term to denote the number of days since 1 January of the current year. 2.12.1. Is there a formula for calculating the Julian day number? ----------------------------------------------------------------- Try this one (the divisions are integer divisions, in which remainders are discarded): a = (14-month)/12 y = year-a m = month + 12*a - 3 For a date in the Gregorian calendar: JDN = day + (306*m+5)/10 + y*365 + y/4 - y/100 + y/400 + 1721119 For a date in the Julian calendar: JDN = day + (306*m+5)/10 + y*365 + y/4 + 1721117 JDN is the Julian day number that starts at noon UTC on the specified date. The algorithm works fine for AD dates. If you want to use it for BC dates, you must first convert the BC year to a negative year (e.g, 10 BC = -9) and make sure that your integer divisions round down, not towards zero: -9/4 should be -3, not -2. 2.12.2. What is the modified Julian day? ---------------------------------------- Sometimes a modified Julian day number (MJD) is used which is 2,400,000.5 less than the Julian day number. This brings the numbers into a more manageable numeric range and makes the day numbers change at midnight UTC rather than noon. MJD 0 thus falls on 17 Nov 1858 (Gregorian) at 00:00:00 UTC. 3. The Hebrew Calendar ---------------------- The current definition of the Hebrew calendar is generally said to have been set down by the Sanhedrin president Hillel II in approximately AD 359. The original details of his calendar are, however, uncertain. The Hebrew calendar is used for religious purposes by Jews all over the world, and it is the official calendar of Israel. The Hebrew calendar is a combined solar/lunar calendar, in that it strives to have its years coincide with the tropical year and its months coincide with the synodic months. This is a complicated goal, and the rules for the Hebrew calendar are correspondingly fascinating. 3.1. What does a Hebrew year look like? --------------------------------------- An ordinary (non-leap) year has 353, 354, or 355 days. A leap year has 383, 384, or 385 days. The three lengths of the years are termed, "deficient", "regular", and "complete", respectively. An ordinary year has 12 months, a leap year has 13 months. Every month starts (approximately) on the day of a new moon. The months and their lengths are: Length in a Length in a Length in a Name deficient year regular year complete year ------- -------------- ------------ ------------- Tishri 30 30 30 Heshvan 29 29 30 Kislev 29 30 30 Tevet 29 29 29 Shevat 30 30 30 (Adar I 30 30 30) Adar II 29 29 29 Nisan 30 30 30 Iyar 29 29 29 Sivan 30 30 30 Tammuz 29 29 29 Av 30 30 30 Elul 29 29 29 ------- -------------- ------------ ------------- Total: 353 or 383 354 or 384 355 or 385 The month Adar I is only present in leap years. In non-leap years Adar II is simply called "Adar". Note that in a regular year the numbers 30 and 29 alternate; a complete year is created by adding a day to Heshvan, whereas a deficient year is created by removing a day from Kislev. The alteration of 30 and 29 ensures that when the year starts with a new moon, so does each month. 3.2. What years are leap years? ------------------------------- A year is a leap year if the number year%19 is one of the following: 0, 3, 6, 8, 11, 14, or 17. The value for year in this formula is the 'Anno Mundi' described in section 3.8. 3.3. What years are deficient, regular, and complete? ----------------------------------------------------- That is the wrong question to ask. The correct question to ask is: When does a Hebrew year begin? Once you have answered that question (see section 3.6), the length of the year is the number of days between 1 Tishri in one year and 1 Tishri in the following year. 3.4. When is New Year's day? ---------------------------- That depends. Jews have 4 different days to choose from: 1 Tishri: "Rosh HaShanah". This day is a celebration of the creation of the world and marks the start of a new calendar year. This will be the day we shall base our calculations on in the following sections. 15 Shevat: "Tu B'shevat". The new year for trees, when fruit tithes should be brought. 1 Nisan: "New Year for Kings". Nisan is considered the first month, although it occurs 6 or 7 months after the start of the calendar year. 1 Elul: "New Year for Animal Tithes (Taxes)". Only the first two dates are celebrated nowadays. 3.5. When does a Hebrew day begin? ---------------------------------- A Hebrew day does not begin at midnight, but at sunset (when 3 stars are visible). Sunset marks the start of the 12 night hours, whereas sunrise marks the start of the 12 day hours. This means that night hours may be longer or shorter than day hours, depending on the season. 3.6. When does a Hebrew year begin? ----------------------------------- The first day of the calendary year, Rosh HaShanah, on 1 Tishri is determined as follows: 1) The new year starts on the day of the new moon that follows the last month of the previous year. 2) If the new moon occurs after noon on that day, delay the new year by one day. (Because in that case the new crescent moon will not be visible until the next day.) 3) If this would cause the new year to start on a Sunday, Wednesday, or Friday, delay it by one day. (Because we want to avoid that Yom Kippur (10 Tishri) falls on a Friday or Sunday, and that Hoshanah Rabba (21 Tishri) falls on a Sabbath (Saturday)). 4) If two consecutive years start 356 days apart (an illegal year length), delay the start of the first year by two days. 5) If two consecutive years start 382 days apart (an illegal year length), delay the start of the second year by one day. Note: Rule 4 can only come into play if the first year was supposed to start on a Tuesday. Therefore a two day delay is used rather that a one day delay, as the year must not start on a Wednesday as stated in rule 3. 3.7. When is the new moon? -------------------------- A calculated new moon is used. In order to understand the calculations, one must know that an hour is subdivided into 1080 'parts'. The calculations are as follows: The new moon that started the year AM 1, occurred 5 hours and 204 parts after sunset (i.e. just before midnight on Julian date 6 October 3761 BC). The new moon of any particular year is calculated by extrapolating from this time, using a synodic month of 29 days 12 hours and 793 parts. 3.8. How does one count years? ------------------------------ Years are counted since the creation of the world, which is assumed to have taken place in 3761 BC. In that year, AM 1 started (AM = Anno Mundi = year of the world). In the year AD 1997 we will witness the start of Hebrew year AM 5758. 4. The Islamic Calendar ----------------------- The Islamic calendar (or Hijri calendar) is a purely lunar calendar. It contains 12 months that are based on the motion of the moon, and because 12 synodic months is only 12*29.53=354.36 days, the Islamic calendar is consistently shorter than a tropical year, and therefore it shifts with respect to the Christian calendar. The calendar is based on the Qur'an (Sura IX, 36-37) and its proper observance is a sacred duty for Muslims. The Islamic calendar is the official calendar in countries around the Gulf, especially Saudi Arabia. But other Muslim countries use the Gregorian calendar for civil purposes and only turn to the Islamic calendar for religious purposes. 4.1. What does an Islamic year look like? ----------------------------------------- The names of the 12 months that comprise the Islamic year are: 1. Muharram 7. Rajab 2. Safar 8. Sha'ban 3. Rabi' al-awwal (Rabi' I) 9. Ramadan 4. Rabi' al-thani (Rabi' II) 10. Shawwal 5. Jumada al-awwal (Jumada I) 11. Dhu al-Qi'dah 6. Jumada al-thani (Jumada II) 12. Dhu al-Hijjah (Due to different transliterations of the Arabic alphabet, other spellings of the months are possible.) Each month starts when the lunar crescent is first seen (by an actual human being) after a new moon. Although new moons may be calculated quite precisely, the actual visibility of the crescent is much more difficult to predict. It depends on factors such a weather, the optical properties of the atmosphere, and the location of the observer. It is therefore very difficult to give accurate information in advance about when a new month will start. Furthermore, some Muslims depend on a local sighting of the moon, whereas others depend on a sighting by authorities somewhere in the Muslim world. Both are valid Islamic practices, but they may lead to different starting days for the months. 4.2. So you can't print an Islamic calendar in advance? ------------------------------------------------------- Not a reliable one. However, calendars are printed for planning purposes, but such calendars are based on estimates of the visibility of the lunar crescent, and the actual month may start a day earlier or later than predicted in the printed calendar. Different methods for estimating the calendars are used. Some sources mention a crude system in which all odd numbered months have 30 days and all even numbered months have 29 days with an extra day added to the last month in 'leap years' (a concept otherwise unknown in the calendar). Leap years could then be years in which the number year%30 is one of the following: 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, or 29. (This is the algorithm used in the calendar program of the Gnu Emacs editor.) Such a calendar would give an average month length of 29.53056 days, which is quite close to the synodic month of 29.53059 days, so *on the average* it would be quite accurate, but in any given month it is still just a rough estimate. Better algorithms for estimating the visibility of the new moon have been devised. One such algorithm is implemented in a program called 'Islamic Timer' by professor Waleed A. Muhanna. Interested readers may find the program on the World Wide Web at http://www.cob.ohio-state.edu/facstf/homepage/muhanna/IslamicTimer.html You may also want to check out the following web site (and the pages it refers to) for information about Islamic calendar predictions: http://www.ummah.org.uk/ildl 4.3. How does one count years? ------------------------------ Years are counted since the Hijra, that is, Mohammed's flight to Medina, which is assumed to have taken place 16 July AD 622 (Julian calendar). On that date AH 1 started (AH = Anno Hegirae = year of the Hijra). In the year AD 1997 we will witness the start of Islamic year AH 1418. Note that although only 1997-622=1376 years have passed in the Christian calendar, 1417 years have passed in the Islamic calendar, because its year is consistently shorter (by about 11 days) than the tropical year used by the Christian calendar. 4.4. When will the Islamic calendar overtake the Gregorian calendar? -------------------------------------------------------------------- As the year in the Islamic calendar is about 11 days shorter than the year in the Christian calendar, the Islamic years are slowly gaining in on the Chistian years. But it will be many years before the two coincide. The 1st day of the 5th month of AD 20874 in the Gregorian calendar will also be (approximately) the 1st day of the 5th month of AH 20874 of the Islamic calendar. --- End of part 2 --- 5. The Week ----------- Both the Christian, the Hebrew, and the Islamic calendar have a 7-day week. 5.1. What Is the Origin of the 7-Day Week? ------------------------------------------ Digging into the history of the 7-day week is a very complicated matter. Authorities have very different opinions about the history of the week, and they frequently present their speculations as if they were indisputable facts. The only thing we seem to know for certain about the origin of the 7-day week is that we know nothing for certain. The first pages of the Bible explain how God created the world in six days and rested on the seventh. This seventh day became the Jewish day of rest, the sabbath, Saturday. Extra-biblical locations sometimes mentioned as the birthplace of the 7-day week include: Egypt, Babylon, Persia, and several others. The week was known in Rome before the advent of Christianity. 5.2. What Do the Names of the Days of the Week Mean? ---------------------------------------------------- An answer to this question is necessarily closely linked to the language in question. Whereas most languages use the same names for the months (with a few Slavonic languages as notable exceptions), there is great variety in names that various languages use for the days of the week. A few examples will be given here. Except for the sabbath, Jews simply number their week days. A related method is partially used in Portuguese and Russian: English Portuguese Russian Meaning of Russian name ------- ---------- ------- ----------------------- Monday segunda ponedelnik After do-nothing day Tuesday terca vtornik Second day Wednesday quarta sreda Center Thursday quinta chetverg Four Friday sexta pyatnitsa Five Saturday sabado subbota Sabbath Sunday domingo voskresenye Resurrection Most Latin-based languages connect each day of the week with one of the seven "planets" of the ancient times: Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn. The reason for this may be that each planet was thought to "rule" one day of the week. French, for example, uses: English French "Planet" ------- ------ -------- Monday lundi Moon Tuesday mardi Mars Wednesday mercredi Mercury Thursday jeudi Jupiter Friday vendredi Venus Saturday samedi Saturn Sunday dimanche (Sun) The link with the sun has been broken in French, but Sunday was called "dies solis" (day of the sun) in Latin. It is interesting to note that also some Asiatic languages (Hindi, for example) have a similar relationship between the week days and the planets. English has retained the original planets in the names for Saturday, Sunday, and Monday. For the four other days, however, the names of Anglo-Saxon or Nordic gods have replaced the Roman gods that gave name to the planets. Thus, Tuesday is named after Tiw, Wednesday is named after Woden, Thursday is named after Thor, and Friday is named after Freya. 5.3. Has the 7-Day Week Cycle Ever Been Interrupted? ---------------------------------------------------- There is no record of the 7-day week cycle ever having been broken. Calendar changes and reform have never interrupted the 7-day cycles. It very likely that the week cycles have run uninterrupted at least since the days of Moses (c. 1400 BC), possibly even longer. Some sources claim that the ancient Jews used a calendar in which an extra Sabbath was occasionally introduced. But this is probably not true. 5.4. Which Day is the Day of Rest? ---------------------------------- For the Jews, the Sabbath (Saturday) is the day of rest and worship. On this day God rested after creating the world. Most Christians have made Sunday their day of rest and worship, because Jesus rose from the dead on a Sunday. Muslims use Friday as their day of rest and worship, because Muhammad was born on a Friday. 5.5. What Is the First Day of the Week? --------------------------------------- It is common Jewish and Christian practice to regard Sunday as the first day of the week. However, the fact that, for example, Russian uses the name "second day" for Tuesday, indicates that some nations regard Monday as the first day. In international standard IS-8601 the International Organization for Standardization (ISO) has decreed that Monday shall be the first day of the week. 5.6. What Is the Week Number? ----------------------------- International standard IS-8601 (mentioned in section 5.5) assigns a number to each week of the year. A week that lies partly in one year and partly in another is assigned a number in the year in which most of its days lie. This means that Week 1 of any year is the week that contains 4 January, or equivalently Week 1 of any year is the week that contains the first Thursday in January. Most years have 52 weeks, but years that start on a Thursday and leap years that start on a Wednesday have 53 weeks. 5.7. Do Weeks of Different Lengths Exist? ----------------------------------------- If you define a "week" as a 7-day period, obviously the answer is no. But if you define a "week" as a named interval that is greater than a day and smaller than a month, the answer is yes. The French Revolutionary calendar used a 10-day "week" (see section 6.1). The Maya calendar uses a 13 and a 20-day "week" (see section 7.2). The Soviet Union has used both a 5-day and a 6-day week. In 1929-30 the USSR gradually introduced a 5-day week. Every worker had one day off every week, but there was no fixed day of rest. On 1 September 1931 this was replaced by a 6-day week with a fixed day of rest, falling on the 6th, 12th, 18th, 24th, and 30th day of each month (1 March was used instead of the 30th day of February, and the last day of months with 31 days was considered an extra working day outside the normal 6-day week cycle). A return to the normal 7-day week was decreed on 26 June 1940. 6. The French Revolutionary Calendar ------------------------------------ The French Revolutionary Calendar (or Republican Calendar) was introduced in France on 24 November 1793 and abolished on 1 January 1806. It was used again briefly during under the Paris Commune in 1871. 6.1. What does a Republican year look like? ------------------------------------------- A year consists of 365 or 366 days, divided into 12 months of 30 days each, followed by 5 or 6 additional days. The months were: 1. Vendemiaire 7. Germinal 2. Brumaire 8. Floreal 3. Frimaire 9. Prairial 4. Nivose 10. Messidor 5. Pluviose 11. Thermidor 6. Ventose 12. Fructidor (The second e in Vendemiaire and the e in Floreal carry an acute accent. The o's in Nivose, Pluviose, and Ventose carry a circumflex accent.) The year was not divided into weeks, instead each month was divided into three "decades" of 10 days, of which the final day was a day of rest. This was an attempt to de-Christianize the calendar, but it was an unpopular move, because now there were 9 work days between each day of rest, whereas the Gregorian Calendar had only 6 work days between each Sunday. The ten days of each decade were called, respectively, Primidi, Duodi, Tridi, Quartidi, Quintidi, Sextidi, Septidi, Octidi, Nonidi, Decadi. The 5 or 6 additional days followed the last day of Fructidor and were called: 1. Jour de la vertu (Virtue Day) 2. Jour du genie (Genius Day) 3. Jour du travail (Labour Day) 4. Jour de l'opinion (Reason Day) 5. Jour des recompenses (Rewards Day) 6. Jour de la revolution (Revolution Day) (the leap day) Each year was supposed to start on autumnal equinox (around 22 September), but this created problems as will be seen in section 6.3. 6.2. How does one count years? ------------------------------ Years are counted since the establishment of the first French Republic on 22 September 1792. That day became 1 Vendemiaire of the year 1 of the Republic. (However, the Revolutionary Calendar was not introduced until 24 November 1793.) 6.3. What years are leap years? ------------------------------- Leap years were introduced to keep New Year's Day on autumnal equinox. But this turned out to be difficult to handle, because equinox is not completely simple to predict. Therefore a rule similar to the one used in the Gregorian Calendar (including a 4000 year rule as descibed in section 2.2.2) was to take effect in the year 20. However, the Revolutionary Calendar was abolished in the year 14, making this new rule irrelevant. The following years were leap years: 3, 7, and 11. The years 15 and 20 should have been leap years, after which every 4th year (except every 100th year etc. etc.) should have been a leap year. [The historicity of these leap year rules has been disputed. One source mentions that the calendar used a rule which would give 31 leap years in every 128 year period. I may have to update this section.] 6.4. How does one convert a Republican date to a Gregorian one? --------------------------------------------------------------- The following table lists the Gregorian date on which each year of the Republic started: Year 1: 22 Sep 1792 Year 8: 23 Sep 1799 Year 2: 22 Sep 1793 Year 9: 23 Sep 1800 Year 3: 22 Sep 1794 Year 10: 23 Sep 1801 Year 4: 23 Sep 1795 Year 11: 23 Sep 1802 Year 5: 22 Sep 1796 Year 12: 24 Sep 1803 Year 6: 22 Sep 1797 Year 13: 23 Sep 1804 Year 7: 22 Sep 1798 Year 14: 23 Sep 1805 7. The Maya Calendar -------------------- (I am very grateful to Chris Carrier (72157.3334@CompuServe.COM) for providing most of the information about the Maya calendar.) Among their other accomplishments, the ancient Maya invented a calendar of remarkable accuracy and complexity. The Maya calendar was adopted by the other Mesoamerican nations, such as the Aztecs and the Toltec, which adopted the mechanics of the calendar unaltered but changed the names of the days of the week and the months. The Maya calendar uses three different dating systems in parallel, the "Long Count", the "Tzolkin" (divine calendar), and the "Haab" (civil calendar). Of these, only the Haab has a direct relationship to the length of the year. A typical Mayan date looks like this: 12.18.16.2.6, 3 Cimi 4 Zotz. 12.18.16.2.6 is the Long Count date. 3 Cimi is the Tzolkin date. 4 Zotz is the Haab date. 7.1. What is the Long Count? ---------------------------- The Long Count is really a mixed base-20/base-18 representation of a number, representing the number of days since the start of the Mayan era. It is thus akin to the Julian Day Number (see section 2.12). The basic unit is the "kin" (day), which is the last component of the Long Count. Going from right to left the remaining components are: unial (1 unial = 20 kin = 20 days) tun (1 tun = 18 unial = 360 days = approx. 1 year) katun (1 katun = 20 tun = 7,200 days = approx. 20 years) baktun (1 baktun = 20 katun = 144,000 days = approx. 394 years) The kin, tun, and katun are numbered from 0 to 19. The unial are numbered from 0 to 17. The baktun are numbered from 1 to 13. Although they are not part of the Long Count, the Maya had names for larger time spans: 1 pictun = 20 baktun = 2,880,000 days = approx. 7885 years 1 calabtun = 20 pictun = 57,600,000 days = approx. 158,000 years 1 kinchiltun = 20 calabtun = 1,152,000,000 days = approx. 3 million years 1 alautun = 20 kinchiltun = 23,040,000,000 days = approx. 63 million years The alautun is probably the longest named period in any calendar. 7.1.1. When did the Long Count Start? ------------------------------------- Logically, the first date in the Long Count should be 0.0.0.0.0, but as the baktun (the first component) are numbered from 1 to 13 rather than 0 to 12, this first date is actually written 13.0.0.0.0. The authorities disagree on what 13.0.0.0.0 actually means. I have come across three possible equivalences: 13.0.0.0.0 = 8 Sep 3114 BC (Julian) = 13 Aug 3114 BC (Gregorian) 13.0.0.0.0 = 6 Sep 3114 BC (Julian) = 11 Aug 3114 BC (Gregorian) 13.0.0.0.0 = 11 Nov 3374 BC (Julian) = 15 Oct 3374 BC (Gregorian) Assuming one of the first two equivalences, the Long Count will again reach 13.0.0.0.0 on 21 or 23 December AD 2012 - a not too distant future. The Long Count was not, however, put in motion on 13.0.0.0.0, but rather on 7.13.0.0.0. The date 13.0.0.0.0 may have been the Maya's idea of the date of the creation of the world. 7.2. What is the Tzolkin? ------------------------- The Tzolkin date is a combination of two "week" lengths. While our calendar uses a single week of seven days, the Mayan calendar used two different lengths of week: - a numbered week of 13 days, in which the days were numbered from 1 to 13 - a named week of 20 days, in which the names of the days were: 0. Ahau 5. Chicchan 10. Oc 15. Men 1. Imix 6. Cimi 11. Chuen 16. Cib 2. Ik 7. Manik 12. Eb 17. Caban 3. Akbal 8. Lamat 13. Ben 18. Etznab 4. Kan 9. Muluc 14. Ix 19. Caunac As the named week is 20 days and the smallest Long Count digit is 20 days, there is synchrony between the two; if the last digit of today's Long Count is 0, for example, today must be Ahau; if it is 6, it must be Cimi. Since the numbered and the named week were both "weeks", each of their name/number change daily; therefore, the day after 3 Cimi is not 4 Cimi, but 4 Manik, and the day after that, 5 Lamat. The next time Cimi rolls around, 20 days later, it will be 10 Cimi instead of 3 Cimi. The next 3 Cimi will not occur until 260 (or 13*20) days have passed. This 260-day cycle also had good-luck or bad-luck associations connected with each day, and for this reason, it became known as the "divinatory year." The "years" of the Tzolkin calendar are not counted. 7.2.1. When did the Tzolkin Start? ---------------------------------- Long Count 13.0.0.0.0 corresponds to 4 Ahau. The authorities agree on this. 7.3. What is the Haab? ---------------------- The Haab was the civil calendar of the Maya. It consisted of 18 "months" of 20 days each, followed by 5 extra days, known as "Uayeb". This gives a year length of 365 days. The names of the month were: 1. Pop 7. Yaxkin 13. Mac 2. Uo 8. Mol 14. Kankin 3. Zip 9. Chen 15. Muan 4. Zotz 10. Yax 16. Pax 5. Tzec 11. Zac 17. Kayab 6. Xul 12. Ceh 18. Cumku Since each month was 20 days long, monthnames changed only every 20 days instead of daily; so the day after 4 Zotz would be 5 Zotz, followed by 6 Zotz ... up to 19 Zotz, which is followed by 0 Tzec. The days of the month were numbered from 0 to 19. This use of a 0th day of the month in a civil calendar is unique to the Maya system; it is believed that the Maya discovered the number zero, and the uses to which it could be put, centuries before it was discovered in Europe or Asia. The Uayeb days acquired a very derogatory reputation for bad luck; known as "days without names" or "days without souls," and were observed as days of prayer and mourning. Fires were extinguished and the population refrained from eating hot food. Anyone born on those days was "doomed to a miserable life." The years of the Haab calendar are not counted. The length of the Tzolkin year was 260 days and the length of the Haab year was 365 days. The smallest number that can be divided evenly into 260 and 365 is 18,980, or 365*52; this was known as the Calendar Round. If a day is, for example, "4 Ahau 8 Cumku," the next day falling on "4 Ahau 8 Cumku" would be 18,980 days or about 52 years later. Among the Aztec, the end of a Calendar Round was a time of public panic as it was thought the world might be coming to an end. When the Pleaides crossed the horizon on 4 Ahau 8 Cumku, they knew the world had been granted another 52-year extension. 7.3.1. When did the Haab Start? ------------------------------- Long Count 13.0.0.0.0 corresponds to 8 Cumku. The authorities agree on this. 7.4. Did the Maya Think a Year Was 365 Days? -------------------------------------------- Although there were only 365 days in the Haab year, the Maya were aware that a year is slightly longer than 365 days, and in fact, many of the month-names are associated with the seasons; Yaxkin, for example, means "new or strong sun" and, at the beginning of the Long Count, 1 Yaxkin was the day after the winter solstice, when the sun starts to shine for a longer period of time and higher in the sky. When the Long Count was put into motion, it was started at 7.13.0.0.0, and 0 Yaxkin corresponded with Midwinter Day, as it did at 13.0.0.0.0 back in 3114 B.C. The available evidence indicates that the Maya estimated that a 365-day year precessed through all the seasons twice in 7.13.0.0.0 or 1,101,600 days. We can therefore derive a value for the Mayan estimate of the year by dividing 1,101,600 by 365, subtracting 2, and taking that number and dividing 1,101,600 by the result, which gives us an answer of 365.242036 days, which is slightly more accurate than the 365.2425 days of the Gregorian calendar. 8. Date ------- This version 1.7 of this document was finished on Maundy Thursday, the 27th of March, anno ab Incarnatione Domini MCMXCVII, indict. V, epacta XXI, luna XVIII, anno post Margaretam Reginam Daniae natam LVII, on the feast of Saint John the Egyptian. The 18th day of Adar II, Anno Mundi 5757. The 18th day of Dhu al-Qi'dah, Anno Hegirae 1417. Julian Day 2,450,535. --- End of part 3 ---