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## 1. The Gregoriancalendar

The calendar in official use today in most Western countries, and also in the Christian faith, is the Gregorian calendar.

Days in the Gregorian calendar start at midnight. The Gregorian calendar is a solar calendar and tries run in step with the beginning of spring in the northern hemisphere (related to the Christian feast of Easter [Steel]) without keeping track of the phases of the Moon. The Gregorian calendar year consists of 12 months that each contain between 28 and 31 days, inclusive), as is shown in the following table.

Table 1: Months of the Gregorian Calendar

month days
1 January 31
2 February 28/29
3 March 31
4 April 30
5 May 31
6 June 30
7 July 31
8 August 31
9 September 30
10 October 31
11 November 30
12 December 31

Most years consist of 365 days, but some years (the leap years) have 366. The average length of the calendar year is 365 97/400 = 365.2425 days, which is about 11 seconds longer than the average time from one spring to the next, and about 11 days longer than 12 synodical months. In the Gregorian calendar, 400 years are exactly equal to 146,097 days, and after such a period the sequence of ordinary and leap years repeats itself. The first day of the year, 1 January, falls just after the middle of the winter half od the year in the Northern hemisphere. The first year of the calendar is tied to the birth of Jesus (from the Christian faith).

The rules for leap years are as follows:

1. a year (after year 1) is a leap year if its year number is a multiple of 4, except when rule number 2 holds.
2. a year is no leap year after all if its year number is a multiple of 100, except when rule number 3 applies.
3. a year is yet a leap year if its year number is a multiple of 400.

Following these rules, the years 1700, 1800, and 1900 weren't leap years, but 1600 and 2000 were.

The Gregorian calendar is a modified version of the Julian calendar. Modification was necessary because the average length of the Julian calendar year, 365 1/4 day, was about 11 minutes too long compared to the March equinox year so that the vernal equinox (the beginning of spring in the northern hemisphere) moved backwards through the calendar year by about one day per 131 years. When the Julian calendar was introduced, in −44, the vernal equinox fell around 25 March. At the time of the Council of Nicaea in 325 the vernal equinox had shifted to 21 March, and the rules for determining the date of Easter were based on that date. Around 1500 the vernal equinox had shifted further back to about 11 March. For this and other reasons, Pope Gregory XIII proclaimed the new calendar rules that define what we now call the Gregorian calendar.

To make the calendar year follow the return of the vernal equinox more closely, the last two leap year rules listed above were promulgated. To get the vernal equinox back to 21 March, the Pope decreed that the day following 4 October 1582 be called 15 October 1582, thereby skipping 10 days of the calendar. Of course, this does not mean that 10 days were taken from everybody's life, but only that something changed in what the days were called.

The Gregorian calendar reform was adopted quickly in the Catholic regions of the world (of which the Pope was the religious leader), but there was a lot of resistance to the reform in Protestant areas. Many Protestant areas started using the Gregorian calendar around 1700. Great Britain and its possessions followed in 1752, and Russia in 1918.

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It is clear that the gradual acceptance of the Gregorian calendar causes no end of problems in accurately dating dates from the period. A date in 1612, for example, is probably on the Gregorian calendar if it occurs on a letter sent from a Catholic country, but probably on the Julian calendar if it is on a letter sent from a Protestant country.

Dating is made even more difficult because not everybody celebrated the beginning of a new year on January 1st. That January 1st is now considered to be the first day of the year is a historical accident. In the past, other dates have been used for New Year's Day, such as 25 March and 25 December, but eventually 1 January was the most popular.

If I'm not mistaken, then in Christianity the 25th of March is associated with the conception of Jezus, the 25th of December with the birth of Jezus, and the 1st of January with the circumcision of Jezus. In pre-Christian times, 25 March was the ascending equinox (in the northern hemisphere the beginning of the summer half of the year), 25 December the southern solstice (the darkest day in the north, the middle of the winter half of the year, the midwinter feast), and 1 January was the New Year of the Romans (at least since Julius Caesar in 45 B.C.).

In the Middle Ages, each civil or religious authority could determine when New Year would be celebrated in their jurisdiction. If you thought that the conception of Jezus or the beginning of the summer half of the year was the most important, then you chose 25 March. If you thought the birth of Jezus or the midwinter feast (Yule) was the most important, then you picked 25 December. If you thought the circumcision of Jezus was the defining moment, or if you preferred to have the first day of the year at the beginning of a month, or if you wanted to keep to the tradition of the Romans, then you used 1 January.

So, if one city celebrated New Year on 25 March but another city on 1 January, then the same day could be called 1 March 1504 in the one city but 1 March 1505 in the other one. There were also differences in the year counts that were used. For the Julian calendar, for example, the eras of the Foundation of Rome (−752) and the start of the reign of Emperor Diocletian (284) were used before the era of Christ (invented in 532) became popular.

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## 2. Five Sundays in February

February can only have five of any particular day of the week in a leap year, because in an ordinary year February has exactly four weeks, so then every weekday occurs exactly four times in February. In a leap year, the five-of-a-kind day of the week occurs on both the first day and the last day of February (the latter is also the leap day or bissextile day). The year 2004 is a leap year and the first day of February in 2004 is a Sunday, so February has five Sundays in 2004.

This situation repeats itself every 28 years, forward until the end of the 21st century and backward until the beginning of the 20th century, so there were and will be five Sundays in February of 1920, 1948, 2004, 2032, 2060, and 2088. There are also months of February with five Sundays before the 20th century and after the 21st century, but not in the same arithmetical progression, because the years 1900 and 2100 are not leap years.

Each other month of the calendar has two or three weekdays of which there are five that month, every year.

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## 3. Week Numbers

ISO standard 8601 defines a calendar that assigns a week number to each week of the year. The week always begins with Monday and the first Thursday of a year is always in week number 1. For the Gregorian calendar, this means that the week that includes 4 January is always week number 1.

In this calendar, all years in which 1 January is a Thursday have 53 weeks, and leap years in which 1 January is a Wednesday do so as well. Other years have 52 weeks.

The explanation for this is as follows:

If 1 January falls on a Monday, Tuesday, Wednesday, or Thursday, then it is in week number 1. If 1 January falls on a Friday, Saturday, or Sunday, then it is in the week preceding week number 1, so it is week 52 or 53 of the preceding year. For convenience, I refer to the week preceding week 1 as week 0.

An ordinary year contains 52 weeks plus one day. If New Year's Day in such a year falls on a Monday, Tuesday, or Wednesday (week 1), then the next New Year's Day falls on a Tuesday, Wednesday, or Thursday (week 1), so the just concluded year then contains 52 weeks (as far as week numbers are concerned). If New Year's Day falls on a Thursday (week 1), then the next New Year's Day falls on a Friday (week 0), so then the year contained 53 weeks. If New Year's Day falls on a Friday or Saturday (week 0), then the next New Year's Day is on a Saturday or Sunday (week 0), so then the just concluded year contained 52 weeks. If New Year's Day is on a Sunday (the last day of week 0), then the next New Year's Day falls on a Monday (the first day of week 1), but there is an extra week boundary between those two (from Sunday to Monday), so the year yet contained 52 weeks.

A leap year contains 52 weeks plus 2 days. If New Year's Day in such a year falls on a Monday or Tuesday, then such a year has 52 weeks. If New Year's Day falls on a Wednesday or Thursday (week 1), then the next New Year's Day is a Friday or Saturday (week 0), so the just concluded year contained 53 weeks. If New Year's Day falls on a Friday, then the next New Year's Day is on a Sunday (week 0), so then the year has 52 weeks. If New Year's Day falls on a Saturday or Sunday (week 0), then the next one is on a Monday or Tuesday (week 1), but there's an extra week boundary in between, so the year yet contained only 52 weeks.

The next table shows how New Year's Day is distributed across the days of the week during a period of 400 Gregorian years. After such a period the distribution of leap years repeats itself, and that period is also equal to a whole number of weeks (namely 20,871) so the next period of 400 years starts with the same day of the week as the previous one.

Table 2: New Year

ordinary leap total
Monday 43 13 56
Tuesday 44 14 58
Wednesday 43 14* 57
Thursday 44* 13* 57
Friday 43 15 58
Saturday 43 13 56
Sunday 43 15 58
Total 303 97 400

For example: During a period of 400 years, there are 15 leap years that have 1 January fall on a Sunday. The years that contain 53 weeks are marked with an *. There are 71 of those, and that fits a total of 71×53 + 329×52 = 20,871 weeks.

After a year with 53 weeks, the next year with 53 weeks is 5, 6, or 7 years later. Usually (43 out of 71 times) this is 6 years later, quite often (27 of 71 times) 5 years later, and once per 400 years it is 7 years later.

The first years since 2000 that contain 53 weeks are 2004, 2009, 2015, 2020, 2026, and 2032. The years 2296 and 2303 also contain 53 weeks, and that is the next time that there are 7 years between two years with 53 weeks.

## 4. The Gregorian/Julian prolepticcalendar

We saw that there was much variation in the precise details of the calendars that have been used in Western Europe, even though they're all called Julian or Gregorian. For dating of historical events (for example, for history books or astronomical tables) it is useful to have an unambiguous calendar. For this unambiguous calendar, one uses the rules of the Gregorian calendar for dates from 15 October 1582, and the rules of the Julian calendar for dates up to and including 4 October 1582, with the era of Christ, even for days from before that era or the Julian calendar were even invented. The application of the rules of a calendar to days when those rules weren't in use (or even invented yet) is called proleptic. So, we can talk about 1 March of the year −300 in the Julian (proleptic) calendar, even though the Julian calendar hadn't been invented yet at that time, and the beginning of the era of Christ was still 300 year in the future.

Outside of astronomy, it is common to designate years before the epoch as "before Christ" (or BC), and years after the epoch as "ante diem" (or AD), such that the first year is AD 1, and the years immediately preceding that year are 1 BC, 2 BC, and so on. With this year count, which we may refer to as the historical year count, one must be extra careful when calculating the length of a period that includes the epoch. For example, the number of years between 1 March of year 3 BC and 1 March of year AD 2 is not 2 + 3 = 5, but one less: one year from 3 BC to 2 BC, another one to 1 BC, one more to AD 1, and the last one to AD 2. Ordinary arithmetic falters because the historical year count does not include a year 0. The most popular Christian date for the creation of the world is 23 October 4004 BC, and some people celebrated the 6000th anniversary of that date in 1996, presumably because 4004 + 1996 = 6000, but because of the absence of a year 0 there were only 5999 years between 4004 BC and AD 1996.

In astronomy there are quite a few calculations with years, and for those it is much more convenient to use an unbroken sequence of numbers for the years, including negative year numbers, and including a year 0. For years after the epoch, the astronomical and historical year counts coincide: the "astronomical" year 2001 or +2001 is equivalent to the "historical" year AD 2001. The years immediately preceding year +1 are years 0, −1, −2, and so on. To figure the astromical year number that is equivalent to a particular historical BC year number, subtract one from the BC year number and then put a minus sign in front of it. The year 3 BC corresponds to the year −2, and the number of years between that year and the year +2 is simply +2 − (−2) = 2 + 2 = 4.

For identifying leap years, the astronomical year numbers are more convenient than the historical year numbers, too. According to the rules of the Julian proleptic calendar the year AD 8 was a leap year, because that year number is a multiple of 4. The calendar has a leap year each 4 years, so if we repeatedly count back 4 years from a leap year, then we find more leap years. In the historical year count, you then find leap years AD 4, 1 BC, 5 BC, 9 BC, and so on. The BC leap year numbers are therefore not multiples of 4. With the astronomical year count, however, those leap years are referred to as +4, 0, −4, −8, and so on, so then leap years before the epoch also have year numbers that are multiples of 4.

## 5. How Many Leap Days Have There Been Already?

Julius Caesar introduced the Julian Calendar and with it the bissextile day in February. The first year of that calendar was the year −44, but that year was not considered to be a leap year, although it matches the rule that says that year numbers evenly divisible by four indicate leap years. That rule was not applied correctly for some time, which meant that leap years were observed more frequently than intended. This was noticed around the year −7, and then the leap year rule was suspended for a while until the excess number of leap years from the recent past had been balanced. From about the year 8 the calendar ran as intended, with leap years for year numbers evenly divisible by four.

I suspect that the counting error came from a cultural difference between the Romans and the advisor (Sosigenes from Egypt) who recommended the leap year rule to Caesar. If you asked a Roman to point out "each fourth year" starting from year I, then that Roman would first count I - II - III - IV and end up at year IV = 4, and then count IV - V - VI - VII and end up at year VII = 7, and then VII - VIII - IX - X and end up at year X = 10. The difference between the successive last years was 3 and not 4, because they counted the last year of the previous period also as the first year of the next period. We would have called that "every third year", and I think Caesar's adviser did, too. That adviser surely explained to Caesar what was meant by "every fourth year" in the leap year rule, but Caesar was murdered long before the first leap year according to the new rule began, and his successor apparently had not gotten the same explanation.

It is not entirely clear which years in that first period of the new calendar were counted as leap years, because the Romans misapplied the rule and no documents have been handed down to us that say which years they had taken to be leap years. However, when they became aware of their mistake, then they suspended the leap year rule for a while until the calendar was back in line with what was originally intended. The Romans themselves did know which years out of the previous 40 years they had taken to be leap years, so they knew also how many bissextile days they needed to skip to get back in line.

For that reason we can assume that the number of leap years between the years −44 and +7 is the number that fits the rule that every year number divisible by four indicates a leap year, with as only exception that the year −44 itself was not a leap year. That means that year 8 was the 13th leap year. After that there was a leap year every 4th year, which means 25 leap years per century. The number $$N$$ of leap years in the Julian Calendar up to and including year $$j$$ ($$j ≥ 8$$) is equal to

$$N = ⌊j/4⌋ + 11$$

The year 100 was the 36th leap year ($$N = ⌊100/4⌋ + 11 = 25 + 11 = 36$$). The year 1000 was the 261st leap year ($$N = ⌊1000/4⌋ + 11 = 250 + 11 = 261$$).

The now most widely accepted proposal for which years were regarded as leap years by the Romans in the first period of the Julian calendar is that of Scaliger from 1583 (see //en.wikipedia.org/wiki/Julian_calendar), who regarded the following years as leap years between the years −44 and +7: −41, −38, −35, −32, −29, −26, −23, −20, −17, −14, −11, −8. After that, the year +8 was the first leap year.

From 1582 (in different years at different locations) the Julian Calendar was replaced by the Gregorian Calendar, which has leap years slightly less frequently. Until 1582 there had been (in the Julian Calendar) $$N = ⌊1582/4⌋ + 11 = 395 + 11 = 406$$ leap years. Since that time (in the Gregorian Calendar) years with year numbers evenly divisible by 100 but not by 400 are no longer leap years. The years 1600 and 2000 were leap years (just like for the rules of the Julian Calendar), but the years 1700, 1800, and 1900 were not.

The number $$N$$ of leap years since the introduction of the Julian Calendar up to and including the year $$j$$ in the Gregorian Calendar ($$j ≥ 1582$$) is equal to

$$N = ⌊j/4⌋ - ⌊j/100⌋ + ⌊j/400⌋ + 23$$

In some countries the Gregorian Calendar was introduced much later than in the year 1582, but the difference between the leap year rules of the Julian and Gregorian Calendars was taken into account then, so that a country that switched later ran in step from then on with a country that had switched already in 1582.

The year 1600 was the 411th leap year (for countries that had switched to the Gregorian Calendar, and also for countries that still used the Julian Calendar then ― the year 1700 was the first year for which the leap year rules of the two calendars said different things about whether that year was a leap year). Since that time the Gregorian Calendar has only 97 leap years per 400 years, instead of 100. The year 2000 was the 508th leap year. The first couple of leap years from the year 2000 are:

$${j}$$$${N}$$
2000 508
2004 509
2008 510
2012 511
2016 512
2020 513

## 6. The Milanković Calendar

At the beginning of the 20th century some Eastern Orthodox Churches wanted to switch from the Julian calendar to a more accurate calendar, but preferred not to use the Gregorian calendar of their religious competitors the Catholic Church. Milutin Milanković (1879-1958) invented a calendar change for these Churches that yields a calendar that follows the average of the seasons more accurately than the Gregorian Calendar does.

This Milanković Calendar (also called the Revised Julian Calendar) was equal to the Gregorian Calendar, except in the determination of which years are leap years. Years of which the number is divisible by 4 but not by 100 are leap years, just like in the Gregorian Calendar, but years of which the number is divisible by 100 are leap years only if the number leaves a remainder of 200 or 600 when it is divided by 900 (while in the Gregorian Calendar the year number must be evenly divisible by 400). With this, the average length of a year in the Orthodox Calendar is 365 109/450 days.

This new calendar was adopted by a synod of Orthodox Church leaders in Istanbul in 1923. //www.friesian.com/russia.htm says that the calendar was used sporadically in the 1920s and not after that, but //en.wikipedia.org/wiki/Revised_Julian_calendar mentions many Orthodox churches that use the calendar and does not say anything about the calendar being abandoned.

## 7. The Islamic calendar

The Islamic calendar is used officially in countries where Islam is the state religion (especially in the Middle East), and in the Islamic faith. The Islamic calendar is a lunar calendar, and tries to follow the phases of the Moon without regards to the seasons.

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For religious use, an Islamic month start with the first appearance of the crescent Moon in the evening sky after a New Moon. Because the beginning of the month depends on observations it cannot be accurately predicted. It may also happen that Muslims in different countries or cities observe the new Moon first on different days, and so put the start of the month on different days. Those differences are ordinarily at most one day, but can cause some problems when making appointments. Therefore, there is also an administrative calendar that is based on fixed rules (and can thus be predicted arbitrarily far into the future) but yet follows the phases of the Moon pretty closely. The difference between the religious and administrative Islamic calendars is usually at most one day.

An Islamic calendar year always has 12 months. In the administrative calendar, the months alternate between 30 and 29 days, and the last month can get an extra day (and so have 30 rather than 29 days), as is shown by the following table

Table 3: Months of the Islamic Calendar

Month days
1 Muḥarram (المحرّم‎) 30
2 Ṣafar (صفر‎) 29
3 Rabīa Ⅰ (ربيع الأوّل‎) 30
4 Rabīa Ⅱ (ربيع الثاني‎) 29
5 Jumādā Ⅰ (جمادى الأولى‎) 30
6 Jumādā Ⅱ (جمادى الثانية‎) 29
7 Rajab (رجب‎) 30
8 Shaban (شعبان‎) 29
9 Ramaḍān (رمضان‎) 30
10 Shawwal (شوّال‎) 29
11 Dhu al-Qadah (ذو القعدة‎) 30
12 Dhu al-Ḥijjah (ذو الحجّة‎) 29/30

With this, a calendar year has 354 or 355 days. In the administrative calendar, a month lasts on average 29 191/360 days, and the year 354 11/30 days. With this, the calendar month is about 3 seconds short of the average synodical month. After 30 years = 10631 days the administrative calendar repeats itself (as far as the order of leap years is concerned).

Because the Islamic calendar year is about 11 days shorter than the tropical year, the beginning of the Islamic year moves through all seasons in about 34 years' time, and the difference between the Gregorian year number and the Islamic year number (now about 580) decreases by one about every 34 years. The beginning of the Islamic year 1420 is around the third day before the Gregorian year 2009. The administrative Islamic and the Gregorian calendar will "intersect" around the 5th month of the year 20,874: the 5th month of the year 20,874 in the Gregorian calendar will start on the same day as the 5th month of the year 20,874 in the Islamic administrative calendar.

The epoch of the Islamic calendar is sunset of 15 July 622 on the Julian proleptic calendar, and is tied to the migration of Muhammad from Mecca to Medina.

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The next table shows at what dates in the Gregorian calendar the Islamic New Year and the beginning of the Islamic month of Ramaḍān fall in the Islamic years 1424 through 1430, according to the administrative calendar.

Year New Year Ramaḍān
1434 2012-11-15 2013-07-09
1435 2013-11-05 2014-06-29
1436 2014-10-25 2015-06-18
1437 2015-10-15 2016-06-07
1438 2016-10-03 2017-05-27
1439 2017-09-22 2018-05-16
1440 2018-09-12 2019-05-06

## 8. The Jewish Calendar

The Jewish calendar is used in Israel and in the Jewish faith. The Jewish calendar is a lunisolar calendar: the days start at sunset, months start with New Moon, and the year follows the seasons. The calendar months have 29 or 30 days, and the calendar year 12 or 13 months. The months of the Jewish calendar and their lengths are listed in the following table.

Table 4: Months of the Jewish Calendar

Month Days
1 Nisan (נִיסָן‎) 30
2 Iyar (אִיָּר / אייר‎) 29
3 Sivan (סִיוָן / סיוון‎) 30
4 Tammuz (תַּמּוּז‎) 29
5 Av (אָב‎) 30
6 Elul (אֱלוּל‎) 29
7 Tishri (תִּשׁרִי‎) 30
8 Ḥeshvan (מַרְחֶשְׁוָן / מרחשוון‎) 29/30
9 Kislev (כִּסְלֵו / כסליו‎) 29/30
10 Tevet (טֵבֵת‎) 29
11 Shevat (שְׁבָט‎) 30
-/12 Adar Ⅰ (‎אֲדָר א׳)‎ 0/30

The Bible names Nisan as the first month, but mentions 1 Tishi as New Year's Day. Leap (bissextile) days can be inserted at the end of Ḥeshvan and Kislev, and an embolismic month Adar Ⅱ can be inserted before Adar, which is then called Adar Ⅰ. With this, there are six possible lengths of the calendar year:

Table 5: Month Types of the Jewish Calendar

year length type name
353 deficient ordinary year
354 regular ordinary year
355 complete ordinary year
383 deficient leap year
384 regular leap year
385 complete leap year

The calendar month on average lasts 29 13753/25920 days, and the calendar year 365 617/2500 days. The calendar year is about 6 minutes longer than a tropical year, and the calendar month about 0.3 seconds longer than a synodical month. The beginning of the Jewish calendar year is around the beginning of autumn in the northern hemisphere, between 3 September and 12 October on the Gregorian calendar.

The epoch of the Jewish calendar is at sunset of 6 October −3760 on the Julian proleptic calendar, and is tied to the (assumed) creation of the world.

## 9. The Julian day count

The Julian day count is a calendar that knows no months or years, but merely counts the days since the epoch (Julian Day 0), which was at noon of 1 January −4712 on the Julian proleptic calendar. This calendar is useful for astronomical calculations, and also as way station for the conversion of a date from one calendar to another calendar. Also, the day of the week is easy to calculate from the number of the Julian day that starts on the desired date: take the Julian day number and determine the remainder after dividing by 7. A remainder of 0 indicates the day is a Monday; 1 means Tuesday, and so on.

The following table shows some correspondences between modern calendars, the Egyptian calendar, the Maya Long Count, and the Julian day count. The correspondences were calculated for noon (12:00) of the dates. For the Jewish calendar, Tishri is month 1. For the Egyptian calendar, the 5 days after the 12 regular months are counted as a single month 13. Dates are given in the order year number, month number, day number.

JD Julian Gregorian Islamitic Jewish Egyptian Long Count
0 −4712 1 1 −4713 11 24 −5498 8 16 −952 10 20 −3968 2 18 8.18.16.17.17
347998 −3760 10 7 −3760 9 7 −4516 8 26 1 7 1 −3015 7 21
500000 −3344 12 4 −3344 11 7 −4087 8 5 417 9 8 −2599 13 3 12.8.5.15.17
584283 −3113 9 6 −3113 8 11 −3849 6 8 647 6 11 −2368 12 1
1000000 −1975 11 7 −1975 10 21 −2676 7 24 1786 8 25 −1229 11 13 2.17.14.13.17
1448273 −747 2 26 −747 2 18 −1411 7 23 3013 12 25
1500000 −606 10 11 −606 10 4 −1265 7 12 3155 7 15 141 9 23 6.7.3.11.17
1948440 622 7 16 622 7 19 1 1 1 4382 5 3
2000000 763 9 14 763 9 18 146 7 1 4524 7 3 1511 8 3 9.16.12.9.17
2415021 1899 12 20 1900 1 1 1317 8 28 5660 11 1 2648 8 19 12.14.5.6.18
2451545 1999 12 19 2000 1 1 1420 9 24 5760 10 23 2748 9 13 12.19.6.15.2
2488070 2099 12 19 2100 1 1 1523 10 19 5860 10 20 2848 10 8 0.4.8.5.7
2500000 2132 8 17 2132 8 31 1557 6 18 5892 6 19 2881 6 13 0.6.1.7.17

At noon on 1 January 2001, JD (= Julian day) 2,451,911 began. The Julian Day Number Calculation Page explains how you can calculate the Julian Day Number for dates in the Gregorian calendar.

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## 10. The Chinese Calendar

The traditional Chinese Calendar is a lunisolar calendar. Calendar days begin at midnight, calendar months begin on the day of New Moon, and calendar years follow the seasons, with Chinese New Year falling between one and two lunar months after the southern solstice, currently at some time between 19 January and 18 February on the Gregorian Calendar. A calendar year has 12 or 13 calendar months, depending on how many New Moons there are between two successive southern solstices. The months get numbers 1 through 12. A leap month gets the same number as the preceding month, but is marked as a "leap month" (闰月 rùnyuè). Not all leap months have the same number.

Besides the calendar months (lunar months) there are also solar months. Those solar months are needed to determine the numbering of months in a leap year. The solar months are tied to the position of the Sun along the ecliptic and correspond to the (Western) astrological signs of the zodiac: Each solar month begins when the ecliptical longitude of the Sun is a multiple of 30°.

The rules for determining the month numbers are:

• The month that contains the southern solstice is always the 11th month of the year.
• In a year with 13 months, the leap month is the first lunar month that is wholly contained in a solar month.

There have been many different ways of defining which year was year 1 (the epoch). Long ago (from −139 through 1912) a new era (with a new year 1) began whenever a new emperor started ruling. Other ways were (according to https://en.wikipedia.org/wiki/Chinese_calendar#Year):

• era of the reign of the legendary (mythical?) Fuxi, −2951; published by Martino Martini in 1658.
• era of the reign of the Yellow Emperor (Huángdì) (黄帝紀年), −2698 or −2697. The era −2698 was published by Martino Martini (1658) and Philippe Couplet (1686), and was promulgated by Sun Yat-sen in 1912 at the beginning of the republic, and is used by many Chinese communities outside of South-East Asia. Modern Chinese chronology usually puts the beginning of the reign of the Yellow Emperor at −2697. Also −2710 and −2490 have been used (at the beginning of the 20th century) as the era of the Yellow Emperor.
• era of the invention of the calendar by the Yellow Emperor, traditionally put at −2636.
• era of Emperor Yáo (唐尧紀年), −2156.
• era of the Gònghé regency (共和紀年), −841.
• era of the birth of Confucius (孔子紀年), −551.
• era of the unity of China under Emperor Qin Shi Huang (統一紀年), −221.
• the Common Era (元始), 1.
• era of the Republic, 1912.

In the traditional Chinese Calendar, years have a name from a set of 60 names that repeat themselves (干支 gānzhī). The first year from the first cycle began on the day that corresponds to 8 March −2636 in the Julian proleptic calendar (CJD 758326). The name of a year is formed from a "celestial stem" (天干 tiāngān) from a list of 10 and a "terrestrial branch" (地支dìzhī) from a list of 12. The names of the heavenly stems are untranslatable, but each pair has a wu xing name that is translatable. The heavenly stems are:

Table 6: Celestial Stems from the Chinese Calendar

#
name wu xing
1 jiǎ wood
2 wood
3 bǐng fire
4 dīng fire
5 earth
6 earth
7 gēng metal
8 xīn metal
9 rén water
10 guǐ water

The terrestrial branches are:

Table 7: Terrestrial Branches from the Chinese Calendar

#
Name English Name
1 Rat
2 chǒu Ox
3 yín Tiger
4 mǎo Rabbit
5 chén Dragon
6 Snake
7 Horse
8 wèi Goat
9 shēn Monkey
10 yǒu Rooster
11 Dog
12 hài Pig

The celestial stem and the terrestrial branch both shift for each year, so after year jǐ-chǒu (stem 6, branch 2) follows year gēng-yín (stem 7, branch 3). The first year from the cycle of 60 is jiǎ-zǐ (stem 1, branch 1). Early in Gregorian year 2000 a Chinese Year of the Metal Dragon (庚辰 gēng-chén) began.

In the 11th century, Shao Yong in his book Huángjíjīngshì (皇极经世) introduced what is now called the yuánhuìyùnshì system (HYSN-systeem). He wrote that 1 yuán = 12 hui; 1 hui = 30 yùn; 1 yùn = 12 shi; 1 shi = 30 years, so 1 yuán = 129600 years. The following table shows the number of years in each of these periods.

 yuán 元 129600 hui 会 10800 yùn 运 360 shi 世 30

A year number in that system is formed from the numbers of hui, yùn, shi, and years (H, Y, S, N). Each number is written with two digits, and between HY and SN a dash (-) is written. H counts the number of hui from the beginning of the only yuán. Y counts the number of yùn from the beginning of the current hui. S counts the number of shi from the beginning of the current yùn. N counts the number of years from the beginning of the current shi. All four numbers begin at 01. The epoch corresponds to the year −67016 of the Julian proleptic calendar. The greater part of the year 2016 of the Gregorian calendar corresponds to HYSN 0712-1003. A few other correspondences:

 −2576 HYSN 0630-0101 −2356 HYSN 0630-0811 1984 HYSN 0712-0901 2013 HYSN 0712-0930 2014 HYSN 0712-1001

Go to the general calendars or the historical calendars.

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Last updated: 2017-12-28