History of the European calendar
Describing all the complexities of the Jewish calendar, the outstanding Khorezm scholar Biruni( 973 1048) exclaimed: "But this is only a net and networks that the priests arranged to catch ordinary people and subordinate them to themselves. They have achieved that people did not do anything unconcerned with their opinions and only started on their destiny, not consulting with anyone else, as if these priests, not Allah  the rulers of the world. But Allah will pay them. .. "
Little is known about the original Jewish calendar. Four months are mentioned in the Bible: the first is Aviv, the month of ears( Exodus, XIII, 4), the second is the winter month of flowering( III Kings, VI, 1), the seventh is Afanim, the month of stormy winds( III Kings, VIII, 2)and the eighth  bul  the month of growth( III Kings, VI, 38).Apparently, the names of these months are related to the agricultural periods. It may be thought that these are the months of the ancient lunisolar calendar, although no mention is made of any rules for inserting( or even inserting) the 13th month in the Bible. On the other hand, it is as if the ordinal account is kept for days on decades( "asor"), and therefore it is possible that the ancient Jews used the solar calendar borrowed from Egypt, in which the months had 30 days.
While in the Babylonian captivity( 586539 BC), ancient Jews borrowed from the Babylonians their lunisolar calendar. This can be seen by comparing the names of the Jewish and Babylonian months. At that time in Babylon, rules for the insertion of the 13th month on the system of octaetridides had already been developed. However, the ancient Jews could hardly use it: the Bible prescribed the Easter feast on the full moon of the first spring month( "on the fourteenth day of the month in the evening").On this day, the ancient Jewish cattlebreeders brought to the gods purifying sacrifices( lambs and pigeons), and later the Passover was timed to the beginning of the harvest of barley and the first sheaf was dedicated to God( Leviticus XXIII, 10).The inset of the thirteenth month was made, conforming exclusively with the state of the loaves. Here is what he wrote, for example, who became in 75 AD.e.the chairman of the Sanhedrin( patriarch) Gamaliel II to the Jews who are in Babylon and Media: "Since the pigeons are still small and the lambs are very young and the time of Aviv has not yet arrived, then. .. we recognized the need to add thirty more days this year".
For the beginning of the month in the Jewish calendar, neomentality was taken  the moment of the first appearance of the moon in the evening sky. Even in the I century.n.e.at the Jerusalem Sanhedrin there was a special commission of three people who on the 29th of every month sent witnesses outside the city to watch the appearance of the sickle of the new Moon. Such witnesses to the appearance of the moon in neomenii should have been no less than two. Every Jew  an eyewitness of the first approach of the Moon  was obliged, under all conditions and even on a holiday  Saturday  to go to Jerusalem for testimony. After hearing the eyewitnesses, the board decided to consider the upcoming 30th day of the current month as the 1st day of the new month( otherwise only the next day was announced).The New Moon was proclaimed with the words "It is consecrated," which all the participants repeated in chorus. About this event, the population of the outskirts of Jerusalem was notified with the help of lights that were lit on the hills, and later  by the sending of messengers.
In the 70th year BCe. Jerusalem was completely destroyed by the troops of the Roman emperor Vespasian, and the Jews scattered throughout many countries of Europe and Asia. And, as already noted, the conditions for the visibility of the new Moon essentially depend on both geographical coordinates and the time of the year. It is not surprising, therefore, that in the calculation of time for Jews in the IIIV centuries.n.e.there was confusion. In particular, "such a holiday as the day of Yom Kippur, in some places because of not received timely information do not know when to celebrate;in other places, each at its own peril, it is celebrated on different days. .. Some cities celebrate two days of Yom Kippur. "Therefore, it was necessary to create a calendar that would not depend on the conditions for the visibility of the new Moon in this or that place, but would be based solely on calculations. The development of such a calendar continued for many years and was completed in 499 AD.e. At the same time, the beginning of the new month was already transferred to the new moon.
Echoes of understandable uncertainty  "whether the sickle of the" new moon "was seen today in the evening sky or not"  and it was that in the Jewish calendar year from 12 months there are. .. 18 days named "Rosh Hashush"( "newmonth. ") The fact is that the last day of the 30day month the Jews also called Rosh Hashush, ie, they celebrated the beginning of every second month for two days. ..
The structure of the Jewish calendar
The basis of the Jewish lunisolar calendar, which is still official in Israel today, is the 19year cycle of cycles. The insertion of the 13th month is carried out in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of the cycle. However, the number of days in the 19year cycle is not the same: it is 6939, 6940, 6941 days. The fact is that for religious reasons( which will be discussed below), the beginning of the new year can not be Sunday, Wednesday and Friday. If according to calculations the New Year falls on one of these days, then it is transferred the next day, and sometimes even two days in advance. Therefore, in the Jewish calendar, both the simple and the embolismic( 13month) year can be:
1) short or inadequate( "hazarin") containing 353 or 383( embolismic) days,
2) correct or complete( "kesidran")  354 or 384 days and
3) redundant( "shalam")  355 or 385 days. The distribution of days by month of the Jewish calendar is shown in Table.
Table. The duration of the months of the Eurasian calendar
sequence  month name  in the month simple annual  in the month of leap years  
month number  
insufficient  right  excess  insufficient  right  excess  
1  Tishri  30  30  30  30  30  30  
2  Cheshvan  29  29  30  29  29  30  
3  Kislev  29  30  30  29  30  30  
4  Tevet  29  29  29  29  29  29  
5  Shvat  30  30  30  30  30  30  
6  Adar 1  29  29  29  30  30  30  
7  Adar 2        Nisan  30  30  30  30  30  30 
9  Iyar  29  29  29  29  29  29  
10  Siwan  30  30  30  30  30  30  
11  TAMUZ  29  29  29  29  29  29  
12  Av  30  30  30  30  30  30  
13  Elul  29  29  29  29  29  29  
Total number of  353  354  355  383  384  385  
days per year 
The additional month is inserted beforeadar and gets the name "adar 1", adar becomes the next month and gets the name "adar 2".For this month, holidays are carried, which are celebrated in adar in a simple year. It is curious that the year with the 13th month the ancient Jews called the "Ibbur", i.e. "pregnant."The 19cycle cycle was called "makhzor", ie, "anticipated".
Jews borrowed from the ancient Babylonians and the sevenday week, but the days of the week( except Saturday) were called the easiest way: 1st in the week, 2nd in the week, etc. The day was divided by them for 24 hours, every hour for1080 heleks, each helik consists of 76 reggae( "moments").It is pertinent to note that the number 1080 is a multiple of all singlevalued divisors, except 7. Begin the day at 6 pm.
Until the end of the III century. BC.e. The new year of the calendar began with the spring month of Nisan, then the beginning of the year was postponed to the autumn month of Tishri. The starting point of the Jewish calendar is the mythical era "from the creation of the world"  October 7, 3761 BC.e. Monday, 5 pm 204 hlek( in calculations it is often taken 6 hours) in the afternoon. Adding to the year number of the Gregorian calendar 3760, we find the year of the Jewish era, ending in this Gregorian year, and adding 3761year of the Jewish era, which begins this year.
So, in the Jewish calendar there are six different options for the duration of the calendar year. But why are there so many?
In the grip of religious injunctions
The reason for the extreme complexity of the Jewish calendar can only be understood if you at least briefly get acquainted with some of the religious precepts that guided its writers. And there are 613 such judgments in Judaism, of which 365 are prohibitions and 248 are commands. A large part of them is "painted" for individual days and hours of the Jewish lunisolar calendar, being an integral part of the annual cycle of rites and holidays that entangle the whole life of believers.
First of all, believing Jews should honor the Sabbath  a day of rest. This is stated in the Bible as follows: "Six days you can do works, and on the seventh day is a Sabbath of rest, a holy assembly;do not do any work "(Leviticus XXIII, 3).Even food, and this is important for calendar calculations, on Saturday had to be prepared the day before, contact with fire on Saturday was considered almost a mortal sin: "After all, anyone who does it will be put to death"( Exodus XXXV, 2).
Further, the Bible prescribed "no work to work" on the first and seventh day of the Passover( Lev. XXIII, 58).It was also forbidden to do anything on the first, the ninth, the fifteenth and the twentysecond days of the month of Tishri( respectively, the "holiday of redemption", the "day of atonement" and the "Feast of Tabernacles"), etc.
And very important here isnext moment. According to the Bible during the holidays, it was necessary to make a sacrifice from large cattle, from sheep and from goats, and sacrificial meat should be eaten on the same day, in extreme cases it could be left the next day. If it remained on the third day, then the victim was considered to be desecrated. In addition, if the Bible instructed to "make a sacrifice" and not work, then it was possible to prepare sacrificial meat for food only the next day, therefore, on that day there could by no means be a Saturday. This was the main requirement for the creators of the Jewish calendar.
It turned out that these requirements for the alternation of holidays and Saturdays can be met only if the beginning of the new year in the Jewish calendar is 1 Monday, Tuesday, Thursday or Saturday, and the Easter holiday on Tuesday, Thursday, Saturday or Sunday.
And how here again do not remember the words of Biruni! Could anyone, except for highly educated priests, understand this web of prohibitions and injunctions?
How to determine the beginning of the year?
To determine the beginning of the year of the Jewish calendar( 1 Tishri) in the dates of our Gregorian calendar, you can directly or by preliminarily calculating the date on which 15 Nisan falls.
Here are the main steps of the first path:
1. Find the number of complete 19year cycles that have elapsed from the initial date of the Jewish era. For this, the number of years of the Jewish calendar should be divided by 19. The remainder indicates the ordinal number of the year within the unfinished cycle. For example, in 1986, 5747 years of the Jewish era begins, before this time will pass 5746 years. Dividing 5746 by 19, we get 302 and in the remainder 8. Consequently, 5747 will be the 9th year of the 303rd cycle.
2. Find the number of simple and leap years separately in the full years of the current cycle. In this case, out of eight years, five simple and three( they were the 3rd, 6th and 8th in the 19year cycle) are leap years.
3. Multiply the number of complete cycles by 1 hour 485 cheleks, that is, the difference between the duration of 19 solar years and the duration of the 19year cycle of the lunisolar calendar:
302 * 1 hour 485 hooks = 18 days 5 hours 670 hleks
4. We multiply the number of complete prime years in an unfinished cycle by 10 days 21 hours 204 hleks, i.е.on the difference between the duration of the solar and lunar year:
5 * 10 days 21 hours 204 chelets = 54 days 9 hours 1020 hleks.
Number of the year in the cycle  Years of the Jewish era
 Date of the Gregorian calendar, accounting for 1 Tishri
 Number of days in the year
 Date of the Gregorian calendar, accounting for 15 Nisana

8  5746 *  16. IX.1985.Mon.  383  24. IV.1986.Th. 
9  5747  4.X.1986. Sat.  355  14. IV.1987.Tues. 
10  5748  24. IX.1987.Th.  354  2.1 V. 1988. Sb. 
11  5749 *  12.1 X. 1988. Mon.  383  20. IV.1989.Th. 
12  5750  30. IX.1989.Sat.  355  10. IV.1990.Tues. 
13  5751  20. IX.1990. Th.  354  ZOL 11.1991.Sat. 
14  5752 *  9. IX.1991.Mon.  385  18. IV.1992.Sat. 
15  5753  28. IX.1992.Mon.  353  6. IV.1993.Tues. 
16  5754  16. IX.1993.Th.  355  27.111.1994.Sun. 
17  5755 *  6. IX.1994.Tues.  384  15. IV.1995.Sat. 
18  5756  25. IX.1995.Mon.  355  4. IV.1996.Th. 
* 19  5757 *  14. IX.1996.Sat.  383  22. IV.1997.Tues. 
1  5758  2.X.1997.Th.  354  11. IV.1998.Sat. 
2  5759  21. IX.1998.Mon.  355  1. IV.1999.Th. 
3  5760 *  11. IX.1999.Sat.  385  20. IV.2000.Th. 
4  5761  30. IX.2000.Sat.  353  8. IV.2001.Sun. 
5  5762  18. IX.2001.Tues.  354  on 28.111.2002.Th. 
6  5763 *  7. IX.2002.Sat.  385  17. IV.2003.Th. 
7  5764  27. IX.2003.Sat.  355  6. IV.2004.Tues. 
8  5765 *  16. IX.2004.Th.  383  24. IV.2005.Sun. 
9  5766  4.X.2005.Tues.  354  13. lV.2006.Th. 
10  5767  23. IX.2006.Sat.  355  3. IV.2007.Tues. 
11  5768 *  13. IX.2007.Th.  383  20. IV.2008.Sun. 
12  5769  30. IX.2008.Tues.  354  9. IV.2009.Th. 
13  5770  19. IX.2009.Sat.  355  on 30.111.2010.Tues. 
14  5771 *  9. IX.2010.Th.  385  19. IV.2011.Tues. 
15  5772  29. IX.2011.Th.  354  7. IV.2012.Sat. 
16  5773  17. IX.2012.Mon.  353  26.111.2013.Tues. 
17  5774 *  5. IX.2013.Th.  385  15. IV.2014.Tues. 
18  5775  25. IX.2014.Th.  354  4. IV.2015.Sat. 
19  5776 *  14. IX.2015.Mon.  385  23. IV.2016.Sat. 
1  5777  Z.X.2016.Mon.  353  11. IV.2017.Tues. 
2  5778  21. IX.2017.Th.  354  31.111.2018.Sat. 
3  5779 *  10. IX.2018.Mon.  385  20. IV.2019.Sat. 
4  5780  30. IX.2019.Mon.  355  9. IV.2020.Th. 
5
 5781
 19. IX.2020.Sat.  353  28.111.2021 Sun. 
Note. An asterisk marked embolism years. 
5. Multiply the number of leap years contained in an unfinished cycle, for a minus 18 days 15 hours 589 heleks;that is, the annual increase for a leap year:
3 *(  18 days 15 hours 589 heleks) = 55 days 22 hours 687 heleks.
6. Add all three numbers found and subtract from the result the accretion of the Jewish era, which is 12 days 20 hours 204 hlek. As a result, for 1986 we have 3 days 20 hours 799 cheleks.
7. The resulting amount is deducted from the date of the Jewish calendar  October 7, 18 hours. If the above amount is more than 7 days 18 hours, then to this last value we preliminarily add 30  the number of days in September.
In this particular example, we have 7 days 18 hours = 3 days 20 hours 799 heleks = 3 days 21 hours 281 helek.
From here it would follow that 1 Tishri 5747 in the Jewish era falls on October 3, that is, that the new moon of the month of Tishri( "Tishri prayer") will come on October 3rd at 21:00 281 helek. However, from the "Eternal Calendarcalendar" it is clear that October 3, 1986 is Friday, Therefore, the beginning of the new 5747 is moved one day ahead  on Saturday October 4, 1986
In addition to the mentioned digressions of the Jewish calendar, it is necessary to bear in mindalso the following:
1. If the new moon of the month of Tishri( "Tishri prayer") comes after 18 hours, the New Year is postponed for one day ahead. But if this next day is Sunday, Wednesday or Friday, then another day ahead.
2. If the new moon of Tishri in the year following the leap one falls on Monday after 15 hours of 589 Hleks( this happens in the 1st, 4th, 7th, 9th, 12th, 15th and 18ththe lunar cycle), then the new year is postponed to Tuesday.
3. If the prayer of Tishri in a simple year comes on Tuesday after 9 hours of 204 Höleks, the New Year is postponed to Thursday.
To avoid errors in calculations, it is useful to check them using the second method  calculating by the Gauss formula the date for which 15 Nisan was accounted for in the previous year.
Since the months of Nisan, Iyar, Siwan, Tammuz, Aw and Elul have a constant number of days, there will be 163 days from 15 Nisan until the next new year( to 1 Tishri), ie 23 weeks and 2 days.
Let R be the designation of the year of the Gregorian calendar, then A = R + 3760 is the designation of the Jewish year. When calculating the date of 15 Nissan, we first find:
1) the remainder of a division of the value 12 A + 17 by 19 and
2) the remainder b from division A by 4.
Further we make the number 32,044,093 3 + 1,554,241 8a + 0.2560.003177 79 A = M + m, where M is its integer part, m is a fractional part.
Finally, we find the remainder from dividing M + 3a + 5b + 5 by 7. Then:
1) if c = 1, and>b and 0,632 870 37, the Jewish Passover( 15 Nisan) will be M + 2 March according to the Julian calendar,
2) if c = 2, 4 or 6, and also with c = 0, and>11 and m & gt;0,897 723 76, it falls on M + 1 March of the art.art., and
3) in all other cases  on March M, art. Art.
In this case, if the remainder of a & lt;12, the year of the Jewish calendar consists of 12 months, with a & gt;11 it is 13month, i.e., embolic.
In particular, for 5746( A = 5746) we find a = 18, b = 2, M = 42, m = 0.2608644, c = 5. Thus, the 15th of Nisan this year falls on April 11 of the art. Art. April 24, 1986 on the Gregorian calendar( Thursday).Counting forward 23 weeks and two days, we find that 1 Tishri 5747 will come on Saturday, October 4, 1986.
The dates of the Gregorian calendar from 1985 to 2021, which are 1 Tishri and 15 Nisan, are given in Table.
Table. Correspondence of the dates of the Jewish and Gregorian calendars
Knowing the days of the week for which 1 Tishri and 15 Nisan fell, it is easy to determine the number of days in a year, that is, to establish whether this year is "insufficient", "right" or "excessive".For this, see Table.in the horizontal line we find the day of the week, which was 1 Tishri of the given year, and on the vertical  the day of the week, which had 15 Nisan Number, located at the intersection, and indicates the number of days in a given year. As can be seen, if a simple year starts on Monday or Saturday, it may be either inadequate( 353 days) or surplus( 355 days), if 1 Tishri is Tuesday, then the year will necessarily be correct.
Table. Number of days in the Jewish calendar year
15 Nisan 1 Tishri  Sunday  Tuesday  Thursday  Saturday  Sunday  Tuesday  Thursday  Saturday 
easy year  embolismichesky year  
Monday 

353
 355 



383
 385 
Tuesday  354  384  
Thursday  355  354  383  385  
Saturday  353  355  383  385 
In our time, New Year's Day of the Jewish lunisolar calendar( 1Tishri) is between 5 September and 5 October, and 15 Nisan  between 26 March and 25 April( 13 March and 12 April Julian calendar).At the same time, the beginning of the year( for the number of all subsequent months from 1 Tishri to 1 Nisan) goes furthest in the 1 st and 9 th years of the 19year cycle, whereas in the 6th, 14th and 17thThe Gregorian calendar is closest to all others to its lower limit.
On the accuracy of the
calendar So, if we are talking about the structure of the Jewish calendar, then, by its complexity, it hardly has rivals. But how accurate is it?
Analysis of this calendar shows that the average duration of the calendar month in it is 29d12h44m3( 1s / 3), which almost coincides with the duration of the synodic month. Therefore, the Jewish calendar practically can not disperse with the average astronomical phases of the Moon. However, the average for 19 years, containing 235 months, the length of the calendar year, equal to
29d 12h44m3( 1s / 3) *( 235/19) = 365d5h55m25( 8s / 19) = 365d, 24682,
for 6 min 39 s longer than the astronomical tropicalof the year. This means that the beginning of the Jewish year for every 216 years moves forward one day relative to the dates of the Gregorian calendar. Therefore, also in our days 15 Nisan in the 8th, 11th and 19th years of the 19year cycle is no longer on the first, but on the second full moon after the vernal equinox. If we compare the Jewish calendar with the Julian calendar, then the average duration of the year of the Jewish calendar is shorter than the Julian calendar by 0.00318 days. Therefore, the beginning of the year( 1 Tishri) in every 314.46 years moves in the Julian calendar for one day "back"( from September to August).
In the Hebrew calendar, there are several methods for calculating the current moments of the equinoxes and solstices. According to one of them  "Tõfaft Adda"  at the above average length of the calendar year, the beginning of the account( epoch) is attributed to Tuesday 6 pm on April 1( according to the Julian calendar) 3760 BC.e."At 0 o'clock Wednesday," when, according to the Bible, the luminaries were created. The era of the era of the Jewish calendar was chosen for the convenience of calculations. It turns out that it was accurate to two minutes coinciding with the moment of the spring equinox in 500 AD.e. On this basis, and concluded that it was in this year and was reformed Jewish calendar. But in connection with the inaccuracy in the adopted duration of the solar year, the Nisan flowing  the calculated beginning of spring  lagged behind the spring equinox in 1000 for 2 days 7 hours, by 2000 it will be almost 7 days after the astronomical beginning of spring.
And one more thing. According to Gauss formulas, 15 Nisan  the date of the spring full moon  is determined quite easily for any year n.e.or before it. Thus, it is much easier to solve some problems of chronology, when it is necessary to know the phase of the moon( for example, if it is necessary to verify the correctness of reporting the allegedly "then" solar or lunar eclipse).It should be remembered, however, that in fact the calendar 15 Nisan is not necessarily the true full moon: if, due to the abovementioned departures, 1 Tishri is transferred one or two days in advance, then 15 Nisan is automatically shifted.
In fact, the molad( new moon) of Tishri is separated from the Nisan moloda by( 6 * 29 days 12 hours 793 helek =) 177 days 4 hours 438 heleks. The full moon of Nisan falls on 14 days 18 hours 396.5 helek after the new moon. Therefore, the full moon of Nisan of the year of the Jewish era with the number A is remote from the astronomical new moon of the autumn month of Tishri( A + 1) th year by 162 days 10 hours 41.5 hlek. In the Jewish calendar from 15 Nisan to 1 Tishri there are 163 days. If, as a result of the calculation of the cycles, it was found that the Tishri prayer occurs on the X( hours, cheleks) of a certain day of the Julian calendar, then the spring astronomical full moon took place 163 days ago( 23 weeks and 2 days), but at the time X + 13 hours 1038,5 helek. In the case of deviation, if the calculations are carried out according to Gauss formulas, appropriate correction should be made.