in a month. The standard year had 12 months. But since the tropical (solar) year consists of 12.36826 synodic (lunar) months, the inaccuracy added up over time and had to be eliminated by adding an extra month. Whether there was a need for the extra month was determined based on the conditions of the grain crops and the age of the sacrificial animals – after all, the Passover law had to be observed: “In the fourteenth day of the first month [Nisan (Aviv)] at even is the LORD’S passover… Ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the LORD, to be accepted for you: on the morrow after the sabbath the priest shall wave it. And ye shall offer that day when ye wave the sheaf an he lamb without blemish of the first year for a burnt offering unto the LORD” (Lev 23:5, 10—12). That is why the year had to be extended by one month if the grains were not ripe, and the lambs too young. The extended 13-month year was usually called embolismic.
Around 500 NE, the Jewish calendar was reformed. The beginning of the year was moved to the month of Tisri, and Molad became the starting point for counting months (the astronomic new moon). Also, specific rules were introduced for alternating the number of days in a month and adding an extra month, which facilitated the process for calculating dates.
The reference point of the Jewish calendar was moved to the foundation of the world, Tisri 1 (“the new moon of creation”), which corresponded to October 6/7, 3761 BNE in the Julian calendar.
The principles of the Jewish calendar
In the Jewish calendar, the 13th month was inserted according to the 19-year cycle: specifically, for years 3, 6, 8, 11, 14, 17 and 19. The extra month was added before Adar and was called Adar 1 (Adar Rishon). Adar, then, became the next month and was named Adar 2 (Adar Sheni, Adar Bet, Beadar). All the religious feasts of the month of Adar were transferred to this month.
The number of days in a year varied from 353 to 385. There were six variants:
a) short, or insufficient year (hasarin) had 353 days (standard) or 383 days (embolismic);
b) the proper, or full year (kesedran) had 354 days (standard) or 384 days (embolismic);
c) the excessive year (shalamim) had 355 days (standard) or 385 days (embolismic).
The rationale behind such a complex system is the desire of the Jews to observe all the Talmudic religious traditions. It is only possible to fulfill the several hundreds of Talmudic prescriptions if the 1st of Tisri (the beginning of the new year) falls on Monday, Tuesday, Thursday or Saturday, and the 15th of Nissan (the Jewish Passover) falls on Tuesday, Thursday, Saturday or Sunday.
So, in any given year, if you know which days of the week fall on the 1st of Tisri and the 15th of Nissan, you can understand whether the year is insufficient, proper or excessive, that is, you will know how many days it consists of.
One other percularity of the Jewish calendar is that 24-hour days are counted from sunset, not from midnight. This was believed to be the pattern of creation in the corresponding Genesis account: “And the evening and the morning were the first day” (Gen 1:5).
The formulas of Gauss
Around the year 1800, the German mathematician Carl Friedrich Gauss (1777 – 1855) introduced formulas for calculating the dates for Christian Easter and the Jewish Passover. These formulas made chonological calculations much easier.
The procedure for calculating the date of the Jewish Passover (Nissan 15) is as follows:
1) A=R+3660,
where “A” is the year according to the Jewish calendar; “R” is the year NE.
2) a= (12A+17) mod 19,
where “mod 19” is the remainder of the division by 19.
3) b=A mod 4.
4) M+m= (32,0440933+1,5542418a+0,25b-0,00317779A),
where “M” is the integer part, and “m” is the fractional part.
5) c= (M+3A+5b+5) mod 7.
So, three variants are possible:
1) is c=1, a> b and m≥0.63287037, then the Jewish Passover (Nissan 15) falls on M+2 March in the Julian calendar.
2) if c=2, 4 or 6, and also when c=0, a> 11 and m≥0,89772376, then the Jewish Passover falls on M+1 March.
3) in all other cases it falls on M in March.
If the resulting value is greater than the number of days in March, you should subtract 31. The result will correspond to a date in April.
For example, let us calculate the date for the Jewish Passover in 2016:
1) A=5776.
2) a=17.
3) b=0.
4) M+m=40,11128886.
M=40, m=0,11128886.
5) c=6.
Since c=6, you should add 1 to M=40 and subtract 31. So, in 2016, the Jewish Passover (Nissan 15) falls on April 10 in the Julian calendar, which corresponds to April 23 in the Gregorian calendar. As noted above, the 24-hour day in the Jewish calendar begins at sunset. That is why the 15th of Nissan in this case begins in the evening of April 9 and ends in the evening of April 10.
The procedure for calculating the date of Christian Easter using the Gauss formula is as follows:
1) a=R mod 19,
where “R” is the year NE; “mod 19” is the remainder after the division by 19.
2) b=R mod 4.
3) c=R mod 7.
4) d= (19a+15) mod 30.
5) e= (2b+4c+6d+6) mod 7.
Three variants are possible:
1) if the sum of (d+e) does not exceed 9, then Christian Easter falls on March (22+d+e).
2) is (d+e) ≥10, then Easter falls on April (d+e-9).
For example, let us calculate the date for Easter in 2016:
1) a=2.
2) b=0.
3) c=0.
4) d=23.
5) e=4.
Since (d+e) exceeds 9, let us calculate (d+e-9) =18. So, in 2016, Orthodox Christian Easter falls on April 18 of the Julian calendar, which corresponds to May 1 in the Gregorian calendar.
The Julian period and calendar
Historians and annalists often deal with calendar calculations which involve various
