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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 types of dates. To simplify the process for converting dates from one calendar to another, the “system of Julian days” or “continuous day count” was introduced. In 1583, the French scholar Joseph Justus Scaliger (1540 – 1609) came up with the idea of the so-called “Julian period”. He named this method of calculation in honor of his father Julius Caesar Scaliger (1484 – 1558), the famous humanist and scholar.
Joseph Scaliger suggested a chronological scale against which any historical date could be aligned. The starting point for counting the “Julian days” (JDN=0) was set to January 1, 4713 BNE, which was the era “from the foundation of the world” according to Scaliger. Then, the JDN value would increase by one every day. So, January 2, 4713 BNE equals to JDN=1 and so on. For example, January 1 of the 1st year NE is JDN=1721424.
In 1849, John Herschel (1792 – 1871) suggested expressing all the dates though the JD value, which is the number of days that passed since the beginning of the Scaliger cycle. The difference between the Julian date (JD) and the Julian day number (JDN) is that the former contains a fractional part which indicates the time of 24-hour day. It was agreed that the beginning of the Julian day would be noontime according to Greenwich Mean Time. So, the midnight of January 1 of the 1st year NE corresponds to JD=1721423.5. Note that the JD=1721424 will accumulate only by the noon of the specified day, because the count was started at noon January 1, 4713 BNE (the “zero point”). To make our calculations easier, we will use the rounded value of the Julian date or the Julian day number (JDN).
The procedure for calculating the Julian day number (JDN) for a specific Julian calendar date is as follows:
1) a= [(14-month) /12].
2) y=year+4800-a.
3) m=month+12a-3.
4) Julian day number:
JDN=day+ [(153m+2) /5] +365y+ [y/4] -32083.
Where “year” is the year of NE; “month” is the number of the month; “day” is the day of the month; value in brackets is the integer part.
Knowing the JDN, you can find the day of the week by calculating the remainder of the division of JDN by 7. Based on the remainder value, the days of the week are distributed as follows: 0 – Monday, 1 – Tuesday, 2 – Wednesday, 3 – Thursday, 4 – Friday, 5 – Saturday, 6 – Sunday.
For example, let us calculate the Julian day number for the Jewish Passover in 2016 (April 10 in the Julian calendar):
1) a=0.
2) y=6816.
3) m=1.
4) JDN=2457502.
Remainder of division (JDN mod 7) =5, therefore, it is Saturday.
Finding dates based on the Julian days
The method of calculation based on the Julian days can be useful, for example, for finding the date for Tisri 1. We don’t know the interval between Nisan 15 and Tisri 1 within one year. But the interval between Tisri 1 of the year to be found and Nisan 15 of the previous year is always 163 days because Nisan, Iyyar, Siwan, Tammuz, Ab, and Elul have an unchanging number of days. If you know the Julian day number for Nisan 15, you can, by adding 163, find the Julian day number for Tisri 1 of the following year.
For example, based on the Gauss formulas, Nisan 15 of the year 5775 in the Jewish calendar corresponds to March 22, 2015, in the Julian calendar. Now we can calculate JDN for this date (JDN =2457117). Consequently, Tisri 1 of the year 5776 corresponds to JDN=2457280. The remainder of division is 0. Therefore, the day is Monday. We have already determined that Nisan 15 of the year 5776 in the Jewish calendar (2016 NE) falls on Saturday. Using Table 3 (see above), we see that the year 5776 of the Jewish calendar is embolismic, that is, its duration is 385 days (an excessive year).
The procedure for converting a Julian day number (JDN) into a Julian calendar date is as follows:
1) c=JDN+32082.
2) d= [(4c+3) /1461].
3) e=c- [1461d/4].
4) m= [(5e+2) /153].
5) day=e- [(153m+2) /5] +1.
6) month=m+3—12* [m/10].
7) year=d-4800+ [m/10].
Where year is the year of NE; month is the number of the month; day is the day of the month; value in brackets is the integer part.
For example, let us convert the Julian day number JDN=2457280 (Tisri 1 of the year 5776 in the Jewish calendar) into a Julian calendar date:
1) c=2489362.
2) d=6815.
3) e=184.
4) m=6.
5) day=1.
6) month=9.
7) year=2015.
Consequently, Tisri 1 of the year 5776 in the Jewish calendar falls on September 1 2015 of the Julian calendar, or September 14 of the Gregorian calendar.
The Julian day and the Egyptian calendar
To convert a Jewish calendar date into a Julian calendar date, we should first find the the Julian day number:
JDN= (N-1) *365+ (M-1) *30+ (D-1) +1448638,
where “N” is the year of the Nabonassar era; “M” is the ordinal number of the Egyptian calendar month; “D” is the date of the month.
The calculation procedure of date is the same as the one described in the previous chapter.
Для For example, let us calculate the day of the Julian calendar corresponding to Pharmuthi 25 of the year 777 of the Nabonassar era.
Julian day number:
JDN= (777—1) *365+ (8—1) *30+ (25—1) +1448638=1732112.
Julian calendar date:
1) c=1764194.
2) d=4830.
3) e=37.
4) m=1.
5) day=7.
6) month=4.
7) year=30.
So, Pharmuthi 25 of the year 777 of the Nabonassar era falls on April 7th 30 NE.
Calculations simplified
Chronology calculations became so much easier now that we have computers. The above algorithms can be effectively implemented using popular computational programs or programming language scripts. Also, there are ready-to-use programs and online services designed specifically for calendar calculations; some of them are described in the Addendum.
Section 2. The Old Testament chronology
Speaking of the challenges of calculating dates in the era “from the foundation of the world”, or “from Adam” in Section 1, we didn’t mention one important reason for the lack of consensus regarding the biblical chronology. There are several versions of the Old Testament text (Jewish-Masoretic, Samaritan, Septuagint). For the most part, they are identical, but there are discrepancies which may affect chronology. Some passages contain significant chronological inconsistencies, so a natural question arises as to which dating is more accurate.
Eastern Orthodox Church regards the Septuagint as the canonical text of the Old Testament (The Septuagint comes from Latin “Interpretatio septuaginta seniorum” – “The translation of the seventy scholars”, for this reason, it is often referred to as LXX). The Septuagint is the translation of the Jewish Scripture into Greek completed at the initiative of Demetrius of Phalerum (350 – 283 BNE), the founder and head of the Library of Alexandria. Demetrius persuaded the Egyptian king Ptolemy II Philadelphus (308 – 245 BNE) to have the sacred books of the Jewish canon translated into Greek. Torah (the Pentateuch) was translated in the 3rd century BNE. The remaining Old Testament books were translated later – in the second and first centuries BNE. Some details regarding the process of translation for the Pentateuch are provided in the “Letter of Aristeas, the bodyguard of Philadelphus, to brother Philocrates”. This letter is cited, for example, by Philo of Alexandria (The Life of Moses, 2,6—7), by Josephus Flavius (Judean Antiquities, 12,1—2), in the Talmud (Megillah, 9), by Clement of Alexandria (Stromata, 1,22), by Irenaeus of Lyons (Against Heresies, 3,21,2), by Cyril of Jerusalem (Catechetical Lextures and Mystagogic Catecheses, 4.34), by Epiphanius of Cyprus (On the Seventy Interpreters), by Augustine the Blessed (The City of God, 18,42).
Below are listed some basic of the arguments in favor of the higher degree of accuracy of the Septuagint:
1) The translation of the Old Testament (Tanakh) into Greek, known as Septuagint, was completed in 3rd-1st centuries BNE and was based on the authentic Jewish text provided by the Jewish high priest. This is a sufficient warranty of the accuracy of the resulting text. One can argue about the nuances of meaning in the translated text, but it is hardly possible to assume that the dates and numbers could have been mistranslated.
2) The translators of the Septuagint were highly educated Jewish scribes. Those seventy two men continued to interact with each other as they were working on the Pentateuch (each Semitic tribe was represented by 6 scribes). One and the same passage was translated by different groups of scribes; then, the results were compared against each other. Thanks to this procedure, the probability of mistakes is very low.
3) The Septuagint was not created just for the Library of Alexandria. The Greek text was distributed far and wide. One copy was always at the disposal of the Jewish high priest:
“After he had arrived in Jerusalem, he [Egyptian king Ptolemy IV Philopator (c. 242 – 203 BNE)] offered sacrifice to the supreme God and made thank offerings and did what was fitting for the holy place. Then, upon entering the place and being impressed by its excellence and its beauty, he marveled at the good order of the temple, and conceived a desire to enter the sanctuary. When they said that this was not permitted, because not even members of their own nation were allowed to enter, not even all of the priests, but only the high priest who was pre-eminent over all – and he only once a year – the king was by no means persuaded. Even after the law had been read to him, he did not cease to maintain that he ought to enter, saying, ‘Even if those men are deprived of this honor, I ought not to be.’ And he inquired why, when he entered every other temple, no one there had stopped him” (3 Macc 1:9—12).
4) The Apostles and early Church Fathers quoted predominantly from the Greek text of the Old Testament. Luke, for example, follows the Septuagint (Gen 10:24; 11:12—13) when he gives his genealogy of Jesus by inserting Cainan (Lk 3:36) between Sala and Arphaxad. In the Masoretic text, Cainan is omitted.
5) From antiquity to the present day, the text of the Septuagint has been preserved almost intact, at least with regard to chronology.
6) The idea that the Jewish-Masoretic Tanakh is an infallible text of the original Old Testament is, obviously, erroneous. There’s enough evidence to the fact that the Jews had several versions of Tanakh with varying chronological data. For example, in the account from Adam through Noah, the Samaritan-Israelite Pentateuch is closer to the Jewish-Masoretic Tanakh, whereas in the partition from Arphaxad through Abraham it is closer to the chronology given in the Septuagint. This indicates an evolutionary accumulation of discrepancies over a long period of time.
7) Accumulation of errors in the Tanakh continued throughout the1st century NE. The Jewish historian Josephus Flavius in his famous book provides the chronological data in the Bible which contains further discrepancies (see Judean Antiquities).
8) The belief in the infallibility of the Jewish-Masoretic Tanakh faded away altogether after the ancient Jewish manuscripts of Qumran had been unearthed in 1947. These manuscripts reflect a whole range of chronological and textual traditions. Based on the paleographic data, external evidence, and the radiocarbon analysis, the main body of these manuscripts date between 250 BNE to 68 NE.
9) The finalization of chronology in the Jewish Tanakh occurred, most likely, around the 2nd century NE. This process must have been caused by historical circumstances, such as the destruction of the Jewish Temple, and the Roman invasion of Judea in 70 NE. Scattered throughout the world, the Jews must have been motivated to start thinking about preserving the uniformity of their religious texts. The fixed Jewish-Masoretic text of Tanakh was first translated into a foreign language in Syria at the end of the 2nd century NE. This translation was later called Peshitta. After some time, in the 4th century NE, the Jewish-Masoretic Tanakh was translated by into Latin by Jerome of Stridon; this translation was termed the Vulgate.
10) The fixed text of the Jewish-Masoretic Tanakh is not identical across various manuscripts and contains multiple discrepancies.
11) The much shorter chronology of the Jewish Tanakh contradicts the current data obtained through independent dating methods. For example, according to the Jewish tradition, the conquest of the Babylonian Empire by Cyrus II happened in 370 BNE (year 3390 from the foundation of the world in the Jewish calendar). But the scientific dating places this event in 539 BNE.
Based on the above considerations, it seems reasonable to use the dates and numbers of the Septuagint as the source for recreating the Old Testament chronology.
The brief research given below is not meant to demonstrate the whole range of the Old Testament datings. Its main purpose is to indicate the general duration of the described events. That’s why only key dates have been included. Let us first note that the period from the creation of the world to the beginning of the new era was 5550 years. So for the sake of convenience, the calculated dates are given in a twofold format: first, the dates from Adam, then the astronomical dates in NE (in parentheses; -5549 NE corresponds to 5550 BNE and so on).
1 (-5549). Creation of Adam and Eve
“So God created man in his own image, in the image of God created he him; male and female created he them… And the evening and the morning were the sixth day” (Gen 1:27, 31; compare Gen 2:7—25).
“And Adam called his wife’s name Eve; because she was the mother of all living” (Gen 3:20).
