The missing years in the Hebrew calendar refer to a chronological discrepancy between Talmudic chronologists for the destruction of the First Temple in 423 BCE (3338 AM) and the modern secular dating for it in 587 BCE.
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Destruction of the First Temple
Both the Babylonian Chronicles and the biblical Chronicles indicate that Nebuchadnezzar captured Jerusalem, but secular scholars have attempted to set a year when the event took place. The Babylonian Chronicles, which were published by Donald Wiseman in 1956, establish that Nebuchadnezzar captured Jerusalem the first time on 2 Adar (16 March) 597 BCE. The Chronicles, with the names of Jewish kings, is derived from British Museum - Cuneiform tablet with part of the Babylonian Chronicle (605-594 BC), (See also Chronicle Concerning the Early Years of Nebuchadnezzar II Reverse, lines 11' - 13')
In the seventh year (of Nebuchadnezzar, 599 BCE) in the month Chislev (Nov/Dec) the king of Babylon assembled his army, and after he had invaded the land of Hatti (northern Syria and southern Anatolia) he laid siege to the city of Judah. On the second day of the month of Adar (16 March) he conquered the city and took the king (Jeconiah) prisoner. He installed in his place a king (Zedekiah) of his own choice, and after he had received rich tribute, he sent forth to Babylon.
2 Chronicles 36:6-10 says:
6 Against him (Jehoiakim) came up Nebuchadnezzar king of Babylon, and bound him in fetters, to carry him to Babylon.
7 Nebuchadnezzar also carried of the vessels of the house of the LORD to Babylon, and put them in his temple at Babylon. 8 ... and Jehoiachin his son reigned in his stead. 9 Jehoiachin was eight years old when he began to reign; and he reigned three months and ten days in Jerusalem...
Before Wiseman's publication, E. R. Thiele had determined from the biblical texts that Nebuchadnezzar's initial capture of Jerusalem occurred in the spring of 597 BCE, while other scholars, including William F. Albright, more frequently dated the event to 598 BCE.
From the date of the first siege of Jerusalem in about 597 BCE to the date of the destruction of the First Temple requires resort to biblical sources.
2 Chronicles 36:11 says:
11 Zedekiah was twenty and one years old when he began to reign; and he reigned eleven years in Jerusalem
There has been some debate as to when the second siege of Jerusalem took place. Though there is no dispute that Jerusalem fell the second time in the summer month of Tammuz (Jeremiah 52:6), Albright dates the end of Zedekiah's reign (and the fall of Jerusalem) to 587 BCE, whereas Thiele offers 586 BCE.
Thiele's reckoning is based on the presentation of Zedekiah's reign on an accession basis, which was used for most but not all of the kings of Judah. In that case, the year that Zedekiah came to the throne would be his first partial year; his first full year would be 597/596 BCE, and his eleventh year, the year Jerusalem fell, would be 587/586 BCE. Since Judah's regnal years were counted from Tishri in autumn, this would place the end of his reign and the capture of Jerusalem in the summer of 586 BCE.
Two-year difference within the Hebrew calendar
The 2nd century CE rabbinic work Seder Olam Rabbah, which formed the basis of the era counting of the Hebrew calendar, interpreted the prophecy of seventy weeks in Daniel 9:24–27 as referring to a period of 490 years, with a "week" being interpreted as a period of seven years, which would pass between the destruction of the First and Second Temple. This is used to date the destruction of the First Temple to 423 BCE (3338 AM) – about 165 years after the current scholarly dating of the event. The discrepancy between these two dates is referred to as "missing years".
Today, Hebrew dating places the creation of the world near the end of "Year One" AM and afterwards the first year of Adam's life as "Year Two" AM. However, Seder Olam Rabba shows that the Hebrew dating originally counted the first year of Adam's life as "Year Zero" AM. This may mean that the Hebrew dating has shifted in the course of history such that traditional dating of ancient events appears two years earlier than the modern Hebrew dating would be (Edgar Frank, Talmudic and Rabbinic Chronology, 1956).
Rabbinic tradition says that the First Temple was destroyed in "year 3338" AM and the Second Temple in "year 3828" AM. If there was no calendar shift, the Common Era equivalents would be 423 BCE and 68 CE, respectively. If there was a calendar shift, the destructions would have taken place in our years 3339 and 3829 AM, or in 3340 and 3830 AM, and the Common Era equivalents would be 422 BCE and 69 CE, respectively, or 421 BCE and 70 CE.
If there was no calendar shift, the length of the missing-years period would be 163 years (586 minus 423). If there was a calendar shift, the length of the missing-years period would be 164 or 165 years.
The missing years and Daniel
A popular explanation for the missing years suggests that the Jewish sages interpreted the prophecy in Daniel 9:24–27 as meaning that there would be 490 years from the destruction of the First Temple to the destruction of the Second Temple and, working backwards from the destruction of the Second Temple (in 3828 AM), wrongly dated the destruction of the First Temple (in 3338 AM).
A variation on this argument states that the Jews deliberately altered the dating so that the true date of the "anointed one" (Mashiah) mentioned in Daniel 9:25 would be hidden. Other apologists have countered with claims that the dating was indeed altered for one or another reason and should be understood as fable, not history.
These explanations come from the ambiguous meaning of the word 'week' in Hebrew, which means 'a heptad', or a group of seven. The Hebrew word for 'week' is used to refer to periods of seven days as well as seven years. The understanding of this number as referring to 490 years can also be found in Seder Olam. Christians also interpreted these verses as years and connect them to Jesus, although Rashi's interpretation is such that it upholds the tradition that the anointed one in question is the Persian king Cyrus.
Mistakes in the Hebrew or secular dating
If traditional dates are assumed to be based on the standard Hebrew calendar, then the differing traditional and modern secular dating of events cannot both be correct. Attempts to reconcile the two systems must show one or both to have errors.
Missing reign lengths in the Hebrew dating
Secular scholars see the discrepancy between the traditional and secular date of the destruction of the First Temple arising as a result of Jewish sages missing out the reign lengths of several Persian kings during the Persian Empire's rule over Israel. Modern secular scholars tally ten Persian kings whose combined reigns total 208 years. By contrast, ancient Jewish sages only mention four Persian kings totaling 52 years. The reigns of several Persian kings appear to be missing from the traditional calculations.
The conclusion one may easily draw from these facts is that the Old Testament merely focuses on chronicling certain Persian kings rather than others. "Focus" of the text being the operative word. Historians have long recognized that many texts, which accurately record history, often do so with a certain focus in mind. This would mean that the author did not find the missing Persian kings to be noteworthy—hence their absence from the text.
Missing years in Jewish tradition
R' Azariah dei Rossi, in Me'or Einayim (c. 1573), was likely the first Jewish authority to claim that the traditional Hebrew dating is not historically precise regarding the years before the Second Temple.
R' Nachman Krochmal in Guide to the perplexed of our times (Hebrew, 1851) points to the Greek name Antigonos mentioned in the beginning of Avot as proof that there must have been a longer period to account for this sign of Hellenic influence. He posits that certain books of the Bible such as Kohelet and Isaiah were written or redacted during this period.
R' David Zvi Hoffman (1843–1921) points out that the Mishna in Avot (1:4) in describing the chain of tradition uses the plural "accepted from them" even though the previous Mishna only mentions one person. He posits that there must have been another Mishna mentioning two sages that was later removed.
It has been noted that the traditional account of Jewish history shows a discontinuity in the beginning of the 35th century: The account of Seder Olam Rabbah is complete only until this time. It has been postulated that this work was written to complement another historical work, about subsequent centuries until the time of Hadrian, which is no longer extant.
It appears that Jewish dating systems only arose in the 35th century, so that precise historical records would naturally have existed only from that time onwards. The Minyan Shtarot system, used to date official Jewish documents, started in the year 3449. According to Lerman's thesis, the year-count "from Creation" was established around the same time (see Birkat Hachama).
It has also been posited that certain calculations in the Talmud compute better according to the secular dating. Two possible harmonizations are proposed by modern rabbis:
Critiques of secular dating
Attempts have been made to reinterpret the historical evidence to agree with the Rabbinic tradition (see Excursus: Rabbinic Tradition, below), however this approach to the discrepancy is problematic. The reinterpretation of the Greek, Babylonian and Persian sources that is required to support the traditional dating has been achieved only in parts and has not yet been achieved in its entirety. Similar problems face other attempts to revise secular dating (such as those of Peter James and David Rohl) and mainstream scholarship rejects such approaches. Where and how the Gregorian or Julian calendric differential gets factored in, remains another argument entirely.