9780899063003 - ArtScroll Mishnah Series Rosh Hashanah, published by Mesorah

Worldcat entry
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The names of the Jewish months in use today are not found in the Torah or in the early Prophets and K'suvim [Hagiographa]. Only eight of the twelve are mentioned in the later books...all of which were written after the exile to Babylon.

In fact, only seven of the current Jewish month names are mentioned in scripture:
1. Nisan (Esther & Ezra)
2. Sivan (Esther)
3. Elul (Ezra)
4. Kislev (12 Prophets)
5. Tevet (Esther)
6. Shevat (12 Prophets)
7. Adar (Esther & Ezra)
The author likely misunderstood Ezekiel 8.14 as a reference to the month-name Tammuz, but in fact the prophet is referring to the ancient Mesopotamian god and not the month.
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In earlier books of Scripture they are referred to only as the first month, etc.

This is incorrect-
1. The Torah calls the first month חדש האביב
2. Kings calls the second month ירח זו
3. Kings calls the seventh month ירח האתנים

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The יוֹם הַחוֹדֶשׁ הַשֵּׁנִי, second day of the chodesh, mentioned in I Samuel (20:27) refers to the second day of Rosh Chodesh

This phrase does not appear in the indicated verse which instead says ממחרת החדש השני.