1) WHAT TYPE OF ROSH HASHANAH IS MENTIONED IN THE MISHNAH
QUESTION: The Gemara (7b) asks why the Mishnah counts only four different days of Rosh Hashanah. The Mishnah should have included an additional two days -- the sixteenth of Nisan (Rosh Hashanah for the Omer and for eating from newly-grown grain) and the sixth of Sivan (Rosh Hashanah for the Shtei ha'Lechem and for bringing Bikurim to the Beis ha'Mikdash). Rav Ashi answers that the Mishnah counts only days of Rosh Hashanah which occur on Rosh Chodesh, and those two days do not occur on Rosh Chodesh. Indeed, when the Mishnah says that there are four days of Rosh Hashanah but it lists five, it is because the Rosh Hashanah for Regalim (the fifteenth of Nisan) does not occur on Rosh Chodesh and thus is not included in the number.
Why does the Mishnah mention the Rosh Hashanah of Regalim altogether? Once it includes a Rosh Hashanah that does not occur on Rosh Chodesh (even though it is not included in the Mishnah's number of days of Rosh Hashanah), it should also include the Rosh Hashanah of the Omer and Shtei ha'Lechem (and just omit them from the number). (PNEI YEHOSHUA)
ANSWER: Perhaps the Mishnah includes the Rosh Hashanah of Regalim because it occurs in Nisan, and Nisan is already mentioned in the Mishnah (Rosh Chodesh Nisan, the Rosh Hashanah for Melachim). In contrast, the Rosh Hashanah of Shtei ha'Lechem occurs in a month which does not have any Rosh Hashanah on Rosh Chodesh. Therefore, the Mishnah does not mention the Rosh Hashanah of Shtei ha'Lechem. Once the Mishnah omits Shtei ha'Lechem it also omits the Rosh Hashanah of Omer (even though it occurs in Nisan), since the nature of that day is very similar to the nature of the Rosh Hashanah of Shtei ha'Lechem and it would look strange for the Mishnah to list one without the other. (M. KORNFELD)
2) ROSH HASHANAH FOR "YEARS"
QUESTION: The Mishnah (2a) states that the first of Tishrei is the Rosh Hashanah for Shanim, years. The Gemara explains in the name of Rav Papa that the first of Tishrei marks the beginning of the year for the reign of a king (in order to write the correct date in a contract). This refers to non-Jewish kings, because the Mishnah says that the years of Jewish kings are counted from the first of Nisan.
The Gemara quotes an independent statement of Rav Chisda who says that the years of non-Jewish kings are counted from Tishrei. In its attempt to explain why Rav Chisda teaches something that is already taught in the Mishnah, the Gemara says that Rav Chisda does not agree with Rav Papa (that the Mishnah means to teach the date for the Rosh Hashanah of non-Jewish kings). Rather, Rav Chisda follows the view of Rebbi Zeira who maintains that when the Mishnah says that the first of Tishrei is the Rosh Hashanah for years, it refers to the count of the "Tekufah" (Tishrei is the month during which the world was created).
Why does the Gemara assume that Rav Chisda follows the view of Rebbi Zeira? The Gemara quotes Rav Nachman bar Yitzchak who also argues with Rav Papa and says that the first of Tishrei is the Rosh Hashanah for Din, judgment, and it is the time when the world is judged. Perhaps Rav Chisda interprets the Mishnah like Rav Nachman bar Yitzchak and not like Rebbi Zeira.
ANSWER: The PNEI YEHOSHUA answers this question based on another statement of Rav Chisda. In the Gemara later (8b), Rav Chisda derives from the verse, "La'asos Mishpat Avdo..." (Melachim I 8:59), that a king is judged before the rest of the people. While this verse does not specifically mention judgment, Rav Chisda apparently relies on Rebbi Yosi's interpretation of the verse. The Gemara later (16a) quotes a Beraisa in which Rebbi Yosi states that a person is judged every day and not just on the first of Tishrei. Rav Chisda explains that the source for Rebbi Yosi's opinion is the phrase at the end of the same verse, "Devar Yom b'Yomo." This is why Rav Chisda understands that the verse refers to judgment.
Since Rav Chisda agrees with Rebbi Yosi that a person is judged every day, he must argue with Rav Nachman who explains the Mishnah here to mean that a person is judged only on Rosh Hashanah (the first of Tishrei). It must be that Rav Chisda follows the view of Rebbi Zeira with regard to the Rosh Hashanah for years. (See also KUNTRUS SHESH SHE'ARIM by RAV YAAKOV D. HOMNICK, who suggests another approach.)