(Our Mishnah): The year for Ma'aser Behemah begins on the first of Elul (R. Meir) or on the first of Tishrei (R. Elazar & R. Shimon).

Explanation #1 (R. Yochanan): Both opinions are derived from a single verse:

"The meadows are clothed with sheep, and the valleys are covered with corn, they shout with joy, they also sing" (Tehillim 65,14).

(R. Meir): The meadows are clothed with sheep (i.e. they conceive) at the time when the valleys are covered with corn, which is Adar. The sheep then give birth in Av, so the Rosh Hashanah is the first of Elul.

(R. Elazar & R. Shimon): The meadows are clothed with sheep (i.e. they conceive) at the time when the crops shout with joy and also sing (i.e. the ripe grains rustle in the stalks), which is Nisan. The sheep then give birth in Elul, so the Rosh Hashanah is the first of Tishrei.

The Gemara discusses difficulties with these opinions:

Question: What does R. Meir do with the phrase "they shout with joy, they also sing"?

Answer: It refers to the minority of sheep that conceive late, in Nisan.

Question: What do R. Elazar & R. Shimon do with the phrase "the valleys are covered with corn"?

Answer: It refers to the minority of sheep that conceive early, in Adar.

Question: R. Meir's explanation works well, as the verse lists the majority of sheep first; but according to R. Elazar & R. Shimon, the verse should have said "they shout with joy, they also sing" (which refers to the majority of sheep) before "the valleys are covered with corn"!?

Explanation #2 (Rava): All agree that the sheep conceive in Adar; the dispute derives from a different verse:

"Aser T'aser"(Devarim 14,22) refers to both Ma'aser Behemah and Ma'aser Dagan (comparing the two by way of a Hekesh):

(R. Meir): Just as Ma'aser is taken from Dagan close to the end of the process (in Tishrei, when the grain is gathered), so too Ma'aser is taken from Behemos close to the end of the process (when the sheep give birth, in Elul).

(R. Elazar and R. Shimon): Just as Ma'aser Dagan has its Rosh Hashanah in Tishrei, so too Ma'aser Behemah has its Rosh Hashanah in Tishrei.

Question: For which Halachos do we need to be told that the new year for counting years is on the first of Tishrei?

Answer #1 (R. Papa): In order to discern that a contract is pre-dated (and thus invalid).

Question: Surely we learned that the first of Nisan was the Rosh Hashanah for kings for the same Halachah?

Answer: For Jewish kings, the first of Nisan is the Rosh Hashanah; for non-Jewish kings, the first of Tishrei is the Rosh Hashanah.

Question: R. Chisda (3a) stated that the first of Nisan was only the Rosh Hashanah for Jewish kings, but the year for non-Jewish kings begins in Tishrei; was he coming to tell us that which had already been stated in a Mishnah itself?

Answer #1: R. Chisda was coming to teach us how it is derived from the Torah.

Answer #2: R. Chisda understands that the purpose of Tishrei being the Rosh Hashanah for years is as stated by R. Zeira:

(R. Zeira): The first of Tishrei is the new year for calculating the cycles of the sun and moon.

This follows R. Elazar's view that the world was created in Tishrei.

Answer #3 (R. Nachman bar Yitzchak): The purpose of the Rosh Hashanah for years is to begin the process of judgment - at the beginning of the year, the fate of the end of the year is determined.

Question: How do we know that this occurs in Tishrei?

Answer: The Torah states "Blow the Shofar at the new month, at the appointed time (Bakeseh) on our festival day" - the festival on which the moon is Keseh, covered (i.e. small and invisible from afar) is Rosh Hashanah. The next Pasuk states that this is the time of judgment.(Tehilim 81,4-5)

(Beraisa): The Pasuk states that "It is a law for Yisrael" and then it states that "It is a judgment for the God of Yakov" - this teaches that the Heavenly Court sits in judgment only after Beis Din have sanctified the new month.(Tehilim 81,5)

(Another Beraisa): "It is a law for Yisrael" only teaches that there is judgment for Jews; "It is a judgment for the God of Yakov" teaches that there is also judgment for all nations.

Question: Why, then, does it need to say "It is a law for Yisrael" (seeing as Jews would anyway be included in the second phrase)?

Answer: To teach that the Jews enter judgment first, in accordance with R. Chisda's teaching;

(R. Chisda): A king is judged before the community.

Question: Why is this so?

Answer #1: It isn't respectful for him to have to wait in line.

Answer #2: He should be judged before the judge grows too angry.

Question: How do we know that the new year for Shemitah is on the first of Tishrei?

Answer: Shemitah is called "the seventh year" (Vayikra 25,4) and we learn (by way of Gezerah Shavah) from the occurrence of the word "year" in the Pasuk "from the beginning of the year" (Devarim 11,12) in reference to Tishrei.

Question: Why not learn from the occurrence of the word "year" in the Pasuk "It is the first for you for the months of the year" in reference to Nisan? (Shemos 12,2)

Answer: We should learn the occurrence of the word "year" without any mention of months from the same type of "year", rather than learn it from the occurrence of the word "year"which is mentioned along with months.

Question: Surely the new year for Yovel is on the tenth of Tishrei, not the first?

Answer: It follows the view of R. Yishmael ben R. Yochanan ben Beroka in the following Beraisa:

Question: Why does the Torah state that "the fiftieth year should be consecrated" (Vayikra 25:10) (seeing as we have already been told that Yovel follows the forty-ninth year)?

Answer: To teach that the whole year is sanctified, from the first of Tishrei, and not from Yom Kipur, when the Shofar is blown.

(R. Yishmael ben R. Yochanan ben Beroka): Therefore, although servants weren't released during the first ten days of Tishrei, they also weren't actively enslaved to their masters; instead they would eat, drink and be merry.

When Yom Kipur arrived, the servants would be released to their homes, and the fields would return to their original owners.

The opposing Rabanan use the word "year" (of the previously mentioned verse) to teach that only years (of Yovel) are consecrated with a proclamation, and not new months.

(Beraisa) Question: What do the words "It is Yovel" teach us?(Vayikra 25,11)

Answer: From the words "the fiftieth year should be consecrated" one might think that just as the beginning of Yovel starts from the first of Tishrei (even before the tenth of Tishrei), so too it should extend to the tenth of Tishrei of the fifty-first year.

The extension beyond the duration of a year is not unusual, as we attach non-holy to holy time-periods in order to lengthen them.

Therefore the Torah states that "It is Yovel"-it, and not the fifty-first year.

The Rabanan (who did not learn that Yovel begins from the first of Tishrei, and therefore do not need "It is Yovel" to teach that it shouldn't be extended into the next year), use "It is Yovel" to teach that the fiftieth year is counted as Yovel alone, and not as both Yovel and as the first year of the next cycle.

This goes against the opinion of R. Yehudah, who holds that it is also counted as the first year of the next fifty-year cycle.