Reference for Rosh Hashanah 1:1
אַרְבָּעָה רָאשֵׁי שָׁנִים הֵם. בְּאֶחָד בְּנִיסָן רֹאשׁ הַשָּׁנָה לַמְּלָכִים וְלָרְגָלִים. בְּאֶחָד בֶּאֱלוּל רֹאשׁ הַשָּׁנָה לְמַעְשַׂר בְּהֵמָה. רַבִּי אֶלְעָזָר וְרַבִּי שִׁמְעוֹן אוֹמְרִים, בְּאֶחָד בְּתִשְׁרֵי. בְּאֶחָד בְּתִשְׁרֵי רֹאשׁ הַשָּׁנָה לַשָּׁנִים וְלַשְּׁמִטִּין וְלַיּוֹבְלוֹת, לַנְּטִיעָה וְלַיְרָקוֹת. בְּאֶחָד בִּשְׁבָט, רֹאשׁ הַשָּׁנָה לָאִילָן, כְּדִבְרֵי בֵית שַׁמַּאי. בֵּית הִלֵּל אוֹמְרִים, בַּחֲמִשָּׁה עָשָׂר בּוֹ:
There are four New Years: The first of Nissan is the New Year for kings and for festivals. [The kings of Israel count from Nissan. If a king began to reign in Shevat or Adar, when Nissan arrives, that year ends and they begin to count a second year for him. This is derived from (I Kings 6:1): "And it came to pass in the four hundred and eightieth year after the children of Israel went out of the land of Egypt, in the fourth year, in the month of Ziv, which is the second month, of the reign of Solomon over Israel, etc." The reign of Solomon is being likened to the exodus from Egypt. Just as the exodus from Egypt is counted from Nissan, so the reign of Solomon is counted from Nissan. And the reign of the gentile kings is counted from Tishrei. As stated below: "On the first of Tishrei is the New Year for years"; that is, for the years of the gentile kings." A (practical) difference: They used to indicate the date on bills (of payment) according to the years of the king's reign, for the sake of shalom malchuth ("the peace of the kingdom"). And if a king began reigning on Adar, and a scribe wrote (on a bill) in Nissan thirteen months later: "Nissan of the second year," since Nissan is the New Year for kings, this is a shtar mukdam ("a pre-dated bill") and is invalid; for that Nissan was the Nissan of the third year. And pre-dated bills are invalid; for they enable the lender to wrongfully seize property bought from the borrower between the date on the bill and the time of the loan, the sale having preceded the loan. He (the lender) was, therefore, penalized to lose out even from the (real) time of the loan by not being allowed to seize bound property (from that time). ("the first of Nissan is the New Year for festivals":) What is meant is that its festival, Pesach, is the New Year for festivals in respect to vows. If three festivals pass from the time one vows and he does not fulfill his vow, he is in transgression of (Deuteronomy 23:22): "You shall not delay" (to fulfill it). And R. Shimon says that he does not transgress unless three festivals pass in order, Pesach being first, viz. (Ibid. 16:16): "On the festival of matzoth, and the festival of Shavuoth, and the festival of Succoth." The halachah: If one festival alone passed and he did not fulfill his vow, he transgresses a positive commandment, viz. (Ibid. 12:5-6): "And you shall come there, and you shall bring there your burnt-offerings, etc." — On the first festival that you come, you must bring. And if three festivals pass, he is in transgression of delaying (his vow). (The halachah is not in accordance with R. Shimon, who requires three festivals in order.)] The first of Elul is the New Year for the tithing of beasts. [i.e., The tithe is not taken from animals born in one year for animals born in another, it being written (Deuteronomy 14:22): "Tithe shall you tithe." Scripture states: "year by year" — and not from this year for another.] R. Elazar and R. Shimon say: The first of Tishrei. [Since the beast tithe is likened to the grain tithe, just as the new year for the grain tithe is Tishrei, so the new year for the beast tithe is Tishrei. The halachah is not in accordance with R. Elazar and R. Shimon, who say the first of Tishrei. And the first tanna holds that since the beast tithe is likened to the grain tithe, it being written: "Tithe shall you tithe" — Just as grain is tithed close to its consummation (its new year being Tishrei, following its consummation; for all of the summer it remains on the threshing floor for drying) — so with the beast tithe, consummation is followed by tithing, on the first of Elul. For (most) animals are born in Av, the end of their period of gestation, viz. (Psalms 65:14): "Lavshu karim hatzon, and the valleys are covered with corn." When (is it that) "lavshu karim hatzon"? (i.e., When is it) that the ewes (karim) are "clothed" (mithlabshoth) by the rams (hatzon), which mount and impregnate them? When the valleys are covered with corn; that is, in Adar, when the sprouting vegetation is clearly visible. And they (the lambs) are born in Av, the gestation period of a small beast being five months. Therefore, the new year for the beast tithe is the first of Elul, close to its (the animal's) consummation.] The first of Tishrei is the New Year for years, [the years of the reign of the gentile kings, as explained above], for shemitah [When Tishrei enters, it is forbidden to plow and to sow by Torah law], for the Jubilee year, for planting [If he planted a tree forty-five days before the first of Tishrei (which comprise two weeks for "taking," the planting "taking" in two weeks, and thirty days after planting to make a year, thirty days in a year being considered a year — thus, forty-five days), when the first of Tishrei arrives, it is counted as (the beginning of) the second year of the years of arlah, and the Tishrei of the next year is counted as (the beginning of) the third year. And after the third year, when the first of Tishrei of the fourth year arrives, if fruits were formed from this planting, they are still forbidden because of arlah. For even though Tishrei is the New Year for panting, the fifteenth of Shevat is the New Year for trees; and this has already become a tree. Therefore, its year is not renewed vis-à-vis leaving the status of arlah until the fifteenth of Shevat. From that point on, if it forms fruits, they are in the status of revai (the fourth year) to be eaten in Jerusalem. And all fruits that are formed on it from the fifteenth of Shevat of the next year and on are not in the status of revai.], and (Tishrei is the New Year) for greens [i.e., for tithing greens. The tithe is not taken from greens picked before Rosh Hashanah for greens picked after Rosh Hashanah.] The first of Shevat is the New Year for trees according to Beth Shammai [vis-à-vis the tithing of fruits. The tithe is not taken from fruits of a tree, which were formed before Shevat for those which were formed after Shevat. For in a tree, (fruit) formation is the criterion. Or, there is a practical difference vis-à-vis the third year of shemitah, when the poor tithe obtains. Those fruits which were formed from Rosh Hashanah of the third year are regarded as fruits of the second year that has passed, and ma'aser rishon and ma'aser sheni obtain with them; and (with those fruits which were formed) from Shevat on, ma'aser rishon and the poor tithe obtain.] Beth Hillel say: (The New Year for trees is) the fifteenth (of Shevat).