Rif Rosh HaShana 1a {Rosh HaShanah 2a; 4a}

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Note: I think the video component will go on hiatus for the duration of masechet Rosh haShana, as I need to focus on some personal obligations.

{Rosh HaShanah 2a}
Mishna:
There are four New Year days, viz.: The first of Nissan is New Year for (the ascension of) Kings and for (the regular rotation of) festivals {regalim}; the first of Elul is New Year for the cattle-tithe, but according to Rabbi Eliezer and Rabbi Shimon, it is on the first of Tishri.

{Rosh HaShanah 4a}
Gemara:
Do then the festivals commence {=Pesach} on the first of Nissan? Do they not begin on the fifteenth of Nissan?
Rav Chisda said: The festival which is in it {Nissan} is the beginning of the festivals.
What halachic difference does this make? To one who vows - that he must fulfill it so as not to violate bal te`acher {not delaying}

{That is, Devarim 23:22:

כב כִּי-תִדֹּר נֶדֶר לַה' אֱלֹקֶיךָ, לֹא תְאַחֵר לְשַׁלְּמוֹ: כִּי-דָרֹשׁ יִדְרְשֶׁנּוּ ה' אֱלֹקֶיךָ, מֵעִמָּךְ, וְהָיָה בְךָ, חֵטְא.

22 When thou shalt vow a vow unto the LORD thy God, thou shalt not be slack to pay it; for the LORD thy God will surely require it of thee; and it will be sin in thee.

}

For they learnt {in a brayta}: Both one who vows and one who consecrates and one who evaluates {=promises to donate the evaluation of X}, once three festivals pass over him, he violates bal te`acher. Rabbi Shimon says: Three festivals in their order, and the festival of matzot is the first. And so would Rabbi Shimon say: Festivals, sometimes three, sometimes five, sometimes four. How so? If he vowed before Pesach there are three. {Namely, the Pesach that is about to occur, then Shavuot, then Succot.} Before Shavuot, there are five. {Namely, Shavuot and Succot do not count the first time, since there was no Pesach leading them off, and then Pesach, Shavuot, and Succot.} Before Succot, there are four. {Namely, Succot does not count since there was no Pesach leading off, but then we have Pesach, Shavuot, and Succot.}

The Sages learnt {in a brayta}:
Devarim 23:24:

כד מוֹצָא שְׂפָתֶיךָ, תִּשְׁמֹר וְעָשִׂיתָ: כַּאֲשֶׁר נָדַרְתָּ לַה אֱלֹהֶיךָ, נְדָבָה, אֲשֶׁר דִּבַּרְתָּ, בְּפִיךָ. {ס}

24 That which is gone out of thy lips thou shalt observe and do; according as thou hast vowed freely unto the LORD thy God, even that which thou hast promised with thy mouth. {S}

מוֹצָא שְׂפָתֶיךָ - this is a positive commandment.
תִּשְׁמֹר - this is a negative commandment.