Rashi's opinion: Rashi explains that Shemini Atzeres is called an independent Regel because we do not sit in the Sukah. We say that it has an independent Korban, because it is unlike the order of bulls offered during Sukos. It is an independent Berachah, for on it we say "ba'Yom Shemini Chag ha'Atzeres ha'Zeh."

Question: The Tosefta connotes that "Berachah" refers to blessing the king - "ba'Yom Shemini Shalach Es ha'Am va'Yvarchu Es ha'Melech." If so (also here, Berachah is blessing the king) our Sugya is unlike Sukah 47a, which says "just like the seven days of Sukos require Korban, Shir, Berachah and Linah, also Shemini Atzeres requires these." The Gemara concludes that Berachah is [mentioning the festival in] Birkas ha'Mazon and Tefilah!

Answer: This is not difficult. However you will explain Berachah, according to the Gemara's Hava Amina there that Berachah is Zman (Shehecheyanu), you must say that it is unlike here, for here we count Berachah and Zman separately! (Therefore, it is not difficult if also in the conclusion Berachah is unlike here.)

R. Tam's opinion: R. Tam explains that Berachah refers to ("ba'Yom Shemini Chag ha'Atzeres ha'Zeh" in) Birkas ha'Mazon and Tefilah, like in Sukah. Regel refers to Linah, that one must lodge the night after the last Yom Tov [in Yerushalayim, just like] the seven days of Sukos require Linah. We learn from Pesach, about which it says "u'Fanisa ba'Boker v'Halachta l'Ohalecha";

V'Halachta l'Ohalecha must refer to Chol ha'Mo'ed [and not leaving on the morning of Yom Tov], for one must enter the Azarah on Yom Tov.

Question: There is no [Chiyuv of] Linah for all seven days!

Answer: (Indeed, it applies all seven days.) If one did not bring Chagigah on the first day, and brought it on one of the other six days, he must lodge [in Yerushalayim] the following night.

Question: (Why do we need a Hekesh to obligate Linah during Sukos?) In any case Linah is required in order to fulfill Shlamei Simchah (eating Kodshim every day), which applies all eight days!

(Mishnah): Hallel and Simchah apply all eight days.

Development of Question: The entire year, one who brings a Korban must lodge [in Yerushalayim]. The Sifri expounds "u'Fanisa ba'Boker" to teach that all Zevachim obligate Linah, i.e. [animal] Korbanos like Pesach; what is the source to include birds, Menachos and [one who brings] wood [for the Mizbe'ach]?

Answer: "U'Fanisa" teaches that whenever you turn (leave the Mikdash), it is only from the [next] morning or later.

Answer: We find this (that Linah is needed only due to the Hekesh) regarding one who did not bring a Korban that day, e.g. he sent a Shelamim yesterday. (Had he brought it himself, he should have finished it the same day - Chachmas Mano'ach Rosh Hashanah 5a.) It is eaten for two days and the night in between.

Support: Also, in Chagigah (8a) a Beraisa expounds v'Samachta b'Chagecha to include all kinds of [Kodesh meat, through which one may fulfill] Simchah. We say in Pesachim (71a) that [one fulfills Simchah] even through clean clothing and old wine! (Si'ach Yitzchak - since it suffices to be b'Simchah, one can be Yotzei through another's own Korban.) This (clean clothing and old wine) is the only way to fulfill Simchah all eight days when the first day of Yom Tov is on Shabbos;

One cannot fulfill it through a Shelamim offered on Erev Yom Tov, for one must slaughter it at a time of Simchah [during the festival]. Rather, he fulfills Simchah even through clean clothing and old wine.

Support: The Yerushalmi (Bikurim 2:3, b'Sof) says that [Bikurim] obligate Linah when there is no Korban. If there is a Korban, Linah is required even without [Bikurim], due to the Korban.

Inference: Zevachim 97a connotes that Linah is obligatory all seven days of Pesach. All of them are considered like one night regarding R. Tarfon's law that if one cooked [Kodshim] in a pot from the beginning of the festival, he may cook in it the entire festival. (We do not say that absorbed taste in the pot of the first day's Kodshim is emitted into the food, and forbids it due to Nosar.)

R. Yitzchak answered that "u'Fanisa va'Boker" teaches that the entire festival is considered like one morning. Even though this is rejected, for we ask that if so, Nosar would not apply during the festival, presumably regarding Linah the entire festival is considered like one morning, like the simple meaning of the verse, that after the festival he returns to his house.

Support #1: This explains how we include Linah for Shemini Atzeres and Bikurim from "u'Fanisa" - whenever you turn, like the Sifri expounds. Rosh Hashanah 4b requires a Hekdesh to obligate Linah during Sukos just like during Pesach;

Had we learned only from "u'Fanisa", one might have thought [that Linah is obligatory] only one night. Therefore we need a Hekesh to teach about all seven days, like Pesach, in which the entire festival is considered one morning.

Support #2: This explains why Sukah 47a says that just like the seven days of Sukos require Korban and Linah...

Question #1: However, this is difficult. If so, why do we say that Shemini Atzeres is an independent festival regarding Linah? Since the seven days require Linah without interruption, why is this an independent festival regarding Linah? (It is like a continuation!)

Question #2: Also, the Sifri includes even birds and Menachos like Linah of festivals. This implies that also Linah of festival is only one day!

Rejection of Inference (k): Zevachim 97a is not difficult [based on what I explained above, that Linah is only one day]. Initially, we understood that R. Tarfon requires Linah all seven days because the Torah considers them like one morning, but we conclude unlike this, rather, like I explained.

R. Tam established in his [Piyut for] Ma'ariv [of Shemini Atzeres] "the eighth [day] is Kodesh for Linah, and pushes aside Sheloshim (the stringencies of a mourner within 30 days) when it consoles Avelim;

However, it does not totally push aside Sheloshim. Rather, it counts like seven days towards Sheloshim, like it says in Mo'ed Katan 24b "if one observed Shiv'ah one day before Sukos, [the first day of Yom Tov cancels Shiv'ah,] and [the seven days of Sukos] and its (Sukos') eighth [day] count like 21."

Observation: Maseches Sofrim (19:2) says that some used to say la'Mnatze'ach Al ha'Sheminis on Shemini Atzeres. Perhaps also the Leviyim used to say it in the Beis ha'Mikdash.

Implied suggestion: The Gemara could have answered that we learn two goats offered for the Tzibur, e.g. on Yom Kipur, when we offer a goat inside [the Kodesh ha'Kodoshim] and a goat outside, from the Milu'im, when they offered two goats for the Tzibur, i.e. the goat of the Milu'im and the goat of Rosh Chodesh. This excludes Shavu'os, on which only one goat is offered!

Rejection: Also on Shavu'os the Tzibur offers two goats, for the goat mentioned in Emor [that is offered with Shtei ha'Lechem] is not the same goat mentioned in Pinchas [which is part of Musaf].

Support: In Zevachim (6b) and Menachos (45b) the Gemara discusses the goats offered on Shavu'os.

Question: Even according to the opinion that that says that two rams [are offered on Yom Kipur], we could say that we learn to Yom Kipur, when we offer one bull, from the Milu'im, when they offered one bull, but not Shavu'os, when we offer three bulls, one for the Chiyuv (obligation) of the day [offered with Shtei ha'Lechem] mentioned in Emor, and two that are part of Musaf, in Parshas Pinchas!

Answer: Since the Chiyuv of the day of Shavu'os is only one bull, we could properly learn from the Milu'im, in which one bull was offered for the Chiyuv of the day;

Implied question: On Shavu'os, two other bulls are offered with Musaf!

Answer: Also in the Milu'im, two other bulls were offered with Musaf, for they offered the Musaf of Rosh Chodesh, for the Milu'im was on Rosh Chodesh Nisan.

Implied question: Why is there no argument about how many rams are offered on Shavu'os, like Tana'im argue about how many rams are offered on Yom Kipur?

Answer: Menachos 45b explains that we are forced to say that on Shavu'os, the ram mentioned in Emor (offered with Shtei ha'Lechem) is not what is mentioned in Pinchas [with Musaf, for two are mentioned there, and only one is mentioned in Emor].

Reference: Rebbi's opinion here is explained below (70b).

Question: Granted, the Korban is [monetarily] of the Tzibur. However, it is Aharon's Korban, and it atoned for him for Chet ha'Egel, like it says in Toras Kohanim - "v'Aseh Es Chatascha v'Es Olasecha v'Chaper Ba'adcha v'Aseh Es Korban ha'Am v'Chaper Ba'adam"!

If so, we can say that we learn Aharon's Chatas bull that atones for him, from Aharon's Chatas bull that atones for him, and we learn his ram on which he does Semichah, from his ram on which he does Semichah!

Strengthening of Question: In Beitzah (20a), "va'Yakrev Es ha'Olah va'Ya'aseha ka'Mishpat" teaches that Olas Chovah requires Semichah, just like the law of Olas Nedavah!

Answer: Indeed, we could have learned this way. The Gemara prefers to clarify the truth, that here all agree that Aharon's brings his own Korban, like the Gemara proceeds to explain.

Question #1: Why didn't the Gemara say so above, when R. Avahu answered that we exclude Shavu'os and Rosh Hashanah, on which the bulls are of the Tzibur? (He should have answered that on Shavu'os and Rosh Hashanah the bulls are Olos, but in the Milu'im and Yom Kipur they were Chata'os!)

Question #2: Alternatively, why didn't R. Avahu answer [that we do not learn Korbanos of the Tzibur from Aharon's Korban] to exclude all of the questions (e.g. also from Regalim), like Ravina [answers all the questions with one answer] below?

Answer: Perhaps R. Avahu needed to answer only regarding Rosh Hashanah. We already answered above (3a) regarding Shavu'os (we do not learn it, because on Shavu'os we offer two rams for Chovas ha'Yom. R. E. M. Horovitz connotes that in Tosfos' text, R. Avahu answered only about Rosh Hashanah. Perhaps he omits "Atzeres" from the Dibur ha'Maschil - PF);

Rav Ashi (below) asked about Rosh Hashanah and Shavu'os together, and gave one answer for both of them. Ravina asked all of the above questions together, and answered all of them with one answer.

Implied question: Why didn't the Gemara say that Amora'im argue about R. Yochanan's opinion? Often, the Gemara says that when Rav Dimi came [to Bavel], he argued with what Amora'im before him said [in the name of R. Yochanan]!

Answer: Since we explicitly find that R. Yochanan said so in the name of R. Yishmael, we answer this (that he taught that his Rebbi expounds two separations, but he himself disagrees).

Question: How can R. Yochanan argue with R. Yishmael, who was a Tana?

Answer #1: We find a Beraisa below (4a) that holds like R. Yochanan's own opinion. The text of the Beraisa that is like R. Yochanan does not say `"La'asos" refers to the actions of Parah', rather, `"La'asos Lechaper" refers to the actions of Yom Kipur.'

Support: Rashi connotes like this. He explained [there] "just like we separate [the Kohen Gadol before Yom Kipur, we separate also the Kohen who will burn Parah Adumah]" is not learned from a verse. Rather, it is a mere stringency.

Answer #2: "R. Yochanan said in the name of R. Yishmael" connotes that Rabanan disagree. He holds like Rabanan.

Question: Granted, we may learn from the Milu'im even to all generations, since it is explicitly written "Ka'asher Asah ba'Yom ha'Zeh Tzivah Hash-m La'asos Lechaper";

However, how can we learn from Sinai? "This is a Banah Av" connotes that we learn from an extra verse, just like "this is a Banah Av, that the Chatas always comes before the Olah", which we learn from an extra verse (Zevachim 90a). What verse is extra here?

Answer: Also here, it would have suffices to write "va'Ya'al Moshe El Har Sinai va'Ychas he'Anan Es ha'Har va'Yikra Hash-m l'Moshe ba'Yom ha'Shevi'i mi'Toch he'Anan";

Question: Why does it say "va'Ychasehu he'Anan Sheshes Yomim"?

Answer: This is to teach to [all] generations.

Question: Also according to R. Yehudah ben Beseirah, six days [of separation] should suffice regarding Parah, since a Tevul Yom is Kosher for Parah. [Even if his wife was found to be Nidah,] he can immerse in the morning of the seventh day, and burn the Parah the same day!

We conclude below (6a) that a Bo'el Nidah may immerse during the [seventh] day.

According to the opinion that holds below that he immerses at night [after day seven], we say that we separate him shortly before Shki'ah. If so (the latest he could have had relations was the day before separation, so) he could immerse on the night [following] day seven [from relations], and he could burn the Parah the next day!

Answer #1: I explained above (2a DH Lehotzi) that a Tevul Yom is Kosher for Parah only regarding a Tum'ah mid'Rabanan. Regarding a Tum'ah mid'Oraisa, such as Bo'el Nidah, Chachamim forbid him;

Implied question: The Torah permits even a Tevul Yom of a mid'Oraisa Tum'ah!

Answer: Chachamim decreed to forbid. This resolves Question (a).

Answer #2: Also, there is a Tana (Yevamos 74b) who holds that even the Torah permits only a Tevul Yom of Tamei Mes, which is written in the Parshah, or of a Sheretz, which is learned from a Kal va'Chomer from Tamei Mes;

According to that Tana, we cannot learn to a Bo'el Nidah from a Kal va'Chomer. We can ask that a Tamei Mes is not Metamei a Mishkav (a Kli special for sitting on it) under him [that he does not touch] to be [a Rishon l'Tum'ah] like a Kli above a Zav, like a Bo'el Nidah is Metamei it;

Perhaps R. Yehudah ben Beseirah holds like that Tana.