Gemara

Beraisa #1 - Suggestion: "V'Inisem Es Nafshoseichem...; V'Chol Melachah Lo Sa'asu" - perhaps one is liable for Melachah or eating during Tosefes Yom Kipur (i.e. he did not refrain before nightfall)!

Rejection: "B'Etzem ha'Yom ha'Zeh v''Ha'avadti Es ha'Nefesh ha'Hi"; "B'Etzem ha'Yom ha'Zeh v'Nichresah" - there is Kares only for the day itself, not for Tosefes.

Suggestion: Perhaps a Lav forbids Melachah or eating during Tosefes Yom Kipur!

Rejection: "V'Chol Melachah Lo Sa'asu b'Etzem ha'Yom ha'Zeh" - the Lav applies to Melachah only for the day itself. All the more regarding Inuy, which does not apply to Shabbos or other Yomim Tovim.

81b - Beraisa #2 - Suggestion: "V'Inisem Es Nafshoseichem b'Sish'ah" - perhaps we fast on the ninth!

Rejection: "Ba'Erev".

Suggestion: Perhaps "Ba'Erev" teaches that we need not begin to fast until night!

Rejection: "B'Sish'ah".

Resolution: We begin to fast before night. This is the source that one must add from Chol onto Kodesh.

Question: This is the source regarding the beginning of the day. What is the source to add from Chol onto Kodesh at the end?

Answer: "Me'Erev Ad Erev".

Question: This is the source for Yom Kipur. What is the source for other Yomim Tovim?

Answer: "Tishbesu".

Question: This is the source for Yomim Tovim - what is the source for Shabbos?

Answer: "Shabatchem".

Question: The Tana of Beraisa #1 can infer the Isur of Tosefes Yom Kipur from the verse that exempts for it. How does he expound "b'Sish'ah"?

Answer: This teaches that if one eats and drinks on the ninth, the Torah considers it as if he fasted on the ninth and 10th.

Shabbos 148b - Assumption: Regarding a transgression mid'Rabanan, we do not protest even if people will not listen. It is better that Yisraelim transgress b'Shogeg than b'Mezid. However, regarding a mid'Oraisa transgression we protest even if they will not listen.

Rejection: Mid'Oraisa, one must add to Inuy. Some people eat until night, nevertheless we do not protest.

Sukah 28a - Beraisa: "Ha'Ezrach" obligates women in Tosefes Inuy.

Rishonim

The Rif and Rosh (8:8) bring Beraisa #2 (with a minor difference from our text) and Beraisa #1.

Rosh (8:8): The Tosefes cannot be during Bein ha'Shemashos, for Bein ha'Shemashos is Safek night. We do not need an extra verse to forbid this time! The Gemara did not specify how long the Tosefes must be. Surely it has a Shi'ur. The Gemara (Shabbos 148) says that we do not protest against women who eat and drink until night. Surely, they do not eat until the last possible moment, for they do not know precisely when night begins! Rather, they stopped slightly before Bein ha'Shemashos; nevertheless they did not fulfill Tosefes. Similarly, one must add at the end of the day.

Rebuttal (Magid Mishnah Hilchos Shevisas Asor 1:7 and Ran Beitzah 16b DH Tanan): It is possible that Tosefes has no Shi'ur. Nevertheless, since the women refrained only due to Safek, they did not fulfill Tosefes. Alternatively, according to the opinion that (mid'Oraisa) Safek mid'Oraisa l'Kula, the Tosefes could be during Bein ha'Shemashos.

Korban Nesan'el (2): Sefer Chasidim (262) says that even regarding a Safek Isur mid'Oraisa we do not protest (unless it is in a Beis ha'Keneses). It is possible that they ate during Bein ha'Shemashos.

Question (Ran DH Tanu Rabanan V'Inisem): The Tosefes cannot be Bein ha'Shemashos (which is Safek Kares), nor can it begin before sundown, for Beis Hillel (Shabbos 17b) permit one to begin Melachah just before sundown!

Answer (Ran ibid.): Bein ha'Shemashos begins long after sundown; the time in between is enough to walk three Mil and a third or quarter (about an hour). One makes all or part of the time in between Tosefes. The Tosfefes after Shabbos or Yom Tov is until one can see three stars right near each other. Really, it is night once one can see three stars even if they are not near each other.

Support - Question: The Gemara (Shabbos 34b) says that Bein ha'Shemashos is after Sheki'ah, as long as Penei Mizrach (the eastern sky) is reddish. Amora'im argue about whether it lasts for three quarters or two thirds of a Mil. After this is Tzeis ha'Kochavim, i.e. night. But Pesachim 94a says that from Sheki'ah until Tzeis ha'Kochavim is four Mil!

Answer (R. Tam): Pesachim 94a discusses the beginning of Sheki'ah, when the sun goes below the horizon. Shabbos 34b discusses the end of Sheki'ah, when the sun leaves the 'window' in the firmament.

Rambam (Hilchos Shevisas Asor 1:6): One must add from Chol onto Kodesh at the beginning and end of the day - "V'Inisem Es Nafshoseichem b'Sish'ah la'Chodesh ba'Erev". I.e., one must begin to fast in the evening of the ninth, shortly before the 10th. Similarly, at the end of the day one must continue fasting a bit into the 11th - "Me'Erev Ad Erev Tishbesu Shabatchem".

Magid Mishnah: The Rambam holds that Tosefes is mid'Oraisa only regarding Inuy, but not regarding Melachah, not on Yom Kipur or Shabbos or Yom Tov. Tana'im argue about this. The Rambam rules like R. Akiva who expounds that if one eats and drinks on the ninth, it is as if he fasted on the ninth and 10th, therefore he rules that Tosefes Melachah is not mid'Oraisa.

Question: The Gemara said that the Tana of Beraisa #1 expounds that '...it is as if he fasted on the ninth and 10the', yet Beraisa #1 implies that also Tosefes Melachah is mid'Oraisa!

Answer #1 (Lechem Mishneh): The Rambam had a different text of Beraisa #1 which excludes Tosefes Melachah from a Lav, but not Tosefes Inuy

Answer #2 (Pnei Yehoshua Rosh Hashanah 9a DH Emnam): The Rambam holds that once we say that one is rewarded for feasting on the ninth as is he fasted. The verse cannot be expounded to forbid Tosefes. We learn Tosefes Inuy from ha'Ezrach (Sukah 28a)

Poskim

Shulchan Aruch (OC 608:1): On Erev Yom Kipur we stop eating before Bein ha'Shemashos, for one must add from Chol onto Kodesh. There is no Shi'ur to this Tosefes. One may add a lot or a little; it must be before Bein ha'Shemashos. Bein ha'Shemashos lasts the time needs to walk 1500 Amos; it is right before night.

Shulchan Aruch (261:2): The Tosefes must be sometime between the beginning of Sheki'ah, when the sun goes below the horizon, until Bein ha'Shemashos, which begins three Mil and a quarter later.

Magen Avraham (9): The Yere'im, Mordechai and Agudah say that Bein ha'Shemashos ends at sundown, and then it is night. Important people refrain from Melachah two hours before (Tzeis ha'Kochavim).

Mishnah Berurah (23): The Shulchan Aruch holds like R. Tam. Many Rishonim and the Gra disagree; they say that Bein ha'Shemashos starts immediately after sundown. Happy is one who refrains from Melachah (or eating on Erev Yom Kipur) 30 or at least 20 minutes before sundown. This fulfills all the opinions (including the Yere'im), even if the time to walk a Mil is 24 minutes, and it fulfills many opinions regarding Tosefes. One may not end Shabbos until three medium stars are visible. Three quarters of a Mil after sundown suffice for the latitude of Bavel in Nisan or Tishrei. In northern areas one must wait much longer.

Shulchan Aruch (ibid.): The entire time between sundown and Bein ha'Shemashos may be made Tosefes, or just part of the time, as long as he adds time which is definitely Chol.

Bi'ur Halachah (DH Eizeh): Some say that there is a Shi'ur Tosefes. In any case it is not more than three quarters of a Mil.