Toras Chayim - the Gemara's answer is weak. In conclusion, all agree that [here] 'Ohr' means day. We are understanding that Rav Huna and Rav Yehudah hold that 'Ohr' has only one meaning, therefore all of the verses or Beraisos are difficult for one opinion. The contradiction between verses (the next verse shows that 'Ohr' is night) did not bother the Gemara, for it holds that 'Ohr' has both meanings.
Even though Rav Huna is refuted, the Gemara seeks more refutations. Subsequent questions against Rav Yehudah try to show that 'Ohr' has both meanings, or that Tana'im argue about its meaning. Alternatively, all these questions were asked in different Batei Medrash. Ravina and Rav Ashi included all of them in the Gemara - Maharam Chalavah.
Any blood that a woman sees between days 15 and 80 days after giving birth [normally] to a girl [or between days 8 and 40 after a boy] is Dam Tohar (postnatal blood, which is Tahor). After this period ends, the next blood she sees is Dam Nidah.
Maharsha - if the clean way is longer, either may be used. Chidushei ha'Ran - the shorter way must be used.
Minchas Chinuch (Mitzvah #14 DH v'Hinei bi'Semag) - Semag holds that the Lav of "Kol Ben Nechar Lo Yochal Bo" applies to Nochrim. This is why they killed him.
The Rambam holds that Yisraelim are commanded not to give to him. Perhaps they found that he did not fulfill his seven Mitzvos.
Kovetz Shi'urim (#6) says that his purchase of a share of the Korban was invalid, hence he transgressed stealing. Also see Insights to Pesachim 3.
Ritva - we did not just say that Aveilus due to an old tiding is only for a short time, because the law that a short time counts like a whole day applies also to Avelus of seven and 30 days. R. Chiya decided to bathe in order to teach these last two Halachos. An Avel may not learn or teach Torah which gladdens the heart, but laws of mourning are permitted.
Rashash - if the reason for Bedikah is lest one come to eat it, the tenant should check. If it is due to Bal Yera'eh and Bal Yimatzei, the landlord should check.
Maharam Chalavah - since we do not ask which of them would transgress Bal Yera'eh and Bal Yimatzei, this implies that if Reuven deposits Chametz with Shimon, mid'Oraisa neither transgresses!
R. Noson (Shabbos 70a) says that it is Mechalek. R. Yosi says that it teaches that burning is not a Melachah, rather, [on Shabbos] it is a Lav without Misah or Kares.
Tosfos - if it were only a Lav, there would be no source to forbid it on Yom Tov.
R. Chananel - a Lav would forbid it on Yom Tov, but not an Aseh. The Aseh of Tashbisu would override the Lav.
Yemenites make soft Matzah resembling Pitah. It is understandable how it could become moldy within a week. It is unreasonable that the hard, dry Matzah eaten by most communities today would become moldy during Pesach, especially before the Gemara said that warm Matzah was placed on top each day!
If a Safek originated in a place where everything is forbidden or everything is permitted, the Isur is called 'Kavu'a'. It is considered an even Safek.
Rashi - another Bedikah is required. Tosfos - surely, we discuss a Safek mid'Oraisa, similar to meat, i.e. is the piece itself permitted, or must one repeat Bedikah in a case when Bitul was not done. Chidushei ha'Ran - first, we rule that the piece is forbidden [as if it were Vadai Chametz]. Therefore, another Bedikah is required.
Rashi (Avodah Zarah) - R. Eliezer does not Metaher because it is a Sefek-Sefeka - rather, we learn from Sotah only to Metamei Safek Tum'ah in Reshus ha'Yachid when one surely entered the place of Tum'ah.
Hazamah is when latter witnesses (Mezimim) testify that the first witnesses were not present where they claimed to see the testimony. They receive whatever punishment they tried to impose on the plaintiff. If a witness did not remember which month it was, he could be Huzam only if witnesses said that they were with him that entire year in another place.
An Av ha'Tum'ah is something that can be Metamei [even] people and Kelim. A Vlad ha'Tum'ah received Tum'ah from another source, and can be Metamei food or drink, but not people or Kelim. An Av makes [something it touches] a Rishon [l'Tum'ah], a Rishon makes a Sheni, and so on.
Mid'Oraisa, if any part of a person is Tamei, his entire body is Tamei. Because hands touch many things, Chachamim decreed that hands are Tamei unless one has guarded them from the last time he washed them.
One may not give it [or anything for free] to a Nochri, due to "Lo Sechanem." One may not sell it to a Ger, rather, one should give it for free, since we are commanded to support a Ger, and it is worth much more to him than to a Yisrael, i.e. in a place that is mostly Yisrael and there is little market for all the Neveilos - Tosfos Avodah Zarah 20a (DH v'Rebbi Meir).
Rashi explains that we discuss Eglah Arufah. Tosfos - the Beraisa is a Tosefta regarding Parah Adumah. Even though Parah and Eglah teach about each other, the Gemara (Sotah 46a) says that even stationary work disqualifies Parah Adumah, but Eglah Arufah is disqualified only if it pulled. Perhaps Rashi understands this to mean any work while walking.
The Gemara does not explain why he made it enter the Revakah, just that he did not intend for it to thresh. Perhaps he intended for it to nurse, or so it will not get lost. The text in the Tosefta says that it entered [by itself]. R. Chananel (Bava Metzi'a) - animals are brought to Revakah to be fattened, and not to thresh. It threshed on the way there.
Rashi - the question is only from Asham Taluy. Really, Lo Sosiru does not apply to Chatas ha'Of brought mi'Safek, since one may not eat it!
Maharam Chalavah - also in Sukah (36b) R. Yehudah learns from a Kal va'Chomer even though the stringency leads to a leniency. Above (27b) he did not really retract. He merely sought a source that Chachamim could agree to. (NOTE: It is interesting why he was silent. He could say "I learn from the Kal va'Chomer. You should learn from Nosar, due to Lo Sosiru!" The Amora'im who bring the coming parables seem to understand that he refutes himself. - PF) In any case, we must explain why Chachamim argue!
Tosfos (28a DH Amru) - Chachamim refute R. Yehudah according to his own reasoning. They themselves do not learn from Nosar because they expound (24a) "v'Sarafta Es ha'Nosar ba'Esh" - Nosar is burned, but other Isurim are not.
Tosfos (27b DH Ein) - the Halachah follows R. Yehudah regarding Chametz. This is reasonable, since the Halachah is that Nosar Asham Taluy is burned, like Chachamim [and we must say that we hold like R. Yehudah who does not expound "v'Sarafta..." to exclude other Isurim from Sereifah].
He pays Chulin to a Kohen (at least the principal must be to the owner); it becomes Terumah. Whenever one adds a fifth, it is a fifth of the total payment, i.e. a quarter of the principal.
This is the smallest amount that is considered money, the weight of half a barley seed of silver (Rambam Hilchos To'en 3:1), which is about 1/14 of a gram, or .002 troy ounces. It used to be relatively more valuable (silver was more expensive with respect to most things). One could buy a loaf for 16 Perutos. At the time of this writing (5773), a troy ounce of silver sells for 17$, so a Perutah is about 13 Agarot, or about 3.5 cents.
Above (24b) the Gemara explicitly learns wood and frankincense from the Reisha "veha'Basar Asher Yiga..." Malei ha'Ro'im asks why Rashi says that we learn from the end of the verse "veha'Basar Kol Tahor Yochal Basar."
Rav Huna says that Lechem ha'Panim was a Tefach thick. The Mishnah in Menachos (96a) gives Shi'urim for the length and width of Lechem ha'Panim, but not the thickness. Ralbag (cited in Kesef Mishnah Temidim u'Musafim 5:9) and Tif'eres Yisrael (preface to Zevachim, Chomer ba'Kodesh 2:51 DH Ulam) prove that [unless there was a miracle] the thickness must have been less than one finger, as follows:
We assume that the volume of Matzah cannot exceed that of the flour used to make it, since the dough may not rise. The water does not increase the volume much. We give a simplified calculation according to R. Meir. (The calculation according to R. Yehudah is similar). Each loaf contained two Esronim of Soles, 3/5 of a Se'ah (an Isaron is a tenth of an Eifah, which is three Sa'im). 40 Sa'im are 1x1x3 Amos (Daf 109a). Each loaf was 3/5 of 3/40 = 9/200 of a cubic Amah. Since each loaf was 5/3 Amos long and 5/6 Amah wide, its (average) thickness was 9/200 ÷ (5/3)(5/6) = .0324 Amah, which is .7776 thumbs.
Two Tefachim on each end were bent up, forming walls on each side (Menachos 96a). Rashi connotes that these walls were a Tefach thick (making the top faces a Tefach). These walls comprised 2/5 of the length of the loaf. Even if they contained almost all of the flour, they would be only 5/2 as thick as calculated above, less than half a Tefach!
Tiferes Yisrael explains that only the corners were a Tefach thick. If so, no miracle was needed. (A thorough illustrated discussion of this is found in Meleches ha'Mishkan v'Kelav p.122-127.)
Something bought with Ma'aser money is less Kodesh than Ma'aser itself. If it became Tamei it cannot be redeemed because its Kedushah is too weak to transfer to something else. Therefore, it is just as valid for Matzah.
Still, Lachmei Todah are permitted only in walled cities in Eretz Yisrael! Sefas Emes - perhaps Moshvoseichem connotes only where we expect Yisraelim to be settled, i.e. walled cities in Eretz Yisrael. However, we often expound this to include Chutz la'Aretz! (Tosfos Kidushin 37b does so regarding Matzah.)
(NOTE: Rav Eliezer Chrysler suggests that we expound "b'Chol Moshvoseichem" to include or exclude what seems appropriate in each case. Alternatively, perhaps Reish Lakish learns from "b'Chol". Alternatively, perhaps we say that Lachmei Todah are permitted "b'Chol Moshvoseichem", because any place could be annexed to Eretz Yisrael and/or surrounded by a wall. This does not apply to Bikurim, for not every place can receive Kedushas Yerushalayim. - PF)
Hagahos Maimoniyos (7:20) brings Yerushalmi (2:5, 18a), that Chazeres is initially sweet, and at the end it is bitter. If Romaine lettuce is left to grow for about a month or so after it is usually harvested, it sends up a stalk several feet high with a new set of bitter leaves while the old leaves remain mild. It then has all the Simanim that Chazal gave. It starts out mild and turns bitter. It has a stem, and a white milky juice (sap) coming out of the bitter leaves. Farmers and botanists call it "bolted lettuce". The wild lettuce mentioned by Chazal can be found growing all over Israel in time for Pesach. It is not Kil'ayim with regular [Romaine] lettuce, for they are the same species. (From Sefer Chidushei Torah of Avroham Shlema Adler, Taste and See... II.)
Rashi explains that the Tana'im argue about whether or not a mixture of vinegar is Cholet. R. Yehonason explains that the Beraisa discusses using flour to seal a crack in a pot. The second Tana forbids putting flour first, lest the vinegar not reach all the flour in the crack, and afterwards water will get there. This explains why it discusses Melilas Kederah (a pot), as opposed to Melilas Tavshil (a cooked food). He does not explain how this relates to R. Yehudah in the coming Beraisa.
Tosfos Rabeinu Peretz - since these are normally consumed over a long time, we decree not to eat even immediately.
Rosh - the Charoses contains water. Perhaps we are concerned lest the flour become Chametz immediately. Mei Peiros with water is Mechametz faster than pure water (Shulchan Aruch 572:2).
Tosfos asks that also salves are Tachshitei Nashim! He explains that Tachshitei Nashim connotes salves applied near the Ervah (no one would eat them). Tipulei Nashim are used for facial hair. Tosfos could agree with Rashi that girls are embarrassed about the hairs. Also facial hair does not normally come until puberty. Perhaps they are embarrassed about pubic hairs, even though the hairs are not visible to others. Alternatively, others see them in the Mikveh, or when bathing, or we refer to hair on other places; R. Shimon ben Yehudah (Nidah 52b) holds that hairs on the fingers are signs of adulthood.
Aruch - girls at the age of Bagrus who have not matured physically use these Tipulim to seal pores in the skin to retain body heat. This prompts development.
When Elul 30 is on Thursday, we must treat it like Yom Tov, in case witnesses will come in time to be Mekadesh the day to be Rosh Hashanah. If they do not come, Friday is Rosh Hashanah.
Rashi says "if he has opportunity to slaughter many birds or Chayos." Presumably, "many" explains why one should be exempt for an entire furrow, even though it is much greater than the amount needed to cover the blood of one bird.
Tosfos - Melachah on Yom Tov is an Aseh and a Lav. It is not permitted for the sake of covering blood! R. Tam - he is liable for plowing for Kil'ayim, but not for Yom Tov.
They argue about which becomes Chametz more easily. It is difficult to say that Chachamim argue about an observable phenomenon!
Perhaps they argue about what is considered Chametz, e.g. barley reaches R. Meir's definition of Chametz before wheat, and wheat reaches R. Yehudah's definition faster than barley.
Alternatively, each of them is more prone to become Chametz in certain conditions (of temperature of the water and oven, the amounts of time it was worked and left idle, etc.).
Rav did not specify which grain he discusses. Perhaps we may assume that he discusses wheat, since barley is primarily for animals (Daf 3b).
Alternatively - Rav surely discusses the Shi'ur of flour, and not of grain. (The Shi'ur for Chalah depends on the amount of flour.) Perhaps two or three Kavim of wheat or barley makes this much flour, according to one of the Tana'im (who discussed the amount of wheat and barley).
Alternatively, Rav argues with them. He is considered a Tana. However, it seems that he argues with R. Eliezer and R. Yehoshua. Neither of them suggested making a dough slightly smaller than the maximum allowable Shi'ur. According to Rav, such a dough is exempt! However, it is very unusual to argue with Tana'im earlier than the previous generation. Perhaps Rav enacted a stringency in his generation, presumably because people were not so zealous to guard from Chimutz - PF.
Rashi (52b Mishum) - Bi'ur is to make the produce Hefker where animals can go. Tosfos (52b Misba'arin) - he keeps three meals worth of the species for each person, and takes the rest outside his house and makes it Hefker for people and animals. (Afterwards, he may take it back in his house.) Rambam - he burns it. Ra'avad - everyone brings the produce to a storehouse of Beis Din, and they give to everyone enough for three meals.
This is a decree lest one rent or lend, and the Yisrael's animal will not rest on Shabbos. Alternatively, perhaps the Nochri will buy just before Shabbos, and have problems, and (on Shabbos) the Yisrael will show the Nochri how to make the animal go. In some places they do not sell small animals, lest people sell large animals. The first Tana forbids animals too young or sick to work due to animals that can work. Ben Beseira permits horses, for they are used for carrying people, which is not a Melachah, since ha'Chai Nosei Es Atzmo - Shabbos 94a.
Nebuchadnetzar made an image and decreed that anyone who does not bow to it be thrown into a giant furnace. Only Chananyah, Misha'el and Azaryah refused to bow.
Rashi - we ask why they were not concerned for "Va'Chai Bahem", that preservation of life overrides (almost all) Mitzvos. Maharsha - Kidush Hash-m did not apply, for there were not 10 Yisraelim present.
Tosfos - it was not a real idol. We ask why they were Moser Nefesh, or why they didn't stay away from there.
It was miraculous that the letters, which were carved out from one side of the Luchos to the other, could be read from any direction, and that the Samech and final Mem did not fall.
Rashi explains Michtav to be the stylus. If the letters were engraved Bein ha'Shemashos, Michtav is unlike everything else listed, which was only needed later. (According to Ohr ha'Chayim, "Kesuvim b'Etzba Elokim" (Shemos 31:18) refers to a fire that emanated from Hash-m, but Rashi connotes that He used a solid writing tool.)
Rashi explains like Sforno (Bamidbar 16:32), that all Korach's supporters were swallowed - we must say that the verses are out of order (verse 35 recounts the burning of the 250 who offered Ketores), and that they were swallowed after they were burned.
Ramban explains that "Kol ha'Adam Asher l'Korach" refers to slaves or others that lived with him (but the 250 were not swallowed).
Although Rashi initially says that mid'Oraisa, the Tamid may be slaughtered anytime during the last six hours of the day, he concludes that it may be slaughtered only after shadows lean, after six and a half hours.
Above, Rava himself (5a) clearly says that mid'Oraisa the time of Shechitah begins after six hours. Rashi there cites Yoma 28b, which explains that we wait an extra half hour, mid'Rabanan, for the shadows of the Heichal to tilt noticeably eastwards. For a more thorough discussion see Insights to Pesachim 58.
Torah Temimah - the verse did not say 'v'Az Hayah he'Chazeh l'Aharon', which would have made receiving Chazah v'Shok contingent on Haktaras ha'Eimurim. Therefore, we learn that they always receive it. Alternatively, since the next verse says "ha'Makriv... v'Es ha'Chelev... Lo Sihyeh Shok ha'Yamin l'Manah", this verse is extra to teach about even when it is not offered. - PF
Rashi mentions that the Arel's brothers died through circumcision, i.e. he is exempt from circumcision because it is dangerous for him. If not, he would be forbidden due to Ben Nechar! Rashba (Yevamos 70a ha'Arel) adds that we circumcise him after his blood is properly absorbed.
The Torah forbids Temurah, i.e. trying to transfer or exchange an animal that has been designated as a Korban for another animal. If one tries, he transgresses, and both animals are Kodesh. The second animal gets the same Kedushah as the first. In most cases, it may be offered.
Matirim are things that are offered that permit other things, e.g. offering the Kometz (handful taken from a Minchah) and Levonah permit the remainder of the Minchah to Kohanim. All agree that if both were done with Pigul intent, the Minchah is Pigul. If one was offered with Pigul intent, Rebbi Meir says that it is Pigul, and Chachamim are Posel mid'Rabanan (Menachos 16a).
For bird offerings, the analog of slaughter is Melikah. The Kohen presses his thumbnail into the neck and cuts the Simanim.
Surrounding the Azarah courtyard (including the Ezras Nashim), ten Amos away from its wall, was a low wooden fence, called the Soreg. The ten Amos in between are called the Cheil.
Tal Torah asks that the Torah forbids Mechamer even regarding birds and Chayos! Perhaps this refers only to birds and Chayos that are usually used for Melachah, e.g. carrier pigeons and Kelavim Kufrin (hunting dogs - Rashi Bava Kama 80a) or elephants. - PF
Rashi - Shevus done unskillfully is more lenient than those forbidden in the Mishnah. Seemingly, it should likewise be permitted to remove a wart or do Haza'ah unskillfully!
Tosfos - a Shevus done through an animal is more lenient. This also answers why one may use an animal, which is Shevus.
A man who had two or three emissions of Zov (which is unlike regular semen) within three consecutive days can become a Zav. He cannot immerse to become Tahor until seven consecutive days without emissions. One who saw any number of emissions of normal (cooked) semen is called a Ba'al Keri. He can immerse at any time, and becomes totally Tahor when the day ends.
A Nidah is Teme'ah for seven days due to menstrual blood. A Zavah is Teme'ah due to post-menstrual blood. A Yoledes (one who gave birth) is Teme'ah like a Nidah.
A Metzora is Tamei due to special appearances on his body. His Tum'ah is never less than seven days.
Seemingly, it would be more proper to learn the entire day to commemorate Matan Torah!
Rabeinu David - one shows that he is happy about Matan Torah by eating and drinking.
Roke'ach (brought in Gilyonei ha'Shas) - he must show Simchah over pardon of sins through Kabalas ha'Torah. Ben Yehoyada - we show that Yisrael receives luxuries in this world (even though Yakov opted for the world to come and ceded this world to Esav) because we sustain the world through Torah. (Without Torah the world would cease. If something is being swept away by a river and the owner cannot save it, it is Hefker, and one who saves it may keep it.)
(For more on the subject, see Insights to Pesachim 68.)
The one who brought the Beraisa for a support understood this to mean that one can slaughter bi'Zman Simchah before the last Yom Tov. Perhaps the rejection explains like the Gra - before Shemini Atzeres, there were Mitzvos of Simchah, Lulav and Sukah - "Ach Some'ach" teaches that only Simchah applies on Shemini Atzeres. But since Simchah does not apply before the first night, it is unreasonable to expound "Ach Some'ach" to teach that only Simchah applies on the first night. - PF
Shomeres Yom k'Neged Yom - the eleven days following the seven days of Nidah are "days of Zivah." If a woman sees blood on one (or two consecutive) of these days, she is Teme'ah. If she does not see blood the following night, she may immerse the next day, but it helps (to Metaher her immediately for Chulin, and after dark for Terumah and Kodshim) only if she will not see blood the rest of that day. She is called a Shomeres Yom k'Neged Yom, because she must watch the following day to see whether or not she sees blood.
If she sees blood for three consecutive days of Zivah, she is a Zavah Gedolah. To become Tehorah she must count seven 'clean' days (without blood). On the seventh clean day she may immerse, but it helps only if she does not see blood the rest of the day. She must bring a Korban (two birds, after seven full clean days) to permit her to enter the Mikdash or to eat Kodshim.
A Zav who had two or three emissions must count seven clean days (without emissions). On the seventh clean day he may immerse (in flowing water), but it helps only if he has no emissions for the rest of the day. If he had three emissions, he must bring birds (after seven clean days) to permit him to enter the Mikdash or to eat Kodshim.
The words of every Gezeirah Shavah are a tradition from Moshe from Sinai, but it was left to Chachamim to decide from which occurrences of those words to learn and what to learn from them.
Hash-m rebuked Miryam and Aharon by telling them that Moshe's prophecy was so direct and constant that he needed to separate from his wife, but other prophets need not! Perhaps we learn from Moshe the attribute for a Navi to separate, and for other Nevi'im this does not warrant divorcing a proper wife, but it does warrant divorcing a harlot (R. Y. Glick Shlita.) Alternatively, perhaps the rebuke was "do you consider yourself a bigger zealot for My honor than Moshe, who opposed My suggestion to destroy Yisrael and start a new nation from him? If so, you should conduct yourself with Kedushah no less than his!" - PF
According to the parable, we would expect the number of converts to be far greater than the number of Yisraelim exiled. Perhaps this is true if we count all descendants of the converts. Perhaps it refers to the number of Mitzvos [that apply only in Eretz Yisrael] lost due to exile. This is far outnumbered by the number of Mitzvos performed by converts [and their descendants for all generations], especially if every word of Torah is equated to all 613 Mitzvos - PF.
Tosfos - R. Zeira (Nedarim 36a) holds that mid'Oraisa, minors are exempt from Pesach, therefore one may be Memaneh them without their consent. Rashi does not address this, presumably because even if it were mid'Oraisa, since their Da'as is not needed, this refutes the proof that Yesh Breirah.
Shulchan Aruch (459:2) defines the time to walk a Mil to be 18 minutes (since one walks 40 Mil in a day of 720 minutes (12 hours)). Gra - this is in Nisan, when the days and nights are equal, i.e. from sunrise until sunset is 12 hours. We shall conclude like a Beraisa (94a) that eight of the 40 Mil are walked before sunrise or after sunset. (To explain Ula, we assumed that one walks 40 Mil from dawn until Tzeis ha'Kochavim; the Gemara never retracted from this.) It follows that one walks only 32 Mil in 12 hours, so each Mil takes 22 and a half minutes. The Shulchan Aruch must explain that when Ula was refuted, we also retract from the 'day' of walking. Rather, one walks 40 Mil between sunrise and sunset. It is proper to begin and end 'b'Chi Tov', while the sun is out (Rashi 2a).
Perush ha'Mishnayos (3:2) says that it takes 24 minutes to walk one Mil. This is like Ula, who says that one walks 30 Mil from sunrise to sunset. Ula was refuted; Gra says that the Rambam had a different text in his Gemara. Also Tosfos (11b, DH Echad) holds like Ula.
In Shabbos (34b-35b), R. Yehudah himself says that Bein ha'Shemashos is the time to walk 3/4 of a Mil (some say, even less) after sunset. After this is Vadai night, i.e. Tzeis ha'Kochavim! How can we reconcile this with his opinion in our Gemara?
Rabeinu Tam - here the Gemara says "mi'Sheki'as ha'Chamah." This refers to the moment that the sun disappears from view, and enters the thickness of the Raki'a. This is the beginning of sunset. In Shabbos, the Gemara discusses mishe'Tishka ha'Chamah", after it passed through the entire thickness of the Raki'a. This is end of sunset. Bein ha'Shemashos begins after the end of sunset, and ends 3/4 of a Mil later, 4 Mil after the beginning of sunset.
Gra (OC 235) - in both places, sunset is when the sun disappears from view. In Shabbos, the Gemara says that it is night when three medium-size stars can be seen. This is 3/4 of a Mil after sunset. Here "Tzeis ha'Kochavim" refers to when it is so dark that all the stars can be seen. This is 4 Mil after sunset, which is long after night.
(For more on the subject, see Insights to Pesachim 94a.)
The Gemara, Rashi and Tosfos cannot literally mean the people who are presently standing in the Azarah. Before we decide that Pesach will be brought b'Tum'ah, a Tamei may not enter the Azarah!
Rambam - it refers to the people who plan to enter the Azarah. We ignore Menuyim who will not enter. (Presumably, when Rashi says that we exclude those who are "ba'Chutz", he also refers to Menuyim who do not plan to enter.) R. Yosi explicitly says so in the Yerushalmi (7:6) - "Ad she'Hen miba'Chutz Mesha'arin Es Aztman" (they count the number of Tehorim and Temei'im while outside).
The last food one eats on Pesach night must be the Korban Pesach. How can they fulfill this if they do not know which is the Pesach? Ha'Gaon R. M. Shapiro Shlita answered that at the end of the meal, one puts a k'Zayis of each Korban in his mouth at once, chews them, and swallows them at once. (One can swallow k'Beitzah at once. All agree that this is at least two k'Zeisim.) Each is a Torah Mitzvah, so neither is Mevatel the other.
Tosfos (99b) says, according to texts [like ours] that read Erev Pesachim, this can refer to Erev Pesach Rishon and Sheni. If so, why was Rav Papa refuted? The Mishnah discusses Pesach to teach about Pesach Sheni!
Tzelach - had it taught Erev Shabbos and Yom Tov, all the more so we would know that one may not eat on Erev Pesach Sheni, for there is a Mitzvah mid'Oraisa to eat on Pesach Sheni!
Zehav Sheva (printed with Tosfos ha'Rashba) - we can say that the Isur to eat after Minchah applies on Erev Pesach Sheni only if the reason (also for Pesach Rishon) is lest one neglect to offer the Korban. (The Gemara will discuss this on 107a-b.) If it is to eat Matzah with appetite, this does not apply on Pesach Sheni. The only Mitzvah to eat Matzah that night is with Pesach Sheni, which is eaten when satiated!
Tosfos (printed in Mordechai, page 36) - at first, Rava and Rav Yakov misunderstood each other. Rava thought that Rav Yakov knew that a torch is the ideal Mitzvah, but asked why one must light one if a lamp is already lit. He answered that one need not. The attendant did so on his own.
Really, Rav Yakov was asking why there is any preference for a torch. Surely, the attendant would not seek one unless Rava said that it is better! Rava now understood, and answered that a torch is the ideal Mitzvah.
One Se'ah is six Kavim, which is 24 Lugim, which is 96 Revi'yos, so 40 Sa'im is 3840 Reviyos. This equals three cubic Amos, so each cubic Amah is 1280 Reviyos. An Amah is six Tefachim, which is 24 thumbs, so a cubic Amah is 24x24x24 thumbs. If we divide this by 1280 (which is 8x8x20), we find that one Revi'is is 3x3x1.2 = 10.8 cubic thumbs, which equals 2x2x2.7.
The Gemara (Yoma 31a) assumes that the Mikveh in Beis ha'Parvah was 1x1x3 Amos; Tosfos - this was to spare the Kohen Gadol from having to crouch down, i.e. if it would be wide and shallow. However, an Amah fits very comfortably in the diagonal, which is over seven fifths of an Amah! The same applies if we assume that the Shulchan was immersed in the stream running through the Mikdash called the Amah, which was an Amah wide.
Rivash (#292) - the thickness of the top of the Shulchan, with the Misgeres (rim) and Zer (crown) was too big to immerse it on a diagonal (see the diagram below), for [the top and bottom] would be more than 1.5 Tefachim from the corner (so the resulting diagonals on the bottom left and top right are less than an Amah). He holds that the Misgeres was connected to the Shulchan, but not to the legs.
The Noda bi'Yehudah (2 OC 112) says that Rav Ashi cannot hold like R. Yehudah, who says that the Amah of the Shulchan is five Tefachim, for all agree that the Amah of a Mikveh [and measure of land, i.e. the floor of the Mikdash] is 6 Tefachim, so there would be no reason to discuss an exact fit. According to R. Meir, Amas ha'Shulchan is six Tefachim. We would have to say that the top and bottom were 12/7 Tefachim from the corner, to say that there was no room to immerse the Shulchan, for then 30/7 Tefachim remain on the side, so their diagonal is (30/7)(7/5) = 6 Tefachim, i.e. 1 Amah. It is difficult to say that the Rivash said "more than 1.5", and he means at least 12/7, which is more than 1.7. Further, why does he assert this distance from the corner without proof? If so, the combined thickness of the top of the Shulchan and Misgeres was (12/7)(7/5)=2.4 Tefachim. Me'il Tzedakah (39), Malbim (Shmos 25:25) and many others say that the thickness of the top of the Shulchan was a Tefach. The Targumim on "Misgeres Tofach Saviv" (ibid.) and the Meforshim say that the thickness of the Misgeres was a Tefach. What accounts for the extra .4 Tefachim?
Even if the Rivash holds like the Pleisi (YD 43:6), that the gold plating on the Shulchan was a third of a finger thick, and that it was covered above and below, this adds only 2/3 finger (1/6 Tefach) to the thickness, far less than .4 Tefachim. Perhaps he holds like Chizkuni, that the rings of the table were attached to it below the thickness of the table, and the Misgeres was above.Minchas Chinuch (95) says that the Torah did not fix a Shi'ur for the size of the rings. They and the staves must suffice to support the weight of the Shulchan. Perhaps the Rivash assumes that they must be at least .4 Tefachim. If he also holds like the Pleisi, it suffices for them to be .24 Tefachim.
If the Rivash holds that Rav Ashi learns from R. Yehudah, who says that Amas ha'Shulchan is five Tefachim, he precisely said that it would be more than 1.5 Tefachim from the corner. Then, 3.5. Tefachim remain on the side, and the diagonal is 7/5 as much, i.e. 4.9 Tefachim, a drop less the Amah width needed. It follows that the combined width of the Shulchan and Misgeres was 1.5(7/5) =2.1 Tefachim. If it were only two Tefachim, it would be only 1.4 Tefachim from the corner, and 3.6 would remain, and their diagonal is 3.6(7/5) = 5.04 Tefachim, i.e. more than the Amah required! It would suffice to say that the Rivash holds like the Pleisi, or like the Chizkuni, and it suffices for the rings to be .1 Tefach.
Most who say that the Misgeres was above the Shulchan hold that it slanted away from the table, or was totally outside the table, for the Lechem ha'Panim occupied the entire area of the table. (However, Sefas Emes says that the Lechem ha'Panim was narrower at the bottom.) However, if so, there was no need for the Gemara to say that an Amah cannot fit in exactly an Amah, for the Misgeres was wider! Likewise, we cannot say that the Zer made it or the Shulchan wider than an Amah. (Some of the opinions cited here are based on Meleches ha'Mishkan u'Kelav.)
Ran - the Hillel cited was an Amora. (Alternatively, one could explain that Hillel, the contemporary of Shamai, taught what should be done after there will be a Churban. Tosfos (Gitin 36a-b) says that Hillel enacted Pruzbul to take effect after the Churban. It was known that Bayis Sheni would stand only 420 years - Nazir 32b.)
The Rishonim argue about how the Mitzvos are Mevatel each other.
Rashbam - the Maror is Mevatel the taste of Matzah.
Ran - one must taste Matzah by itself (but we are not concerned for taste of another Torah Mitzvah mixed with it).
Maharam Chalavah citing Ramban - we are not discussing Bitul of taste. Rather, Chachamim enacted to perform each Mitzvah by itself.
Tosfos - since Maror is only mid'Rabanan, it would be Mevatel Matzah if they were eaten together at first. After fulfilling both Mitzvos separately, they are both Reshus, so they are not Mevatel each other. If the Halachah were fixed like Hillel, we could do so right after eating Matzah. It would be a Mitzvah mid'Rabanan to eat them together, so Matzah would not be a Reshus that can be Mevatel the Mitzvah mid'Rabanan of Maror.
Rashi explains that Kapa is a poisonous sap. Perhaps he explains that the Kapa "dies" from the smell of the Charoses, i.e. it is rendered impotent. Tosfos favors R. Chananel's explanation, that Kapa is a worm. Death of the worm removes the danger. Perhaps the danger is lest a live worm make a hole in the esophagus or stomach...
If Romaine lettuce is left to grow after it is usually harvested, it grows new, bitter leaves, with a white milky juice (sap). These leaves are generally free of bugs, for the sap is a bug repellent. (From Sefer Chidushei Torah of Avroham Shlema Adler, Taste and See... II.)
Perhaps Rashi holds that one is Yotzei only with the bitter leaves, so worms are not a problem. This forced Rashi to explain that Kapa is the sap, even though it is not so proper to discuss "death" of the sap. R. Chananel holds that one is Yotzei even with the initial leaves. This enabled him to explain simply that Kapa is a worm. There is no sap to be concerned for.
Maharsha - he was not Yotzei eating them. (Note: Perhaps "va'Amartem Zevach Pesach Hu" (Shmos 12:27) teaches that one must discuss it, and Matzah and Maror are equated to Pesach. However, why did the Tana need to say 'three'? The Torah did not command to eat anything else! Perhaps one might have thought that the same applies to Chiyuvim mid'Rabanan, e.g. the four cups or Karpas - PF.)
Me'iri - he was not Yotzei Hagadah. (Note: Why is it more essential to mention Maror than Keri'as Yam Suf, or the Makos, about which it says "u'Lema'an Tesaper b'Aznei Bincha... v'Es Ososai Asher Samti Bam"?! Perhaps all are Me'akev, but one might have thought that eating Pesach, Matzah and Maror suffices in place of mentioning them. However, if so, why did the Tana need to say "three"? - PF.)