1) HALACHAH: USING A FLAMMABLE MATERIAL FOR HYGIENIC PURPOSES
OPINIONS: The Gemara (81b) says that a person should not use a piece of Cheres (potsherd) for Kinu'ach in the bathroom because of Sakanah (because of its sharpness, it can cause damage to the rectal musculature orifice), and because of Keshafim (sorcery; sorcerers are able to cast spells on those who use it). The Gemara (82a) then says that one should not use a material that is flammable, like grass, because it can cause damage to the rectal musculature.
The REMA (OC 3:11) mentions that it is the prevalent practice today to use paper for Kinu'ach. Why are we not concerned for the danger that the Gemara depicts?
(a) The REMA in DARCHEI MOSHE and in the Shulchan Aruch cites the AGUDAH who says that "since our bathrooms are not in the fields, we are not concerned about using a material that is flammable." The Rema in Darchei Moshe declares that he does not understand the Agudah's reasoning: Why does the fact that our bathrooms are not in the fields permit the use of a flammable material for Kinu'ach? Nevertheless, the Rema concludes that since people commonly use paper for Kinu'ach, it falls into the category of "Shomer Pesa'im Hash-m" -- "Hash-m protects the fools" (Tehilim 116:6).
(b) The MAGEN AVRAHAM explains that the Agudah means that the reason a flammable material is dangerous is because of Keshafim (it enables sorcerers to cast spells on those who use it); it is not physically dangerous by itself. (This is not the simple meaning of the Gemara on 81b that seems to distinguish between the two reasons of Sakanah and Keshafim.) Since there is no fear of Keshafim nowadays, one is permitted to use paper.
(c) The PERISHAH explains that the Rabanan prohibited using a certain object for Kinu'ach only when there are two reasons not to use it (such as Sakanah and Keshafim). Nowadays, there is only one reason (Sakanah, and not Keshafim), and therefore one is not prohibited to use a flammable material.
(d) The CHASAM SOFER (81b) explains that we rule in accordance with Rava, who says that one is prohibited to use Cheres only because it may tear off hairs on Shabbos, and not because of Sakanah or Keshafim. Although the Gemara refutes Rava when it cites a teaching of Rebbi Yochanan in which he is not concerned for a Davar she'Eino Miskaven, the Halachah apparently follows Rava. (With regard to Rebbi Yochanan's statement, we find in many places that there were two opinions as to whether or not Rebbi Yochanan ruled like every "Stam Mishnah," and therefore it is not clear that he ruled that Davar she'Ein Miskaven is permitted.) Accordingly, the only reason one is prohibited to use Cheres is because it rips off hairs. Since Rava maintains that there is no concern for Sakanah and Keshafim with regard to Cheres, perhaps there is also no concern for Sakanah and Keshafim with regard to all other items, and that is why we may use flammable paper for Kinu'ach. (See CHASAM SOFER there.)
2) PEREK "AMAR REBBI AKIVA"
QUESTION: The previous chapter concludes with a Mishnah that deals with the minimum size of a clay potsherd for which one is Chayav for Hotza'ah when he carries it into Reshus ha'Rabim. The new chapter begins by discussing the Tum'ah that the Rabanan enacted for objects of idolatry. What is the connection between this chapter and the previous one?
(a) RASHI says that the Mishnah here is unrelated to Shabbos. It is included here only because the Mishnah later (86a) teaches that a certain law of Shabbos is derived from an allusion in a verse, and therefore the Mishnah teaches other laws that are inferred from verses. All of the Mishnayos that ask "Minayin" -- "from where [do we learn this law]" and answer with a verse are included here because of the one Mishnah that is relevant to Shabbos, in which it asks "Minayin" and answers with a verse.
(b) RABEINU CHANANEL says that the end of the previous Mishnah cites a verse as an Asmachta for a Halachah mid'Rabanan. This chapter continues with Asmachtos from verses for Halachos mid'Rabanan.
(c) TOSFOS gives the answer of Rabeinu Chananel and adds that the source for the Asmachta in this Mishnah is a verse in the same place in Yeshayah as the source for the Asmachta in the previous Mishnah.
3) "EVEN MESAMA"
OPINIONS: Rebbi Akiva and the Rabanan argue about the extent of the Tum'ah of Avodah Zarah. Rabah and Rebbi Elazar argue about the exact point of dispute between Rebbi Akiva and the Rabanan. According to Rabah, Rebbi Akiva and the Rabanan argue whether Avodah Zarah is compared to Nidah with regard to being Metamei with Even Mesama. Rebbi Elazar maintains that neither Rebbi Akiva nor the Rabanan maintain that Avodah Zarah is Metamei with Even Mesama.
What is the difference between being Metamei with Even Mesama, and being Metamei with Maga and Masa? It seems that Even Mesama and Masa are the same type of transmission of Tum'ah.
(a) According to RASHI in Nidah (55a, DH Even Mesama), Even Mesama is an extension of Tum'as Masa (and applies even to objects which are not suited for sitting upon). It differs from the normal Tum'as Masa, since generally only something which is fit to be moved by the Nidah becomes Tamei through Masa, while something which lies on a very heavy stone which is above the Nidah cannot be moved by her (since the stone is too heavy to carry).
(b) According to RASHI here (DH b'Even Mesama), Even Mesama refers to a large stone which was suspended on poles, on which a Nidah or Zav sits, while utensils are underneath the stone. The utensils are Tamei even though no pressure is put on them by the stone on which the Nidah or Zav sits. This seems to apply even to objects that are not made for sitting upon (see Tosfos), and it is a completely new type of Tum'ah.
(c) According to RABEINU TAM (cited by Tosfos here and in Eruvin 27a, DH Kol), an Even Mesama is such a heavy stone that the weight of the Nidah above it in no way adds any noticeable pressure to any of the objects beneath it. Objects beneath the stone become Tamei with Rishon l'Tum'ah. It is not the same as Tum'as Masa, since Masa requires moving the object and not just holding it (as Rashi explains in Chulin 21b).
(d) TOSFOS (here and in Eruvin 27a, DH Kol, and Nidah 55a, DH Even Mesama) cites Rabeinu Tam's definition of the Even Mesama, but explains that any objects that are fit for sitting upon that lie below the stone will become Tamei. (That is, Even Mesama is an extension of Mishkav u'Moshav.) (See also Insights to Nidah 55:1.)