1) THE MINIMUM AMOUNT REQUIRED FOR CHALAH
QUESTION: The Gemara quotes Rav Huna who says that there were three occasions on which Shamai and Hillel argued. The first involved the minimum amount of dough required for Chalah to be separated from the dough. Shamai required that there be at least one Kav of dough in order for one to be obligated to separate Chalah from it. Hillel required two Kavim. The Chachamim argued with both Shamai and Hillel and said that there must be at least one and a half Kavim of dough.
The Chachamim's opinion is simple to understand. One and a half Kavim is equivalent to the quantity of Man that fell for each person every day when the Jews traveled through the wilderness. (There are 432 eggs in one Eifah. Hence, 1/10 of an Eifah equals 43.2 eggs, which equals 1.8 Kavim in Midbariyos units (24 eggs = 1 Kav), which is 1.5 Kavim in Yerushalmiyos units.) The Chachamim derive from a verse that the requirement of Chalah is equivalent to the amount of Man that fell for each person. What, though, is the reasoning of Shamai and Hillel?
(a) TOSFOS (DH Shamai Omer) cites the Yerushalmi that explains that Shamai and Hillel also derived their amounts from the Man, but they measured the Man differently from the way the Chachamim measured it. Even though the Man that fell each day was 1/10 of an Eifah, they both understood that the tenth was a tenth "from the outside" (that is, if it is added with another nine parts, it equals ten parts), or, in our terms, 1/9 of an Eifah. (The Man, therefore, measured 48 eggs (1/9 of 432), which equals 2 Kavim.)
Hillel maintained that Chalah must be separated from dough of that quantity, and thus he required at least 2 Kavim. Shamai said that Chalah needs to be taken from one meal's worth of the quantity of Man that fell. Since the Man fell in quantities of two meals' worth, and the total amount of Man was 2 Kavim, Shamai required that Chalah be separated from 1 Kav of dough. (Accordingly, Hillel and Shamai are not discussing the same type of Kavim that the Chachamim are discussing. They are discussing Kavim of Midbariyos units, and the Chachamim are discussing Kavim of Yerushalmiyos units.)
(b) The RASHBA cites RABEINU TAM who explains the respective reasons of Hillel and Shamai. Hillel and Shamai agree that the amount of Chalah that the Torah requires us to separate is a significant amount, which means a measure of at least one egg. The Torah, therefore, requires us to separate Chalah only when there is enough dough from which to separate one whole egg unit of dough as Chalah.
The Mishnah (in Maseches Chalah) teaches that a Ba'al ha'Bayis (one baking bread for his own personal use) separates 1/24 of his dough for Chalah, and a professional baker (who uses large amounts of dough) separates only 1/48 for Chalah. This fair percentage must have been the practice before the Rabanan officially instituted it as law. Hillel maintained that one Kav of dough is the minimum amount that can be obligated in Chalah, because when the Ba'al ha'Bayis separates 1/24, he has one egg's worth (because 1 Kav = 24 eggs). Shamai maintained that the Torah's directive to separate Chalah also refers to a professional baker, and thus the minimum amount of dough that is obligated is 2 Kavim (such that when one separates 1/48 of it, he separates one egg's worth).
2) SPEAKING IN THE DIALECT OF ONE'S REBBI
QUESTION: Hillel said that a "Melo Hin" of drawn water invalidates a Mikvah. Hillel concluded his statement by saying, "... because a person is required to speak the way his Rebbi speaks." What was Hillel's intention in making that statement?
(a) RASHI (DH she'Chayav) explains that Hillel was explaining why he used the name for the unit of measure that the Torah uses ("Hin"), and not the name of the measure used by the Rabanan ("Lug").
(b) The RAMBAM (in Perush ha'Mishnayos, Eduyos 1:3) explains in the name of his father that since Hillel's teachers, Shemayah and Avtalyon, were converts (Berachos 19a), they spoke with the inflection of the language of the land in which they were raised. Instead of "Melo Hin," they pronounced the words, "Melo Ein." Hillel would therefore repeat the words with the same mispronunciation of his teachers and excuse himself, "I know that I mispronounced it, but that is the way my Rebbi spoke!"
(c) The VILNA GA'ON (Eduyos 1:3) explains that it does not seem reasonable to suggest that a person is required to mispronounce words the way his Rebbi did. Instead, he explains that since Shemayah and Avtalyon could not pronounce "Hin" properly and said "Ein" instead, they feared that their students might misunderstand what they were saying. They might think that their teachers were saying "Ein Mayim She'uvim Poslim Es ha'Mikvah" -- "drawn water does not invalidate a Mikvah," instead of "Hin Mayim She'uvim Poslim..." -- "a Hin of drawn water invalidates a Mikvah." Therefore, they would phrase their Halachah by saying, "Melo Hin Poslim Es ha'Mikvah" -- "a full Hin of drawn water invalidates the Mikvah," so that the students would not misunderstand it. When Hillel repeated it, although he pronounced "Hin" properly and it was not necessary for him to say "Melo Hin...," he nevertheless added the word "Melo" because that is the way his Rebbi taught the law!
(d) The DOROS RISHONIM explains that all three Tana'im -- Shamai, Hillel, and the Chachamim -- had received through oral tradition their opinions as to how much drawn water invalidates a Mikvah. Their oral tradition was common in that it said that a Mikvah is invalidated by 3 of some measure (Shamai said 9 Kavin, which is 3 Hin; the Chachamim said 3 Lugin; and Hillel said a Hin, which is 3 Kavin). Shamai and Hillel did not specifically use the number 3 with the appropriate unit of measure, even though that was the tradition they had received, because they were saying it the way they had heard it from their teachers.

15b----------------------------------------15b

3) THE TWO STAGES OF THE DECREE OF GLASS VESSELS
QUESTION: The Beraisa says that Yosi ben Yo'ezer and Yosi ben Yochanan were the first to enact the decree that land outside of Eretz Yisrael ("Eretz ha'Amim") makes a person Tamei. The Gemara challenges the Beraisa from sources that clearly state that later Chachamim instituted that decree. The Gemara answers that Yosi ben Yo'ezer and Yosi ben Yochanan originally made the decree, and the later Chachamim added to it.
However, this does not answer the question regarding the second decree enacted by those two Tana'im, the decree that glass vessels can become Tamei, which the Gemara also asserts was made by later Chachamim. If the same answer applies (that the later Chachamim merely added to the original decree), then what did they add? They could not have added that the Tum'ah of glassware requires Terumah to be burned, because the Gemara (16a) explicitly says that it causes Terumah to be Toleh and not burned (as a reminder that the Tum'ah of glassware is not mid'Oraisa).