Beitzah, Chapter 1, Mishnah 1. Welcome to Maseches Beitzah. We will begin with a few words about the name of the tractate.
The name of the tractate:
'Maseches Bei'a' - the Magen Avraham holds that the tractate should be called by its Aramaic name, 'Bei'a', since a person should not let a coarse expression leave his mouth, and in Hebrew the word 'beitzah' also serves as a term for a man's testicle.
'Maseches Beitzah' - others, as found in the standard text, hold that it should specifically be called 'Beitzah', since that is its true name in Lashon HaKodesh and the Torah itself uses it.
Throughout our study we will use both versions interchangeably - sometimes 'Beitzah' and sometimes 'Bei'a'.
One more point about the name: the subject of the tractate is the laws of Yom Tov, and some of the Rishonim indeed call it 'Maseches Yom Tov'. The name 'Beitzah' was adopted because that is the opening word of the tractate. Our Mishnah opens with the laws of muktzeh as they apply to an egg laid on Yom Tov.
The Mishnah is about to list several halachos in which Beis Shammai and Beis Hillel disagree regarding the laws of Yom Tov. It is worth noting that here, unlike the common pattern in which Beis Shammai are the stringent ones, it is Beis Hillel who are stringent.
"Beitzah shenoldah b'Yom Tov" - an egg that was laid on Yom Tov:
"Beis Shammai omrim tei'acheil" - it is permitted to eat the egg on Yom Tov.
"U'Beis Hillel omrim lo tei'acheil" - it is forbidden to eat it.
The Gemara first clarifies what case we are dealing with: our Mishnah is discussing a hen that is designated for eating, which may be slaughtered on Yom Tov. For that reason the mother hen itself is not muktzeh, since it is designated as food and may be eaten on Yom Tov, and seemingly the egg too should be permitted.
Rather, the case is a Yom Tov that falls right after Shabbos. Beis Hillel hold that every egg laid today was already completed yesterday, so it turns out that Shabbos is, to some degree, preparing for Yom Tov. And the Torah established that a weekday may prepare for Shabbos and a weekday may prepare for Yom Tov, but Shabbos does not prepare for Yom Tov. Therefore an egg laid on a Yom Tov that follows Shabbos is forbidden, even though it happens naturally, by the hand of Heaven, and it is not an action that a person actually performs - and nonetheless it requires preparation from a weekday and not from Shabbos. And since this is the law on a Yom Tov that follows Shabbos, the Sages decreed that it should be the law on every Yom Tov. Beis Shammai, by contrast, do not accept this principle.
The measure of chametz and se'or:
From here the Mishnah moves to another dispute between Beis Shammai and Beis Hillel, concerning chametz. There is ordinary chametz, and there is something called se'or - the term for leavening agent, even though some of today's leavening agents are not chametz at all. Se'or is a more concentrated form of chametz, something that has the power to leaven other things, a leavening agent.
The measure for which a person is liable for eating chametz on Pesach, to the point that he must bring a korban, is a kezayis. The question in our Mishnah concerns the prohibition of bal yera'eh and bal yimatzei, which forbids keeping chametz in one's possession: with what measure does a person transgress the prohibition and bear the responsibility to destroy the chametz?
"Beis Shammai omrim: se'or b'chazayis v'chametz b'chakosseves" - se'or, which is the stronger of the two, has a measure of a kezayis, while ordinary chametz has a measure the size of a kosseves (a date), which is between a kezayis and a kebeitzah.
"U'Beis Hillel omrim: zeh v'zeh b'chazayis" - Beis Hillel are stringent, and equate the measure of se'or and chametz at a kezayis, which is smaller than a kosseves. That is the measure with which a person becomes obligated to destroy his chametz.
In summary: We began Maseches Beitzah - its name, which reflects the word that opens it, and its subject, which is the laws of Yom Tov. We learned the dispute of Beis Shammai and Beis Hillel regarding an egg laid on Yom Tov, and the principle of preparation - that Shabbos does not prepare for Yom Tov, from which the prohibition was extended to every Yom Tov. We also examined their dispute concerning the measure of se'or and chametz with regard to bal yera'eh and bal yimatzei.