Daf 22
1) Originally, Beis Din accepted testimony from everyone. However, when they found out that the Baitusim were trying to ruin their calculation (since they wanted to testify that the new moon of Adar was seen on the thirtieth when it fell out on Shabbos, so Shvuos should fall out on Sunday, since they Darshened “the next day after Shabbos” is the day after the regular Shabbos, which is Sunday. [Tosfos explains: although, by this false testimony, they're ruining the day that Pesach should fall out on; they must have also Darshened (about establishing the month like we do) “you,” whatever you do, even if you purposely change it from what it's supposed to be.] Therefore, they enacted not to accept testimony only from people they know. If they don't recognize them, you need two witnesses to vouch for them.
2) [Tosfos: and they can desecrate Shabbos to testify that the witnesses are reliable.] Even a single witness may desecrate Shabbos since he may find someone to combine with to testify.
3) Everyone is believed to say when Beis Din established Rosh Chodesh. After all, they won't lie for something that the truth will eventually be revealed.
4) Originally, they informed the rest of the country and beyond with torches of cedar wood on a mountain. They would light it and wave it up and down, and left to right. [Tosfos quotes Yerushalmi: so it's not confused to be a shooting star that only goes in one direction.] Then the next mountain lights until it seen on the next mountain etc.
5) They only made this on a short month. After all, if they would do it by both types of months, then you'll have a problem when Rosh Chodesh falls out on Friday since you can't light it that night on Shabbos. You can't say that you can just not light that month and people will know it was because of Shabbos, since people might say that it really fell on Shabbos, and they didn't light on Moitzie Shabbos because they were too drunk from Shabbos. [Tosfos explains: however, now that we don't light if it's a full month, they'll assume that the reason they didn't light on Moitzie Shabbos is because it's a full month, and not that it was short, but they got drunk. After all, it's regular not to light for a full month.]
Daf 23
6) The reason that they didn't just make it only for a full month, and not for short months; since you'll have a problem of unnecessarily keeping people from working for two days. [Rashi explains: regarding Rosh Hashana. If they only light by a full month, then they would always need to wait two days to go back to their work from a Safeik. However, now that they light by a short month, they know the night of the thirty-first that Yom Tov was the day before, and they can do work the second day. Tosfos adds: this reason is really applicable by every Rosh Chodesh, as the Gemara in Megila says that Rosh Chodesh is a day that there is four Aliyos since it won't impede people from working. It must be that they had a custom not to work on Rosh Chodesh. Also, it's commonly said that they added Rosh Chodesh as a Yom Tov to women as a reward of not giving away their gold for the Eigel.]
7) There was a large courtyard in Yerushalayim, and all the witnesses come into it. Beis Din interrogated them there. They made a great feast for them to give an incentive for them to come back.
8) Originally, they didn't leave this place since they left their T'chum (and only have four Amos, or, if they're within an enclosure, they can walk through the whole enclosure). [Tosfos says: this was after the wall of Yerushalayim was breached. After all, a city that's enclosed for people to live there is considered as four Amos, even regarding people that left the T'chum. However, without an enclosure, the city is not considered as four Amos regarding people that left the T'chum, although it's considered as four Amos regarding acquiring your T'chum in the first place at the beginning of Shabbos.] However, then R' Gamliel enacted for them to have two thousand Amos in all directions. (The same goes for midwives who leave their T'chum to assist a birth. This is also for all the people who come to help save from a fire, an army, a rising river, or if houses caved in.)
9) The Gemara has an inquiry: when they originally had to stay in place, [Tosfos: if it was crowded, or they had room. (Mahrsha says: i.e., if they had other courtyards that this courtyard was open to.) The reason that they would be allowed to go from courtyard to courtyard, since, before the breaches, he would be able to walk in the whole Yerushalayim, so now, after the breach, they could at least go to the courtyards and alleys that are open to it.]
10) They interrogated the witnesses if the moon's crescent is facing the sun, or is it facing away from the sun. (Since the sun never sees the crescent side, (in order not to see the missing part of the moon, which it caused), if someone claims that it faced the crescent, he's obviously lying.) They also interrogated: when the sun is in the west right before sunset; if the moon is to its north, or south. (As, during the winter, when the sun sets in the south, the moon is north of it. However, in the summer, when the sun sets in the north, the moon is south of it. [Tosfos says: although, at times the crescent is facing the sun, that's because the sun's cycle is running above the moon's cycle, even though the crescent sees the sun, the sun doesn't see the crescent.]
11) They interrogated the witnesses how high was the moon in the sky. If one says that it's as high as two Mardos (a shepherd's staff), and the other said it's three high, it doesn't invalidate the testimony. However, if one said it was three high, and the other said it was five high, it invalidates the testimony. However, either one can combine to another witness who says like him. [Tosfos says: this fits well according to the opinion that holds that contradicting witnesses don't become invalid witnesses, and they may say other testimony from this point onwards. However, this could also fit well according to the opinion that they become invalid witnesses, but here, since he says this testimony with a second witness, he's believed since a single witness is never believed against two witnesses.]
12) They first interrogate one witness, and they they bring in the second witness and interrogate him. If their testimonies don't contradict each other, they accept their testimony. However, they still ask rudimentary questions of the other witnesses in order that they shouldn't be disappointed that they came for nothing, so that they should come again next time.
Daf 24
13) If they saw the moon's image from a reflection in water, or through clouds, or through a lantern, it can't be accepted. Even if you see half of the moon in the water, cloud or lantern, and the other half you saw in the sky; it's still can't be accepted.
14) If they said that they happened to noticed the moon, but then, when they wanted to purposely see it, they didn't see it; you can't accept it since they might have seen a round cloud.
15) To sanctify the month, the head of the Beis Din says “Mekudash,” and everybody there responds “Mekudash, Mekudash.”
16) The Tanna Kama holds that they said this Mekudash ceremony both when it's a full or short month. R' Elazar b. Tzadok says that it's only for a short month, since heaven already made it Kodesh by a full month. (After all, once they didn't Mekadash the month on the thirtieth, it will definitely be Mekudash on the thirty-first day.) Plimo says that they only said Mekudash by a full month, since, when it comes on the expected day, i.e., on the thirtieth, it doesn't need strengthening. R' Elazar b. Shimon says that you never Mekadesh the month, since he Darshens that you only Mekadesh years, and not months. The Halacha is like R' Elazar b. Tzadok.
17) Someone is forbidden to make his house the same as the Heichel. He can't make a porch the same as the Ullam. He can't make a courtyard like the Azara, a table like the Shulchon, a candelabra like the Menorah, but he can make it with five, six, or eight lamps. The Tanna Kama says that it's only forbidden to make a metal candelabra like a Menorah, since the Torah only allows to make them from metal. The makeshift Menorah that they made when the Chashmanaim came back was made from iron spits. [Tosfos says that they were called 'spits' because of the Gemara in Menachos that says they didn't have engraved cups, apples, or flowers, since those were only said by a gold Menorah.] R' Yossi b. Yehuda held that you can't make a candelabra out of wood, since a wooden Menorah in the Mikdash is Kosher, and the Chashmanaim made the Menorah out of wood.
18) The Torah forbade making the image of the four-faced Chayos Hakodesh. It also forbade making a human image. [Tosfos says: although one of the faces of the Chayos was human, the reason we needed a Drasha for Chayos and we don't say it's anyhow forbidden because of the human face; I might think that, if it had other faces besides the human's, it's better than having the human by itself. Alternatively, if you found an image of a human, you can't finish it off by making three more faces to make a Chaya.]
19) You also can't make the image of the moon, sun, stars and constellations. Although R' Gamliel made moon images in order to interrogate witnesses how they saw the moon, [Tosfos: and even if it was a flat image, it would be forbidden since the moon seems flat in the sky]; we must say that he got non-Jews to make them. [Tosfos says: even though we have an unresolved inquiry in Bava Metzia if there's a prohibition to ask a non-Jew to do for you any other Lav (besides Shabbos); we must say that it's different when it's for a Mitzvah.] However, this is only permitted by R' Gamliel who was the Nassi and the public is common to be around him. Therefore, he has the status of a community that we don't suspect, so we're not worried that people will suspect him that he made it himself. This is the same for some image made for a Shul. However, individuals can't show any images made by non-Jews. Thus, a signet ring with a protrusing image can't be worn, but can be signed (since it makes an indented image). However, an indented image can be worn, but can't sign since it makes a protrusing image. [Tosfos says: the suspicion here is that he's using it as an idol, therefore, we don't suspect a community. The Bahag says that we're referring to a signet of an image of a Darchon, a snake, which is more suspicious to be an idol than other images.]
20) Alternatively, R' Gamliel's moon was made from pieces (and wasn't a full image). Alternatively, since it was to teach, it's permitted. This is like the Drasha about witchcraft, that you can't learn it to perform witchcraft, but you may learn it to be able to understand it so you can Paskin on it.
21) If the witnesses saw the moon in the morning in the East, and towards evening in the West; R' Yochanan b. Nuri held that they must be false witnesses. R' Gamliel had a tradition that the moon sometimes travels in a longer route, and sometimes in a shorter route. [Rashi explains R' Yochanan b. Nuri's complaint: that it couldn't be that the new moon that was seen in the East in the morning can travel across the sky in one day. Tosfos disagrees since it's simple that it can cross the sky in one day. Rather, it's as the Yerushalmi explains that the new moon was seen within twenty-four hours of the old moon, so R' Yochanan b. Nuri says it's invalid. R' Gamliel had a tradition that it might pass a shorter route and make it within twenty-four hours. Although we say that it can't be seen within twenty-four hours, and you can use that to disprove testimonies; we must say that R' Yochanan b. Nuri held that it must be twenty-four larger hours, like a twelfth of a day during the summer. However, R' Gamliel held that it may come early, and it may be only invisible for twenty-four average hours, or even smaller ones, like a twelfth of a winter day.]
22) If they see the image of the moon on the twenty ninth night, they can't make it Rosh Chodesh. After all, you can never have a cycle of the moon less than 29 days and a half, plus two hours and 73 Chalakim.
23) If witnesses say that they saw the new moon on the thirtieth night, but they tried seeing it on the next night, but it wasn't there; R' Gamliel held to accept them. R' Yehoshua and R' Dosa b. Hurkonus held that they're false witnesses. After all, how can someone testify that one day that a woman gave birth, and the next day, she's very pregnant?
24) The Chodesh depends on what the Nassi makes it, whether he made a mistake in his calculation, or purposely made it on the wrong day, or was led astray with false witness.