Rosh Hashana 20b

1. The Ba'al Hameor has a beautiful mehalech in explaining the statement about "layla v'yom min hachodesh" and "nolad kodem chatzos...". His premise is that there is a point in the world where the clock is 18 hours behind Yerushalaim (Far East). The halacha is that at the time of the molad in Yerushalaim there must be a place in the world where that entire day is going to be after the molad is seen in Yerushalaim. Therefore, if the molad is before midday in Yerushalaim, that day can be Rosh Chodesh since there is a place in the world that is 18 hours behind and will have the entire day, beginning with sunset, after the molad. But if the molad is after midday then even in a place that is 18 hours behind, the sun would have already set for that day and you don't have the "layla v'yom min hachodesh". This is the source for the opinion of the Chazon Ish about the dateline. I recently gave a shiur about this - click to download: http://www.torahmedia.com/downloadlink.php?fid=22678&bw=low

2. The Brisker Rav (in his sefer on kiddush hachodesh) claims that the gemara challenging shmuel's expertise regarding the 'sod ha'ibur' and Rabbeinu Chananel's commentary is the source for the Rambam (kiddush hachodesh 5:1) that there is a Halacha l'moshe misinai that beis din can establish the rosh chodesh through cheshbon when there is no sanhedrin. Rabbeinu Channanel says that shmuel could make a cheshbon until the coming of mashiach when there would be a sanhedrin, which implies that with the return of the sanhedrin we can no longer use cheshbon and will have to be mekadesh through witnesses. Furthermore, from Rabbeinu Chananel's commentary on the challenge to shmuel, he proves that the expertise is not just in scientific knowledge but to do proper 'cheshbon' one must be an expert also in the traditions passed on from Moshe mi'sinai.