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1) [line 19] YAD'EI CHAVRIN BAVLA'EI MAI TIVUSA AVDINAN BA'HADAIHU - our Babylonian colleagues realize the great service that they (the Beis Din in Eretz Yisrael) did for them (by making Elul of that year a thirty-day month)
*2*) [line 19] MAI TIVUSA AVDINAN BA'HADAIHU? - the Amora'im wanted to point out to the people of Bavel that *they* stood to gain from the extra day of Elul, since otherwise the Babylonian Jews would complain that the extra day just caused them anguish. (Had there never been an extra day to Elul, they would only have had to keep one day of Rosh Hashanah every year, TOSFOS. Alternatively, their distress was caused by the fact that they only observed one day of Rosh Hashanah, and it turned out to be the wrong day, since Beis Din added a day to Elul, see Background to 19:11)
3) [line 20] MISHUM YARKAYA - so that the vegetables picked on Erev Shabbos should not spoil, leaving nothing for Yom Tov (or so that those picked on Erev Yom Tov should not spoil before Shabbos). Had Elul been a twenty-nine- day month, Yom Tov would have been adjacent to Shabbos. The extra day in between allows fresh vegetables to be picked for each day.
4) [line 20] MESAYA - so that the dead should not decompose
8) [line 26] IY HACHI, MAI SHENA L'DIDAN? - if so, why is it only a benefit for us (in Bavel) when they add a day to Elul?
9) [line 27] CHAVIL LAN ALMA - our part of the world is hot
11) [line 32] ME'AIMIN AL HA'EDIM ... L'ABRO - we intimidate the witnesses [so that they do not testify that they saw the moon] in order to declare Rosh Chodesh on the thirty-first day
12) [line 38] EIN BEIN ATZERES L'ATZERES V'EIN BEIN ROSH HA'SHANAH L'ROSH HA'SHANAH ELA ARBA'AH YAMIM BILVAD The day of the week on which Shavuos and Rosh ha'Shanah fall, respectively, is always four days after its day of the week of the previous year (The Halachah does not follow this opinion.)
13) [line 2] YACHILNA L'TAKUNEI L'CHULAH GOLAH - I could prepare a calendar for the Diaspora [without the aid of witnesses]
14) [line 4] SOD HA'IBUR - a Beraisa that teaches astronomy as it applies to the synchronization of the lunar year with the solar seasons, and that is only understood by the initiated
15) [line 4] NOLAD KODEM CHATZOS - if the Molad (conjunction, or the calculated time at which the moon passes between the sun and the earth) occurs before noon
16) [line 8] TZARICH SHE'YEHEI LAILAH V'YOM MIN HA'CHADASH - the old moon must no longer be visible for an entire night and the following day in order for that day to be counted as part of the new month
17) [line 9] TOLADTO - the time of the Molad
19) [line 13] L'AKCHUSHEI SAHADEI - to contradict [the testimony of] the