One Long Day or Two Days?

One does not say the Shehecheyanu [over the lulav and etrog] on the second day - Because either way one has fulfilled one's obligation with saying Shehecheyanu on the first day even if it were a weekday, since one does not detract from anything when one says Shehecheyanu at the time of the construction of the lulav, and one would fulfill the obligation. This is not similar to the Kiddush when one says Shehecheyanu on the second evening, because that is for the day and not the for the sukkah. And even though with regard to the shofar our custom is to say Shehecheyanu even on the second day, that situation is different because sometimes the essence of the holiness is in the second day.

(7) When the fixing of the month was accomplished by the observation method, the majority of the inhabitants of Eretz Yisrael observed Rosh Hashanah for two days, because they were in doubt as to which day had been declared as new-moon day, since the messengers could not go forth on a holyday [and Rosh Hashanah coincides with the first day of the month].

(8) Moreover, even in Jerusalem itself, which was the seat of the court, it frequently happened that Rosh Hashanah was observed for two days. If witnesses failed to arrive on the thirtieth day of Elul, the people were accustomed to celebrate it as Rosh Hashanah, while waiting for witnesses to appear, observing the following day likewise as a holyday. Since they used to keep two days of Rosh Hashanah even when the method of observation was in use, the sages ordained that even the citizens of Eretz Yisrael should keep it for two days always, at the present time when the calendar is determined by computation. You may thus infer that even the second day of Rosh Hashanah is based on the authority of the sages.

On the Second Day - of Rosh Hashanah because in the days of Ezra they intercalated [the month of] Elul and made two days of the holiday for Rosh Hashanah, according to the rabbis.