"From The Second Day Onward"
Moshe David Tokayer commenting on Second Day Rosh Hashana - A Question:
Reb Tzadok HaCohen writes about the second day of Rosh HaShana in Pri Tzadik on Rosh Hashana os 27. Unfortunately, I don't have the time just now to properly translate. The general idea though, is based on a Midrash Shocheir Tov that brings the pasuk in Tehillim, "נר לרגלי דברך ואור לנתיבתי / Your word is a candle for my feet and a light for my path." The Midrash asks about the significance of ner vs or. The Pri Tzadik says that it's also a remez to the time from Rosh Hashana until Yom Kippur. Rosh Hashana represents a candle - the beginning of Torah enlightenment from the source. From the second day onward there is more and more revelation of Torah light until the revelation of salvation on Yom Kippur.
The Pri Tzadik then discusses the significance of the second day of Rosh Hashana based on the pesukim in Nechemia describing the reading of the Sefer Torah on Rosh Hashana during the time of Ezra. The pasuk says that "on the second day the heads of the families ... gathered together to Ezra the scribe, to delve into the words of the Torah."
This, according to the Pri Tzadik, refers to a revelation of Torah SheBeal Peh and it is the source of holiness of the second day of Rosh HaShana. Tosfos in Maseches Rosh HaShana states that from the time of Ezra throughout Bayis Sheni, Elul was never me'ubar. During the entire time of Bayis Sheni there was only one day Rosh Hashana. The exception was the very first Rosh Hashana as this pasuk from Nechemia describes.
It is particularly significant that Bayis Sheni began with a two day Rosh Hashana, the second of which, represents Torah SheBeal Peh since during Bayis Sheni there was the institution of Anshei Knesses HaGedola , the source of revelation of Torah SheBeal Peh.
Only after the destruction of the Second Beis Hamikdash did Rabban Yochanan ben Zakai decree two day Rosh Hashana since it was during his time that Torah SheBeal Peh became fully established with the students of Shamai and Hillel. Obviously, Torah SheBeal Peh was around before from ma'amad Har Sinai. Apparently, though, Reb Tzadok is saying that the machlokes lesheim shamayim of the schools of Shamai and Hillel were the main Torah SheBeal Peh in the sense of elu ve'elu divrei Elokim chaim. The idea of chidush came to the fore during the period of Bayis Sheni and it is for this reason that from that time forward we have two days of Rosh Hashana.