"On Rosh Hashanah, we read 'on the seventh month on the first day', and the maftir reads, 'is not Ephraim my precious son'; and some say we read 'and G-d remembered Sarah', and the maftir with Chana (Shmuel 1,Chapter 1-2). And now that we have two days, on the first day, we do like 'some say', and on the morrow, 'and G-d tested Abraham'; and the maftir, 'is not Ephraim my precious son'" (Megillah 31a). Interestingly, the ruling of the Mishnah that we read "on the seventh month on the first day (a day of blowing it should be for you)" is ignored. That reading has been relegated to the reading of the maftir from a separate Torah scroll, something that apparently was not yet in practice in Talmudic times; the maftir portion referred to in this Talmudic passage is what we call today the haftarah.
The ruling of the Mishnah and the first view in the Gemara are easily understood. We are to read the one and only portion of the Torah that makes reference to Rosh Hashanah. The 'remembering of Sarah' seems to have little to do with Rosh Hashanah. Yet that is what we read.
"On Rosh Hashanah, Sarah, Rachel, and Chana were remembered (conceived)". And Rosh Hashanah is "Yom Hazicharon", the day of remembering. While Rosh Hashanah marks the creation of man, unfortunately, man failed and failed again, leading G-d to choose the Jewish people to carry out His mission. And that mission begins with Abraham and Sarah.
Our Sages explain that our matriarchs had great difficulty conceiving because, "G-d desires the prayers of the righteous". On the surface, this sounds most cruel. Yet prayer is the realization that man is dependent on G-d. We cannot achieve what we want ourselves and must turn to the Master of the Universe. By demonstrating to man that reproduction is much more than a biological function, we can begin to appreciate the miracle of childbirth--something that is so natural, it is easy to forget how miraculous it really is.
It is Chanah's tefillah that serves as the paradigm of prayer for the Jewish people. "Rav Hamemunah said: How many great laws can we learn from Chana? 'And Chana was speaking on her heart'; from here, we learn that one must focus one's heart in prayer" (Brachot 31a). This is the just the first of many laws derived from Chana's prayer for a child. Our mussaf Amidah on Rosh Hashanah consists of nine blessings, corresponding to the nine times Chana invoked G-d's name in her prayer.
Hamelech, the King. So begins Shacharit, highlighting the theme of the day--the coronation of G-d as our King. The theme of kingship found throughout the Rosh Hashanah davening is so central that one who concludes the third blessing of the Amidah with the standard HaE-l hakadosh and not Hamelech hakadosh must repeat the Amidah. Not to mention G-d as King is to miss the essence of prayer at this time of year.
Only one who truly feels dependent recognizing that He is the Master of the Universe can pray. Thank G-d, this is a most difficult belief to master. Modern man has accomplished so much that we often feel no need for G-d. We must continue to achieve, to master nature even at the risk of losing that sense of dependence on G-d. But we must work hard to minimize and eliminate that risk.
Unfortunately, this past year has made prayer easier for many. War in Israel, the increasing threat of radical Islam, even the emerging Ebola health crisis all combine to highlight the limits of our power.
Let us pray that conditions should be such that prayer be most difficult and may we not need any reminders that it is the King of world who provides us with all our needs.
 The Mishnah lists no reading for the second day; apparently Rosh Hashanah, at that point in time, was still only one day.
 This verse actually appears twice in the Torah: once in Parshat Emor where the laws of the holidays are spelled out, and once in Parshat Pinchas where the laws of sacrifices are listed. Fascinatingly, the Torah Ohr, which gives the Biblical source for each verse quoted in the Talmud, references "the seventh day" in the Mishnah to Parshat Emor (Vayikra 23); yet when that verse is quoted in the Gemara, he references us to the verse in Parshat Pinchas (Bamidbar 29). I await enlightenment re: the above.