On the eighth day there shall be an assembly ("atzeret") to you -- from the holiday Torah reading
All of the Divine revelations and G-dliness which are brought down from Above during the seven days of Sukkot in an "encompassing" manner are absorbed and internalized on Shemini Atzeret; indeed, the word "atzeret" itself implies absorption a nd assimilation.
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'And you shall rejoice' -- from the holiday Torah reading
On the first day of Sukkot a Jew fulfills three separate mitzvot: eating in the sukka, making a blessing on the Four Kinds, and simcha -- rejoicing in the holiday. On the subsequent days of Sukkot he fulfills two mitzvot: sukka and rejoicing [as the main mitzva of the "four kinds" is on the first day]. On Shemini Atzeret, however, only one mitzva remains: "And you shall rejoice."
Genuine Desire on Simchat Torah
There is a unique dimension to Simchat Torah in that every Jew is given an aliya to the Torah. Generally, on a festival, five aliyas are called to the Torah; on Yom Kippur, six; and on Shabbat, seven. Simchat Torah is the one and only time in which every person in attendance is given an aliya. Although it involves time and one might think that it would be improper to delay the entire communal prayers for this reason, this practice is followed. Why? Because it is the genuine desire of each member of the community that every person present receive an aliya.
(The Rebbe, Simchat Torah, 5752-1992)
A Torah Which is Always New
We begin the Torah anew on Simchat Torah to show that "the Torah is beloved to us like a new object and not like an old command which a person does not submit to. It is like a new one toward which everyone runs.