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.
'And you shall rejoice' -- from the holiday Torah reading
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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.
A Single Heart
On Simchat Torah we finish reading the Torah and begin reading it once again. The last letter of the Torah is "lamed" (found in the word Yisrael -- Israel). The first letter of the Torah is the "beit" in B'reishis ("In the beginning").
These two letters together spell the word lev, heart.
The Torah is the heart of the Jewish people and demands that we view each other as one singular heart, pulsating, beating and bringing life to our world and every one of its inhabitants.
You have been shown to know, that the L-rd is G-d, there is none else aside from Him -- from the verses recited on Simchat Torah
All To Be Known
The entire month of Elul, Rosh Hashana, the blowing of the shofar, the Ten Days of Repentance, Yom Kippur, Sukkot, the Four Kinds and Hoshana Rabba are only preparations for the "You have shown to make it known" of Simchat Torah.
(Rabbi Moshe of Kobrin)
Have you ever seen people singing and dancing for hours and hours on Simchas Torah?
The people who are celebrating are humans, not angels.
They each have their own array of worries and troubles.
But on Simchas Torah they are not concerned with these matters at all. They are not thinking of themselves. As they sing and dance, they are connecting to a deeper dimension that exists within their being.
That is where the simchah comes from.
The Frierdiker Rebbe says in Sefer HaSichos 5704, p. 36, that on Simchas Torah, the Torah itself wants to dance. However, since a Torah scroll has no feet, the Jews must function as its feet and carry it around the reader's platform.
This analogy enables us to understand why a person can be so happy on Simchas Torah.
Because he has gone beyond his own identity, he is no more than the Torah's feet, and he can rejoice with complete abandon. And yet, his life will be filled with the meaning and purpose that stems from the Torah he is carrying.