(לו) שִׁבְעַ֣ת יָמִ֔ים תַּקְרִ֥יבוּ אִשֶּׁ֖ה לַיקוק בַּיּ֣וֹם הַשְּׁמִינִ֡י מִקְרָא־קֹדֶשׁ֩ יִהְיֶ֨ה לָכֶ֜ם וְהִקְרַבְתֶּ֨ם אִשֶּׁ֤ה לַֽיקוק עֲצֶ֣רֶת הִ֔וא כָּל־מְלֶ֥אכֶת עֲבֹדָ֖ה לֹ֥א תַעֲשֽׂוּ׃
(36) Seven days ye shall bring an offering made by fire unto the LORD; on the eighth day shall be a holy convocation unto you; and ye shall bring an offering made by fire unto the LORD; it is a day of solemn assembly; ye shall do no manner of servile work.
(לט) אַ֡ךְ בַּחֲמִשָּׁה֩ עָשָׂ֨ר י֜וֹם לַחֹ֣דֶשׁ הַשְּׁבִיעִ֗י בְּאָסְפְּכֶם֙ אֶת־תְּבוּאַ֣ת הָאָ֔רֶץ תָּחֹ֥גּוּ אֶת־חַג־יקוק שִׁבְעַ֣ת יָמִ֑ים בַּיּ֤וֹם הָֽרִאשׁוֹן֙ שַׁבָּת֔וֹן וּבַיּ֥וֹם הַשְּׁמִינִ֖י שַׁבָּתֽוֹן׃
(לו) וְהִקְרַבְתֶּ֨ם עֹלָ֜ה אִשֵּׁ֨ה רֵ֤יחַ נִיחֹ֙חַ֙ לַֽיקוק פַּ֥ר אֶחָ֖ד אַ֣יִל אֶחָ֑ד כְּבָשִׂ֧ים בְּנֵי־שָׁנָ֛ה שִׁבְעָ֖ה תְּמִימִֽם׃
(36) but ye shall present a burnt-offering, an offering made by fire, of a sweet savour unto the LORD: one bullock, one ram, seven he-lambs of the first year without blemish;
The bull offering brought for Shmini Atzeret
(ב) וַעֲשִׂיתֶ֨ם עֹלָ֜ה לְרֵ֤יחַ נִיחֹ֙חַ֙ לַֽיקוק פַּ֧ר בֶּן־בָּקָ֛ר אֶחָ֖ד אַ֣יִל אֶחָ֑ד כְּבָשִׂ֧ים בְּנֵי־שָׁנָ֛ה שִׁבְעָ֖ה תְּמִימִֽם׃
The bull offering brought for Rosh Hashana
(ח) וְהִקְרַבְתֶּ֨ם עֹלָ֤ה לַֽיקוק רֵ֣יחַ נִיחֹ֔חַ פַּ֧ר בֶּן־בָּקָ֛ר אֶחָ֖ד אַ֣יִל אֶחָ֑ד כְּבָשִׂ֤ים בְּנֵֽי־שָׁנָה֙ שִׁבְעָ֔ה תְּמִימִ֖ם יִהְי֥וּ לָכֶֽם׃
The bull offering brought for Yom Kippur
We can see that Shmini Atzeret was originally created as an agricultural holiday, marking the end of the agricultural year. How then do we relate to this holiday in the diaspora?
"No such historical reference appears in the Torah regarding Shavuot or Shemini Atzeret. As a result, these holidays ran the risk of having their unique qualities lost throughout the years of national homelessness. The Oral Law, however, proceeded to elucidate an additional aspect latent in these festivals. As opposed to the "Biblical Jew," for whom agriculture stood at the center of existence, the "post-Temple Jew" concentrates his religious life in the proverbial "four cubits of Halakha." Therefore, the agricultural calendar was replaced by the "Torah calendar." In this system, the festival of the first harvest (Shavuot), which commemorated the first opportunity to benefit from one's produce, was transformed into the holiday of Matan Torah, the first step in the nation's acceptance of the Torah. Shemini Atzeret, which, for the agrarian society, was celebrated as the end of the year, evolved into the celebration of the completion of the Torah reading, Simchat Torah."
-Rav Amnon Bazak, Translated by David Silverberg
(א)... (שם) ויבא ירושלם ויעמד לפני יקוק ויעל עולות ושלמים ויעש משתה לכל עבדיו א"ר יצחק מכאן שעושין סעודה לגמרה של תורה:
(1)..."'The people of Jerusalem came to stand before God, and [the king] offered burnt and peace offerings and made a feast for all his servants' - Rabbi Yitzchak said: This shows that one makes a feast when completing the Torah."
We know that we celebrate the completion of the Torah at the end of the cycle, but those cycles ended at different times in Israel and in the Diaspora.
"There [in Cairo] were two synagogues, one for those from Israel and one for those from Babylonia… They observed different customs with regard to the reading of the portions in the Torah. The communities from Babylonia read a portion each week, as they do in Spain, thus completing the Torah each year. But the communities of Israel do not follow this practice. Rather, they divide each portion into three sections and finish the Torah every three years. There is among them a custom to join all together and pray on the day of Simchat Torah and on the day of Matan Torah."
-Binyamin of Tuleda, 13th Century
(א) המנהג הפשוט בכל ישראל שמשלימין את התורה בשנה אחת מתחילין בשבת שאחר חג הסוכות וקורין בסדר בראשית בשניה אלה תולדות בשלישית ויאמר יקוק' אל אברם וקוראין והולכין על הסדר הזה עד שגומרין את התורה בחג הסוכות ויש מי שמשלים את התורה בשלש שנים ואינו מנהג פשוט.
... There are those who complete the Torah every three years, but this is not the widespread practice."
ביו"ט הראשון של חג קורין בפרשת מועדות שבתורת כהנים ובשאר כל ימות החג בקרבנות החג
On the first day of the Festival [of Sukkot] they read from the portion of the festivals in Leviticus, and on the other days of the Festival [of Sukkot] the [sections] on the offerings of the Festival.
There is no specific mentioning of a reading for Simchat Torah
On the last day of the festival we read, “All the firstlings” [but first we read] the commandments and statutes [which precede it], and then “all the firstlings” (Deuteronomy 15:19-16:17) and for haftarah, “And when Solomon had finished” (I Kings 8:54-9:1). On the next day we read, “And this is the blessing” (Deuteronomy 33:1-34:12), and for haftarah, “And Solomon stood” (I Kings 8:23-53).
In the writings of the Geonim we find an additional reason for this reading, one which is relevant even according to the custom of the communities of Israel: "In order to juxtapose the blessing of the king [Shelomo] to the blessing of Moshe" (Machzor Vitri).
(א) וַיְהִ֗י אַחֲרֵ֛י מ֥וֹת מֹשֶׁ֖ה עֶ֣בֶד יקוק וַיֹּ֤אמֶר יקוק אֶל־יְהוֹשֻׁ֣עַ בִּן־נ֔וּן מְשָׁרֵ֥ת מֹשֶׁ֖ה לֵאמֹֽר׃...
Our custom, of reading the opening chapter of Yehoshua as the haftara on Simchat Torah, appears for the first time in the writings of Ritz Gi'at (R. Yitzchak ibn Gi'at). The reason is well understood, as this chapter immediately follows the end of the Torah, which is read on this day. Once the Babylonian custom of completing the cycle each year was accepted by communities throughout the world, and, consequently, "Ve-zot Ha-berakha" is read on Simchat Torah even in Israel, the opening chapter of Yehoshua has evolved as the standard haftara for Simchat Torah. This phenomenon reflects in strictly halakhic terms this conceptual transition of the day, from Shemini Atzeret to Simchat Torah.
The Transformation of Shmini Atzeret into Simcha Torah
"It is customary on this day, the day on which we complete the reading of the Torah ... to sing all types of praises of the Torah and to rejoice in all types of celebration, and this day was called, "The Day of Simchat Torah."
-Ritz Gi'at
In the diaspora we do not feel this transformation as much as those in Israel might feel it because this day is split into two here, but the essence of Shmini Atzeret, according to the sages, is that it has turned into a day of rejoicing and celebrating the completion of the Torah cycle.