Based on a shiur by Harav Yehuda Amital
"And you shall take on the first day the fruit of the hadar tree, branches of palm trees ... and you shall rejoice before the Lord your God seven days." (Vayikra 23:40)
This verse seems to refer to both a one-day and a seven-day obligation. Chazal (Yerushalmi Sukka 3:11; see also Rashi, Sukka 41a) derive from here a distinction between the mitzva of lulav pertaining within the mikdash (Temple) and the mitzva as practiced in the "gevulin" (places other than the Temple). In the Temple, the obligation to take lulav lasts all seven days of the festival, whereas elsewhere the obligation is limited to the first day. According to several Rishonim, there are many other differences between the first day and the rest of the holiday. The Rambam, for instance, believes that the disqualifications listed in chapter 8 of his Hilkhot Lulav render a lulav unfit only on the first day, but do not apply on the rest of the days.
Having established the uniqueness of the first day of Sukkot with respect to the other days, the question can be asked: "What underlies this distinction?"
Rav Joseph B. Soloveitchik (in his Kovetz Chiddushei Torah) explains that the taking of the lulav fulfills different functions on the first day and on the rest of the holiday. On the first day (in the gevulin), the mitzva is a formal one of performing a specific act; however, the seven-day, mikdash-centered obligation is to rejoice before the Lord by means of taking the lulav. Rav Soloveitchik points to the verb usage in the source text as support for his theory, specifically, the shift from "and you shall take" regarding the one-day obligation to "and you shall rejoice" in reference to the seven-day obligation. Rav Yerucham Fischel Perlow (in his commentary to Rav Sa'adia Gaon's Sefer Ha-mitzvot) makes a similar distinction and draws practical ramifications from it. For example, if a person were to take lulav on the first day outside of the Temple, and later he were to arrive at the Temple, he would be required to take lulav again, since the mitzva in the mikdash is one of rejoicing and not merely of lifting.
However, a mishna in Sukka (4:1) presents a difficulty to the above distinction. The mishna discusses the rabbinic decree that when the first day of Sukkot coincides with Shabbat, one should fulfill the mitzva of lulav at home and not at the Temple. (When the first day fell on Shabbat, people would bring their lulavim to the Temple before Shabbat and leave them there. When they came back on Shabbat, there was such a commotion surrounding the lulavim that people would be injured, and the rabbis decreed that people should leave their lulavim at home.) If indeed the requirement of lulav in the mikdash and in the gevulin are completely separate, how could performing the mitzva at home replace the mikdash-oriented obligation? (This, of course, assumes that Yerushalayim was not considered part of the "mikdash" for the purpose of this mitzva.)
Rav Soloveitchik resolves this difficulty by qualifying his original position; the distinction between the one-day and seven-day obligation exists only on the final six days. On the first day itself, however, the two obligations are identical.
Nevertheless, the dual-obligation approach to the mitzva of lulav can be attacked on a more fundamental level. The Rambam, both in Sefer Ha-mitzvot and in the heading to Hilkhot Shofar Ve-lulav, counts lulav as a single mitzva. Moreover, he cites lulav as an example of a single mitzva which is performed at separate times (Sefer Ha-mitzvot, Shoresh 13).
From this it can be concluded that the mitzva of lulav is a single mitzva that consists of two intertwining obligations: the requirement to perform the act of taking a lulav, and the aspect of rejoicing. On the first day, both obligations are present, as it had been in the mikdash. Only the rejoicing aspect remains during the final six days, although it is itself a fulfillment of the mitzva of lulav.
Now perhaps we can answer a question which has perplexed many. Mishna Sukka (3:1) ascertains the minimum length of a lulav as three tefachim (handbreadths) with which to shake the lulav. This implies that the obligation to shake the lulav is of Biblical authority, since it is the criterion used to regulate the minimum length of a proper lulav. This presents a difficulty for the Ba'al Ha-ittur, who explains that the obligation to shake the lulav (as opposed to lifting it) is of rabbinical authority.
This difficulty can be resolved in light of our conclusions. The obligation to shake the lulav falls within the rejoicing aspect of the mitzva, which essentially applies only in the mikdash, although rabbinically it has been extended to the gevulin as well. Thus, nowadays, the requirement of shaking the lulav is rabbinical in origin, and it is specifically to this situation which the Ba'al Ha-ittur referred.
(Adapted by Aviad Hacohen; translated by Eli Fischer.)
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