1) THE KORBAN CHAGIGAH OF THE FOURTEENTH OF NISAN, AND EATING THE KORBAN PESACH "AL HA'SOVA"
OPINIONS: The Mishnah (69b) teaches that one brings a Korban Chagigah along with the Korban Pesach on the fourteenth of Nisan. This Korban is known as the "Chagigah of the Fourteenth." There is no verse in the Torah that commands one to bring such a Korban. What is the source for the obligation to bring this Chagigah of the Fourteenth?
(a) The Gemara implies that this Korban is brought in order to be eaten before the Korban Pesach, so that one will fulfill the Mitzvah to eat the Korban Pesach "Al ha'Sova," while satiated.
What is the source for the Mitzvah to eat the Korban Pesach "Al ha'Sova"? RASHI explains that eating the Korban Pesach while one is satiated makes the act of eating into one which is more significant and meaningful. Eating while satiated enables a person to better experience and appreciate the taste. Enhancing one's appreciation of the taste of the Korban is a form of beautifying the Mitzvah. (The MECHILTA, Parshas Bo, ch. 6, derives from the verse "Al Matzos u'Merorim Yochluhu" (Bamidbar 9:11) that "the Korban Pesach is eaten 'Al ha'Sova,' but not the Matzos and Maror." The Mechilta may be describing the Hidur Mitzvah that Rashi mentions.)
Perhaps the reason why this requirement applies to the Korban Pesach and not to any other Korban is because the Korban Pesach is the only Korban that is brought solely for the purpose of being eaten. All other Korbanos are brought primarily for Kaparah, atonement.
(b) TOSFOS (DH Lav) cites the Yerushalmi (Pesachim 6:4) that seems to conclude that the requirement to eat the Korban Pesach "Al ha'Sova" is only mid'Rabanan. Tosfos suggests that the Rabanan enacted such a requirement as a Gezeirah to prevent people from breaking the bones of the Korban Pesach (which is forbidden by the Torah). If a person eats the Korban Pesach when he is hungry, he might break the bones of the Korban due to his haste. Therefore, the Chagigah is brought so that a person will eat it first and be satiated before he eats the Korban Pesach. (According to Tosfos, the verse cited by the Mechilta might be merely an Asmachta.)
(c) The Gemara mentions another opinion, that of Ben Teima, who maintains that the obligation to bring the Chagigah of the Fourteenth is mid'Oraisa. It is derived from a verse that compares the Chagigah to the Korban Pesach. Tosfos (DH Lav) adds that even the Rabanan who argue with Ben Teima (and who do not compare the Chagigah to the Korban Pesach, and who maintain that the Chagigah is eaten for two days, unlike the Pesach) also derive from a verse (see 71b) the obligation to bring the Chagigah. Accordingly, they both maintain that the Chagigah is an obligatory, and not just optional, Korban, and as such it is brought even on Shabbos, and even when the Tzibur is Tamei.
The Halachah, however, follows the opinion of a third Tana, the Tana of the Mishnah here, who says that the Chagigah is not brought on Shabbos. His source for the requirement to bring the Chagigah of the Fourteenth is apparently one of the sources mentioned above (in (a) and (b)).
(d) The RAMBAM (Hilchos Korban Pesach 10:12) cites a verse as the source for the obligation to bring the Chagigah of the Fourteenth. This implies that he maintains that the obligation to bring the Korban is mid'Oraisa. Nevertheless, the Rambam explicitly writes that it is an optional Korban, and that it is not brought on Shabbos. (TOSFOS 59b, DH b'Arba'ah, also concludes that the Chagigah of the Fourteenth does not override Shabbos, even if it is mid'Oraisa.)
The Rambam seems to understand that even according to the Mishnah, the Chagigah of the Fourteenth is mid'Oraisa. Why, then, does the Gemara say that it is brought merely in order to eat the Korban Pesach "Al ha'Sova"? According to the Rambam, perhaps the Gemara is suggesting why a person would want to bring it if it is not obligatory. The Gemara explains that he benefits from the Korban because it enables him to eat the Korban Pesach "Al ha'Sova." Nevertheless, even without that reason, the Torah gives the option to bring the Korban, perhaps in order for the Beis ha'Mikdash to be filled with Korbanos on Erev Pesach.
2) EATING THE KORBAN PESACH DURING THE DAY AND NIGHT
QUESTION: Ben Teima derives from a verse that the Korban Chagigah is similar to the Korban Pesach. Accordingly, the Chagigah of the Fourteenth is eaten only "for one day and one night," and not for two days and a night like a normal Korban Shelamim.
If Ben Teima compares the Korban Chagigah to the Korban Pesach, why does he say that the Chagigah is eaten "for one day and one night"? The Korban Pesach is eaten only at night, and not during the day. One who eats the Korban Pesach during the day transgresses a Mitzvas Aseh (41b).
ANSWERS:
(a) TOSFOS (89a, DH d'Ilu) seems to understand that the phrase "for one day and one night" means until the end of that period. This phrase does not express the time period during which the Korban may be eaten (for a full day and a full night). Rather, it expresses only the end of the time period during which the Korban may be eaten; the Korban may be eaten only at night, until the end of the night. The time period does not begin during the day, but only at the beginning of the night, and it lasts until the end of the full "day and night" period. (The reason the Gemara expresses the time period in this manner may be because the Korban is slaughtered during the day, and thus it may be eaten until the end of that "day and night" period.) (This is also the understanding of the TOSFOS HA'ROSH in Berachos 9a and the RAMBAN to Vayikra 7:15.)
(b) RAV YECHEZKEL ABRAMSKY zt'l (in CHAZON YECHEZKEL to Pesachim) suggests that perhaps it is appropriate to say that the Korban Pesach may be eaten during the day and night with regard to its status as a Korban. The Halachos of the Korban dictate that it may be eaten during the day and night, like all other Korbanos. However, the specific laws of Pesach require that the Korban be eaten only at night.
The Chagigah of the Fourteenth, which is compared to the Korban Pesach, also has the Halachos of the Pesach element of the Korban Pesach, and thus it may be eaten only at night. Similarly, other Halachos of the Pesach element of the Korban Pesach may apply to it (for example, it must be roasted, and perhaps its bones may not be broken).

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