Peah refers to the section of the field that must be left to the poor. While the first Mishnah teaches that on a biblical level there is no minimum, the second Mishnah teaches that the Chachamim required one sixtieth of the field.
The third Mishnah however discusses where Peah must be left. A simple reading of the pasuk would suggest that it must be at the end of one's field: "...you shall not complete your reaping to the corner of your field" (Vayikra 19:9). The way the pasuk is constructed however places the word "likzor" (to cut) at the end, which implies that a section can be defined as peah even if there is more of the field yet to be cut. This point is behind the debate in the third Mishnah.
The Tana Kama teaches that if peah is left near the beginning or middle alone, it is considered peah (Bartenura). R' Shimon however understands, that even if sections are left early in the process, one has not fulfilled their obligation of leaving peah unless they leave a section at the end "of the required size". Exactly what size that is, we shall discuss. It is important to understand that if the obligation of leaving peah was not fulfilled, then some of the cut wheat would need to be separated as peah. For completeness, R' Yehuda understands that for the earlier sections to be considered peah, it is sufficient to leave one stalk at the end of the field (which would satisfy the biblical obligation). While it is true that one could gift wheat to the poor in any case, if produce has the status of peah then it is exempt from terumot and maaserot, while a gift would not.
Let us return to the position of R' Shimon. The Yerushalmi asks whether the peah at the end needs to be for the entire field or only for "that which remains" and concludes that the latter is true. What is the meaning of "that which remains"?
The Bartnura explains, R' Shimon would require that the amount left at the end is enough to ensure that the total amount of peah left for the field is one sixtieth, including the sections that were left early, i.e., "that which remains" of the total sixtieth. It would appear then that the novelty of the pasuk is that the total amount of peah can be left piecemeal in the field, provided some is left at the end.
The Bartenura however continues that the Rambam, in his commentary to the Mishnah, understands that one sixtieth of the entire field would need to be left at the end. According to this understanding, the basic demand of the pasuk is unchanged. One must leave peah at the end of the field. The novelty then is that anything left on the way, could also have the status of peah. How do we understand this explanation in light of the Yerushalmi. The Bi'ur Gra (Yerushalmi) explains that "that which remains" means that a sixtieth must be left at the end proportional to entire size of the field, excluding those sections that have been left as peah. In other words, since the earlier sections will adopt the status of peah, peah need not be left for it.
The Tosfot Yom Tov however notes that the Rambam provides a different understanding in the Mishnah Torah which he understands follows the Yerushalmi. The Rambam explains that the amount of peah that must be left at the end is equal to the one sixtieth of the size of the field that was harvested after the previous peah was left in the middle of the field. The Derech Amuna (2:107) adds that if the section that was left in the middle was not equal to the sixtieth of what harvested until that point, then an additional amount would need to be added at the end also. The Mishnah Rishona adds that even if a sixtieth for the entire field was left in the middle, one would still need to leave a sixtieth for the section harvested after that point. It would appear that according to this understanding, the pasuk is teaching that peah must be left at the end of the field. The novelty is that if one leaves peah in the middle of the field, it is considered as if the field is being split into multiple fields, with peah being left at the end of each one.
One practical difference between these two explanations may be found in the debate regarding whether the poor may start taking the peah from these early sections prior to the end of harvest. According to the Mishnah Rishona, they cannot. Everything is considered peah only once peah is left at the end. This would align with the second understanding that the full amount of peah must be left at the end, and only then do the early section attain the status of peah. The Derech Emuna (Tziyon Halacha 225) however cites the Ridbaz that the poor can take the peah, since those earlier sections have exempted that which preceded it. This clearly aligns with the final understanding that each early peah is consider like the end of the field of that had been harvested.
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