The Mishnah (4:2) discusses a case of an animal, e.g. a cow, that is have difficulty in its first birth. If the calf is delivered it has the sanctity of a bechor, and if it dies during delivery, no benefit may be gained from the carcass and instead it must be buried. Nevertheless, the Mishnah teaches that as limbs exit the body, they can be severed and feed to dogs. Once however a majority of the calf is delivered, it must be buried, as it then has the sanctity of a bechor. The Mishnah adds that the next calf born would not be considered a bechor.
The Bartenura explains the when the Mishnah states that if a majority was delivered it must be buried that is when a majority of the body was deliver in one go and only after cut up.
The Tifferet Yisrael adds that if the limbs were cut as they exited and not immediately fed to the dogs, but instead collected, then only those parts of the animal after a majority exited would have the kedusha (sanctity) of a bechor. That is because the kedusha only begins from the point that a majority has been deliviered.
The Tosfot R' Akiva Eiger (on the Mishnah) however cites that Gemara that concludes that even if parts of the calf were cut off and collected, then as soon as a majority has exited the animal it all must be buried. In other words, since a majority of the calf is present, it is considered as if a majority was delivered at once. This is indeed the position Rambam (Bechorot 4:14) who explains that since a majority of the animal is present, irrespective of how it was delivered, it all become retroactively kadosh.
Tosfot R' Akiva Eiger (on the Rambam) notes that according to the Rambam if the limbs work collected, and not immediately fed to the dog, then the person would have violated the prohibition of inflicting a blemish on kodshim.1 He continues by citing the Tosfot that it is only those limbs that are present that become retroactively kadosh once a majority has exited the cow.
The Rambam also rules that in the case where the limbs were cut and immediately fed to the dogs, the next animal born would be considered the bechor. This means that the final statement in the Mishnah that the next animal would not be the bechor was only referring to the last case in the Mishnah the a majority of the animal was born (or present). It would appear then that in the case where the limbs were immediately fed, it is not considered as if the first animal was born and therefore not considered the first born. Consequently, even the limbs that follow would not have the kedusha of a bechor.
The Chazon Yechezkel finds the Rambam position difficult based on the Tosefta (Nida 4:6) that explains that if a woman give birth, whether the child is born whole or in pieces, once a majority has exited, the child is considered born and mother because tameh. The Rambam rules accordingly and does not differentiate whether the limbs are present or not. Why in the case of the bechor must the limbs be present whereas for childbirth there is no mention of this requirement?
The Chazon Yechezkel suggest that there is a difference between kedushat bechor and leida (birth). Kedushat bechor is a law that applies to the calf, whereas the law of leida applies to the mother. For leida, once a majority of the child is delivered, it is considered that the mother has given birth and she is tameh. For bechor, since the parts of the calf have been consumed by the dogs by the time a majority has been delivered, a majority of the calf is not present to be deemed a bechor.
1 Based on this, the Tosfot R' Akiva asks how the Mishnah could permit cutting the limbs even if they were immediately fed to the dog. One should be concerned that perhaps, before having the chance to the feed the limb, the rest of the animal might be delivered which would result in the violation of the prohibition of causing a blemish to kodshim.
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