The Mishnah (9:1) records the debate regarding a situation where a drop of water falls into mei chatat. R’ Eliezer maintains that one should perform hazaya (sprinkling) twice, instead of once and the water is valid. The Chachamim however explain that all the mei chatat is invalid and may not be used for purification from tumat met.
The Tosfot (Zevachim 79b) explain that on a biblical level, with respect to mei chatat, the invalid water would be batel (annulled) in the overwhelming majority of valid water. Consequently they explain that this debate is regarding the extent of the rabbinic imposed stringency on mei chatat.
The Gemara (Zevachim 80) discusses this debate in great detail and the opinion of R’ Eliezer in particular. Rava understands that R’ Eliezer maintains the concept of “bila”; meaning that one treats the foreign drop is if it as has mixed evenly throughout the water. Furthermore, R’ Eliezer maintains that there is no minimum shiur (measure) of water that must come into contact with the person undergoing the purification. Consequently, one hazaya would be enough. Nevertheless the Chachamim instituted a knas (fine) requiring sprinkling twice in order that one should not benefit from mixing in the foreign water.
Rav Ashi however understands that R’ Eliezer does not hold by the principle of bila. Consequently there is a concern that the first hazaya will consist completely of the foreign water. Therefore two hazayot are required guaranteeing that some mei chatat will fall on the person at least once.
One may be tempted to ask that performing hazaya on a tahor person would cause them to be tameh. Consequently, according to both answers, it is possible that the extra hazaya would cause the person to be tameh. The Tifferet Yisrael pre-empts this question and answers that indeed the person would be required to immerse in a mikveh following the hazayot to rid himself of this lower level of tumah.
The Chachamim however argue that the water is invalid. The Bartenura explains the Chachamim require a full shiur in one hazaya. Maintaining the principle of bila, this would not be possible.
Returning to the opinion of R’ Eliezer, the Rambam explain that the two hazayot are not placed on the person, but rather cast on the ground prior to performing hazaya. After that, all the remaining water would be valid. The Bartenura is at a loss for the source of the opinion of the Rambam, as he understands the Gemara as it was explained above.
The Tosfot Yom Tov explains that the Rambam is holding like the opinion of Rav Ashi who does not hold by the principle of bila. Consequently once the two drops are cast aside, the foreign drop is assumed to be removed and the remaining water is valid (talinan). The Mishnah Achrona explains further. Since R’ Eliezer does not by the concept of bila, it is comparable to a case of mixed dry product (yavesh b’yavesh) and the foreign drop is really batel in a majority (as mentioned above). Now ordinarily the concept of talinan is not employed from the outset. It is only used in specific situations, after the fact, for example if part of the mixture fell in the ocean (see Shulchan Aruch YD 140). This case is different. The Mishnah Achrona explains that since this case does not involve food or items offered in the Beit Ha’Mikdash, R’ Eliezer employs a leniency within the law of talinan.
The Tosfot Chadashim however explains that the Rambam holds like Rava. Rava maintained that the two hazaya requirement was simply a knas preventing one from benefitting from the mixture. Consequently casting the water to ground would satisfy. Note that that solution would avoid the requirement of following the hazaya with tevilah mentioned earlier in the name of the Tifferet Yisrael.
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