Gemara

(Mishnah): If Dimu'a (a mixture of Terumah and Chulin) became mixed with Chulin, it is Medame'a (forbids the mixture to Zarim or Teme'im, as if it was Terumah) according to the calculation (of how much Terumah it contains);

If Mayim She'uvim (water that was in a Kli) fell into an (incomplete) Mikveh, it forbids it according to the calculation.

(R. Chiya bar Aba): Our Mishnah is like R. Eliezer ben Yakov;

(Beraisa - R. Eliezer ben Yakov): If there are 21 Sa'im of rainwater in a Mikveh, one may pour 19 Se'ah into a furrow so they will fall in, and the Mikveh is Kosher;

Objection: Chachamim agree to this!

(Rabah): Rather, the Mishnah means that Mayim She'uvim forbid based on the number of Kelim. It is like Yosef ben Choni:

(Beraisa - Yosef ben Choni): If three Lugim were poured into a Mikveh from at most three Kelim, they disqualify the Mikveh, but not if they were from four or more Kelim.

Avodah Zarah 73a (Ravin citing R. Yochanan): If Yayin Nesech fell into a pit of wine, and also water fell in, we ignore the Heter wine. If there is enough water to be Mevatel the Yayin Nesech, it is permitted.

Version #1 (Rav Shmuel citing R. Yochanan): This is only if the water fell in first. If the Yayin Nesech fell in first, it forbids the whole pit.

Version #2 (Rav Shmuel citing R. Yochanan): The Mishnah says that Yayin Nesech forbids b'Mashehu, but if also water fell in, we ignore the Heter wine, and if there is enough water for Bitul, it is permitted. (end of Version #2)

73b (R. Yochanan): If two cups of diluted wine, Chulin and Terumah, got mixed, we ignore the Chulin wine, and the water (if there is enough) is Mevatel the Terumah.

Chulin 108b (Beraisa - R. Yehudah): If a drop of milk fell on a piece of meat in a pot, and the milk gives Ta'am to the piece, it becomes like Neveilah, and forbids all the pieces, for they are Mino (the same species);

Chachamim forbid them only if it gives Ta'am to the sauce, Kipah (thick gravy with bits of meat) and pieces.

Rebbi: I agree with R. Yehudah when the pot was not stirred or covered. I agree with Chachamim when the pot was stirred or covered.

Rishonim

Rambam (Hilchos Ma'achalos Asuros 9:8): If meat fell into milk or vice-versa and they became cooked, if it absorbed taste, it is forbidden. If he did not remove the piece before it emitted the milk inside, we estimate if there is 60, for the absorbed milk that became forbidden leaves and mixes with the rest of the milk.

Magid Mishneh: The Rambam applies CHaNaN (Chatichah Atzmah Na'asis Nevelah, i.e. a piece forbidden due to absorptions is considered like pure Isur) only to meat and milk. He did not mention it regarding other Isurim. In Hilchos Terumos (13:3) he said that Meduma forbids according to the calculation. He did not distinguish it from other Isurim.

Rosh (7:38): R. Efrayim applies CHaNaN only to meat and milk.

Ran (43a DH Tanan): We learn CHaNaN from meat and milk, so we apply it only similarly to meat and milk, i.e. through Bishul (cooking in water).

Question: When a piece of Nevelah or Tamei fish fell on a piece, it gives taste to it. That piece becomes Nevelah, and forbids all of the pieces. If so, when Yayin Nesech fell into wine, it should make the wine like Nevelah, and we should require enough water to be Mevatel all of the wine!

Answer (Ran Avodah Zarah 36a DH v'Omar): CHaNaN applies only to things cooked, like meat and milk. The Torah forbade them Derech Bishul, or if cold fell into hot, for also this is Derech Bishul, but not in other ways. CHaNaN for other Isurim is only mid'Rabanan. They decreed due to meat and milk, for which it is mid'Oraisa, therefore they decree only in similar cases. This is why in Temurah we say that Meduma forbids only based on the calculation (of how much Terumah there is), and similarly regarding fermenting.

Ran (Sof 44b): In Version #1 in Avodah Zarah, we say CHaNaN even for a mixture of wet and wet. The Ramban says that Tana'im argue about this in a Mishnah. R. Yehoshua says that Mayim She'uvim disqualify based on the calculation. Chachamim consider the entire mixture like She'uvim. They argue about wet and dry Isurim. The Gemara did not ask about the contradiction between the Sugyos.

Rebuttal (Ran): That Mishnah does not discuss a mixture of She'uvim and Kosher water, rather, the number of Kelim from which the water fell. It says 'according to the calculation' for parallel structure with the clause about Meduma. Rather, CHaNaN is mid'Oraisa for meat and milk, for each of them becomes Isur.

Poskim

Shulchan Aruch (YD 92:4): We say CHaNaN only for Basar v'Chalav, but not for other Isurim.

Beis Yosef (DH u'Pasak): CHaNaN applies only when milk fell onto meat, for each of them is permitted by itself, and together they are forbidden. The meat itself becomes Isur, and one is lashed for half a k'Zayis of the meat and half a k'Zayis of the milk.

Gra (15): The Yerushalmi says that for all Isurim, if more Heter was added b'Shogeg, they are permitted.

Rema: Some apply CHaNaN to all Isurim. This is the custom. One should not deviate.

Beis Yosef (ibid.): The Ri said that we should be stringent, for there is no clear proof. Semag said that we should not be lenient like R. Efrayim, for all the early Chachamim disagree. The Mordechai (697) applies CHaNaN to other Isurim only to the piece itself, due to its importance. It is a recognized, separate Isur, and became forbidden by itself. How can Heter help to be Mevatel Isur absorbed in it?! We need enough to be Mevatel the entire piece if it is not Chatichah ha'Re'uyah Lehiskaved or Davar sheb'Minyan. However, regarding something wet, e.g. Terumah in Chulin, wine in water of idolatry, or gravy in gravy, there is no importance to the Isur. The second Heter joins with the first. CHaNaN does not apply at all. The Rashba, Ran and Ra'avad apply CHaNaN to Isur absorbed in a piece. The Isur does not leave the piece; it remains forbidden like a Nevelah. If the Isur and Heter are both wet, e.g. wine or water, even though it was initially forbidden, when more Heter falls in, it totally mixes. Therefore, it forbids only according to the calculation. They rejected those who say oppositely, that CHaNaN is when both are wet and they totally mix.

Shach (12): The Agur says that all agree that CHaNaN is only mid'Rabanan for other Isurim.

R. Akiva Eiger: Tosfos (100b DH beshe'Kadam) asked how they Kashered big pots taken in the war against Midyan. This shows that he holds that it is a Torah Isur!

Rema: This is only if the Isur clings to the Heter, or it is totally outside of the liquid and Isur fell on it. If part of the piece was in the liquid, and the Isur does not stick to it, we do not say CHaNaN, and the entire pot joins to be Mevatel the Isur. In any case one should be stringent to forbid the piece.

Shach (13): The Rema made a compromise about when to say CHaNaN. This requires investigation. The Shulchan Aruch learns from Isur v'Heter that CHaNaN applies when part of the piece is outside the liquid. Isur v'Heter explicitly says that the entire pot joins to permit the piece! The Shulchan Aruch applies CHaNaN to meat and milk when part of the piece is in the liquid. All the Poskim disagree. The Gemara says that the piece on which milk fell forbids (up to 60 others) only if the pot was stirred. The Ri and R. Tam says that this is when the entire piece is outside the liquid. If part is in the liquid, the entire pot helps to permit even the piece itself! Even though Rashi says that part is in the liquid, if becomes like Nevelah to forbid others only through stirring. It seems that the Rema discusses when it was stirred and the piece was totally covered. The Rema is stringent about other Isurim because he is unsure whether we follow Rashi or the Ri. He was lenient about the rest of the pot, for in any case some say that CHaNaN is only for meat and milk. All the Mechabrim and Acharonim hold like the Ri.

Shach (14): This opinion holds that CHaNaN applies only when wet Isur was mixed with dry Heter. We say CHaNaN because the pieces are important. We do not say it if wet Heter and Isur mixed, even if later the mixture was mixed with dry or wet Heter.

R. Akiva Eiger: Based on this, even if all the pieces were shredded and dissolved, we require 60 times as much as all of them, since they became Nevelah, since they were important. However, the Rashba (Toras ha'Bayis ha'Aruch p.97) says that CHaNaN applies to piece because the Isur remains, but a liquid mixes uniformly, so everything joins for Bitul. If so, here it is like a liquid, for it mixes uniformly.

Rema: This applies to other Isurim. For meat and milk, even if the Isur does not cling, and part is in the liquid, we say CHaNaN.

R. Akiva Eiger: The Shach (87:3) says that we are stringent even for meat and milk mid'Rabanan.

Rema: Some do not apply CHaNaN if a wet Isur and Heter mixed, and later it was mixed with more Heter. We need 60 only based on the original Isur. One may rely on this for other Isurim to avoid a big loss, but not for meat and milk. If dry Isur and Heter mixed, we do not say CHaNaN for any Isur.

Taz (16): The Rema discusses a dry mixture without cooking. If it was cooked, the Isur would be Batel is the majority. If it was initially mixed with less Heter than itself, and later it was mixed with more Heter, a majority against the Isur suffices. If it was cooked, CHaNaN applies, and we need 60 times as much as the piece. The Tur says that R. Tam applies CHaNaN only if it was forbidden through taste, but not through Mashehu, even if the piece was forbidden due to Min b'Mino according to R. Yehudah. All the Poskim bring R. Tam. The Levush says that when Chametz or Yayin Nesech forbids b'Mashehu, it does not become Nevelah. If it falls into another pot, it is Batel is the majority. If one recognizes the Isur, he must discard it. The Shulchan Aruch (OC 447:1) says that Chametz forbids b'Mashehu. This refers only to the first mixture. This is why the Tur said that CHaNaN does not apply to Chametz. The Bach said that he discusses Chametz after Pesach. This is wrong.

R. Akiva Eiger: Mishbetzos Zahav (100:1) discusses whether CHaNaN applies to something that became forbidden through an absorption less than one part in 60, e.g. the Isur could be tasted, or to was to spice or ferment.

Shach (16): This opinion is lenient even for meat and milk if more Heter was added afterwards.

R. Akiva Eiger: If later the piece was cooked with other pieces, then we say CHaNaN.

Gra (18): The Rema is stringent like Rashi for meat and milk, and lenient like the Ri for other Isurim. The other Poskim totally rule like the Ri.