Machshirin, Chapter 2, Mishnah 3. This chapter deals with waters whose status is in question: moisture that collects in houses and bathhouses, water that has been used and discarded, and water of one status that has become mingled with water of another. Our mishnah is built around one recurring rule that appears in case after case: when something tamei and something tahor come together, we follow the majority, and when neither side holds the majority, the tumah that we know is present remains in force.
Two pools in one house
"Shtei breichos": two pools of water stand inside a house, "achas taharah ve'achas temeiah": the water of one is tahor, while the water of the other has already contracted tumah. A utensil left in that room begins to sweat, collecting droplets from the vapor that rises off the surface of the water.
"Hamazi'a karov latemeiah, tamei." If the object that is beading with moisture is standing nearer to the tamei pool, the moisture on it is tamei. We assume that the perspiration forming on it comes from the water right beside it, and whatever little may have drifted over from the other pool is nullified in the majority, bitul berov, and is disregarded.
"Karov lataharah, tahor." If it stands nearer the tahor pool, the moisture is tahor, for exactly the same reasoning in reverse.
"Mechtzah lemechtzah, tamei." And if it is standing precisely midway between the two? Then it is tamei. Some of the moisture on it certainly came from the tamei pool, and the tahor side is not a majority, so there is nothing to nullify what is tamei. Since we know tumah is there, it stays there.
Three further cases with the same structure
From here the mishnah moves on to cases that are not directly about machshirin at all. They are recorded here because they follow the very same pattern as the opening case: majority tamei, majority tahor, and an even split.
"Barzel tamei shebalo": iron bearing tumah that was fused together, "im barzel tahor": with iron that is tahor. To grasp what is being discussed, one needs a foundation in the laws of keilim: tumah takes hold of a utensil only while it remains intact and serviceable. Once it breaks, it is no longer capable of contracting tumah, and a utensil that had already become tamei will ordinarily shed that tumah upon breaking. Metal vessels, however, carry an added stringency: if a tamei metal kli shattered and thereby became tahor, and it was later recast into a utensil again, Chazal ruled that the tumah it formerly bore returns to it.
Our case, then, is a fragment of such a kli: it had been tamei, it broke and became tahor, and now it is being melted together with fresh iron in order to make a new kli. The Rav explains that the tahor iron here is iron that was never fashioned into a kli at all, so it never had any opportunity to receive tumah in the first place.
"Im rov min hatamei, tamei": should the greater part of the newly formed utensil derive from the metal of the earlier tamei kli, the whole vessel is tamei by force of that rabbinic ruling, notwithstanding the tahor iron included in it. Where the bulk of it consists of the fresh iron that was never a utensil, the vessel is tahor: "im rov min hatahor, tahor". And "mechtzah lemechtzah, tamei": an equal mixture leaves it tamei, just as with the two pools, since tumah is present here and there is no preponderance of tahor material to swallow it up.
"Gzistros sheYisrael veGoyim matilin letochan": chamber pots into which both Yidden and non-Jews would relieve themselves, that having been how people attended to their needs. "Im rov min hatamei, tamei": on the reading adopted by most commentators, this means that when non-Jews formed the greater part of those who used the vessel, it contracted tumah. When Yidden were the majority of the users, "im rov min hatahor, tahor", the pot is tahor. And an even division, "mechtzah lemechtzah, tamei", still leaves it tamei. All of this rests upon a rabbinic ruling that assigns a non-Jew the status of a zav, and since a zav is tamei, his urine conveys tumah as well. Thus a pot whose contents came largely from non-Jews is tamei throughout; one whose contents came largely from Yidden is tahor; and where the two sides are balanced, the tumah is not displaced.
"Mei shefichos": waste water, "sheyardu lahem mei geshamim": into which rain has come down. The Rav defines mei shefichos as water that a person washed with and then poured off. Water like this bears a chezkas tumah, an assumed status of being tamei, because nobody troubles to protect it: he has no intention of drinking it, he merely bathes with it. Rain has now descended into it, and the two have combined.
"Im rov min hatamei, tamei": where the larger portion of the liquid is the discarded washing water, all of it is tamei. Where the greater share is the rain that has just come down, which is tahor, the entire mixture is tahor: "im rov min hatahor, tahor". And "mechtzah lemechtzah, tamei": equal amounts leave it tamei once more, because tamei water is present here with no preponderance of tahor water to nullify it.
"Eimatai?" Under what circumstances can a majority of tahor rainwater override the tumah? The mishnah answers: "bizman shekadmu shefichos", solely where the used water was in place first and the rain fell into it afterward. "Aval kadmu mei geshamim, afilu kol shehein lemei shefichos, tamei": but if the rainwater was there first and the waste water was poured in later, even a minute quantity of that waste water renders the whole thing tamei, however much rainwater may be standing alongside it.
So the mishnah has given us one principle in four garments: moisture beading in a house between two pools, iron melted into a new kli, a chamber pot used by many people, and discarded water into which rain has fallen. In each one we go after the majority, and where there is no majority, the tumah remains.