Before us is the third Mishnah in chapter one of Maseches Eduyos - the last of the three cases in which Hillel and Shammai disagreed and the Sages side with neither of them. The question at hand: how much drawn water, water that had been inside a vessel, invalidates a mikvah when it mixes in before the mikvah has become valid?
Background: the mikvah in the Torah:
The Torah establishes that in order to bring about purity one needs a mikvah, that is, a gathering of water, as it says: "But a spring or a cistern, a gathering of water, shall be pure". Either a natural spring, or a pit of water that collected together on its own - rainwater, melted snow, runoff, a river or floodwater - provided that it was not gathered in a vessel. The very word "mikvah" means a gathering together.
The Rishonim disagree about what the Torah-level requirement actually is:
Rabbeinu Tam and the Rashbam: By Torah law the majority of the mikvah must be rainwater; if the minority is drawn water, water drawn in a vessel, the mikvah is valid.
The Ri and the Rambam: By Torah law, as long as the immersion is not inside a vessel the mikvah is valid - even if one filled the pit in the ground with buckets or through a pipe. It is only by rabbinic law that this was invalidated, out of concern that a person might come to immerse in a bathtub, which is not a valid mikvah at all.
The measure of forty se'ah:
It is a halachah leMoshe miSinai that the mikvah must contain enough water for an average-sized person to immerse his entire body in it comfortably. The Sages fixed the measure precisely: forty se'ah. A se'ah is about six two-liter bottles, let us say about twelve liters (and there is a range of opinions on this), so forty se'ah come to roughly four hundred and eighty liters, about half a cubic meter of water - not a lot, but more than enough.
Once there is a valid mikvah of forty se'ah, nothing can invalidate it any longer: water poured into it touches the water of the mikvah, and through that very contact it becomes purified itself. One may therefore add to it any water in the world and it remains valid. But before it has reached the measure of forty se'ah, the Sages said that a certain quantity of drawn water invalidates it. And the whole question is: what is that quantity?
The three opinions in overview:
Hillel: A full hin. The hin is not a Mishnaic measure but a Torah measure, and it was chosen because it is the largest liquid measure in the Torah. Hillel used this expression because that is how he heard it from his teachers, Shemayah and Avtalyon, and he repeated their words word for word. It is like a person today saying "a gallon," because that is the largest of the volume measures we commonly have, meaning: do not use a liquid measure larger than this.
Shammai: Nine kabin. The hin is half a se'ah, which is three kabin, about three two-liter bottles, while Shammai triples the amount. So, ironically, it is specifically Hillel who is the stricter one: according to him three such bottles are enough to invalidate, whereas Shammai does not invalidate until nine kabin. Shammai's position is rooted in the tradition that the measure for washing the body, like a shower, is nine kabin, a measure that comes up elsewhere in halachah as well, such as regarding a person who has no mikvah.
The Sages: Three lugin. The Sages accepted neither the three kabin nor the nine kabin, and required a stricter measure. A log is about a five-hundred-milliliter bottle, a quarter of a kav, so three lugin are a quarter of the measure Hillel gave.
Let us look at the language of the Mishnah itself:
"Hillel omer: melo hin mayim she'uvin poslin es hamikvah" - as long as the mikvah does not contain forty se'ah, pouring in a full hin, half a se'ah, which is three kabin, about six liters of water, is enough to invalidate it.
The Mishnah itself questions this wording: a hin is a unit of measure that belongs to the Chumash, not to the Mishnah. And its answer: "ela she'adam chayav lomar bilshon rabo" - a person is obligated to transmit the tradition he received in the very same words he heard from his teacher. Hillel heard from his teachers Shemayah and Avtalyon in the language of "hin," and so that is how he transmitted it.
On this point the Bartenura brings the tradition of the Rambam, which he received from his father: Shemayah and Avtalyon, who were converts, did not pronounce the word "hin" correctly, and Hillel pronounced it the way they did, rather than in its precise form.
"V'Shammai omer: tishah kabin" - about nine two-liter bottles, the amount required for washing the body.
"Va'chachamim omrim: lo k'divrei zeh v'lo k'divrei zeh" - not like Hillel and not like Shammai, since they had no precise measure; they knew only that the water invalidates. And when did the measure become known? "Ad sheba'u shnei gardiyim mi'sha'ar ha'ashpos sheb'Yerushalayim" - two weavers who came from the Dung Gate in Jerusalem.
Why are their trade and their place of residence mentioned?
The trade of the gardi, the weaver, was considered a very lowly station: at the end of Maseches Kiddushin the weaver's occupation is listed among the inferior trades, whose practitioners are not appointed to positions of authority because they carry no dignity. Their place of residence also tells us about their situation: the Dung Gate, which literally means the gate of refuse (and it is still called that today, that southern gate in Jerusalem), was the place to which garbage was taken out, to the low part of the city. It was not a desirable place to live, and someone who lived there was poor and of little worth in people's eyes.
And this is the point of the Mishnah, as the Bartenura explains: even though they were poor and simple people, standing at the bottom of the social and economic ladder does not exempt a person from coming to the beis midrash and learning like everyone else. What is more, the halachah was decided like them, even though in terms of this world they were considered nothing at all - while in terms of the World to Come, their names are recorded in the Mishnah to this very day.
The two weavers came "v'he'idu b'shem Shemayah v'Avtalyon: shloshes lugin mayim she'uvin poslin es hamikvah". From their testimony it emerges that Shemayah and Avtalyon reconsidered and changed their opinion after the discussion with Hillel. And if we picture the kav as a two-liter bottle, Hillel said three large bottles, while they testify to only three small bottles - a quarter of the amount.
"V'kiymu chachamim es divreihem" - the Sages ruled like them. And according to many, even Hillel and Shammai themselves, when they heard the testimony, conceded and said: indeed so, we forgot that our teachers had reconsidered. And although Hillel transmitted his teachers' words word for word and in that very same language, he afterwards understood that Shemayah and Avtalyon had reconsidered and adopted the measure of three lugin instead of the hin.
To summarize: the halachah was decided in accordance with the testimony of the two weavers in the name of Shemayah and Avtalyon - three lugin of drawn water poured into a mikvah before it has reached the measure of forty se'ah invalidate it.