Eduyos, Chapter 1, Mishnah 1. The first chapter of Eduyos deals mainly with the disputes between Hillel and Shammai - not Beis Hillel and Beis Shammai, but Hillel and Shammai themselves. In the first three mishnayos we will look at cases in which the two of them disagreed, and the Sages ruled neither like the one nor like the other.
Background: How a Woman Becomes a Niddah:
The first mishnah deals with matters of tumah and taharah, and it revolves around a woman who has become a niddah. The place where a baby develops is called the womb, and when there is no baby in it the lining may come out in the form of blood. The moment the blood leaves the womb and enters the corridor - the outer passage leading out of her body - the woman becomes tamei. If this happened as a normal part of her cycle, she is a niddah; if it happened in the middle of her cycle, she is a zavah. In any case, the blood's leaving the womb and entering the corridor is the moment at which the woman becomes tamei.
The Question Before Us:
The concern the mishnah addresses is a woman who was handling taharos - Terumah or sacred foods - and during her lunch break discovered that there was blood there. She is certainly tamei, but the question is how we can know when the blood reached its place: perhaps it arrived much earlier and remained in the corridor without emerging.
Strictly speaking, in any situation of doubt we first turn to the chazakah, the halachic status quo, known as 'chazakah d'meikara' - the previous state, in which the woman was tehorah. According to this we should assume she was tehorah until the moment the blood was seen. The mishnah, however, indicates that the matter is not so simple.
Shammai's Position:
"Shammai omer: kol hanashim dayan sha'atan" - according to Shammai, until a woman sees the blood she may consider herself tehorah. If a woman saw blood during her lunch break, the taharos she handled five minutes earlier are tahor. "Dayan sha'atan" means that it is enough for us that she becomes tamei from the moment she saw it and onward, and we have no concern at all for what happened beforehand.
Hillel's Position:
"Hillel omer: mipekidah lipekidah" - Hillel agrees that strictly speaking she has a chazakah d'meikara and is presumed tehorah until the time she saw the blood, but a further stringency should be decreed when it comes to Terumah and sacred foods. We therefore push the time of tumah backward to the limit of the doubt: if the last time she examined herself was five days earlier, we assume that the blood may have been there already for five days, and all the taharos she touched during those days are tamei. In his words: "afilu l'yamim harabbeh" - and it makes no difference if we are speaking of many days.
The Ruling of the Sages:
"V'chachamim omrim: lo k'divrei zeh v'lo k'divrei zeh, ela mei'eis l'eis mema'etes al yad mipekidah lipekidah, umipekidah lipekidah mema'etes al yad mei'eis l'eis". The Sages disagree with both of them: Hillel is too stringent, while Shammai is too lenient. They too agree that one must be stringent in handling sacred foods lest they were exposed to tumah, but they adopt the lesser of the two time periods:
Mippekidah lifkidah: the span of time that has passed since the last examination in which she verified that she was tehorah.
Me'es le'es: even if she did not examine herself at all, we do not go back further than twenty-four hours.
It emerges that according to the Chachamim, a woman who handles taharos and touches items that must remain tahor is obligated to examine herself twice a day: in the morning and in the evening. And if she did not do so, there is a kind of penalty here - we go back twenty-four hours, but no further than that.
A woman with a fixed veses:
That is the essence of the halachah. The Mishnah then adds: "Kol ishah sheyesh lah veses - dayah sha'atah" - every woman who has a fixed veses, her time suffices. The Chachamim agree that a woman whose cycle is fixed and known to her - for example, that every thirtieth day of the month her cycle returns and she sees blood - and who examined herself and found the blood precisely at the time she expected to see it, her time suffices: we assume the blood came only at that moment and not before. But if she saw at a time other than her veses, we return to the rules explained above: twenty-four hours back, or mippekidah lifkidah, from the last examination she performed.
"Hamshammeshes be'edim harei zeh kifkidah":
Finally the Mishnah teaches that when a woman examines herself after relations with her husband, this examination is included in the reckoning - "mema'etes al yad me'es le'es ve'al yad mippekidah lifkidah" - it reduces both the twenty-four hour period and the period from examination to examination. The Mishnah used the plural, "be'edim," because she must examine herself both before relations and afterward; however, the examination beforehand does not count, since we assume she may not have performed it properly because her mind was elsewhere.
The examination after relations, by contrast, does count. One might have thought that it would not count, for some reason - perhaps the blood was concealed by another fluid - but this is not so: once she has examined herself, her examination counts for her, and from it we measure the stopping point of the pekidah, the last time she verified that she was tehorah.
In summary: in this Mishnah, Shammai and Hillel disagree about a woman who was handling taharos and found blood: according to Shammai, "dayan sha'atan" - their time suffices, and we have no retroactive concern; according to Hillel, "mippekidah lifkidah" - from examination to examination, even over many days. The Chachamim ruled neither like this one nor like that one, but rather like the lesser of the two time spans - me'es le'es or mippekidah lifkidah. We also learned that a woman with a fixed veses who saw at her expected time, her time suffices, and that the examination of a woman who has relations be'edim counts as a pekidah in this regard.