Niddah, Chapter 1, Mishnah 1. Our masechta opens with the laws of a woman who becomes a niddah, and the very first question it raises is a question of time: from which moment is she temeiah?
The basic halacha is that a woman receives the status of niddah, becoming temeiah herself and rendering tomei whatever objects she touches, from the moment that blood leaves her uterus. It is not necessary that the blood already emerged from her body entirely; the moment it has left the uterus, her status has changed.
That principle creates the very problem our mishnah addresses. A woman sees blood now, and she knows that from this moment onward she is temeiah. But how long has the blood been on its way? Perhaps it left the uterus hours earlier, and she has already been temeiah for a stretch of time without knowing it, touching objects and handling taharos all the while. Is there a retroactive tumah, and if so, how far back does it reach?
Shammai: Dayan Sha'atan
The first view belongs to Shammai, who speaks of "kol hanashim," all women without distinction, and rules "dayan sha'atan": her own hour suffices for her. Her tumah begins at the instant she discovers the blood and not a moment sooner. No earlier stretch of time is called into question, and any vessel or food she handled before that instant retains its taharah.
Hillel: Mipkida Lifkida
Hillel rejects that approach completely. His yardstick is "mipkida lifkida," measuring from one internal examination to the next. We look back to the most recent bedikah in which she checked herself and found nothing, and from that examination until the present sighting she is treated as having been temeiah the whole time, so that anything she touched during that interval is tomei. Hillel places no ceiling on this reckoning: "va'afilu l'yamim harbei," even should a great many days separate that earlier examination from today, the entire span is included.
The Chachamim: Mei'es L'es
The Chachamim answer that we follow neither of these views: "lo chedivrei zeh," not the ruling of the one, "velo chedivrei zeh," and not the ruling of the other. Their halacha joins the two factors together so that each one curtails the other. A "mei'es l'es," a single full day counted from that hour to the same hour, is the furthest limit of the retroactive tumah, which puts a stop to Hillel's open ended count; in the words of the mishnah, that day diminishes (mema'eses) the reckoning that runs from one bedikah back to another. And the day itself is diminished by a bedikah: if she examined herself and found nothing at some point inside those twenty four hours, the retroactive tumah stretches back no further than that examination.
A concrete illustration makes it clear. Suppose a woman sees blood at four o'clock on Tuesday afternoon. According to the Chachamim, whatever she touched from four o'clock Monday afternoon onward is tomei. If, however, she had performed a clean bedikah at some point during that twenty four hour span, then she is temeiah retroactively only from the moment of that bedikah, and everything she handled before it remains tohor.
This ruling of the Chachamim is the halacha that we follow.