Eduyos, chapter seven, mishnah 5. This mishnah presents two testimonies that are unrelated to one another: the first from Rabbi Yehoshua and Rabbi Yakim of Hadar, and the second from Rabbi Papyas. (Rabbi Yakim of Hadar - his name was Yakim, and he came from the city of Hadar, where he served as the central halachic authority.)
The first testimony - storing the ashes of the Parah Adumah:
The Torah states: "Ve'asaf ish tahor es efer haparah" - a pure man gathers the ashes of the burned red heifer, "vehiniach... bemakom tahor" - and places them in a pure place.
This raises the question: what is the law when the ashes themselves remain pure, but they are stored on top of something that is not pure? For example, the ashes are placed inside an earthenware vessel, and an earthenware vessel does not contract tumah through its outside, only through its inside. If the vessel is pure, and it is resting on top of a source of tumah - a dead sheretz, a dead mouse - is this acceptable, or have the ashes been disqualified and treated as impure?
This is the subject of the dispute among the Tannaim. The wording of the mishnah: "Al kelal shel chatas shenasno al gabei hasheretz shehu tamei" - a "kelal" is the earthenware vessel that holds the ashes of the chatas, the ashes of the burned red heifer; and a "sheretz" is one of the eight creeping creatures listed in the verse, which died and became a source of tumah. In the view of Rabbi Yehoshua and Rabbi Yakim, even though the vessel itself is pure, that is not enough, since the verse requires placing the ashes in a pure place, and this is not a pure place but an impure one - and therefore the ashes became impure.
And why did they need to testify about this? Because "Rabbi Eliezer metaher" - Rabbi Eliezer disagrees and holds that in this situation there is no problem at all, since the vessel is pure, and it is the vessel that the verse is referring to - and therefore placing it on top of a dead mouse does not disqualify the ashes.
The second testimony - counting two terms of nezirus:
Rabbi Papyas's testimony deals with counting the days of nezirus. One who says "I am hereby a nazir," without specifying, has committed himself to thirty days, and this is the default when no other length of time is stated. And one who says "I hereby accept two terms of nezirus" has taken upon himself two consecutive periods of thirty days. During his days of nezirus the nazir is restricted in three ways:
He may not become impure through contact with the dead.
He may not eat grape products.
He may not cut his hair.
At the end of a full thirty days, on the thirty-first day, he must perform his "tiglachas" - a term for the entire process of completing the nezirus, and not merely for the cutting of the hair.
That is the ideal procedure. However, there is an accepted principle that "a part of the day is like the whole of it" - part of a day is equivalent in certain respects to a full day. Therefore, after the fact, if he performed his tiglachas on the thirtieth day of his nezirus, he has fulfilled his obligation, since he had part of the thirtieth day, and that part counts as the full day.
Accordingly, one who accepted upon himself two consecutive terms of nezirus and shaved for the first one on the thirtieth day - one day earlier than is ideal - must count another thirty days: the thirtieth day is the first of them, the fifty-ninth day is the last, and on the sixtieth day he performs the tiglachas of the second nezirus.
And this is Rabbi Papyas' novel ruling: after the fact, if he performed the first shaving on the thirtieth day and the second on the fifty-ninth day, he has fulfilled his obligation, since the fifty-ninth day is the thirtieth day of the second nezirus. In practice there are two novel points here:
That one may apply the principle of 'part of the day is like the whole day' at both the beginning and the end at once: in the first nezirus, day one is only part of a day, and day thirty is likewise only part of a day, and even so the shaving is permitted to him.
That this very thirtieth day counts both as the last day of the first nezirus and as the first day of the second nezirus - a double use, and even so it is valid.
And the wording of the Mishnah: "He'id Rabbi Papyas al mi shenazar shtei nezirus" - concerning one who accepted upon himself two consecutive periods of nezirus, "she'im gilach es harishonah yom sheloshim" - that he performed the concluding procedure on the thirtieth day, "shegam megaleiach hasheniyah yom shishim". For ideally he must count thirty full days: the thirtieth day is the first, the fifty-ninth day is the thirtieth, and the sixtieth day is the proper time for the second shaving.
"Ve'im gilach yom shishim chaser echad" - if he performed the shaving of the second nezirus on the fifty-ninth day, literally sixty minus one, "yatza" - he has fulfilled his obligation after the fact, "sheyom sheloshim oleh lo min haminyan" - for the thirtieth day, although it was the last day of the first series, also counts as the first day of the second series.
In summary: this Mishnah contains two testimonies. In the first, Rabbi Yehoshua and Rabbi Yakim of Hadar testified about a kalal of chatas ashes that was placed on top of a dead creeping animal - that the ashes are tamei, even though the earthenware vessel itself remains tahor, since this is not a 'tahor place'; and Rabbi Eliezer rules that it remains tahor. In the second, Rabbi Papyas testified regarding the law of two periods of nezirus, and the halachah follows him: ideally the shaving should be done on the thirty-first day and on the sixty-first day, but after the fact a person may do it earlier, on the thirtieth day and on the fifty-ninth day, and he has fulfilled the obligation of both of his periods of nezirus.