This mishnah continues the collection of Tannaim who listed additional leniencies of Beis Shammai. Here the statements are taught in the name of Rabbi Yishmael, but many have the reading "Rabbi Shimon" - this is the reading in the early manuscripts, and it is also the version in Maseches Yadayim. That version is also more consistent, since the entire chapter revolves around the students of Rabbi Akiva, and just before this we learned the statements of Rabbi Yosi and Rabbi Yehudah. In any case, three additional matters are listed here in which Beis Shammai were lenient.
1. Koheles - whether it renders the hands tamei:
There is a general rule that the sacred writings (Tanach) are a rabbinic source of tumah. The reason for this decree, which is not intuitive, lies in the storage arrangements of the mishnaic period: space was limited, and people would keep all their sacred items in a single cabinet - the Terumah together with the scrolls. Since the food would sit there until a kohen came along, mice would come, eat the food, and gnaw at the sacred writings as well. The Sages said: let us decree tumah on the sacred writings, and then people will not place them next to Terumah, and in this way the writings will be preserved.
This principle is agreed upon and is not disputed. The question is which books are included in this status of being a source of tumah, on account of which they may not be placed beside Terumah. There are several books about which there is a dispute: one of them is Shir HaShirim, which is not mentioned in our mishnah, and the second is Koheles, which is the subject of our mishnah.
Koheles was written by King Shlomo, and the question is: is it merely an expression of Shlomo's extraordinary wisdom, containing no ruach hakodesh - and therefore not a source of tumah? Or are these not merely words of wisdom, but words spoken with divine inspiration, with ruach hakodesh, and therefore part of the canon and part of Tanach.
The wording of the mishnah: Beis Shammai hold that Koheles is a work of wisdom and not ruach hakodesh, and therefore "eino metamei es hayadayim" - it does not render the hands tamei. And Beis Hillel say that it does render the hands tamei, because it was written with ruach hakodesh and is part of the Kesuvim - and this is how we rule.
2. Mei chatas that have fulfilled their purpose:
This topic is discussed in Maseches Parah as well. It refers to the purifying waters made from the ashes of the red heifer mixed with spring water for purposes of purification. The verses in the Chumash call them "mei niddah" (waters of sprinkling), and Chazal call them "mei chatas" - from the root meaning to cleanse, to purify and to clean.
Even before they are used, and in a manner that is entirely counterintuitive, mei chatas are themselves a source of tumah, even though they are used to purify people from the tumah of a corpse. What is more, they are a rather severe source of tumah: they transmit tumah to anyone who touches them and even to one who carries them without direct contact, by touch and by carrying, when this is not being done for the sake of the purification process. Even moving them from one storage place to another - lifting the jug of mei chatas and setting it down elsewhere - renders the one who lifts it tamei.
The question is what their status is after they have been sprinkled on a person, that is, after they have fulfilled their purpose and are no longer in use: are they still included in the "mei chatas" and "mei niddah" of the verse, and therefore a source of tumah? For example, when they were sprinkled on a person and a drop of those waters drips from his nose and touches something - does that thing become tamei? Or perhaps once they have fulfilled their purpose and purified the man, they are no longer "mei niddah" and are not an av hatumah.
In the wording of the mishnah: "Mei chatas she'asu mitzvasan - Beis Shammai metaharin uBeis Hillel metamin" - mei chatas that have fulfilled their purpose: Beis Shammai declare them pure and Beis Hillel declare them a source of tumah. Beis Shammai hold that once they have done their job they are no longer the mei chatas and mei niddah of the verse, and they are not an av hatumah. And Beis Hillel say that they retain their original status as the mei niddah of the verse, and therefore they are a source of tumah that transmits tumah by touch and by carrying.
3. Ketzach (black cumin):
The Rishonim disagree about the identity of the ketzach: according to Rashi, the Baal HaTurim and many other Rishonim it is fennel, while according to the Rambam and other Rishonim it is cumin (cumin seeds and fennel seeds look remarkably alike). For the sake of our explanation we will take it to be fennel. Eating it in large quantities is not good for a person, but people were accustomed to mix it into their food as a sort of remedy, for example against a headache.
The question is this: since it is a type of plant that grows from the ground and comes in the form of a seed, and even though it is not good for a person when eaten on its own or in large quantities, in small quantities it does have medicinal effects and people did in fact eat it as part of their food - is it considered food or not?
If it is considered food, it is susceptible to the tumah of foods and is obligated in Terumah and tithes. And if it is not considered food, like the law of a medicine, whose consumption is not considered eating - for we would all agree that a paracetamol tablet is not food, even though one swallows it - then it is not susceptible to tumah and is not obligated in tithes. And here we are not dealing with a separate medicine, but with something sprinkled into the food itself.
Beis Shammai are lenient and say that it is tahor, it is not a source of tumah and it is not food, and therefore it is not obligated in tithes. Beis Hillel are stringent and say that it is susceptible to tumah, because it is food, and therefore it is also obligated in Terumah and tithes.
In the words of the Mishnah: "HaKetzach - Beis Shammai metaharin uVeis Hillel metam'in, vechen lema'asros" - regarding the ketzach, Beis Shammai declare it tahor and Beis Hillel declare it susceptible to tumah, and the same applies to tithes. Beis Shammai hold that it is not susceptible to the tumah of foods, since it is not food but a sort of remedy sprinkled onto food that happens to grow from the ground. Beis Hillel hold that people do eat it - in small quantities it is fit to eat and it is part of the meal - and therefore it is food that is susceptible to the tumah of foods. And so too regarding tithes: since it is a food that grows from the ground and is eaten, one must first separate Terumah and tithes from it as required.
Here too, then, in each of the two directions, a leniency and a stringency are built in together.