Taharos, Chapter 1, Mishnah 4. This chapter began with the laws of a carcass, setting the neveilah of a kosher bird beside the neveilah of a non-kosher bird and working out which shiurim and which halachos govern each one. Now the Mishnah widens its scope from birds to animals, turning to every part of an animal that is not meat: do such parts get added into the shiur of tumah, and if so, for which category of tumah?
Here is the list as the Mishnah gives it: "U'vivheimah, ha'or, v'harotev, v'hakipah, v'ha'alal, v'ha'atzamos, v'hagidin, v'hakarnayim, v'hatlafayim." That is a dense string of terms, so each one deserves a translation. Ha'or is the skin. Harotev is the gravy that comes off the meat. Hakipah means the seasoning cooked along with it. Ha'alal is the meat scraps left over. Ha'atzamos are bones, hagidin are tendons, hakarnayim are horns, and hatlafayim are hooves.
And the ruling? "Mitztarfin l'tamei tumas ochlin, aval lo tumas neveilos" - every item here joins in for the tumah of food, yet none of them joins in for the tumah of a neveilah. What these parts have in common is that they serve as shomer: the hide, the horns, the hooves and their like form a protective covering around the meat inside. Not one of them is something a person would make a meal of on its own. My understanding is that people do consume them along with the rest of the animal (a serving of horns has never come my way, so anyone with firsthand experience is welcome to write in). Since they are not foods in their own right, they cannot carry food tumah independently. They can, however, attach to the meat, so that their bulk is measured together with it toward the shiur needed for tumas ochlin.
With regard to tumas neveilos they contribute nothing. That tumah operates on a different principle: it is tied to the carcass as such, as the Torah writes in Vayikra chapter 11, verse 39, about a person who comes into contact with the carcass of an animal that has died. There the tumah rests on the dead animal's own flesh, and it follows logically that it would not spread to the shomer, that is, to any of the items enumerated above.
"Kayotzei bo" - the Mishnah now supplies a parallel case, and it brings us back to the arresting picture of a Jew doing shechitah for a non-Jew: "hashocheit behemah teme'ah l'nochri v'hi mefarkeses." A Jew slaughters a non-kosher animal on behalf of his gentile neighbor. The shechitah has been carried out, the animal has died, and still the carcass goes on jerking and shuddering. Where does that leave its halachic standing?
"Metamei tumas ochlin" - food tumah applies to it already. True, a Jew has no food here, but the non-Jew who commissioned the slaughter does, and that suffices to establish the machshavah, the intent that designates something as food, which is a prerequisite for food tumah. "Aval lo tumas neveilos, ad shetamus o ad sheyatiz es roshah" - neveilah tumah, though, waits: it takes hold only once death is complete, meaning once the spasms have ceased, or alternatively once the head has been chopped off, which resolves the question just as decisively.
The Mishnah signs off with the rule that binds the whole piece together: "Ribah l'tamei tumas ochlin mimah sheribah l'tamei tumas neveilos." Scripture drew the boundaries of food tumah more generously than it drew those of neveilah tumah. Each of our two cases demonstrates this: skin, bones, horns and hooves are counted in for food tumah and left out of neveilah tumah, while an animal still in its death throes already transmits food tumah though it has not become a neveilah yet.