Keilim, Chapter 13, Mishnah 2. This mishnah introduces us to a whole family of metal implements built on a single principle: one long metal shaft with a different tool fixed at each end. Two working parts, one object. The question the Mishnah asks about each of them is what happens when one of the two ends is lost. Does the object remain a kli, capable of receiving tumah, or has it become a broken piece of metal?
The Koligrifon
The first item is the koligrifon. One end of it was fitted with prongs for pulling meat or loaves out of an oven, while the opposite end held a scoop shaped like a spoon, used to rake out ash and embers. The Mishnah rules: "Koligrifon shenitlah kapah, teme'ah mipnei shinah", when the scoop end of a koligrifon has been detached, the implement is nonetheless susceptible to tumah on account of the prongs that remain. And the same holds in the opposite case: "Nitlah shinah, teme'ah mipnei kapah", once the prongs are gone, the scoop alone is enough to keep it susceptible.
The logic is straightforward once we see the object properly. This is not one utensil with a decorative extra; it is genuinely two utensils sharing one handle. Therefore, as long as either one of them remains intact and usable, the whole object is still a functioning kli, and a kli receives tumah.
The Mikchol
The Mishnah turns next to the mikchol. One end came to a slender point and served for applying cosmetics around the eye, roughly the function of a mascara wand today. The other end was formed as a miniature shovel, whose job was to clean the inside of the ear. The ruling: "Mikchol shenital kaf, tameh mipnei hazachar", should the shovel end be broken off, the pointed end keeps the mikchol susceptible to tumah. "Nital hazachar, tameh mipnei hakaf", and should it be the point that is lost, the little shovel that survives keeps it susceptible.
The Machteiv
Third in the list is the machteiv, the stylus with which one wrote on a tablet coated in wax. Its one end tapered to a sharp point that carved the letters into the wax, and its other end was broad and flat, meant for erasing: pressing that side down against the wax smoothed the letters out of existence. The Mishnah rules: "Machteiv shenital koteiv, tameh mipnei hamocheik", if the writing point is broken away, the stylus retains its status as a utensil and remains susceptible to tumah thanks to the erasing end. "Nital mocheik, tameh mipnei hakoteiv", and in the reverse case, where the erasing end is lost, the writing point that remains keeps it susceptible.
The Zomalistera
Fourth comes the zomalistera, an implement put to use at the stove. Its one end was a ladle, for skimming away the foam and bubbles that gather on top of a simmering pot, and its other end was a fork, for drawing pieces of meat up out of that pot. The Mishnah teaches: "Zomalistera shenitlah kapah, teme'ah mipnei hamazleg", once the ladle end is gone, the fork keeps the implement susceptible to tumah. "Nital hamazleg, teme'ah mipnei kapah", and where the fork is what broke off, the ladle end suffices to keep it susceptible.
The Maded
The Mishnah then extends the ruling: "Vechen hashein shel maded", the identical halachah governs the tooth of a maded, an implement used in working the soil. Its one end bore a tooth for breaking up and turning the ground, and its other end carried a piece used in fitting together the components of a plow. Here as well, should that tooth be broken away, the maded stays susceptible to tumah by virtue of the working end that survives at the opposite side.
The Minimum Length of the Handle
When one of the two working ends snaps off a double implement, a stretch of the shaft frequently comes away with it. The Mishnah therefore concludes with a principle that governs every case on the list: "Shiur kulan kedei la'asot melachtan", in all of them the required amount is however much allows each to carry out the task it was made for.
In other words, the length of handle that must remain is measured by the needs of the tool that survived. The remaining shaft must be long enough that the surviving end can actually be used for its purpose. If too little of the handle is left, the surviving tool cannot be worked with at all, and then nothing usable remains. Such a piece is no longer a kli, and therefore it cannot become tameh.
The thread running through this entire mishnah is the same one that runs through so much of Masechet Keilim: tumah attaches to function. A double utensil that loses half of itself is still a utensil, because half of it still works. But a fragment that can no longer do any job at all has left the category of kli altogether.