Challah, Chapter 2, Mishnah 8. In Chapter 1 we established the rule that challah, like Terumah, is taken min hamukaf - from dough that is adjacent. A person may not separate challah on behalf of dough that is distant from him. For this reason we mentioned there an opinion that one does not take from the pure on behalf of the impure: when impure dough is before him, he does not take challah for it from the pure dough, out of concern that he will bring the two too close to one another and the pure challah will become impure.
The View of Rabbi Elazar:
In our Mishnah, Rabbi Elazar disagrees and presents a case in which it is permitted to take from the pure on behalf of the impure. In his words: "Nitteles min hatahor al hatamei" - challah may be taken from pure dough on behalf of impure dough.
How can the two be brought close to one another, and even placed right up against each other, without the pure dough becoming impure?
"Isah tehorah v'isah temei'ah" - before him are two batches of dough, one pure and one impure.
"Notel k'dei challah mei'isah shelo hurmah challasah" - first he sets aside an amount sufficient for challah, taking it from the dough whose challah has not yet been removed, which is the pure one.
"V'nosen pachos mik'beitzah ba'emtza" - he takes a piece of dough that is less than the volume of an egg and places it in the middle, between the pure dough and the impure dough.
The underlying principle: dough that is less than the volume of an egg cannot transmit impurity to something else. True, there is a dispute as to whether something less than an egg's volume can become impure at all, but even if it does become impure, it has no power to pass that impurity further along, since transmitting impurity to something else requires at least an egg's volume. It emerges that the small piece resting between the impure dough and the pure dough does not render the large batch of pure dough impure.
Therefore the pure dough remains in its purity, and challah may be taken from it even on behalf of the impure dough, since they are all connected to one another. And this is the reason for placing the piece in the middle: "K'dei sheyitol min hamukaf" - so that the taking should be from an adjacent place.
The View of the Sages:
"Vachachamim osrin" - as we mentioned above, the Sages forbid this, out of concern that he will ultimately cause the impure dough to touch the pure dough. Therefore the Sages said that one should not do this.
In summary: in this Mishnah, Rabbi Elazar and the Sages disagree about taking challah from the pure on behalf of the impure. Rabbi Elazar permits it by means of placing a piece of dough smaller than an egg's volume in the middle, which cannot transmit impurity to what is next to it, and in this way the taking is done min hamukaf without the pure dough becoming impure. The Sages forbid it, out of concern that the two batches of dough will come into contact.