Before us is Mishnah 4 in the first chapter of Tractate Temurah. It lists three cases, and all three actually depart from the subject of the tractate and do not deal with temurah at all. The reason they appear here lies in the next Mishnah, where we will learn that a temurah does not create another temurah.
The principle from which the cases are borrowed:
Offering A: consecrated with full sanctity, let us say as an olah.
Offering B: if one attempts to substitute B for A, a derived sanctity takes effect on B. This is the temurah, and it too is considered an offering and is consecrated.
Offering C: if one attempts to substitute C for B, the derived status of B does not pass on to C, and C remains chullin.
In other words, a temurah cannot extend its status and pass it along: even though it holds a special status derived from the power of the original offering A, it has no power to transmit that status to a third item. This abstract concept is extended by way of analogy to the three cases in our Mishnah, and the connection itself is abstract as well.
The first case - a medumah mixture:
Medumah is a mixture of Terumah, the food that goes to the kohen, together with chullin, ordinary permitted food. Since Terumah carries certain stringencies, the Sages were stricter than usual regarding the rules of a mixture containing Terumah. The basic rule, put simply, is that one hundred times as much chullin as Terumah is required for the mixture to be permitted.
For example: one had a hundred cups of chullin flour before him, and one cup of Terumah flour fell in - the entire mixture is permitted to be eaten, since there is a hundred times as much chullin as Terumah. It is enough to remove one cup of flour and give it to the kohen, since that is his property, and the rest of the mixture is permitted.
But if he had only ninety-nine cups of chullin flour before him, and one cup of Terumah flour fell in - the entire mixture takes on the status of "medumah", a mixture of chullin and Terumah. The term is derived from the language of the verse that calls Terumah "dim'ah".
And here is the question: this medumah mixture, which is only one percent Terumah, fell into ordinary chullin flour. So the Terumah is A, the medumah mixture is B, and the new flour is C. Let us say one cup of the medumah fell into C. Is a hundred times as much chullin required against the entire cup of medumah, meaning a hundred cups, or is a hundred times the amount of Terumah that was in the mixture enough - and since the cup is only one percent Terumah, only a little more than one cup of chullin flour is needed to permit the whole thing?
The bottom line of the Mishnah is that medumah has no halachic status of its own. We do not make medumah calculations; everything is derived from the Terumah: we calculate how much Terumah there is in the third mixture, and if there is a hundred times as much chullin against it, it is permitted.
The Rishonim disagree here: when the cup of medumah fell into C, it contains both chullin and Terumah mixed together, and the question is whether the chullin portion within it joins the calculation. Rashi and the Rosh, of the Ashkenazic Rishonim, hold that it does; the Rambam holds that it does not. This is a technical point.
In any case, this is the content of the Mishnah: "Ein hamedumah medamea ela lefi cheshbon" - the medumah mixture does not transmit its questionable and limited status to a third item, "except according to the calculation", according to the calculation of the amount of Terumah found within it. Conceptually this is the very same principle as in temurah.
The second case - leavening agent:
A similar case, but a different one: machmetzes, that is, a leavening agent. This has nothing to do with Pesach or chametz on Pesach; it is an entirely separate matter. The general rule in mixtures is that from the Torah, as long as we have a mixture of two things, the majority determines the status of the entire mixture, and the minority loses its status in the majority. This is nullification, based on the verse "acharei rabim lehatos" - we follow the majority, and the mixture takes on the status of the majority.
In certain cases the Sages required more than a simple majority, and the underlying idea is that as long as something is still identifiable, it is hard to say that it has lost its identity - after all, one can, for example, taste it. Therefore the Sages said that when it comes to a mixture of tastes, when flavors blend together, a ratio is required at which the taste is nullified.
But a leavening agent is not a taste at all; it is a rising agent. In the times of the Mishnah there was no purified yeast as we have today, and so they used fermented dough, sour dough, which served as the rising agent and the starter for the new dough. If the starter, the se'or, is Terumah, and it is now leavening a new batch of dough, we have before us a mixture of Terumah and chullin, and according to the rules of meduma that we discussed, one hundred times as much chullin is required against it.
But a special law applies to leavening agents: if the bread is leavened, it is evident that a rising agent was put into it, and a person cannot fool himself into saying that it is not there - it is clearly present here, since the bread is leavened bread. Therefore, even if there is one hundred times as much chullin against the unit of Terumah, if the Terumah was the rising agent, the mixture remains forbidden.
So we have A, which is the Terumah, and it is both the source of the prohibition and the leavening substance, the se'or. It fell into B, a mixture of chullin, and leavened it - and the whole thing is forbidden in any amount. This is a meduma mixture from which one may not eat, since the leavening agent is not nullified, because it announces its presence by having leavened the bread.
And if B then entered C, another batch of dough - how much is required? Do we need one hundred times against the amount of B that fell into C, or one hundred times, proportionally, against only the Terumah that was inside B? The answer is the second option, exactly as we said earlier: according to the calculation. Neither the meduma nor the leavening agent has an independent status; the Terumah is the prohibition, and we do the math and calculate how much Terumah is present here - if there is one hundred times as much against it, it is nullified; if not, it is not nullified.
And this is the language of the Mishnah: "Ein hamechametz mechametz ela lefi cheshbon" - we do not view the leavening prohibition as though it leavened the forbidden portion, and we do not treat it as the prohibiting portion, "ela lefi cheshbon" - we calculate proportionally how much of the amount that fell in is actually forbidden. If there is one hundred times as much against the actual Terumah, it is all permitted; if there is not one hundred times against it, it is all forbidden as meduma.
The third case - drawn water and a mikvah:
The third case departs from the topic entirely, and does not fit at all into that same analogy of A, B and C. It was brought here because of the phrase "ela lefi cheshbon": another case, from a different source in the Mishnah, in which matters are determined according to the calculation. The case deals with drawn water that enters a mikvah.
The basic rule regarding mikvaos is that forty se'ah are required. For illustration only, and not as a halachic ruling: one can picture a se'ah as a pack of six two-liter water bottles, roughly twelve liters. Forty of these are required for a valid mikvah.
However, the water must not come by way of a vessel, but must flow in on its own, such as rainwater. If it came by way of a halachic vessel, it is drawn water. And here is the distinction:
A mikvah that is already valid, meaning that it contains forty se'ah - one may pour as much drawn water into it as he wishes and it will never be invalidated. Even if he were to pour the entire ocean into it, it remains valid, since whatever falls into the mikvah is purified together with it and becomes part of it.
A mikvah that does not contain forty se'ah - it is invalidated by even a small amount of drawn water, by three lugin alone, about a liter and a half.
Therefore, someone who wants to speed up the filling of his mikvah and has no rainwater cannot pour water in, but one path is open to him: hamshachah, drawing the water along the ground. How does this work? A mikvah that already contains the majority, twenty one se'ah out of forty, of valid rainwater - the remaining nineteen se'ah he fills in a vessel, and they are drawn water, but he does not pour them into the mikvah itself. Instead he pours them onto the floor next to it, at a distance of at least three tefachim, and they roll and flow inside. This pouring is called hamshachah, and as long as this water is the minority, nineteen out of forty, the mikvah is valid.
There are two approaches to understanding the words of our Mishnah here:
The first explanation of the Bartenura, which is based on Mishnayos in other places: a mikvah that is a combination of rainwater and water that came through hamshachah, water that was in a vessel and was poured onto the ground and flowed in from there - its validity depends on a simple calculation. If twenty one se'ah of rainwater gathered there first, the remaining nineteen se'ah may come through hamshachah along the ground, and it is valid.
The explanation of the Gemara in Temurah, which is the one the Rambam focuses on, and it stands to reason, since this is the Gemara on our Mishnah: even though the rule is that three lugin - each log about half a liter - that enter a mikvah that does not contain forty se'ah invalidate it, if those three lugin came from four separate vessels rather than from three vessels, the mikvah is valid.
None of this is meant as practical halachah, and it goes beyond the scope of what we are dealing with here. The point is the language of the Mishnah: "Ein hamayim she'uvin poslin es hamikvah ela lefi cheshbon" - the amount of water that invalidates a mikvah is determined by mathematics. One must work out the numbers and see: are there twenty one original se'ah and nineteen others, or four vessels rather than three - and then everything is valid.
Admittedly, these cases have been gone through quickly here, without the thoroughness usually given to them, but since they are complicated and not connected to the subject of our tractate, and half an hour cannot be devoted to each case as it truly deserves, this approach will have to suffice.