Kinnim, Chapter 3, Mishnah 2. Our chapter goes on discussing a kohen who took hold of birds and brought them without pausing to clarify which bird had been assigned to which korban. The mishnah before us returns to a case already set out in Chapter 1, Mishnah 3, and out of that case it draws a broad rule: when the kinim of two women became mixed with each other, and the kohen went ahead and offered them, placing one portion on the upper part of the mizbe'ach and the rest beneath the scarlet line, what is the halachah?
Begin by recalling how the case was framed earlier. "Achat lazo" (one ken belonging to this one), "ushtayim lazo" (two belonging to that one), "veshalosh lazo" (three belonging to a third). And the figures can climb much higher: "ve'eser lazo" (ten kinim for the first), "ume'ah lazo" (a hundred for the second). Whatever the numbers, the birds became confused with one another, and the kohen, rather than halting to establish whose bird was whose, acted on his own initiative.
All Above or All Below
"Asa kulan lema'alah": should the kohen have performed the entire mixture on the upper portion of the mizbe'ach, the area marked off by the scarlet line as the place of an olat ha'of, the verdict is "mechtza kasher umechtza pasul." One part of the birds is fit and the other part is disqualified. The reasoning is plain: every ken consists of a bird set aside as an olah together with a bird set aside as a chatat. Those that happened to be the olot received exactly the avodah they required, so they stand; those that were chata'ot were lifted to a spot that is not theirs, and they are lost.
Reverse the picture and nothing changes. "Kulan lematah": if every one of them was offered underneath the line, at the spot reserved for a chatat ha'of, the verdict is once again "mechtza kasher umechtza pasul." The chata'ot among them are fit, having been brought precisely where the Torah places them, while the olot that were brought down there are unfit.
Half Above and Half Below
"Chetzyan lema'alan vechetzyan lematan": suppose instead that the kohen split the mixture, sending one part of it up and the rest of it down. The ruling is "hamirbeh kasher." A quantity equal to the largest group of kinim contributed by any single one of these women emerges fit. Practically speaking, we identify the woman whose contribution was the greatest, and her total is the count of birds that come out valid.
Why does it work out this way? Because when the birds are split evenly, half above and half below, a certain quantity is necessarily placed correctly no matter how the mixture actually fell out. The birds are being divided two by two, and once the arithmetic is worked through, the amount that is guaranteed to have landed in its proper place matches the largest number of pairs contributed by any one of the women.
Our mishnah and the one just before it both treat kinim of two women that became mixed and were then split by the kohen between the upper area and the lower one. There the two had each brought an identical quantity, and the halachah came out that one portion is kasher while the matching portion is pasul. Here the quantities are unequal, and the halachah is that the count of the woman who brought the most is what emerges fit.
Zeh Haklal
At this point the mishnah turns the two outcomes into a formula. "Zeh haklal": here is the governing rule for a kohen who took kinim belonging to two women, mixed as they were, and performed part of them on the upper section of the mizbe'ach and the remainder beneath the line.
"Kol makom she'ata yachol" (in every situation where you are able) "lachlok et hakinim velo yihyu mishel isha achat bein milma'alan bein milmatan": wherever the split can be construed so that no individual woman had birds landing in both locations, each woman's kinim having gone entirely upward or entirely downward, the ruling is "mechtza kasher umechtza pasul." One portion stands and the corresponding portion is ruined.
The logic is that we are obliged to reckon with the least favorable possibility: all of one woman's kinim went up and all of the other woman's kinim went down. On that assumption, among the birds placed above, exactly one part of them is acceptable, since within her kinim only one bird of each pair had been designated an olah; the rest of what went up were chata'ot, brought where they do not belong, and they are pasul. The identical difficulty runs the other way underneath the line: only the chatat of each of her pairs is fit, while the olah of each pair was brought down where it had no place at all, and those are pasul.
"Kol makom she'ein ata yachol" (in any situation where you cannot) "lachlok et hakinim ad sheyihyu" (divide them without there being some) "mishel isha achat bein milma'alan bein milmatan": wherever no division can be imagined without kinim of one and the same woman inevitably ending up in both places, in the upper area as well as in the lower one, the halachah is "hamirbeh kasher." The woman with the greatest count sets the measure of what is valid, exactly as the mishnah declared at the start.
An enormous amount of computation lies behind these halachot, and laying it all out properly would take far more room than we have here. Still, the upshot is plain. Two women who brought equal quantities, whose birds were mixed and then divided by the kohen between the two areas of the mizbe'ach: one portion of the birds comes out kasher. One woman who brought considerably more kinim than her fellow: the count of the greater contributor becomes the measure of validity, and whatever exceeds it is pasul.