Before us is Mishnah 2 in the first chapter of Maseches Eduyos. This mishnah belongs to that group of mishnayos whose details many people never really focus on, because their character is distinctly mathematical. We will therefore begin by presenting the general content of the mishnah while skipping the technical details, and afterwards we will return and lay out the calculations in full, for those who wish to go into the matter in depth.
The content of the mishnah in brief:
The Tannaim disagree about how much flour a dough must contain in order to become obligated in the separation of challah - the portion taken from the dough and given as a gift to the kohen. Three opinions are stated:
Shammai: One kav of flour. (For the sake of illustration, a kav can be pictured as a two liter bottle.)
Hillel: Two kabin of flour.
The Chachamim: Neither like the words of this one nor like the words of that one, but rather a kav and a half.
It is worth noting that this mishnah, together with the mishnah before it and the mishnah after it, is one of the places where Hillel and Shammai disagree and the Chachamim do not rule like either of them, but instead adopt an intermediate measure of their own.
Three stages in the size of the measures:
Here an additional level of complexity enters, because three stages of development took place in the measure of the kav, which is a type of measuring vessel:
The desert measures - the measuring vessels that were in use in the desert.
The Jerusalem measures - when the Sanhedrin sat in Jerusalem they enlarged the measuring vessels.
The Tzippori measures - after the destruction of the second Beis HaMikdash, when the Sanhedrin convened in Tzippori, they enlarged the measuring vessels a second time.
Chazal refer to the addition at each stage as a "shishis" (a sixth), and for our purposes we will render this as twenty percent. It emerges that the measuring vessel called a "kav" grew by twenty percent in Jerusalem relative to the desert, and again by twenty percent in Tzippori relative to Jerusalem. The actual quantity of flour that obligates in challah does not change at all; rather, as the measuring vessels grow larger, the number of kabin stated grows smaller.
The mishnah therefore continues and says: although the Chachamim stated a kav and a half according to the Jerusalem measures, once they moved to the Tzippori measures the amount drops to a kav and a quarter, which is five quarters. And Rabbi Yosi disagrees and says that it is not exactly a kav and a quarter, but a kav and a quarter and a bit more - an addition of about five percent - because the foundation of the measures is the volume of an egg, and the egg has become slightly smaller.
The language of the mishnah:
"Shammai omer mikav lachallah" - one kav of flour creates the obligation to separate challah.
"V'Hillel omer mikabayim" - a measure of two kabin is what creates the obligation.
"VaChachamim omrim lo k'divrei zeh v'lo k'divrei zeh, ela kav vamechetzah chayavim b'challah" - in the opinion of the Chachamim, neither of them has stated the correct measure, and the obligating amount is a kav and a half.
Here a first difficulty arises that needs to be addressed: the measures of a kav and two kavin given by Shammai and Hillel are based on the desert measures, while the kav and a half of the Sages was stated in the measures of Jerusalem. It turns out that the opinions are not speaking in precisely the same terms, and nevertheless this is how the matter was taught.
"U'mishehigdilu hamiddos amru chameishes revi'im chayavim" - once the measuring vessels were enlarged a second time, in Tzippori, the measure is five quarters of a kav, that is, a kav and a quarter instead of a kav and a half.
"Rabbi Yosei omer chamishah patur, chamishah v'od chayavin" - five quarters of a kav are exempt, while five quarters plus a bit more - an addition of about five percent - are obligated in the separation of challah.
The practical halachah: The opinions today range from about nine to thirteen cups of flour for a fresh dough. In general, from nine cups one separates without a berachah, and from thirteen one separates with a berachah. The precise practical procedure should be clarified with a rav who is a posek.
The source of the law in the Torah:
The foundation of the entire discussion is the verse: "V'hayah ba'achalchem milechem ha'aretz tarimu terumah laHashem, reishis arisoseichem challah tarimu terumah" - the first of the dough shall be separated as challah and given as a gift. Note that the word "kohen" does not appear in the verse; rather, the giving is stated as being to Hashem, and the kohen is the one who comes in His place.
However, the verse gives no measure for how large that dough must be. The source of the measure is the manna, the bread from heaven that came down for Israel in the desert, from which they would prepare their food, and the amount each person received per day was an omer per head. The omer is a tenth of an eifah, and the eifah is a very large measure. From here, by comparison, we derive the measure of flour that becomes obligated in challah.
The rule of 'Askalav Gudo':
In order to remember the ratios between the measures of volume there is a rule of thumb: 'Askalav Gudo.' 'Askalav' is written with the letters aleph, samech, kuf, lamed, beis - eifah, se'ah, kav, log, beitzah. 'Gudo' is written with the letters gimmel, vav, daled, vav - three, six, four, six. One writes the five letters of 'Askalav' in a row, and between every two neighboring letters one places one of the four letters of 'Gudo,' which teaches the conversion ratio between them:
Between the eifah and the se'ah - gimmel: three se'ah in an eifah.
Between the se'ah and the kav - vav: six kavin in a se'ah.
Between the kav and the log - daled: four lugin in a kav.
Between the log and the beitzah - vav: six eggs in a log.
If one wants to picture it: the kav is a two liter bottle; the se'ah, which contains six kavin, is a six pack of bottles; and the eifah, which contains three se'ah, is three such packs stacked one on top of the other. By simple arithmetic - three times six - we find that there are eighteen kavin in one eifah.
Since the omer they received in the desert is a tenth of an eifah, we divide eighteen by ten and get 1.8. It turns out that the measure of the manna, and by comparison the measure of flour that obligates in challah, is 1.8 kav in the desert measures, which is a kav and a half in the measures of Jerusalem.
Converting the measures from the desert to Jerusalem:
Let us picture the desert measuring vessel as divided into five equal parts. The measure of a tenth of an ephah comes to 1.8 vessels, that is, one full vessel plus four fifths of a second - nine parts in all. In Jerusalem they added a "shetut" (a sixth) to the size of the vessel, which comes to twenty percent in our terms, so that instead of five parts in every kav there are now six parts of that same size. For that reason a kav and a half is now enough: in one kav there are six parts, and in the additional half another three - nine parts, exactly as before. This is the sense of the conversion from 1.8 to 1.5, which is a multiplication by five sixths. This step may not be intuitive mathematically, but the two pictures we have presented help make it clear. This is the view of the Chachamim.
'Milgo' and 'Milbar':
As for rendering "shetut" as twenty percent: we find in the words of Chazal in many places two ways of measuring a ratio - 'milgo' and 'milbar', loosely translated as from within and from without. Our own way of reckoning is the one Chazal call 'milgo'.
Let us illustrate this with a more familiar case: the added chomesh that appears in the Torah in many places. "Chomesh" ostensibly means a fifth, but in practice it is an addition of twenty five percent. How so? If a person stole four items and must return them with an added chomesh as a penalty, he has to add a fifth item for every four he stole. Once he has handed over all five, one out of the five - the one on the outside, 'milbar' - is the addition. But relative to the debt itself, which was four, this is an addition of twenty five percent. And that is the difference: 'milbar' - from without, and 'milgo' - from within, the way we are accustomed to measure.
The View of Hillel:
Applying this rule to the approach of Hillel and Shammai runs against intuition. Even though the omer is described as a tenth of an ephah, Hillel understands that this "tenth" was stated milbar, and it is not one part out of ten but one part out of nine - just as the chomesh is really a quarter. Accordingly one does not divide by ten but by nine: the eighteen kavin in an ephah divided by nine, and the result is two. This is the basis of Hillel's ruling that the measure is two kavin, and he is speaking in the measures of the desert, without converting to the measures of Jerusalem.
The View of the Chachamim:
The Chachamim, who say a kav and a half, disagree with Hillel on two counts: first, in their view "tenth" was stated milgo, and therefore eighteen divided by ten comes to 1.8. Second, they are not speaking in the measures of the desert but in the measures of Jerusalem, and therefore they multiply by five sixths and move from 1.8 to 1.5. That is the kav and a half of our Mishnah.
The View of Shammai:
Shammai fundamentally holds like Hillel, except that he reads the wording of the verse precisely: "reishis arisoseichem" - in the plural. Since a person eats roughly two meals a day, one in the morning and one in the evening, each dough is half the measure, and that is the amount that becomes obligated in challah. Shammai therefore says that the measure is not two kavin but one kav - half the daily portion - in the measures of the desert.
The View of Rabbi Yosei:
Rabbi Yosei follows the path of the Chachamim: at its source the measure is 1.8 kav in the measures of the desert; in the measures of Jerusalem it drops to 1.5 (multiplying by five sixths); and in the measures of Tzippori it drops again, by a further multiplication by five sixths, to five quarters - a kav and a quarter. In his view, however, since the basis of all these measures is the volume of an egg - as emerges from the rule of 'askalav gudo', that from the kav one moves to four lugin and from the log to six eggs - and the egg has become slightly smaller, five quarters of a kav no longer suffice in the kav of Tzippori; rather one needs "chamishah ve'od" - five and something more, an addition amounting to about five percent.
In summary: In this Mishnah, Shammai, Hillel and the Sages disagree about the amount of flour that obligates one to separate challah - a kav, two kavim, or a kav and a half. The root of the dispute lies in the measure of the omer that fell in the wilderness, a tenth of an eifah, and in the question of whether that "tenth" is measured from within or from without, and also in which system of measures is being spoken of: the wilderness measures, the Jerusalem measures, or the Tzippori measures. Once the measures were enlarged in Tzippori, the amount dropped to five quarters, and Rabbi Yose added a small increment beyond that because the egg had grown smaller.