The rabbis of Caesarea provide a formula for measuring a circle within a square and a square that is inside a circle.
The circle inside a square has a circumference that is 1/4 less than the square. This matches what we said before. So if the perimeter of the square is 16, than the diameter of the inscribed circle is 4. Multiply that by 3 and you get 12, which is 1/4 less than 16.
A square that is inside a circle has a circumference that is half that of the circle.
The Talmud rejects the rabbis of Caesarea’s last formulation. If the square is 4 square cubits, then its 5.6 cubits, which is also the diameter of the inscribing circle. The circumference of the circle is 3 times this amount, meaning 16.8. Indeed it is difficult to imagine that the rabbis of Caesarea (or R. Yohanan) made such a gross error in calculation
The Gaon of Vilna explained that in actuality this was not an error at all. What these rabbis meant to say was that if you put another square around this circle, that square would have a circumference that is 50 percent larger than the inner square. This outside square would have sides that are 5.6. Its circumference would be 24 (5.6 x 4). This square is now 50 per cent larger than the inner square, whose circumference is 16. This, according to the GRA is what these rabbis really intended to say.
This matches R. Yohanan’s number above. R. Yohanan meant to say that this circular sukkah must be large enough so that when you put a kosher sukkah inside (16 square cubits), the square formed outside of the sukkah would be 24 cubits.
Interestingly, we see that R. Yohanan’s math was actually correct. It was later rabbis that seemed to have misunderstood his words. Perhaps, we might surmise that in Eretz Yisrael, where R. Yohanan and the rabbis of Caesarea lived they understood math better than they did at a later period in Babylonia. This makes sense considering the fact that the Greeks were well known to have been excellent mathematicians.
Today’s section begins a new discussion about various forms of sukkot.
R. Levi discusses a two-part structure that houses a potter and probably serves as his shop as well. There is an outer hut and an inner hut. The inner hut is not valid as a sukkah because this is the potter’s regular home. Even if this hut fulfills all of the requirements of a sukkah it is not valid because he lives there all year round. A person can’t just live in his sukkah all year round without doing something to make it valid for Sukkot. However, it is obligated in the mitzvah of mezuzah because it is his regular domicile (big word!).
The outer sukkah is valid as a sukkah (provided it is made of proper dimensions and material) but is exempt from the mitzvah of mezuzah because he doesn’t live there.
The Talmud now criticizes this—why should the outer hut be exempt from mezuzah? After all, even a gate-house is obligated to have a mezuzah. Why can’t we say that the outer hut is like a gate-house to the inner hut and therefore should have a mezuzah?
The answer is that neither huts are permanent. A gate-house is obligated to have a mezuzah because it protects a permanent hut. But this outer hut serves an inner hut which itself is not a permanent structure. Therefore the outer hut is exempt from the mitzvah of mezuzah, even though the inner one is.
This baraita begins with a mnemonic device. It teaches that it doesn’t matter who built the sukkah, even a gentile or a Samaritan, or whether it was primarily built for cattle, it is valid. As long as it was “covered according to the rule.”
R. Hisda answered: Provided that [the covering] was made [with the intention of providing] the shade for the sukkah.
R. Hisda explains that “according to the rule” means that it was made with the intention of providing shade. Note that this doesn’t mean that it had to be built with the intention of using it as a sukkah for the festival of Sukkot. The rule is that the skhakh has to have been put up with the intention of it being used for shade. If it was made for some other reason, such as privacy, it is not a valid sukkah.
The baraita cited above ended with the words, “any sukkah whatsoever”—is valid. Obviously this doesn’t mean any sukkah is valid. There are some types of sukkot that are invalid. So the Talmud asks—what other types of sukkot are valid that weren’t mentioned in the other baraitot.
It includes the sukkot [whose mnemonic is] rakbash, as our rabbis taught: The sukkah of shepherds, the sukkah of fig-watchers, the sukkah of city guards, and the sukkah of orchard-watcher, and any sukkah whatsoever is valid, provided that it is covered according to the rule.
The Talmud connects the previous baraita to another baraita based on a mnemonic device. This baraita states that sukkot built for people who are basically guarding property is valid. These sukkot were obviously not made for Sukkot. Nevertheless, they are valid, as long as they were covered “according to the rule.”
This is the same explanation of “according to the rule” found in the previous section. See there for an explanation.
This section continues to discuss the previous two baraitot that basically allowed any sukkah to be used as long as it was constructed for shade.
The second baraita the one that included “rakbash” added “any booth whatsoever” to include ganbak. Of course, this means that the first baraita (ganbak) included “rakbash” and the second baraita “rakbash” included “ganbak.” This will lead to the question why does each baraita begin with a different list and then add the other list at the end.
The author of the first baraita thought that the “ganbak” sukkot (those built by Gentiles, Samaritans or for cattle) were of greater validity, therefore he listed them and then included the rakbash sukkot with “any sukkah whatsoever” because rakbash sukkot (made for people temporarily filling jobs) are impermanent.