1) VIEWING A CROOKED "KORAH" AS STRAIGHT
OPINIONS: The Tana Kama in the Mishnah (end of 13b) says that a Korah must be strong enough to hold a brick. Rebbi Yehudah argues and says that even if the Korah is made out of straw, we view it "as if it were metal."
The Mishnah continues and says that if the Korah is bent, we view it "as if it were straight," and if it is cylindrical, we view it "as if it were a rectangular solid."
Is the last statement of the Mishnah, regarding a bent Korah and a cylindrical Korah, a continuation of Rebbi Yehudah's opinion, or does the Tana Kama agree that "we view it as if..." in these cases?
(a) The ROSH points out that when the Gemara asks questions on the statements in the Mishnah, it uses different phrases in each question. First, when it cites Rebbi Yehudah's statement that we view a Korah made out of straw as if it were made of metal, the Gemara asks, "Mai Ka Mashma Lan?" That is, since Rebbi Yehudah already said that the Korah does not have to be strong, what is this additional statement teaching us? Next, when it cites the statement that we view a bent Korah as if it were straight, the Gemara asks, "Peshita" -- "this is obvious." Finally, when it mentions the third statement that we view a cylindrical Korah as if it were rectangular, the Gemara asks, "Ha Su Lamah Li" -- "why is it necessary to mention this case as well?"
The Rosh explains that the first question -- "what is it teaching us?" -- addresses specifically the first statement in the Mishnah, the opinion of Rebbi Yehudah. The second question addresses the second statement in the Mishnah, that we view a bent Korah as if it were straight, because this statement is "obvious" -- that is, it is even more logical to apply the reasoning of "we view it as if..." in this case than in the first case. Since it is more logical, it is reasonable to suggest that the Tana Kama agrees with it. Moreover, the Gemara concludes that the Mishnah is teaching a Halachah that was also taught by Rebbi Zeira, and it is logical to assume that Rebbi Zeira was explaining the Halachic opinion, which is that of the Tana Kama. When the Gemara discusses the third statement of the Mishnah (that we view a cylindrical Korah as if it were rectangular), the Gemara asks "Ha Su Lamah Li?" -- because this Halachah is identical to the previous one. Accordingly, in this case, too, the Tana Kama agrees that we apply the reasoning of "we view it as if...." (This is also the ruling of the RAMBAM, Hilchos Shabbos 17:26.)
(b) RASHI here (DH Peshita) explains that the reasoning of "we view it as if" applies equally to a bent Korah and to a weak Korah. According to Rashi's explanation, it is only Rebbi Yehudah who made these statements in the Mishnah; the Tana Kama never applies the logic of "we view it as if." (This is also the approach of RABEINU YEHONASAN MI'LUNIL; see, however, the BACH there.) The RITVA cites a Yerushalmi that seems to support Rashi's interpretation of the Mishnah.
HALACHAH: The SHULCHAN ARUCH (OC 363:20) rules like the Rosh and the Rambam (a), and applies the logic of "we view it as if" to a bent Korah and to a cylindrical Korah.
2) THE VALUE OF "PI"
QUESTION: The Gemara quotes the Mishnah (end of 13b) which says that the circumference of a circle is three times greater than its diameter (if the circumference of the circle is 3, then the diameter is 1). How do we reconcile this statement with the mathematical fact that the ratio of the circumference of a circle to its diameter is slightly more than three (Pi = 3.141592...)?
(a) The TOSFOS HA'ROSH explains that the Gemara itself addresses this issue. The Gemara asks "from where do we learn" that the circumference of a circle is three times greater than the diameter. Why does the Gemara need a source to teach the ratio of the circumference of a circle to its diameter? We do not need a verse to teach us a mathematical, observable fact! It must be that the Gemara is asking for the source that teaches us that we may use a slightly inexact value to determine the circumference of a circle. The Gemara answers that the verse that describes the circumference of the pool that Shlomo ha'Melech constructed (the "Yam Shel Shlomo") as three times its diameter (Melachim I 7:23) teaches that for all Halachic purposes we may use the approximate ratio of three to one.
Similarly, the RAMBAM (Perush ha'Mishnayos; see also Hilchos Tum'as Mes 12:7) points out that Pi is an irrational number, and "the exact relationship of the diameter to its circumference cannot be known and it is not possible to speak of it... its actual value cannot be perceived." He writes that the value which is commonly used in calculations is 3 1/7 (3.142857...). The Tana'im of the Mishnah rounded this number and expressed it in terms of the nearest whole integer (3).
(c) A fascinating insight regarding the value of Pi is attributed to the Vilna Ga'on. (Actually, there is no source to substantiate the claim that the Vilna Ga'on said it. The actual source for the insight may be credited to Matityahu ha'Kohen Munk (Frankfurt-London), who published the thought in the journals "Sinai," Tamuz 1962, and "ha'Darom," 1967.) In the verse that the Gemara cites as the source for the ratio of the circumference to the diameter (Melachim I 7:23), there is a "Kri" and a "Kesiv" -- a word that is pronounced differently than it is spelled. The word in the verse is written "v'Kaveh" (with the letter "Heh" at the end), but it is pronounced "v'Kav" (with no "Heh" at the end). The Gematriya of the word "Kav" is 106, and the Gematriya of the word "Kaveh" is 111. The ratio of the Kesiv (111) to the Kri (106), or 111/106, is approximately 1.0472. This value represents the ratio of the value for Pi to 3 (3.141592.../3, or approximately 1.0472).
Hence, the difference between the actual value of Pi and its practical, Halachic value is expressed by the difference between the Kesiv (the actual, but unread word) and the Kri (the word as we read it) of the verse that discusses the value of Pi!
3) A LECHI PLACED THREE TEFACHIM AWAY FROM THE WALL OF A MAVOY
OPINIONS: Rava says that a Lechi placed more than three Tefachim away from the wall of a Mavoy is invalid, because "Gediyim Bok'in Bah" -- young goats walk though the area.
Why exactly is the Lechi invalid?
(a) TOSFOS (10b, DH v'Oseh) explains that the Lechi is invalid when distanced more than three Tefachim from the wall because the rule of Lavud cannot be applied to it. Lavud functions only within a distance of less than three Tefachim. Therefore, the Lechi is not connected to the wall (which it must be in order to serve as a valid Lechi), and hence it is invalid.
(b) RABEINU YEHONASAN MI'LUNIL explains that the Lechi is invalid because the airspace on each side of the Lechi is larger than the Lechi, and therefore the Lechi becomes Batel to the airspace ("Asi Avira d'Hai Gisa ud'Hai Gisa v'Ka Mevatel Lei").
The difference between these two reasons is evident in a case in which a wide Lechi (such as a Lechi four Tefachim wide) is placed more than three Tefachim away from the wall, but the distance between the Lechi and the wall is less than the width of the Lechi itself. According to Rabeinu Yehonasan, the Lechi is valid, because the airspace between the Lechi and the wall is less than the width of the Lechi and thus the Lechi is not invalidated. In contrast, according to Tosfos, the Lechi is invalid because Lavud never applies when the gap is greater than three Tefachim.
The TEVU'OS SHOR cites proof from the Gemara for the opinion of Tosfos. Tosfos says that a Lechi that is far away from the wall is invalid because Lavud cannot function to connect it to the wall. We see that Tosfos requires that the Lechi be attached (either physically or Halachically) to the wall. Rabeinu Yehonasan objects and does not require that the Lechi be attached to the wall, and therefore he says that the problem is that the airspace on each side of the Lechi invalidates it.
According to Rabeinu Yehonasan, however, why does the Gemara say that the problem is that young goats can walk through the space (and thereby render it unattached to the wall)? According to Rabeinu Yehonasan, the Lechi does not have to be attached to the wall!
REBBI AKIVA EIGER answers that the reasoning of "Gediyim Bok'in Bah" is necessary in order to explain why Raban Shimon ben Gamliel invalidates this Lechi. According to Raban Shimon ben Gamliel, Lavud up to four Tefachim makes it as if the airspace between the Lechi and the wall does not exist, and thus there is no airspace to annul the Lechi. Why, then, is the Lechi invalid (in a case where the Lechi is more than three but less than four Tefachim) according to Raban Shimon ben Gamliel? The Gemara explains that it is invalid because young goats are able to walk through that space and thus Lavud is unable to take away the airspace. Once Lavud does not apply, the reasoning that the airspace on the sides annuls it applies and invalidates the Lechi.