1) MEASURING THE "TECHUM" OF A BOW-SHAPED CITY
QUESTION: The Gemara says that in order to determine the Techum of a city when a protuberance (such as a house) extends beyond the straight line which forms the boundary of the city, we "square" the city, making each side of the Techum of the city tangent to the farthest protrusion on that side of the city. We measure the Techum from these extended boundaries and not from the actual boundaries of the city.
Why, then, does the Gemara discuss how to measure the Techum of a city in the shape of a bow? Why do the normal guidelines of how to measure a Techum not apply to a bow-shaped city? We should simply "square" the city, making it a rectangle and not bow-shaped! (RASHBA)
(a) The RA'AVAD and RABEINU TAM (cited by the Rashba) answer that it is apparent from this Sugya that there is a limit to the situations in which we may square a city to establish its Techum. We may not square a city when the distance from the far end of the Pagum (protruding house) to the city proper is greater than 4000 Amos. Therefore, when a city is shaped like a bow with more than 4000 Amos between its ends, we cannot apply the rule of squaring the city.
(b) The RASHBA suggests that even when a Pagum protrudes more than 2000 Amos from a city, we do not square the city -- even though it is within a 4000-Amah line from the city. (The Rashba eventually disproves this suggestion and retracts it.)
(c) The Rashba suggests further that when a city is shaped like a bow, it is unlikely that the area between the ends of the bow will be built up and settled. When the residents of the city build more houses, they build them at the ends of the bow. Since the area between the ends of the bow will not become part of the city, we do not square the city and include that area in the city. We square the city only in the case of a Pagum, since it is possible that the city will be extended to fill in the entire area adjacent to the Pagum. (See also Insights to 57:1:d.)
2) MEASURING THE "TECHUM" FROM THE "KESHES"
OPINIONS: Rav Huna says that when the two ends of a city shaped like a bow are less than 4000 Amos away from each other, the city's 2000-Amah Techum is measured from the imaginary line between the two ends (the "bowstring"). When, however, the two ends of the bow-shaped city are 4000 Amos or more away from each other, then the 2000-Amah Techum is measured from the inhabited part of the city (the "bow").
From exactly which point in the bow is the Techum of a house measured when the two ends of the city are 4000 Amos or more from each other?
(a) RASHI (DH Pachos; 61a, DH Ir ha'Asuyah) writes that "each person measures the Techum from the entrance of his house." Rashi implies that one may walk only 2000 Amos from his house, even though he is still within 2000 Amos of the border of the city. Rashi understands that we ignore the city limits entirely when the city is shaped like a broad bow and we view the house as though it is in an uninhabited desert. (This is the understanding of the RASHBA and MAGID MISHNEH (Hilchos Shabbos 28:8) in Rashi's words.)
(b) The RITVA and RASHBA disagree with the above approach. They point out that in every city, the entire area of the city is considered like one's four Amos with regard to the Shabbos Techum, and the 2000-Amah Techum is measured from the borders of the city. Similarly, for the people living in the bow-shaped city, when the Techum cannot be measured from the "bowstring" it is measured from the borders of the "bow" part of the city (i.e., from the place along the edge of the "bow" that is nearest to the person's house). The RITVA asserts that this is also the intention of Rashi -- we measure each person's Techum from the edge of the bow. For those living further inside the bow, we measure 2000 Amos from the point along the bow that is closest to them. (According to this understanding, Rashi mentions "each person's house" only to negate the following two opinions.)
(c) TOSFOS and other Rishonim that the people within the bow may walk 2000 Amos in the direction of the bowstring, starting from the place where the two separate sides of the bow have not yet begun to spread apart from each other, so far that an imaginary bowstring of 4000 Amos can be drawn between the two ends of the bow.
(d) The TUR (OC 398) adds that those within the bow may also walk 2000 Amos in the direction of the bowstring starting from an imaginary bowstring 2000 Amos away from the innermost point of the bow.
(e) The Rashba quotes the RA'AVAD who points out that one who walks in the direction opposite that of the bowstring may walk not only 2000 Amos from the border of the city, but also 2000 Amos from an imaginary square that is drawn around the bow's-bend of the city. (That is, this bow-shaped city is measured the same way the Techum of any city, whose borders are not straight lines, is measured. A square is drawn around the city, from which the Techum is then measured; see previous Insight.)
HALACHAH: The SHULCHAN ARUCH (OC 398:4) writes only that the Techum is measured from the "bow" part of the city ((b) above). The REMA adds that it is measured from where the bow widens to more than 4000 Amos ((c) above).

55b----------------------------------------55b

3) HALACHAH: THE "TECHUM" OF A CITY IN THE SHAPE OF A BOW
OPINIONS: Rav Huna (55a) says that when the two ends of a city shaped like a bow are less than 4000 Amos away from each other, the city's 2000-Amah Techum is measured from the imaginary line between the two ends (the "bowstring"). We view the empty space between the "bowstring" and the inhabited part of the city (the "bow") to be filled with houses. When, however, the two ends of the bow-shaped city are more than 4000 Amos away from each other, then the 2000-Amah Techum is measured from the inhabited part of the city (the "bow").
Rabah bar Rav Huna and his son Rava disagree about the maximum distance between the "bow" part of the city and the imaginary "bowstring." Rabah bar Rav Huna maintains that the distance must not be more than 2000 Amos. If it is more than 2000 Amos, then even if the distance between the two ends of the city is less than 4000 Amos, the Techum is not measured from the "bowstring" but from the actual city (the inhabited area) itself. Rava, the son of Rabah bar Rav Huna, maintains that even if the empty area between the city and the "bowstring" is more than 2000 Amos, since one can walk from the "bow" part of the city to the ends of the city, the 2000-Amah Techum is measured from the ends of the city and from the "bowstring." Abaye sides with Rava. What is the Halachah?
(a) TOSFOS (DH Pachos) and the ROSH (5:2) point out that Rava maintains that as long as the two opposite sides of the city are "Muvla'a" within each other's Techum (i.e., the city is less than 2000 Amos in either length or width), the Techum is measured from the "bowstring." Thus, even when there are more than 4000 Amos between the two ends of the city, if there are less than 2000 Amos from the "bow" part of the city to the "bowstring," the Techum is still measured from the "bowstring."
(b) The RITVA and RASHBA argue with Tosfos and the Rosh and assert that when there are more than 4000 Amos between the two ends of the city, the Techum cannot be measured from the "bowstring" even if there are less than 2000 Amos between the "bow" and the "bowstring."
HALACHAH: The SHULCHAN ARUCH (OC 398:4) writes only that "if there are more than 4000 Amos between the ends of the city, the Techum is measured from the actual city border," which implies that he does not rule like Tosfos. The REMA, however, writes that we may be lenient when the distance between the "bow" part of the city and the "bowstring" is less than 2000 Amos, and we may measure the Techum from the "bowstring."