THE MIGRASH AROUND A CITY
Answer #2 (to Question 3:h, 56b - Abaye): We can say that the city is 1000 by 1000 Amos;
The Techum [without the corners] is eight [square Mil, i.e. a rectangle one by two in each of four directions]. The corners are another [four squares of two by two, i.e.] 16; altogether it is 24.
The Migrash [itself] is four [a square one by one in each of four directions]. The corners are another four [such squares]. In all it is eight.
Question: The Migrash is not a quarter, rather, it is [eight divided by 24, which is] a third!
Answer: You assumed that the city and Migrash are square. Really, they are circles;
A circle is three quarters of the square. Therefore, the Migrash is not eight, rather, six. This is a quarter of 24.
Answer #3 (Ravina): (Really, the city is 2000 by 2000. The Migrash is square. The calculation of the area of the Techum excludes the city itself.) There is no Migrash in the corners, therefore it is only eight, a quarter of the Techum (32).
Answer #4 (Rav Ashi): There is Migrash (some say - only) in the corners, the Beraisa teaches that the area of the Migrash in the corners is [four square Mil,] a quarter of the area of the corners in the Techum (16. This is true no matter how big the city is.)
Question (Ravina): It says "Soviv" (around, and the Tana cited this. Surely, this means around the edges)!
Answer (Rav Ashi): This refers to the corners, which are around the city.
Support: Regarding Olah it says "v'Zarku Es ha'Dam Al ha'Mizbe'ach Soviv." It does not mean around the edge. (The blood is thrown only on corners!)
Rather, you must say that "Soviv" refers to the corners. The same applies to the Migrash!
Question (Rav Chavivi of Chuzna'ah): Above (56b), the Beraisa said that the Migrash is circular. The Migrash is not 1000 Amos around the corners of the city!
Answer (Rav Ashi): The case is, the city is circular.
Question: We square the city before making the Migrash!
Answer: Indeed, we consider the city to be square, but in truth it is not. (It suffices that the Migrash is 1000 Amos around the actual city).
Question (Rav Chanilai of Chuzna'ah): The Beraisa (56b) said that [since we add a square Techum in each corner, and not a circle of 2000,] the Techum gains 800 Amos in each direction.) A square is a quarter more than the circle inside. (The excess is a quarter of the square, which equals a third of the circle.) If so, the addition in the corner of the Techum is only 666 and two thirds (a third of 2000)!
Answer (Rav Ashi): The area [or perimeter] of a square is a quarter more than that of the circle inside, but the diagonal is more than this;
It was taught that if the side of a square is one, the diagonal is one and two fifths. (A square of side 2000 surrounds a quarter circle of radius 2000, so the diameter is an extra two fifths of this, i.e. 800.)
WHEN DO WE GIVE A KARFEF TO A CITY?
(Mishnah - R. Meir): We give [room for] a Karfef to a city [before measuring the Techum];
Chachamim say, we give a Karfef only between two cities;
If when we give 70 Amos and a fraction to each city, these areas overlap, the cities are considered like one.
The same applies to three cities that are Meshulashim (evenly spaced in a line). If the outermost cities are less than 141 and a third Amos apart, the middle city makes them like one.
(Gemara) Question: What is the source of this?
Answer (Rava): "Mi'Kir ha'Ir va'Chutzah" - we give extra room outside a city before measuring.
(Mishnah - Chachamim): We give a Karfef only...
(Rav Huna): Two Karfifos are given for two cities;
(Chiya bar Rav): We give only one Karfef.
Question (against Rav Huna - Mishnah - Chachamim): We give a Karfef only between two cities.
Answer: It means that the concept of Karfef applies only between two cities, but we give one to each city.
Support (Seifa): If when we give 70 Amos and a fraction to each city, they overlap, they are considered like one city.
Question: This refutes Chiya bar Rav!
Answer: R. Meir taught the Seifa.
Question: R. Meir already taught in the Reisha that we give a Karfef to a city! (Surely, two cities get two Karfifos.)
Answer: He must teach both of these:
Had he taught only the Reisha, one might have thought that one Karfef is given for one or two cities. The Seifa teaches that two cities get two Karfifos;
Had he taught only the Seifa, one might have thought that we give Karfifos to two cities [near each other], for they are cramped, but a lone city gets no Karfef. The Reisha teaches that this is not so.
THE MIDDLE CITY JOINS THE OUTER ONES
Question (against Rav Huna - Mishnah): The same applies to three cities that are Meshulashim. If the outermost ones are less than 141 Amos apart, the middle one makes them like one.
Inference: If not for the middle city, they would not be like one!
Answer: Rabah explained that they are not evenly spaced in a line. (Rather, they are in a triangle. See Perush Chai diagram, Perek 5 number 64, in the English Charts section.) If when we project the third city between the other two, it is within 141 Amos of each of them, then all are like one.
Question (Rava): How far can the third city be [from the others, and still we may consider it to be between them]?
Answer (Abaye): It may be up to 2000 Amos away (since it is within the Techum of the others, we project it as if it is between them).
Question (Rava): You yourself said that presumably, Rava brei d'Rabah is right! [If a city is shaped like a bow and the ends are less than 4000 apart,] the bow may be more than 2000 Amos from the Yeser [and still we measure from the Yeser]!
Answer (Abaye): There, there are houses along the bow (one can traverse it without leaving the city). Here, there are no houses between the cities.
Question (Rava): What is the maximum distance between the outermost cities?
Answer (Abaye): We are not concerned for this! As long as when we project the middle city between them it is within 141 of each of them, all are like one!
Question: Is this even if the outer cities are 4000 Amos apart?
Question: Rav Huna taught that if a city is shaped like a bow and the ends are less than 4000 apart, we measure from the Yeser. If not, we measure from the bow!
Answer: There, we cannot view the Yeser as if it were full. (Rashi - there is nothing to fill it with. Tosfos - if we would project the houses middle to the Yeser, this would deprive them of their Techum from the bow. Alternatively, we do not project, for the Yeser cannot contain everyone in the bow.) Here, we can project the middle city.
Question (Rav Safra): People of Akistefun measure from the end of Ardeshir and vice-versa, even though the Diglas river separates them, and it is more than 141 and a third wide!
Answer (Rava): Part of the city wall is in the river within 70 Amos and a fraction [of the Ibur of the other city].
HOW WE MEASURE THE TECHUM
(Mishnah): We measure with a rope 50 Amos long, no more and no less. It must be held next to the heart;
If along the way there is a canyon or wall [that fell, and the resulting rock pile is an incline that can be traversed], we 'swallow' it (people hold the ends of the rope from opposite sides. For a wall, they hold it up on poles. It counts like 50 Amos, even though the 'walking' distance along the incline is much greater);
We resume measuring, as long as we need not leave the Techum. (This will be explained.)