1) THREE CITIES WHICH FORM THE VERTICES OF A TRIANGLE
The Gemara says that when three cities lie at the vertices of a triangle, the middle city is able to "link" the two outer cities, allowing a person to walk from one to the other even if the distance between them is more than 4000 Amos. The Gemara adds that this rule applies only if two conditions are met. First, there may not be more than 2000 Amos between the middle city and each of the outer cities. Second, the middle city must be large enough so that if positioned on a line connecting the two outer cities, the edges of the middle city would lie within 141 1/3 Amos (two Karpifos) of each of the outer cities. In such a case, we view the three cities as if they were one large city and allow a person to walk from one outer city to the other even if they are separated by a distance of more than 4000 Amos.
The Gemara asks how this case differs from the case of a city in the shape of a bow. In that case, the two ends of the city must be within 4000 Amos of each other in order to permit one to walk from one end to the other. Why do we not say in that case as well that the houses in the "bow" part of the city may be viewed as situated between the two tips, and thus those houses join the ends to permit one to walk even farther than 4000 Amos to get from one city to the other? Why do we not look at the space between the tips as filled up with houses?
The Gemara answers that we cannot view the empty space between the two tips of the bow-shaped city as filled up with houses ("Mali").
Why not? The Gemara does not explain the difference, nor does Rashi seem to shed any light as to why the part of the city in the "bow" cannot be used to fill up the space between the two ends of the bow!
(a) TOSFOS (DH Hasam) first answers that the Gemara does not mean that there is a difference between the case of a city in the shape of a bow and the case of three cities which form a triangle. Rather, the Gemara means, in order to determine the Techum Shabbos from the "bow" part of the city, one must measure from the actual physical location of his house, and not from the "bowstring" (i.e., the corresponding point between the two ends of the bow). The reason for this is as follows: If we were to imagine that he lives at a point between the two ends of the bow, then we would have to make his Makom Shevisah at that place (between the two ends of the bow). Consequently, he would lose the Techum in the direction in which his house is actually located (towards the "bow"). We cannot say "Mali" if, by adding to the Techum between the two populated areas, we will detract from the Techum on the other side. (See MAHARSHA and MAHARSHAL.)
(b) In his second approach, Tosfos says that Abaye simply means that it is only when the "bow" part of the city is more than 2000 Amos from the "bowstring" that we do not view the area between the two ends as filled up with houses from the "bow." However, when there are less than 2000 Amos between the bow and the bowstring, then we indeed apply "Mali." (That is the ruling of Rava brei d'Rabah bar Rav Huna mentioned earlier in the Sugya here and on 55b. He says that even if the distance between the two tips is greater than 4000 Amos, we still measure the Techum from the "bowstring" if the distance between the "bow" and the "bowstring" is less than 2000 Amos. Abaye himself supports this opinion on 55b.)
(c) Tosfos gives a third answer in which he suggests that we apply "Mali" only when the city at the top of the triangle fits between the two base cities. However, if the top city is wider than the distance between the two cities at the base, then we cannot apply "Mali," for the large city will not fit there. Similarly, in the case of the bow, there are more people living in the bow than can be placed in the direct line between the two tips.
(d) The RITVA and RASHBA (55a) focus on the issue of normal population expansion. They explain that "Mali" can be applied only if the normal growth of the city at the top of the triangle would eventually encompass the area between the two base cities. Since the normal manner is for cities to expand, we view the city as if it is already between the two cities at the base. However, in the case of the bow-shaped city, there is no reason to assume that it will grow along the center of the city. In fact, if more houses are added, they probably will be built along the ends of the bow. Since the city is not likely to grow along the center of the bow to meet the line between the two ends, "Mali" does not apply.