According to Abaye, the Beraisa is saying that the Migrash of a city 1000*1000 is one quarter of the Techum+Keranos around the city minus the area of the city itself.
Tosfos points out that the Gemara's calculation is that the Techum+Keranos = 25 boxes of 1000*1000, minus the city which is one box of 1000*1000. But we established that the city is round, and not 1000* 1000 - and this was a necessary part of the calculation of the area of the Migrash! If so, we should subtract a round city of 1000 diameter, the are of which is 3/4 of 1000*1000!
Tosfos answers that the calculation is Lav Davka, and indeed the Techumim+Keranos is a bit more than 24 boxes of 1000*1000.
The Vilna Gaon on the Daf argues with Tosfos, and writes that the Gemara's calculation works out exactly if the city is actually 1,250 and not 1000.
I can't figure out what he means (even if he is using the true measure for pi, and not the round number of 3). The Yad Binyamin cites Nachal Aravim who explains the Gaon, but the way he quotes the Nachal Aravim it seems to be riddled with mistakes. The only thing I can figure out from there is that if the city is oval , and 1,250*1000, then the calculation will be exact.
But if this is the Gaon's understanding, that means that an oval city is 3/4 the area of a square city in which each side of the square city are equal to the longest diameter of the oval. That doesn't seem to make sense either - is there any source for that? What would be in an oval city 1000*10?!
If you can help me out with this I would very much appreciate it.
Thanks!
Uri Wolfson, Jerusalem,Israel
I, too, was not able to understand the Nachal Aravim brought by the Yad Binyamin. However, I would like to suggest a different understanding in the words of the Vilna Ga'on.
When the Ga'on says that the calculations work perfectly when the city is "Elef u'Revi'a" what he means is that the area of the circular city is one-quarter larger than that of such a city 1000 Amos in diameter. The reason for this is to make up for the difference between the area of the square city which the Gemara first understood Abaye to be suggesting and the circular city the Gemara understands Abaye to be referring to in the Maskanah. Since a square city with a side of 1000 Amos is one-quarter larger than a circular city with 1000 Amos, adding an additional one-quarter of area to that circular city evens the difference. The Gemara's final understanding of Abaye is that he was referring to a circular city with an equivalent area of a square city with a side of 1000 Amos.
Given this, we now can make the following calculations. The area of a circle is pi multiplied by the radius squared. The radius of the circle is 500 Amos. Using 3 for the value of pi (although if one uses the value of 3.1415.... the calculations result in virtually the same final answer), the area of a city with a radius of 1000 Amos is 750,000 square Amos. One-quarter of 750,000 is 187,500. Add the two values and the area of the city which the Vilna Ga'on says our Gemara is referring to is 937,000 square Amos. In order to get the radius of such a city, we must make the same calculation backwards. Dividing 937,500 by 3 gives us 312,500. The square root of this value is almost exactly 559. Therefore the city under discussion has a diameter of 1118 Amos.
The total area of the Techum of such a city including the city (5118 x 5118) is 26,193,924 square Amos. After subtracting the area of the city itself (3/4 of 1118 x 1118 = 937,443), we arrive at a total Techum of 25,256,481 square Amos, or approximately 25.256 squares of 1000 x 1000 Amos. One quarter of this is 6.314 such squares. The Techum of the city is 3/4 the Techum of a square city with a side of 1118 Amos. The Techum of the square city is 8.4472 [4 + 4(1118)] squares of 1000 x 1000 Amos. 3/4 of this is 6.335 squares of 1000 x 1000 Amos - a value extremely close to the quarter of total Techum. According to this understanding, the Ga'on must be of the opinion that rounding off by .023 is so negligible that one need not even say that the Gemara is Lo Dak (thereby avoiding the problem of Tosfos).
I hope that you find this helpful!
Moshe Binyamin Cohen