Abaya gave the dimension of the round city as being 1000 amos in diameter. Tosfos D"H Abaya asks, according to the cheshbon the Techum is not exactly 4 times the Migrash. Tosfos answers that the techum is calculated as though the city had been square. The Hagaos Ha'Gra writes that this is dochak and says that the Gemara was lav davka and meant (instead of 1000) to say 1250.
I could not understand what the Gra meant. My chaveirim and I made our own cheshbon and calculated that the city needs to be exactly (8 - the square root of 48) * 1000, which comes out to about 1072 amos across the diameter.
Why would the Gra give a less exact number than the Gemara, while saying that it is more exact?
I saw the Nachal Ha'aravim, brought by the Yad Binyamin, but I find that even harder to understand. He calculates the Techum and Migrash as though the city was only 1250 x 1000, and he calculates the area of the round city as "3/4 of 1250 square, which comes out to 1000 square" which is not accurate. I even tried to cheshbos this with an oval city but it still does not come out exact, whichever way you measure the Techum and Migrash. The Nachal Ha'aravim writes that his cheshbon is "exact and precise" which it is if you make the cheshbonos the way he does, but considering how the Gra called Tosfos' pshat dochek, the above mentioned method seems even more so.
The only answer for the Gra that I could think of is that he wasn't being midakdeik either, and he only meant to say that the real city size is 1000 and a bit. What does this Gra means?
Bumy Goldson, Beitar Illit, Israel
What the Gra means is that since the Gemara later on says that the city is round, it follows that the techum of the city is one quarter bigger since a square is a quarter bigger than a circle. Tosfos asks on Abaye that it follows that the techum is a quarter bigger than Abaye said, so the Migrash is not a quarter. The Gra answers the question of Tosfos by saying that Abaye did not really mean the city is 1000 but to be accurate he meant 1250. Therefore the city is indeed bigger than the literal figure of 1000 that Abaye gave, because Abaye really meant 1250.
KOL TUV
Dovid Bloom
I'm not sure I understand your answer but let me try to repeat it back in a nutshell.
When we turn the city into a circle we lost a quarter of the city's area (and since the city was 1000 x 1000 so we lost 250).
To compensate for that we add to the city's width 250.
Is that right?
It's just a svara bi'alma and not an actual calculation?
Abaya was discussing the width. The amount lost from the Techum was an area.
Yes, I think that is right.
According to the simple explanation of the Gemara Abaye said we are referring to a square city 1000 x 1000.
The Gra said that Abaye was actually referring to a square city 1250 x 1250.
The Gemara top 57a says that we are in fact referring not to a square city but to a round city whose width is 1250, accroding to the Gra. That means the radius is 625.
So it is not a svara b'alma but is in fact a calculation.
Abaye was discussing the width but clearly the width affects the area.
Yasher Koach
Dovid Bloom
But then Tosfos's question comes back!
If the round city is 1250 wide then...
The city's area is 1,171,875 (1250 squared, times 3/4)
The migrash area is 6,750,000 (3250 squared, times 3/4, minus the city area)
The Techum is 26,390,625 (5250 squared, minus the city area)
Dividing the Techum by the Migrash gives us 3.9097222...
Which is not exactly 4, which is what Tosfos's question was in the first place!
I think there may be corrections to be made here.
1) The migrash area is not 3250 squared. It is in fact 3750 squared. This is because the Gemara top 57a stated that the migrash is 8 boxes of 1000 x 1000, which the Gra said really means 8 boxes of 1250 x 1250. The city itself is in the middle and is not counted, so we have 3 boxes by 3 boxes, not counting the city in the middle. 3 boxes each of 1250 is 3750, which means the migrash is 3750 squared times 3/4 minus the city area.
This is 14,062,500 x 3/4= 10,546,875 - 1,171,875 = 9,375,000.
So the migrash area is 9,375,000.
2) The techum is not 5250 squared. It is in fact 6250 squared. This is because the Gemara bottom 56b states that the techum is 8 boxes and the keranos are 16 boxes which makes 24 boxes, plus one box for the city itself in the middle, so we have 25 boxes. That means it is 5 x 5 boxes. The Gra says that the boxes are 1250 wide so that means the width of the entire city with the techum is 5 x 1250 = 6250.
So the techum is 6250 squared minus the city area = 39,062,500 - 1,171,875 = 37,890,625.
So the techum is 37,890,625.
3) Dividing the Techum by the Migrash gives us 4.04166....
because 37,890,625/9,375,000 = 4.04166..
4) So the Migrash is not exactly one quarter but it is actually 1/4.04166..
I think Tosfos would also agree that this is close enough to one quarter for the Gemara to call it one quarter.
Yasher Koach
Dovid Bloom
39,
With that method, one would get a Techum/Migrash ratio of 4.041666 regardless of the size of the city. If everything is infalted together then the ratio stays the same.
This inaccuracy of 4.04166 is the exact number that Tosfos was bothered by in the first place.
OK, but the Gra is not bothered by a small innaccuracy. The Gra has support for his approach from the Gemara Sukah 8a which states that even though Rabbi Yochanan said (7b) that a round Sukah is Kosher if it holds 24 people, he actually means 16 and 4/5 people, but he was not accurate. The Gemara asks on this that do we not say "Lo Dak" only for a small difference, but not for such a large discrepancy?! We do learn from this, however, that it is acceptable to say "Lo Dak" for a small discrepancy. Therefore the Gra learns that even though the Gemara here states that the Migrash is 1/4, this really means 1/4.041666..
KOL TUV
Dovid Bloom
So let the Gra say "lo dak" on the 1000 and leave it at that.
What does he gain by saying that Abaya meant 1250 if that doesn't change anything?
By saying that Abaye meant 1250 the Gra answered the question of Tosfos because it indeed works out that the techum is a quarter bigger, as Tosfos asks that it should be.
The Gra also incidentally says quite a big chidush. The Gemara Sukah 7b tells us that Rabbi Yochanan said that a round Sukah must be able to seat 24 people around its circumference. The Gemara 8a challenges that one does not need 24, but rather 16 and four fifths should be sufficient. The Gemara attempts to answer on this challenge 'Lo Dak" but then the Gemara rejects this because the difference between 24 and 16 and 4/5 is too large to be able to say lo dak.
However the Gra says here that for the difference between 1000 and 1250 it is still possible to say lo dak.
Now 24 is 0.3 bigger than 16 and 4/5.
And 1250 is 0.25 bigger than 1000.
So the Gra is saying that a differnce of 0.3 is too wide to be able to say lo dak, but 0.25 is still small enough to say lo dak.
It is an interesting point how the Gra knew this.
Purim Sameach
Dovid Bloom