I just wanted to share two thoughts I had about Tosphot in Bava Batra page 27A.
One thought is about the "Other path" that Tosphot brings in the middle. The way that "other path" is understanding Ula is this: When Ula say a tree needs 16 yards around it and the Gemra then add that means as a square is a fourth larger than a circle and that Ula meant 16 2/3, the way Tosphot understands that is that Ula was describing a square with one side being 33 1/3. Thus the whole square is 33.3^2 and a fourth is the size of the ground around the tree that the Mishna gives which is 33.3* 25.
That is the opposite of the way Tosphot was thinking up until that point that Ula was describing a circle with radius 16.6
The other thought I had was about the very first way that Tosphot understands Ula which is his winding a string method. The idea I had was that even without looking up the geometric formulas what Tosphot says makes a lot of sense. That is if you have a circle around the tree with radius 16 and wrap a string around it for 2/3 a yard you get the length of the inner string to be 2*r*pi=32*pi and the length of the outer circle 2*r*pi=33.3*pi.
So then if you flatten the whole thing out you get a rectangle pi*2r1*2/3=64 with a triangle at the top. And that from what I recall the area of a triangle is 1/2 base*height,- which is exactly what Tosphot says there 1/2*4*2/3. And all that brings up from the 768 square yards of Ula up until the 833.3 of the Mishna.[difference of 65].
Does all that work?--you might ask. I mean what would be the regular way of figuring it out? Normally you would take the area of the large circle (pi*r^2=pi* 16.67^2)=[833]-the area of the small circle (pi*r^2=pi*16^2). [827-768] Does that come out the same as Tosphot?There is a slight discrepancy. But in any case Tosphot is making an approximation as I mentioned before.
[The Gemara in this section is using an approximation of 3 for pi and the difference between a aquare and a circle to be 4/3]
In any case what looks important here is that in fact the Tosphot string method is not exact.
One thought is about the "Other path" that Tosphot brings in the middle. The way that "other path" is understanding Ula is this: When Ula say a tree needs 16 yards around it and the Gemra then add that means as a square is a fourth larger than a circle and that Ula meant 16 2/3, the way Tosphot understands that is that Ula was describing a square with one side being 33 1/3. Thus the whole square is 33.3^2 and a fourth is the size of the ground around the tree that the Mishna gives which is 33.3* 25.
That is the opposite of the way Tosphot was thinking up until that point that Ula was describing a circle with radius 16.6
The other thought I had was about the very first way that Tosphot understands Ula which is his winding a string method. The idea I had was that even without looking up the geometric formulas what Tosphot says makes a lot of sense. That is if you have a circle around the tree with radius 16 and wrap a string around it for 2/3 a yard you get the length of the inner string to be 2*r*pi=32*pi and the length of the outer circle 2*r*pi=33.3*pi.
So then if you flatten the whole thing out you get a rectangle pi*2r1*2/3=64 with a triangle at the top. And that from what I recall the area of a triangle is 1/2 base*height,- which is exactly what Tosphot says there 1/2*4*2/3. And all that brings up from the 768 square yards of Ula up until the 833.3 of the Mishna.[difference of 65].
Does all that work?--you might ask. I mean what would be the regular way of figuring it out? Normally you would take the area of the large circle (pi*r^2=pi* 16.67^2)=[833]-the area of the small circle (pi*r^2=pi*16^2). [827-768] Does that come out the same as Tosphot?There is a slight discrepancy. But in any case Tosphot is making an approximation as I mentioned before.
[The Gemara in this section is using an approximation of 3 for pi and the difference between a aquare and a circle to be 4/3]
In any case what looks important here is that in fact the Tosphot string method is not exact.
