I do not really have a question but rather a kind of comment on Tosphot in Bava Batra page 27a concerning the value of pi. This was noticed by my learning partner David Bronson in a different context where Tosphot was giving his winding a string or rope method where the Maharshal has a diagram showing what Tosphot meant.
My comment is this. That Tosphot is going with an approximation that Pi is 3. And so he defends the idea that a tree really would need 16 2/3 to get up to the value of the Mishna. [That how he explains one Girsa in the Gemara which says "there is lacking 2/3"] On the other hand the Rashbam defends the idea of a tree needing 16.5. But with a more accurate value of pi to be about 3.141 the actual radius around the tree would have to be 16.288. [I.e., (833.3/pi)^1/2. ]
Still it makes no difference in terms of the Gemara which says that Ula was simply being a drop strict in saying a tree with less than 16 amot from a neighbors's field is not allowed to bring Bikurim.
The background information here is this: Ula says a tree needs 16 amot [yards] radius of area areound it. The Gemara asks from where does he get this law? It suggests from the Mishna that gives three trees a space of 2500 square amot/yards. Thus each tree is getting 833.33 square amot.
[Tosphot has around five different ways of explaining the Gemara, but as far as I have gotten so far, it seems Tosphot is using a rough approximation for pi.]
[The winding method of Tosphot is to wind a string around a circle of 32 diameter and to keep winding until the diameter is 33.33. So the low circumference is 32*pi. The big one is 33.3*pi. Then you take the area which is 65 and that brings from the area of Ula up to the area of the Mishna. That works fine. See Tosphot for the exact calculations. But still Tosphot is using a very rough approximation for pi.
[I recall that there are places like in tracate סוכה where the Gemara gives a much more accurate approximation for pi, but I guess here it was thinking that that degree of accuracy was not necessary.]
I should mention that the winding method of Tosphot is quite ingenious. It does not require measuring the all the string but merely the inner string and the outermost string.
My comment is this. That Tosphot is going with an approximation that Pi is 3. And so he defends the idea that a tree really would need 16 2/3 to get up to the value of the Mishna. [That how he explains one Girsa in the Gemara which says "there is lacking 2/3"] On the other hand the Rashbam defends the idea of a tree needing 16.5. But with a more accurate value of pi to be about 3.141 the actual radius around the tree would have to be 16.288. [I.e., (833.3/pi)^1/2. ]
Still it makes no difference in terms of the Gemara which says that Ula was simply being a drop strict in saying a tree with less than 16 amot from a neighbors's field is not allowed to bring Bikurim.
The background information here is this: Ula says a tree needs 16 amot [yards] radius of area areound it. The Gemara asks from where does he get this law? It suggests from the Mishna that gives three trees a space of 2500 square amot/yards. Thus each tree is getting 833.33 square amot.
[Tosphot has around five different ways of explaining the Gemara, but as far as I have gotten so far, it seems Tosphot is using a rough approximation for pi.]
[The winding method of Tosphot is to wind a string around a circle of 32 diameter and to keep winding until the diameter is 33.33. So the low circumference is 32*pi. The big one is 33.3*pi. Then you take the area which is 65 and that brings from the area of Ula up to the area of the Mishna. That works fine. See Tosphot for the exact calculations. But still Tosphot is using a very rough approximation for pi.
[I recall that there are places like in tracate סוכה where the Gemara gives a much more accurate approximation for pi, but I guess here it was thinking that that degree of accuracy was not necessary.]
I should mention that the winding method of Tosphot is quite ingenious. It does not require measuring the all the string but merely the inner string and the outermost string.
