[Avodah] Malachim & mistakes

[hankman]

RMB wrote:

While we're open to that amud, I'm more impressed with the proto-calculus
in the previous Tosafos, in their proof that the area of a circle is
pi * r ^ 2. (See also my hesped for R Dr Eliezer/Leon Ehrenpreis, where
I open describing my first-impression of REE -- his using this Tosafos
to open the teaching of calculus. I give the proof in English.
<http://www.aishdas.org/asp/2010/08/r-dr-eliezer-ehrenpreis-zl.shtml>)

CM notes:

I too liked this “proof.” I quoted the word proof, because part of the detail is missing. We of course make the “right” assumptions to fill out the proof automatically since we know the answer. The piece of the proof that is lacking (though of course true) is the assumption we all make that the resulting hypotenuse of the triangle (after you cut and “roll out” the circle) is in fact a straight line forming the third edge of the triangle. We need to prove that it in fact turns out to be a straight line and not some other curve enclosing the area, leaving you without a triangle as assumed by Tos. What needs to be added is that the length of all the strings as you move inward to the apex are all 2*pi*r (or pi*r for the half triangle) so that these lengths vary linearly and this is what guarantees the fact that the “hypotenuse” is actually a straight line as assumed and not some other curve.

Kol Tuv veChag Sameach

Chaim Manaster
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