[Avodah] Tosafos's "Proof" of the Area of a Circle

[Micha Berger]

On Tue, Oct 11, 2011 at 10:06:25AM -0400, hankman 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...

: I too liked this "proof." I quoted the word proof, because part of
: the detail is missing. ....                           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...

And in fact, when it comes to ellipse this assumption is not true, and
the parallel "proof" would give you the wrong area for an ellipse.

The area of an ellipse is pi * a * b, where a and b are the two radii.
Not that when a = b, i.e. the ellipse is a circle, we have pi * r ^ 2.

But calculating the circumferance is a very complex thing.
(4 * a * E(sin(phi)), where a is the axis that runs through both focci
and the center, E is the complete elliptic integral of the second kind,
and phi is the angle of exxcentricity of the ellipse. And don't ask me
to translate all of that, because I can't.)

Tir'u baTov!
-Micha