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Re: Isn't it the limit?





On Sun, 30 Mar 1997, John Mallinckrodt wrote:

I'd *like* to be able to say something like, "the effects of curvature
don't become negligible since the area is only singly infinitesimal." In
other words, relate the error to the fact that even the infinitesimal
triangle "samples" a large enough portion of the sphere's surface to
retain artifacts of its curvature. An argument like this would seem to
suffice for explaining why we can get the surface area by adding up the
Euclidean areas of flattened out, *doubly* infinitesimal patches even
though larger patches don't work. But then there is the fact that singly
infinitesimal latitudinal strips *do* work.

Would it be adequate to point out that in one case, the error introduced
is an infinitesmal fraction of the (infinitesmal) area of the strip, but
that in the "triangle" that isn't the case. In choosing patches, you need
to be able to show, or at least convince yourself, that the error becomes
an infinitesmal fraction of the area of each patch.

Al