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Bill's quadrature
Posted:
Jan 2, 2002 5:33 PM



First, it's clever and I like it, though I had to read it several times to get the pictures right (but I'm getting old and slow).
However, one hidden assumption is that the area under a curve whose heights are k times the heights of a second curve is k times the area under that second curve. We think this is obvious geometrically because we think of areas as limits of areas of rectangles. Archimedes would prove such a statement by using similarity and exhaustion, and we would use limits of Riemann sums, but it does require a proof of this nature. Thus, the limits are hidden but there.
A similar assumption is made in computing the area of (half) the hill, namely that if two figures have equal crosssections, their areas are equal (even if their shapes aren't). In some of his later and more recently (1908, but we're taking the long view) discovered work (the famous palimpsest containing the "method" for finding volumes of cones cylinders and spheres), Archimedes himself uses this kind of argument (buttressed with a clever "weighing")  see, e.g., Simmons' "Calculus Gems"  guess or derive the formulas which he proves using exhaustion.
Calculus is still about limits.
 Mark
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