Hippocrates' quadrature of the lune depended on the fact that areas of circles are in proportion to the squares of their diameters (or the squares on their diameters, depending on your point of view). This result was certainly known long before Hippocrates' day, for it is equivalent to knowing that the area of a circle is some constant times the square of its diameter, but it received its earliest proof in the literature in Euclid's Elements, the geometrical encyclopedia of its day. This proof has long been attributed to Eudoxus of Cnidos (408?--355? BCE), an accomplished philosopher, especially in mathematical astronomy (the study of the geometry of the sphere), and said to have been a student of Plato in Athens. While the result is certainly important in the development of the ideas we have been discussing here, what is even more important for our later work in calculus is the method he uses in his argumentation.
Eudoxus was the first to employ the method of exhaustion in the quadrature problem, a method that later geometers would return to again and again. The basic idea is this: to show that region R has the same area as region S, we use the logical device of the double reductio ad absurdum to structure an indirect proof. That is, to show that R = S, we show instead that it is impossible for R > S or for R < S; this is the double reductio. It requires two separate arguments, one to show that the assumption R > S leads to a contradiction, and another to show that the assumption R < S leads to a contradiction. Since neither assumption can then be true, it follow that the only other alternative, namely R = S, must hold.
Consider the case R > S.
Then, since region R typically has a curved boundary, we get a handle
on the size of R by inscribing within it a polygon P, whose
area will satisfy R > P. But we will choose P so that
its boundary closely approximates the curved boundary of R, so closely
that the area of P satisfies R > P > S. Notice that P
is chosen after regions R and S are known, so one must be
sure that it is always possible to select a polygon P whose area
can be squeezed between those of R and S regardless how close
together the areas of R and S might be to each other. The
ability to find such a P testifies to the fact that one can "exhaust"
the area of region R by filling in the region with triangles contiguous
to each other so that their union can be taken as the polygon P.
And no matter what tiny slivers of area may be found to lie between P
R, we can always fit some even tinier triangle within it, one
of whose sides is an edge of P. As a result, P can be improved
as an even better approximation to R. In other words, no matter
An entirely similar
argument works under the complementary assumption that R < S,
for here we circumscribe R with a closely approximating polygon
so that R < P < S. It is this spirit of approximation by polygons
that characterizes the method of exhaustion and was used successfully by
the ancients in dealing with the area problem. Compare this with the notion
of convergence of infinite series we discussed earlier: the approximating
polygon takes the role of the converging partial sums of the series that
get closer and closer to the actual sum of the series.
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last modified 8/30/02
Copyright(c) 2000. Daniel E. Otero