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One of the endearing things about mathematicians is the extent to which they will go to avoid doing any real work.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Loci: Convergence
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James Gregory and the Pappus-Guldin Theorem
Gregory's Proof Revealed
With all of this proportion theory in hand, Gregory's proof of the Pappus-Guldin Theorem falls into place relatively easily. Suppose that AB is the geometrical figure which is to be rotated around an axis and that a is its center of gravity. The central idea of his proof is to use the proportional version of the theorem given in the last section to compare AB with another, easy-to-understand 2-dimensional figure. In this case, that figure is a rectangle HIJK and its axis of rotation is simply the side HI of the rectangle.
For a rectangular figure HIJK, we have area(HIJK) = HI×HK. Since the solid of revolution obtained by revolving HIJK around the the line HI is a cylinder with height HI and radius HK, we get rev(HIJK) = πHI×HK2. Finally, the center of gravity h of a rectangle is the geometrical center of the rectangle, so the distance from h to HI is (1/2) HK and thus circum(h) = 2π×(1/2)HK = πHK. With some algebraic simplification, the proportional version of the Pappus-Guldin theorem from the last section then becomes
\eqalign{ {rev(AB) \over \pi HI\times HK^2 } &= {area(AB) \over HI\times HK} \times { circum(a) \over \pi HK} \cr &= {area(AB) \times circum(a) \over \pi HI \times HK^2} \cr }
rev(AB) = area(AB) \times circum(a)
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Leahy, Andrew, "James Gregory and the Pappus-Guldin Theorem," Loci (January 2009), DOI: 10.4169/loci003262