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

In particular, the denominators on both sides of the equation are the same. Consequently, the numerators must be equal as well. That is,

which is precisely the Pappus-Guldin theorem.