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