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Gregory treats separately the ratio between the trunk and the solid of revolution and the ratio between the trunk and the cylinder. The key to understanding the first ratio is Cavalieri's Principle: "If two plane (or solid) figures have equal altitudes, and if sections made by lines (or planes) parallel to the bases and at equal distances from them are always in the same ratio, then the plane (or solid) figures also are in this ratio". (See [10, pp. 315-321], cited in [11, p. 516].) Once Gregory establishes a fixed ratio between corresponding slices of the trunk and the solid of revolution, this principle will imply that the solids themselves have the same ratio. He begins with the trunk, slicing it by an arbitrary plane *OPV* perpendicular to the axis of rotation.

Let *V* denote the point of intersection of *OPV* with the axis of rotation and let *O* and *P* denote the intersections of *OPV* with* l* and *k*, respectively. Then

OP = \hbox{height of the cylinder} \qquad \hbox{and} \qquad PV = \hbox{radius of rotation}

{OP \over PV} = {GF \over FV} \qquad \hbox{and} \qquad {OP \over PV} = {HE \over EV}

\eqalign{ {OP \over 2 \pi PV} = {GF \over 2 \pi FV} & \Rightarrow {OP \over 2 \pi PV} = {GF \over 2 \pi FV}\cdot {{1 \over 2} FV \over {1 \over 2} FV } \cr & \Rightarrow {OP \over 2 \pi PV} = {{1 \over 2} GF\cdot FV \over \pi FV^2} \cr & \Rightarrow {OP \over 2 \pi PV} = {area(\Delta GFV) \over area(circle(FV))} \cr }

where Δ*GFV* denotes the triangle *GFV* and *circle*(*FV*) denotes the circle on radius *FV*.

The second equality shows similarly that

{OP \over 2 \pi PV} = {area(\Delta HEV) \over area(circle(EV))}

{OP \over 2 \pi PV} = {area(GHEF) \over area(annulus(FV - EV))}

where *area*(*GHEF*) denotes the area of the trapezoid *GHEF* and *area*(*annulus*(*FV - EV*)) denotes the area of the annulus obtained by revolving the segment *EF* around the axis of rotation. Note that the numerator on the right is a slice of the trunk and the denominator is the corresponding slice of the solid of revolution. Since *OP* and *PV* do not change, it follows by Cavalieri's principle that the ratio of the volume of the trunk to the volume of the solid of revolution is equal to the ratio of *OP* to 2π*PV*.

To be more specific, if *AB* is any planar figure let *rev*(*AB*) denote the volume of the solid of revolution obtained by revolving *AB* around an axis of revolution and *trunk*(*AB*) denote the volume of a trunk of a right cylinder over *AB*. Then

{trunk(AB) \over rev(AB)} = {OP \over 2 \pi PV}

{ trunk(AB) \over rev(AB)} = { height(AB) \over circum(AB)}

This is the sought after ratio between the solid of revolution and a trunk constructed from the same 2-dimensional figure.

This formula also yields a way to describe the ratio between the volumes of two solids of revolution--something quite important for someone brought up to appreciate Euclidean proportion theory. Suppose, for instance, that *AB* and *EF* are two planar figures extended into 3-dimensions to form cylindrical figures which by assumption have *the same height*.

Using the notation from above, the previous result implies that

{trunk(AB) \over rev(AB)} = {height(AB) \over circum(AB)}\qquad \hbox{and}\qquad {trunk(EF) \over rev(EF)} = {height(EF) \over circum(EF)}

{rev(AB) \over rev(EF) } = {trunk(AB) \over trunk(EF)} \cdot {circum(AB) \over circum(EF)}

Andrew Leahy (Knox College), "James Gregory and the Pappus-Guldin Theorem - A Ratio Between the Trunk and the Solid of Revolution," *Convergence* (February 2010), DOI:10.4169/loci003262