Archimedes' Method for Computing Areas and Volumes - Proposition 2 of The Method

The applet below illustrates the balancing relationship in Proposition 2 of Archimedes' The Method.

Since this balance holds for every cross section, it follows that if the entire sphere and cone are moved so that their centers of gravity are at H, they will exactly balance the cylinder, left where it is. By symmetry, the center of gravity of the cylinder is at K, so by the law of the lever,

\[\left(\mbox{Volume of Sphere}+\mbox{Volume of Cone}\right)\times|AH|=(\mbox{Volume of Cylinder})\times|AK|.\]

Since \(|AH| = 2|AK|,\) it follows that

\[2\left(\mbox{Volume of Sphere}+\mbox{Volume of Cone}\right)=\mbox{Volume of Cylinder}\] or \[\mbox{Volume of Sphere}=\frac12\left(\mbox{Volume of Cylinder}\right)-\mbox{Volume of Cone}.\] Finally, using the fact that the volume of the cone is 1/3 the volume of the cylinder, we find that the volume of the sphere is 1/6 the volume of the cylinder. Now since the cylinder's height and radius are both \(2r\) (where \(r\) is the radius of the sphere), the volume of the cylinder is \[\pi(\mbox{radius})^2\cdot (\mbox{height})=\pi(2r)^2\cdot(2r)=8\pi r^3\] so the volume of the sphere is \[V_{\mbox{sphere}}=\frac16\cdot 8\pi r^3=\frac43\pi r^3.\]