
Cavalieris proof of the volume formula for the sphere compares the half sphere
of radius L with the complement of a cone in a cylinder of radius L and height L.
The cross section of each body at height z has area (L^{2}z^{2})^{1/2}.
Since the cylinder has volume pi L^{3} and the cone has a third of this volume,
the hemi sphere has 2/3 of the volume of the cylinder.
 
Please send comments to maths21a.harvard.edu 
Oliver Knill, Maths21a, Multivariable Calculus, Summer 2005, Department of Mathematics, Faculty of Art and Sciences, Harvard University 