C J 53 1 
VII. Demonstration of a Theorem , by which such Portions of 
the Solidity of a Sphere are assigned as admit an algebraic 
Expression. By Robert Woodhouse, A. M. Fellow of Cains 
College , Cambridge. Communicated by Joseph Planta, Esq . 
Sec. R. S. 
Read February 12, 1801. 
In 1692, a problem was proposed by Viviani, (Act. Erud. 
Lips.) to the geometricians of his time, in which it was required 
to separate from the surface of a sphere, such portions, that what 
remained should be quadrable. 
In the second volume of the Memoirs of the National Insti- 
tute, M. Bossut announces a theorem relative to the solidity of 
a sphere, very simple, he says, and as remarkable as Viviani’s, 
but depending on an integration much more complicated : the 
theorem is this. 
“ If a sphere be pierced perpendicularly to the plane of one 
of its great circles, by two cylinders of which the axes pass 
through the middle points of two radii that compose a diameter 
of this great circle, the two portions, thus taken away from the 
whole solidity of the sphere, leave a remainder equal to two- 
ninths of the cube of the sphere’s diameter. 1 ’ 
M. Bossut withholds the analysis that led to this result. I 
have obtained it in the subjoined process, in which the integra- 
tion is not at all more complicated than what is used in the 
solution of the Florentine problem. 
MDeccr. X 
