Fig. 1 
C 310 3 
XV. Of the Penetration of a Hemisphere by an indefinite Number 
of equal and similar Cylinders. By Thomas Knight, Esq . 
Communicated by Sir Humphry Davy, LL. D. Sec. R. S. 
Read March 19, 1812. 
he well known theorems of Viviani and Bossut, respect- 
a ing certain portions of the 
surface and solidity of a he- 
misphere, form, together, a 
single case of the following 
problem ; which is one of 
the most remarkable, for 
generality and simplicity of 
result, in the whole compass 
of geometry. 
1 
n\ A 
Kill \ 
1 c 
S\ N 
N. / a/Tt 
/ 
3 
Problem. 
To pierce a hemisphere , perpendicularly on the plane of its base , 
with any number of equal and similar cylinders ; of such a kind , 
that, if we take away from the hemisphere those portions of the 
cylinders that are within it, the remaining part shall admit of an 
exact cubature : and if we take away, from the surface of the 
hemisphere , those portions cut out by the cylinders, the remaining 
shall admit of an exact quadrature. 
Let fig. 1 represent the nearest half of the hemisphere, 
where a is the pole, bdcf a quadrant of the great circle form- 
