Fig. I. 



[ 516 ] 



Lemma. 



If a fphere be perforated by a right cylinder, the axis of which 

 paffes through the centre of the fphere, the folid content of the 

 remainder of the fphere is to ^ of a cube, the fide of which is 

 the length of the cylinder entirely within the fphere, as the cir- 

 cumference of a circle to its diameter. _ 



Dem. Let AMB m be a fedion of the fphere through the 

 axis of the cylinder Mm. Let G M.H /img be a fedlion of the 

 cylinder, G^ = H/& its length wholly within the fphere, and 

 let A B and D F be drawn parallel to G H, g b which will 

 therefore be diameters of fedions of the fphere perpendicular 

 to M w. 



The area of the fedion through D F : fed of cylind. :: D S' t 

 K S' therefore dividendo 



fed. of remainder : fed. of cylind. :: D S^ — K S* : K S» 

 but fed. of cylind. : great circle : : K S' : A C^ 

 whence 



Sed. 



