To cohfider more particularly forrie of thefe cafes, 



I. Let a 



~ z 



>2 _rir\ 



vT' 



andthefluentof^;''^^^^="•!^^=''■" = ^"''*°' = '■)"' 



I \ax r'—z'\^= &. -J-aXr'— s» =-|- r'z— f jjJ = (when z=r)$r5. 



Becaufe in this hypothefis a — z or a r = r z' 



therefore a r = circ. arc rad. r and fine 5f, or /, A R = C », or cofine 

 of BR = C «. Confequently the angle B « C is a right angle, 

 and the locus of « or the cufve C n m is a femicircle defcribed on 

 the diam. B C. Thts cafe therfefore furni'fhe's the folution which 

 Viviani himfelf gave of the Florentine problem. 



If on the radius which is the continuation of BC another femi- 

 circle be defcribed and the fame conftrudion made, the furface 

 of the hemifphere infifting perpendicularly between the femi- 

 circle and thp quadrants = 2 !x 2 r^ = the fquare of the diame- 

 ter of the fpherte. 



Alw, if a circle he defcribed on each radius of a diameter of a 

 fphere, and the fphere be perforated by cylinders infifting perpen- 

 dicularly 



