Mifcellanea Curiofa. 175 



The Quadrature of a "Portion of 

 the Epicycloid. 



By Mr. CafwelL 



(Eg. si.) 



SUppofe D P V to be half of an exterior 

 Epicycloid, V B its Axis, Kthe Vertex, 

 VLB half of the Generant Circle, E its 

 Center , D B the Bafe, C its Center : Bi- 

 fe& the Arc of the Semicircle V B in L 1 and 

 on the Center C through L draw a Circle 

 cutting the Epicycloid in P : Then I fay the 

 Curvilinear Triangle VLP will hz — BEq 

 C E 



in ; that is, the Square of the Semidia- 



C B 



meter of the Generant Circle will be to the 

 Curvilinear Triangle VLT y as CB the Se- 

 midiameter of the Bafe, to CE - 7 which CE 

 in the exterior Epicycloid is the Sum of the 

 Semidiameters of the Bafe and Generant, 

 but in the interior Epicycloid Dpu, 'tis the 

 difference of the faid Semidiameters. 



COROLLARY. 



In the interior Epicycloid, if CE is i 

 the Epicycloid then degenerating into a right 

 Line, the Quadrature of the Triangle / u p 

 will be in efFeft the fame with the Quadrature 

 of Hippocrates Chita, 



COROL 



