ij% Mifcellanea Curio fa. 



Proportion, by the help of which all forts of* 

 Curvilineal Spaces, as well of the Cycloidal 

 as Epicycloidal kind, as well the whole 

 Spaces as the parts, are meafur'd. And fur- 

 ther, not only the Spaces VPL, but alfo in- 

 numerable others, QTP and VQTL, are de- 

 mon fixated to be capable of an exact. Qua- 

 drature *, and this not only in the Primary 

 Epicycloids, but but alfo in the Prolate and 

 contracted ones. 



The Proportion is as follows. 



Proportion. 



The Area of any Cycloid or Epicycloid, 

 either Primary, Prolate, or Contracted, is 

 to the Area of the Generating Circle^ and 

 alfo the Areas of the generated parts in 

 thofe Curves, are to the Areas of the Ana- 

 logous Segments of the Circle ^ as the Sum 

 of twice the Velocity of the Center, and 

 the Velocity of the Circular Motion, to the 

 Velocity of the Circular Motion. 



Demonstration. * 

 (Pig. 32.) 



Let YPQRSUB be any Epicycloid defcrib'd 

 by the Revolution of the Circle ULB, up- 

 on the Circular Bafis YMNB. Let the Cen- 

 ter of the Generating Circle be in c, and 

 drawing cMK, let the Circle ftand upon the 

 Bafis in the point M, and let the describing 

 point be S. Now diftinguiihing the Motions, 

 let the point S grit of all be carried by the 

 Circular Motion into R, fo that the Arch 



SM 



