Mifcellanea Curio fa, a j j 



COROL IV, 



When the point F is in A, or when the 

 Vertex is defcrib'd by the Evolution, that is, 

 when x == 0, then the Value of the Evoluent 

 Line (or the Radius of the Curvature) UF 

 (which in this Cafe coincides with KA) viz* 



— z 



fH~*. , becomes only a. - That is, the point 

 , a ' '.i : ; - ' ' v ttjMMiX' 

 K where the Curve UK meets the Axis, is 

 as much above the Vertex of the Catena, A, 

 as C is below it. Whence the Diameter of 

 a Circle that has the fame Curvature with 

 the Catena at the Vertex, is equal to the 

 Axis of the Conterminal Hyperbola AH. 

 And confequently the Catena AD and the 

 Hyperbola AH, have the fame Curvature in 

 the Vertex A. For it is known that the 

 foremention'd Circle has the fame Curva- 

 ture with the Equilateral Hyperbola AH, in 

 the Vertex A. But this follows alfo from 

 the Property of the Catenaria^ demonftrated 

 at Prof option II. For the Nafcent FH or (AP 



r= the Nafcent BP =) Vw, is double the 



Nafcent BH or (V \ax\xx, that is, xx va«* 

 nifhing, when x is very fmall) *liax. And 

 therefore the fame point is as well in the 

 JSJafcent Hyperbola^ as in the Nafcent Cate- 

 narian that is, the one is coincident with the 

 other at their firft arifmg, and confequently 

 thefe Curves have the fame Curvature at the 

 Veitex A. 



CORO t. 



