1S0 Mifcellanea Curiofa. 



the Velocity of the Circular- Motion RS, to 

 double the Velocity of the Center, adding 

 the Circular Motion, or iCc + RS : As al- 

 fo the Area UBZ to the Area QUBN, and 

 confequently the Semicircle ULB to the Cur- 

 vilineal Space UQYNB. Wherefore the 

 Trofofition is manifeft. 



And there is no other difference in the 

 manner of demonftrating, if the generating 

 Circle moves upon the Concave fide of the 

 Arch, except only that the Angle cMC, in 

 this cafe, is the difference of the Angles 

 MCN, MKN. But if the Bafis were a right 

 Line, then MKN vaniming, and RM, QN, 

 being parallel, the Conftru&ion will be ea- 

 fier. 1 forbear drawing Corollaries from this 

 Propofition, flnce they are obvious. But 

 now in all thefe Curves, the Portions that 

 are Analogous to thofe Portions which Do- 

 dor Wallis has found capable of a perfect 

 Quadrature in the Primary Cycloid, are here 

 alfo equally fquarable j which eafily follows 

 from what has been laid. 



Upon the Center K, thro' the point Q_, 

 draw the Circular Arch Q_Z, and draw ZB 

 cutting off the Segment ZLB == the Segment 

 QTN. Then bifeft the Semicircle UB in L, 

 and thro' the point L and on the Center K, 

 defcribe the Arch PL cutting the Epicycloid 

 in P, the generating Circle in T, and the 

 Chords QN, ZB, in y and X. Let the Arch 

 VZ — a, its Sine = s, the Radius of the 

 generating Circle ±= r, the Radius of the 

 Bafe = R, and the Arch CE or the Motion 

 of the Center = m. It is plain that the 

 Sedor CKE, is to the Space XyNB, as the 



Square 



