Mifcellanea Curio fa. 1 79 



SM is increafed by the indivifible Particle 

 RS. Next fjppofe the Center c to be tranf- 

 ferr'd to C} by which Motion the Segment 

 RSM being brought into the Pofition QTN, 

 the point Q. will touch the Curve. 'Tis 

 plain that the Triangle RSM is the Mo- 

 mentum or Fluxion of the Segment of the 

 Circle, and that the Trapezium QSMN is 

 the Fluxion of the Curvilineal Space gene- 

 rated in the fame time. And fince SM, 

 RM, QM, are fuppos'd to differ but by a 

 point from one another, let the little Area 

 QSMN be conceiv'd to confift of the three 

 Seders RMS, RMQ., MQN; and fo the 

 little Area RMS to be to the little Area 

 QSMN, as the Angle RSM to the Sum of the 

 three Angles RMS+RMQ_+MQN. But the 

 Angles RMQrl-MQN, are equal to the Angles 

 MCN + MKN, or to the Angle cMC \ be- 

 caufe of the Lines RM, QN, inclin'd to one 

 pother in an Angle equal to MKN, and 

 becaufe of the Angle MQN equal to i MCN 

 (by EncL 3. 20.) confequently the Angle 

 RMS is to the Angles RMS + cMC, that 

 is (by the fame Propofition mention'd) the 

 Arch j RS to the two Arches Cc + i RS, or 

 RS to 2Cct~RS, as the little Area RSM, 

 to the little Area QSMN, or as the Mo- 

 mentum of the Circular Segment QJN, to 

 the Momentum of the Epicycloidal Segment 

 QSYMN generated in the fame time. And 

 fince thefe Momenta are ever in that fame 

 Ratio, where-ever the point Q, be taken, 'tis 

 manifeft that the Areas QJN, QSMYN 

 themfelves, generated from thefe Momenta, 

 have alfo the fame conftant Ratio, viz.. of 

 N 2 the 



