Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/282

 of the curve AP, the right line CP, the circular arc BP, and the right line VP, will be the ame as of the lines PV, PF, PG, PL, repectively. But ince VF is perpendicular to CF, and VH to CV, and therefore the angles HVG, VCF equal; and the angle VHG (becaue the angles of the quadrilateral figure HVEP are right in V and P) is equal to the angle CEP, the triangles VHG, CEP will be imilar; and thence it will come to pas that as EP is to CE o is HG to HV or HP, and o KI to KP, and by compoition or diviion as CB to CE o is PI to PK, and doubling the conequents as CB to CE o is PI to PV and o is Pq to Pm. Therefore the decrement of the line VP, that is the increment of the line BV - VP to the increment of the curve line AP is in a given ratio of CB to 2 CE, and therefore (by cor. lem. 4.) the lengths BV - VP and AP generated by thoe increments, are in the ame ratio. But if BV be radius, VP is the coine of the angle BVP or $$\scriptstyle \frac 12$$ BEP, and therefore BV - VP is the vered ine of the ame angle; and therefore in this wheel whoe radius is $$\scriptstyle \frac 12$$ BV, BV - VP will be double the vered fine of the arc $$\scriptstyle \frac 12$$ BP. Therefore AP is to double the vered fine of the arc $$\scriptstyle \frac 12$$ BP as 2 CE to CB. Q. E. D.

The line AP in the former of thee propoitions we hall name the cycloid without the globe, the other in the latter propoition the cycloid within the globe, for ditinction ake.

Hence if there be decribed the entire cycloid ASL and the ame be biected in S, the length of the part PS will be to the length PVT (which is the double of the line of the angle