of the curve AP, the right line CP, the circular arc BP, and the right line VP, will be the ſame as of the lines PV, PF, PG, PL, reſpectively. But ſince VF is perpendicular to CF, and VH to CV, and therefore the angles HVG, VCF equal; and the angle VHG (becauſe 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 ſimilar; and thence it will come to paſs that as EP is to CE ſo is HG to HV or HP, and ſo KI to KP, and by compoſition or diviſion as CB to CE ſo is PI to PK, and doubling the conſequents as CB to CE ſo is PI to PV and ſo 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 thoſe increments, are in the ſame ratio. But if BV be radius, VP is the coſine of the angle BVP or BEP, and therefore BV - VP is the verſed ſine of the ſame angle; and therefore in this wheel whoſe radius is BV, BV - VP will be double the verſed fine of the arc BP. Therefore AP is to double the verſed fine of the arc BP as 2 CE to CB. Q. E. D.
The line AP in the former of theſe propoſitions we ſhall name the cycloid without the globe, the other in the latter propoſition the cycloid within the globe, for diſtinction ſake.
Cor. 1. Hence if there be deſcribed the entire cycloid ASL and the ſame be biſected in S, the length of the part PS will be to the length PVT (which is the double of the line of the angle
