Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/286

 therefore two pendulums APT; Apt be unequally drawn aide from the perpendicular AR, and let fall together, their accelerations will be always as the arcs to be decribed TR, tR. But the parts decribed at the beginning of the motion are as the accelerations, that is, as the wholes that are to be decribed at the beginning, and therefore the parts which remain to be decribed and the ubequent accelerations proportional to thoe parts, are alo as the wholes, and o on. Therefore the accelerations, and conequently the velocities generated, and the parts decribed with thoe velocities, and the parts to be decribed, are always as the wholes; and therefore the parts to be decribed preerving a given ratio to each other will vanih together, that is, the two bodies ocillating will arrive together at the perpendicular AR. And ince on the other hand the acent of the pendulums from the lowet place R through the ame cycloidal arcs with a retrograde motion, is retarded in the everal places they pas through by the ame forces by which their decent was accelerated, 'tis plain that the velocities of their acent and decent through the ame arcs are equal, and conequently performed in equal times; and therefore ince the two parts of the cycloid RS and RQ lying on either ide of the perpendicular are imilar and equal, the two pendulum will perform as well the wholes as the halves of their ocillations in the ame time. Q. E. D.

The force with which the body T is accelerated or retarded in any place T of thr cycloid, is to the whole weight of the ame body in the highet place S or Q, as the arc of the cycloid TK is to the arc SR or QR.