therefore two pendulums APT; Apt be unequally drawn aſide from the perpendicular AR, and let fall together, their accelerations will be always as the arcs to be deſcribed TR, tR. But the parts deſcribed at the beginning of the motion are as the accelerations, that is, as the wholes that are to be deſcribed at the beginning, and therefore the parts which remain to be deſcribed and the ſubſequent accelerations proportional to thoſe parts, are alſo as the wholes, and ſo on. Therefore the accelerations, and conſequently the velocities generated, and the parts deſcribed with thoſe velocities, and the parts to be deſcribed, are always as the wholes; and therefore the parts to be deſcribed preſerving a given ratio to each other will vaniſh together, that is, the two bodies oſcillating will arrive together at the perpendicular AR. And ſince on the other hand the aſcent of the pendulums from the loweſt place R through the ſame cycloidal arcs with a retrograde motion, is retarded in the ſeveral places they paſs through by the ſame forces by which their deſcent was accelerated, 'tis plain that the velocities of their aſcent and deſcent through the ſame arcs are equal, and conſequently performed in equal times; and therefore ſince the two parts of the cycloid RS and RQ lying on either ſide of the perpendicular are ſimilar and equal, the two pendulum will perform as well the wholes as the halves of their oſcillations in the ſame time. Q. E. D.
Cor. 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 ſame body in the higheſt place S or Q, as the arc of the cycloid TK is to the arc SR or QR.
