Proposition LIII. Problem XXXV.
Granting the quadratures of curvilinear figures, it is required to found the forces with which bodies moving in given curve lines may always perform their oſcillations in equal times.
Let the body T (Pl. 20. Fig. 2.) oſcillate in any curve line STRQ, whoſe axis is AR paſſing through the centre of force C. Draw TX touching that curve in any place of the body T, and in that tangent TX take TY equal to the arc TR. The length of that arc is known from the common methods uſed for the quadratures of figures. From the point Y draw the right line YZ perpendicular to the tangent. Draw CT meeting that perpendicular in Z, and the centripetal force will be proportional to the right line TZ. Q. E. I.
For if the force with which the bod is attracted from T towards C be expreſſed by the right line TZ taken proportional to it, that force will be reſolved into two forces TY, YZ, of which YZ drawing the body in the direction of the length of the thread PT does not at all change its motion; whereas the other force TY directly accelerates or retards its motion in the curve STRQ. Wherefore ſince that force is as the ſpace to be deſcribed TR, the accelerations or retardations of the body in deſcribing two proportional parts (a greater and a
