Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/293



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 ocillations in equal times.

Let the body T (Pl. 20. Fig. 2.) ocillate in any curve line STRQ, whoe axis is AR paing 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 ued 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 expreed by the right line TZ taken proportional to it, that force will be reolved 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 ince that force is as the pace to be decribed TR, the accelerations or retardations of the body in decribing two proportional parts (a greater and a