Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/112

 why may we not be allowed in the following dicoure, to ue the equable decription of areas as an indication of a centre, about which all circular motion is performed in free paces?

The centripetal forces of bodies, which by equoble motions decribe different circles, tend to the centres of the ame circles; and are one to the other, as the quares of the arcs decribed in equal times applied to the radii applied the circles.

Thee forces tend to the centres of the circles (by prop. 2. and cor. 2. prop. 1) and are one to another as the vered ines of the leat arcs decribed in equal times (by cor. 4. prop. 1.) that is, as the quares of the ame arcs applied to the diameters of the circles, (by lem. 7.) and therefore ince thoe arcs are as arcs decribed in any equal times, and the diameters are as the radii; the forces will be as the quares of an arcs decribed in the ame time applied to the radii of the circles. Q. E. D.

Therefore, ince thoe arcs are as the velocities of the bodies, the centripetal forces are in a ratio compounded of the duplicate ratio of the velocities directly, and of the imple ratio of the radii inverely.