why may we not be allowed in the following diſcourſe, to uſe the equable deſcription of areas as an indication of a centre, about which all circular motion is performed in free ſpaces?
Proposition IV. Theorem IV.
The centripetal forces of bodies, which by equoble motions deſcribe different circles, tend to the centres of the ſame circles; and are one to the other, as the ſquares of the arcs deſcribed in equal times applied to the radii applied the circles.
Theſe forces tend to the centres of the circles (by prop. 2. and cor. 2. prop. 1) and are one to another as the verſed ſines of the leaſt arcs deſcribed in equal times (by cor. 4. prop. 1.) that is, as the ſquares of the ſame arcs applied to the diameters of the circles, (by lem. 7.) and therefore ſince thoſe arcs are as arcs deſcribed in any equal times, and the diameters are as the radii; the forces will be as the ſquares of an arcs deſcribed in the ſame time applied to the radii of the circles. Q. E. D.
Cor. 1. Therefore, ſince thoſe 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 ſimple ratio of the radii inverſely.
