# Note on the Leibniz Controversy

If several bodies revolve about a common center & the vis centripeta be reciprocally as the square of the distance \of the body/ from y^{e} center: the orbs wil described by the bodies will be all of one & the same kind but may differ from one another in species; & such are the Conic Sections. And if the f vis centri{illeg}|p|eta be reciprocally as the cube of the distance the Orbs described will be all of another kind but differ{illeg} from one another in species, such as are the Spiralis Logarithmica, the Spiralis Hyperbolica, & three other species of Curves. And if in either case a man should shew how to describe the Curve which a body shall describe when projected from any given place with any given velocity in any given determination of motion: he would shew how to at the same time shew how to describe all the curves in w^{ch} a body can move in that case. M^{r} Newton in the XVII^{th} Proposition of his first Book of Principles has done this in the first case, that is, when y^{e} force is reciprocally as the square of the distance, & has found that the Curve is always one of the three Conic Sections. If M^{r} B{illeg}ernulli {sic} had done this in the second case & found that the Curve described will be always a Logarithmic Spiral & will not comprehend all the Curves which can be described by such a force