every direction it ſtrikes the circle, will be as its velocity: and therefore the ſum of the forces, in a given time, will be as that velocity and the number of reflexions conjunctly; that is, (if the ſpecies of the polygon be given) as the length deſcribed in that given time, and increaſed or diminiſhed in the ratio of the ſame length to the radius of the circle; that is, as the ſquare of that length applied to the ratios: and therefore if the polygon, by having its ſides diminiſhed is increſead, coincides with the circle, as the ſquare of the arc deſcribed in a given time applied to the radius. This is the centrifugal force, with which the body impells the circle; and to which the contrary force, wherewith the circle continually repells the body towards the centre, is equal.
Proposition V. Problem I.
There being given in any places, the velocity with which a body deſribes a given figure, by means of forces directed to ſome common centre; to find that centre. Pl. 3. Fig. 1.
Let the three right lines PT, TQV, VR touch the figure deſcribed in as many points P, Q, R, and meet in T and V. On the tangents erects the perpendiculars PA, QB, RC, reciprocally proportional to the velocities of the body in the points P, Q, R, from which the perpendiculars were raiſed, that is, ſo that PA may be to QB as the velocity in Q to the velocity in P, and QB to RC as the velocity in R to the velocity in Q; Thro' the ends A, B, C, of the perpendiculars draw AD, DBE, EC, at right angles,
