Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/345

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Cor. 2. By a like reaſoning if each of the bodies of the ſyſtem A, B, C, D, &c. do ſingly attract all the reſt with accelerative forces, which are either reciprocally or directly in the ratio of any power whatever of the diſtances from the attracting body; or which are defined by the diſŧances from each of the attracting bodies according to any common law; it is plain that the abſolute forces of thoſe bodies are as the bodies themſelves.

Cor. 3. In a ſyſtem of bodies whoſe forces decreaſe in the duplicate ratio of the diſtances, if the leſſer revolve about one very great one in ellipſes, having their common focus in the centre of that great body, and of a figure exceeding accurate; and moreover by radiu drawn to that great body deſcribe area's proportional to the times exactly; the abſolute forces of thoſe bodies to each other will be either accurately or very nearly in the ratio of the bodies. And ſo on the contrary. This appears from cor. of prop. 68. compared with the firſt corollary of this prop.

Scholium.

Theſe propoſitions naturally lead us to the analogy there is between centripetal forces, and the central bodies to which thoſe forces uſe to be directed. For it is reaſonable to ſuppoſe that forces which are directed to bodies ſhould depend upon the nature and quantity of thoſe bodies, as we ſee they do in magnetical experiments. And when ſuch caſes occur, we are to compute the attractions of the bodies by aligning to each of their particles its proper force, and then collecting the ſum of