Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/106

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Proposition II. Theorem II.

Every body, that moves in any curve line deſcribed in a plane, and by a radius, drawn to a point either immoveable, or moving forward with an uniform rectilinear motion, deſcribes about that point areas proportional to the times, is urged by a centripetal force directed to that point.

Case 1. For every body that moves in a curve line, is (by law 1.) turned aſide from its rectilinear courſe by the action of ſome force that impels it. And that force by which the body is turned off from its rectilinear courſe, and is made to deſcribe, in equal times, the equal leaſt triangles SAB, SBG, SCD, &c. about the immovable point S, (by prop. 40. book 1. elem. and law 2.) acts in the place B, according to the direction of a line parallel to cC, that is, in the direction of the line BS; and in the place C, according to the direction of a line parallel to dD, that is, in the direction of the line CS, &c. And therefore acts always in the direction of lines tending to the immovable point S. Q. E. D.

Case 2. And (by cor. 5. of the laws) it is indifferent whether the ſuperficies in which a body deſcribes a curvilinear figure be quieſcent, or moves together with the body, the figure deſcrib'd, and its point S, uniformly forwards in right lines.