which an arc equal to the radius ſubtends, as SH (Pl. 14. Fig. 3.) the diſtance of the foci, to AB the diameter of the ellipſis. Secondly, a certain length L, which may be to the radius in the ſame ratio inverſely. And theſe being found, the problem may be ſolved by the following analyſis. By any conſtruction (or even by conjecture) ſuppoſe we know P the place of the body near its true place p. Then letting fall on the axis of the ellipſis the ordinate PR, from the proportion of the diameters of the ellipſis, the ordinate RQ of the circumſcribed circle AQB will be given; which ordinate is the ſine of the angle AOQ ſuppoſing AO to be the radius, and alſo cuts the ellipſis in P. It will be ſufficient if that angle is found by a rude calculus in numbers near the truth. Suppoſe we alſo know the angle proportional to the time, that is, which is to four right angles, as the time in which the body deſcribed the arc Ap, to the time of one revolution in the ellipſis. Let this angle be N, Then take an angle D, which may be to the angle B as the ſine of the angle AOQ to the radius; and an angle E which may be to the angle N - AOQ + D, as the length L to the ſame length L diminiſhed by the co-ſine of the angle AOQ, when that angle is leſs than a right angle, or increaſed thereby when greater. In the next place take an angle F that may be to the angle B, as the ſine of the angle AOQ + E to the radius, and an angle G, that may be to the angle N - AOQ - E + F, as the length L to the ſame length L diminiſhed by the co-ſine of the angle AOQ + E, when that angle is leſs than a right angle, or increaſed thereby when greater. For the third time take an angle H, that may be to
Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/220
