Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/220

 which an arc equal to the radius ubtends, as SH (Pl. 14. Fig. 3.) the ditance of the foci, to AB the diameter of the ellipis. Secondly, a certain length L, which may be to the radius in the ame ratio inverely. And thee being found, the problem may be olved by the following analyis. By any contruction (or even by conjecture) uppoe we know P the place of the body near its true place p. Then letting fall on the axis of the ellipis the ordinate PR, from the proportion of the diameters of the ellipis, the ordinate RQ of the circumcribed circle AQB will be given; which ordinate is the ine of the angle AOQ uppoing AO to be the radius, and alo cuts the ellipis in P. It will be ufficient if that angle is found by a rude calculus in numbers near the truth. Suppoe we alo know the angle proportional to the time, that is, which is to four right angles, as the time in which the body decribed the arc Ap, to the time of one revolution in the ellipis. Let this angle be N, Then take an angle D, which may be to the angle B as the ine 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 ame length L diminihed by the co-ine of the angle AOQ, when that angle is les than a right angle, or increaed thereby when greater. In the next place take an angle F that may be to the angle B, as the ine 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 ame length L diminihed by the co-ine of the angle AOQ + E, when that angle is les than a right angle, or increaed thereby when greater. For the third time take an angle H, that may be to