Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/221

 the angle B as the ine of the angle AOQ + E + G to the radius; and an angle I to the angle N - AQQ E - G + H, as the length L is to the ame length L diminihed by the co-ine of the angle AQQ + E - G when that angle is les than a right angle, or increaed there by when greater. And o we may proceed in infinitum. Latly, take the angle AOq equal to the angle AOQ + E + G + I + &c. and from its co-ine Or and the ordinate pr, which is to its line ine as the leer axis of the ellipis to the greater, we hall have p the correct place of the body. When the angle N - AOQ + D happens to be negative, the ign + of the angle E mut be every where changed into -, and the ign - into +. And the ame thing is to be undertood of the igns of the angles G and I, when the angles N - AOQ - E + F, and N - AOQ - E - G + H come out negative. But the infinite eries AOQ + B + G + I &c. converges o very fat, that it will be carcely ever needful to proceed beyond the econd term E. And the calculus is founded upon this theorem, that the area APS is as the difference between the arc AQ and the right line let fall from the focus S perpendicularly upon the radius OQ

And by a calculus not unlike, the problem is olved in the hyperbola. Let its centre be O, (Pl. 14. Fig. 4.) its vertex A, its focus S, and aymptote OK. And uppoe the quantity of the area to be cut off is known, as being proportional to the time. Let that be A, and by conjecture uppoe we know the poition of a right line SP, that cuts off an area APS near the truth. Join OP, and from A and P to the aymptote draw AI, PK parallel to the other aymptote