4 PHILOSOPHICAL TRANSACTIONS. [aNNO 17 13. 



the former be carried on, there will be found another Aq ; which in like manner, 

 by repeating the same process, will give another a^; and thus we may approach 

 as near as we please to the truth. 



The angle ac^ being found, we shall easily have the angle as^, since in the 

 triangle ^cs are given the sides cq and cs, and the angle ^cs. Thence will be 

 given the angle esq, whose tangent is to be lessened in the ratio of the less 

 axis of the ellipse to the greater, that at length may be had the tangent of the 

 angle asp. Or perhaps the angle asp may be found more easily thus : let p be 

 the number expressing the length cs in such parts as ca is 100000 : from the 

 point q draw qr perpendicular to the axis, which will be the sine of the arc ao, 

 and cr will be the cosine of the same, and sr will be equal to the sum or differ- 

 ence of the right lines cr, cs, that is, sr = f + cos. Acq : therefore in the 

 right angled triangle rsq, sr and rq being given, there may be found the angle 

 rsq. Hence, if there be added into one sum the log. sine of the angle ao^, 

 and the arith. complement of the log. of sr, and the log. of the ratio of the 

 less axis of the ellipse to the greater, there will be obtained the tangent of the 

 angle asp. 



But the facility of this method is such, that it requires rather to be illustrated 

 by examples, than any further explained. Therefore we may try it in the mo- 

 tion of the planet Mars, in whose orbit, according to the Caroline tables, the 

 excentricity is to the mean distance, as 14100 to 1 52369, and therefore the log. 

 of the arc b, which is equal to the right line sc, will be 0.7244451. Also in 

 this example l will be 1080631 of such parts as the radius is 100000 : find the 

 angle acq, where the mean motion, or the arc proportional to the time com- 

 puted from the aphelion, is 1 degree. Because cs is here nearly one tenth part 

 of CA, I suppose the arc aq to be 0.9 degrees, that is, one tenth part less than 

 the mean motion. Let there be added the log. sine of the arc Aa to the log. of 

 B, and the sum 8.9205471 is equal to the log. of the number 0.083281, which 

 number expresses an arc equal to the right line sp = np. And if the arc Aa 

 had been rightly assumed, it would be an — np = aq, and qp = O. But 

 here it is qp = 0.01 67 1 9, from whence if we take away its 1 ith part, since as 

 exceeds ac by about the 11th part of itself, there will remain q^ = 0.0152; 

 which being added to aq, gives aq = O.9152, which does not differ from the 

 true Aq by a thousandth part of a degree. Secondly, let the arc an or the 

 mean motion be 2 degrees. I make aq = 1.83, almost double the former a^, 

 and to its log. sine let be added the log. of b. The sum will be 9.2286997, 

 which is equal to the log. of the number 0.1 693 1. Whence it will be 

 QP = 0.00063, and Aq = 1 .83063, which does not differ from the true Aq by 

 the ten thousandth part of a degree. After the same manner let the motion. 



