2 ■*' PHILOSOPHICAL TRANSACTIONS. [aNNOIJIS. 



to which the area of the whole circle is in the same ratio: for if from o. the per- 

 pendicular QH be drawn, meeting the ellipse in p, drawing sp, it will give the 

 elliptic area required, and the point p will be the place of the planet at the given 

 time. For the elliptic semisegment aph is to the circular semisegment auh, as 

 HP to HQ, that is, as the area of the whole ellipse is to the area of the whole 

 circle : but the triangle sph is to the triangle sqh also in the same ratio of ph 

 to QH : therefore the area asp is to the area of the whole ellipse, as the area Asa 

 is to the area of the whole circle. So that if we had a method of cutting the 

 area of the circle in a given ratio, by a line drawn through the given point s, 

 it would be easy to cut the elliptic area in the same ratio. 



Kepler himself, who first proposed the problem, had no direct method of 

 computing the planets places, from the time being given : but he was obliged 

 to proceed through the several degrees of the semicircle aqb, from the given 

 arc AG, called the excentric anomaly, and both to calculate the time by the 

 area Asa, which is proportional to the mean anomaly, and the angle asp, that 

 is the planet's place, or the coequate anomaly corresponding to this time. 



Since then the solution of this problem was difficult, astronomers had recourse 

 to other hypotheses, assuming some point for that about which the motion is 

 equable, or proportional to the time, and thence the mean anomaly being given, 

 they determined the coequate anomaly. But computations founded on these hy- 

 potheses were found not to agree with the observations. Therefore geometers 

 had recourse to various approximations, by which, from the given area Asa, 

 which is analogous to the time, the angle asp, or the place of the planet, may 

 be had very nearly. Now the easiest of all these, and most ready for practice, 

 seems to be that method which is taught by Mr. Newton in his Principia, 

 p. 1 1 1 and 112, of the first edition, which is very much like that method, by 

 which analysts extract the roots of affected equations ; and indeed is so much 

 the more to be esteemed, as that it not only exhibits the places of the planets, 

 whose orbits approach very nearly to the form of circles, but almost with the 

 same facility may be applied to comets, which move in orbits that are very 

 excentric. Therefore I thought it not amiss to explain that method here, for 

 the sake of such artists as are desirous of constructing astronomical tables, ac- 

 cording to the true laws of motion, and not by any fictitious hypotheses. 



Therefore let aqb, fig. 2, be a semicircle described on the greater axis of an 

 ellipsis, whose centre is c, and s the focus in which the sun is placed. Let ca 

 be drawn, on which, produced if necessary, let fall the perpendicular sp. The 

 area Asa is equal to the sector acq, added to the triangle csq = -^cq X aq -f" 

 ^CQ X SF ; and therefore, because of -J^cq being given, the area asq will 

 always be proportional to the arch aq added to the right line sf, when the mo- 



