The GEORGIUM SID US. 317 



centre ; and let A, B, C, D, E, be the places of the Planet in its 

 fucceffive oppofitions to the Sun ; draw the chords AB, BC, 



CD, DE, AC, CE, and the radii veclores AS, BS, CS, DS, ES. 

 We may fuppofe that the points x and y, where the chords AC, 



CE, are interfered by the radii veclores BS, DS, are in the 

 middle of thofe chords. For, let us fuppofe that thofe chords 

 are bifecled in x and y by radii SB and SD, the rectilineal tri- 

 angles ABS, BCS are equal, and the fegments cut off by the 

 chords AB, BC are very nearly equal ; thefe fegments are very 

 fmall in comparifon with the triangles ABjc, Bj^C, and thefe 

 triangles are very fmall in comparifon with the triangles AxS> 

 X CS. Therefore, the elliptical feclors ABS, BCS, are very near- 

 ly equal, and B is very nearly the place of the Planet at the fe- 

 cond oppofition. 



LET the angles ASB be = u, BSC = i>, CSD = *, DSE =jr 

 ASC = w, CSE = as, AxS = j, and CyS = y. 



Then, AS : A^ Jin. % :Jin. u, 



and Cj, or Aj : CS -=.Jin. -v :Jin. %. 

 therefore, AS : CS Jin. v '.Jin. u t 



alfo, ES : CS fin. x :fin.y. 



THUS, we have obtained the ratio of the three diftances AS, CS, 

 ES, and we have the angles ASC, CSE, given by obfervation. 

 This is all that is necefiary for conftrucling the ellipfe, by 

 means of the 2ift prop, of NEWTON'S Principia, B. I. or of a 

 theorem to be delivered afterwards. 



THIS ellipfe will be found to have its femitranfverfe axis 

 about nineteen times the earth's diftance from the Sun, and its 



excentricity about of its femitranfverfe axis, and the angle 



PSC about 73. As it approaches very near to the form of the 

 ellipfe really defcribed by the Planet, we may difcover, by its 

 means, the errors which have arifen from the fuppofition that 

 the fectors ASB, BSC, are equal, when AJC is equal to xB. 



FOR 



