THE 



PHILOSOPHICAL TRANSACTIONS 



OF THE 



ROYAL SOCIETY OF LONDON; 



ABRIDGED. 



The Newtonian Solution of Kepler's Problem, of finding the True Motion of the 

 Planets, describing Areas proportional to the Times, in Elliptic Orbits, about 

 one of the Foci ; Demonstrated and Illustrated with Examples. By Mr. John 

 Keill, SaviU Profes. of Astr. Oxford, and F.R.S. N° 337, p. 1, Art. \, 

 Vol. XXFIII. Translated from the Latin. 



JA^EPLER was the first who demonstrated that the planets do not revolve in 

 circular orbits, but in elliptical ones ; and that they go round the sun placed in 

 one of the foci of the ellipsis, in such a manner, that a radius extended from 

 the planet to the sun's centre, always describes elliptical areas, which are pro- 

 tional to the times of description. This divine discovery of the sagacious 

 Kepler, was owing to the accurate observations of Tycho Brahe; and is so 

 much the more to be esteemed, that by help of it Newton has perfectly ex- 

 plained the laws of motion, and the philosophy of the system of the universe. 

 Since therefore the planets revolve about the sun by such a law, that their 

 places in their own orbits may be determined to any given time, it is necessary 

 that the following problem should be solved, viz. 



To find the Position of a Right Line, which passing through either focus of an 

 ellipse, may cut off an area described by its Motion, whichmay be to the whole area 

 of the ellipse, in a. given ratio. — Let the ellipse be apb, fig. 1, pi. 1, a focus of 

 of which is s: there is to be found the position of the right line sp, which may 

 cut off the trilinear area asp, to which the area of the whole ellipse has the same 

 ratio, as the periodic time of the planet describing the ellipse, has to any other 

 given time: which being found, the point p will be given, where the planet will 

 be found at that given time. Or, let agib be a semicircle described on the greater 

 axis of the ellipse; a line sq is to be drawn through s, cutting off the area asq, 



VOL. VI. B 



