NEWTON S PRINCIPIA. 71 



18 a 2 D 2 

 y - ^ s an( ^ making B L = y } L P parallel to 



A M, cuts A P in the point P required. Sir Isaac 

 Newton gives a very elegant solution geometrically by 

 bisecting A S in Gr, and taking the perpendicular G K 

 to the given area as 3 to 4 A S, or to S B, and then 

 describing a circle with the radius R S ; it cuts the para 

 bola in P, the point required.* This solution is infinitely 

 preferable to ours by the hyperbola, except that the 

 demonstration is not so easy, and the algebraical de 

 monstration far from simple. 



It is further to be observed, that the place being given, 

 either of these solutions enables us to find the time. 



3 D 2 



Thus, in the cubic equation, we have only to find ^-. 



It is equal to - - 5 - ; and as D 2 is the given integer, 



or period of e. g. half a year, the body comes to the point 

 P in a time which bears to D 2 the proportion of unity to 



if -f 3 a 2 ?/ 



Sir Isaac Newton proceeds to the solution of the same 

 important problem in the case of the ellipse, which is 

 that of the planetary system, and is termed Kepler s 

 problem from having been proposed by him when he had 

 discovered by observation that the planetary motions were 

 performed in this curve, and that the areas described by 

 the radii were proportional to the times. In the parabola 

 which is quadrable and easily so, the area being two- 

 thirds of the rectangle under the co-ordinates, the solution 

 of this problem is extremely easy. But the ellipse not 



* The most singular relation subsists between the hyperbolas and pa 

 rabolic areas, giving rise to very curious Porisnis connected with Quadra 

 tures. See Phil Trans. 1798, part ii. 



F 4 



