NEWTON S PEINCIPIA. 65 



on one segment of the line joining the given points, and 

 another on the given line.* These are comparatively 

 simple problems ; in the more difficult cases of the conic 

 sections this embarrassment is often inextricable, f 



To illustrate the application of these important pro 

 blems, let us suppose that by observation we obtain three 

 points in the orbit of any planet, and would ascertain from 

 those points the position of the greater axis, and the focus 

 in which the sun is placed, the eccentricity of the orbit or 

 distance of the focus from the centre of the ellipse, and the 

 aphelion, or greatest distance to which in its course the 

 planet ever is removed from the sun ; this is easily done by 

 means of Prop. XVIII. (Book I.), for that enables us to 

 find the elliptical and hyperbolical trajectories, which pass 

 through given points, when one focus and the transverse 

 axis are given ; and thus to find the other focus, and the 

 centre of the curve, and the distance from the given focus 

 to the further extremity of the axis, which is the aphelion. 



In like manner the problem which Sir Isaac Newton 

 calls by far the most difficult of any, and says that he 

 had tried to solve in various waysj, that of finding the tra- 

 jectory_of a comet from three observations, supposing it 

 to move in a parabolic orbit, is reduced by an elaborate 

 an3^3irBcult process of reasoning to describing a parabola 

 through two given points, which are found in its own orbit 

 from the observations. /Now Prop. XIX. of Book I. 

 gives an easy solution of this problem. It is only to 



* The above algebraical solution is that of Prop. 43. of the Arith. Univ., 

 where Props, 59, 60, and 61. are also solutions of the three first problems 

 of Sect. V. of the Principia, B. I. 



f Maria Agnesi s Instituzioni Analitiche abounds in elegant alge 

 braical investigations of geometrical problems, but affords no grounds 

 for modifying the above remark. 



J Problema hocce longe difficillimum multimode aggressus (Lib. III. 

 Prop. 41.). 



Several other propositions are given in the first book for the purpose 



