NEWTON S PRINCIPIA. 69 



bola through given points, or touching given lines, be 

 side gratifying a curiosity purely geometrical, leads us 

 to calculate within 20&quot; of the truth the place of bodies 

 revolving round the sun in orbits so eccentric that the el 

 lipse which they describe coincides with a parabolic line, 

 instead of being nearly circular like the path of our globe, 

 although our own distance from that luminary is near a 

 hundred millions of miles. 



iii. We are next (to consider the motion of bodies in 

 conic sections which are giveny and &amp;lt; ascending or de 

 scending in straight lines under the influence of gravity ; 

 that is, the velocities and the times of their reaching given 

 points, or their places at given times. This branch of the 

 subject, therefore, divides itself into two parts, the one 

 relating to motion in the conic sections, the other to the 

 motion of bodies ascending or descending under the in 

 fluence of gravitation. The Sixth Section treats of the 

 former, the Seventh of the latter. 



(1.) In order to find the place of a revolving body in its 

 trajectory at any given time, we have to find a point such 

 that the area cut off by the radius vector to that point 

 shall be of a given amount ; for that area is proportional/^^ 

 to the time. Thus, suppose the body moves in a parabola, 

 and that its radius vector completes in any time a certain 

 space, say in half a year moves through a space making 

 an area equal to the square of D ; in order to ascertain 

 its position in any given day of that half year, we have 

 to cut off, by a line drawn from the centre of forces, an 

 area which shall bear to D 2 the same proportion that 

 the given time bears to the half year, say 3 to m 2 , or we 



3 

 have to cut off a section A S P = ^ D 2 , A P being the 



parabola and S the focus. This will be done if A B 



F 3 



