NEWTON S PRINCIPIA. 



51 



are given for determining the central force required to 

 make the body move in a given orbit of one of the four 

 conic sections. Secondly, the inquiry becomes material 

 how curves of a given kind, namely, the conic sections, 

 may severally be found by merely ascertaining certain 

 points in them, or certain lines which they touch, because 

 this enables us to ascertain, among other things, the 

 whole of a planet s orbit, from ascertaining certain points 

 by actual observation. This branch of the subject is 

 purely mathematical, consisting of the rules for drawing 

 those curves through given points, or between, or touching 

 given straight lines ; and it is subdivided into two heads 

 according as one or neither focus is given. The third 

 object is to ascertain the motion, place, and times of bodies 

 moving in given trajectories generally ; and, among others, 

 also of bodies descending, or retarded in ascending, by 

 gravity. The fourth branch treats of the converse inquiry 

 into the figures of the trajectories, and the places, times, 



and motion, when the nature of the centripetal force is j &amp;lt; 



known. 



It is thus manifest that the great importance of motion 

 in the Conic Sections made Sir Isaac Newton consider 

 those curves in particular, before discussing the general 

 subject of trajectories. 



i. In exemplifying the use of the formulas we have 



E 2 



