64 NEWTON S PRINCIPIA. 



instances, especially where the curve is given, the for 

 mula -^ will be found most easily dealt with. 



ii. The next branch of the inquiry relates to the de 

 scribing the conic sections severally, where certain points 

 are given through which they are to pass, or certain lines 

 which they are to touch. The subject is handled in two 

 sections, (the fourth and fifth,) the first of which treats the 

 case where one of the foci is given; the second the case 

 where neither focus is given. This whole subject is purely 

 geometrical; and exhibits a fertility of resources in treating 

 these difficult problems, as well as an elegance in the manner 

 of their solution, which has few parallels in the history of 

 ancient or modern geometry. This portion of the Prin- 

 cipia, however, is incapable of abridgment; and there is no 

 advantage whatever in resolving the problems analytically, 

 but rather the contrary; for with the exception of one of 

 the lemmas, in demonstrating which Sir Isaac Newton 

 himself has recourse to algebraical reasoning in order to 

 shorten the proofs, the geometrical process is in almost 

 every instance extremely concise, in all cases much more 

 beautiful, and less encumbered than the algebraical. The 

 superiority of the former to the latter method of in 

 vestigation in such solutions is apparent on trying al 

 gebraically some simple case, as that of describing a circle 

 through three points, or through two points and touching a 

 line given in position ; no little embarrassment results from 

 the number and entanglement of the quantities in the solu 

 tion. Even so great a master of analysis as Sir Isaac Newton, 

 in solving the problem of describing a circle through two 

 points, and touching a given line, could find no better ex- 



- e 2 b V e 2 b 2 + e 2 a 2 - d 2 a 



pression than x = 3 although 



ct 2 a? 



geometrically the construction is easy by drawing a circle 



