Review of the Principia of Newton. 313 



cal solution into one which was algebraical ; his formula for 

 the increment, or differential of the arc Z, the measure of 



a 2 c 2 x' 

 the angular motion is Z= ,^=— ■- — .which 



& Vabx i ~x i \/^x — a 2 c 2 x 2 



is precisely the same as that of Newton, if a 6 — \/ t x be 

 substituted for the area ABGE, a c for Q,, and a for the radius 

 of the circle measuring the angular motion. But this mathe- 

 matician affects to consider Newton's solution as incomplete, 

 because he had not applied it to the most important cases of 

 a body acted on by a force varying in the inverse duplicate 

 ratio of the distance. In this, he appears not to have recol- 

 lected, or regarded the copious investigations of the direct 

 problem of centripetal forces in the 2d and 3d sections of 

 the Principia, and of the converse of this particular case in 

 corol. 1, prop. 13, and the more general solutions in proposi- 

 tions 16 and 17. The same author, for the purpose of show- 

 ing the necessity of his own solution, says, that many other 

 curves besides the equi-angular spiral may be described by a 

 force varying in the inverse triplicate ratio of the distance. 

 This subject has also been fully developed, in a general man- 

 ner, in the 3d corollary of the proposition now under conside- 

 ration, which, on account of its profundity, and most curious 

 results, we would gladly exhibit ; but of this, a review would 

 hardly be possible, on account of the mutiplicity of diagrams 

 and symbols, necessary for its illustration. Though it be simply 

 a corollary, it is susceptible of expansion to volumes, and 

 comprehends an immense portion of the theory of motion : 

 we can do little more than state its substance, and some of its 

 results. 



If a body be projected perpendicularly to the radius vector, 

 and be acted on by a force varying in the inverse triplicate 

 ratio of its distance from the center of force, the curve in 

 which the body will move, will be determined by taking the 

 angular motion as the sector of a conic section whose center 

 is the center of force, and the distance equal to the distance 

 from that center to the point of intersection of the tan- 

 gent with the axis of the figure. The particular section as- 

 sumed for the construction of the trajectory, will depend on 

 the initial velocity. If this be such as would be acquired by 

 a body falling from an infinite height, and the projection be 

 perpendicular to the radius vector, the differential or fluxion 

 of the trajectory becomes equal to the element of the tan- 



Vol. XIIL— No. 2. 15 



