28 KEPLER'S PROBLEM. 



to a curve of the third order, called, if I rightly remember, 

 by Sir Isaac Newton, in his " Enumeratio Linearum Tertii 

 Ordinis" a parabolism of the hyperbola. Now, although this is 

 extremely simple, in comparison of the complex equation 

 given by the direct method first mentioned, it has manifestly 



/2 

 one impossible case, viz., when $ is equal to ax ^ / -, or 



when the given area is to two-thirds of the square of the 

 diameter of the curve, as m -f- n to m : In this case, no para- 

 bolism of the hyperbola can be drawn, which will intersect 

 the given curve in the point required ; and this is an impos- 

 sibility affecting every possible value of c ; that is, every 

 position of the given point, in this particular magnitude of 

 the given area. But this circumstance makes no difference 

 on the resolution of the problem by the direct method. Thus, 

 when the eccentricity vanishes, or the given point is in the 

 punctum duplex, the solution is derived from a cubic equation 



/2 



equally resolvable when < a A / - as when <b is either < 



V 3 



/2 

 or > 



The method of resolving this interesting problem by loci, 

 is the source of an immense variety of the most curious pro- 

 positions concerning the properties and mutual relations of 

 curve lines ; and, more especially, leads us to the discovery 

 of various porisms, which we otherwise should never have 

 found out. In order to generalize and extend these, it is 

 necessary that, instead of considering merely the case of 

 Kepler's problem, where an area is cut by a straight line, 

 we should consider also the far more difficult problem of 

 cutting the area of one curve by another curve, in a given 

 ratio ; and then the problem may be extended to the section, 

 not of one curvilinear area, but of an infinite number of areas, 

 contained between two given lines, or of the areas of all the 

 curves of a particular kind which can be drawn between 

 those given lines. It is easy to perceive, that the same 



