p6 PHILOSOPHICAL TRANSACTIONS. [aNN0 1714. 



omitted the demonstration of this case; since he himself has often proposed 

 theorems, without any where giving their demonstrations; and why may not 

 Newton do the same, when in haste to proceed to other matters? But now in 

 the new edition of the Principia is his demonstration of this very thing, which, 

 though very short, is yet much easier and clearer than that of Bernoulli. 



Lastly, that Bernoulli might show the necessity of his demonstration of the 

 inverse problem in this particular case, he thus adds: it must be considered, 

 says he, that the force which causes a body to move in the logarithmic spiral, 

 must be reciprocally as the cube of the distance from the centre; but it does not 

 hence follow, that such curves must always be described by such forces, since 

 the like forces may also be the cause that the body may move in the hyperbolic 

 spiral. 



Now it is truly surprising how this great man could imagine, that Newton 

 ever drew such a consequence. For, besides the logarithmic spiral, Newton 

 shows how other curves, different and infinite in number, may be formed, all 

 of which may be described by the same centripetal force as the logarithmic spiral, 

 and among these may be reckoned this very hyperbolic spiral, as we shall show 

 hereafter. 



And from hence Newton concludes, that the conic sections only can be 

 described by a centripetal force which is reciprocally proportional to the square 

 of the distance: because the curvature of any orbit is given, by having given 

 the velocity the centripetal force, and the position of the tangent. And that 

 having given the focus, the point of contact, and the position of the tangent, a 

 conic section may always be described, which shall have a given curvature, is 

 what I have shown in the Philos. Trans. Anno 17O8. Therefore by virtue of 

 this/orce the body shall move in this curve and no other: since a body setting 

 out from the same place, in the same direction, with the same velocity, and 

 urged by the same centripetal force, cannot describe diverse courses. 



Dr. Keill then gives another solution of this problem, of the inverse method 

 of centripetal forces, by means of a fiuxionary process; and he also applies it 

 to a particular case, in which the force is reciprocally as the cube of the distance, 

 and at the same time produces a demonstration of cor. 3, prop. 41, of Newton's 

 Principia. After which, he then adds as follows. 



Concerning the areas described by bodies, by means of a centripetal force 

 which is reciprocally as the cubes of the distances, my worthy colleague, the 

 excellent geometrician, professor Halley, observes, that if bodies by this law 

 describe different circles, or different hyperbolic spirals; the areas of the sectors, 

 both in the circles and in the spirals, will always be equal when described in 

 equal times. For the velocities of bodies moving in circles by this law, ought 



