58 NEWTON S PRINCIPIA. 



tions answer this condition. Therefore no other curves 

 can be the trajectories of bodies moving by a centripetal 

 force inversely as the square of the distance.* 



It may be remarked that J. Bernouilli objects (Mem. 

 Acad. des Sciences, 1710) to Sir Isaac Newton that he 

 had assumed the truth of this important proposition 

 without any demonstration. But this is not correct. He 

 certainly gives a very concise and compendious one ; but 

 he states distinctly that the focus and point of contact 

 being given, and the tangent given in position, a conic 

 section may be described which shall at that point of 

 contact have a given curvature; that the curvature is 

 given from the velocity and central force being given; 

 and that two orbits touching each other with the same 

 centripetal force and velocity cannot be described. This 

 is in substance what we have expounded in the above 

 demonstration. But it must also be observed, as Laplace 

 has remarked, that Newton has in a subsequent problem 

 shown how to find the curve in which a body must move 

 with a given velocity, initial direction, and position ; and 

 since, when the centripetal force is inversely as the square 

 of the distance, the curve is shown to be one or other of 

 the conic sections, he has thus demonstrated the proposition 

 in question ; so that if he had not done so in the corollary 

 to one problem, he has in the solution of another, f 



J. Bernouilli objects also to a very concise and elegant 



* The equation may be resolved and integrated ; there results, in the first 

 instance, the equation d x = ^-=y and therefore the integral is this 



quadratic, c 3 a- 2 2 cy 1 2 e C .r + C 2 + D 2 =0. Another demonstration is 

 given in the Appendix, No. 2. 



f Systeme dn Monde, liv,, v. chap. 5. It is to be observed, that the 

 Seventeenth Prop. Book I., is exactly the same in the first as in the subse 

 quent editions, except the immaterial addition of a few lines to the demon 

 stration. Consequently, Bernouilli must have been aware of it when he 

 wrote in 1710. 



