94 PHILOSOPHICAL TRANSACTIONS. [aNNO 1714. 



are always found equal at equal distances. This is the sum of Newton's demon- 

 stration, which he explains so clearly, that few easier can be found, even among 

 elementary propositions. But Bernoulli does not proceed thus. He is satisfied 

 with saying, that mechanics show that the force in de is to the force in ik, as 

 IK to DE ; and that mechanics show the increments of the velocities to be in tiie 

 ratio of the forces and times conjointly; also that at the beginning of the mo- 

 tion supposing the velocities to be equal, the times are as the spaces described 

 DE, ik; and hence, by a reasoning exactly like Newton's, he concludes that 

 the increment of velocity, acquired by the body in describing ik, is to the 

 increment of velocity in describing de, as de X ik to ik X de; and therefore 

 that the increments of the velocities will always be equal at equal distances. 



But if he wished to give an easy demonstration for the sake of novices, he 

 ought to have cited the mechanical proposition, and have accommodated it to 

 the present case. And indeed there was occasion for amplifying, that this might 

 be done by the theorem which he seems to intend, in which is treated the de- 

 scent of bodies on inclined planes: for here no plane is given which may impede 

 the direct descent of bodies: nay, so far is the body from being impeded by a 

 plane, that on the contrary it is continually attracted by a certain force from the 

 plane or tangent. Doubtless therefore the force of his reasoning would have 

 been more plain, if, omitting his mechanical propositions, he had demonstrated 

 the whole matter from its own first principles, as Newton has done. For by 

 resolving the right angled triangle kni into two equiangular triangles, it is ki 

 to IN as in to it, and therefore, instead of the ratio in to it, he might have 

 put the ratio of ki to in or to de. 



If the body fall from any place a in the right line ac ; and from its place e a 



perpendicular eg be always raised, which may be proportional to the centripetal 



force; and if bfg be the curve line which the point g always touches; Newton 



demonstrates (prop. 39 and 40 of the Principia) that the velocity of the body 



at any place e, is as the square root of the curvilineal area abge. Therefore if 



the velocity be called v, then v^ will be as the area abge. And if p denote the 



greatest altitude to which the body revolving in the trajectory can ascend, when 



projected upwards from any point of it, with the velocity which it has there; 



and if a denote the distance of the body from the centre, in any other point of 



its orbit; and if the centripetal force be always as any power of a, suppose as 



A"-' ; then the velocity of the body, at every altitude a, will be as y'np"— wa". 



In like manner M. Bernoulli shows, that if the distance from the centre be 



called Xj the velocity r, and the centripetal force <p, then will v = v^a^— flu.fi-; 



where it is plain from quadratures, that the area abge =z ab — flu. ^x. It is 



therefore all the same, whether the square of the velocity be expressed by the 



