VOL. XXIX.] rHILOSOPHICAL TRANSACTIONS. Q5 



area abge, or by the quantity ab — flu. (p.i, which is equal to it. And if the 

 centripetal force <p be as wa"-' or nx"-^ , it will be ab = v , and flu. (p.v = a ; 

 so that ab — flu. <pi is as the quantity p „ — a". 



Let the body describe the curve vk by a centripetal force tending to c, and 

 let there be given the circle vxy, described with the centre c and any radius cv. 



Let Q be a constant quantity, and put -= z; also let ki be an element of the 

 curve, IN or de an element of the altitude, and xy an element of the arc: then 

 Newton demonstrates, that the element of the arc, or xy, may be expressed by 



O ^ IN X cx 



this formula » . ==. In like manner, from the premises, M. Bernoulli, 



A VaBGE — 2* 



putting the arc vx = z, and the altitude or distance = ar, reduces the element 



a^cx 



of the arc to this formula, z = / , , Tl ^ • o o ^ Now even at first sight 



V abx*—x* flu. (Px— arc^x^ ° 



Newton's formula would seem rather more simple than Bernoulli's, as consisting 

 of fewer terms; but on examining the matter more carefully, it is found that 

 the two formulas exactly coincide, the diff^erence being only in the notation of 

 the quantities. For if for ab — flu. <pi be put abge, for ac put q, and x for 

 A, also a for cx, and <t for in; then is 



a^CX QXCXXIN QXCXXIN . ^ o o o 



— = . ~^ o n = 2 , ■= =, having put a^z = a^. 



which Newton does for a more commodious notation. Hence it appears that 

 the two formulas differ no otherwise than as any thing written in Latin characters 

 would diff'er from the same thing written in Greek characters. 



After having delivered the general formula, M. Bernoulli descends to a parti- 

 cular case, in which the centripetal force is reciprocally as the square of the 

 distance; and by various reductions and troublesome operations, he shows the 

 construction of curves which may be described by that centripetal force, and by 

 reducing them to equations, he proves they are the conic sections. After which, 

 he complains that Newton supposes, without any demonstration, that curves 

 described by such a force would be conic sections. 



But it is impossible he could believe that Newton was unacquainted with the 

 demonstration of that fact; for he very well knew that Newton was the first 

 and only person who had treated of this doctrine of centripetal forces in a geo- 

 metrical manner, and had brought it to such perfection, that after more than 

 20 years, very little has been added to it by the most excellent geometricians. 

 Bernoulli knew also, that besides giving the general solution of the inverse 

 problem, Newton had showed how curves might be constructed, which are 

 described by a centripetal force decreasing in the triplicate ratio of the distance; 

 and that therefore he could not be ignorant of that other case. Nor indeed 

 can I understand with what reason Bernoulli objects to Newton, that he had 



