VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. 



Observations on Mr. John Bernoulli's Remarks on the Inverse Problem of Centri- 

 petal Forces, in the Memoirs of the Academy of Paris for the Year 17 10; 

 with a Neiv Solution of the same Problem. By John Keill, M. D. N° 340, 

 p. 91. 



To determine the curve described by a body, which is urged by a given law 

 of centripetal force, when projected with a given velocity from a given place, 

 in the direction of a given right line, is a most noble problem. In the Prin- 

 cipia, Newton long since gave a complete solution of it, granting the quadra- 

 ture of curve figures. Since that, the celebrated Mr. John Bernoulli has also 

 undertaken the same problem, in the Memoirs of the Paris Academy for 17 10. 

 Now having compared his solution with that of Newton, I have made the fol- 

 lowing remarks upon them. 



M. Bernoulli premises the same proposition as employed by Newton, for 

 demonstrating his problem, which is the 40th in his Principia, and is no less 

 elegant than easy to be demonstrated. It is this, viz. if a body be any how 

 moved by a centripetal force, and another body ascend or descend directly; and 

 if their velocities be equal in any case of equal altitudes, then will their velo- 

 cities be equal at all equal altitudes. 



M. Bernoulli says, the demonstration of this proposition is delivered by 

 Newton in too complex a manner, and therefore he substitutes his own instead 

 of it, which he calls a more simple one. But permit me to say, without offence 

 to so great a man, that if there be any difference between their demonstrations 

 it is this, that Newton's seems to be much the easier and less complex of the 

 two. For with the centre c, fig. 1, pi. 4, let the two circles di and ek be 

 described, at the very small distance de from each other, and let the velocities 

 of the bodies at d and i be equal ; and if there be drawn nt perpendicular to ik; 

 then Newton fully shows that the accelerating force in de is to that in ik, as in 

 to IT. For if the force in de or in be represented by the right line de or in, 

 then the force in in is resolved into the two ti, tn, of which, that only which 

 is as Ti accelerates the motion in the direction ik. But the accelerations, or 

 the increments of the velocities, are as the forces and as the times in which they 

 are generated conjointly; and, because of the equal velocities in d and i, the 

 times are as the spaces described de, ik : therefore the acceleration in the mo- 

 tion of the bodies through the lines de and ik, are as de to it and de to ik 

 conjointly; that is, as de^ or in^ to the rectangle it X ik: and therefore, be- 

 cause in^ = it X IK, the increments of the velocities are equal. Therefore 

 the velocities in e and k are equal. And by the same argument the velocities 



