440 PHILOSOPHICAL TRANSACTIONS. [aNNO 1708. 



delivered by Newton, on the centripetal force of a body moved in the same 

 curve, in pr. 44 of the Principia. oelfi ad 



Because the centripetal force tending to the point s, by which the body can 



move in a curve, is always as — ^ ; hence, from the law of the centripetal 



SP^SA 



force being given, the relation between sa and sp may be found ; and therefore, 

 by the inverse method of tangents, the curve may be exhibited, which shall be 

 described by a given centripetal force. 



For instance, let the force be reciprocally as any power m of the distance, 



that is, let -^—r = - — r,, it will be — r = -r— - ; then the fluents give 



^.sp"* = — — ^^^ and hence , *"TJ^ ' 'Mi 'sp^; and here multiplying the 

 numerator and denominator by sa"^', and putting d^ for m — -^a^ it becomes 

 , = SP*; therefore sp = d'/ rn. ^zr, or = d^—r— only, when e is 



6 q: csA*"-' 6 ^: csa"*-' ^ b 



equal to nothing. 



Thus, if the force be reciprocally as the square of the distance; it may be 



either sp = c?i/-r> or do/ -r-^, or d>/-rr—\ the curve, in the first case, 



h ' 6 — SA 6-1-8A ' 



being the parabola whose latus rectum is — , in the second case it is an ellipsis, 



and in the third an hyperbola. 



If the force be reciprocally as the cube of the distance, it maybe either 



dSA rfSA rfsA ' .\ c L » 



SP = -r-> or = ,^ , or = ., ; in the first case the curve 



h v6 — esA Vi + esA 



being the nautical spiral ; in the second case the curve is the same as that 



which Newton has constructed by the sector of an hyperbola; iand the third 



case the same as he constructed by elliptic sectors, in cor. 3, pr, 1, lib. 1, 



IVincipia. ' ' - 



If the centripetal force be reciprocally as the distance, the relation between 



8A and SP cannot be defined by any algebraic equation ; yet the curve may be 



constructed by the logarithmic line, or by the quadrature of the hyperbola ; for 



... d 



it is then SP = 



V'6 — log. of sa' 



All these things follow from the now so much celebrated method of fluxions^ 

 of which our Newton was doubtless the first inventor, as will be evident to any 

 one who shall read his letters published by Dr. Wallis. Yet afterwards the same 

 method was published by Mr. Leibnitz in the Acta Eruditorum ; only changing 

 the name and the manner of notation.* 



* This is the sentence which gave such offence to Mr. Leibnitz, that he made a formal complaint 



