I02 



On Newton's two practical methods of Jolving Kepler's Problem. 



Both thefe methods* are given without demonftration. The for- 

 mer confifts in obtaining fucceffive approximations to the excentric 

 anomaly. Keil has given a demonftrationf of it, but has not aflign- 

 ed the rate of convergence of the Nevptonian feries, from which arifes 

 the great value of this method. 



It may be explained and demonflrated fi-om the confideration of the 

 equation c—m — e s, c by which the rate of converging will be pointed 

 out. 



Let a near affumed value of ^ be c and let c=c+<: 



/ 



// / 



let alfo c=OT — es, c 



then becaufe c-\-c'=-ni — es, c-\-c 



we have c +£— c=f XJ'j c — s, c+c 



I I 



I / ' 



now s,c^c=s^c')(.cs^c-\-cs.,c'%s,c 



but cs, c—i — |f''+&c. and s, c=c — i c^ +&c. 



' ' 71' 



Therefore c-\-c~c~-ecxcs,c — \cs, c not regarding c' &c. 



Newton's 



* PriD. Math. Nat. Phil. Lib. i. Sefl. 6, Schol. 

 t Phil. Tranf. Keil's Aft. 2J Seft. Horfley's Newt, vol. 2, page 133. 



