292 ON NEWTON'S SOLUTION OF KEPLER'S PROBLEM. [38 



None of the subsequent authors, however, mentions this method as being 

 Newton's, the unusual form in which Newton's solution is given having, no 

 doubt, caused them to overlook it. 



In the first edition of the Principia a modification of the method is 

 given which was, I have no doubt, intended by Newton to be equivalent to 

 the second method given above; but by some inadvertence, instead of the 

 denominator of So/ being 



/ 1 

 le cos (x 1 + - 



\ * 



when expressed in the above notation, he takes it to be what is equiva- 

 lent to 



le cos \x' + - e sin x'} , 

 \ / 



which is only true for the first approximation when a3 is taken =z. 



In the second and third editions this error is corrected, but Newton 

 contents himself with the more simple expression given by the first method. 



We need not be surprised that Newton should have employed this 

 method of solving the transcendental equation 



x e sin x z, 



since the method is identical in principle with his well-known method of 

 approximation to the roots of algebraic equations. 



For convenience of calculation, the approximate values x a , x', x", &c., 

 should be so chosen that their sines may be taken directly from the tables 

 without interpolation ; and, since each approximation is independent of the 

 preceding ones, this may always be done if x' be taken equal, not to 

 x + 8x itself, but to the angle nearest to x + 8x which is contained in 

 the tables, and if similarly x" be taken equal to the tabular angle which 

 is nearest to x 1 + Sx 1 , and so on. In the first approximation it will be 

 amply sufficient to use 5 -figure logarithms, but in the subsequent ones tables 

 with a larger number of decimal places should be employed. 



A first approximate value of the eccentric anomaly corresponding to 

 any given mean anomaly may be found by a very simple graphical con- 

 struction, provided we have traced, once for all, a curve in which the 

 ordinates are proportional to the sines of the angles represented on any 

 given scale by the abscissae. 



