324 



EDWARD SANG ON THE APPROXIMATION TO THE ROOTS OF 



and a still more rapid convergence is obtained by putting b = 4:C + d; we then 



find 



c*-20c 2 d- 9cd 2 -(P=0, 

 while 



a = 9c + 2d, b=4c + ld. 



Here r = l , q = $, p = 20, and the rapidly converging series is 



2 9 182 3 721 76 067 185 805 



1 ' ' 4' 



81 ' 1 656 ' 33 853 



82 691 



, &c. 



Enneagon. 



If we contract an isosceles triangle, having each angle at the base quadruple 

 of the angle at the vertex, and if we lay off along the side two parts, each equal 

 to the base, and from the vertex one part, the three measures overlap by a 

 distance easily shown to be the fourth term of a continued proportion, of which 

 the side and the base are the first and second terms. 



Hence, if a be the long diagonal of an enneagon, and b the side, 



or 



a 2 : b 2 : : b : 35 — a , 



a?-8a 2 b + W=0. 



This equation gives at once a series having r— — 1, q = 0, j9 = 3 for the 

 multipliers, viz. — 



, , 1 , 3 , 9 , 26 , 75 , 216 , 622 , 1 791 , 5 157 , &c, 



which converges pretty rapidly to the ratio of the base to 

 the long diagonal. Here, from thrice the term last found, 

 we subtract the ante- penult, in order to get a new term ; 

 that is, from thrice AB we subtract PN to obtain AF. 

 From thrice AF we should subtract KB to get the 

 long diagonal of an enneagon having FA for its side. 

 and so on, the distances PN , KB, BA , AF being in 

 continued proportion. 



The figures FMNP and FABK are evidently similar ; 

 so if in the continued direction FB we measure from K, 

 twice FB, to M', we shall obtain an enlarged edition of the 

 figure FMNP. 



The convergence becomes more rapid if we put a = 3b 

 — c, so as to get the equation 



I— 9S s c+6&c-c s =0. 



