CUBIC EQUATIONS BY HELP OF RECURRING CHAIN-FRACTIONS. 313 



Here, in order to find the ratio of a to h, we, following Brouncker's plan, 

 try how often b is contained in a. Clearly it is only once, with something 

 over. We therefore write a = lb + c, and get, by substitution, 



l&3_ 3& 2 f _ 36c 2_ c 8_ 0; 



an equation easily managed. The ratio of c to b is now obtained from a 

 progression regulated by the multipliers r=l , q = S , p = 3 ; thus 



0, 0, 1, 3, 12, 46, 177, 681, 2 620, 10 080, &c; 



so that if any one term — say 177 — be assumed for c, the succeeding term, 681, 

 is approximately the corresponding value of b; but a = lb + c, wherefore 858 is 

 the corresponding value of a. In this way we form the series — 



(0) 



(1) (2) 



(3) 



(4) (5) 



(6) (7) 



(8) (9) 



1 



1 4 



15 



58 223 



858 3 301 



12 700 44 861 



' 0' 



1' 3' 



12' 



46' 177' 



681' 2 620' 



10 080' 38 781 



(10) 



(ID 



(12) 



(13) 



(14) 



(15) 



187 984 



723 235 



2 782 518 



10 705 243 



41 186 518 



158 457 801 & 



149 203' 571032' 2 208 486' 8 496 757' 32 689 761' 125 768 040 



approaching very rapidly to the cube root of 2. 



Among these we notice that each member of the terms (3) , (6) , (9) , (12) , 

 (15) is divisible by 3. On simplification, these terms, with the prefixes 



i_ , ^ , form a series progressing according to the scheme r = 1 , q — — 3 



p = 57 ; thus 



(3) (4) (5) 



16 287 927 506 52 819 267 

 12 927' 736 162' 41922 680' 



converging still more rapidly to the required root. The term (2) is true to 

 within the accuracy of five-place logarithms, the defect being -000 0032. The 

 next term (3) passes beyond the exactitude of seven-place tables, its loga- 

 rithm being in excess by '00000 00157 . The excess in the case of (4) is 

 •00000 00000 31409 , which could not be detected with the ten-place tables ; 

 while (5) gives a defect of "00000 00000 00053 , as tested by my manuscript 

 tables to fifteen places. The errors are two in defect, two in excess, and so on. 





(0) 



(1) 



(2) 



+1 







5 



286 



-i' 



0' 



4' 



227' 



The roots of numbers immediately above or below a cube are very readily 

 found. Thus for the cube root of 9 the equation becomes a? — 9b 3 = 0; whence 

 a = 2b + c, and 



b s -12h 2 c-6bc 2 + lc 3 =0. 



