NUMERAL EQUATIONS, 



593 



By dividing the original equation by x — 432 = 0, the other 

 roots may be found, but in this case they are imaginary. 

 But, instead of thus approximating to a root by single figure;-, 

 we can (after the root has been sufficiently diminished) find, 

 by a Table of Logarithms, as many places of figures at one 

 operation, as there are places of figures in the logarithms; as 

 in the following 



EXAMPLE III. 



Suppose S3 — 2x =5. 

 then j'°-2x'=4 



By the Rule, 



divisor. Resolvend 1 



3x' — 2=10} 

 3x' = 6$. 



1 =13 



gives 10 x" + 6 x" J + x"3 



near Root x' = 2 



near Root x =,09 

 0,949329 Subtrahend. 



Resolvend 0,050671 = v. 



This being now sufficiently reduced we proceed thus :■ 



By the Rule, 

 3x"* + 12 x" + 10 = 11,1043 (A) 



New Equation. 



, , c ' , _ _ ") 11,1043s'" + 6,27x"'* + x'"3=,050671 = V and by 



+ 6 = 6,27 — B( V B 



= 6,27 = B ? 



r l ~ c $ 



Then, by logarithms, 



reversion of series, x'"=s — — — V* -+• &c. 

 A A3 



v — a 



d — a = 

 6,27 = 



b + c + d = 



V = ,050671 

 A = 11,1043 



X =.00456319 

 A 



V 



A 4 



B 



j* v =,00001176 

 A3 



Log. 

 Log. 



Log. 



8.704760 

 1.045492 



7.65926S 



6.613776 

 0.797268 

 5.070312 



V B 



_ _ _y j = x'" = ,00455143. whence x = 2.09455143 



A A3 



And if ,004551 be put for x'", in the above new equa- 

 tion, the value and root would be again reduced, so that we 

 should obtain x"" == ,0000004815424, and consequently the 

 root x = 2,0945514815424, true to the last figure. 



