CUBIC AND HIGHER EQUATIONS. ^ft 



RULE.* 



1. Find by trial two numbers, as near the true root as pos- 

 sible, and substitute them separately in the given equation^ 

 instead of tlie unknown- quantity j marking the errors, 

 which arise £ron> each Qf them. 



2. Multiply the difference of the two numbers, found 

 by trial, hf the l^a-st etror, and divide the product- by the 

 difference of tlie errors, when they are alike, but by their 

 sum, when they are unlike. Or say, as the. difference or 

 sum of the errors is to the difference of the two numbers, 

 so is the least error to the correction of its supposed number, 



3. Add the quotient last found to the nyijiber belonging 

 to the least error, when that number is too little, but sub- 

 tract it, when too great ; and the result will give the true 

 root nearly, 



4. Take this root and the nearest of the two former, or 

 any other, that may be found nearer ; and, by proceeding 

 in like manner as above, a root will be had still nearer 

 dian before ; and so on, to any degree of exactness requir-* 

 ed. Each new operation commonly doubles the number 

 of true figures in the root. 



Note i. It is best to employ always two assumed num- 

 bers, that shall differ from each other only by unity in 

 the last figure ow the right hand , because then the differ- 

 ence, or multiplier, is only i. 



EXAMPLES. 



* This rule may be uced for solving, the questions of Double 

 Position, as well as, that given in the Aritlimetic, and is prefera- 

 ble for the present purpose. Its truth is easily deduced from the 

 same supposition. 



For, by the supposition, r : j : : rv — a : x — h, therefore, bt" 

 division, r — ^ \ s : : h — a : x — b j which is ths riilc 



