Mr. P. Nicholson on derivative Analysis. 437 



same, it is quite useless to place them in the operation: I will 

 therefore give another example, and will omit them in the 

 work. 



Extract the cube root of the number 3. 



Here we have only to extract the root of the cubic equa- 

 tion ^54-Oa^'-fOj:- 3=0. 



Now the nearest cube root to 3 is 1 ; therefore 



Divide 20 by 3, and the quotient 6 would be the next 

 fiaure ; but as the work has not yet acquired a state of con- 

 vero-ency, 6 will be found to be too much, and upon trial it 

 wilfbe found that no higher number than 4 will succeed; 

 therefore 



30 300 -2000 



The work having now acquired a state of convergency, di- 

 vide 2560 by 588, and the quotient 4 is the new figure of the 

 root, therefore 



420 58800 -2560qq_ 



60496 241984 [ 14016 

 62208 



Divide 140160 by 62202, and the quotient 2 is the next 

 figure of the root ; therefore 



4320 6220800 -14016000 



1)002 1 4322" 8644 I 6229444 12458888 \ 1557112 

 0002 4324 8648 | 6238092 

 0002 I 4326 

 And therefore the root as far as the work is extended is 1-442; 

 and if we divide the absolute number 1557112 with four ci- 

 phers annexed by the coeflScient 62208, we may have the torn- 

 additional figures 2496, and thus extending the root to the 

 eiffht figures 1-4422496. If we omit the first, thu'd, htth,&c. 

 columns, and unite the parts of the other columns remain- 

 ing, the work will sUmd as follows : 



1 



