436 Mr. P. Nicholson on derivative Analysis. 



To show the application, 



Let it now be required to extract the square root of the 

 number 3856724-. 



Divide the number into periods of twos from right to left 

 thus: 3,85,67,24. 



Take the square root of the last figure 3, which is 1, and 

 proceed as if it were required to diminish the root of the 

 quadratic equation .r* + 0.r— 3 = by 1. 



1 xl = l 

 1 



1 xl = l I -2 

 2 



20-4-2 gives 10 

 9 liowever will only succeed, therefore 



20 —285 by annexing 85 to 2 



1 X9 = 9 

 9 



29x9 = 261 

 38 



380 



— 24 



1 X6 = 6 

 6 



386X6 = 2316 | —151 

 392 



240-^38 gives 6, therefore 

 2467 annexing 67 to 24 



3920 



1510-f-392 gives 3 

 — 15124 



1 X3 = 3 I 3923x3 = 11769 j —3355 

 3 I 3926 



and so on. 



And hence the common method of extracting the square 



root. 



3,85,67,24(1963 

 1 



•3355 



The preceding work is placed in such a manner as to ren- 

 der it obvious to mspection, and therefore description unneces- 

 sary. To apply my method of transformation to the extrac- 

 tion of roots, as the multipliers for each distinct figure are the 



same, 



