145 



which would be derived by extracting the Square Root of 1.331 by the 

 known method which depends upon trial. 



In like manner, we extract the Cube Root of 1+ -'3 or of 1.4641, 



which may be otherwise expressed 1+-*-+ -|_+— V-+~i~, viz. 



Given Power is 1 + tV + tkt: + t »V » + t o fro 



3 

 1 =1 



Remainder zr * 4x-'-+&c. 



3)4x^Va>^T'o 



Given Power is 1 + 4 x -'- + 6 x t-- + 4 x -r:r'---+-r^i-r~ 



' lO' loo' lOOO'lOOOQ 



3 



1+|xt'> = l+4x---+Vx-i- + |4xT^V^ 



Remainder = * * yX ~- + &c. 



Given Power is 1 +4 x -^'- + 6x TiT+4 x --'_-. + -_i-_ 



3 



i+ixV- + ixTi^ Z= 1 +4 X ,V+6 X ^i--+ V-^ X T3-Vo + &c. 



Remainder =: * * * ~-^\ x t^W ^^• 



*^J T~ X ^-^-^-^ [ Ti X TS'o'o 



Therefore 1 + tV =l+TXTo+-tx .ioV-x toVo+^c. and substi- 

 tuting for the Fractions the equivalent Decimals, we shall have the 

 above expression =1.135&c. the same which would result from the 

 usual mode of extracting the Cube Root of a number by trials. 



I have applied the general Example to the extraction of 

 the Square and Cube Roots, merely to elucidate the method by 



which 



