144 



To elucidate this Example, by applying it to the Extrac- 

 tion of Roots of numbers, we shall extract the square root 

 of 1.331, or of the cube of 1.1, where t—^-^ and 1.331 is 

 arranged according to the Indices of powers in the Decimal 

 series, and may thus be otherwise expressed 1 + 4^ + t-o + -Io-s-t- 

 Now III being = 3, and r=z2, we shall apply the foregoing Ex- 

 ample as follows : 



3 



Given Power [l + -\) = 1 + tV + tIo + t oVo 



2 

 1 =1 



Remainder * — + ^ "^^ • 



2)tV(4x-.'o 



Given Power 1+^ = 1+ 3 x -U + 3 x -i- + __>_- 



2 



Remainder * * i^ ~i'^'^^^- 



""/T X T'S^'S'It X T'ST'o 



Given Power 1 + -.-^ = 1 + 3 x -V + 3 x -i- + -r'~- 

 2 



i+iXr'z + iXrW rrl+3xVo+3x-i-+|x^-'~ + &c. 



■iaiiii\ Remainder * * * -^x~--- — &c. 



g '\-J. y .' f~_'_ V I 



• 8'^lOOoV 161000 



Hence 1 + ^ = 1 + |x -r'- + | x -J._— J- x -iVo -> &c. and putting 

 for 4 x -V> I X -i-, — -■- X --'--, &c. the Decimals, 0.15, 0.00375, 



3 



—0.0000625, &c. we have l+-'3=: 1.1536&C. and this is the result 



which 



