EVOLUTION. l^i 



^. TliSn, twice tL^ supposed cube acklecl. to the given 

 number, is to twice the given nuir/oer added to the sap- 

 posed cube, as the root of the supposed cube is to the too* 

 required nearly. Or as the first sum is to the diff'-rence 

 of the given and supposed cube, so h the supposed toot to 

 the dificrence of the roots nearly. 



3. By talcing the cub6 of the roOt thus found for the 

 supposed cCibeV aftd repeating the operation, the root Vv'ill 

 be had to a still greateii- degrse of exactness. 



example:. 

 i. It is required to find the cube root of 9800344^, 



Let 



Let A^=: given number, a'= supposed cube, and x:=z cor- 

 rection. 



Then 2a' -{-N : iN-^a'^ :: a : a-{-x by the rule, and con- 



sequently 2a' +Nxa-\-xz=2N-{-a^ Xa, or 2a' +a'-^x^ X a-^x 



z=2N-{-a'^ Xa, 



Or 2a'^ + 2a'x-4-a*-f 4a^Ar -{- Ca^x"- -f ^x"^ + .v'*=2aiV+a^ 

 and by transposing the terms, and dividing by 2a 



A^=a'-f-3a*x-{-3a.v*-fx'-f-Ar' -{-■—, V/hich by neglecting the 



terms x'^ -f ..., as being very small, becomes Nzza'' ■\-'ia'^x-\' 

 2a 



3'T;v''+^:'= the known cube of a-\-x. Q. E. L 

 W 



