362 On extractfng the Cube Root hi Nuinbers. 



•Again divide 22125 hy 16567, and the quotient 1 is the next 

 figure of the root ; therefore proceed with the next itep 

 705 165675 221^5 ... 



705 1 1657lo'5 1 ^ 55j04417 



7052 lt;>81603 

 70)3 



And so on ; so that the root is 2 351. 



This process being sufiiciently understood, the learner may 

 then work the whole cf the s>teps in One continued «.peraiionj 

 thus 



6 \2 5 ...(2.351 root 



111 this operation we 

 mav ol)«ervethat the mul- 

 tipHcations and additions, 

 as also the multiplications 

 and subtractions may be 

 performed in one line, as 

 shown in ^onie of our best 

 systems of arithmetic. 

 It may now be o])served, that wherever the operation is ter- 

 minated without having obtained the correct root, as many more 

 figures except one as the number of figures in the root already 

 obtained may be found by dividing the iast remainder by the 

 second coefficient, wanting as many of its last figures as the num- 

 ber of figures to be found. 



Thus in the present instance 5550449 divided by 16581 gives 

 334, which annexed to the part 2-35 1 of the root already found 

 gives 2-351334, which is true to the last figure. 



This operation will admit of a proof at every step, which may 

 be done by the following rule : 



Consider the coefficients and remainder from which the step 

 to be proved is found as whole numbers, and the figure of the 

 root as a decimal in the place of tenths ; then add into one sum 

 the cube of the new figure, the product of the first coefficient and 

 the square of the new figure the product of the second coefficient, 

 and the new figure itself togetiier with the last remainder ; then, 

 if the work is right, the sum will be equal to the preceding re- 

 mainder or absolute number. 



Example. — The coefficients and absolute number by which 

 the third figure of the root 5 in the example given are 69, 15S7 

 and 833 considered as whole numbers ; then 

 (-5)5= -125 

 69 X (-5) = = 17-25 

 1587 X (-5) =793-5 

 22- 125 = 22-125 



S33 000 LIX. De- 



