[ 65 3 



Having found thefe nine terms, and negleding all the reft 

 for the reafon given above, let them be added according to their 

 figns, and their fum is the cubic root of the binomial i — ,. 



And fince 216 — 6(= 210) : i — ^ : : 216 : i, it will be 



V210 : Vi — :^'- : 6 : I. Therefore let the cubic root of i -t 



3" 36 



found above, be multiplied into 6, and the number refulting, which 



(negleding the three laft figures*) is 5,9+3921952763, is the 



cubic root of 210, true to twelve places of decimals, as was 



required. 



The given numbers in thefe two examples, were purpofely 

 chofen fuch as fhould make the operation eafy. But in 

 other examples, the difference between the given number and 

 the cube number next greater or next lefs, may be fo great 

 in proportion to that cube, as to make the fraction, (the fe- 

 cond member of the binomial,) too large : In thefe cafes 

 the feries will converge fo flowly, that the labour will be almoft 

 intolerable. Thus, if the given number were 13, it muft be 

 made equal either to 8-15, or to 27 — 14, and the binomial will- 

 be either i+g-, or i — --. If the given number be 3, the bino- 

 mial will be I 1; and if the given number be 2, the binomial 



v/ill be I — ^. In all thefe cafes, and fuch like, the feries 



* As before, the numbers 0,009346341206142 and 0,00934634 1 2061 50 are the 

 lefs and greater limits of the fum of tlie negative terms ; and the numbers 

 0)990^53'5s87938s8 and 0,990653658793850 are the greater and lefs limits of 

 the fum of all the terms added together according to their figns. Confequently 

 S.94392i9527'>3'48 and -5,943921952763100 are the greater and lefs limits of 

 the cubic root of the number 210, which agree eve«_ to thirteen places of deci- 

 mals ; and therefore the root itfelf is fo far accurately calculated. 



I will 



