[ 7° ] 



In the fame manner, may be found as many anfwers as any 

 one fhall pleafe, and in every fucceeding anfvver the deno- 

 minator of the fradion, (the fecond member of the binomial,) 

 becomes greater ; but the fradion itfclf, (which is ultimately 

 fought by this inveftigation,) becomes lefs. Now by the help 

 of any of the above equations marked Jirji, fecond, third, &c. 

 anfwers, may the values of a and b be found. Thus, in any of 

 thofe equations make the lefs quantity =o and the greater =i, 

 and from thefe, by going backward, may fuccelTively be found 

 the values of all the letters that had been thrown off by the 

 feveral fubftitutions, until we come to b and a ; and the value 

 of J' will be the coefficient of the term on the left fide of that 

 equation, where the greater and lefs quantities were made 

 =1 and o. Thus, in the fourth anfwer, if we make /=o and 



e—i, then will 



d—(2e—f=) 2 

 c={6d+e=.) 13 

 b={ic—d=) 37 

 a=.{2b+c=) 87 



Now the cube of 37 is 50653, which multiplied by 13 gives 

 658489, and the cube of 87 is 658503 j and the difl^erence is 

 14, the coefficient of the term on the left fide of that equation : 

 and when the fign of j is +, as in the firft equation, a' will 

 be greater than the multiple of b\ and therefore in the bino- 

 mial, the fign of the fradional part will be negative : But when 

 the fign of J is +, a' will be lefs, and the fradional part of 

 the binomial will have a pofitive fign. 



In 



