«4 



ferting fufScient numbers of mean proportionals between their terms : 

 for if between two numbers, a and b, it be required to infert a feries 

 of mean proportionals whofe number is w, the firft of them will be 



_r I 



^^^"■^'^t ai^d the lafl will be'J^-4-i j in each cafe one of the given 

 terms muR be involved to a power whofe index is the number of 

 means required ; and this, we may fafely fay, would be imprafticable in 

 the prefent cafe, if that term be different from i, on account of the 

 greatnefs of the number m : but if that term be i , this trouble is 

 wholly avoided, every power of i being dill i. 



Again, the other term of every ratio is propofed under this form 



I + f , that is as a binomial or refidual of which the firfl member is 

 I ; the reafon of which will appear from this, that if the ot"" power 



of a binomial or refidual a "^ b, h& to be found by the binomial 

 theorem, the firft, fecond, third, and following terms of the feries 

 will contain the powers of a whofe indices are m, m- i, m - z &c. 

 that is a" will be the firft term, a"" - ' will be one faftor of the fecond 

 term, a"" - * one of the third terra, and fo on ; and if (in the cafe of 

 calculating logarithms) a be different from i , it may fafely be pronounced 

 imprafticable to find thofe powers of a : whereas if a be made equal 

 to I, all that trouble vaniflies, every power of i being ftill i. 



And lalHy, e, the fecond member of the binomial or refidual, is fup- 

 pofed to be a proper fraftion ; for otherwife the feries of art. 9, would 

 either perpetually diverge, or after converging flowly for fome time, 

 would afterwards diverge ; or laftly, would converge perpetually, but fo 

 flowly as to be totally ufelefs : but we need not infift further upon 

 thofe particulars, becaufe in the feries of art. 17, which is the only 

 one that we can ever have occafion to reduce to numbers, the quan- 

 tity e muft always be a proper fraftion, its numerator being the 

 difference, and its denominator the fum of the terms of the given 

 ratio. Some ufeful cautions however may be given, relating to that 

 fraftion; as that it is convenient to have i for its numerator ; for each 



fucceeding 



