33° NEW SERIES for the 



in the firft place, that each term of our feries, taking its co-ef- 

 ficient into account, js greater than one-fourth of the term be- 

 fore it. 



t' i t° i 



Again, becaufe -=- (i-\-t n ); and, fimilarly, 7 =-(i-f t' n ), 



t 2 I 2 



and it having been proved that *' < t, Co that fimilarly, t" < t\ 

 therefore - (i + 1^) < I (i -f ***}_, and confequently -r < -„ and 



t n 



t" < — -. Thus it appears, that each of the quantities ^, &c. is 



lefs than a third proportional to the two immediately before it, 

 and the fame muft alfo be true of the terms of the feries. 



6o. Upon the whole, then, our firft feries for the calculation 

 of logarithms may be expreffed as follows : 



JL JL JL 



I _I*+I (I , I I I 



— \t — ; r o — * tz — 



log*"' 2*- 1 4^t +i * x i +I l6 ** + I 



+ T( m ) -f- T(«+i) + R); 

 and here, as in the former feries, T( OT ) and T( m +i) denote any 



two fucceeding terms, and R is a quantity greater than _T( m + i), 



but lefs than a third proportional to T(» — T( m + 1) and Tm + ijj 

 or it is lefs than 



i T( - + ' )+ 3(T(.)-T(,:,) T( * +,> 



6i. That 



