3 i 8 NEW SERIES for the 



being greater than the former, but lefs than the latter. This 

 laft limit may alfo be otherwife exprefTed thus, 



I rp - (l6Tf w +i) T m) Tfm+l) 



15 ' 15 (r ( «)— 1(„+,0 



Or compute the feries of quantities t, t\ t\ &c. one from 

 another by thefe formulae 



- abj — r / - r "~ V ' ~~ t t f > - I ~" /l —t 7 

 l '^ ab-j+i' -i + Vi-f/ ~ 1 + s/F+7' 



Then fhall 



j ab j- — 1 3 v 4 4 A- 



the fymbols T( ra ), T( w+1 ), and R, being put to denote the 

 fame as before. 



49. We might now inveftigate other feries for the quadra- 

 ture of an hyperbolic feclor, fimilar to the third and fourth fe- 

 ries we have found for the rectification of an arch of a circle ; 

 but this inquiry would extend the Paper to too great a length. 

 For this reafon, and alfo becaufe the manner of proceeding in 

 the one cafe is exactly the fame as has been followed in the 

 other, it feems unneceffary, in the cafe of the hyperbola, to ex- 

 tend our inquiries farther. I fhall therefore now proceed to 

 the third and laft object propofed in this Paper, namely! the in- 

 veftigation of formulae for the calculation of logarithms, be- 

 ginning with a few remarks that may ferve to conned: thefe 

 formulae with the common theory. 



50. It is ufually fhewn by writers on this fubjec~t, that all 

 aumbers whatever are confidered as equal, or nearly equal, to 



one 



