[ 391 1 



On the APPLICATION of a CONVERGING SERIES to tbe 

 CONSTRUCTION of LOGARITHMS. By Mr. WILLIAM 

 ALLMAN, A. B. Trinity College, Dublin. 



X ROM a due confideration of Newton's binomial theorem, it R«ad May 



may be (hewn, that the logarithm of the ratio of one number 



to another, according to Napier's fyftem, is equal to the fum 



2d 2d^3 id^ 2d'' 

 of the feries, 1" 777 + "77 + _,T + &^- d reprefenting the 



difference, and s the fum of the numbers : which logarithm of 

 the ratio, added to, or fubtraded from, the logarithm of the 

 antecedent, according as the antecedent is lefs, or greater, than 

 the confequcnt, gives the logarithm of the confequent. 



In any fyflem whatever, the logarithm of the ratio of 

 one number to another is equal to the fum of this feries,, 



ipd 2pd^ ipd^ ipd'' 



— !■ 'fl rT "^ ~7T "^ "^TT" "^ ^'^^ where d expreffes the difference, 

 J JJ^ 5J yj 



and 



