[ 430 ] 



Call the fum of this feries, s. Then, (= 



d + a ab 



~. ) = s '.• b — a=- sb -\- sa, b — sb =■ a -V sa^ b ■=. a^ 



ab I — s 



and a ~ b. 



I + J 



Here then we have a feries confifling of but half the 

 number of terms of either of the feries firft propofed ; but a 



divifion is afterwards requifite to find the natural number _, 



a 

 a 



Let us endeavour to render this feries more convenient for 

 pradice and apply it to find the natural numbers of logarithms 

 of any fyftem whatever, without the trouble of reducing the given 

 logarithm to Napier's fyftem. If we affume the firft term 



. I ■ + ? 



for the whole feries, i. e. — = / ; then will — =. and 



2 a / 



/ ^ z 



I I 



__ = making Z,to denote the logarithm of — in any fyftem 



o t 



whatfoever, 



