IS 



fucceeding term of the ferles is to be derived from the preceding, and 

 if its numerator be not i, there will be a neceffity of both multiply- 

 ing and dividing; whereas by making its numerator i, the multiph- 

 cation is avoided. And for this rcafon, if a ratio be propofed whofe 

 terms will not immediately give a fraaion of this kind, it is to be re- 

 folved into others, in each of which the difference of the terms is either 

 I, or a meafure of their fum. Thus, if the logarithm of the ratio 

 of 5 to 8, were required, from what has hitherto been explained, tlie 

 quantity e would be -^ ; but, inftead of immediately finding that Iqga- • 

 rithm, the ratio is to be refolved, either into the ratios of 5 to 6 and 

 of 6 to 8 and then the fraftions become Vt and \ ; or into the ratios 

 of 5 to 7 and of 7 to 8, and then the fraftions become i and fs ; 

 or laftly into the ratios of 5 to 6, and of 6 to 7, and of 7 to 8, and 

 the fraftions become -rV. tt. and ,-l : and the logarithms of any of thefe 

 fets of ratios being found, their fum will be the logarithm of the ratio 

 of 5 to 8. 



There is alfo frequently another reafon for refolving the ratio firft 

 propofed, into others ; and that is in order to diminiOi the fraftion 

 e% for as it is diminiflied, the feries converges the fafter, and it may 

 frequently be eligible to find two or three or more logarithms by fe- 

 ries that converge faft, rather than one by a feries that converges 

 flowly. 



3. In art. 15, by adding two feries together, a third feries refults 

 Biore fimple than either of them. The feveral fteps, by which this is 

 effefted, are now to be explained. 



After the given ratio is refolved into two, it is ordered that one 

 (and one only) of thefe ratios be inverted ; for if neither of them, or 

 both, were inverted, they would ftill be, either both afcending, or both 

 defcending ; and, in either cafe, the two feries produced would have their 

 correfpondent terms (i. e. terms that involve the fame power of the 

 literal quantity e) affefted by like figns, and therefore no term would 

 vanifli by addition. Whereas by inverting one ratio only, one feries 

 has all its terms affefted by the fame fign, and the other has its terms 



alternately 



