12 



this analogy b : a '. : i '• i — f, which gives this equation b- be=a, whence 



l>e=b — a, and e=z — here e mud neceflarily be a proper fraftion, and 



the logarithm of the ratio of i to i — e (by art. lo.) is -e—\e'' — le^ 

 — \e* — \e^ &c., which is therefore the logarithm alfo of the ratio of b 

 to a. 



13. From art. ii, we may obferve that when the given ratio is 

 afceoding, or of leffer inequality, the value of e is the difference of the 

 given terms divided by the leffer of them : and from art. 1 2, that when 

 the given ratio is defcending, or of greater inequality, the value of e is 

 the difference of the fame terms divided by the greater. 



14. Either of the above feries might be fufficient for finding all loga- 

 rithms ; but by joining the two together a third feries refults, much more 

 convenient for the purpofe, as it converges twice as fall as either of 

 them; the method of doing it (which jnufl; be carefully attended to) is 

 as follows. 



15. Between a and b, the terms of the given ratio, place;) an arithme- 

 tical mean ; the whole ratio of a to i Is thereby refolved into two, that 

 of a to /», and of / to ^ : invert the former, and it becomes the ratio of 



ptQ a; and if we make i : \—e : : p : a, then (by art, 13 j will e-^-^^ 



and the logarithm of the ratio of i to 1 — e, or of/ to a, is the feries 

 _f_^f^_^e'_if*_Je' &c. by art. 10. 



16. Again, if we make i : \-^e : : p : b, e will be tt by art. 13, but, 



Ijnce p is an arithmetical mean between a and b, ■— —— ; therefore e 



r t P 



has the fame value as in the laft article ; and the logarithm of the ratio of 

 I to i4£', orof /> to b, isf— 5f*+-J(f'— i^* + if' &c. by art. 9. 



17. In art. 15, the logarithm of the ratio of /> to a was found to be 

 _^_ie' — ^tf3__i^4_-!f> &c. Invert this ratio again, and it becomes 

 the ratio of a to p, and its logarithm is the fame as before, only its fign 

 is changed : that is, the logarithm of the ratio of a to /> is the feries 

 ^4.if>4-|f^-l-ie*4-Je' &c., and by art. 16, the logarithm of the ratio 

 of /* to * is e—\€'--Y\e'—\e'-^\e' &c., and therefore the logarithm 

 of the compound ratio, or of a to b, is the fum of thefc two feries, which 

 is 2<'+aX|fH2Xi(f' &c., or aXf+^f'+t^ &c'. 



