II 



9. Let now e and y be proper fraaions, and (by lemma 4) nxi+^f -i 



I 



■ ■ n 



— g_^g^ ^^e'—ie* +^c' &c. and alfo«xi+y -i r:)' -s^'+i)"— 47*+ 

 iyj &c.; and thus by feries of this kind we can find Neper's logarithms of 

 all mixed numbers between i and 2, 



10. In like manner, if the number whofe logarithm is fought, be lefs than 

 1, that is, if the ratio be that of i to i—c, having placed a fufficient 

 number of mean proportionals between i and i—e, he fubtrafted i from 

 i « 



1—e, the firft or greateft of them, and made the remainder i— ^ -i (which 

 now becomes negative) to be the logarithm of one ratiuncula, and there- 



fore nxi—e -i will be the logarithm of the ratio of i to i — e, or of 



i 



the number i — e; and hence as before, ?zxi — y -i niuft be the loga- 

 rithm of the ratio of i to i— y, or of the number i — y. But (by lemma 



I — 



4) nxi—e"-i = -e—^e'—ie^—ie*^\e' &c., and alfo nxi—y -1 



— -y ^y^ lyi — ^y* — ^y' &c., and therefore, by feries of this kind, 



we can find the logarithms of all numbers lefs than i . 



1 1 . Let now any two numbers be propofed, a the leffer, and b the 

 greater, the logarithm of whofe ratio is required. We mud firft find 

 a ratio whofe antecedent is i, and which {hall be equal to the ratio of 

 atob: this is done by finding the value of e in the following analogy, 

 a: b : \\ : i-\-e; which being changed into an equation, becomes a-\-ae 



= bf whence ae = b — a, and therefore £■ = — : and if .? be a proper frac- 

 tion, we may then find the hyperbolic logarithm of the ratio of i to !+£•, 

 which (by art. 9.) appears to be e — ^e''+^e^ — ie^+ff' Sec; and fmce 

 equal ratios have the fame logarithm, that feries will alfo be the loga- 

 rithm of the ratio of a to b. 



12. And if the ratio be that of b to a, we muft ftill find an equal 

 ratio whofe antecedent is i, which is done by finding the value of e in 



( B 2 ) this 



