13 



i8. In art. 15 f was found to be ^, and in art. 18 e was found 



P 



to be ^; and it was there obferved, that — = — , becaufe f is an arith- 



metical mean between a and b ; from which it alfo follows that either nu- 

 merator, f-a or b-f, is equal to \b-\a, and alfo that the common deno- 

 minator /> is equal to \bArka., therefore e is always equal to -r,~^!'', or 



to -r— -> that is, the value of e is always a fraftion whofe numerator is 



the difference of the terms of the given ratio, and whofe denominator is 

 their fum : and the logarithm of the ratio is the double of a feries formed 

 by the following laws : 1 ft. the feveral terms of the feries contain the 

 powers of that fraftion or quantity whofe indices are the odd numbers ; 

 adly, every term is divided by the index of the power of the quantity 

 e in it ; and 3d]y, the terms are all affirmative, when the ratio is that 

 of a to b, or afcending ; but would all be negative if the ratio were that 

 of b to a, or defcending. And by thefe feries may be found the logarithms 

 that are called Neper's Logarithms, and fometimes the Natural Loga- 

 rithms, but mofl ufually the Hyperbolic Logarithms of Numbers or 

 Ratios. 



This is Doftor Halley's method, as far as it relates to logarithms in 

 general. But it may be neceflary to add fome obfervations upon it, and 

 particularly to alTign the reafons of the feveral operations where thefe rea- 

 fons are not fufKciently obvious of thcmfelves. 



1. I have throughout fuppofed that the logarithms of an afcending ratio 

 (or of lelfer inequality) is affimative, and that the logarithm of a defcend- 

 ing ratio (or of greater inequality) is negative ; but this is a matter in 

 its own nature abfolutely indifferent : the logarithms of a ratio of either 

 inequality may be made affirmative ; but then the logarithms of ratios of 

 the other inequality mufl be negative, and reciprocal ratios muft have 

 logarithms equal in quantity, but with unlike figns. 



2. In art. 11 and 12, I have fuppofed every ratio to be fo reduced, as 

 that its antecedent may be i. I might have reduced them fo as to 

 make i the confequent of each. But the neceffity of one or other 

 will appear from hence, that the logarithms of ratios are found by in- 



fening 



