Thefe two (Neper and Briggs, whom we may call the firft authors 

 of logarithms) ufed two methods in their computations : the one by in- 

 volution, or raifing the number whofe logarithm they fought to a cer- 

 tain power; the other by evolution, or extracling out of it a root the 

 denominator of whofe index was fufficiently great ; and this latter they were 

 obliged to do by repeated extractions of the fquare root, no eaficr method 

 being then known. 



But afterwards, when Sir Ifaac Newton's famous binomial theorem 

 was made public, Doftor Halley took advantage of that invention, and 

 fliewed a method of calculating logarithms by throwing the root re- 

 <iuired (or rather the logarithm derived from it) into a converging feries : 

 and this method is as eafy and expeditious as can ever be expefted, 

 or indeed defired, the law of the feries being obvious, the terms eafily 

 reduced to numbers, and a very few of them fufficient. 



This difference in the manner of extrafting the root makes the prin- 

 cipal difference between the methods of finding the logarithms ufed by 

 the firft authors and by Halley. 



There are fome properties of thefe roots which are neceffary to be 

 known, and which are obvious enough when the roots are found out 

 in Briggs's way, but which require proof when they are found by the 

 binomial theorem. Thefe properties are here premifed in the form of 

 Lemma's, that the explanation of Halley's method may not be inter- 

 rupted by proving them afterwards. 



Lemma, i. 



" Let f be a proper fraftion, and n any whole number confifting of 

 many places of figures ; if out of the binomial i +e a root is to be extracted 



whofe index is — , it is required to find the mofl firaple feries that fliall 



give that root true to a number of places of decimals lefs by four or 

 more than twice the number of places of figures in «." 



CJJu 



