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tional to thefe terms, and to the leffer term add the Icfler of thefe 

 parts, and make that fum the intermediate term, and then invert the 

 latter of the two ratios. But fince this method has no advantage over 

 the other, and fmce the proof of it is not fo obvious, Doftor Halley 

 juftly paffed it over in filence. 



The doftrine delivered in art. 7. may perhaps become clearer by 

 being divided into feveral propofitions, as follows. 



Prop. i. 



The logarithms of two different powers of the fame number 

 are to each other as the indices of the powers. 



For the logarithms of thefe powers are the products of the logarithm 

 of the root into the refpeftive indices ; and therefore are to each other 

 as the indices. 



Prop. 2 



If out of two numbers be extracted roots, whofe indices are 

 fuch that the roots themfelves may be equal, the logarithms 

 of thofe numbers will be to each other as the denominators 

 of the indices of the roots. 



For if the common root be raifed to a power, whofe index is the 

 greater denominator, that power will be the greater number; and if 

 the fame root be raifed to a power whofe index is the leffer denominator, 

 that power will be the leffer number ; and therefore (by the preced- 

 ing prop.) the logarithms of the numbers will be as the denominators. 



Prop. 3. 



If out of a number, which ftands between i and 2, be ex- 

 tracted different roots, the denominators of whofe indices are 

 Vol. IX. ( C ) numbers 



