i8 ' 



numbers confifling of many places of figures, the exceffes of 

 thefe roots above unity will be to each other as the indices 

 of the roots, or reciprocally as the denominators of the indices ; 

 provided that the roots be calculated only to a number of places 

 of figures lefs by 2 or 3 than twice the number of places in 

 the lelTer denominator. 



This propofition is the fame with lemma 3, and has been proved before. 



Prop. 4, 



If out of two numbers, both (landing between i and 2, be e.r- 

 trafled roots, whofe indices are the fame, and of which the 

 denominator is fufEciently great, the exceffes of the roots above 

 unity will be to each other as the logarithms of the numbers 

 themfclves. 



For fuppofe i ft, that roots are cxtrafted out of the numbers whofe in- 

 dices are fuch that the roots themfelves may be equal ; then (by prop, 

 ad) the logarithm of the greater number will be to the logarithm of 

 the leffer as the greater denominator is to the leffer. Suppofe 2dly, 

 that out of the leffer number another root is extrafted, whofe index 

 is the fame with the index of the root extrafted out of the greater 

 number; there are now extracted out of the leffer number two diffe- 

 rent roots, and (by prop. 3.) the exceffes of the greater and leffer of 

 thefe roots above unity will be to each other as the greater and leffer 

 denominator; that is (as was proved above) as the logarithms of the 

 greater and leffer number. But the greater of thefe roots is equal (by 

 fuppofition) to the root extracted out of the greater number ; therefore 

 the exceffes above unity of the roots extrafted out of the greater and 

 leffer number when the index is the fame, are to each other as the 

 logarithms of the numbers themfelves. 



The fame argument in fymbols is in the next page. 



Let 



