244 LOGARITHMS. 



or lo, Sec. of the - geometrical series of whole numbers i 

 and by interpolation the whole system of numbers may be 

 made to enter the geometric series, and receive their pro- 

 portional logarithms, whether integers or decimals. 



It is also apparent from the nature of these scries, that 

 if any two indices be added together, their sum will be 

 the index of that number, which is equal to the product 

 of the two terms in the geometric progression, to which 

 those indices belong. Thus, the indices 2 and 3, being 

 added together, make 5 ; and the numbers 4 and 8, or the 

 terms corresponding to those indices, being multiplied to- 

 gether, make 32, which is the number answering to the 

 index 5. 



In like manner, if any one index be subtracted from an- 

 other, the difference will be the index of that number, 

 which is equal to the quotient of the two terms, to which 

 those indices belong. Thus, the index 6 minus the index 

 4rz2 ', and the terms corresponding to those indices are 

 64 and 16, whose quotient r:;4 j which is the number an- 

 swering to the index 2« 



For the same reason, if the logarithm of any number 

 be multiplied by the index of its power, the product will 

 be equal to the logarithm of that power. Thus, the index 

 or logarithm of 4, in the above series, is 2 ; and if this 

 number be multiplied by 3, the product will be zz6 ; 

 which is the logarithm of 64, or the third power of 4. 



And if the logarithm of any number be divided by the 

 index of its root, the quotient will be equal to the logar- 

 ithm of that root. Thus, the index or logarithm of 64 is 

 6 ; and if this number be divided by 2, the quotient will 

 be 1=3 ; v/hich is the logarithm cf 8, or the square root 

 Di 64. 



The logaritlims most convenient for practice are such, 

 3S are adapted to a geometrical series, increasing in a ten^? 

 fold proportion, as in the last of the above forms ; and are 



those, 



