Memoirs of John Napier of Merchistoii. 281 



second in like manner, by deducting from the second the ten- 

 millionth part of its value, or 0.9999999, according to our deci- 

 mal notation ; and, progressing in this manner, we obtain by 

 simple subtraction as many terms as we wish, all following each 

 other in geometrical progression, according to the ratio selected. 

 The correspondence of the terms, and the indices marking their 

 rank, compose the table exhibited below, in which the succession 

 is indicated to the hundredth term after the first, pushing the 

 value of each term to the seventh decimal. 



Index of the rakk Numerical value of 



OF teems of the geo- the successive terms of 



metrical progression, the geometrical progres- 



starting from the first. sion. 



100000000.0000000 



1.0000000 



1 9999999.0000000 



0.9999999 



2 9999998.0000001 



0.9999998 



3 9999997.0000003 



0.9999997 



4 9999996.0000006 



And so on to the 100th term, which 

 will be 



100 9999900.0004950 



Here is precisely the first table formed by Napier ; I have only 

 found it necessary to copy it out in order to give an exact idea 

 of his method. We may apply to the terms which compose it 

 all the properties demonstrated by Archimedes in his geometri- 

 cal progressions, and obtain the same simplifications of the mul- 

 tiplying and dividing them together. But slow as is the ratio 

 of the progression here employed, it is still but the expression 

 of an intermitting change, while the definition of a logarithm 

 requires that we determine the indices of the rank, which cor- 



