Baron of Merchiston. 169 



tion, a very great and material difficulty. To form the 

 successive terms of the geometrical proportion, it is neces- 

 sary to multiply them successively by their common pro- 

 portions, as many times as there are units in the index of 

 their rank. Hence, we fall into the calculations of multi- 

 plication, which we wished to avoid. Napier relieves us 

 from the difficulty by a very simple method ; he forms his 

 geometrical progression by descending from the greater to 

 the smaller number, instead of rising from the small to 

 the great as Archimedes did, and he employs for the con- 

 stant proportion of the successive terms that of 10 to 9, or 

 of 100 to 99, or of 1000 to 999, or generally, by an entire 

 power of 10 to this same power diminished by a unit. Then 

 each term may be deduced from the preceding by simple 

 subtraction, for if the first term is, for example, 10000000, 

 and the second 9999999, the latter will be obtained by 

 subtracting unity from the former, which is the millionth 

 part. The third will be deduced from the second in taking 

 from the latter the ten millionth of its value, or 0*9999999, 

 according to our actual decimal notation, and in continuing 

 this proceeding we obtain as many terms as we wish by 

 simple subtraction, which will follow in the same geometri- 

 cal proportion as we shall have chosen. The correspondence 

 of the terms and indices which mark their rank will form 

 the following table, in which the succession is indicated to 

 the hundredth term after the first, carrying the value of 

 each term to the seventh decimal place. 



Index of the rank of the terms of the geome- Numerical value of the successive 

 trical progression departing from the first, terms of the geometrical progression. 



100000000-0000000 



1-0000000 



1 .* . 9999999-0000000 



0-999999 



2 9999998-0000001 



0-9999998 



3 . . 9999997-0000003 



0-9999997 



4 • 9999996-0000006 



and so on to the 100th term which will be 



100 0999900-0004950 



