280 M. Biot's Abstract of Mr Napier's 



bers, so that the progress of multiplying and dividing those 

 numbers together may be superseded by the mutual addition 

 and subtraction of the corresponding indices. But how is iivery 

 number to be comprehended in the same geometrical series con- 

 tinually progressing by equal ratios ? It is precisely in this 

 question that Napier's fundamental idea consists. It was only 

 necessary to make that common ratio of increase so little removed 

 from equality that the progression would march by steps exces- 

 sively slow ; whereby any given number, if it did not fall exactly 

 upon one of the terms of the progression, would at least be found 

 comprehended betwixt two terms so slightly differing from each 

 other that the error might go for nothing ; or, still better, as 

 Napier did, it was only necessary to conceive the idea of the 

 corresponding geometrical and arithmetical progressions being 

 engendered by the continuous motion of two moveable points, 

 starting together at the same time, but the one marching by a 

 geometrical acceleration, the other with a movement always 

 equi-different and uniform. The simultaneous position of these 

 two moveable points at any given instant of the progressions, 

 will give, in the geometrical progression, the number, in the 

 arithmetical, the corresponding index or logarithm. 



But this simple idea presented, in the attempt to realize it, a 

 great practical difficulty. In order to form the successive terms 

 of the geometrical progression, it is necessary to multiply them 

 successively by their common ratio, as often as there are units 

 in the index of the terms ; and here we are again plunged into 

 calculations by multiplication, precisely what we wished to escape 

 from. Napier extricated himself from this embarrassment by an 

 expedient very simple, and replete with ingenuity. He formed 

 his geometrical progression in the descending scale, from large 

 numbers to less, instead of mounting from small numbers to 

 great as Archimedes did ; and he took for the constant ratio of 

 the successive terms, that of 10 to 9, or of 100 to 99, or of 1000 

 to 999, or generally that of a whole power of 10 to the same 

 power diminished by unity. Thus each term could be derived 

 from the preceding one by simple substraction ; for, if the first 

 term be, for example, 10000000, and the second 9999999, this 

 last is obtained by cutting off a unit from the former, that is to 

 say, its millionth part. The third term is derived from the 



