168 Memoir of John Napier, 



which express only the rank of each term, increase only 

 by one unit, and always by one unit in passing from one 

 term to the following term, which constitutes another kind 

 of progression called progression by equi-difference, or arith- 

 metical. Archimedes recognized and proved all that we 

 have explained respecting the relations of two similar pro- 

 gressions, when these terms were thus placed in corre- 

 spondence. And, in order to show that these properties 

 took place for any terms of the two series, he thought of 

 representing generally these terms by letters employed only 

 as signs of quantities without any peculiar numerical value, 

 giving thus the first example of reasoning applied to figured 

 symbols, representing abstractions in which algebra consists, 

 that powerful instrument of the mind for discovering the 

 general relations of great sums. 



From this to logarithms there is only a step, and even 

 logarithms are only indices employed after the manner of 

 Archimedes to express the rank of each number in an in- 

 definite geometrical series which comprises them all ; so 

 that their multiplication and division with each other may 

 be re-placed by addition, or the mutual subtraction of indices 

 which correspond with them. But how can we comprehend 

 all the numbers in the same geometrical series, proceeding 

 continually by equal proportions? It is in this that the 

 fundamental idea of Napier consists. To make this propor- 

 tion common, if near equality, it is necessary only that the 

 progression should proceed by very slow steps, so that any 

 number given, if it does not fall upon one of the terms of 

 the progression, is found at least comprised between two 

 terms differing so little from each other, that the error may 

 be neglected ; or better still, it is only necessary to repre- 

 sent, as Napier did, geometrical progression and the cor- 

 responding arithmetical progression, as produced by the 

 continued motion of two moving bodies proceeding from a 

 state of rest, and advancing, the one with a geometrical 

 acceleration, the other with an equi-different and uniform 

 motion. The simultaneous positions of the two moving 

 bodies at any time will give in geometrical progression the 

 number, in arithmetical progression the index or the loga- 

 rithm which corresponds to it. 



But this idea, simple though it is, presents, in the execu- 



