Baron of Merchiston. 171 



the geometrical progression between which it is included. 

 If the proposed number exceeds the limits of the table, he 

 shews how we may obtain the logarithm by causing it to 

 re-enter. 



The general problem, therefore, of determining all num- 

 bers, exactly or approximately, in the same geometrical 

 progression is thus completely resolved ; and then, by mul- 

 tiplying and dividing these numbers the one by the other, 

 whatever they may be, we obtain the same facilities as 

 Archimedes had found for the particular geometrical pro- 

 gression of which he has made use in his work on the num- 

 ber of the sand. Such is the invention of Napier. He has 

 rendered continuous and general for all numbers the ad- 

 vantages which Archimedes had only obtained for inter- 

 mittent and particular numbers. If we ask why Archimedes 

 did not attain this second step, which now to us appears so 

 near the first, we may find, in our opinion, a plausible 

 reason in the nature of the symbolic letters employed in his 

 time to distinguish the numbers. For the signification of 

 these characters being absolute, numbers differing very 

 little from each other were often expressed by characters 

 which had no apparent relation to each other, or, if their 

 expressions possessed common elements, the proportion 

 of the size of the latter to similar ones was not indicated 

 by the same numerical expression ; in place of these two 

 kinds of indication existing and striking, so to speak, our 

 views in our actual mode of writing the number; above all, 

 when generalizing the idea which gives a value of position 

 to the cyphers, we advance in an inverse sense to the sub- 

 divisions of units, by the employment of decimal cyphers. 

 Hence, this is one of the examples of the influence of signs 

 upon the extension of ideas with which the history of ma- 

 thematics abounds. 



We may remark on this subject, that Napier first em- 

 ployed, in Europe, this generalization so simple in the 

 mode of writing decimal sub-divisions, which was indispen- 

 sible to produce successive subtractions. If we wish to con- 

 vince ourselves that this idea was not so easy to discover, 

 as we might think at present, when it has become familiar 

 to us by use, we have only to inspect the complicated and 

 almost impracticable means l»y which Steven, an able and 



