J 70 Memoir of John Napier, 



This is precisely the first tahle formed by Napier, which 

 has been copied for the purpose of giving an accurate idea 

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

 all the properties demonstrated by Archimedes in reference 

 to geometrical progression, and obtain the same simpli- 

 fications by multiplying and dividing them by each other. 

 But, however slow the progression here employed may be, 

 it is still only the expression of an intermittent movement, 

 while the definition of the logarithm requires us to deter- 

 mine the indices of the rank which shall correspond to the 

 same terms produced by a motion completely continued. 

 Napier did not obtain the absolute expression for this cor- 

 rection, as we now can do by our differential methods, 

 which enables us to pass, without error, from intermission 

 to continuity. 



But, in comparing the essential conditions of continued 

 motion with those of intermittent motion, he established 

 measurable limits between which the logarithm of a given 

 number is always comprised ; so that if these two limits 

 exceed the order of the decimals which are to be preserved, 

 we may legitimately take any one, or, what is still better, 

 their mean, as a sufficiently near approximation to the 

 logarithm. Applying this to his table, he shews that the 

 logarithm of the first term 9999999 is necessarily com- 

 prised between 1,0000000 and 1,0000001, so that it becomes 

 equal to 1 ,00000005 ; now, the exact value of the logarithm 

 calculated by our actual methods is 1,00000 00500 00003 

 333, so that the valuation of Napier only errs by one-third 

 of unity above the 14th decimal of this logarithm. It is then 

 the first term of the arithmetical progression corresponding 

 to the geometrical progression which he has adopted ; then 

 by multiplying by the successive numbers 1, 2, 3,&c. which 

 mark the rank consecutively of the terms of this geometri- 

 cal progression, we obtain the indices, that is to say, the 

 logarithms of all these terms. It is in this way that he 

 operates, and with some abbreviations carries on his table of 

 correspondence from 10000000 to 5000000, where its pro- 

 gressive decrease indicates the proportion of 2 to 1. Then, 

 if we point out any number comprised between these limits, 

 he shews how we may obtain its logarithm with the requi- 

 site approximation, by comparing it with the two terms of 



