282 M. Biof s Abstract of' Mr Napier s 



respond to the same terms engendered by a continuous move- 

 ment. Napier did not obtain the absolute expression of that 

 rectification, as in the present day we can do by means of 

 our differential methods, which enable us to pass without error 

 from discontinuity to continuity. But in comparing the essen- 

 tial conditions of the continuous movement with those of the 

 intermitting change, he establishes assignable limits between 

 which the logarithm of a given number is always comprised ; 

 so that if the difference betwixt these two limits is only beyond 

 the order of decimals which we care to keep, either of these 

 limits, or still better, their medium may be legitimately taken 

 for an expression sufficiently approximating to the logarithm 

 sought. Applying this principle to the table, he shews that 

 the logarithm of the first term 9999999 is necessarily com- 

 prised betwixt 1.0000000 and 1.0000001, wherefore he takes 

 it as equal to 1.0000005 ; now the precise value of that loga- 

 rithm, calculated by the methods now in use, is 1.00000 00500 

 00003 333, so that Napier's valuation of it is only in error by 

 the third of a unit on the fourteenth decimal of that logarithm. 

 This, then, is the first term of his arithmetical progression 

 corresponding to the geometrical progression which he adopts ; 

 and in multiplying them by the series of numbers, 1, 2, 3, &c., 

 which mark the successive rank of the terms of that geometrical 

 progression, he obtains the indices, that is to say, the logarithms 

 of all those terms. This substantially is his mode of operation, 

 and, with some abbreviations, he leads his table of corresponding 

 progressions from 10000000 to 50000000, so as to obtain a 

 numerical progress decreasing in the ratio of 2 to 1 Then, any 

 number being assigned comprised within these limits, he shews 

 how its logarithm may be directly obtained of the requisite 

 approximation, by a comparison with the two terms of the geo- 

 metrical progression between which it lies. If the given 

 number is without the limits of the table, he shews how to bring 

 it within, and obtain its logarithm. Thus the general problem 

 of intercalating every number precisely, or by approximation, 

 in the same geometrical progression, is completely solved ; and 

 thus for every possible multiplication or division of these num- 

 bers with each other, there is obtained the same facilities, the 

 same simplifications, which Archimedes had discovered for the 



