TRANSACTIONS OF THE SECTIONS. 



25 



By this multiplication we happen to have destroyed the third signilicant figure as 

 well as the second. To destroy the fourth, multiply again by 1 — '0006 ; in other 

 words, subtract six times the number from itself four places in advance. We should 

 next multiply by 1 — -00006 and 1 - -000003 ; and, after what has been said, the 

 process will be intelligible without further explanation : 



l-OOOIGG. 37160 000 

 10 00.3198 2.56 



1-000 J 3361 j 74 4 



6000.3180 2 



1-000 0|33o7 9412 

 13000 010 



1-000 00 0357 932 



The next factor required would be 1 - -0000003 ; but it is evident that multiplica- 

 tion by this factor would not aftect the twelfth place of decimals, and consequently 

 the last six significant figures tlius obtained represent, without any further work, 

 the remaining factors required. 

 It is thus shown that 



or that, to the requisite degree of accuracy, 543839 can he expressed as a fraction, 

 the numerator of -which is 



10= X 5, 

 and the denominator 





The method of applying the positive table to find antilogarithms is better known, 

 and need not here be explained. 



It is further evident that we may by similar means express in terms of the neo-a- 

 tive factors the concluding figm-es of any number, or any decimal addition made to 

 a given number. Thus, suppose we know the logarithms of 543 to 12 places, and 

 wish to know that of 543-839, we operate on the latter number as follows : — 



543 8 3 9;000|0 00;0 



is 438 391 I 



543 2 



9516 

 7164 



543023 5134 



2|l7209 



10 000 

 75 80^5 

 195 

 405 



X-999 

 X -999,5 

 X -999,96 



643 00 1,7 924 79 X -999,997 

 ll6 2 90 0514 



5430001 

 1 



6 3 4 73 6 

 629000 



X3 



X-0,1 



5 736 

 5 430 



30 6^ X-0,5 



272' 

 34; X6 



