1881.] as a Means of Calculating Logarithms, §c. 



399 



constructed to the requisite number of decimal places, logarithms and 

 anti-logarithms can be calculated by various methods. The improved 

 bimodular method requires a much less extended radix for the same 

 number of decimal places than any other method. Thus, to find tab. 

 log N=6 tab. log 76=tab. log 192 699 928 576, the example used 

 in my former paper. The requisite radix logarithms are assumed 

 from Gray and Thoman. The details of division and multiplication 

 are omitted for brevity. 



First with the positive numerical radix, 



1 -92 699 



928 



576 





a = N^10 n 



9) '63 499 



642 



880 





b = 5a 



1 -07)055 



515 



875 555 555 555 



556 



c=l+9 



1 -000 5)18 839 958 463 136 033 



d=c+l-07 





16 



364 179 999 652 



896 



e=2M(d-l -0005) 



2 '001 



18 



839 958 463 136 



033 



f=d + I -0005 



o -ooo 



08 



177 924 001 951 



217 





•0l5 





241 



645 



h= correction 



•ooo 



217 



092 972 230 208 



282 



=tab. log 1 -0005 



•029 



383 



777 685 209 640 



835 



=tab. log 1 -07 



•301 



029 



995 663 981 195 



214- 



■l==compt. tab. log 5 



•954 



242 



509 439 324 874 590 



— tab. log 9 



11 -o 









=tab. log 10 11 



11 -284 



881 



553 684 748 111 



783 



sum = tab. log N 



For the details of the method see my former paper. The pre- 

 paration is here carried a step further than there, before the inter- 

 polation. We first multiply by 5 and then divide by 9, as in Ex. 2 

 of my former paper, but the division by 9 is now carried to twenty- 

 three places of decimals, and then 1'07, a number in the radix, is 

 separated off as a new divisor of c, giving d, where 1*0005 is separated 

 off as the next less number in the radix. But we might have used it 

 as a divisor, and should have then found l'OO 001 833 078 307 160 

 . . . from which still more decimal places could be obtained in the 

 final result. But stopping at d, we form the dividend e and divisor/, 

 and then find the quotient g. The correction is obtained from the 

 formula in my former paper, Table II, No. 5. But as these correc- 

 tions involve the use of other tables, they would be illegitimate in 

 first constructions, which would give only fourteen decimal places 

 correct in place of twenty. The rest is as usual. 



The disadvantage of this method by the positive numerical radix, is 



