384 Mr. A. J. Ellis. Improved Bimodular Method of [Feb 3, 



Hearn's processes given by Mr. Peter Gray in the introduction to his 

 Tables for twelve-place logarithms, 1865 (first published in 1845), and 

 is also very much simpler than that proposed by Mons. Thoman in his 

 " Tables de logarithmes a 27 decimales," 1867. 



Interpolation. — The finding of log N is thus made dependent on 

 finding log (n + d), where n is a tabular number and d is less than 

 •001. We then find 2Md-i-(2n + d), which gives the "uncorrected" 

 logarithm of n + d, or the "quotient" x or z. The multiplication 

 2M X d is effected by the multiples of the bimodulus given in the 

 tables, when M is not 1, the unit place of each multiple of 2M being 

 placed immediately below the determining figure of d, care being taken 

 to preserve as many places as are necessary for the final result. The 

 division is a single contracted division. The resulting x or z has to 

 be "corrected " by the equations (5) and (6), as shown in Section III. 



Completion. — Having found log (n + d), we add the logarithm of the 

 power of 10 by which we first divided, and the logarithm of any other 

 divisor, and the arithmetical complement of the logarithm of the 

 power of 10 or any other multiplier. All these logarithms are given 

 in the table. The result is the complete log N to the number of 

 decimal places for which the table is adapted. 



Anti- Logarithms. — A logarithm being given we have to reduce it to 

 the logarithm of a number between 1 and 11. This is most con- 

 veniently done by subtracting from it (or adding to it) the logarithm 

 of the largest power of 10, which will make the result lie between 

 and log 10, and afterwards subtracting the next least logarithm of an 

 integer between 1 and 10, and then the next least logarithm of a number 

 between 1*1 and 2. The logarithms of all these numbers are given in 

 the table. The result will be the logarithm of a number less than 1*1. 

 We then subtract the next less logarithm in the table of interpolation, 

 and obtain the equivalent to the corrected quotient % + c or 2 + Mc of 

 (5) and (6). We find the correction in the same way as for the 

 quotient, and subtract it, thus obtaining x or z. Then we divide the 

 bimodulus increased by this x or z, by the bimodulus decreased by this 

 x or z, as in (7), and thus find which is the number correspond- 



ing to the "quotient" in the direct method. For " completion" this 

 has to be multiplied by the numbers corresponding to all the logarithms 

 subtracted in the preparation. 



Section III. — Calculation of the Bimodular Corrections. 



The principal peculiarity of this improved bimodular method con- 

 sists in the calculation of the corrections and the determination of the 

 number of places which can be trusted in any case, as assigned in the 

 tables. 



The repetition of any digit m times within the same number will 



