1881.] computing Natural and Tabular Logarithms, §c. 397 



The preparation a, b, c, is similar to that in Ex. 1, but 5 is used as 

 the multiplier by way of variety. The difference c — 1*070 being 

 *0 3 55 . . ., which lies between *0 3 653 and *0 3 303, we cannot be certain 

 of more than ten places without correction (Table II, No. 4). As only 

 twelve places are wanted, we use the short corrections and work to 

 thirteen places. The chief peculiarity relates to the multiplication of 

 c — 1*070 by the bimodulus by means of the multiples in Table II, 

 No. 3, omitting all the decimal points. The integer of the multiple is 

 placed under the determining figure of the multiplicand, and the 

 multiple is then written out as far as necessary, neglecting the point, 

 but regulating the last figure. It is best to write in the integer 0, as 

 in line e, to preclude error. As the quotient must begin with *0 3 , only 

 ten significant places are wanted, and hence only eleven places in 

 the divisor g, the four underlined 5556 are therefore rejected. The 

 correction h is found from Table II, No. 6, as belonging" to a quotient 

 between *0 3 217 and 3 232. The rest is sufficiently explained in the 

 notes. The result is accidentally correct to thirteen places. 



Here a is the given tab. log to eighteen places. We first subtract 

 11 log 10, or the characteristic. Next, if the remainder were greater 

 than any logarithm in the lower part of Table II, No. 2, we should 

 subtract that. But in this case it is not, and hence we proceed to 

 the upper part of No. 2, and subtract the next less, or tab. log 1-9. 

 This completes the preparation, as the difference c=a — b, lies between 

 the tab. logs of 1*014 and 1*015 in No. 1, the table for interpolation. 

 Hence, subtracting tab. log 1*014, we find tab. log e, of which the 

 number e has to be found. Now, the formula (7) applies only to an 

 uncorrected z— tab. log e', which cannot differ from tab. log e in the 

 three first significant figures. In the direct process, tab. log e is found 

 from tab. log e' by adding the correction found by Table II, No. 5. 

 Hence we have only to subtract this correction /, which is calculated 

 from the same first six significant figures in both cases, as shown in 

 the example. Having found this uncorrected tab. log e', we add it to 

 and subtract it from, the bimodulus, obtaining k and I respectively, 

 and thus find e=Jc-^-l. Now, tab. log e' cannot be greater than 'the 

 greatest difference between two tab. logs in Table II, No. 1, "for 

 interpolation," that is, it cannot be greater than *000 434 077 . . and 

 hence than *001 X modulus. Hence the result of this division A*-*-/, 

 must be less in any case than (2M + *001 x M)-^(2M— *001 x M) = 

 1*0010005 . . . , and must be greater than 1, hence it must com- 

 mence with 1*000. As the modulus divides out, this conclusion holds 

 for all systems of logarithms. As the last divisor in the contracted 

 division must have two digits for safety, it follows that the number of 

 digits in the quotient k-^l = e, will be one less than the number of 

 digits in the divisor, that is, than the number of decimal places in the 

 given logarithm. And as the first of these digits is a whole number, 



