1881.] computing Natural and Tabular Logarithms, Sfc. 393 



tions, twelve places with, short corrections, and fourteen to eighteen 

 places with full corrections. 



Rule. — Proceed precisely as for natural logarithms, except that 

 instead of multiplying by 2 it is necessary to multiply by the tabular 

 bimodulus, by help of the multiples given in No. 3. 



Tables I and II. Rule to find the number from the logarithm. — Sub- 

 tract the logarithm of the next lower power of 10, and then, in order, 

 the next lower logarithm in the lower, and then that in the upper 

 part of the table " 2. For Preparation," and afterwards the next lower 

 logarithm in the table for interpolation. ■ « 



Considering this as an approximate logarithm of a reduced number, 

 find the correction as if it were a quotient by No. 5 or 6, and subtract 

 (instead of adding) the correction, which reduces it to the form of a 

 quotient or approximate logarithm. 



Add the resulting number to and subtract it from the bimodulus 

 (which is 2 for natural logarithms) and divide the sum by the 

 difference. 



Multiply the quotient in succession by the numbers corresponding 

 to the logarithms subtracted. The result is the number required. 

 Examples, fully worked out, with explanations. 



Let N=192 699 928 576 = (76) 6 . 



Then calculating the value of 6 nat. log 76 from Wolfram's tables 

 appended to Vega's, and multiplying the result by the tabular 

 modulus we find to twenty places — 



nat. log N= 25-984 400 041 717 986' 473 06 

 tab. log N -=11*284 881 553 684 748 111 78 



These numbers serve as checks to the correctness of the folio wing- 

 work. 



Here a, b, c form the "preparation" of N. As a begins with 1*9, 

 where the first decimal place is more than 3 times the integer, a is 

 multiplied by 6 to produce 11*56 . . ., a decimal fraction of which the 

 integer 11 is less than 13 and more than twice the first integer 5. 

 Both 5 and 4 would have also answered. The divisor 11 is separated 

 off by ), and in the quotient c the next less number 1*051 in the table 

 for interpolation is similarly separated. This leaves c— 1*051 to the 

 right of ), with the decimal point already omitted. Then this diffe- 

 rence is multiplied by the bimodulus 2, to obtain the dividend d. The 

 whole of c is added to the separated part 1*051, and then the decimal 

 point is omitted, giving e. As the difference c = *0 4 905, lies between 

 *0 3 106 and *0 4 493, we can certainly obtain twelve places without cor- 

 rection (Tahle I, No. 4), and as it lies between *0 3 1 and *0 4 894 we can 

 obtain seventeen places with full corrections (Table I, No. 5). We 



2 f 2 



