Instead of commencing with 3141 as the first factor, 3140 or 3139 could 

 have been used. In general, any number may be used as the first factor pro- 

 vided the first eight digits of the resulting quotient fall within the interval 

 of the argument for the second column of the table. When checking the result 

 obtained from one series of values, there is a definite advantage in employing 

 an entirely dififerent series. This provides for checking the work of the com- 

 puter and the accuracy of these tables. 



In finding the common logarithm of a number, the procedure is much the 

 same as that for finding the natural logarithm. Table 2, containing the com- 

 mon logarithms, will be used. The characteristic of the common logarithm is 

 obtained in the customary manner employed when tables involving interpola- 

 tion methods are utilized. The only significant difference is found in the com- 

 puted value of the logarithm of the fourth factor. 



To emphasize the modification required when obtaining a common logarithm, 

 the following example will illustrate the method in finding the log of tt. The 

 four factors are the same, and the logarithms of the first three factors are 

 found in the first, second, and third columns, respectively, of table 2. In the 

 previous example the In of the fourth factor was found to be 0.0^^5638,28195, 

 88052. This result must be multiplied by the modulus, 0.43429 . . . , and 

 their product rounded to 23 decimal places. Hence : 



log 1.0^5638,28195,88209= (0.0^^5638,28195,88209-0.0-157) 

 (0.43429,44819,03251,82765,11829) =0.0^^2448,67474,212. 



Therefore, the log of tt follows : 



log 3.141 =0.49706,79363,98504,77113,023 



log 1.0n886= 8,19002,16339,29522,736 



log 1.0^8307= 360,76841,11325,155 



log 1.0^^5638...= 2448,67474,212 



Thus the log TT =0.49714,98726,94133,85435,126 

 in which there is an error of one unit in the twenty-third decimal place. 



INVERSE FACTORIZATION 



An antilogarithm is found by reversing the process. From the logarithm 

 proposed, the natural logarithms of the numbers XXXX, l.O^XXXX, etc., 

 are subtracted in succession. The product of these numbers gives the number 

 sought. 



In most cases the value of In N should first be brought into that part of 

 the table where N lies between 1.000 and 10,000. This may be accomplished 

 by adding or subtracting a multiple of In 10 which only affects the position 

 of the decimal point in the value of N. The proper multiple of /;/ 10 to be 

 added or subtracted may be readily determined if the computer knows only 

 the approximate value for the In 10 = 2.3, In 10- = 4.6, In 10^ = 6.9, In 10* = 9.2, 

 etc. 



In the example that follows it is again assumed that the computer has 

 access to a calculating machine. 



