In both tables of this volume, the error in the twenty-third place does not 

 exceed 0.5. The maximum error, therefore, which can occur in a result 

 obtained by adding together three logarithms from the table and the com- 

 puted logarithm of the fourth factor will be less than 4(0.5), i.e. less than two 

 units in the twenty-third place. 



DIRECT FACTORIZATION 



When the natural logarithm of a number of more than four significant digits 

 is desired, the number must be broken up into factors, the first of which con- 

 sists of the number formed by the first four digits. The others are the three 

 numbers l.O^XXXX, 1.0'XXXX, and l.O^^XXXX . . . formed from the 

 first 8, 12, and 23 digits, respectively, of the quotient that results when the 

 division is performed with the preceding factor as divisor and commencing 

 with the first four digits. The logarithms of the first three factors are taken 

 from the first, second, and third columns, respectively, of table 1. The loga- 

 rithm of the last factor, 1.0"XXXX . . . , is the decimal O.O^^XXXX . . . , 

 to 23 places, minus a correction factor that may be obtained by a simple 

 mental calculation. The sum of these logarithms is the desired logarithm. 



It is assumed, in the example that follows, that the computer has access to 

 a calculating machine. 



Example : To find btir, given the value 



TT = 3. 141 59,26535,89793,23846,264 



Solution : 



7r= 3.141 59,26535,89793,23846,264 

 = 3.141 X 1.0n8,86830,91306,34780,72588 

 = 3.141 X 1. on 886 X 1.0^830,75638,28242,71930 

 = 3.141 X 1.0n886x 1.0^8307 X 1.01^5638,28195,88209 

 = 3141 X 1.0^886 X 1.0^8307 X 1. 0^5638,28195,88209 X 10-3 



The above are the factors of ir identical, respectively, with 



XXXXxl.O^XXXXxl.O^XXXXxl.Oi^XXXX . . . 



In practice it is obviously not necessary to record all the data given above. 

 The /« of the first three factors may be obtained from the three columns of 

 table 1. The /«( 1.01^5638,28195,88209) =0.0i^S638,28195,88209-0.0--157 = 

 0.0i'5638,28195,881, from the series /M(l+.r)=.v-,r72+ . . . in which only 

 the first two terms are significant. From table 1 : 



In 3141 =8.05229,64995,38646,54598,997 



In 1.0^886= 18,85822,17255,84856,087 



In 1.0^8307= 830,69996,54968,774 



In 1.0115638....= 5638,28195,881 



8.05248,5 1648,3 1 537,22619,739 

 (minus) In 10^ =6.90775,52789,82137,05205,397 



Therefore, In n = 1.14472,98858,49400,17414,342 



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



