1881.] computing Natural and Tabular Logarithms, Sfc, 383 



(5), 



M2 



And putting for x and z their values from (2) we find 



(7) 



And by expanding the first of these equations (7) 



(8). 



Subtracting (8) from (3), and multiplying by M to find the Mc of 

 (6), we have 



a converging series of which the limits are the first term and the first 

 two terms. 



Preparation. — To insure p being small in all cases, I have invented 

 the rule of preparation, founded on the fact that if N be the number 

 whose logarithm is sought, and a and b any two numbers of which 

 the logarithms are known, such that ~Na-l-b=n + d, where n is the 

 next less number to ~Na -4- b in the table, and d, the difference, is 

 less than the difference between two numbers in the table, then 

 log N = log (n + d) +log b— log a. In Tables I and II the difference 

 between two consecutive numbers is "001, and as there are 100 entries, 

 all the numbers lie between 1 and 1*1 ; so that if Na-^& is less than 

 1*1, the required reduction is effected. 



Preparation is accomplished in two lines of simple multiplication 

 and division, as follows : — 



The given number N" is divided or multiplied by such a power of 10 

 as will leave the quotient or product as a decimal fraction between 1 

 and 10. This is effected by simply shifting the decimal point. 



If the first decimal place is Jess than 3 times the integer (which is 

 always the case when the integer exceeds 3), divide by the integer and 

 divide the quotient by 1*1 or 1*2. The result is less than 1*1. 



If the first decimal place is more than twice the integer, then it is 

 always possible, generally in several ways, to find an integer between 

 1 and 10 which, used as a multiplier, will give a product of which the 

 integer is less than 13, and the first decimal place less than the integer. 



The following rule embraces every case : — Multiply any of the numbers 

 1-30 to 1-340 by 4; 1-340 to 1-80 by 7 ; 1-80 to 1-960 by 5 ; 1-960 to 

 1-99 by 6 ; 2*50 to 2'99 by 4 ; 3-80 to 3-99 by 3. Then dividiug this 

 product by the integer the quotient is less than 1*1. 



This preparation is very convenient also for starting Weddle's and 



