1878.] 31}' Glaisher, On factor -table?, 137 



The principle of the method is to multiply the given prime (sup- 

 posed to consist of 4, 5 or 6 figures) by such a factor that the 

 product may be a number within the range of the factor tables, 

 and such that when increased by 1 or 2 the prime factors may be 

 all within the range of the logarithmic tables. The logarithm is 

 then obtained by use of the formula 



d d? d^ 

 log (a? + 1?) = log ^ + - - 1 -:, + Ks - &c., 



tjU li/ lA^ 



d 

 in which of course the object is to render -.as small as possible. 



If the number be incommensurable, or consist of more than 

 seven figures, we can take the first seven figures of it (or multiply 

 or divide the number by any factor, and take the first seven 

 figures of the result), and proceed as before. An application to 

 the hyperbolic logarithm of tt is given by Burckhardt in the In- 

 troduction to his second million (see § 15). 



I have often employed this method, and even with the gap from 

 three millions to six millions I have never found any difficulty in 

 rapidly obtaining the logarithms I required. Of course there are 

 methods which may be more expeditious, if the calculator is 

 thoroughly conversant with them and accustomed to their use. 

 But if it is only occasionally that a logarithm has to be calculated, 

 the factor method possesses great advantages ; for there is no 

 rule to be remembered, and the reasoning is so elementary that 

 there can be no doubt as to whether the principles have been 

 correctly applied. Whenever a rule or method is but rarely 

 used, great care is necessary in applying it, unless the reason for it 

 is so self-evident that the work of itself shows that no error in 

 principle has been committed. 



In the calculation of a table that is to extend to n places it is 

 usual to calculate the values to n+ 2 places, and if the last two 

 figures are 50, and it is desired that the last figure retained shall 

 be always the nearest to the truth, it becomes necessary to extend 

 the calculation in this case to w + 3 or w + 4 places, which probably 

 exceeds the number of places for which the logarithmic tables are 

 available. Recourse must then be had to some method of cal- 

 culating logarithms, and it is in occasional cases of this kind or 

 such as that mentioned above that I have found the factor method 

 so convenient. 



CONTENTS OF THE FOREGOING PAPER. 



PAGE 



§ 1. Determination of whether a number be prime. Construction and value 



of a factor table .99 



§ 2. Use of a factor table 102 



§ 3. List of factor tables prior to 1811 103 



