1878.] Mr Glaisher, On factor tables. 101 



all the prime factors, or only the least factors of the numbers, are 

 to be recorded in the table. 



Assuming the necessity for occasionally requiring the factors of 

 a large number, it will be seen that the existence of a factor table 

 of considerable extent is a very important matter ; for the process 

 of determining, without a table, the factors of a number, unless 

 one of them happens to be small, or of proving a number to be 

 prime, is excessively laborious ; the process is also an unsatis- 

 factory one, as in the event of no factor being found that divides 

 the number without remainder, a duplicate calculation will be 

 required in order to render it certain that no error has been 

 made in the divisions. Thus to determine, for example, whether 

 the number 8,559,091 is or is not a prime would require a long 

 day's work, as the number is a prime ; and even then the determi- 

 nation would be less satisfactory than if the result were obtained 

 directly from a table. 



The intrinsic value of a table, qua table, may be regarded as 

 proportional to the actual amount of time saved by the table, 

 whenever there is necessity for consulting it : thus, ex. gr. a table 

 of square roots to 10 decimals is intrinsically more valuable than 

 a table of squares, as the extraction of the root would occupy more 

 time than the multiplication of the number by itself \ There are 

 not many tables that exceed a factor table in intrinsic value. 

 It should be remarked also that the intrinsic value of a factor 

 table is not appreciably diminished if only the least factor be 

 given, instead of all the factors, for we have only to divide the 

 number by the least factor, and enter the table with the quotient 

 as argument, and so on. It is more convenient in some cases to 

 have all the factors given, but the purpose for which the table is 

 constructed is equally fulfilled by giving the least factor as by 

 giving all, and the saving of space, when room has to be found 

 for one factor only, is very great. 



1 The fact of whether a table is easy or difficult to construct does not affect 

 its intrinsic value when completed. When a table is made, it is made for ever, 

 and as far as the user is concerned it is indifferent whether its construction 

 occupied the original calculator for one year or ten years. The value of a table 

 of logarithms, for any given person, consists in the amount of time he saves by its 

 use ; and the work originally devoted to the calculation of logarithms two centuries 

 and a half ago in no way concerns him. 



Eeferring to factor tables, Lambert wrote : " Universalis finis talium tabularum 

 est: ut semel pro semper computetur, quod scepius de novo computandum foret ; 

 et ut pro omni casu computetur, quod in futurum pro quovis casu computation 

 desiderabitur." (Supplementa Tabularum, 1798, p. xv.) 



The theory of the construction of a factor table is simple, but in practice 

 the table is not an easy one to form, the chief difficulty consisting in the incessant 

 care required to avoid making errors, (as an error made, cannot be readily dis- 

 covered and corrected), and the absence of any complete method of verification, 

 such, for example, as is supplied by differences in the case of a table containing 

 the values of a continuous quantity. 



8—2 



