100 Mr Glaisher, On factor tables. [Feb. 11, 



Start from 2, and strike out every second number ; we thus 

 reject all the multiples of 2, excejDt 2 itself; start from 3, and 

 strike out every third figure, and we thus reject multiples of 3, 

 and similarly for 5, 7, 11, &c. In this way, all the numbers not 

 divisible by a factor less than any given prime p can be found ; 

 or in other words, we thus form a list of primes up to p^. The 

 same process, which is quite mechanical, also gives all the prime 

 factors of each composite number. For, take a pair of compasses, 

 and open them to such an extent that the points are separated by 

 a distance equal to twice the distance between two consecutive 

 numbers ; if, starting with the number 2, the compasses be 

 'stepped' along the line of numbers, we have all the multiples 

 of 2 marked, and have only to write down 2 opposite to each 

 number; similarly, if the points of the compasses be separated by 

 a distance equal to 3, we have all the multiples of 3 marked, and 

 so on. In this way we obtain a factor table as shown below, the 

 primes being distinguished by their having blanks opposite to 

 them^: 



iro CO !>. io CO »o Jt- 



(M lO G<f «<l CO (M G<r (m" CO" (M (m" G<f CO" 



p-iG<>CO'*»0 01>.OOCiOr-lMCO'*»OC01>-COCiOT-i... 

 i-Hi— li— li— IrHrHr— li— li— lr-l(MCM 



This is the principle of the mode of construction of all factor 

 tables and lists of primes ; but the process is very greatly modified 

 in the actual methods that have been used. In practice the 

 compasses could not be employed, as the natural numbers must 

 be arranged in columns, instead of being written in one straight 

 line, but methods have been devised in which their use is replaced 

 partly by measurements and partly by calculations. 



In the case of large factor tables, numbers divisible by 2, 3, or 

 5 are omitted, so that every pth number of those in the table is 

 not divisible by p. The law regulating the occurrence of the 

 multiples of ]J is still periodic, but is more complicated, and it 

 is convenient to use, for each prime p, a screen or sieve, i. e. a 

 piece of paper or card, of the proper size, from which certain 

 squares have been cut out according to such a law, that when 

 suitably laid upon the columns of squares representing the argu- 

 ments of the table, those squares which correspond to multiples 

 of the number p shall appear through the holes in the sieve : and 

 the factor p is accordingly to be entered in each square, or in 

 each empty square, that appears through the sieve, according as 



1 Considering only the theory of the table, it would be better, in using the 

 compasses, to always start from unity, so that a prime would be recognised by 

 its having only one factor opposite to it, and that factor equal to the number : 

 but, practically, the insertion of these factors in the table would be useless, and 

 very inconvenient. 



