1878.] i/r Glaisher, On factor tables. 107 



If, therefore, we liad met with K)S among the tabuljar results 

 (as would first have been the case with the number 992,551 

 had the table extended so far), we should have looked for the 

 block A), which we should have found to be the block 141, and 

 looking for >S in this block we should have found 793 — S, showing 

 that A) 8 denoted the prime 141,793. Thus for primes above 

 20,353 the table has not only to show that the number is a prime, 

 but also to define it by two letters, and this in such a way that if 

 the letters be given, we can readily obtain from them the corre- 

 sponding prime. 



On the first page of the principal table (1 to 6,000) the last 

 three figures of the argument are always given with the tabular 

 result (the primes are distinguished by the three figures being- 

 followed by a hyphen, and also by their having only one letter given 

 as tabular result), so that this first page might be used in place of 

 table A ; also the contents of table A are, in fact, included in 

 the principal table if we choose to use it to find the primes repre- 

 sented by the pairs of letters up to 20,353 in the same way 

 that it is used when the prime exceeds this number. 



The method of arrangement is very ingenious, but it will be 

 seen that the mode of entry is complicated, and that as the factors 

 need interpreting afterwards there is great danger of error. There 

 are four operations needed, (1) divide by 6, (2) enter table B, 

 (3) enter the principal table, (4) interpret the factors. Also, as 

 the table proceeds, the symbols for the primes become embarrass- 

 ing, as when all the simple combinations are exhausted, brackets, 

 &c. are used as in {^^, [F] a, (f) 33. The appearance of the pages 

 of the table, containing figures and letters from so many alphabets 

 in juxtaposition, is very remarkable. The arrangement in the 

 eight columns a, b, ... g, h depends on the fact that all numbers 

 not divisible by 2, 3 or 5 must be of the form ZOq-\-r, where 

 r= 1, 7, 11, 13, 17, 19, 23 or 29, the a column containing numbers 

 of the form 30^' + 1, the b column numbers of the form '^Oq + 7, 

 &c. ; and this explains the division by 6. A division by 3 would 

 suffice, but the division by 6 shows in addition on which half of 

 the page the result will be found in the principal table. The 

 pages of the principal table are numbered from 1 to 24, corre- 

 sponding to the numbers 1 to 144,000, and then the half-pages are 

 numbered 1 to 88, corresponding to the numbers 144,000 to 408,000 \ 



1 The arrangement in eight columns headed 1, 7, 11, 13, 17, 19, 23, 29, so 

 that the first column contains multiples of the form SOg + 1, the second multiples 

 of the form 30(j' + 7 &c. is also that adopted hy Euler in his memoir : " De tabula 

 numerorum primorum, usque ad millionem et ultra contluuanda ; in qua simul 

 omnium numerorum non jDrimorum minimi divisores exprimautur." {Opera 

 minora coUecta vol. ii. pp. 64 — 91. ) The memoir originally appeared in vol. xix. 

 of the Novi Commentarii PetropoUtani (1774), but I do not know whether the 

 arrangement — a very natural one for a table from ^Yhich multiples of 2, 8 and 



