524 MR SANG'S NEW TABLE OF LOGARITHMS TO 200 000. 



From this example, the general principles which regulated the actual course 

 of proceeding may be understood. Two things have to be kept in view when 

 seeking for a convenient way of getting at the logarithm of a proposed prime 

 number, one to get an easy divisor, the other to obtain by a change of sign the 

 logarithm of some other number not previously found, preferably a prime 

 number. 



For example, we have filled the list of prime numbers up to 29, the logarithm 

 of which has now to be found. Our first business is to search for some 

 multiple of 29 which ends in 0001, or in 9999, in order that the divisor terminate 

 in zeroes, the more the better ; 29 ends in 9, and therefore we may use the divisor 

 30, which would also give us the logarithm of 31 ; this divisor, however, is too 

 slow, so we carry on our search thus : — 



1 x 29 = 29 

 30 x 29 = 870 



31 x 29 = 899 

 900 x 29 = 26 100 



931 x 29 = 26 999 



There the divisor 900 would have clone, provided the logarithm of 31 had 

 been known ; wherefore we proceed another step, which brings us to the divisor 

 27 000 ; this divisor is available if the logarithm of 931 be known. On turning 

 to the filled-in table of natural numbers, we find the logarithm of 931 there; it 

 had come from the product 7719. From 27 000 we also get 27 001, and there- 

 fore inquire whether this be a prime or a composite number. This research in 

 itself would have been enormously tedious, so much so that any saving from the 

 discovery of the factors would have been but a small set-off against the labour 

 expended. That most admirable table, however, of the Divisors of Numbers 

 constructed by Burkhardt makes the matter easy;* it shows us that 27 001 is 

 the product of 1331 67; so that once the logarithm of 31 is found, that of 67 



also may be obtained. Wherefore, making ,#=-——, we obtain log 29 and log 

 27 001. 



Subsequently making &=— — we have log 31, log 901, and log 53; and 



thence again log 97. 



Here it may be proper for me to bear testimony to the great value of 

 Burkhardt's work, which contains the divisors of all numbers up to three 

 millions. The prodigious amount of labour, in the face of an expected small 



* la this particular instance we might have done without Bukkhardt's help, because 27 001 

 = 30 3 -J-l 3 and so is divisible by 30 + 1. 



