136 Mr Glaislier, On factor tables. [Feb. 11, 



identical with those employed by Burckhardt, and the calculation 

 of the million suggested no improvements upon them, except in 

 a few matters of detail. The construction of the table, though 

 very simple in theory, required such continual care at every 

 step, and such constant supervision, that it could not be under- 

 taken by any one who was not prepared to devote a great portion 

 of his time to the work. 



§ 21 (Appendix to § 2). I have always found that the factor 

 tables of Burckhardt and Dase afforded the best practical method 

 of obtaining an isolated logarithm to more places of decimals than 

 can be obtained directly from the tables. The principle of the 

 method is best exhibited by an example. I required the hyper- 

 bolic logarithms of the first eleven Bernoullian numbers to 24 

 places of decimals. Of these the numerators and denominators 

 are all composed of prime factors less than 10,009 (the limit of 

 Wolfram's table ^), except the numerator of the ninth Ber- 

 noullian number, which is the j)rime number 43,867. Now 

 50 X 43,867= 2,193,350, and on looking in Burckhardt for a num- 

 ber near to this, which shall have no prime factor greater than 

 10,009, it appears that 



2,193,349 = 23 x 47 x 2,209, 

 and thus 



4.3,867 = sV (23 x 47 x 2,029 + 1), 

 and therefore 



log 43,867 = log 23 + log 47 + log 2,029 - log 50 



111 1 1 



^ 2,193,349 2 (2,193,349)^ ^ ^ (2,193,349/ 



The first term of the series in the second line 



= 0-0000 0045 5923 7950 7319 6286; 

 dividing this by 2 x 2,193,349 we obtain 



0-0000 0000 0000 1039 3325 3457, 

 and the third term is 



0-0000 0000 0000 0000 0003 1590, 

 so that the series 



= 00000 0045 5923 6911 3997 4419; 

 whence, taking out the logarithms from Wolfram's tables, 



hyp. log 43,867 =10-6889 1760 7960 5681 0191 3661. 



1 Wolfram's table gives hyperbolic logarithms of all numbers up to 2,200, and 

 of primes (as well as of a gi-eat many composite numbers) up to 10,009, to 48 

 decimal-places. It first appeared in Schulze's Sammlung (1778), and was reprinted 

 in Vega's Thesaurus (1794). 



