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



the entry is somewhat inconvenient. The real improvement 

 effected by Burckhardt was to so print the table that t'le three 

 groups could all appear on one page, and that therefore the end- 

 figures of the arguments were the same for the same line on each 

 page of the volume. 



The arrangement in Burckhardt's table, by which the least 

 factors for 9,000 numbers are given in the space of half a square foot, 

 is so compendious and clear that it seems scarcely to admit of any 

 further improvement. 



The least factor being given, in order to find the other factors 

 it is necessary to divide the number by the least factor; and to 

 facilitate this division Burckhardt gives (on one page) the first 

 nine multiples of the primes from 1 to 1,423. It is to be 

 observed that with the aid of Chernac's table only one division 

 will ever be required, as a single division will bring any number 

 not exceeding seven millions within the limits of Chernac's table. 



In the introduction to the million Burckhardt explains very 

 briefly the mode of construction of the table. He refers to Hin- 

 denburg's method as described in his book (see § 9), and points 

 out that, although it is very simple, it is open to the objection 

 that half the work is absolutely wasted, for, in the table as printed 

 the multiples of 3 and 5 are rejected, and it is therefore necessary 

 to copy again the part of the work that is retained. To avoid 

 this Burckhardt employed a form containing 77 x 80 squares, and 

 a series of sieves or perforated cards; in this way he obtained 

 his table in the exact form in which it is printed. This method 

 will be fully described in § 20, which contains an account of the 

 mode of formation of the factor table for the fourth million. 



In 181 G Burckhardt published the continuation from 2,028,000 

 to 3,036,000, There is no introduction, and the million is exactly 

 similar to the previous one. 



§ 16. In the following year (1817) Burckhardt published the 

 first million (1 — 1,020,000), the arrangement being the same ; 

 so that the three millions (1 — 3,036,000) form a handsome quarto 

 volume of about 350 pages \ All the prime factors for the first 

 million had been given by Chernac, and Burckhardt states in the 

 preface that he had intended to finish his volume with the fourth 

 million, but that ' reasons of every kind ' obliged him to prefer 

 the first million. He adds in a note, " Si la vente de ces trois pre- 

 miers millions paraissait assez favorable au Libraire pour qu'il 

 crut pouvoir se charger de I'impression des 4^ 5® et 6^ million, 

 je n'aurais que peu de chose a faire pour achever le manuscrit." 



^ The three millions were also issued together in one Tolume, with the title-page 

 " Table des diviseurs pour tons les nombres des l^"", 2" et 3® million." In these 

 copies the introduction to the second million immediately follows the preface to the 

 first million at the beginning of the volume. 



