as EEPORT— 1873. 



sufficiently favourable to induce the bookseller to undertake them. There 

 are three pages on the use of the tables. This work, though containing the 

 first million, was published after the second and third. 



Five errors are pointed out at the beginning of Case's ' Seventh Million.' 



Burckhardt (Second Million), 1814. The arrangement is the same as for 

 the first million ; and the table extends from 1,020,000 to 2,028,000. This 

 was the first published of the three millions ; and the method of calculation &c. 

 is explained in the introduction, the least factor alone being given. If the 

 others are required, the process is of course to divide the number by this factor 

 and enter the table again with the quotient. To facilitate the division, on 

 the first page (p. viii) a table is given of the first 9 multiples of all primes 

 to 1423. 



Burckhardt (Third Million), 1816. The arrangement is the same as in 

 the other millions : the table extends from 2,028,000 to 3,036,000. 



Rees'sCyclop8edia(vol.xxviii. Art. 'Prime Numbers'), 1819. Attached 

 to the article "Prime Numbers" in liees's ' Cyclopfcdia,' is a table of 23 pp., 

 giving a list of primes up to 217,219 arranged in decades — a very convenient 

 table, as there are 910 primes on each page. It is stated (and truly) that the 

 primes are given to twice the extent that they are to be found in any previous 

 English work. In the course of the article the author says, "And a work lately 

 published in Holland, not only contains the prime numbers up to ] ,000,000, 

 but also the factors of all composite numbers to the same extent — a performance 

 which, it must be allowed, displays the industry of its author to much more 

 advantage than either his genius or judgement." Tliis can only refer to Chbh- 

 NAc's table, which was published at Dcvcnter (Davcntria) in 1811 ; and it is a 

 matter of regret that an English writer on mathematics should have thought 

 only deserving of a sneer a work the performance and extension of which 

 had been consistently urged by Euler and Lambert and afterwards by Gauss. 

 One would expect the article of such a writer on the theoiy of numbers to bo 

 very poor ; and such is the case. He has not thought it worth while to 

 state where the table he gives has been copied from ; it is no doubt taken 

 from Vega (' Tabulje '), 1797, Vol. II. T. I. 



Dase (Seventh Million), 1862. The least divisor of all numbers from 

 6,000,001 to 7,002,000 (multiples of 2, 3 and 5 excluded), and therefore 

 also a table of primes between these limits. 



The arrangement is as in Bueckhaedt, there being 9000 numbers to the 

 page. 



This work was undertaken by Dase at the suggestion of Gauss ; and the letter 

 of the latter is printed in the preface. In it Gauss adverts to, and expresses 

 his concurrence in, Felkel's desire Ihat the factorial tables should be extended 

 to ten millions ; he states that a manuscript containing the fourth, fifth, and 

 sixth millions (viz. 3,000,000 to 6,000,000) was some years before presented 

 by Crelle to the Berlin Academy, and he expresses a hope that it will soon be 

 jrablished ; he therefore suggests that Dase should complete the portion 

 from 6,000,000 to 10,000,000. Dase accordingly undertook the Avork, and 

 at the time of his death in 1862 had finished the seventh miUion entirely 

 and the eighth million nearly ; while many factors for the ninth and tenth 

 millions had been determined. The seventh million (as also the two follow- 

 ing) were published after Dase's death by a committee of his fellow-towns- 

 men as a memorial of his talent for calculation. 



Dase (Eighth MiUion), 1863. The arrangement is the same as in the 

 seventh million ; and the table extends from 7,002,001 to 8,010,000 ; the 

 paging runs from 113 to 224. 



