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



the periods of the reciprocals of primes. This table is reprinted 

 by Jacobi at the beginning of his Canon Arithmeticus (Berlin, 1839), 

 who found it very useful in constructing the tables of which the 

 Canon consists. By a ' tabula Burckhardiana ' is generally meant 

 a table of this kind, and not a factor table. 



§ 17. Gauss always took the greatest interest in factor tables. 

 In a letter to Encke, dated December 24, 1849, and printed on 

 pp. 444 — 447 of the second volume of his Werke, he mentions the 

 enumerations of primes that he made in 1792 or 1793 from Lam- 

 bert's Zusdtze and in 1796 ^ from Vega's tables. He was much 

 rejoiced at the appearance of Chernac's table in 1811, and he and 

 Goldschmidt made enumerations from Chernac's and Burckhardt's 

 tables. 



After speaking of these enumerations he proceeds, " Konnten 

 Sie nicht den jungen Dase veranlassen, dass er die Primzahlen in 

 den folgenden Millionen aus denjenigen bei der Academie befind- 

 lichen Tafeln abzahlte, die wie ich fllrchte das Publicum nicht 

 besitzen soil ? " This sentence is interesting as being probably 

 the first mention of Dase in connexion with factor tables. Zacha- 

 rias Dase had great natural talent for computation, and calculated 

 a large table of seven-decimal hyperbolic logarithms of numbers, 

 arranged like an ordinary seven-figure table of Briggian loga- 

 rithms, which was published at Vienna in 1850. 



On December 9, 1850, Gauss wrote to Dase, to repeat in writing, 

 he states, what he had previously told him by word of mouth. This 

 letter, which forms the chief portion of the introduction to Gauss's 

 seventh million, is interesting and valuable. Gauss first points 

 out the difiiculty of resolving large numbers into their factors, 

 and he then gives a brief history of the previous tables^. He 

 refers to the Berlin manuscript of the fourth, fifth, and sixth 

 millions (from which he had wished Dase to make the enumera- 

 tions, as mentioned in his letter to Encke) as follows : " Die wich- 

 tigste Arbeit dieser Art ist aber das Manuscript, welches die 4te, 

 5te und 6te Million (also von 3000000 bis 60000^0) enthalt 

 und von Herrn Crelle vor mehreren Jahren der Berliner Akade- 

 mie zur Verwahrung ubergeben ist. Gegen TJntergang ist es also 

 geschiitzt, und ich zweifle nicht, dass es fiber kurz oder lang 

 auch publicirt werden wird." The next sentence, in which Gauss 



1 See pp. 49, 50, of the present volume of Proceedings. 



2 The account is veiy correct; only two points need notice, (1) Felkel's table 

 is given as 1 — 336,000, and (.2) in regard to Felkel's project for a table to 24,600,000 

 Gauss doubts if an additional cipher has not been added by a misprint. But Felkel's 

 account shows that the number is right as printed: probably Gauss never saw 

 the Suppleinenta itself, and only knew of the proposed table from the passage in 

 the Monatliche Correspondenz, quoted in the note near the end of § 11. The 

 mention of Eosenthal's manuscript has been referred to at the end of § 7. 



