352 History of the Theory of Numbers. [Ch.u'. xiii 
table to 100 million; the manuscripts^ has been in the library of the Vienna 
Royal Academy since 1867. Lehmer^- gave an account of the first of the 
eight volumes of the manuscript, listed 226 errors in the tenth million, and 
concluded that Kulik's manuscript is certainly not accurate enough to 
warrant publication, though of inestimable value in checking a newly 
constructed table. Lehmer^^ gave a further account of this manuscript 
which he examined in Vienna. Volume 2, running from 12 642 600 to 
22 852 800 is missing. The eight volumes contained 4,212 pages. 
B. Goldberg*^" gave all factors of numbers prime to 2, 3, 5, to 251 647. 
Zacharias Dase,^^ in the introduction to the table for the seventh million, 
printed a letter from Gauss, dated 1850, giving a brief history- of previous 
tables and referring to the manuscript factor table for the fourth, fifth and 
sixth milUons presented to the Berlin Academy by A. L. Crelle. Although 
Gauss was confident this manuscript would be pubhshed, and hence urged 
Dase to undertake the seventh million, etc., the Academy found the manu- 
script to be so inaccurate that its publication was not ad\'isable. Dase died 
in 1861 lea\'ing the seventh million complete and remarkably accurate, 
the eighth nearly complete, and a large part of the factors for the ninth and 
tenth millions. The work was completed by Rosenberg, but ^vith numerous 
errors. The table for the tenth million has not been printed ; the manuscript 
was presented to the Berlin Academy in 1878, but no trace of it was found 
when Lehmer^- desired to compare it with his table of 1909. 
C. F. Gauss^- gave a table showing the number of primes in each thousand 
up to one million and in each ten thousand from one to three million, with a 
comparison with the approximate formula jdx/log x. 
V. A. Lebesgue^^ discussed the formation of factor tables and gave that 
to 115500 constructed by Hoiiel. 
W. H. Oakes^ used a complicated apparatus consisting of three tables on 
six sheets of various sizes and nine perforated cards (cf. Committee, ^^ p. 39). 
W. B. Da\as^s considered numbers in the vicinity of 10^, and of 10^^ 
E. MeisseP^ computed the number of primes in the successive sets of 
100 000 numbers to one million and concluded that Burckhardt's*^ table 
gives correctly the primes to one million. 
••Cited by Kulik. Abh. Bohm. Gesell. Wiss., Prag, (5), 11, 1860, 24, footnote. A report on the 
manuscript was made by J. Petzval, Sitzungsberichte Ak. Wiss. Wien (Math.), 53, 1866, 
II, 460. Cited by J. Perott, I'interm^iaire des math., 2, 1895, 40; 11, 1904, 103. 
•"Primzahlen- u. Faktortafeln von 1 bis 251 647, Leipzig, 1862. Errata, Cunningham." 
•'Factoren-Tafeln fur alle Zahlcn der siebenten MilUon . . . , Hamburg, 1862; . . .der achten Mil- 
Uon, 1863;. . .der neuhten MilUon (erganzt von H. Rosenberg), 1865. 
•^Posthumous manuscript, Werke, 2, 1863, 435-447. 
••Tables diverses pour la decomposition des nombres en leurs facteurs premiers, M6m. soc. sc. 
phys. et nat. de Bordeaux, 3, cah. 1, 1864, 1-37. 
•♦Machine table for determining primes and the least factors of composite numbers up to 
100 000, London, 1865. 
••Jour.de Math., (2), 11, 1866, 188-190; Proc. London Math. Soc, 4, 1873, 416-7. Math. 
Quest. Educ. Times, 7, 1867, 77; 8, 1868, 30-1. 
••Math. Annalen, 2, 1870, 63&-642. Cf. 3, p. 523; 21, 1883, p. 304; 25, 1885, p. 251. 
I 
